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' ' '1 mm , ‘ I III .:.' m QWV‘M‘M , -T.‘ "if” “A“. .4 I Pl 9 (I) LIBRARY Michigan State University This is to certify that the dissertation entitled Risk Reduction Capabilities of Hedging Techniques in the Financial Futures Market: A Comparison Test presented by Bruce 8. Berlin has been accepted towards fulfillment of the requirements for Ph . D . degree in Finance ' aw Date March 1, 1985 MS U it an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LlBRARlES .— \1 RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. RISK REDUCTION CAPABILITIES OF HEDGING TECHNIQUES IN THE FINANCIAL FUTURES MARKET: A COMPARISON TEST BY Bruce S. Berlin A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Finance and Insurance 1985 j}5/—-éo.3f © Copyright by BRUCE S. BERLIN 1985 ABSTRACT RISK REDUCTION CAPABILITIES OF HEDGING TECHNIQUES IN THE FINANCIAL FUTURES MARKET: A COMPARISON TEST BY Bruce S. Berlin Hedging of interest rate risk has been an impor— tant goal of financial managers and investors who must leave their funds at risk for a specific period of time. A number of approaches have been attempted to determine the best method of reducing interest rate risk. The advent of active futures markets for government and quasi government securities has opened another avenue for hedging activities that attempt to immunize the investment port- folio against interest rate changes. Prior to the advent of futures markets in securi— ties, immunization was accomplished by changing the compo- sition of the portfolio, altering both its overall risk and its expected rate of return. Hedging by taking posi— tions in contracts for future delivery of government secu— rities allows the investor to change the risk of the portfolio without changing the expected return. Research in the area of hedging effectiveness has developed along two lines. Bruce S. Berlin 1. Take positions in futures based on the correlation between spot and futures prices of the same security. The corre— lation approach has also been used for cross-hedging. 2. Take futures positions based on the rela- tive durations of the investment security and the hedging vehicle. Both of these approaches require restrictive assumptions about the shape and movements in the yield curve. This study develops hedge positions based on investor expectations of future yields. No assumptions are made about the dynamics or future shape of yield curves. Since it deals with changes in asset prices as a result of changes in forward interest rates, this approach should be equally useful in direct and cross—hedging. The tests were done as cross hedges between government bonds and Treasury Bill futures contracts. When risk was measured by variance of wealth change, there was no significant difference between the expecta- tions hedge method and the duration method or the correla— tion adjusted method. When risk was measured as the possi— bility of earning below—target returns (R < 0) the expec— tations hedge provided better protection against interest rate changes. ACKNOWLEDGMENTS I would like to thank the members of my disserta— tion committee: Dr. John R. Brick and Dr. George W. Kutner for their help and support. The committee chairman, Dr. Larry J. Johnson was especially helpful in the original development of the idea for this dissertation. .His help throughout the research and writing has been of inestim- able value. I owe a debt of gratitude, as well, to Dr. John L. O'Donnell who acted as Chairman of the Department of Finance and Insurance during the preparation of this research. I am grateful for his support. iii TABLE OF CONTENTS Page LIST OF TABLES. . . . . . . . . . . . . . . . . . . Vi LIST OF FIGURES . . . . . . . . . . . . . . . . . . viii Chapter I. INTRODUCTION . . . . . . . . . . . . . . 1 II. REVIEW OF THE LITERATURE . . . . . . . . 5 Duration and Immunization. . . . . . . 5 Hedging Methods and Strategies . . . . 15 The Term Structure of Interest Rates. . . . . . . . . . . 23 Tests of Term Structure Theories . . . 28 Estimation of Yield Curves . . . . . . 31 III. THE HEDGING MODELS . . . . . . . . . . . 34 An Expectations Hedging Model. . . . . 36 Comparison to the Kolb and Chiang Hedge. . . . . . . . . . . . . 39 Comparison to the Kolb Hedge . . . . . 41 IV. HYPOTHESES . . . . . . . . . . . . . . . 43 V. METHODOLOGY. . . . . . . . . . . . . . . 48 Data 0 O O O O O O O O O O O O O O O 48 Yield- -to-Maturity. . . . . . . . . . . 50 Forward Rates. . . . . . . . . . . 52 Forecasting Forward Rates. . . . . . . 53 Simulation . . . . . . . . . . . . . . 59 Wealth Change. . . . . . . . . . . . . 61 Tests of the Hypotheses. . . . . . . . 67 Summary of Steps in Analysis . . . . . 70 VI. RESULTS AND CONCLUSIONS. . . . . . . . . 74 Independence of Forward Rates. . . . . 74 Hedging Activity . . . . . . . . . . . 76 Tests of Lower Partial Moments . . . . 78 iv Chapter Page Conclusions. . . . . . . . . . . . . . 79 Further Research . . . . . . . . . . . 80 Appendix A COMPUTER PROGRAMS . . . . . . . . . . . 83 B FORWARD RATE FORECASTS AND FORECAST VARIANCES. . . . . . . . . . . 104 C CONTRACTS SOLD . . . . . . . . . . . . . 120 LIST OF REFERENCES. . . . . . . . . . . . . . . . . 131 Table A1 A2. A3. A4. A5. A6. A7. A8. A9. B1. B2 B3 LIST OF TABLES Annualized Yields as of November 30, 1976 and November 30, 1978 . . . . . ARIMA models . . . . . . . . . . Regression Coefficients. . . . . . . . Haugh Statistics . . . . . . . . . . Wealth Changes . . . . . . . . . . . . F-Ratios . . . . . . . . . . . . . . Fishburn a-t Ratios. . . . . . . . . . Yield to Maturity Calculation. Forward Rate Calculation . . . Lagged Cross-Correlation Calculation Simulated Yield Curve Calculation. . Expectations Hedge Ratio Calculation . Duration Hedge Calculation . . . . Duration Hedge Calculation Adjusted for Regression . . . . . . . . . . . Comparative Wealth Change Calculation. Fishburn Test Calculation. . . . . . . Forward Rate Forecasts First Test Period . . . . . . . . . . Forward Rate Forecast Variances First Test Period . . . . . . . Forward Rate Forecasts Second Test Period. . . . . . . . vi Page 52 55 67 75 77 78 79 83 87 89 91 93 96 98 100 103 104 108 112 Table B4. C1. C2. C3. C4. C5. C6. Forward Rate Forecast Variances Second Test Period. . . . . . . . . . Contracts Sold. Duration Hedge First Period. . . . . . . . . . . . Contracts Sold. Adjusted Duration Hedge. First Period. . . . . . . . . Contracts Sold. Expectations Hedge. First Period. . . . . . . . . . . . . Contracts Sold. Duration Hedge Second Period . . . . . . . . . . Contracts Sold. Adjusted Duration Hedge. Second Period . . . . . . . Contracts Sold. Expectations Hedge Second Period . . . . . . . . . . vii Page 116 120 121 122 125 126 127 LIST OF FIGURES Figure Page 1 Illustration of Differences Between Duration and Expectations Hedging Approaches . . . . . . . . . . . 63 viii CHAPTER I INTRODUCTION Investors who must put funds at risk for a known period of time are faced with a number of alternatives that would allow them to mitigate that risk. The investor who is averse to risk is expected to take such action that will reduce the risk of the investment portfolio until the expected return of the investment portfolio is at the appropriate level given the level of risk accepted. One component of risk that is faced by the investor in such a situation is interest rate risk. This is the risk that the market value of the investment will have changed over the investment period because of changes in the interest rate structure and levels that will have taken place over the planning horizon or time the funds are at risk. The other component of risk is reinvestment risk. This is the risk that an investor faces when investing in an asset whose maturity is shorter than the planning horizon. The principal will be reinvested at a rate that is unknown at the time of the original commitment. This interest rate risk can be mitigated by matching the duration of the investment to the planning horizon. An example of this might be a corporate treasurer 1 2 investing corporate funds that will be needed for opera— tions in three months. If these funds are invested in 91 day Treasury Bills the expected value of each bill is $1,000,000 and it is default free. The expected return is a function of current interest rates. There is no interest rate risk because the receipt of $1,000,000 per bill is assured at the end of the planning horizon which corre— sponds to the maturity of the T—Bill. Contrast this with the investment behavior of the corporate treasurer in the same situation of having funds available for a short time who invests those funds in a Treasury Bond with 24 months to maturity. This investment might be undertaken to take advantage of higher expected returns offered by the longer maturity investment. The interest rate risk in this position derives from the sto- chastic nature of the interest rate structure in the future. The value of the T-Bonds three months hence is a function of the interest rates that will prevail at that future time for the period from that point to the date of maturity of the bonds. The investor does not have to accept that risk. The longer maturity bonds could be replaced by securities whose durations match the planning horizon. This would defeat the goal of higher expected returns. Alternatively, the investor could take a position in the futures con— tracts of a financial instrument such that changes in the value of the futures position would offset changes in the 3 value of the investment position.l Both values are affected by changes in forward rates between the inception of the position and offsetting hedge and the end of the planning horizon. This activity has the benefit of changing the duration of the investment position without changing the expected return on the assets since the pres- ent value of the futures contract is zero. So, hedging allows the investor to reduce the risk of the investment position without reducing the expected return. This is optimizing behavior for the risk—averse investor. In a perfect market hedging would not be necessary. The valuation of an asset would appropriately reflect the expected cash flows and the risk of those cash flows for each asset in the market. Investors would be able to pro- vide their own hedging combinations so it would not be incumbent upon the firm to do so. Even in the situation where the firm could use futures contracts to immunize its investment portfolio against interest rate risk without changing the expected return, there would be no incentive for the firm to do so. Where there are imperfections in the market, such as indivisibility or differential taxes or differential access, there would be a need for hedging. Where there are inefficiencies such as differential infor- mation availability hedging might also be beneficial. Hedging would be a trivial activity if all assets 1Bankers attempt to immunize their assets and liabilities by some similar form of matching. L_____ _ 4 experienced the same price changes as a result of interest rate changes. An investor would take a position in the futures contracts calling for delivery of securities of the same face value as the value of the original invest- ment. Because of differences in duration, coupon and default risk there is not a one-to-one relationship of those price changes. Even with default risk differences abstracted there is still considerable difference in the price effect of interest rate changes. Hedges that are different in this way are referred to as cross-hedges. The determination of the number of futures con- tracts to acquire in order to hedge a unit of investment is the hedge ratio. Hedge ratios calculated using the durations of the investment and the asset underlying the hedging vehicle (futures contract) have been shown to be superior to the naive or one-to-one hedge in reducing wealth changes over the planning horizon. The present study improves upon the hedging activity suggested by the duration—based hedges by taking into consideration that yield curves will be changed by any number of additive and multiplicative shocks that will change forward rates independently of one another. The duration-based hedges are limited to allowing a single additive shock to the entire yield curve. The tests of effectiveness of the hedges are tests of the variance and semi - variance of the wealth changes resulting from each of the hedging behaviors tested. CHAPTER II REVIEW OF LITERATURE This study of hedging effectiveness of alternative hedging strategies is based on developments in three areas of research: 1. Duration and immunization; 2. Hedging methods and strategies; and 3. Term structure of interest rates. Duration and Immunization The concept of duration is central to hedging posi- tions suggested by those whose hedging activities and hedge ratio calculations will be compared with the approach suggested in this study. Duration is a measure of the sensitivity of asset prices with respect to interest rate changes. In that way it is a measure similar to that mea- sure referred to as maturity, however duration is defined as dimensionals. Both are measured in units of time. The term duration was coined by Macaulay (1938). The measure takes into consideration that the receipt of coupon payments or other periodic distributions and their subsequent reinvestment, reduce the average amount of the investment. Hicks (1946) developed the same measure 6 independently. He referred to it as the "average period" of investment. Duration calculations use averages of expected flows weighted by the length of time until receipt of those flows. Maturity measures do not consider the timing of expected intermediate cash flows. Rather, all expected flows are equally weighted so the maturity becomes the time to the last expected cash flow with no considera- tion of the effects of the intermediate expected flows. Duration, as expressed by Macaulay, is n CFtlt) Z t D t=1 (1+R) n CF t z t t=1 (1+R) cash flow at time t maturity yield to maturity. where CF ll II II 73:25 The duration of a pure discount, zero coupon, bond is equal to its maturity because there are no intermediate cash flows. When the duration of a coupon bond is calculated, that duration will be shorter than the maturity of the bond. Macaulay has shown that for long maturity bonds selling below par, the duration decreases with maturity in some cases. He also shows that all durations reach a finite limit even though the maturities may be limited. Investors with an identifiable planning horizon have an interest in assuring an expected return over that planning horizon. Fisher and Weil (1971) define an 7 immunized portfolio as one whose ...value at the end of the holding period, regard— less of the course of interest rates during the holding period, must be at least as large as it would have been had the interest rate function been consistent throughout the holding period. (99- 415) Redington (1952) provided an early statement of this idea. Fisher and Weil relax some of Redington's restrictive assumptions about interest rates to make their model more general. One obvious way that an investor can insure an expected return over a planning horizon would be for the investor to purchase pure discount instruments that mature at the end of the planning horizon. There is no problem with reinvestment of coupons because there are no coupons. The expected value of the bonds at maturity is the amount the borrower is expected to repay. In the case of the U.S. government as borrower, that value is the face value of the bonds. A policy of purchasing zero coupon bonds to match the planning horizon is difficult to implement because of a dearth of such zero coupon securities over a range of maturities.2 An investor who chooses a coupon bond is subject to two different risks because of interest rate fluctuations. 