A SECURITY VALUATION MODEL
FOR A SECURITY IN
V OPTIMAL PORTFOLIOS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
LEROY DAVIS BROOKS II
1971

 

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thesis entitled
A SECURITY VALUATION MODEL FOR A

SECURITY IN OPTIMAL PORTFOLIOS
presented by

LeBoy Davis Brooks II

has been accepted towards fulfillment
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ABSTRACT

A SECURITY VALUATION MODEL FOR A

SECURITY IN OPTIMAL PORTFOLIOS

by

LeRoy Davis Brooks II

Investors who add a security to their investment portfolio
should be concerned with the value this new security could contribute
to that portfolio. A security valuation model should be able to
measure this potential contribution. The purpose of this study,
therefore, was to develop a security valuation model where value
would be determined by the security's contribution of utility to

portfolios.

The study was only concerned with security valuation in
optimal portfolios, since it is assumed that rational investors
would only hold optimal portfolios, i.e., obtainable portfolios
that maximize investors' utility. Also,a portfolio selection
technique that determined the set of optimal portfolios was needed
in the formation of the valuation model.1 The quadratic

programming technique by Markowitz was used to meet this need.

Brooks

In developing the security valuation model, one that came
closer to being applicable to a real world situation was sought.
The objective of real world application required the model to use
observable data for input information; such items as the utility

functions of each and every investor could not be required.

The valuation process as invisioned by the author led to
the assumptions adopted in the model. Since the real-world
application requirement prevented the adoption of some assumptions
that more closely portrayed the invisioned valuation process. The
set of assumptions used differed from preceding contributions.

Some of the more important ones are now briefly reviewed.

In the model, both the rate and variance of returns were
the only two moments of a return distribution that affected
investors' utility functions. Investors were viewed as being
risk averse wealth maximizers. Any two substitute items could
have been used for return and variance without major model modi-
fications. Further, the basic valuation model could have been
constructed for any n items that affect utility; however, the
ability to apply the model in a real world situation would have

been lost by doing this.

The model was constructed under the assumption that more
than one optimal portfolio could exist. The condition of multiple
optimal portfolios came from the assumptions that both risk related
debt costs and market imperfections in the borrowing and lending

of funds existed. The condition of multiple optimal portfolios

Brooks

increasedmodel complexity over what would have been needed with

a single optimal portfolio valuation model.

Return outcomes of a security were viewed to be a function
of multiple factors rather than just a single systematic factor,
like the economy. This multi-factor assumption increased model

complexity.

The assumptions mentioned above together with other
assumptions, were presented and discussed in the study. A selected
set of these assumptions including those discussed above, were
used in formulating the security valuation model. The derivation
of input information required for the model, the model's practical
limitations, and a discussion of some alternative assumptions

were included in the study.

An operationally feasible computational algorithm that
can use observed data in determining estimates of the final
equilibrium value of a security was outlined in an appendix.

A theoretical discussion on large security offerings and the
relationship between equilibrium price and the quantity supply

of a security was briefly discussed in another appendix.

Primarily.a normative model which could be applied to aid
decision making was formulated in the study. As an example this
model might be used by management decision makers in planning. One
could form expectations of the change in both a security's value

and the optimal portfolios in which it had membership if the firm

Brooks

affected changes in the security's expected distribution of

cash returns. It might also be used by portfolio managers in
adjusting their portfolios when their expectations change on the
returns of different securities. As another alternative use,
investment bankers could use the model in deriving estimates

of a subscription price on a new "untested" market offering.

 

1Harry M. Markowitz, Portfolio Selection, Monograph 16
of the Cowles Foundation for Research in Economics at Yale
University, John Wiley and Sons, Inc., (New York, 1959), p. 102.

 

A SECURITY VALUATION MODEL
FOR A SECURITY IN

OPTIMAL PORTFOLIOS

By

LeRoy Davis Brooks II

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Accounting and Financial Administration

1971

(:) Copyright by
LeRoy Davis Brooks II

1972

TO KANDI

ACKNOWLEDGMENTS

The following people deserve my thanks and sincerest

appreciation for their contributions to this study.

Professor Myles Delano, Chairman of my dissertation
committee, for his helpful direction and insightful suggestions
that proved invaluable in both examining the theoretical under-

pinnings of the model and organizing the presentation of results.

Professor R. Hayden Howard, for the numerous intellectually
stimulating and personally enjoyable discussions that contributed

immeasurably to both my education and this study.

Professor Ronald Marshall, whose valuable suggestions
provided the impetus to convert my views on valuation into a
more clearly defined model, and whose knowledge provided me

with the tools to accomplish this task.

John Zdanowicz, whose friendship and knowledge provided

many enjoyable evening discussions.

To numerous teachers and associates who offered both time

and helpful suggestions in all phases of the study.

To my parents, whose encouragement and self-sacrifice in

earlier years made all of this possible.

ii

Most of all, to Kandi, my wife, for being with me when
I needed her and for her remarkable patience during her typing

of numerous drafts on many weekends and vacations.

Of course, all errors and omissions are my own.

iii

CHAPTER

CHAPTER

CHAPTER

TABLE OF CONTENTS

1: INTRODUCTION

The Nature of the Study ........................
A Brief Historical Review ......................
The Emphasis of the Study ......................
Possible Uses for the Model ....................
The Type of Model Constructed ..................
Plan of the Study ..............................

2: MODEL FORMULATION

Parameters .....................................
Relevant Sets ..................................
Portfolios and Their (e,v) Values ..............
The Feasible Decision Set ......................
Shareholder Utility Functions ..................

Location of the Utility Function on an (e,v)

Cartesian Plane ...........................
Maximization of Investor Utility ...............
Valuation of a Security ........................

Determining the Final Equilibrium Value

of a Security ........ . ....................
Computational Procedure for Model Solution .....

3: MAJOR THEORETICAL DECISION AREAS CONSIDERED
DURING MODEL CONSTRUCTION

Equal Expectations, Perfect Markets, Riskless
Debt, and the Single Optimal Portfolio ...
Systematic Risk, The Common Factor Model and
Multi-Factor Models .......................
Utility Functions and Utility Measurement .....
An Initial Market Equilibrium Exists With
Respect to Every Security's Price ........
There Were a Large Number of Portfolio Holders
and Securities ...........................
Probability Distributions Pertained to Real
Dollar Returns ...........................
Shareholders Have a One Period Time Horizon ...
There Were No Transaction Costs or Taxes ......
Shareholders Made Rational Decisions ..........

iv

Page

hmmNI—H—I

34

43
47

48
49
SO
50

50
51

page
CHAPTER 4: ADDITIONAL CONSIDERATIONS AND CONCLUSIONS

Securing Input Information for the Application

of the Model ..................... .... ..... 52

Eliminating Some of the Model's Assumptions ..... 58
Conclusions ...................... ,.... ........... 59
FOOTNOTES ............................................... 61
BIBLIOGRAPHY ............................................ 68

APPENDIX A: The Computational Procedure for Model
Solution ................................... A-l

APPENDIX B: Pricing Large Share Offerings .............. B—l

LIST OF FIGURES

page
The Feasible Set for Three Securities .................. 8
Determining the Optimal Set with Three Securities ...... 10
The Feasible Set in the Multi-Security Case ............ 11
The Feasible Set and Utility Indifference Curves ....... 15
The Feasible Set and the Locus of (e,v) Points of a
Specific Portfolio ............ ... .............. 19

The Feasible Set with No Certainty Return Securities ... 23

The Feasible Set with a Certainty Return Portfolio ..... 23
The Feasible Set Before j* and the Equilibrium

Feasible Set Containing j* ..................... 29
The Condition of a Single Optimal Portfolio ............ 35

Multiple Optimal Portfolios with Borrowing and
Lending Imperfections .......................... 37

Multiple Optimal Portfolios with Variations in
Expectations ................................... 38

Multiple Optimal Portfolios with a Risk Related

Debt Cost ...................................... 39
An Optimal Portfolio Zone .. ............................ 40
The Optimal Decision Set and a Market Risk-Return

Trade-Off Curve ................................ 41
A Flow Diagram of the Algorithm ........................ A-2
The Locus of Points of a Category 2 and 3 Security ..... A-ll

The Risk-Return Trade-Off Curve When a Certainty
Return Portfolio Does Not Exist . ............... A-l7

Turning Point (e,v) Locations and the Locus of
Points of a Portfolio .......................... A-18

vi

Chapter 1

INTRODUCTION

The Nature of the Study

 

The study deve10ps a security valuation model that considers
a security's value to be a function of both its expected return
distribution and the return distributions of other securities. The
value of a security is in part a function of the value it can
contribute to a portfolio of securities and this value is not solely
a function of the single security's expected return distribution;
the expected return distributions of other securities must also be

considered.