2Lately, securities brokers have been marketing U.S. government bonds with the right to receive coupon payments stripped from the security. The right to receive coupon payments is marketed separately. If this new type of security enjoys wide market acceptance in the future, investors may be able to make more use of the direct matching policy. 8 First is the reinvestment risk. Yield—to-maturity calcu— lations assume reinvestment of coupon payments such that the average return will be the yield—to-maturity. Interest rates change so the risk of reinvestment is always present. The second risk is a price risk. Because of interest rate changes, the price received for a bond whose maturity is beyond some planning horizon will not be the expected price at maturity. Interest rate changes affect the value of coupon bonds in Opposite directions through the reinvest- ment and price effects. Immunization occurs when the effects are equalized, leaving the wealth of the investor unchanged as a result of the interest rate changes expe- rienced. Hopewell and Kaufman (1973) provide an explanation of the observation that bonds selling at a discount behave differently from what would be expected as a result of the belief that the market prices of longer term bonds exhibit greater changes with a change in interest rates than do shorter term bonds. They show that the relationship is better described using duration as a measure of price sensitivity than it is using maturity. They point out that the duration of a discount bond increases with maturity but then reaches a maximum and actually begins to decrease. This confirmed the earlier work of Macaulay. Fisher and Weil (1971) provide a review of the development of the relationship between duration and immu- nization. Redington (1952) was the first to use the term 9 immunization to refer to protection of the value of a portfolio from the ravages of interest rate changes. Indeed, Redington's mean term is duration. Hicks (1946) showed that the change in present value of one investment relative to another as the interest rate changes is a function of the duration of the payment streams of the investments. Samuelson (1945) extended this argument by looking at an asset and a liability.3 He noted that the effect on the net investment value will be different depending on the duration of the two positions. If the duration of the liability is greater than that of the asset, an increase in interest rates will result in an increase in the present value of the position. The effect on the longer duration liability will be greater than that on the shorter duration asset. The implication of this for planning is that a position can be fully hedged (immunized) by matching the durations of the assets and the liabilities. An investor can take a speculative position by adjusting durations according to expectations of the direction of interest rate changes. Redington's immunization is the banker's or asset— liability approach to immunization. The banker is seen as managing the size and duration of both the assets and the liabilities of the institution. The banker inten— tionally can leave a gap between durations of assets and 3"Positive flow" and "negative flow" in the Hicks context. 10 liabilities, the size and sign of which are a function of the banker's expectations of interest rate fluctuations. In order to immunize the bank's assets, their duration should equal the duration of the liabilities. This is accomplished by adjusting the asset and liability holdings. An approach to immunization that is similar to asset-liability matching is the matching of asset duration to a planning horizon. When asset durations do not match the investor's planning horizon, that investor is subject to reinvestment risk and price risk as a result of the effect of interest rate changes until the planning hori- zon is reached. The hedging activity considered in the present study is a method of adjusting the duration of assets so is especially useful in the planning horizon matching goal. This planning horizon goal strategy is assumed in this study. Fisher and Weil (1971) test an immunization strat- egy based on duration that implies matching duration to planning horizon and adjusting the duration to the im— pending planning horizon through appropriate reinvestment of the coupon payments. Earlier immunization strategies assumed that there could be only parallel shifts in a flat yield curve for each investment. A yield curve demonstrates a parallel shift when a single shock affects all yields in the same magnitude of change. Thus, if the one month yield increases by 100 basis points, so do each of the longer yields increase by 100 basis points. The ll shape of the yield curve does not change, rather the whole term structure is displaced. This is Redington's approach. Fisher and Weil show that a modified immunization strategy can be effective when there are multiple shocks to the term structure. However, they require periodic rebalancing of the portfolio over the planning horizon to achieve maxi- mum benefit from duration-based immunization. Grove (1966) points out that an immunization ap- proach is valid for small changes in the interest rate. If interest rate changes are large, immunization based on duration is less effective. He also put duration in a Pratt-Arrow risk analysis context (1974) and argues that for utility functions increasing in wealth with decreasing absolute risk aversion, the investor will maximize utility by attempting a perfect hedge only if no change is expected in interest rate variability. Otherwise the investor will take some risk by exposing either a net asset or net lia— bility position. Fisher and Weil also show in their proof of the im— munization theorem that immunizing with a discount bond whose maturity is equal to the planning horizon is inferior to immunizing with coupon bonds of appropriate duration because coupon bonds provide an expected return that is no less than that required but may be more. Bierwag and Kaufman (1977) extend the work of Fisher and Weil to show that the measure of duration will be more complex when the random effects on the yield curve are 12 multiplicative rather than additive. Bierwag (1977) provides proofs of these theorems which also point up the differences for immunizing choices depending upon the effect of the shocks that change interest rates. He shows that an immunization strategy in the case of combined additive and multipli- cative effects requires a more complex approach than a linear combination of assets for immunization. Kaufman (1978) shows duration as an important com- ponent of bond risk. Risk measures that have been developed for the equity market are not appropriate for measuring bond risk. Such measures would only be appro- priate when there is a completely immunized portfolio. An expected duration can be calculated for equity securities. The implication of this is that risk of bond portfolios cannot be thought of in the same context as equity risk. Duration comes into play as risk and expected return must be considered in terms of both the terminal price of the bond and of the income from reinvestment of the coupons. Appropriate use of duration to immunize the portfolio will offset the risks. Bierwag and Khang (1979) show that immunization provides for maximization of the minimum expected return. They see immunization as an optimization technique. They also show that an immunized portfolio is a zero—beta portfolio since it is riskless with respect to interest rate changes. Included in their paper is a capsule 13 history of the immunization—duration relationships beginning with the contributions of Macaulay and con- tinuing through portfolio approaches. Bierwag (1979) develOps an immunization policy that takes into account successive changes in the yield curve rather than just one shock. This is an extension of his earlier work. In it, immunization is achieved by adjust- ing the portfolio to reflect changes in the length of the planning horizon and changes in the yield curve which derive from changes in the forward rates from the new deci- sion point forward. This is a multi-period approach. It extends the single-period analysis done for prior tests. However, it requires that portfolio decisions be made periodically rather than allowing immunization to be achieved with a single decision. The single decision could be expected to be the goal of a multi-period approach to the investment decision. Cox, Ingersoll, and Ross (1979) have developed a measure of duration from their continuous time model of the term structure of interest rates. Their method allows the yield curve to attain any shape and sustain more than one shock or change. The immunization proce- dures developed with this model could be more generally applicable for risk reduction because of the less restricted nature of the interest rate generating process. Weil (1973) presents an early history of duration. Bierwag, Kaufman, and Khang (1978) have written a complete l4 description of the uses of duration. Ingersoll, Skelton, and Weil (1978) have done a survey of the properties and uses of duration for immunization of portfolios. These papers survey the field of duration. Bierwag, Kaufman, and Khang in their criticism of the uses of duration, say that duration is not a measure of risk because the way risk is usually defined implies the use of a measure of utility. Duration measures do not, in themselves, use utility measures. They also note the limitations imposed by the restrictive assumption of flat yield curves and the single additive shock. They discuss the relationship be- tween duration and beta as developed by Kaufman (1978). Beta relates the duration of the portfolio and the market portfolio and the period over which beta is measured. If that period is equal to the planning horizon and the port- folio is default-free, the portfolio beta will be equal to zero. In light of the restrictive nature of the assump- tions and the conflicting conclusions of previous studies, a more generally useful approach to immunization is needed. The shape of the yield curve will be irrele- vant in this approach. No assumptions need be made about the allowable number or effects of shifts in the yield curve. Other, earlier immunization strategies based on duration were not expected to be completely effective because of the duration assumptions that were necessarily of a simplifying nature. 15 Hedging Methods and Strategies In order to immunize the bond portfolio, it is necessary to match its duration to the planning horizon. This would apparently require that a portfolio of coupon bonds, or for that matter, one coupon bond, be chosen such that the combination of coupon payments and maturity pro- vide a duration that matches the planning horizon. In the case where the investments are already held, some adjust- ment would be necessary that could alter the expected return of the investment because bonds would have to be replaced by others with durations that would allow for the matching. The investor is forced to give up his chosen level of expected return in order to reduce the risk of the investment. The existence of an organized futures market for U.S. Treasury securities, GNMA securities, and CDs pro- vides the investor with another alternative. For small transactions cost4 the investor can take a position in futures contracts that will hedge the position and provide the interest rate risk protection that comes with immuni— zation. Portfolio immunization can be achieved without material effect upon its expected return. Hedging strategies have been proposed by a number of authors in the past few years. The strategies are distinguished by their underlying assumptions about the 4Round—turn (buy and sell) commissions are less than $25 per contract. l6 shape and movement of yield curves and whether they take a portfolio or a duration approach. The commodity futures markets have long provided hedging opportunities for participants in cash markets for commodities. From the commodity trading standpoint, the holder of a position in the cash market who wants to hedge that position takes a position in the futures market that is opposite to and equal in volume to the cash posi- tion. This is the naive hedge. As long as the futures price and the spot or cash price move in the same direction and by the same amount, the position is effectively hedged. The spot price and the futures price are not expected to be equal until the delivery date. The difference between spot and futures price is called basis. That is, B = FP - P. If a market participant has a position in the cash market, the gain or loss realized is the price change of the com- modity from the time the position is initiated until the time the position is liquidated. Symbolically, G = A(Pl - PO) where A is the size of the position and P is price. If the position is hedged in the futures market, the gain or loss on the overall position, cash and futures, is the net of the gains and losses in the two offsetting positions. l7 Letting GH represent the net gain, GH = A(P1 - PO) - A(FPl - FPO). A perfect hedge occurs when the change in basis is zero so (Pl - FP - (P - FPO) = 0. 1) o In the commodity markets both the change in basis, known as basis risk, and the basis itself are considered small. Where this is the case, most of the risk associated with the position can be hedged naively. Ederington (1979) provides an analysis of three hedging theories. 1. Traditional Theory. Hedgers take futures market positions equal and opposite to their cash positions. This begs the question of basis change. The traditional theory holds that basis and basis change are small be- cause of the possibility of delivery on the futures contract. Traditional theory pro- vides an approximation of an optimal hedge ratio. As noted above, the hedger attempts to reduce the variance of the position through hedging. 2. Working's Hedging Hypothesis. Working (1953, 1962), considers investors to be speculators 18 who will hedge only if they expect the basis change to be unfavorable to their position. Thus, he deals directly with the problem of basis change. Investors are con- sidered to be, at best, risk-neutral profit maximizers. Working argues that futures trading is not done primarily to reduce risk but in expectation of a favorable basis change. He calls this selection or anticipatory hedging (1962). 