A Brief Historical Review

 

Markowitz was one of the first to recognize explicitly that
a security's value was a function of its contribution to a portfolio
and that this was mainly a function of the security's interaction
with other securities:

In portfolios involving large numbers of
correlated securities, variances shrink in importance
compared to covariances. A security adds much or little
to the variability of a large portfolio, not according
to the size of its variance, but according to the sum
of all its covariances with the other securities of the
portfolio.1

Many authors have acknowledged that a security's value was
partiallydetermined by the expected distribution of returns of
other securities.2 Yet, most security valuation models dropped from
the valuation process the influence of other securities' expected
return distributions on portfolio formation and failed to consider
this effect on a security's value. These theoretical models were
designed to determine value and were not intended for use in actual
security valuation. The effects of other securities on valuation
were excluded, and simplifying assumptions were made; such procedures
aided in using the model to examine other items that might effect
valuation and equilibrium pricing. No assumptions were made that

other securities' effects on valuation would be nominal.3

The Emphasis of the Study

 

The emphasis of this study was directed toward the formation
of a security valuation model that continued to include the expected
distribution of returns of other securities as a determinant of value
for each security. An underlying aim was to construct a model that
came closer to being applicable to real world situations. Such an
objective required the model to use input information that could be

specified from observed data.

A model was constructed that moved closer to application
even though the complexity of the problem required a number of
restrictive assumptions. These assumptions and their effects on the

model are discussed in a later chapter together with a tentative

examination of the possible consequences of alternative

assumptions.

While this study differed from previous works in the approach
it used,it drew heavily from the basic work already done in port-

folio theory.

Possible Uses for the Model

 

The model could be used to form expectations of the change
in both a security's value and the optimal portfolios it entered
when the firm affected changes in the security's expected distri-
bution of cash returns. The information requirements of the model
allow it to be used in a real situation. Application valuation
models should be of interest to management decision makers in
planning, to portfolio managers in adjusting their portfolios, and
to investment bankers who must anticipate the market's pricing

of an untested security.

The Type of Model Constructed

 

The model was developed for use as a guide to decision
making with the assumption that the valuation of a given security
should include other securities' distributions of returns as an item
effecting value. Furthermore, the intent was to develop a normative
model which could be applied to aid decision makers. The predictive

ability of the model was not tested.

Plan of the Study

 

The formation of the model is discussed in Chapter Two.
First the basic portfolio selection model is presented and then the

security valuation model briefly discussed above is developed.

In Chapter Three, the underlying assumptions of the model
are presented and discussed. The derivation of input information
required for the model, the model's practical limitations, and a
discussion of some alternative assumptions are covered in Chapter

Four. Possible model contributions are also reviewed.

A computational algorithm for the determination of a
security's value is presented in Appendix A while Appendix B is

concerned with equilibrium pricing of large share offerings.

Chapter 2

MODEL FORMULATION

Parameters

 

The set so = {B} = {(61, 62,...,6m)} is a given ordered
m-tuple of elements representing the parameters of the portfolio
selection decision. The parameter set includes: 1) the expected
cash return outcomes, ca., of each security j = l, 2,..., n, in

J

each outcome, aj = l,2,...,bj; 2) the probability of occurrence,

ya , attached to each possible expected cash return outcome of each

J
security; 3) ya, the joint probability of occurrence of ca

Ji

with ca for each possible joint outcome, a'ji = l,2,...,b'ji
i

for each pair if securities i = l,2,...,n, and j = l,2,...,n;1
and 4) the price pj, of each of the n securities. The assumption
is made that all shareholders hold equivalent expectations with

respect to the parameter set.
Relevant Sets

The symbol R is used to represent the set of all real numbers,
that is, R = {r : r is a real number}. R2 is the set of all possible
ordered pairs of real numbers, that is R2 = {(r1,r2) : r1 eR and
r2 eR}. R2 is the Cartesian plane of analytic geometry. Let SCR2
such that S = {(r1,r2)eR2 : r1 eR and r2 3 0}. S corresponds to

the first two quadrants of the Cartesian plane. Let E = {6 5R: e

is a possible expected return of a portfolio} and V = {v eR; v is
a possible expected variance of returns of a portfolio and v a 0}.

Let T = EXVCS.

Portfolios and Their (e,v) Values

 

Every given portfolio has a determinable (e,v) value in set

T. The expected return, e, of a given portfolio is

n
2-1) e = 2

where “j is the percentage weight of the portfolio in security j

and the conditions

2-2) a. 3 O
n
2-3) X a. = 1
hold. Thus, a specific portfolio, k, is defined by an n-tuple of

aj values (j = 1,2,...,n) satisfying (2-2) and (2-3). The expected

variance of returns of a portfolio is

n n
2-4) V = .2 X ajai Oji
3:1 1:1
where
bi bj 1 1
2-5) oji = E 2 (ca. pj -Cj)(ca. pi -ci)ya,
ai=l aj=1 J 1 31

and

2-6) c. = p

In the above, Oji, is the expected variance of returns of a security
j when j = i and the covariance between securities for j # i. Ej
is the expected return of security j. Equation (2~l) can be

alternatively expressed in a more general functional form

ll
H
v
N
v
v
:3
U

2-7) 8 = ge (k I Pj, Ca.a Ya.3 j
J J

ll
H
U
N
a
O
b
O‘
V

a
where k is the n-tuple of aj values, that is k = (al,a2,...,a ).

Variance, from (2-4), can be eXpressed as

a_. ya. ; J = l,2,...,n;

2-8) v = gv (k | pj, c ..
J 31

i = l,2,...,n;a'.. = l,2,...,b' .).
)1 J1

The parameter set for (2-8) can be used for (2-7) although the

reverse condition does not hold. Now a new rule g is formed.

2-9) g : K + 2
such that
2—10) g ( kl. ) = (e,v) = 2 £2

for all k 5K where the dot notation represents the parameters shown

in equation (2-8).

The Feasible Decision Set

 

For every possible portfolio there is an (e,v) value. Denote
this value by z and let Z represent the set of all feasible (e,v)

values. Owing to the stated conditions we also have ZCT = EXVCS.

The feasible region of T will now be described. A two

security and three security environment will be assumed to simplify

1/

explanation. Figure 2.1 shows the (e,v 2), the return-standard

deviation space, location for three securities, 21, 22, and 23.
The explanation is more easily accomplished by using the standard

1/2

deviation, v , as a variable and by later returning to the use

1/

of variance, v. First, the possible (e,v 2) values of all possible

Figure 2.1

The Feasible Set for Three Securities

v1/2

 

 

 

combinations of two securities, 21 and 22, is covered. The feasible

1/

locus of (e,v 2) values of two securities is represented by a line

connecting the two (e,vl/z) locations of the two securities. If the

two securities are perfectly positively correlated, covij = vll/val/z,

1/

2) locations is a straight-line segment, 2' of

1/

the locus of (e,v

figure 2.1, whose end points are the (e,v 2) locations of securities

21 and 22 in Figure 2.1.2 At the opposite extreme, the perfect

. . 1/2 1/2
inverse correlation case, covij = - vi vj

two straight line segments, 212" and z”z2 of Figure 2.1, that go to

a common location on the e axis.3 Between the perfect correlation

, has a locus of

case and the perfect inverse case, the feasible locus will be a line

segment convex to the e axis, denoted z"'in Figure 2.1.4

Expanding into the three security case, let lines zlz"'z2 ,

 

of Figure 2.1 represent the line segments between

1/

2223, and 2123
the securities (e,v 2) locations. All points bounded by

_____ 1/

I” .
212 22 , 2223, and 2123 are attainable (e,v

any combination of two feasible portfolios will also be feasible.

 

2) locations since
5

The boundary that maximizes e while it minimizes V“2 will
be convex with respect to the e axis in (e,vl/z) space. A possible

diagram of the feasible combinations of a security 21 with 22 and

22 with 23 is shown in Figure 2.2. The curvilinear line segments can

cause an inward bending kink as shown in the figure. In such a case
there will always be some combination of two feasible portfolios

that will have a higher e than the portfolio at the kink, 22, for

1/2.

the same value of v Thus, in Figure 2.2, one could take the

IO

portfolio at 24 and combine it with the portfolio at 25, and with a

perfect correlation between the two new portfolios one could form

1/

a portfolio having an (e,v 2) point somewhere on the straight line
segment 2425. With anything less than perfect correlation the line

segment 2425 would be convex to the e axis.

1/2

D

If the standard deviation variable, v is converted

Figure 2.2

Determining the Optimal Set with Three Securities
1/2
v

 

 

 

to variance, v, the optimal boundary will also be convex through—
out.6' All points bounded by the convex line segments will be'

feasible‘as in the standard deviation‘case.