3. Hedging in a Portfolio Context. Johnson (1960) and Stein (1961) see futures transactions as another investment in the portfolio. When analyzed in this manner, futures contracts are included in the portfolio to the extent that the expected change in the value of the port- folio is greater than zero with the addition of the futures contracts. Ederington (1979) shows that for risk minimization the propor- tion of the portfolio to be hedged is a func- tion of the covariance of the spot and futures prices and the variance of the futures price. Hill and Schneeweis (1982) use the portfolio risk and expected return analysis to develop a hedging strategy. Their hedge is a cross-hedge. The investment portfolio is represented by a corporate bond index. The hedging vehi— cles are GNMA futures contracts and Treasury Bond futures l9 contracts. Their hedge ratio is Johnson's (1960) pr0por- tion of the portfolio to be hedged. Letting HR represent the hedge ratio, number of futures contracts bought or sold per unit of investment. HR = cov (Cs,Cf)/Var(Cf) where Cs and Cf are price changes over the planning horizon for spot and future prices. The measure of effectiveness they use is the reduction in portfolio variance. They also show that risk can be reduced more effectively if there is a higher correlation between futures and spot prices. They found that this hedge strategy gave greater risk reduction than did a naive hedging strategy. D'Antonio and Howard (1982) also used a portfolio approach to test the effectiveness of financial futures for hedging. Their analysis fits futures positions into the classical CAPM framework. Their optimal hedge ratio is a function of a risk and expected return relationship between the expected price change for futures contracts and the risk premium for the risky security. Their measure of hedging effectiveness is the change in expected return of the hedged portfolio over the expected return of the unhedged. They used a direct hedge. The assumed invest- ment was in T-Bills and T-bill futures were the hedging vehicles. D'Antonio and Howard found only moderate im- provement in the portfolio risk and return relationship by 20 using futures hedges. This is to be expected because the portfolio concepts they use assume that the financial mar- kets are in equilibrium. The T-Bill and T-Bill futures markets are apparently efficient and themselves in equil— ibrium. Their work is an extension of the analysis pro— vided by Fischer Black (1976). Black, working with com- modity futures contracts in a CAPM model, finds that the change in futures price is related to the beta of the futures contract. But the investor makes no investment in a futures contract so the beta cannot be measured in terms of a rate of return, rather it is measured in dollars. D'Antonio and Howard point out that their model is derived from Black's CAPM. To do this they must assume that their investment portfolio (T—Bills) is the market portfolio. They also need to be able to place Black's model in terms of returns so they compare their model to Black's by dividing through by the futures price. The difficulty with this comparison is that Black begins by setting the initial price at zero because the value of a futures contract at its inception is zero and the con- tract is revalued to zero each trading day.5 5The process is called mark—to—market. At the end of each day's trading the delivery (settlement) price of every contract is adjusted to the market closing price. Holders of contracts make adjustments with the clearing corporation, usually through their brokers, in cash. Thus, any net change in the settlement price is offset by a net change in cash position. 21 D'Antonio and Howard's finding of little or no effectiveness of their hedge is not surprising. For the hedge to be effective, there would have to be signifi— cant covariance. Dusak (1973) in a study of commodity futures contracts for wheat, corn, and soybeans found a zero beta, that is, covariances between futures price changes and the return on the Standard and Poor's Stock Index were found to be close to zero. While portfolio theory hedges and their effectiveness is an appropriate area for further study, the present study does not con- sider hedges based on portfolio concepts. Two duration based hedge ratios will be included in this study. Kolb and Chiang (1981) have developed a duration based hedge ratio. They recognize that duration is affected by the asset's interest rate sensitivity. Their hedge ratio is expressed as, N = -fi. P. D./R. FP. D. j 1 1 1 j j where, R. = l + the yield to maturity expected 3 for the asset underlying the futures contract Ri = l + the yield to maturity expected for the asset being hedged P. = Price at the termination of the hedge of the asset being hedged FP. = Price at which title to the under- 3 lying asset will pass when the futures contract matures 22 Di = Expected duration at termination of the hedge of the asset being hedged D. = Expected duration at termination 3 of the hedge of the underlying asset. In the derivation of this hedge they find that the hedge ratio will be a function of an elasticity of the expected rate of return on the asset being hedged with respect to the expected rate of return on the asset under— lying the futures contract. They assume this elasticity is unity for the purpose of developing their hedge ratio. They assume a flat yield curve for each instrument and parallel shifts in the yield curve to test their hedge ratio. In a later paper, Kolb and Chiang (1982) develop hedge ratios while relaxing the assumption of a flat yield curve for each instrument. The parallel shift assumption is still in force. They also develop a hedge ratio for risky assets. Kolb (1982) combines the duration approach and the portfolio approach by regressing asset yields on the yields on the assets underlying the futures contract. He then includes that regression coefficient in the duration based hedge ratio replacing the assumed relation of unit elasticity: N = (-Rj Pi Di/Ri FPj Dj)rij where r = the regression coefficient estimated. That 23 regression coefficient is a measure of the elasticity dRi/de' For the present study, each of the yields over the investment periods were regressed on the three month yield to determine the appropriate regression coefficients for incorporation in the hedge ratio calculation. The Term Structure of Interest Rates As has already been noted, hedging activities in the eyes of many analysts are determined by the relation- ship among yields over a number of relevant maturities and by how those relationships are expected to change. Hedging activities are an attempt to protect the investor from the effects of those changes. There have been a number of explanations of the term structure that have had some acceptance. These are attempts to explain how interest rates on securities of one maturity relate to rates for securities of other maturities. The expectations theory was developed by Irving Fisher (1930). In simplest form it states that any long- term rate of interest is an average of expected future ‘short-term rates and the current spot rate. The pure expectations theory requires the assumptions of perfect markets with no differential taxes, no transactions costs, and homogeneous expectations. With these assump- tions and expected return maximizing behavior on the part of investors, the current forward rates equal expected future short-term rates. A forward rate is usually 24 defined as a one period rate on an investment made in the future. Thus (1+tRn)“=<1+tRl)<1+ )(1+ )(1+ ). t+lr1,t t+2r1,t t+n-lrl,t where R r spot rate forward rate The forward rate under the certainty and perfect market con— ditions noted above is the future spot rate. If, in a two period case, the investor's expectation of the second period spot rate was lower than the forward rate, the two-period investment would be undertaken. If the expected future rate was higher it would behoove the investor to make a one period investment and reinvest at the higher expected rate, which under the assumptions, is certain. In equilibrium the forward rate would equal the future spot rate. E (1 + 1R2,t) = (1 + lr1,t) = (1 + R2)2 / (1 + R1) Under this expectations theory we would conclude that bonds of any maturity will have the same expected return over a given holding period. The expected return will also be the realized return. Any one bond with any matu- rity is a perfect substitute for any other over that holding period. The ability of investors to make cost- less transactions will allow the arbitrage mechanism to 25 work as investors attempt to maximize returns over their holding periods. With perfect markets and costless arbitrage opportunities, bond prices would reflect all relevant information in an unbiased manner. When new information becomes available expectations adjust instan- taneously as do prices. Liquidity preference theory allows for uncertainty. Instead of the future short-term rate being equal to the forward rates, those future rates are uncertain. Hicks (1946) developed the liquidity preference theory.6 He sees investors forming expectations of the uncertain future rates. Borrowers are assumed to desire to borrow long to avoid interest rate risk, that is, the need to refinance at a higher rate. Lenders want to lend short-term in order to maintain the stability of their wealth. Short- term investments are less sensitive to interest rate fluc— tuations than are long-term investments. If risk aversion is assumed on the part of inves- tors, they must be offered a premium in return if they are to invest in the less desirable long-term bonds. Forward rates are no longer unbiased estimates of the future short-term rates. They are biased and exceed the the expected future rates by this premium for liquidity. For any forward rate: 6While Hicks is the developer of liquidity prefer- ence theory, its roots can be traced back to Keynes' "normal backwardation." 26 t+nrl,t) = Et (1 + t+an,t) + t+nLl,t' Thus, the perfect substitutability from the expectations theory is lost. A long-term investment would have a higher expected return than would a series of short-term investments. Empirical tests of the liquidity preference theory show that it is more consistent with the efficient markets hypothesis than is the pure expectations theory. Liquidity preference and risk aversion are part of the information set that is available to the market and, as such, are impounded in security prices instantaneously and in an unbiased manner. An argument against liquidity preference and sup— portive of the expectations hypothesis is that there are speculators who seek risk and investors who are risk indifferent (expected return is their sole decision cri- terion) in the market. This would imply that there are investors who would pay a premium for the longer maturity investments. Through their arbitrage activities, these investors are seen to eliminate the liquidity premium. Where these investors are present, all maturities of bonds would provide the same liquidity because of investor in- difference to risk (maturity) and because of risk seeking investors, so liquidity premiums disappear. Market segmentation is a third approach. It is presented in opposition to expectations theory. 27 Proponents of this theory argue that the investor is risk averse. There are structural or institutional reasons that limit the participation of certain investors to cer- tain segments, by maturity, of the market. Investors with long planning horizons, such as life insurance firms, would invest primarily in long maturity assets since they are concerned with maintaining a level of income and not exposing themselves to reinvestment risk. Similarly, short-term securities are attractive to investors with short planning horizons. This theory recognizes that in- vestors try to match planning horizon and portfolio matu- rity. Bonds of different maturities are not substitutes under this theory. Culbertson (1957) is recognized as having promulgated the concept of market segmentation. In such a segmented market, the term structure results from the supply and demand for securities within each segment. Changing the maturity structure of government securities, for example, would change the yield curve for government securities. Proponents of the expectations hypothesis would not accept this conclusion. Modigliani and Sutch (1966) developed another approach to segmentation. This is their preferred habitat theory. The theory is a synthesis of liquidity preference and market segmentation. The habitat referred to is maturity that equals a planning horizon or perceived need for funds in the future. Unlike Culbertson, though, they allow for investors with a preferred habitat to be 28 willing to invest in securities of other maturities to take advantage of higher expected returns. There is not, then, only one precise maturity which will be acceptable to an investor. This theory provides for a continuity in the term structure that is not present under the market segmentation approach. It relates to the liquidity pref- erence theory under the circumstance where all investors' preferred habitat is the shortest maturity possible. Like the market segmentation hypothesis, this theory would lead to the conclusion that a change in the relative supply of bonds in a specific maturity range would cause shifts in the yield curve. Tests of Term Structure Theories Two of the theories discussed relate observed for- ward rates to expected future rates. Tests of the expec— tations theory were based on the market activity that forced forward rates and future rates to be equal. In order to perform such a test, assumptions need to be made about the forming of expectations on the part of the in- vestors. Usually this is done by making market efficiency assumptions that allow the investigator to accept past levels of interest rates to be used as the only information necessary to derive expectations of future rates. Then the hypothesis as usually tested, t+1r1,t ' + e t+1Rl t+l ' 29 is a test of a joint hypothesis. The first argument is that market expectations are formed as assumed above. The second is that the term structure comes about from a martingale model of prices as hypothesized. The test of the joint hypothesis falls prey to an inability to dis- tinguish which statement is being rejected. A rejection of the hypothesis of the expectations theory tested in this manner may just as well be due to a misspecification of the way expectations are formed or to the term structure not being developed from the expectations hypothesis. Jarrow (1981) notes that testing of term structure theories generally begins with the expectations hypothesis as the null. He points out that there have been three statements of the expectations hypothesis. They are: 1. Forward rates equal expected future spot rates; 2. Yield equals average expected future spot rates; and 3. Over a given holding period, bonds with different maturities have the same expected return. Macaulay (1938), Meiselman (1962), and McCulloch (1975) use the first approach. Indeed, Meiselman develops an error learning hypothesis to explain the way expecta- tions are formed. He argues that expectations do not have to be correct to form yield curves. Meiselman deter- mined that forecast errors could be used to explain 30 significant portions of changes in forward rates. Accord- ing to Meiselman, investors adjust their expectations based on the new information contained in the realized short-term interest rates being different from those which were forecasted. This adjustment causes the entire yield curve to shift. The second approach has been used by Modigliani and Shiller (1973) and by Dobson, Sutch, and Vanderford (1976) in their distributed lag average model. Tests of the expec- tations hypothesis using the third statement form were done by Santomero (1976) and by Fama (1976). These three statements of the expectations hypoth- esis appear to be consistent. In a later work Cox, Ingersoll, and Ross (1978) demonstrate certain inconsisten- cies. They use a continuous time model based on contingent claims in a rational expectations framework. They show that the only statement of the expectations hypothesis that is consistent with their model is that of equality of ex— pected returns of bonds of differing maturities. The other two are shown to be inconsistent with this one. Theirs is an equilibrium model which they use in a later work (1981) to evaluate the liquidity preference, expectations and preferred habitat models. They find that all statements of the expectations hypothesis except on instantaneous returns on bonds are the same for all maturities which they call "Local Expectations" imply a term premium that is inconsistent with expectations theory. Cox, Ingersoll, and 31 Ross also conclude that the preferred habitat theory which encompasses the liquidity preference as a special case requires that risk aversion rather than consumption plans as used by Modigliani and Sutch be the determinant of habitat. No theoretical specification of the term struc- ture is necessary for the development of an immunization strategy for an investment portfolio. A portfolio is seen to be immunized when its expected return is made secure from the effects of interest rate changes over the in- vestor's planning horizon. Estimation of Yield Curves Estimation of yield curves has been done frequently, using regression analysis. Cohen, Kramer, and