The 2* region of Figure 2.3 bounded by 21 through 24
will be used to represent the locus of points of all small 2 eZ. For
any (e,v) point in the Z* region of Figure 2.3 there corresponds
a 2 62 that has the same value.7 Thus, the 2* region of Figure
2.3 will be entirely filled with points, i.e., attainable (e,v)
value portfolios. The highest possible variance in Figure 2.3 is

at location 21 = (e3,V4). In any set

11

Figure 2.3

The Feasible Set in the Multi-security Case

 

 

 

 

 

 

e1 0 e2e3 e4
the (e,v) value of the single security portfolio with the highest
variance will be the 2:2 with the maximum v. Combining this
security with any other security will result in a portfolio
weighted variance less than the variance of the single security.
This can be confirmed through equation (2-4). Likewise, the
security having the highest expected return, e4, will have the

262 with the maximum e, z = (e4,v3) of Figure 2.3. The security

2

with the lowest expected return, e , will have the 232 with the

1

minimum e, z = (e1,v2) of Figure 2.3. The minimum variance

3

portfolio, at z of Figure 2.3, is not as easily derived.8 The

4

variance of a portfolio is derived from equation (2-4). The

minimum variance portfolio could be found by determining the variance
of all possible portfolio combinations, which are infinite in

number, or by a quadratic programming technique. Thus, from.Figure

2.3,ZC {zzz = (e,v) where e1 s e 6 e4 and v1 s v s v4}. e1,e4,

and v4 are easily found by looking at the expected returns and

variance of the n securities. v will be equal to zero if a

1
certainty return security exists. The points on 2224 minimize v for
given values of e where e2 5 e s e4. The points on 232122 are

those that maximize v for given values of e where e1 s e 3 e2. The

12

line EEEZEE— is convex with respect to the E axis. This is also
the case for the line 233:: As shown in Figure 2.3, 2:33. is
concave with respect to the e axis. The (e,v) boundary for any
set 2 is determinable. The variance, expected return, and

covariance terms of the n securities determine this boundary.

Shareholder Utility Functions

 

Every investor has a utility function, say F1, that relates

utility, u, to wealth, in this case expected cash returns, c. Thus,
2-11) u = F1(c)

The assumption is made that investors have positive utility for

wealth with decreasing marginal utility. Thus,

2-12) F1 > 0

2-13) Fl" < 0.
Wealth is to be measured in real dollar terms.

Investors are assumed to be risk averters; they would be
willing to pay more to receive the expected value of an investment
rather than the investment itself. Investors are by definition
risk averse, when from (2-13), F1" < O. This is equivalent to
requiring the utility function to be strictly concave.9 Returning
to the first two statistical moments of a portfolio's return

distribution, (2-1) and (2-4), a new utility function is formed,

13

2-14) u = F2(e,v).

Since there is positive utility for wealth from (2-12), it follows

that

2-15) BFZ > 0.

Be
With the assumption that investors are risk averse, there will be

disutility for variance of returns,

2-16) 3F2

3v < 0'

 

Location of the Utility_Function on an (e.v) Cartesian Plane

 

The current analysis is concerned with investments and

there return so that

2-17) -m < e < + m.

Since v is the second moment of a distribution of expected returns,

by definition

2—18) v

\V

If utility is held at a constant arbitrary value, say d, we can

express equation (2-14) in the form

2-19) v = Gd(e).

Gd is a mapping from E to V,

14

2-20) cdze + v 2 Cd CT

From conditions (2-17) and (2-18) we have TCS and thus, now viewing
Gd as a set, GCS. The investor will be indifferent among (e,v)
values with the same utility, i.e., any (e,v,)e Gd. The utility
functions are in positive real (e,v) space. Further, given
investor utility for e, from (2-15) and disutility for v, from
(2-16), the utility indifference function, Gd, is strictly concave.

That is,
2-21) Gd" < O.

Geometrically, Figure 2.4 shows three possible utility indifference
curves, 61, G*, and G2 where the utility corresponding to the curve

G1 is greater than 6* which is greater than 62.

The Optimal Attainable Decision Set

 

Of the attainable (e,v) combinations, 2 eZ, only a subset

of possible 2 values would be desired by the investors. The subset,
called the optimal decision set, includes those elements z 52 that
fulfil both the conditions of minimizing v for every given e value
and maximizing e for any given v value. Thus, a new set is formed,
M = {z eZ:z that minimizes v for an attainable e value while also
maximizing e for any given v value}, corresponding to 2332' of
Figure 2.3. We have MCZ. A quadratic programming technique could

be used to determine the optimal subset , M, of (e,v) values.10

15

Maximization of Investor Utility,

Given the condition that investors have utility for e, from
(2-15) _,and disutility for v, from (2-16), the investor maximizes
his utility by selecting an element, or 2 value, from the Optimal
subset M. That is,

2-22) max F2(z) = F2(z*), 2* 5M.
2 eZ

The market equilibrium set of Optimal 2 values is defined
by this study as being identical to the investor's optimal
decision set, M, represented by line 3123 of Figure 2.4. From
(2—20) and (2-21), the nature of utility indifference curves, G1,
6*, and 62 of Figure 2.4, were specified; they were strictly
concave. The intersection of M with a shareholder's highest

utility indifference curve, 6*, of Figure 2.4, gives the single

Figure 2.4

The Feasible Set and
Utility Indifference Curves

V

 

 

 

l6

optimal 2 value that maximizes the shareholder's utility, i.e., it
fulfils the requirements of (2-22). In the figure it is the 2

location at 2*.

In summary, the portfolio selection process for an investor

has the decision structure

1) H = (909 M: F2)
with the maximization problem
ii) max F2(z).

zeM

Valuation of a Security

 

The relationship between a securityls price and aygiven
portfolio: the single security_case. To simplify explanation the
initial assumption is made that only single security portfolios

are held. Thus,

Otj=0

Ho
‘I‘klIM:

Ho

OLj=l

in any given portfolio. A single security portfolio's e, derived

from (2-1) can be expressed as

2-23) 6 = —- c.

17

and (2-4) can be expressed in the form

bi
2-24) v =-I1;2 331 (ca - 33-)2 ya
3' J J' 3
where
. b1
2-25) cj = a-g 1 Ca. ya.’
3 J J

From (2-23) and (2-24) we can see that e + t w and v + w as p + 0
while e + O and v + O as pj + w for real non-negative values of
pj. For any real non-negative pj there are both a real e and a
non-negative real v. j* will be used to represent the specific
security whose equilibrium price we are seeking. The Cartesian
product, T, of E and V has already been established: T = {(e,v):
eeE and veV}. We had previously established the condition TCS.
Let the set P = {pzp is the possible price of security j* and
p > 0}. Since there is a mapping of P into E as shown in (2—23)
and of P into V as shown in (2-24), there exists some rule O which

is a mapping of P onto the set L,

2-26) Q : P + L

such that

_ o o ’= o O = I O 0': °
2 27) Q( pIk 9 caj yavji’ 3 192300.911: 3 ji 1:23-009b ji’l 1’2’--°9n’

and pj for all j#j*) = (e,v) e LCT

for all peP. The rule Q may also be identified as the following

set:

18

2‘28) Q = {(Pse,V)€w : ezp-lej, vzp-z

ya for all p > 0}

j

where WCR3 such that W = {(x1,x ' x > 0 and x a O}. Q is

2’x3) ' 1 3

the set consisting of all ordered 3-tuples (p,e,v) such that for
peP we have e and v, the corresponding elements of E and V
respectively. We now have the image set of Q, denoted L = Q(P),
that has the set of ordered (e,v) pairs designated by (2-27). In
this study, the market equilibrium price for a security in a specific
portfolio is defined to be the price of security j* to which corres-
ponds an (e,v) pair, t, in the optimal decision set M. In the
section on the optimal decision set it was established that

MCZCTCS. A security's equilibrium price is any price peP

for which Q(p) = te X where X = LnM. The set denoted fik of

equilibrium prices for the security j* corresponding to portfolio

k is:

2-29) fik = {peP : Q(pl. ) = t e LnM}.

If X = c then Pk = ¢ and there are no real non-negative equilibrium

p values. Diagramatically, Figure 2-5 shows attainable set, Z,

the locus of points teM, represented in the figure by 2122, and the

locus of a subset of the values teL, represented by lltz of

Figure 2.5. z*eZ would be the 25X. The unique peP corresponding

19

Figure 2.5

The Feasible Set and the Locus
of (e,v) Points of a Specific Portfolio

 

 

 

to z*eX is the equilibrium price of j* in this specific portfolio

k.

The relationship between a security's price and a given

 

portfolio: the multi-security portfolio case. Modification of the
previous system of equations will enable the determination of the
equilibrium price of a security in any specific portfolio k.

Looking at the e and v of a multi-security portfolio from (2—1)

n-l bi _1 _1 A
2'30) e = 1:1 3,21 (aipi Cai yai) T aj*pj Cj*
1
i#j*

and from (2-4)

20

n-1 n-1 n-1 2
2-31) v = 1:1 £21 (dial Gig) + 2 1:1 (oi 3*0j*1) + oj*j* aj*
i#j* 1#j
where from (2-25)
-1 bj* b1 -1 ‘ “
2-32) Oj*i = p _ _ [(Ca. pi - Ci) (ca.* -Cj*)ya'. .1
aj*—l ai-l 1 j 3*1

2-33) e = 81 + 82 P.1
and,
2-34) v = 83 + 84p"1 + esp'z.

The 81 for i=3,4, and 5 are all positive given the previous require-
ments of the values of the parameters: 81 and 82 could be negative.
Thus, with (2-33) and (2-34), just as with (2-23) and (2-24),

e + t w and v + w as p + O and e + 0 and v + O as p + m. For any
real non-negative p there are both a real e and non-negative real

v. Again as in the single security portfolio, there exists some
rule, Q, which is a mapping of P onto the set L. The rule Q

is identical to the set

2-35) Q = {(p,e,v) w : e = 81 + sz'l. v = 83 + s4p'1 +

85p }

such that

21

2-36) Q (pl- ) = (e,v) a LCT.

W in (2-35) is as defined for (2-28) and the dot notation in (2-36)

represents the same parameter set as in (2-27). The image set of

Q is
L=Q(P).

The security is equilibrium priced at any peP for which Q(p) =

(e,v) is in the set
2-37) X = LAM.

As defined in (2-29) the arguments of Q, peP related to the ordered
pairs xeX are the equilibrium prices, Pk, for security j* in
portfolio k. The definition corresponds to that on page 18

with the single security case. If X = ¢ then Pk = o and no real
non-negative price exists. This more general multi-security
equilibrium pricing system will also work with single security
portfolios. The equilibrium set of prices, Pk, is for the

specific portfolio k. This is the equilibrium price of j* only

in this particular portfolio k.

There is a possibility of multiple elements zeX. A

diagram of a hypothetical segment of the locus of points

representing the values of the set M is shown by mzm' of

Figure 2.6a. The locus of points representing the t values

of the set L for a given portfolio k is shown by tt'. In

22

Figure 2.6a there are two possible equilibrium pk values

A

corresponding to the points 21 and 22. These pk values are

the equilibrium prices for security j* in this particular portfolio.

Investors whose utility functions 111 and u2 of Figure 2.6a are

tangent at points z or 2 would want to hold this portfolio k

l 2
with j* at the price corresponding to their particular tangency

points, z , or z , of Figure 2.6a.

1 2

If the conditions represented in Figure 2.6a arose, equilibrium
prices would not be at the pkePk already determined. The shareholders
with the utility indifference curve, ul, could go to a higher utility

indifference Curve, ul', by holding portfolio k at a peP correspond-

ing to the keL represented by’t1 in Figure 2.6a. This is a pePk.

The assumption is made that a certainty return security,
t = (e,v) where e > O and v = 0, either exists or can be constructed.
This condition prevents the possibility of obtaining an equilibrium
peP max . Pkas was obtainable in Figure 2.6a. In Figure 2.6b

2* . pkePk

represents the attainable set of 2 values. When cer-
tainty return securities exist then there will be at least one
element 2 E (e,v)eZ where v = O and e > 0, represented by location
e* of Figure 2.6b.11 First, e + O and v + O as p + w for the locus
of points teL corresponding to real non-negative p values. Second,

the locus of points zeM form a convex curve with a positive e

intercept. Any pcP > max Pk will thus have a corresponding

pkePk

23

tel and t¢M, for example 2 in Figure 2.6b, that cannot be

2

optimal.

Figure 2.6a

The Feasible Set With No
Certainty Return Securities

V

 

 

 

Figure 2.6b

The Feasible Set With A
Certainty Return Portfolio

V

 

 

 

24

A further assumption is made that there are both a large
number of investors and securities. The effect that any one
investor or security's cash return distribution, specifically j*,
would have on the equilibrium market condition represented by the
values,zeM, is therefore assumed to be nonexistant. Accepting
the assumption, M remains unchanged when j* is traded. Thus,
for this study, set M is defined as being constant and invariant

to a new security j*.

The nominal effect assumption just given also implies
that no shares of j* would sell or trade below the maximum
pkePk. There would be a large enough number of investors and
small enough number of shares that the supply of j* shares would

be fully subscribed at the highest psP corresponding to a zeX.

The previous assumptions lead to the definition that the
final equilibrium price for j* in a particular portfolio would

be

A

2-38) pk = max pk.

A

pkePk

Only one element of the set P called pk, is the possible price

k

of the security.

Price discrimination or no price discrimination. The

 

possible condition of price discrimination is assumed away with
the nominal effect assumption; all shares sell at the maximum

obtainable price fik. If the assumption of no price discrimination

25

were maintained and the nominal effect assumption dropped, the
security would have a price equilivalent to only one element
of the set Pk; this need not be fik. For Pk greater than a
single element set if pk were not the equilibrium price a
reformation of the set M would be required for the investor
portfolio selection decision. The condition causing this is
similar to the one disclosed in the discussion of Figure 2.6a,
where some investors can obtain higher indifference utility
curves by holding j* at a peP corresponding to a location teL
and t¢Z and thus ttM. If the assumption of price
discrimination were made a reformation of set M would also be

required for the same reasons stated above.

Determining;the Final Equilibrium Value of a Security»

 

Other possible k portfolios that might offer a larger
fik could be found by looking at each possible portfolio where

uj* > O. X represents the set whose elements are an ordered

n-tuple fulfilling the conditions of (2.2), (2-3), and °j* > O;

that is, K = {(a1,az,...,an) : ai is the percentage of the

portfolio held in security i such that ai#j*
n

aj* > O; and .21 ai = 1}. For each keK we have a pk from (2-38).
1:

Let Pk represent the set of pk elements. Then let

a O, for i : l,2,...,n;

2—39) p* = max pk.

pkepk

26

p* is the final equilibrium price for security j*. Further,
for any keK for which there is a corresponding p*, the n-tuple, k,
will disclose the “i weights of the securities in the portfolio(s)
that maximize the price of j*. K* will be used to represent the

set of keK that has j* at the price p*.

In summary, the security pricing process has the decision

structure

2-40) I = (91) K, I)

where the parameter set 91 is identical to the set 90 used in the
portfolio selection decision, covered on page 16 , except that

the probabilities attached to the cash return outcomes for security
j* have been changed. The set K is the decision or policy space

for investors in the market. In the equilibrium condition we

have

2-41) 13k = I (k)

In the general case equations (2-33) through (2-38) comprise
the system of equations and relations that define the rule L,
whereby to each element keK there corresponds a unique peP
called pk. The above decision structure (2-40) has the maximi-

zation problem

2-42) max I (k) = max fik.
keK - -
pkePk

27

Computational Procedure for Model Solution

 

It is not possible to obtain a value for (2-42) unless
a method of satisfying (2-42) can be achieved without looking

at all attainable portfolios containing j*.

An approach that is possible would have to either find an
analytical solution to the problem of deriving p* for security
j* or a method that would reduce the number of computations to

a finite number.

The analytical approach. First, the analytical approach

 

will be discussed. D is used to represent the feasible set of
(e,v)eT corresponding to all possible portfolios containing j*,
i.e., aj* > O, for all peP where p > 0. Thus, there exists some

rule, F3, which is a mapping of P onto the set D,
2-43) F : P +-D.
such that

2-44) F3 (plca ; ya,.‘ ; j = l,2,...,n; i = l,2,...,n;

J 31

.'= l,2,...,b ; '.. = l,2,...,b'..,and . for all. .
3J J a 13 13 p3 J#J*)

= DPCD.
For any peP there is made to correspond a unique set DPCD of t
elements representing the t locations of all possible portfolios
keK. Equilibrium is defined to exist for Dp* = DPCD where both

the conditions hold that

28

2-45) Dpn M 7‘ ¢
and
2-46) Dp = {dpeT: dp = (e,v) E t such that for any

given v in dp if there is an element meM with

the same v it has an e a to the e in dp}

The portfolios containing j* are either optimal, i.e., teM,
or are less than optimal. This is the condition previously
defined as the equilibrium condition; that is, the peP correspond-
ing to Dp where both (2-45) and (2-46) are satisfied is the

p*eP of (2-39) and (2-42). Dp* is the DpCD corresponding to p*.