Waugh (1966) found that although there are a number of government bonds with the same maturity because of continual issuing of new securities, the yields are different because of coupon dif- ferences, tax effects, and preference and institutional differences. A group of regressions were run on yields as they existed on certain specific days. They found that there was a model regressing before—tax yield and one regressing after-tax yield on maturity and log of maturity that provided significant coefficients and explained a large proportion of the variability. They concluded that OLS regression methods were useful for estimating yield curves. McCulloch (1971) argues that linear regression on direct observations is inappropriate because the yield 32 calculations are a weighted average of the principal and coupon weights and tend to generate errors at the longer maturities. This makes precise estimation of that part of the yield curve impossible. Another difficulty he sees is that errors in fitting will be magnified when forward rates are calculated from the yield curve. McCulloch, instead, develops a discounting or present value function first that gives instantaneous forward rates. The average of these rates is then used in the regression analysis. He contends that this method eliminates the difficulties encountered in using direct measurements. He found the best fit to be in segments, that is, discontinuous, and quadratic in form. Echols and Elliott (1976) point out that the data (bond prices) that are available for term structure analysis are incomplete since not all bonds are always traded nor are the data homogeneous because of coupon differences or call provisions. Their regression equation includes a term for the coupon yield. They find the coef— ficient of this term to be significant. Because of continuing difficulty in the estimation of yield curves by econometric techniques, hand plotted curve-fitting, or merely "eyeball" fits statistical mea- surement has not been particularly useful for reproduction or evaluation. Yield curves will not need to be estimated for the present study. As will be explained in the methodology section of this paper, we need only calculate 33 an average of current prices plus accrued interest for all government bonds extant at any point in time of interest to the study. Durand (1942) provided yield curves based on corpo- rate bond yields. He fit the curves by hand through the lower portion of a scatter diagram in an attempt to adjust for risk. The U.S. Treasury Department publishes yield curves that are fit by hand through the middle of the scatter of yield data for Treasury securities. Carleton and Cooper (1976) add another approach to term structure estimation using regression techniques. Instead of an average at each maturity, they use all bond prices. Their model uses periodic rather than continuous interest pay- ments and does not require the discount rate to be the same over all periods. CHAPTER III THE HEDGING MODELS The goal of the hedging activity is to minimize the variability of wealth over the investor's planning horizon. An unhedged position is subject to all the risk inherent in changes in the term structure and level of interest rates during the planning horizon. Hedging strategies that have been proposed earlier have been direct hedges where the investment asset position has been hedged by taking positions in futures contracts for delivery of those same securities. U.S. Treasury Bills would be hedged with T-Bill futures contracts. This approach limits the types of assets that could be hedged. Active futures markets exist for only a few securities. They are: l. U.S. Treasury Bills, 2. U.S. Treasury Notes, 3. U.S. Treasury Bonds, 4. GNMA securities, and 5. Bank CDs. There are also active futures markets in a number of foreign currencies and a fairly recent development of futures contracts for common stock indexes such as the S & P 500, New York Stock Exchange Composite Index, and 34 35 Value Line Index. An investor who wanted to be directly hedged would be limited to three U.S. Government obliga- tions. The other two debt related futures contracts are based on groups of securities. A cross hedge approach to hedging provides a broader range of possibilities. A cross-hedge exists where a position in one asset is hedged with a futures position in contracts of a different asset.7 The hedging activity tested in this study can be characterized as cross-hedging. Positions in U.S. Treasury securities are hedged with T— Bill Futures. The model presented here is an extension of the cross-hedging models developed by Kolb and Chiang (1981) and by Kolb (1982). The difference between the proposed Expectations Hedge Model and those developed by Kolb and Chiang and by Kolb rests in how the investor is seen to judge the future shape of the yield curve. The two dura- tion based hedge ratios assume the investor sees a fixed structural relationship that exists at the time of the institution of the hedge. That relationship is described by flat and/or parallel shifting yield curves over the holding period. These are simplifying assumptions that are unrealistic both in their view of how investors use information and in how interest rates have changed in the past. 7Those differences could be in type of asset, maturity, delivery date, or risk. 36 An Expectations Hedging Model The expectations hedge model uses a stochastic statistical model of the formation of interest rates for developing hedge ratios. The hedge ratio calculated is an expected hedge ratio rather than an exact ratio developed from a relationship between expected price changes of an asset and a corresponding hedging instrument. The expectations hedge uses the term structure at the time of the hedge as one piece of information in an information set that includes prior realizations of and changes in forward rates. The term structure that is relevant to the decision maker is the unknown term structure that will exist at the end of the planning horizon. The duration based hedges project the yield curve that exist at the time of the initiation of the hedge to the end of the planning horizon. When the yield curve is allowed to shift in a more realistic manner, it is expected that the duration based hedges would be less effective than would be an expectations hedge that recognizes that expec- tations are formed by risk averse investors using more information. A model for an expectations hedge ratio was developed that allows for multiple random shocks to the term structure of interest rates. The investor uses the expectations hedge: 37 where, "O ll Price of the investment at termin- ation of the hedge, l + forward rate N-periods ahead, Price of the asset underlying the futures contract at delivery, and Cruz?) H ll 1 + forward rate in period p to try to ensure that AP. = -HAP. 3 SO H = APi/APj The expectations hedge is derived in the following manner: I C s+J C. Pi = Z lE-s ' P' = Z jt-s t=s+l (R1) 3 t=s+l (Rj) ap. ap. N = _1 _J dRN dRN (Note that this measure has no time dimension and is similar to Macaulay's original conception.) s+J II M dP./dR l N t=s+l t=s+J+l s+J d P /dR = Z 3 t=s+l where, J’zwro” c. N Then, E(H) 38 t ~ R Cit t 1T I9 t fi 2 Z p=s+l psg;l p N=S+l R N I- t ~-I| . c. ,1 )1 It H R t ~ 2 s+J p=s+l p p=s+l N=s+l N r- ~q C t it 7T 9 ~ t = + Ttr R 2 2: p51 p=s+l p N=s+l N planning horizon, maturity of investment, number of periods to be hedged, l + forward rate in period p, l + forward rate N-periods ahead, 1 + vield to maturity of the investment, and l + yield to maturity of the asset underlying the futures contract. dPi ~ ~ ARp dRN : —E ~ dP. ~ _Tl ARP dRN — J 39 Ri and Rj are yield calculations that are the product of a series of forward rates. The change in any one forward rate that is part of the product will change the expected yield, but obviously will not change forward rates for periods prior to the change. The yield curve can be shifted or twisted8 or both and this expectations based hedge will still be appropriate. Santoni (1984) relates the elasticity measure of interest rate sensitivity used in the expectations hedge to the duration measure. He shows that the duration of a portfolio of assets and liabilities (a firm) taken together is not simply some weighted linear combination of the dur- ations of each of the assets and liabilities. Rather, the duration of the portfolio can be a value outside the range of durations of the assets and liabilities and may even be negative. He concludes that duration is not as good a measure of interest rate sensitivity as is the elasticity measure . Comparison to the Kolb and Chiang Hedge Kolb and Chiang (1981) calculate a hedge ratio as: H = -RjPiDi/RinDj, E(ARi/ARj) = l , where all symbols are as above and D = Duration. 8A yield curve shifts when the yields over all the observed maturities maintain their relative sizes. Twisting of the yield curve occurs when the relative sizes are not maintained. 40 The yield to maturity of the investment, Ri' and the yield to maturity of the asset underlying the futures contract, Rj’ are expected to change in the same proportional manner over the planning horizon. That is, E(ARi/ARj) = 1. This is the parallel shift in the yield curve that is assumed. This also implies that the shape of the current yield curve is preserved, although the level of rates may change. Kolb (1982) recognizes that a complex yield curve assumption ...is important in making a conceptual advance over the ordinary bond pricing equation (but) the attempt to apply it to all aspects of bond pricing generates more heat than light. (pg. 58) It is hoped that the tests in this study will show that the amount of light generated will justify the attendant heat. The expectation hedge ratio, dPi ~ —— AR aiiN p E(H) = -E , ap ~ ":3" 4R dRN p .L. _ in contrast, uses expectations and variances of the ex- pected yield curves. The investor considers the first two moments of the distribution of hedge ratios calculated from the stochastic process generating forward rates. The hedge ratio is an expected hedge ratio in the sense that it is the mean of a distribution of possible hedge ratios that could occur under various combinations of forward 41 rates that might occur. The investor is not constrained to assuming only a single additive shock to the term structure, a parallel shift. Comparison to the Kolb Hedge Kolb (1982) expands on the Kolb and Chiang hedge to allow a measured relationship between ARi and ARj. He regresses Ri on Rj. This results in the hedge ratio: R P D _ E((ARiI (ARi, AR.) tn) _3__ ,, 3:; R.P D. 3 3 E (AR.|(AR., AR.) 3 1 t1 t = point where hedge is instituted and l+n = previous observation period for est1mat1ng r. So the yield curve expected at the termination of the hedge is the yield curve that exists at the inception of the hedge. The two duration based hedges are calculated from the extrapolation of the existing yield curve. No consider- ation is given to the variability of possible yield curves in the future. Indeed that variability is assumed way. The expectations hedge reflects a distribution of hedge ratios. Variability is a factor in the development of this hedge ratio. If the variance of forward rates forecast further in the future is greater than the variance of near term forecasts, a hedge ratio that considers variance 42 should be more effective than one that does not. The hedge ratios based on duration explicitly mea- sure the duration of the assets and the assets underlying the futures contracts. This duration measure requires the assumption of flat or parallel shifting yield curves. No such assumptions are necessary for the expectations hedge because duration, while implicit in the calculation of the hedge ratios, need not be measured explicitly.9 Both of the hedges based on duration use a calcu- lation of duration that has a time dimension. This is true to Macaulay's measurement of duration but not to the definition. Macaulay's definition would use spot and forward rates rather than yield to maturity. When spot and forward rates are used in the calculation, the time dimension disappears. 9The duration based hedges can be calculated with- out the explicit measure of duration, but they were developed from the theoretical duration concepts. CHAPTER IV HYPOTHESES The goal of hedging is risk reduction. It is use- ful to consider the ways risk is perceived. The usual surrogate for risk that is used to judge risk changes or risk reduction is the variance of returns or, in the case of hedging, the variance of wealth changes. Kolb and Chiang (1981) and Kolb (1982) both use the variance of wealth changes as a result of hedging activities to mea- sure the effectiveness of their hedges. The present study also uses variance of wealth change as a measure of effec- tiveness. Variance as a measure of risk has been accepted in the economics and finance literature at least as far back as the work of Irving Fisher. Markowitz (1952) uses variance of returns as the risk measure in the E — V approach to portfolio selection and Hirshleifer (1965), attributing early statement of a mean - variability analy- sis to Fisher in 1912 says, The mean, variability approach to investment decision under uncertainty selects as the objects of choice expected returns and variability of returns from investments. In accordance with the common beliefs of observers of financial markets, the assumption is made that investors desire high values of the former and low values of the latter - 43 44 as usually measured by the mean (u) and standard deviation (0) respectively of the probability distribution of returns.... (P9. 518) Fisher and Weil (1971) use the standard deviation of wealth changes as a measure of hedging effectiveness. There is much to commend the use of variance as a measure for risk. On the other hand, researchers have also noted that investors see risk as the possibility of loss or of returns below some expected level. Markowitz (1959) states that the semi-variance would be a more appropriate measure to use in the mean-variance analysis as the measure of risk. He then returns to the use of variance because of the computational problems he sees in using the semi- variance. The idea of risk as some below - target variability is a pleasing one. Domar and Musgrave (1944) in a paper discussing the effect of an income tax on risk—taking be- havior recognized risk as probability of actual yield (from an investment) being less than zero.... (pg. 396) And they defined risk more specifically as the sum of each possible loss weighted by the probability of occurrence of that loss. Subsequently, Grayson (1960) studied attitudes to- ward risk among managers engaged in the business of oil drilling which is an endeavor characterized by small prob- abilities of large returns and large probabilities of losses. He described decison-making processes and inferred 45 utility curves for the individuals who were interviewed. The utility curves developed were all steeper for losses than for gains. The consequences of losses were seen to be much greater than the benefits for similar sized gains. Halter and Dean (1971) found similar derived utility func- tions in agricultural pursuits. Kahneman and Tversky (1979) studied choices of risky gambles and found risk averse behavior for gains and risk seeking behavior for losses. This was interpreted as the possibility of loss being a more appropriate measure of risk than other measures. They developed a theory of risk-taking behavior that they call "Prospect Theory." Mao (1970a) calculated semi—variances and com- pared the investment decisions made under expected return - variance to those made under expected return semi-variance and concluded that expected return - semi-variance choice objects are more consistent with utility functions that are not concave at all levels of wealth. He found that these utility functions are more descriptive of investor behavior than are those that are concave downward over the entire range of wealth. Mao (1970b) has also surveyed executives respon- sible for capital budgeting about their attitudes toward risk. He learned that the executives surveyed see risk as the possibility of not meeting some target or required return. Mao concluded that although variance is the mea- sure of risk most used in capital budgeting analysis, 46 risk, in the eyes of decision-makers, emphasizes down- side possibilities. This is more consistent with the semi- variance measure. An approach to calculating a measure of the risk of below-target returns was developed by Fishburn (1977). It is a two parameter model which incorporates a factor for differing attitudes toward risk with risk measured as the probability of not reaching a targeted return. The Fishburn measure is: t Fa(t) =/ (t - Y)adF(Y) -CD where, t = target return, Fa(t) = probability of return below t, Y = observed return < t, and a = risk aversion measure. This measure will be used in judging the efficacy of the hedge ratios in mitigating risk. The Hypotheses l. The variance of the net wealth change is smaller using the expectations based hedge ratio than is the net wealth change using either of the duration based hedges. 