Graphically, Figure 2.7 shows a representation of the
set Z, designated as Z in the figure. Dp* is used to represent
the set Dp*' Dp* of the figure, due to (2-45) and (2-46), is
tangent and to the left of the market equilibrium set M, repre-
sented by the line between points 21 and 22 of Figure 2.7. The

final portfolio held by investors could be those keK corresponding

to the t c Dp* M, at location 2* in Figure 2.7.

The problem arises that the rule F3 is not easily specified.
Any set Dp is defined Dp = {dpzdp = (e,v) E t and is derived from
equations (2-1) and (2-4) for a keK, and subject to the further
condition that aj* >0}. The value of an element dp is a function
of the unique portfolio keK that dp is calculated from. Thus, one

can only define the location of Dp by looking at each element dp

29

Figure 2.7

The Feasible Set Before j* and the
Equilibrium Feasible Set Containing j*

 

 

 

corresponding to each keK. We are back to the original problem

of an infinite number of elements and computational infeasibility.12

A numerical approach. A computationally feasible alternative

 

employs a numerical approach. The quadratic programming technique
used in determining the efficient set M, of (e,v) values is

employed.

Our concern is to find the increase in value of a security

n

that comes from diversification, when 2 a. > O. The equilibrium
i=1
j#j*

value of the security held by itself, i.e., the keK where

aj* = l. and Z a. = O, is used as the point of departure in

j#j*
determining additional value attributed to diversification,

Z a. > 0. First the numerical procedure uSes as a starting price
j#j*

30

the peP for which Q(p) e X = LAM for a portfolio containing only
security j*. Set M contains all the ordered pairs (e,v) E t

of the optimal set. M is a mapping of E into V,

M : E + V

The optimal decision set can be represented by

2-47) v = M(e)

L is the image set of Q such that for each ordered pair

BEL we have 2 = (e,v) where from (2-23)

2-48) e = p-lcj*
and from (2-24)
-2”
2—49) v = p j*
where
b
m A 2
c.* = Z (c - c.*) y
J 3:1 aj* J aj*

c.* and E.* are constant for security j*. The function for the
J J

(e,v) values for the portfolio keK where aj* = 1 can be repre-

sented by
2-50) V = L(e)

However, the function L can be identified. From (2-48) form the

new equation

2-51) e L p

31

Letting

2-52) b1 = j* / c.*

2-53) e = p

Now from (2-49) and (2-53) we find

e2 = vb

and
2-54) v = e2 b

This is the equation for the relationship between e and v as the
price of security j* changes. Bringing (2-54) and (2-47)

together
2-55) M (e) = ezbl

Equation (2-55) can be solved for e. By substituting this e in
either (2-54) or (2-47) one can find v. Now either (2-48) or
(2-49) can be used to determine 5k for keK where aj* = l. The
(e,v) pair is the element xeX and the derived pk the equilibrium
price of the security when k represents a portfolio only contain-
ing j*. The pk so derived is the initial price for security j*
used in the numerical approach. Using this 5k the quadratic
programming technique is used to find a new optimal (e,v) decision
set M. M0 is used to represent the original (e,v) optimal set

defined as the market equilibrium set by the study, previously

32

referred to as M. Now M1 is formed, the optimal two-space set
derived from all n (or n + 1) securities.13 Security j* enters
into the formation of M1 at the pk first derived. Now the check

is made to see if
M0 = M1.

If there is any element m1 = (e,v)eMl such that v equals the v

of the ordered pair m0 6 M0, and we have e from m1 greater than

e from mg, the security j* is undervalued and not equilibrium
priced. This will be called condition (1). Condition (2) will
exist if the e from every element m1 is less than the e from

an element m0 having the same v as m1. In this condition (2)
security j* is overvalued and not equilibrium priced. If condition
(1) holds, the security j*‘s price is incremented and M2 calculated;
if condition (2) holds, security j*‘s price is decreased for the
calculation of M2. By either decreasing the incremental price
change everytime a switch from condition (1) to (2) or (2) to (1)
occurs, or by some optimal seeking technique this process is

repeated from M3, M4,..., Mn until Mn is reached where
2-56) M = M .

An estimate of the equivalance situation, (2-56), would have to be
used. This would decrease the number of iterations to a finite
number, say n, with nominal error in the final equilibrium derived
security value for j*. By increasing n the degree of error is

decreased.

33

When (2-56) occurs we have
2-42) p* = max I (k) = max p .
keK p 5P k
k k
This is the price of j* used in the formation of Mn' The quadratic
programming techniques discloses the keK corresponding to the

Inn 2 Mn so that the optimal portfolio(s) j* entered at the

equilibrium price, p*, can be determined.

The numerical approach just covered can give an estimate
of the final equilibrium value, p*, of security j* -— as close
as desired to the actual value assuming that the computational
process converges, i.e., lim Mi = M0. An estimate can thus be

i-rco
derived with a finite number of computational Operations.

Chapter 3
MAJOR THEORETICAL DECISION AREAS

CONSIDERED DURING MODEL CONSTRUCTION

The assumptions of the model explicit and implied that were
not covered previously are discussed in this Chapter. The choice
of assumptions was influenced by the desire to achieve a more
operationally feasible and useful model; such assumptions, however,
tended to make the model more complex and computation more diffi-
cult.

Equal Expectations, Perfect Markets, Riskless Debt,
andvthe Single Optimal Portfolio

 

With a proper set of assumptions, it can be shown that only
one optimal portfolio would be Obtained. One set of assumptions
that provided this conclusion are now given. First, markets
were perfect. Second, riskless debt borrowing and lending
existed. Thirdly, shareholders held exactly equivalent expecta-
tions with respect to the parameter set given in the portfolio

selection decision.1

Diagramatically, this portfolio corresponded
to the 2 at location A of Figure 3.1. The portfolio was found at
the point of tangency to the Optimal decision set of a straight
line that originated from the riskless rate of interest, r* at

a zero standard deviation. There was no desire to hold any other

portfolio in the optimal decision set, BAD of Figure 3.1.

34

35

Figure 3.1

The Condition of a Single Optimal Portfolio
1/2
v

< A

r*

 

 

The holding of exactly identical expectations, allowed
a single curvi-linear optimal decision set represented by BAD of
Figure 3.1 to represent the set used by all shareholders. The
existence of a perfect market in the purchase and sale of
securities precluded the existence of differential costs or rates
of returns on securities having the same risk. This condition
enabled a single borrowing-lending trade-off line, f7C, to be
used by all investors. The riskless lending and borrowing

1/2)

assumption defined this trade-off line as being straight in (e,v
space. Since the Optimal decision set was curivi-linear and
convex throughout, and there was a single borrowing-lending trade-off
line which is straight throughout, there was only one point, A of
Figure 3.1, of tangency between the two and therefore only one
Optimal portfolio. Thus, when the market was in equilibrium the
weight Of a security in the Optimal portfolio, corresponding to

location A, could be interpreted as the ratio of the aggregate

market value of that security to the aggregate market value Of

36

all stocks.2 This single Optimal portfolio condition was
accepted as the equilibrium situation in analytically deriving
a security's valuation.3 In such a case value was determined
more directly and with a simpler procedure than the one used in

Chapter Two.

Now a new set of assumptions were used to see what would
happen to the optimal decision set, the borrowing-lending line,

and the optimal portfolio.

Market imperfections. If the condition of the existence

 

of market imperfections in the borrowing and lending of funds

were allowed the line segment r*AC might have been different

4

for different investors. In Figure 3.2 investor one might be

able to operate on the line segment r*lAlC1 when he loaned and

borrowed at his available riskless rate and invested in his optimal
portfolio at location A1 while investor two would have optimally

Operated on line segment r*zAZC2 . Different investors would

have different optimal portfolios, at location A1 in one case

and A2 in another. The condition of a single Optimal portfolio

would not have existed in such a market.

37

Figure 3.2

Multiple Optimal Portfolios With
Borrowing and Lending Imperfections

v1/2

 

(D

 

Equivalent expectations. If the assumption of exactly

 

equivalent expectations were dropped, shareholders would have

had different expectations about the expected dollar stream

of benefits attached to each security. In Figure 3.3 one
investor's optimal decision set, constructed from his expectations
might have appeared as BIC; , while the second investor's appeared
as BEG; .5 Given the same pure rate of interest borrowing and
lending rate for both investors they could have both had

different optimal portfolios. The single optimal portfolio condi-

tion would not have held when expectations were not identical

for all shareholders.