2. The Fishburn measure of below-target wealth change is smaller for the expectations hedge ratio than it is for either of the duration based hedges. 47 In order to develop the forecasts of forward rates necessary for this study, the independence of adjacent forward rate series needed to be established. This step was required so forward rates could be forecasted inde- pendently. A secondary hypothesis was tested: 81. Forward rates are independent across time. CHAPTER V METHODOLOGY The effectiveness of the hedging methods used in this study is tested by testing the hypotheses relating to wealth change. The risk that is being hedged is interest rate risk. In order to control for default risk, both the investment and the assets underlying the hedging vehicles are U.S. Treasury securities. Data The source of the bond price data is the CRSPlO Government Bond File. The data used for this study are the month-end bid and asked prices and accrued interest. Bond yields were calculated over two overlapping five-year periods: December 1971 - November 1976 and December 1973 - November 1978, for which the asked prices were the source of price and accrued interest data for the simulation of purchase and subsequent sale of assets at the end of the planning horizon. The investor's asset position was based on the 10Center for Research in Security Prices, Univer- sity of Chicago. 48 49 asked price plus accrued interest at the inception of the hedge. The bonds were ”sold" at the bid price plus accrued interest at the end of the planning horizon. The hedging instrument is the 91-day U.S. Treasury Bill Futures Contract. Trading in T-Bill futures began on the International Monetary Market (IMM) of the Chicago Mercantile Exchange in January of 1976. Prices used to calculate the wealth changes resulting from the hedge positions are the closing prices of relevant contracts on the last day of each month that the contracts are traded. The data is published daily in The Wall Street Journal. The hedges being tested are generated as a result of investors' expectations of the term structure of interest rates. The duration based hedges and the expectations hedge differ in their recognition of the way in which investors form those expectations. The duration based hedges are a function of the term structure that exists at the inception of the hedge. The term structure at in— ception is the hedger's forecast of the term structure that will exist at the end of the planning horizon. The expectations hedge comes about from the in- vestor's explicit forecast of the term structure. The investor is faced with an efficient market and forms expectations rationally. Thus, the rational investor will act as if all past information and all relevant current information and expectations are impounded in current bond and futures prices. 50 In order to forecast future yields, the investor will forecast the single-period forward rates, which are independent for adjacent months, based on past forward rate information contained in past yields. The fore- casted forward rates and forecast variances are used to simulate possible realizations of the forecasted forward rates. Hacket (1978) used a simulation approach to develop yield curves. Rather than simulate individual forward rates from time series model, though, Hackett used both multiplicative and log models for the term structure and simulated single shocks to the yield curve. His asset portfolio is restructured after each shock. In essence, the same shock value is applied to each forward rate that is inferred from a yield structure at a specific point in time. Yield—to-Maturity Yield curves are calculated from the CRSP Govern- ment Bond Price data. For each five year calculation period, monthly annualized yields-to-maturity are calcu- lated for holding periods from one month to 24 months in monthly increments. A new yield curve was calculated at the end of each of sixty months represented by each of the two overlapping five-year periods. This results in sixty yield curves for each five—year period. These calculated yield curves are used to calculate a series of 51 sixty single-period forward rates of from one to 23 months forward. Each yield that is calculated represents an average of the yields implied in the prices of all the U.S. Treasury securities that were outstanding over each yield period.11 The yields are calculated as: m c 5:13:22 “fl—E io it=l (1+§) where, = number of securities outstanding, = number of cash flows to maturity, coupon payment frequency, and 23:33P- II = yield-to-maturity. A computer program for calculating the yields is found in Appendix A, Table A1. The term structures at each inception period for the hedge tests are listed in Table 1. The yields are those used to develop the duration based hedges. They are the yields that are projected forward as the fore- casted yields at the end of the planning horizons. llCallable bonds, flower bonds and other non- standard instruments are excluded as the yields on these bonds are not amenable to direct calculation but require simplifying assumptions to make a calculation of their yields. 52 Table l Annualized Yields Egigggg as of November 30, 1976 1978 1 .04165 .09079 2 .04443 .09212 3 .03910 .09584 4 .04555 .09662 5 .04677 .09792 6 .04247 .09992 7 .04775 .10079 8 .04764 .10309 9 .04836 .10218 10 .04903 .10149 11 .04926 .10229 12 .04961 .10009 13 .04956 .10143 14 .04993 .10140 15 .05075 .09781 16 .05184 .09678 17 .05273 .09971 18 .05264 .09975 19 .05364 .09784 20 .05396 .09849 21 .05483 .09538 22 .05728 .09449 23 .05416 .09766 24. .05425 .09507 Forward Rates In order to calculate the hedge ratios under the expectations method, it is necessary to calculate the forward rates implied in each of the yield curves. This calculation results in sixty forward rates in each of the two five year calculation periods for each of the 23 months forward. The single period forward rates are 53 ( )n+l l + R 12 F = n+1 1. £_ (1 + Rn)12 R = Yield to Maturity and calculated as: where, n = number of months forward These forward rate series are used to forecast forward rates. These forecasts and the forecast variances are used to simulate the investor's forecasting process for hedging interest rate risk. The series of forward rates that are calculated represent a separate time series for each of the 23 months forward. For example, there is a six-month forward rate implied in the yield curve calculated as of December 31, 1971, and another six-month forward rate implied in the yield curve that exists on January 31, 1972, and 58 more six—month forward rates covering the yield curves from February 28, 1972 through November 30, 1976. A com- puter program to calculate the forward rates is included in Appendix A, Table A2. Forecasting Forward Rates The time series of forward rates provide the investor with information necessary to forecast succeeding forward rates. One method of forecasting is that develOped 54 by Box and Jenkins (1970) where a variable observed is described in terms of previous values of the variable and a series of random shocks occurring at previous times. Box and Jenkins show a method of estimating values for coefficients of previous values and previous and cur- rent random shocks that expresses autocorrelated time series in terms of autoregressive and/or moving average components. The models so expressed can be used to pro- duce forecasts whose variances are minimized. The models of forward rates are generally referred to as ARIMA (Auto- regressive, Integrated, Moving Average) models and take the general form d \ .— : ¢(B,(1 B) zt so + 81(B)at where, _ 2 P ¢(B) — 1 - ¢lB - 62B - ... ¢pB 9(a) = 1 - e B — e 32 — e Bq 1 2 ... 8 B = Backshift operator, d = number of observation periods of difference; zt - zt-d’ a = random shock at time t, and 2t = observation at time t, Box and Jenkins (1970). This model can also be expressed in terms of the random shocks which is useful for modeling and fore- casting because the procedure assays the current obser- vation of a time series as the result of a series of un- correlated shocks, each shock carrying a weighted value. 55 The mean of the previous shocks is constant. Each of the two sets of modeled using the ARIMA process. is zero and the variance 23 forward rates is Table 2 lists the models. Tmfle2 Month ARIMA Models For- ward Period 1 Period 2 6 2 l (1-B)zt=(l-.375B )a (l-B)(l+.458B )z =(1-.298B)at (2.75) (-3.82) (2.27) 2 (l-B)zt=(l-.502B)a (l-B)z =(l—.579B)a (4.15) (5.16) 3 (l-B)zt=(l—.453B)a (l-B)zt=(l-.545B)a (3.88) (4.91) 4 (l-B)z =(l—.39SB)a (l-B)z =a t (3.28) t t 5 (l-B)zt=(l-.459B)at (l-B)zt=(l-.579B)a (3.72) (5.28) 6 (l—.453B)(z -2.897)=a (l—B)zt=(l-.527B)a (3.93) (4.53) (4.74) 7 z -5.335=a z -5.634=a (26.93) (28.14) 8 z -5.l4l=a zt—5.509=a (21.29) (23.97) 9 z —5.628=(l+.33OB+.413B3)a zt-5.923=at (18.06) (-2.99) (-3.71) (33.20) 10 z -5.569=a (1-.346B2)(z -3.850)=a (26.55) (2.74) (5.08) 11 z —5.640=a z -5.960=a (20.11) (24.51) 56 Table 2--Continued Month ARIMA Models For- ward Period 1 Period 2 6 2 12 (1-.3l3B )(z -5.166F41+.289B )a z -5.792=a (2.43) (13.65) (—2.21) (34.65) 13 z -5.419=a z -6.040=a t (24.96) t t(32.36) t 14 zt-5.4l9=a z -5.884=a (25.61) (32.63) 15 (1-.3128)(z —3.947)=a z -6.181=a (2.56) (5.50) (24.95) 16 z —5.886=a zt-6.171=a (37.26) (30.29) 17 (l-B)z =(l-.741B)a Zt-5.899=(1+.438B5)at (8.11) (23.87) (-3.02) 18 (l-B)Zt=(l—.757B)a Z -6.27l=a (8.21) (28.94) 19 z -5.9l8=(1+.358B6)a z -6.391=(1+.323B3)a (22.81) (-2.71) (21.99) (-2.38) 20 Z -5.842=(l+.4OlB)at Z —5.9l3=at (23.16) (-3.35) (34.21) 21 zt-5.888=at z -5.8ll=a (29.71) (29.11) 22 zt-6.040=at z -6.034=a (31.76) (28.70) 23 zt-4.823=a z -5.288=at (12.06) (16.17) t - values in parentheses. 57 It is of interest to note that 25 of the series are simply randomly distributed about some constant value. There are no significant autocorrelations so the best forecast of these processes is the constant. These ran- dom processes are consistent with the notion of market efficiency. As will be explained below, the planning horizons for the hedge calculations will be limited to 24 months. Thus it will be necessary to allow for forecasting of forward rates up to 24 months or steps ahead. The fore- cast variances are also calculated since these variances are used to develop the simulations from which the expec- tations hedge ratios are calculated. The ARIMA modeling process can be extended to the forecasting step. Box and Jenkins show that the forecasts can be seen as a weighted sum of past and current random shocks, wjat+2-j' to = l where Q = periods ahead to be forecast. The w-weights need to be calculated. They are a function of the ¢-values and e-values estimated in the original modeling steps for the series: (1+¢1B+¢2B2+¢3B3....) = 2 q 1 61B 923 ... qu The forecast variance at any lead (Q) is calculated as: The forecasts and forecast variances used in this study were generated by an SPSSx program which uses this W—weight method of forecasting. The original modeling of the forward rate series was also accomplished using SPSSx. Before the forecasts of the individual forward rates could be attempted, it was necessary to determine the independence of adjacent forward rates. If adjacent forward rates, i.e. the two month forward rate and the three month forward rate, were not independent they would have to be forecasted jointly. Thus, the secondary hypothesis of independence of forward rates. The test of independence of adjacent forward rates is the test suggested by Haugh (1976). It is a test of the lagged cross-correlations of the white-noise residuals of two time series modeled by the ARIMA method. In this case, the time series in question were series of adjacent forward rates. Haugh shows that the cross-correlation function over a given number of lags for the residual series from two ARIMA models are normally distributed. This makes a test statistic available which can be tested as distributed chi—square. The Haugh statistic is calculated: 59 M 2 I -1" H =N z (N-(kl) r.. (k)2,d.f.=2M+l, k=-M 13 where, N = length of the series, lkl 1 would indicate greater effectiveness of the expectations hedge in reducing risk of negative returns. This test is performed for levels of risk aversion, a = l to a = 4. A similar test was used by Johnson and Walther (1984) to determine hedging effective— ness in the foreign exchange market. A program to calcu— late the Fishburn measure is included in Appendix A, Table A9. 70 Summary of Steps in Analysis 1. Calculate bond yields from CRSP Government Bond Files. a) This is an average yield assuming equal investment in each bond. b) Two overlapping 5-year periods were chosen: 1) December 1971 - November 1976 2) December 1973 - November 1978 c) Each month in each 5-year period was used as a starting point and a 24 month yield curve was calculated for each. The yield calculations were made using average bid prices + accrued interest. The number of securities used for each calculation ranged from one to nine for any month. 2. Calculate forward rates from the calculated bond yields. a) This results in two sets of forward rates. b) There are 60 calculated forward rates for each of one to 23 months forward. 3. Model each series of forward rates using ARIMA process. a) This results in two sets of ARIMA models for each of from one to 23 months forward. b) The residuals from these models are used to test for independence of adjacent month forward rates. 4. Adjacent forward rates are tested for independence. a) The Haugh test is used which measures C) 71 cross-correlation functions of white noise residuals. This test is based on the significance of a chi-square like statistic. Most of the adjacent forward rates were shown to be independent. 5. Forecast forward rates. a) b) The ARIMA models are used to develop forecasts of from one to 24 months (steps) ahead for each of the starting months. There are 24 starting months for each testing period. Forecasts and forecast variances are gener- ated from the ARIMA models of forward rates. 6. Simulate yield curves. a) b) Simulation of yield curves for each of 24 steps ahead is done by randomly selecting a value from the range of the forecast variance for each forward rate (one month forward to 23 months forward) and adding that value to the forecast value. This process is repeated 100 times for each step ahead yield curve. These simulated yields form the basis for calculating the forward rates used in the hedge ratio calculations. 