38

Figure 3.3

Multiple Optimal Portfolios With
Variations in Expectations

v1/2

 

 

 

Riskless borrowing and lending, The assumption of
borrowing and lending at the riskless rate was dropped. With
differences in the "safeness" of loans or debt Obligations a
risk-related borrowing or lending rate would more closely duplicate
the real world situation. The risk increment of debt cost was
assumed to be associated with the risk inherent in meeting the
required debt payments of interest and principle. This would
have in turn been a function of the expected earnings level and
variance in earnings of the particular borrower. If an investor
borrowed, the earnings rate and variance would have been a
function of the investment portfolio held by the investor. The
amount Of borrowing used by an investor would have changed both
the expected earnings level and variance; the risk associated with
the borrowing would have been effected. This situation would have

given rise to the backward bending debt leverage line, line

39

segment AC1 or possibly Ac2 of Figure 3.4.6 If the backward bending

Figure 3.4

Multiple Optimal Portfolios With
A Risk Related Debt Cost

v1/2

 

 

 

AC;— intersected the Optimal decision set, where i = 2 in Figure
3.4, there would have been many possible Optimal portfolios.

Any investor who desired a return greater than that available

at point B2 of Figure 3.4, would have had to select unlevered
portfolios on the optimal decision set BED of Figure 3.4. In this
possible situation there would have been more than one optimal
portfolio. For any desired return risk combination less than 82’

portfolio A would have been optimal and would have been used with

borrowing or leading to get the desired return level.

Bringing the revised set of assumptions together. If the
assumptions of unequal expectations, a risk related debt cost, and
market imperfections were all taken together there would have
been at least one and possibly several optimal portfolios available

to each investor. If the 2 values of all optimal portfolios of all

40

individual investors were plotted, an optimal zone in T space would
have been formed for the market. All of the securities in the
portfolios corresponding to a Z in the optimal zone could have had
a market, and not just the securities in a single optimal portfolio.
In Figure 3.5 the zone is represented by the shaded area of the

set of all possible attainable 2 values from portfolios. This
shaded subset of attainable portfolio 2 values is composed of the
area bounding all of the 2 values of the optimal portfolio for
every investor. Given the modified assumptions, a given portfolio,

k, could have had multiple 2 values.

Figure 3.5

An Optimal Portfolio Zone

V

4 I

 

 

The return of the assumption of exactly equivalent expecta-
tions simplifies analysis by eliminating the condition of both
multiple 2 locations for a given portfolio and by eliminating the
optimal zone in return-variance space, or return-standard deviation
space as shown in some of the figures. The plotting of every

individual investor's optimal portfolio's or portfolios' z value(s)

41

would then have formed an optimal attainable boundary, and not a
zone, that would have included portfolios' 2 values that could
be held in an equilibrium situation. The 2 locations of this
boundary would have been either part or all of the optimal set,

M, described in the portfolio selection model of Chapter Two.

In a perfect knowledge environment the subset of set M
that represented 2 values desired by investors could be determined.
The market required risk-return indifference or trade-off curve
would be needed. This curve represents the t E (e,v) e T from
O s v < m where for each v there corresponds an acceptable e.
The term "acceptable" can be described more easily through an

example. An attainable set where certainty return securities are

Figure 3.6

The Optimal Decision Set and A
Market Risk-Return Trade-Off Curve

 

 

 

42

available is represented by Z* of Figure 3.6. The locus of points

of set M are represented by EIEE’. 23?; represents the locus of point
teT that would be acceptable, i.e., purchased, if portfolios Offering
these t values were available. In this situation the subset of the
Optimal decision set M, the locus Of points zeM represented by line
segment 2233—; would not be acceptable or optimal for investors

even though the values represented by 2223 would maximize e while

minimizing v.

If one could specify the market required risk-return
indifference curve the model of Chapter Two could be modified to
value a security given this new condition. The model was formulated
with the assumption that the set M represented the markets risk-
return indifference curve; Appendix A considers valuation when M
and the locus of points of the indifference curve are not the same

set.

This study's assumptions. The portfolio selection model of

 

the last chapter determined the optimal portfolio combinations
currently available in the market. The 2 values of each of these
portfolios comprised the market equilibrium set, M. This set

was formed with the assumption that debt cost had a risk component
and that market imperfections existed in the investment of funds.
The assumption of unequal expectations was not adopted. The
application Objective of the study would not be obtainable if

this assumption were adopted.

43

Systematic Risk, The Common Factor Model
and'Multi-Factor Models

 

 

In portfolio analysis the one common-factor model has
been used to predict or describe the possible return outcomes
of a security. It accomplished this by limiting variations in
return outcomes to only one variable, the general economic
condition. Thus, the expected rate of return from a security, ri,

was viewed to come from

3-1) r. = a. + b.x + u.,
1 1 1 1

where ai was a constant for security i, bi was a coefficient repre-
senting the responsiveness of firm i's return to the measure of

the general performance of the economy, x. Last, ui represented
the random error term associated with firm i. A given firm's

risk was bizox2 + Ou 2 , where the first term was the systematic

i
risk portion attached to the general factor and the second term
was the residual risk portion associated with the given firm. For

any two firms the covariance term was bibjoioj.

Diversification through the holding of several securities
could have been used to decrease risk. This could have been
accomplished if the ui were truly random. By investing in several
securities the 111 error terms of the several firms would have
tended to "cancel each other out." For example, assume that each
of several securities had a 111 Of plus or minus one percent with

a 50% probability. If y dollars worth of one security were

44

purchased a plus or minus one percent variation in return with
equal probability could have been expected. If y/z dollars of each
of z securities were in a portfolio there would have been only one
chance in 22 of having either a plus or minus one percent variation

0 7
1n returns.

The spreading Of risk did not occur when there were per-
fectly correlated variations in returns. This occurred with the
common factor, x, of equation (3—1). If a portfolio contained
2 securities each with bi = k, where k was some constant, there
would have been no change in the overall risk of the portfolio
regardless of how many securities were held.8 Thus, systematic

risk, bizoxz, could not be diversified away.

A simplifying assumption made by most of the models
relating security valuation to portfolio selection was that the
overall movement of the economy was the single common factor
that affected the returns of securities.9 For an operationally
useful model, the single common-factor approach avoided some of
the factors that could have affected both securities' returns and
shareholders' expectations. A multi-factor model could have been
used to insure that these factors were included in the valuation

model. This might be represented in equation form by
3-2) r. = a. +

where cij represented the factor multiplier for security i in the

jth of m factors, and yj represented the value of the factor j.

The other variables were the same as in equation (3-1). Every j

45

would have represented a factor common to some specified subset
greater than one of the universal set of securities. One of these
factors might have been the overall movement of the economy. If
all securities were affected by this factor every security would
have entered this set, cij#0 for every security.10 An example
might help clarify this approach. Assume that a major auto strike
was imminent and that it could have occurred during the time
horizon being considered for holding the portfolio. For the

auto companies, their dependent suppliers and their franchises,
Operations would have halted or been severely restricted. This
would have affected the profitability Of these firms and the cash
return to their shareholders. One factor could have been used to
represent this event; possibly a more general factor representing
the level of production in the auto industry could have been

used. The weight of the multiplier, Cij’ would have been a
function of the dependence of a given firm's returns on the
operations of the auto industry. Next, due to the size of the auto
industry and its related industries as a component of the GNP

the entire economy might have been affected. This "systematic
effect" of the general economic condition would have been repre-
sented by another factor. To be as accurate as possible further
common factors affecting any subset of the securities greater than
one and less than the universal set would have been included in
the return function, equation (3-2).- If this were done, the 111
terms of all securities would have been non-correlated error terms.

In the single factor approach the residual or error term, ui, for

46

each security was viewed as being independent of other securities'
error terms, i.e., cov (ui, uk) = O for the security i f k.

If the single factor model was used to evaluate the returns of
General Motors, Borg-Warner, Federal Mogal, and U. S. Steel, by
investing in all four one would not have been "spreading the risk"

to the extent predicted. The assumption tht ugm’ ubw’ ufm, and

uuss were all independent would have been faulty. Within this

subset of securities, if the single factor model had been used we
would have had u1 = ciy + e and a would have been quite small. Thus,

the spreading of risk would have been very small if one invested

in all securities in this subset.

A further bias in the estimation of returns existed with
the use of only the general economic condition factor. In the
multi-factor model the yj that represented the value of the
economic factor would have been the same as x in equation (3-1).
However, the cij multiplier would not have equaled bi’ the multi-
plier for the same schrity and factor in equation (3-1). This
would have occurred because of the multicolinearity existing
between many factors and the general economic condition. With
these other factors included in the model we would have expected
cij 5 bi' The earnings in the building sector could serve as
an example. Housing starts and the general level of building and
earnings in the industry are functionally related to both the
general economic condition and the interest rate structure. A

correlation exists between these two factors that would upwardly

bias the mulplier bi of the single factor model over C1 in the

47

multi—factor case. This is true of many factors common to subsets
of securities. The upward bias in the weighting of the general
economic factor and the lack of recognition of factors affecting
only a subset of the universal set of securities both bias away
from diversification that would actually spread risk. Faulty

security valuation could result.

A multiple factor model that considers general factors
and some factors that only affect subsets of the universe of
securities was adopted by this study's model. The construction
of the valuation model viewed investors as using multi-factor

models in forming their return expectations.

The adoption of the multi-factor assumption had the
effect of requiring the entire variance-covariance matrix of
returns of the universe of securities in the formation of both
the optimal portfolio set and the equilibrium pricing of a
security. The adoption of only a single common factor model
would have allowed the use of only the diagonal variance matrix
in the determination of the equilibrium pricing of a security. This
condition eliminated most of the valuation effect that portfolio
entry had on a security. Since one of the major concerns of this
study was in the portfolio effect on security valuation, the multi-

factor assumption was adOpted.

Utilipy_Functions and Utility Measurement

 

The arithmetic mean and variance were the only two moments

of a distribution of returns assumed to affect utility by this

48

study. A concave utility function in (e,v) space was assumed

by the conditions given in the last chapter.

It is recognized that the valuation of a security might
be divergent and in error if the two moment utility function were
adOpted and a cubic or higher degree function were used by share-
holders.11 Empirical evidence indicated that actual return distri-
butions were skewed and leptokurtic with greater probability mass
in their tails than the distribution of a comparable normal variate.12
Yet, the addition of skewness and kurtosis as items did not add
appreciably in the selection of securities for optimal portfolios.13
This result gave some support to the view that the first two
moments, arithmetic mean and variance, might be sufficient in
supplying the necessary information about the return distributions
of securities. The use of the assumptions of binomial utility
functions and/or normal distributions has been well established
in the literature and was adopted by this study.

An Initial Market Equilibrium Exists With Respect
to Every Security's Price

 

 

The assumption of equilibrium pricing was necessary for
developing a model that could be used to estimate value in a
situation where the utility functions and wealth position of
every investor was not known. The requirement for information
on the wealth and utility functions of every investor was
unobtainable in a real situation; a model requiring this
information could not be practically applied. The adoption of the

market equilibrium pricing assumption together with the other

49

assumptions to be presented shortly',implied that all investors
were maximizing their utility. Unless the parameters affecting
utility functions were changed equilibrium pricing would be main-
tained and security prices would not be changed. A new or

modified security could be introduced, and by holding the current
equilibrium condition constant, set M in the last chapter, the
value that this new security would have in this given equilibrium
state could be found. Thus, the valuation was accomplished
indirectly by looking at the equilibrium condition and not directly
by looking at the utility functions of every investor. This
approach enabled the avoidance of requiring the utility information
on every investor.

There Were a Lapge Number of Portfolio
Holders and Securities

 

 

The assumption of having a large number of securities and
portfolio holders implied both that neither the addition, deletion,
or change of one security and its expected future flows of bene-
fits, nor any given portfolio holder could alter the equilibrium
condition. If the condition that any given security could not
affect the equilibrium state were adopted then secondary and
later level valuation effects coming from this new security's
effect on the equilibrium state need not be considered; by
definition, there was no effect. Accepting this condition, it
would be valid to assume that the current equilibrium condition

could be held constant when we either added a new security or

 

changed the probabilities attached to the outcomes of an old

50

security. Both the nominal effect assumption and market equilibrium
assumption were needed to accomplish the indirect valaution
approach that did not need utility information.

Probability Distributions Pertained to
Real Dollar Returns

 

 

The probability distributions of cash returns on securities
pertained to real dollar cash returns rather than dollar money
returns. A money illusion in the measurement of risk and return
could exist if money returns were used instead of real returns.14

A generally accepted price index would be used.

Shareholders Have a One Period Time Horizon

 

A one period time horizon existed and no pre-time horizon
liquidation was allowed. This simplifying assumption was made
to avoid the added complexities that would have come from having
to consider Sequential portfolio strategies. This was viewed as
a different dimension in portfolio analysis that could have been
considered independent from the different one period portfolio
selection models. Different sequential portfolio strategies were

not investigated by this study.

There Were No Transaction Costs or Taxes

 

Transaction costs and taxes were assumed not to exist.
These simplifying assumptions made the model formulation and
manipulation easier without a loss in the validity of the

theoretical model. The theoretical soundness of the model could

51

be questioned since transaction costs and taxes do affect valuation.
Thus, dropping the assumptions of no transaction costs and taxes
will be discussed in the chapter that deals with the possible

usefulness of the model.

Shareholders Made Rational Decisions

It was assumed that shareholders attempted to maximize
their utility when forming portfolios. This involved both the
selection of securities for portfolio entrance and the percentage
of the portfolio in each security selected. With the other
assumptions already adopted, this assumption presumed an accurate
selection of the portfolio from the optimal decision set of 2

values, set M.

Chapter Four
ADDITIONAL CONSIDERATIONS AND

CONCLUSIONS

The derivation of the input information required for the
model, the model's practical limitations, and the relaxation of
some of the model's limiting assumptions are covered in this chapter.

The last section reviews the possible model contributions.
Securing Input Information for the Application of the Model

This section briefly reviews the type of information that
could be used in forming the expected return distributions for both
the new security and the currently existing market equilibrium priced
securities, j = 1,2,...,n. The computational constraint limiting

the usefulness of the model must first be disclosed.

Computational limitations. Both the computational require-

 

ments for the formation of the optimal decision set and the information
gathering task necessitated that the model could be applied only to a
subset of the universe of securities. To the extent that the sample
selected was not representative of the universe, error in valuation
could result., This problem could be in part eliminated by determining
the new security's valuation in different samples. Though this would
not limit the information gathering task it would meet the compu-

tational constraint of the optimal decision set section of the model.

52

53

With enough samples a distribution of possible new security values
could be estimated. Standard statistical analysis could be applied

to this distribution.

Alternative approaches satisfying the computational constraint.

 

Other models have used the diagonal variance matrix rather than the
full variance-covariance matrix, S from Appendix A, and could more
rapidly determine an optimal decision set. This could enable the
inclusion of a much larger sample of securities when using a model.
This study recognized this approach as an alternative means of
generating the optimal decision set in the valuation model. Empiri-
cal evidence indicated that the securities in the optimal decision
set were not materially different between the two approaches. How-
ever, this portfolio generation technique failed to consider the
covariance, or interaction effects, between securities. Theoreti-
cally these items could have had an important effect on the valuation
of a security. A useful empirical study could determine if material
valuation differences occur by applying the two different optimal
decision set models. This test was beyond the scope of the current

study.

The full variance-covariance approach was selected on
normative grounds; the covariance terms should materially affect

a security's valuation.

The sample of securities included in the model. Many of the

 

approaches to sampling including completely random and various

stratified methods could be used. Certain situations, or required

54

conditions, might limit the degree of randomness allowed. An

example will indicate this situation. Assume that there were either
a very large issue of a new security or an offering of a security
having a large outstanding market value. To obtain full subscription
to the new offering the requirement that institutional investors'
portfolios contain the security would be great. Thus, the sample
selected should probably contain several of the securities held
heavily by institutions. With the model one could determine with
which of these securities the new share would combine, and more
specifically, one could determine which institutions would want to
hold the new security. Alternatively, one might desire to select

the sample from securities that have widely disseminated information.
This could attempt to account for the limited information system
of most investors by concentrating on issues where information is

readily available to a large segment of the investment population.

The discussion of the selection of a sample has been
purposely brief. The previous paragraph was given to indicate
that care must be exercised in selection of the sample. Samples
containing securities not often held in multi-security portfolios

could materially distort the valuation of the new security.

The input information available. There would be many
alternative means of specifying the different cash returns obtainable
from each of the securities. The selection would cover several

different dimensions.

55

First, either historical or expected information could
be used to determine the data for the cash return parameter set,
C of Chapter Two. In most of the empirical testing of portfolio
mix models the expected return and variance of each security in
the sample was derived from historical time series data. A problem
exists in having time series data (which was usually collected
for several periods to generate the expected return and variance)
represent the future expected return and variance for a one period
time horizon. The historical record is not a good information
surrogate for future performance. Secondly, the historical variance
and covariance calculations could be materially different for the
securities depending on the length of the time intervals between

the historical time series sample points.

If future dividends and price appreciation, or a reliable
surrogate for these items, could be estimated the shortcomings
of the historical information could be avoided. The requirements
of projecting information on each security in the sample could impose

a time or resource constraint.

Moreover, the one period time horizon needs to be established.
If the time horizon used in the model is materially different from
that used by many investors, the resultant new security valuation
could be faulty. With the model in this study one could select
different time horizons and determine the differential effects on the
new security's valuation. The input information required to fulfil

this test would be substantial. This study did not investigate

56

this area further; it would be the model user's function to both

specify and collect information on the time horizon chosen.