7. Select bond maturity - planning horizon combinations. a) One hundred combinations (maturity < planning horizon) are selected at random from the bonds 72 available for each of the two five-year periods. b) The first set of combinations represents shorter periods of exposure to interest rate risk than does the second set. During the first testing period there were only five or six futures contracts traded. Only 15 - 18 months could be hedged with the available contracts using the expectations hedge. 8. Calculate hedge ratios for each bond - planning horizon combination for each of the 100 realizations of the yield curve simulation for the number of months ahead represented by the planning horizon. a) Duration hedge. b) Expectations hedge. 0) Adjusted duration hedge. 1) Duration hedge ratio is adjusted for the correlation between the three-month rate and the appropriate rate for the maturity of the bond. 2) Each of the 24 month's annualized yields is regressed on the three-month yield and the regression coefficients are used in the hedge ratio calculation. 9. Calculate the average number of futures contracts entered into for the 100 realizations of the yield curve simulation. 10. 11. 73 Calculate the mean wealth change and variance of wealth change based on a $10,000,000 investment and sale and closeout at the end of the planning horizon. An F—test is used to measure the difference in vari— ance. Calculate the Fishburn statistic which considers the probability of earning below-target returns as the risk. For a hedge, this means that negative wealth changes are to be minimized. There is no parametric test. CHAPTER VI RESULTS AND CONCLUSIONS Independence of Forward Rates The Haugh statistics, M . H=N2 z (N- Ikl) l ri. (k)2 K=-M 3 N=6O M=24 1123 'l24 1 ' 3 2 d.f.=49 are shown in Table 4 for each of the two five—year periods used as a base for forecasting. For the first five-year period, December 1971 - November 1976, independence can be rejected for four sets of adjacent forward rates. The Haugh statistics for 7 and 8 months forward and 8 and 9 months forward show strong evidence of dependence. Two others are barely significant. For the second five-year period, December 1973 - November 1978, four different sets of forward rates cannot be considered independent. There are two that show a strong dependence measure 18 and 19 months forward and 19 and 20 months forward. 74 75 Table 4 Adjacent Haugh Statistic Months Period 1 Period 2 1,2 62.446 46.245 2,3 54.813 43.286 3,4 46.314 35.050 4,5 53.025 40.600 5,6 60.184 63.218 6,7 69,865* 53.197 7,8 87.290* 51.050 8,9 73.350* 62.678 9,10 52.890 55.433 10,11 60.642 65.633 11,12 48.138 36.010 12,13 62.177 29.398 13,14 50.824 38.786 14,15 51.055 41.698 15,16 52.466 51.731 16,17 57.673 37.583 17,18 47.045 68.064* 18,19 47.335 74.689* 19,20 56.332 78.594* 20,21 72.951* 59.790 21,22 52.134 65.246 22,23 37.659 69.856* * - significant at a = .05 Because of the small number of significant depend- encies, the forward rate forecasting was done independently. The independence of adjacent forward rates is of secondary importance to this study since the forward rates are used to provide a number of realizations of yield curves which represent simulations of investor perceptions. These simulations are meant to provide a range of possible per— ceived outcomes and not to be precise forecasts. The forecasts of forward rates and the attendant forecast 76 variances are listed in Appendix B. It can be seen that the investor is making forecasts up to 24 steps ahead. The forecast variances increase with increasing leads. Hedging Activity Each of the hedging calculations determines a hedge ratio which is converted into a number of futures contracts sold to hedge the $10,000,000 initial investment. As was noted earlier, the duration based hedges use con- tracts for only one delivery month while the expectations approach allows use of contracts calling for delivery in a number of months. Under the expectations method as many as eight different delivery months may be used in the hedge. The number of contracts sold for each of the maturity/planning horizon combinations for each of the inception points is included in Appendix C, Tables C1 to C6. Each individual hedging decision results in a change of wealth over the period until the end of the planning horizons. The mean wealth changes for each of the hedging methods are listed in Table 5. 77 Table 5 WEALTH CHANGES Inception 11/30/76 Regression Expectations Duration Adjusted Hedge Hedge Hedge Mean Wealth $-52,363 $-73,307 $-74,227 Change Wealth Change 2.297 x 1010 2.330 x 1010 2.361 x 1010 Variance Inception 11/30/78 Regression Expectations Duration Adjusted Hedge Hedge Hedge Mean Wealth $91,723 $74,199 $71,936 Change Wealth Change 2.784 x 1010 3.315 x 1010 3.268 x 1010 Variance The F-ratios in Table 6 show the comparisons of the variances of wealth change for the tested hedging procedures. 78 Table 6 F - Ratios F(100,100) Regression Inception Duration Adjusted Hedge Hedge 11/30/76 Expectations Hedge 0.986 0.973 11/30/78 Expectations Hedge 0.840 0.852 There is no evidence that the expectations hedge reduces the wealth change or its variance any more than do the other hedging approaches. The variance of wealth changes was somewhat less under the expectations hedges than either of the others but not significantly so. Test of Lower Partial Moments The test statistic M is calculated for each of the three hedges for each of the inception points. Table 7 illustrates the results of those calculations. At all levels of risk aversion measured, repre— sented by a = l to a = 4 it can be seen that M > 1 for com- parisons between duration based hedges and the expectations hedge. Increasing values of a represent increasing levels of risk aversion with d = 1 corresponding to risk neutrality and higher values describing risk aversion.12 12An a = 2 corresponds to the semi-variance measure used in earlier analysis, such as that of Markowitz (1959). 79 Table 7 Fishburn a - t Ratios Inception 11/30/76 Dur/Exp Regr.Adj/Exp Regr.Adj/Dur a = 1 1.09361 1.10389 1.00939 8 = 2 1.18563 1.20625 1.01739 a = 3 1.24886 1.27739 1.02284 a = 4 1.28212 1.31663 1.02691 Inception 11/30/78 Dur/Exp Regr.Adj/Exp Dur/Regr.Adj d = 1 1.13208 1.15594 1.02107 a = 2 1.32001 1.28691 0.97492 a = 3 1.66940 1.49204 0.89376 a = 4 2.25932 1.80195 0.79756 Conclusions The results obtained from the simulation of invest- ment and hedging behavior over a large number of maturity/ planning horizon combinations leads to the conclusion that hedging activity based on forecasts of forward rates that consider the possibility of complex shifts in the term structure of interest rates over a planning horizon provide better hedges if not better immunization. Such hedging activity reduces the interest rate risk of an investment position of the type described in this study to a greater extent than does hedging activity based on the duration 80 models. This is so when risk is measured in terms of the achievement of a stated goal. The stated goal in this study is a minimum wealth change of zero over the planning horizon. The risk is one of not achieving this minimum goal. When risk is measured as variance of wealth change over the planning horizon there is no support for the dominance of the expectations hedge over the duration hedges. The investor whose attitude toward risk is described by aversion to the possibility of earning below-target returns is able to reduce that risk best by making forecasts of the course of interest rates using all available infor- mation and including those forecasts in the calculation of the hedge ratios. The simplifying assumption of parallel shifts in the yield curve that is used for the duration hedge calculations appears to be so restrictive as to pre- vent the investor from realizing the most effective risk reduction. Further Research The current study does not consider transactions costs. Negotiated commission structures have resulted in round—turn commissions of less than $25 per contract for financial futures contracts. This study attempts to com— pare different hedging strategies where a similar number of contracts are used for each hedging method. Thus, the differential effects of transactions cost would be small. They were disregarded. Further study could test the 81 absolute effects of transactions costs on the hedging results. This study considers the use of 91—day T-bill futures contracts as vehicles for hedging interest rate risk over periods up to two years. This could represent an extreme example of cross-hedging. Similar studies could be undertaken using futures contracts in longer maturity instruments to hedge positions over longer planning hori- zons. The hedging vehicle could be T-bond futures or GNMA futures which call for delivery of securities of maturities of eight years or longer. If nothing else, use of these contracts which will closer match the durations or maturi- ties of the investment being hedged should simplify the hedge ratio calculation and reduce the variety of contracts that need to be bought or sold to hedge a position. An added difficulty, though, in using T-bond futures contracts is that there are a number of Treasury bonds that are deliverable on any contract. Each of these bonds repre- sents a different coupon and duration combination. The optimal bond to deliver on a contract needs to be deter- mined. This cannot always be accomplished as a unique solu— tion at the time of the inception of the hedge, so the hedge ratio is not determined either. Further analysis of the relationship between bond duration and the optimum bond for delivery on a futures contract might mitigate this difficulty. 82 Further study could also include differential levels of default risk between the investment and the asset underlying the hedging vehicle. The default risk cross-hedge would consider the variability of the interest rate spread for assets of differing levels of perceived risk. It could be possible to develop appropriate hedging behavior to offset the interest rate risk in the risky bond portfolio of a financial institution using futures positions in default risk-free U. S. Treasury securities. The entire area of risk measures has been opened to question by the works of Kahneman and Tversky (1979), Coombs (1975), Swalm (1966), and Williams (1966) to name a few. There is reason to expect that future studies will move away from utility maximization as the accepted goal of investors. The new approaches may well change some long-held ideas of normative investment behavior. APPENDIX A 20 30 40 50 6O 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 Calc 83 APPENDIX A Table A1 ulates YTM and counts number cf times each is entered. OPTION BASE 1 DIN DIM DIH DIM DIE DIM 092 IF IN? 601 PRI PR1 rru C(I 601 093 rap cro PRI INP rap IF BHE cus HT su cor INP IF INP IF( IF INP IF IP 1c cr 601 INP INP YTH(84,25) C(84,25) PI(25) 11(25) 50(25) YH(84,25) N "ytm.sum" FOR INPUT AS #1 EOP(1) THEN CIOSE: GOTO 190 UT #1, rmr.1,a,cc.nvrin 0 160 , NT "AVYLD ("I","J") is" AVYLD; NT "COUNT ("I","J")is" cc (I,J) = YHT ,J) = cc 0 110 N ”BNE.CHS" FOR INPUT AS #2 UT #2,BlE,CNS SE #2 NT " HIGHEST BHE IS ",BHE UT " session start maturity ",IH UT " session start start ",IS BEE > 0 THEN GOTO 280 = IS - 552 = In - IS = IN = 15 O 360 UT " next bond maturity ",5? HT 0 THEN GOTO 1180 UT “ next bond start month ",SH HI=HTT) AND (SH=SHT) THEN TQ = 1 ELSE T0 = 0 HT > SH THEN 6010 360 ELSE 6010 310 UT " T-Bill = 1 ",8Q 80 = 0 THEN GOTO “20 = O = 0 = 1 O “90 UT " interest payment ",1? UT " INITIAL CASH ELGHS “,ICF 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910 920 930 940 84 CF = ICF IF CE = 1 THEN GCTO 490 FOR D = 1 IO CF INPUT " BEGINNING DAYS ",N(D) NEXT D CH = HT - SM Z = HT - SM BET = SH - 552 60588 1460 HTT = HT SHT = SM IF cm > cns THEN cns BM = sn - 552 IF BH > BHE THEN Bar = B! IF CF > 1 GOTO 610 INPUT " BEGINNING DAYS ",J IF 2 = 1 THEN 0010 630 FOR G = 1 T0 2 IF Bar + G > 85 THEN G = z: GOTO 1160 PP = PI(G) IF 80 = 1 THEN GOTO 800 AC = AI(G) IF CF = 1 THEN GCTO 800 s = SU(G) IF N(1) - s > 0 THEN GOTO 760 CH IF CF > 1 THEN CE = CF — 1 IF CF = 1 THEN J = N(2): GOTO 810 FOR I = 1 TO CF w = I + 1 N(T) = N(H) - 5 NEXT T GOTO 820 FOR T = 1 TC CF N(T) = N(T) - 5 NEXT T GOTO 820 s = SU(G) J = J - S PV=PP§AC PFS=O LP=IP+1OO V=365*((((IP*CF)+100)/PV)-1) IF CF=1 THEN R=V/J FLsr R~= V/N(CF) IF CF)! THEN GOTC 910 P1=J/365 PFS=LP/((1+R)**P1) GOTO 990 FOR A=1 TO CF E(A)=N(A)/36S IF A=CF THEN GOTC 960 PP (A)=IP/( (1‘3) “‘5 (M) 950 960 970 980 990 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 85 GOTO 970 PP(A)=LP/(l1+R)**H(A)) PFS=PPS+PF(A) NEXT A IF ABS(PFS-PV) <= .010001 THEN GOTO 1070 IF ABS(PFS - PV) < .05 THEN GOTO 1030 IF PPS > PV THEN B =,B + .0001 IF PFS < Pv THEN B = E - .0001 IF PPS > PV THEN B=B+.00001 IF PPS < Pv THEN B=H-.00001 PFS=0 GOTO 870 YN‘BN,CN) = R YTH(BN.CH) = YTN(BN,CH) + YM(BM,CN) C(BN.CN) = C(FN,CN) + 1 IF 2 = 1 THEN GCTO 1170 IF NT - (PM + 552) = 1 THEN GOTC 1160 IF EN = 84 THEN GOTO 1170 EN = BN + 1 IF BN > BNE THEN ENE = PM OH = CH — 1 NEXT G GOTO 310 OPEN "ytm-sum" FOR OUTPUT AS #1 FOR I = 1 TO ENE FOR J = 1 TC CNS YNT = 0 INT = YTN(1,J) CC = C(I,J) IF YET = 0 THEN GOTO 1310 AVYLD = YNT/CC WRITE #1, YBT,I,J,CC,AVYLD GOTO 1310 PRINT 'AVYLD(”I","J")is" AVYLD; PRINT "("I","J")count is" CC; PRINT "YTN is" TNT NEXT J,I OPEN "BHE-CNS" POE OUTPUT AS #2 HBITE #2. ENE, CNS PRINT " ENE UNITE " BHE " CNS HRITE " CNS CLOSE 51,82 END INPUT " F ".P INPUT " c ",Q IF F = 10 THEN GOTO 1430 C(F'Q’ = CIFOQ) '1 YTN(F,Q) = YTfllva) * YH(F¢Q) GOTO 1370 INPUT ” bl ",BN INPUT ” cm ”,CN STOP 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 86 FOR E = 1 To 2 IF ENT + E > 85 THEN GOTO 1500 INPUT " pp ", 91(8) NEXT B IF EQ = 1 THEN GOTo 1550 FOR U = 1 To 2 IF ENT o H > as THEN GOTO 1550 INPUT " ac ". AI(E) NEXT H IF TQ = 1 THEN GOTO 1600 FOR P = 1 TO 2 IF ENT + F > 85 THEN GOTO 1600 INPUT " subt ", SUtF) NEXT F RETURN 10 20 30 40 50 6O 70 420 87 APPENDIX A Table A2 FORUABD RATE CAICULLTION OPTION BASE 1 DIN rantzu.eu) DIN HYL(84,25) DIN ILD(84,25) open "ytm.sum" for input as #1 IF EOF(1) THEN CLCSE: GCTC 110 80 input ‘1, ynt.i,j.cc,avyld 9o ILD(I,J) = NVTLU 100 GOTO 70 110 I = 0 120 J = 0 130 I = I + 1 140 J = J + 1 150 IF TLD(I.J) = 0 THEN A = J — 1: GOTO use 160 IE J = 1 THEN GOTO 200 170 D = J - 1 180 IF YLD(I,D) = 0 THEN T12 = YLD(I,J): GOSUB suo 190 IF SE = 1 THEN GCTC 290 200 HYL(I,J) = ((1 + YID(I,J))**(J/12)) - 1 210 IF (J = 25) AND (I = 84) THEN 6010 230 220 IF J = 25 TEEN GCTO 120 ELSE GOTO 140 230 FOR P = 1 T0 84 240 FOR H = 1 To 25 250 IF H = 1 THEN GOTO 340 260 K = H - 1 270 N = P — 1 280 IF P = 1 THEN N = P + 1 290 PNU(N.E) = ((1 + NYL(F,H))/(1 + BYL(F,K))) - 1 300 IF FND(K,F) < 0 AND 55 = 0 THEN GOTO 720 310 IF PED(K,P) < 0 THEN END(K,P) = PHD(K,N) 320 35 = 0 330 SE = 0 340 NEXT 3,? 350 open "fed.rat" for output as #2 360 FOR L = 1 TO 24 370 FOR N = 1 T0 84 380 PR = PHD(L.B) .390 write #2. fr, 1, n 400 FOR I = 1 T0 3 410 IF H = T*L THEN GOTO 460 NEXT T 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 NEXT H,L Close #1,#2 END 88 PRINT " FCEHAED RATE (”I",”fl")IS" FE GOTO 430 IF SW = 1 AND J 25 THEN GOTO 330 IF SE = 0 AND J = 25 THEN GOTO 120 IF YLD(I,A) = 0 THEN GCTC 140 TY1 = ILD(I,A) AA = A GOTO 140 PRY = IYZ/TY1 DD = J 2 = DD - AA HPRY = PRY**(1/z) Q = z - 1 IF Q = 1 THEN 6010 670 FOR G = 1 TO Q N = AA * G P = HPRY**G TLD(I,N) = ETL(I.U) = NEXT G EETUEN U = AA + p = NPET ILD(I.U) NYL(I,N) GOTO 660 I = F J = a YLD(P,H) EN = 1 $5 = 1 GOTO 150 10 ((TY1 + 1)**P) - 1 1(1 + YLD(I.W))**(H/12)) ((TI + 1)**p) - 1 ((1 1 YLDKI.H))**(fi/12)) 1 1 89 APPENDIX A Table A3 20 LAGGED CROSS-CORRELATIONS (AFTER EAUGH,1976) 3o OPTION BASE 1 no DIN 330150) 50 DIN RH02(50) 60 DIN CBI(SO) 7o OPEN ”CHISQ.L24" EUR INPUT NS :1 80 IP EOP(1) TEEN CLOSE: GOTO 120 90 INPUT #1, x2.I,J 92 PRINT " CHISQUARE ALREADY CALCULATED FOR " I N." J " IS " x2 100 CRI(I) = x2 110 GOTO 80 120 INPUT " NUNRER CF LAOS + 1 ", R 130 INPUT " NUNBER CF CROSS CORRELATIONS ", N 132 INPUT " SESSION START EONTR P, s 140 I = S 150 J = I 0 1 152 T = 0 170 L = (2*K) - 1 172 PRINT "INPUT CCF FOR HCNTHS " I "AND" J 180 FOR A = 1 T0 L 190 INPUT ' RBO ", RHO(A) 192 IE RHO(A) = 9999 THEN GOTO 272 194 RHO‘A)=RHO(A)/1000 200 NEXT A 210 FOR B = 1 T0 L 220 a = K-8 230 RHOZ = RHO‘E)**2 zuo T = T + (RHOZ/(N - ABS(N))) 250 NEXT B 260 CHI(I) = T * N**2 270 I = J: GOTO 150 272 PRINT " HAUGH TEST STATISTICS FIRST FIVE YEARS " 273 PRINT " 24 LAGS 274 PRINT " 276 PRINT " 280 OPEN "CHISQ.L24" FOR OUTPUT AS 81 288 P = I - 1 290 FOR I = 1 T0 F .300 J = I O 1 310 12 = CHI(I) 320 WRITE t1. X2.I,J 330 PRINT " CHI SQUARE FOR TESTING (" I "," J "1 IS " X2 " DF = " I 340 NEXT I 350 CLOSE #1 360 END 370 STOP 9O U'IDGO 11 77 39 80 30 33 40 89 88 25 20 91 APPENDIX A Table A4 A PROGRAM To CALCULATE SIHULATED YIELD CURVES AND THEIR VARIANCES- 100 YIELD CURVES PILL RE PORECASTED FOR EACH STEP AHEAD FORECAST. PORHAT(P10.8) DIEENSION F(552),VF(552),S(552),VS(552),FRD(100,24) DIHENSION SAVILD(24).TYLD(100,24),VARNCE(24) DINENSION ISN‘SO) INTEGER 2,A,R,Y,C,D c=2u D=C+2u Do 11 L=1.so READ(1,77)ISN(I) CONTINUE PORHAT(I12) ISEED=ISN(D) VRITE(6.39)ISEED.C,D POREAT(' 1,112.1 IS THE SEED NUHRER 1.12.1 PROH Pos ‘,I2) CALL INTGEN(ISEED) READ(u,3o)E READ(5,40)VF FORNAT(F6.4) PORHAT(P10.6) FORNAT(SX.F6.4) Do 20 I=1,1oo Do 25 Y = 1,23 J=C+((I—1)*24) CALL ANORH(REAL) FRD(I,Y)=((SQRT(VF(J))*REAL) + F(J))/1000 IF(I.GT.1) GOTO 89 HRITE‘12,33)VF(J) NRITE(13,33)P(J) VRITE(11.50)TUD(I,Y) IF (Y.EQ-1) GOTO 99 SYLD = SYLD*(FHD(I.Y)+1) T=12.0/Y TYLD(I.Y) = (SYLD**T)-1 VRITE(8,50)TYLD(I.Y) CONTINUE CONTINUE CALCULATE AVERAGE YIELD CURVES AND VARIANCES D0 ‘70 A=1.2u YLDSUH = 0.0 60 55 70 99 200 92 Do 60 R=1,100 YLDSUH = YLDSUR o TYIU(E,A) CONTINUE SRVYLD(A) = YIDSUH/100.0 VRITE(9.50)SAVYLD(A) VARSUH = 0.0 NOR CALCULATE VARIANCE DO 55 L=1.100 VARSUN = VARSUH + ((TILD(L,A) - SAVYLD(A))**2) CONTINUE VARNCE(A) = VARSUH/100.0 NRITE(10.50)VARNCE(A) CONTINUE GOTO 200 SYLD = EVD(I.Y)+1 GOTO 88 STOP END 93 APPENDIX A Table A5 C EXPECTATIONS HEDGE RATIO CALCULATION DIEENSION FND(24,2300),COUP(100),NP(100),NAT(100),PS(100) DIHENSION ACS(1001,1T(5).NC(100),HR(8,8).P(24).RN(8,8),HD(8,8) DIMENSION C(8).PU(100).ACP(100),HE(8),CEA1100,8).CR(100,8).CD(8) DO 30 L = 1,100 READ(1.1S) NP‘L),NAT(L),COUP(L).PS(L),ACS(I),NC(L).PU(L),ACP(L) 30 CONTINUE DO 10 I = 1,24 DO 20 J = 1,2300 READ(2,25) FHD{I,J) 20 CONTINUE 10 CONTINUE HRK = 0 DO 40 H = 1,100 II = (NATlN) - NP(N))/6 NC]? = NC (11) IPwCF .NE. 6) II = II +1 D0 60 JJ = 1,11 200 C(JJ) = COUP‘N)*5 IF(JJ .EQ. II) C(JJ) = C(JJ) + 1000.0 HF(JJ) = HCP + (JJ - 1)*6 60 CONTINUE NR = NP(N) D0 64 NA = 1.100 SN = 10000000.0/((PU(H) v ACP(N))*10) 100 DO 62 HG = 1.11 D = 1 IB 1 + {NA - 1)*23 IL NP(HG) 0 (KA - 1)*23 DO 66 KL = IE,IL 000 D = D*(1 + PUD(NR,RL)) 66 CONTINUE 050 = D**2 CD(NG) = C(NG)/DSQ NEXT NE CALCULATE THREE AT A TIRE (NONTHS FCRHARD) AND CALCULATE APPROPRIATE HEDGE RATIO TO BE USED WITH EACH FUTURES CONTRACT DO 90 NP = 18.11 NR = KP - 18 + 1 600 P(KB) = 0/11 + PUD(NR,NP)) 90 CONTINUE LC = NF(NG)/3 nnn 94 IF(NCF .NE. 3) IC = IC + 1 IF(HCF .EQ- 6) LC = LC - 1 DO 92 JC = 1,LC 800 JD = 1 + (JC - 1)*3 JE = JD + 2 IF(JE .GT. HF(NG)) JE = NFMG) HN(NG,JC) = 0 PX = 1 DO 12 JH = JD,JE PX = PX*P(JR) 12 CONTINUE HN(HG.JC) = CD‘HG)*PX 92 CONTINUE C NEXT NE CALCULATE THE DENOHINATOR DPF/DR FOR EACH CONTRACT IN C SUCCESSICN D0 22 NN = 1,LC 202 NE = ID + (NN-1)*3 NL = IR 1 2 + (NN-1)*3 DD = 1 DO 2a EN = NP,NL DD = 00*(1 + FNDtNR,NN)) 24 CONTINUE DDSQ = DD**2 DDH = 1000000.0/DDSQ PP = 1 D0 26 LL = NF,NL PF = PF*(1 + EUD1NR,LL)) 26 CONTINUE HD(NG,NN) = DUE/DUN DNN = DD/PF RR(HG,NN) = HN1NG,NN)/HD(HG,NN) HRK = HRN + 1 IF(HRK .EQ- 100) HRITE(U,35)E,RA,HR(NG,NN),NG,NN IF(HRK .E0. 100) ERR = 0 002 CR(RA,NN) = CR(RA.NN) o HR(NG,NN)*SN 22 CONTINUE 62 CONTINUE 64 CONTINUE Do 72 HQ = 1.LC 103 CEA(N,HQ) = 0 Do 7a NC = 1,100 CEA(N.NQ) = CBA(H,HQ) + CR(NC,HQ) 7a CONTINUE CEA(N,NQ) = CRA(N,HQ)/100.0 NRITE(3,45)CEN(N,NC),N,HQ 72 CONTINUE DO 32 HX = 1,100 DO 34 NY = 1,LC CB(HX,NY) = 0 34 CONTINUE 32 CONTINUE 95 no CONTINUE 3s PORNAT(3X.I3,1x,13.1x,r15.1o.1x,11,lx.11) us POBNAT(3X,F12.u,1x.I3.1x.12) 15 FORHRT{3X,212,ES-3,F8.S,P7.6,I1,F8.5,F7.6) 25 PORNAT(F9.8) sou STOP END 35 30 “O 25 100 200 10 20 96 APPENDIX A Table A6 DURATION HEDGE CALCULATION BASED ON FLAT YIELD CURVE ASSUHPTION DINENSION ILC12U).COUP(100),Np(100),NAT(100).DUN(100) DIUENSION G(5),IK(100).NC(100),N(S),PS(100),AC5(100) DIHENSION CB(100),YLCH(100),YLC3(100),F(S),PU(100),ACP(100) DO 30 L=1,100 BEAD(1,15)NP(L),HAT(I),COUP(L),PS(L),ACS(L),HC(L),PU(L),ACP(L) F0NNAT(3X,212,I5.3.PU.5.37.6.11.28.5,E7.6) CONTINUE no no K = 1,2u BEAD(2,25) YLC(K) CONTINUE PORHAT(3X.F8.6) DO 20 IH=1,100 KK(IH) = o XNUN=0 IDEN=0 NC? = ucxlfl) I = (HAT(IH) - NP(IH))/6 IF(HCF .NE. 6) I=I+1 KK(IH) = I N = UAT(IU) D0 10 J = 1.1 c = COUP(IH)*S IP(J .EQ. I) C=c+1000.0 N = UCF + (J-1)*6 CN = N * C D = (1 + YLC(H))**(N/12) YLCH(IH) = 1 + ILC(U) 213) = CN/D XNUH = XNUN 0 E(J) G(J) = C/D XDEN = XDEN f GfJ) CONTINUE DUB(IH) = XNUN/(XDEN*12.0) EN = (-u.0)*((1 + YLC(3))**1.25)*DUB(IH)*XDEN YLC3(IH) = 1 + YLC(3) HD = (1 1 ILC(U))*1000000.0 E(IN) = NN/HD PP = {PU(IH) + ACP(IH))*10.0 CB(IH) = (10000000/PP)*H(IH) CONTINUE DO 50 JJ = 1,100 50 35 ’45 97 UNITE(8,35)H(JJ) UNITE(9.u5)CU(JJ) CONTINUE EONUNT:3X,F8-6) PORHAT(3X,f9.u) STOP END 15 30 U0 25 100 200 10 98 APPENDIX A Table A7 DURATION EEDGE CNLCUINTION BASED ON FLAT YIELD CURVE NSSUNPTION ADJUSTED EOE DEI/DNJ B! REGBESSICN ON 3 NONTE YIELDS DINENSION YLC‘2Q),COUP(100).NP‘IOO),HAT(100),DUR(100) DINENSION 6(5).BITA(24),KK(100),BC(100),H(S),PS(100),ACS(100) DINENSION CB(100),ILCH(100),YLC3I100),F(5).PU(100).ACP(100) DO 30 L=1,100 EEAD(1,15)NP(L).NAT(1).COUP(L).pS(L),ACS(L),NC(L),PU(L),ACP(L) EOENAT(3X.212,I5.3,E8.5,?7.6,I1,EU.5,E7.6) CONTINUE DO no N = 1.24 READ(2,25) YLCtK) READ(3,55)EETA(N) CONTINUE FORHAT(3X.F8-6) D0 20 IN=1,100 KK(IH) = o XNUN=0 XDEN=O NOT = HC(IH) I = (NAT(IN) - NP(IH))/6 IE¢NCE .NE- 6) I=I+1 KK(IH) = I N = NET(IN) DO 10 J = 1.1 C = COUP‘IH)*5 IF(J .EQ. I) C=c+1ooo.o N = NCE + (J-1)*6 CN = N t C D = (1 + YLC(H))**(N/12) ILCH(IH) = 1 + YLC(H) E(J) = CN/D INUN = INUN + E(J) 510) = C/D IDEN = XDEN + G(J) CONTINUE DURtIH) = XNUH/(XDEN*12.0) EN = (-u.o)*((1 + YLC(3))**1.25)*DUR(IH)*XEEN ILC3(IN) = 1 + YLC(3) ND = (1 + YLC(H))*1000000.0 E(IH) = HN/HD PP = (PU(IH) + ACP(IH))*10.0 CB(IH) = ((10000000/PP)*H(IH))*BETA(fl) 20 50 35 SS HS 99 CONTINUE DO 50 JJ = 1,100 UNITE(8,35)N(JJ) HEITE19,US)CE(JJ) CONTINUE EONNAT(3X,E8-6) PORNAT(P6.5) FOBHAT(3X,F9.Q) STOP END 100 110 20 10 30 120 H0 :- «Umw'J—"fl “ t 100 APPENDIX A Table A8 A PEOGELN T0 CONPUTE COHPABATIVE HEALTH CHANGES INTEGER X.Y,Z DINENSION NP1100),NET(100),PS(199),ACS(100),NC(100),PU(100) DINENSION ACP(100),COUP(100),CBDlIOO),JC(100),CBE(100,8),PFB(1OO DINENSION NIP(505),NTC(505),PP1505),N(505),PED(100),NCD(100) DINENSION UCUA:100),PPE(100),NCE1100),NCEN(100) D0 10 I = 1,100 NELD(1,15)NP(I),NAT(I).COUP(I),PS(I).ACS(I),NC(I),PU(I),ACP(I) BBAD(2,25)CBD(I) Jx = NATu) - NP (I) JC(I) = (JX/B) + 1 IE (JX .29. 3) GO To 100 IF(JX .89. 6) GO TO 100 IE (JX .E0. 9) GO To 100 IF (ax .Ec. 12) GO To 100 IF (ax .EC. 15) GO TO 100 IF (JX .EQ. 18) GO To 100 IE (JX .EO. 21) GO TO 100 GO TO 110 JC(I) = JC(I) - 1 31 = JC(I) DO 20 K = 1,JI READ(3.35)CBE(I,K) CONTINUE SPT = PS(I) + Acst) PUT = PU(I) + ACP(I) PFB(I) = ((SPT - PUT)*I0.0)*(IOOOOO0.0/PUT) CONTINUE DO 30 NK = 1,505 EEAD(U,N5)NTP(NK),NTC1NN),PP(NN) CONTINUE JNP = 611 L = 0 DO 00 J = 1,505 IF (JNP -EQ. NIP(J)) GO To 120 ONE = HYPtJ) L = L o 1 E(J) = L UNITE(10,17)J,N(J),JNP,NIP(J),NIC(J),EP(J) CONTINUE Do 22 NE = 1,100 PFD(KZ) = 0 101 PFE(KZ) = O 22 CONTINUE DO 50 M = 1,100 Z 0 1&0 2 z + 1 IF {NF(H) -EQ. N(Z)) GO TO 150 GO To 100 150 II = HYC(Z) - BIP(Z) IF (II .EQ. 2) GO TO 1000 Y 0 170 I I + 1 IF (NYC(Z) .EQ- HYC(Y)) GO To 180 GO TO 170 180 PFDtU) = (F912) - FP(Y))*2500*CBD(N) 200 UBITE(12,25)CBB(H) HCD(H) = PFDtu) + PFB(H) WCDA(M) = ABS(HCD(H)) J5 = JC(N) DO 90 X = 1,JH IF(II .NE. 2) GO TO 310 IF‘X .NE. 1) GO To 310 PFE(H) = PFE(N) + (FP(HH) - FP(LL))*2500*CEF(U,X) NEITE(11,25)C8E(N,I),N,I,I,2,EP(NN),EP(LL) I = 0 300 Y = Y + 1 IF (NYC‘Z) .EQ. HYC(Y)) GO TO 90 GO To 300 310 PEE(N) = PEE1N) o (FP(Y) — PP(Z))*2500*CBE(H,X) HBITE(11,25)CBE(U,X),n,X,Y,Z,FF(Y),FP(Z) H N Z = Z + 1 Y = Y + 1 9O CONTINUE PCE(N) = PEE(N) + PEE(N) 900 BCEA(H) = ABS(HCE(H)) Go To 50 1000 LL = z - 1 JET = HYP(LL) 1001 LL LL - 1 IF HYP(IL) .BQ. JHT) GO TO 1001 LL LL + 1 an 0 1070 an an + 1 IP (HYC(LI) .EQ. HYC(HH)) GO TO 1080 GO To 1070 1080 PED(N) = (FP(L1) - FP(HH))*2SOO*CBD(N) GO TO 240 50 CONTINUE AWCD = 0 AWCE = 0 Do 80 an = 1,100 AUCD ANCD + NCD(NN) N 1| "A H 80 88 15 35 Q5 75 85 95 17 102 ANCE = AWCE + UCE(NN) HEITE(7,75)NCD(NN),NCE(NN),ECDN(NN),NCEA(NN) HRITE(9,9S)PFB(HH),PPD‘HH),PFE(HH) CONTINUE ANCD = ANCD/100.0 ANCE = ANCE/100.0 YARD = 0 VABE = 0 DO 88 RV = 1,100 VAPD = VAND + (UCD(NV) -AWCD)**2 VABE = VINE + (NCE(NV) - ENCE)**2 CONTINUE VNED = (VARD/100) VEEE = (VARB/IOO) iNITE18,85)ANCD,ANCE,VAND,VAEE FOBHAT(3X,212,F5.3,F8.5,?7.6,I1,P8.5,F7.6) EOENAT(3x,E9.U,2I,I3,2I,13,21,IS,ZI.IS,2X,E8.2,2x,E8.2) EOENAT(3x,E12.U) ' EONNAT1213,Pu.2) EONNAT(3x,E15.3.2x,E15.3,2x,P15.3,2x,r15.3) FORMAT(3X,F15.3,2X,P15.3,2X,F15.3.2X,F15.3) PONNAT(6X,E15.3,2x,E1S.3,2x,r15.3) EOENNT(31,13,2x,12,2x,13,2x,13,2x,I3,2X,E5-2) STOP END 103 APPENDIX A Table A9 FISHBURN TEST FOR DONNSIDE RISK BASED ON RISK AVERSION WHERE ALPHA GOES PROH 1 - u DIMENSION PFDI100) PFE(100).DPF(H) ,EPP(Q), PE(Q).FD(Q). FIS(Q) 1O 20 3O 25 35 DO 10 I = 1,100 READ(1, 3S) PED(I) READ(3.35)PFE(I) CONTINUE DO 30 DPF(K) EPF(K) RR = 0 LL = 0 D0 20 IP‘PPD(J) .LT. 0) DPF(K) IFlPEE(J) .LT. 0) EPP(N) K J = 1,0 0 0 = 1,100 DPE(N) + (PFD(J)**K) EPF(K) + (PEE(J)**K) IP(PFD(J) .LT. 0) KK = KK + 1 IF!PFE(J) .LT- 0) LI CONTINUE ED(K) EE(K) FIS(K) DPF(K)/KK EPE(K)/LL FD(K)/PE(K) NRITE (2,25)FIS(K) CONTINUE FORHAT(3X,F10-S) FORMAT(3X,F15.3) STOP END LL 0 1 APPENDIX B 104 APPENDIX B Table B1 FORWARD RATE FORECASTS FIRST TEST PERIOD 1 - 24 STEPS AHEAD 1 - 6 MONTHS FORWARD MONTHS FORWARD 1 2 3 4 5 6 3.7845 3.4316 4.8671 4.4110 3.0788 5.8131 3.9831 3.4316 4.8671 4.4110 3.0788 5.5325 3.8724 3.4316 4.8671 4.4110 3.0788 5.4053 3.9596 3.4316 4.8671 4.4110 3.0788 5.3477 3.4409 3.4316 4.8671 4.4110 3.0788 5.3215 4.0500 3.4316 4.8671 4.4110 3.0788 5.3097 4.0500 3.4316 4.8671 4.4110 3.0788 5.3043 4.0500 3.4316 4.8671 4.4110 3.0788 5.3019 4.0500 3.4316 4.8671 4.4110 3.0788 5.3007 4.0500 3.4316 4.8671 4.4110 3.0788 5.3002 4.0500 3.4316 4.8671 4.4110 3.0788 5.3000 4.0500 3.4316 4.8671 4.4110 3.0788 5.2999 4.0500 3.4316 4.8671 4.4110 3.0788 5.2999 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 300788 502998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 4.0500 3.4316 4.8671 4.4110 3.0788 5.2998 7 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 5.3348 105 FORHARD RATE FORECASTS FIRST TEST PERIOD 1 - 24 STEPS AHEAD 7 - 12 MONTHS FORWARD MONTHS FORWARD 8 9 10 5.1414 4.8912 5.5691 5.1414 5.5006 5.5691 5.1414 5.2668 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 5.1414 5.6275 5.5691 11 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 5.6395 12 5.3700 4.7469 5.3137 5.2313 5.4220 4.7990 5.2296 5.0349 5.2120 5.1863 5.2459 5.0512 5.1857 5.1249 5.1802 5.1722 5.1908 5.1300 5.1720 5.1530 5.1703 5.1678 5.1736 5.1546 13 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 5.4187 106 FORWARD RATE FORECASTS FIRST TEST PERIOD 1 - 24 STEPS AHEAD 13 - 18 MONTHS FORWARD MONTHS FORHARD 14 15 16 5.4187 5.6729 5.8855 5.4187 5.7200 5.8855 5.4187 5.7347 5.8855 5.4187 5.7393 5.8855 5.4187 5.7407 5.8855 5.4187 5.7412 5.8855 5.4187 5.7413 5.8855 5.4187 5.7413 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 5.41871 5.7414 5.8855 5.4187 5.7414 5.8855 5.4187 5.7414 5.8855 17 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 5.4960 18 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 5.0401 19 6.0108 6.4682 6.1607 5.5954 7.3248 5.4080 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 5.9175 107 FORWARD RATE FORECASTS FIRST TEST PERIOD 1 - Z4 STEPS AHEAD 19 - 23 MONTHS FDRHARD MONTHS FORWARD 20 21 22 6.0897 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 5.8424 5.8875 6.0398 23 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 4.8234 1 0.4502 0.9004 1.3505 1.8007 2.2509 2.7011 2.8769 3.0528 3.2286 3.4045 3.5803 3.7562 3.9320 4.1079 4.2837 4.4596 4.6354 4.8113 4.9871 5.1630 5.3388 5.5147 5.6905 5.8664 108 APPENDIX B Iable 82 FORWARD RATE FORECAST VARIANCES 1 FIRST TEST PERIOD - 24 STEPS AHEAD 1 - 6 MONTHS FORWARD 2 0.6141 0.7667 0.9193 1.0718 1.2244 1.3770 1.5296 1.6821 1.8347 1.9873 2.1398 2.2924 2.4450 2.5976 2.7501 2.9027 3.0553 3.2079 3.3604 3.5130 3.6656 3.8181 3.9707 4.1233 3 0.4801 0.6236 0.7671 0.9107 1.0542 1.1977 1.3412 1.4848 1.6283 1.7718 1.9153 2.0589 2.2024 2.3459 2.4894 2.6330 2.7765 2.9200 3.0635 3.2070 3.3506 3.4941 3.6376 3.7811 MONTHS FORWARD 4 0.5161 0.7052 0.8944 1.0836 1.2727 1.4618 1.6510 1.8401 2.0293 2.2184 2.4076 2.5967 2.7859 2.9750 3.1642 3.3533 3.5425 3.7316 3.9208 4.1099 4.2991 4.4882 4.6774 4.8665 5 1.2580 1.6267 1.9954 2.3641 2.7328 3.1015 3.4702 3.8389 4.2076 4.5763 4.9450 5.3137 5.6824 6.0511 6.4198 6.7885 7.1572 7.5259 7.8946 8.2634 8.6321 9.0008 9.3695 9.7382 6 2.7744 3.3447 3.4619 3.4860 3.4909 3.4920 3.4922 3.4922 3.4922 3.4922 3.4922 3.4922 3.4922 3.4922 3.4922 3.4922 3.4922 3.9922 3.4922 3.4922 3.4922 3.4922 3.4922 3.9922 7 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 2.3544 109 FORWARD RATE FORECAST VARIANCES FIRST TEST PERIOD 7 8 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 3.5000 1 - 24 STEPS 12 MONTHS FORWARD 9 2.0050 2.2231 2.2231 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 2.5652 AHEAD MONTHS FORWARD 10 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 2.6400 11 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 4.7198 12 2.1739 2.1739 2.3555 2.3555 2.3555 2.3555 2.5678 2.5678 2.5855 2.5855 2.5855 2.5855 2.6063 2.6063 2.6080 2.6080 2.6080 2.6080 2.6100 2.6100 2.6102 2.6102 2.6102 2.6102 13 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 2.8275 110 FORWARD RATE FORECAST VARIANCES FIRST TEST PERIOD 1 - 24 STEPS AHEAD 13 - 18 MONTHS FORWARD 14 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 2.6857 15 1.6702 1.8333 1.8492 1.8508 1.8509 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 1.8510 MONTHS FORWARD 16 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 1.4970 17 0.9855 1.0514 1.1172 1.1831 1.2490 1.3149 1.3808 1.4467 1.5125 1.5784 1.6443 1.7102 1.7761 1.8420 1.9078 1.9737 2.0396 2.1055 2.1714 2.2373 2.3031 2.3690 2.4349 2.5008 18 1.4509 1.5429 1.6289 1.7150 1.8010 1.8870 1.9730 2.0590 2.1450 2.2310 2.3171 2.4031 2.4891 2.5751 2.6611 2.7471 2.8331 2.9192 3.0052 3.0912 3.1772 3.2632 3.3492 3.4352 19 2.3362 2.3362 2.3362 2.3362 2.3362 2.3362 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 2.6353 111 FORWARD RATE FORECAST VARIANCES 19 - 23 20 1.9707 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 2.2874 21 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 FIRST TEST PERIOD 1 - 24 STEPS AHEAD MONTHS FORWARD MONTHS FORWARD 22 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 2.1702 23 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 9.6059 1 7.1961 7.1949 7.3214 7.3219 7.2640 7.2638 7.2903 7.2904 7.2783 7.2782 7.2838 7.2838 7.2812 7.2812 7.2824 7.2824 7.2819 7.2819 7.2821 7.2821 7.2820 7.2820 7.2820 7.2820 112 APPENDIX B Table 83 FORHARD RATE FORECASTS SECOND 1 TEST PERIUD - 24 STEPS AHEAD 1 - 6 MONTHS FURHARD 2 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 7.4588 3 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 7.7067 MONTHS FORWARD 4 8.2432 8.2729 8.3026 8.3323 8.3620 8.3916 8.4213 8.4510 8.4807 8.5104 8.5401 8.5698 8.5995 8.6291 8.6588 8.6885 8.7182 8.7479 8.7776 8.8073 8.8370 8.8666 8.8963 8.9260 5 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 8.1402 6 8.0010 8.0010 .8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010. 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 8.0010 7 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 5.6336 113 FORWARD RATE FORECASTS SECOND TEST PERIOD 1 - 24 STEPS AHEAD 7 - 12 MONTHS FORWARD MONTHS FORWARD 8 9 10 5.5086 5.9227 6.6532 5.5086 5.9227 6.8758 5.5086 5.9227 6.1487 5.5086 5.9227 6.2256 5.5086 5.9227 5.9744 5.5086 5.9227 6.0009 5.5086 5.9227 5.9141 5.5086 5.9227 5.9233 5.5086 5.9227 5.8933 5.5086 5.9227 5.8965 5.5086 5.9227 5.8861 5.5086 5.9227 5.8872 5.5086 5.9227 5.8837 5.5086 5.9227 5.8840 5.5086 5.9227 5.8828 5.5086 5.9227 5.8829 5.5086 5.9227 5.8825 5.5086 5.9227 5.8826 5.5086 5.9227 5.8824 5.5086 5.9227 5.8824 5.5086 5.9227 5.8824 5.5086 5.9227 5.8824 5.5086 5.9227 5.8824 5.5086 5.9227 5.8824 11 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9593 5.9593 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 5.9598 12 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 5.7918 13 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 6.0398 114 FORWARD RATE FORECASTS SECOND TEST PERIOD 1 - 24 STEPS AHEAD 13 - 18 MONTHS FORHARO MONTHS FORHARD 1‘0 15 16 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 5.8836 6.1805 6.1707 17 7.5771 4.3598 7.2410 4.8663 6.5934 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 5.8986 18 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 6.2711 19 6.8715 5.4905 7.1631 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 6.3914 115 FORWARD RATE FORECASTS SECOND TEST PERIOD 1 - 24 SYEPS AHEAD l9 - 23 MONTHS FORWARD MONTHS FORHARO 20 21 22 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 5.9129 5.8113 6.0340 23 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 5.2879 1 1.0884 1.6254 1.6905 1.8484 2.2278 2.5313 2.7346 2.9656 3.2427 3.5056 3.7475 3.9956 4.2533 4.5081 4.7585 5.0102 5.2640 5.5171 5.7693 6.0218 6.2747 6.5275 6.7801 7.3171 116 APPENDIX B Table 84 FORWARD RATE FORECAST VARIANCES SECOND TEST PERIOD 1 - 24 STEPS AHEAD 1 - 6 MONTHS FORWARD 2 0.8142 0.9584 1.1025 1.2466 1.3908 1.5349 1.6790 1.8232 1.9673 2.1114 2.2556 2.3997 2.5438 2.6880 2.8321 2.9762 3.1204 3.2645 3.4086 3.5528 3.6969 3.8410 3.9852 4.1293 3 0.9079 1.0959 1.2838 1.4717 1.6596 1.8475 2.0354 2.2234 2.4113 2.5992 2.7871 2.9750 3.1629 3.3509 3.5388 3.7267 3.9146 4.1025 4.2904 4.4784 4.6663 4.8542 5.0421 5.2300 MONTHS FORWARD 4 0.6825 1.3651 2.0976 2.7302 3.4127 4.0953 6.7778 5.4604 6.1429 6.8255 7.5080 8.1906 8.8731 9.5557 10.2380 10.9210 11.6030 12.2860 12.9680 13.6510 14.3330 15.0160 15.6990 16.3810 5 1.7044 2.0067 2.3090 2.6112 2.9135 3.2157 3.5180 3.8203 4.1225 4.4248 4.7271 5.0293 5.3316 5.6339 5.9361 6.2384 6.5407 6.8429 7.1452 7.4475 7.7497 8.0520 8.3543 8.6565 6 2.4846 3.0403 3.5961 4.1518 4.7075 5.2632 5.8189 6.3747 6.9304 7.4861 8.0418 8.5975 9.1532 9.7090 10.2650 10.8200 11.3760 11.9320 12.4880 13.0430 13.5990 14.1550 14.7100 15.2660 7 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 2.4032 117 FORWARD RATE FORECAST VARIANCES 7 8 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 3.1702 SECOND TEST PERIOD 1 - 24 STEPS AHEAD 9 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 1.9091 12 MONTHS FORHARD MONTHS FORHARO 10 1.7176 1.7176 1.9226 1.9226 1.9471 1.9471 1.9500 1.9500 1.9503 1.9503 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 1.9504 11 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 3.5478 12 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 1.6767 13 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 2.0903 118 FORHARD RATE FORECAST VARIANCES SECOND TEST PERIOD 1 - 24 STEPS AHEAD 13 - 18 MONTHS FORHARD 14 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 1.9510 15 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 3.6818 MONTHS FORHARO 16 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 2.4903 17 1.8987 1.8987 1.8987 1.8987 1.8987 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 2.2622 18 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 2.8173 119 FORWARD RATE FORECAST VARIANCES SECOND TEST PERIOD 1 - 24 STEPS AHEAD 19 - 23 MONTHS FORHARD MONTHS FORHARO 19 20 21 22 2.9818 1.7926 2.3908 2.6519 2.9818 1.7926 2.3908 2.6519 2.9818 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 3.2937 1.7926 2.3908 2.6519 23 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 6.4163 APPENDIX C tomqoxmch—u .5 O 11 12 13 1Q 15 16 17 18 19 20 21 22 23 24 25 -26.7171 -10.2573 -26-4225 -30.5979 -36-5887 -30-4946 -3-3850 -16.3709 -27.2970 -30.1335 ~16.7965 -3.2795 -3.3291 -27.1172 -13-1181 -19.9748 -13.3166 -6.6654 -10.1861 -19.9443 -9.9610 ~16.6386 -13.6195 -9-9721 -16.6070 26 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 120 APPENDIX C Table C1 CONTRACTS SOLD DURATION HEDGE FIRST PERIOD -26-9294 -10.2581 -26.4884 -6.7700 -6-5590 -6.6281 -10.2200 -38.2105 -16.7655 -13.4124 -6.8669 -23.0828 -3.4283 -38.4915 -33-5499 —19.9833 -6.6409 -6.6428 -10.2126 -10.0779 -9.9421 -16.5702 -3.3140 -23.8035 ~13.2562 51 52 53 54 55 57 58 59 60 61 63 64 65 66 67 68 69 7O 71 72 73 74 75 -38.6494 -33.2000 -23.5773 -13.5400 -17.1673 -02.2514 -31.1144 -10.2343 -19.9227 -13.6458 ~47.7456 -AQ.9573 -29.8941 -6.6611 -10.1824 -33.9532 -45.4172 '20.3099 -16.9249 -49.0439 -41.8526 ~3.Q164 -6.7883 ~20.5696 -26.8344 76 78 79 80 81 82 83 84 85 86 87 88 90 91 92 93 94 96 97 98 99 100 -37.1797 -52.4402 ’29.6867 -34.0787 -26.3159 -29.8113 -3.4044 -82.0333 -13.4372 -30.7905 -30.3011 -10.2146 -10.1550 '19.6451 '36.9663 -26.8618 -23.6365 -19.9961 -34.2840 -23-3814 -33.2998 -23.8637 dufiuuuu—s qmmcwwuommqmmuwM-A 18 19 20 21 22 23 24 25 -21.1244 -9-6146 -21.1718 -26.0067 -29.3178 -20.2551 -2.7721 -13.1177 -22.3546 -23.8256 -14-2762 -2.4150 -3.0407 -23.0483 -9-6602 ~18.2440 -12.1627 -5-7543 -9-9136 -13.247Q ~8.7862 -11.0517 -12.7661 -6.6237 -14.7063 26 27 28 29 3O 31 32 33 3Q 35 36 37 38 39 40 Q1 42 43 44 45 46 47 48 49 50 121 APPENDIX C Table C2 CONTRACTS SOLD -23.2486 -9.6153 -19.5061 -5.5442 -n.8301 -5.2406 -9.5796 -28.1382 -11.1360 -8.9088 -6-Q6QQ -16.9982 -3.1312 -30.4340 -26.5269 -14.3304 -5-8575 -5-8825 ~9.5727 -8.5657 -7.8609 -13.1016 -2.6203 -19.4936 -10-4813 51 52 53 54 55 56 57 58 59 61 62 63 64 65 66 67 68 70 71 72 73 7Q 75 ADJUSTED DURATION HEDGE FIRST PERIOD -27.7163 -26.6025 415.6605 -11.0884 -16.1611 -28.0642 -26.8617 -8.8355 -17-5724 ~11.7807 -33.6334 -33.1065 -22.0140 -Q.7768 -9.9100 -22.5524 -32.5696 -16.6326 -13-8605 -32.5759 -33.0916 -3.4164 -6-6067 -18.7873 -23.7632 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 -26.6623 ~34.8318 -20.9122 -28.9651 ~18.5377 -40.5884 -23.8872 -3.4044 -30.1429 -11.4209 -25.2156 -21-7296 -9.5746 -8.3163 -15.7412 -29.2281 -19.2632 -20-0898 -17-2630 -28.0766 -20-7054 -24.5219 -6.3830 -21.0485 -7.1652 122 APPENDIX C Table C3 CONTRACTS SOLD EXPECTATIONS HEDGE FIRST PERIOD 10.7446 1 1 10.9474 14 1 10.1299 10.4455 1 2 10.5837 14 2 10.4574 10.1633 1 3 10.2915 14 3 10-9622 10.4956 2 1 10-1465 15 1 10.9693 10.6538 3 1 9.8931 15 2 10.6061 10.3113 3 2 10.3546 16 1 10.6168 10.0417 3 3 10.3657 16 2 10.4372 11.0133 u 1 10.2679 17 1 10-2751 10.6446 4 2 10.0205 17 2 10.3888 10.6745 4 3 10-1706 18 1 10.1325 10.8426 5 1 10.4004 19 1 10.4112 10.8422 5 2 10-3887 20 1 10.6617 10.5216 5 3 10.3895 20 2 10.2798 10-0328 5 4 10-1923 21 1 9-8250 10.9678 6 1 10.3582 22 1 10.4420 10.6458 6 2 10.1973 22 2 10.9903 10.6818 6 3 10.4875 23 1 10.9953 10.3284 7 1 10.2479 23 2 10.7059 10.1433 8 1 10-2589 24 1 10.7058 10.0032 8 2 10.2780 25 1 10.8657 10.9810 9 1 10.1353 25 2 10.8625 10.6542 9 2 10.9041 26 1 10.5895 10.3825 9 3 10.4999 26 2 9.7658 10.8077 10 1 10.2131 26 3 10.3962 10.5001 10 2 10.4963 27 1 10.4032 10.5326 10 3 10.7313 28 1 10-1182 10.4063 11 1 10.3440 28 2 10.1383 10.2707 11 2 10.0706 28 3 10.4498 10.0197 12 1 10.3344 29 1 10.3284 10.1400 13 1 10.0218 30 1 10.2012 dd...ddeCWNdCUNddwNudN—hN-‘c“Nd-Ad 123 CONTRACTS SOLD EXPECTATIONS HEDGE FIRST PERIOD (CONT.) 10.2870 47 1 10.8227 57 3 10.5488 68 1 10.1302 47 2 10.5005 58 1 10.5593 68 2 10.1139 48 1 10.3353 59 1 10.5056 69 1 10.9329 49 1 10.3399 59 2 10.3417 69 2 10.6089 49 2 10-5322 60 1 11-6738 70 1 10.1205 49 3 10-2892 60 2 11.3531 70 2 10.2405 50 1 11.3928 61 1 11.3851 70 3 9.9944 50 2 11.0628 61 2 11.0485 70 4 11.0779 51 1 11.1001 61 3 11.0626 70 5 11.0827 51 2 10.7621 61 4 11.3512 71 1 10.7794 51 3 10-7620 61 5 11.0504 71 2 10.7871 51 4 11.4551 62 1 11.0584 71 3 10.7760 52 1 11.0814 62 2 10.7638 71 4 10.7743 52 2 11.0932 62 3 9.6135 71 5 10.4591 52 3 10.7178 62 4 10.3242 72 1 9.6463 52 4 10.0772 62 5 10.3285 73 1 10.8562 53 1 10.7807 63 1 10.6629 74 1 10.5401 53 2 10.4027 63 2 10.6744 74 2 10.0558 53 3 10.4282 63 3 10.8456 75 1 10.4589 54 1 10.1933 64 1 10-4555 75 2 10.1973 54 2 10-3967 65 1 10.1612 75 3 10.5660 55 1 11.0460 66 1 11-0184 76 1 10.4291 55 2 11.0359 66 2 11.0212 76 2 11.5331 56 1 10.7518 66 3 10.7212 76 3 11.2149 56 2 9.8937 66 4 10-2426 76 4 11.2366 56 3 11.5431 67 1 11.7257 77 1 10.9278 56 4 11.2153 67 2 11.7304 77 2 9.7493 56 5 11.2365 67 3 11.4356 77 3 11.2092 57 1 10-9023 67 4 11.4212 77 4 10.7994 57 2 10.2491 67 5 11.1109 77 5 124 CONTRACTS SOLD EXPECTATIONS HEDGE FIRST PERIOD (CONT.) 9.5814 77 6 10.7533 86 3 9.6683 97 10.7007 78 1 10.9139 87 1 10.3677 98 10.3730 78 2 10.5769 87 2 10.9756 99 10.4067 78 3 10.6112 87 3 10.6134 99 11.0581 79 1 10-0519 88 1 10-1109 99 11.0610 79 2 10.4232 89 1 10.2628 100 10.7178 79 3 10.2123 90 1 9.8921 79 4 10.2109 90 2 10.6412 80 1 10.9331 91 1 10.3096 80 2 10-9357 91 2 10.0377 80 3 10.6533 91 3 11.5771 81 1 10.1717 91 0 11.5894 81 2 10.8331 92 1 11.2743 81 3 10.5039 92 2 11.2861 81 4 10.2377 92 3 10.9360 81 5 10.8838 93 1 10.9371 81 6 10.5179 93 2 10.7122 82 1 10.0451 93 3 10.3644 82 2 10.3729 94 1 10.3993 82 3 10.3770 90 2 10.2879 83 1 11.1196 95 1 11.0737 84 1 11.1163 95 2 11.1484 84 2 10.8193 95 3 11.1637 84 3 9.9502 95 4 10.8489 84 4 10.8022 96 1 9.6610 84 5 10.4057 96 2 10.3888 85 1 9.9107 96 3 10.1156 85 2 10.8402 97 1 11.0492 86 1 10.8383 97 2 10.7223 86 2 10.4898 97 3 HWN—J-ac \omqmmcwmu ...; O 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 -10.5329 -14.6128 -61.6973 '28.6937 -36-4543 ~36.6288 -7-3064 -10.6483 ~14-2882 -43.6320 ~60.8635 -14.2480 -53.8951 -17.6268 -14-4527 -39-8064 ~39.2245 -59.9070 -10-5633 -40.0959 -14-2374 -6-9858 -10.5545 -54.6716 -3-5594 26 28 29 30 31 32 33 34 35 36 37 38 39 ‘40 41 42 43 44 45 46 47 48 49 50 125 APPENDIX C Table C4 CONTRACTS SOLE DURATION HEDGE SECOND PERIOD -45.9357 -39.1218 -10.6781 -7.0251 -21-2967 -57-4182 -28.7363 -28-4739 -36.3277 '-10.7162 -14.2041 -56-5811 -73.2150 -51.5242 -36.0812 -39.4306 -68-0879 -50.7044 -17.9673 -7.1307 -3.5233 -42.6296 -28-7025 -47.4806 -39-3059 -3.6711 -40.2809 -10-8012 -13.9777 -43-3260 -69.5234 -24.6976 -65.4377 -3.5125 -36.5325 -20.9617 -6.9889 -7.1492 -62-1403 -24.5873 -49.6196 -21.1089 -10.7693 -10.5700 -14.1978 -57-3946 -29.4685 ~42-6618 -3.5254 -29.0075 7.6 78 79 80 81 82 83 84 85 86 87 88 89 91 92 93 94 95 97 98 99 100 -10-6961 -21.6025 ~25-1562 -55.1048 -32.3852 -36.5414 -46-6985 -46.1565 ~39-6911 -39.9618 -56.5456 -21.4477 -32.8067 -24.6611 -24.7008 -17.S413 ~40-3662 -47.1786 -7.0467 -10.6531 -51.0761 -40.7797 -54.2638 -7.0165 -13.9333 mmqo‘mauma -36.7450 -18.0578 -26.1421 -29.9968 -4.5377 -6-6131 -8.7374 -31-2894 -37-7993 -13-4130 -38.7641 -13-9370 -11.8359 -29-3134 -31.4298 -39-4140 -10.2807 -28-7536 -10-4844 ‘6-9858 -7.4349 *32-5608 -2-6211 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 126 APPENDIX C Table C5 CONTRACTS 5010 -32.3585 -27-5586 -7.8633 -6.8372 -13.2263 -41.2980 -21.1614 -24-5821 -22.2147 -6.5530 -8.9391 -37-2258 -45.4702 -31.9991 -22.7070 -25.9421 ~42.8497 -36.3611 ~12.8847 -6.7128 -3-0418 -33.7059 -24.7795 ~28-2780 -25.8601 51 52 53 54 55 S6 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 ADJUSTED DURATION HEDGE SECOND PERIOD -3.4559 -23.9901 -7.1744 -11-2001 -31.1622 -43.1775 ~19.7897 -38.9727 -3.4186 -21.7577 -14.7661 *5.6000 -6.0765 ~38-5922 -19.7013 -34.9S35 -14.8697 -6.4139 -9.1253 -8.8175 -36.1201 -18.3014 -28.0680 -2.7874 -20.8019 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 -10.0692 -14-3488 -15.3833 -34-2228 -20.1128 -24-2715 ~34.3888 -32.5140 -29.2285 -24.4370 -39.8324 -18.2294 -21.7908 -21-7518 -16.2511 -11-5407 -33.0575 -31.3370 -6.0835 -6-7043 -30.4194 -25.3262 -36.0431 -4.6163 -12-7260 127 APPENDIX C Table C6 CONTRACTS SOLD EXPECTATIONS HEDGE SECOND PERIOD 10-8142 1 1 12.9118 11 2 11.7945 11.2928 2 1 12.3702 11 3 10.8043 10.7992 2 2 12.3719 11 4 12.1133 13.1238 3 1 11.8996 11 5 12.1359 13.1569 3 2 11.8885 11 6 11.6511 12.5603 3 3 11.0485 12 1 11-6528 12.5684 3 4 10.5857 12 2 11.0308 12.0659 3 5 12-8236 13 1 10.5358 12.0558 3 6 12.8343 13 2 10.5542 11.7285 4 1 12.3294 13 3 10.8226 11.3319 4 2 12.2867 13 4 12.9515 10.6523 4 3 11.9018 13 5 12.9588 11.9714 5 1 9.7717 13 6 12.3966 11.9836 5 2 11.0308 14 1 12.3385 11.5097 5 3 10.7320 14 2 11.9046 10.1228 5 4 11.2140 15 1 9.7533 12.0530 6 1 10.7430 15 2 10.7771 12.0630 6 2 11-9456 16 1 12.4004 11.6112 6 3 11.9557 16 2 11.9935 10.2133 6 4 11-4845 16 3 11.8875 11.0262 7 1 10.7053 16 4 11.4882 10.8672 8 1 11-7977 17 1 10.5162 11.0657 9 1 11.8128 17 2 11.7519 10.5819 9 2 11.3645 17 3 11.7651 12.5779 10 1 10.5990 17 4 11.2611 12.2134 10 2 12.7725 18 1 10.4998 12.0816 10 3 12.8012 18 2 10.9294 11.7197 10 4 12.2630 18 3 10.6157 9.9561 10 5 12.2789 18 4 11.1553 12.8876 11 1 11.8143 18 5 11-1713 Nd...chN-fimcuwd—Jmmcwwdddwfichd—bc‘ 128 CONTRACTS SOLD EXPECTATIONS HEDGE SECOND PERIOD (C0 NT. ) 12.9156 31 1 12.5329 38 6 10.6958 12.9227 31 2 12.5154 38 7 12.3909 12.4177 31 3 12.8671 39 1 12.0226 12.4082 31 4 12.5045 39 2 11-9029 11.9877 31 5 12.4201 39 3 11.5551 10.7917 31 6 12.0528 39 4 9.8057 11.7353 32 1 12.0490 39 5 11.7058 11.3480 32 2 11.8435 40 1 11.3848 10.6960 32 3 11.8351 40 2 10-7367 11.6806 33 1 11-3363 40 3 12.7801 11.2698 33 2 9.9993 40 4 12.3243 10.6282 33 3 11-9629 41 1 12.2161 11.9939 34 1 11.9897 41 2 11.7473 12.0098 34 2 11.4499 41 3 10.7746 11.4619 34 3 11.4523 41 4 11.8128 10.0574 34 4 13.5059 42 1 11-8296 10.9742 35 1 13.0953 42 2 11.3162 10.9744 36 1 12.9808 42 3 10.5619 10.5178 36 2 12.5736 42 4 11.1403 12.7220 37 1 12.5279 42 5 12.1158 12.7380 37 2 12.1229 42 6 12.1378 12.2078 37 3 10.6634 42 7 11.5589 12.1870 37 4 12.7291 43 1 10.7649 11.7598 37 5 12.3498 43 2 11.0797 10.5820 37 6 12.2631 43 3 10.8493 13.7238 38 1 11.8698 43 4 10.4016 13.3729 38 2 11.8636 43 5 12.6045 13.2791 38 3 11.2135 44 1 12.2003 12.9153 38 4 10.8760 44 2 12.0764 12.9085 38 5 10-8126 45 1 11.6986 46 47 47 47 47 47 48 48 48 49 49 49 49 49 50 50 50 50 51 52 52 52 52 53 54 54 55 55 55 chdedawNudaWN—Amc“NAWN‘mDuN-AA 129 CONTRACTS SOLD EXPECTATIONS HEDGE SECOND PERIOD (CONT.) 9.9168 55 5 12.6378 60 0 9.7291 13.5903 56 1 12.2075 60 5 10.7166 13.2538 56 2 12.2361 60 6 11.8178 13.1011 56 3 11.0852 65 1 11.0505 12.7938 56 0 11.1297 65 2 10.7901 12.7607 56 5 10.0516 65 3 10.9703 12.3753 56 6 12.0903 66 1 11.3076 10.9031 56 7 12.1161 66 2 11.3589 11.5210 57 1 12.0378 66 3 11.7808 11.1820 57 2 11.6276 66 0 11.3010 10.0992 57 3 11.6301 66 5 10.2358 13.7008 58 1 11.0926 67 1 12.9378 13.2391 58 2 11.1267 67 2 12.9535 13.1021 58 3 10.9929 68 1 12.0980 12.6665 58 0 10.8737 69 1 12.0591 12.5807 58 5 10.9721 70 1 12.1290 12.1527 58 6 10-0925 70 2 9-9000 9.5090 58 7 12.7951 71 1 11.8005 10.6230 59 1 12.8058 71 2 11.3858 12.0171 60 1 12.2938 71 3 11.2903 12.0160 60 2 12.2603 71 0 12.0035 11.0620 60 3 11.8386 71 5 12.0339 10.0980 60 0 10.6550 71 6 11-5568 11.0152 61 1 11.9569 72 1 10.1902 11.0091 61 2 11.6115 72 2 12.5510 10.6138 62 1 10-9027 72 3 12.1708 10.8559 63 1 12.0150 73 1 12.0686 13.0752 60 1 11.9912 73 2 11-6951 13.0997 60 2 11.8698 73 3 10.7118 12.6359 60 3 11.0769 73 0 12.0202 73 74 75 75 75 76 77 77 78 78 78 79 79 79 79 79 79 80 80 80 81 81 81 81 82 82 82 82 82 83 amcww—IcdewNuO‘mch—aWNde-fiww-fidm 12.0453 11.9404 11.5688 10.5901 12.0161 12.0441 11.5410 11.5509 12.1030 12.1151 11.5584 11.5631 12.6294 12.6456 12.1672 12.1544 11.7660 10.5770 11.3064 11.3181 11.9621 11.5746 11.4889 11-4734 11.1674 10.1140 11.5646 11.1652 10.0563 10.9709 Amman-aw.-uwamummcuwaaumucww...-mew» 130 CONTRACTS SOLD EXPECTATIONS HEDGE szconn PERIOD (CONT-1 10.6473 91 12.1222 92 12.1293 92 11-6692 92 10.8979 92 12.7349 93 12.3563 93 12.2384 93 11.8443 93 10.8542 93 10.7020 94 10.8854 95 12.8681 96 12.4139 96 12.3277 96 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