Third, the information surrogates must be specified. In
the case of the portfolio models reviewed, historical time series
dividends and price appreciation were used as a surrogate item for
a future period's expected distribution of dividends and price
appreciation. A small number of surrogates of the future flow of
benefits relationship and surrogates that were available, measurable,

and employed by investors would be desired.

The input information recommended my this study. The

 

selection of the information used to determine the different cash
return outcomes of the securities in the sample would be based on
the desire to obtain a theoretically sound situation while fulfilling

the application requirement.

The use of expected rather than historical information would
be recommended. The aim would be to generate an array of possible
price gains and losses or dollar return outcomes for each security.
This array would be generated under the assumption of a multi-
factor return model like the one reviewed in Chapter Three. One
factor would represent the market's reaction to security prices
in different conditions of the economy, the systematic factOr.
Factors affecting the cash returns of subsets of the sample would
also be included. As an example, separate factors on the building,
auto, and banking sector might be needed. Different possible events

that affect these particular sectors and the effect on the securities

57

within each of these sectors should be examined. Finally, at the
extreme end of looking at subsets, one would be interested in
particular items that might materially affect the return of just
one security. A pending FTC ruling, strike settlement, or expro-
priation of foreign properties are examples of items that could
materially affect an individual sample member. Special interest
should be directed toward the factors that affect the security

being valued.

Estimates of each security's outcomes, joint outcomes with
each others security, and the probabilities of occurrence of these
outcomes would need to be formed. The model could be recalculated
using different probability estimates to find the sensitivity of
the new share's price to differences in the probability estimates.
This would only increase calculation requirements arithmetically.
Further, this would not substantially increase the information
gathering task. Since the probability estimates would tend to be
subjectively determined and subject to a high degree of error,

the test of valuation sensitivity could prove useful.

The future earnings distribution's mean and variance were
viewed as being acceptable information surrogates for the future
dividend and price appreciation distribution's mean and variance.
These surrogates would be subject to partial management bias. A
satisfactory defense on normative grounds could be made for using
a company's future cash flow distribution's mean and variance.

More weight would be placed on more reliable surrogates, like

58

dividends or dividend growth, to the extent that the particular
company's future earning's mean and variance prove to be

unreliable surrogates.

The long-run performance of a firm and the wealth maxi-
mization of its investors is a function of the earnings distri—
bution. The ability to sell the security at any time horizon and
achieve one's expected return is a function of the variance of
an earnings distribution. In the study the earnings distribution's
mean and variance at a specific time horizon were viewed as
acceptable surrogates for the dividends and price appreciation
gained by the terminal date. The use of current or historical

earnings information would not be recommended.

The above appraoch was adopted since it required an
explicit consideration of many different possible outcomes, and
segregated uncertainties into specific situations. A variant

to this approach has been used by investment research analysts.1

Eliminating Some of the Model's Assumptions

This section examines the effect of removing some of the
simplifying assumptions used in the development of the model.
The assumption that equivalent expectations were held by all can
be eliminated. The optimal decision zone in (e,v) space that
existed when the assumption was dr0pped was already covered in
Chapter Three. A method of evaluation offered earlier in this

Chapter could be applied when this assumption is not accepted.

59

The model could be recalculated using different probability
estimates, or expected dollar returns, to establish a value range
for the new share. This approach would approximate the condition
of diverse expectations by determining value with different

probability distributions.

Transaction costs and taxes can easily be included in
the model. Since a one period time horizon with no intra-period
trading is used, the purchase transaction costs can be added
to the investment value and the sales transaction costs and taxes
subtracted from the proceeds. The choice of the tax rate would
be left to the model user with the proviso that an incorrect

rate would lead to valuation error.

CONCLUSIONS

This study developed an application oriented security
valuation model that included other securities' effects on valuation.
The aim was to develop a model that maintained theoretical sound-
ness subject to the condition that one must be able to apply the
model to a real situation. This required both new assumptions
and an approach to valuation different from previous models. Both
theoretical and empirical areas that need further investigation have

been suggested.

The major limitation of the study came from the particular
portfolio mix model that was a constituent part of the basic

valuation model. This constraint limited the number of securities

60

considered by the model and thus required that samples of the
universe of securities be used in actual model use. The
constraint was computational in nature. The possibility of using
another portfolio mix model that would alleviate this problem

was discussed earlier.

61

FOOTNOTES
Chapter 1

1Harry M. Markowitz, Portfolio Selection, Monograph
16 of the Cowles Foundation for Research in Economics at Yale
University, John Wiley 6 Sons, Inc., (New York, 1959), p. 102.

2See: John Lintner, "The Valuation of Risk Assets and
the Selection of Risky Investments in Stock Portfolios and
Capital Budgets," The Review of Economics and Statistics, Vol.
XIVII, No. 1. (February, 1965), pp. 13-37; see also John Lintner,
"Security Prices, Risk, and Maximal Gains from Diversification,"
The Journal of Finance, Vol. XX, No. 4, (December, 1965), pp.
587-615; Stewart C. Myers, "A Time State-Preference Model of
Security Valuation," Journal of Financial and Quantitative Analysis,
Vol. III, No. 1, (March, 1968), pp. 1-33; James C. T. Mao and
John F. Hellewell, "Investment Decision Under Uncertainty: Theory
and Practice," The Journal of Finance, Vol. XXIV, No. 2, (May,
1969), pp. 323—38; and Nevins D. Baxter and John G. Cragg, "Corporate
Choice Among Long-term Financing Instruments," The Review of
Economics and Statistics, Vol. LII, No. 3, (August, 1970), pp.
225-35.

 

 

 

3For a discussion of this point see: Charles W. Haley,
"The Valuation of Risk Assets and the Selection of Risky Invest-
ments in Stock Portfolios and Capital Budgets: A Comment", Ih§_
Review of Economics and Statistics, Vol. LI, No. 2, (May, 1969),
pp. 220-1; and John Lintner, "The Valuation of Risk Assets and
the Selection of Risky Investments in Stock Portfolios and Capital
Budgets: A Reply," The Review of Economics and Statistics, Vol.
LI, No. 2, (May, 1969), pp. 222-4.

62

FOOTNOTES
Chapter 2

1There are bj times bi possible joint outcomes.

2For a mathematical proof see: William J. Baumol,
Portfolio Theory: The Selection of Asset Contributions, the
McCaleb-Seiler Publishing Company, (New York, 1970), p. 23.

3Baumol, p. 24.
4Baumol, p. 24.

5William F. Sharpe, Portfolio Tmeory and Capital Markets,
McGraw-Hill Book Company, (New York, 1970), p. 52.

6Markowitz, p. 154-55.
7Sharpe, Portfolio Theory, pp. 52-58.
81f the assumption is made that a certainty return portfolio

either exists or can be constructed, the minimum v portfolio would
have a v = 0 and would be the minimum variance portfolio.

 

9Gordon Pye, "Portfolio Selection and Security Prices",
The Review of Economics and Statistics, Vol. XLIV, No. 1, February,
1967, pp. 111-15.

10The Markowitz method could be used in deriving the
optimal set. See: Markowitz, pp. 170-86, 309-312.

11An alternative assumption not requiring a certainty
return security, that permits an estimate of an equilibrium psP
where paPk is given is the section on page A17 of Appendix A.

12Other assumptions which permit an analytical sOlution
will be discussed in Chapter Three.

13If we have a previously existing security there are n
securities and if j* is a new previously unmarketed security we
have (n + l) securities.

63

FOOTNOTES
Chapter 3

1This set of assumptions were employed by Lintner: John
Lintner, "The Valuation of Risk Assets and the Selection of Risky
Investments in Stock Portfolios and Capital Budgets," The Review
of Economics and Statistics, Vol. XIVII, No. 1 (February, 1965),
pp. 13-37. I

 

 

2Lintner, p. 25.
3For a mathematical presentation see: Lintner, p. 13-37.

4Regulation Q's rate restrictions on bank deposits are
often divergent from other market required rates on debt securities
of less or equivalent risk. This condition serves as an example.
In a perfect market such conditions would not exist. From an
empirical point of view one might even question the ability to
either determine or secure the riskless rate.

SC and C in the specific case of Figure 3.3 might be
explained By a systematic divergence between two shareholders
expectations of risk related returns where shareholder one's
expectation of returns in higher risk situation were higher than
shareholder two.

6James C. T. Mao and John F. Hellewell, "Investment
Decision Under Uncertainty: Theory and Practice", The Journal
of Finance, Vol. XXIV, No. 2, (May, 1969), pp. 325-327.

 

7For twelve securities the chances for either occurrence
would be one in 2048, which is 212.

8For a discussion of systematic and residual risk and
equilibrium portfolio and security price conditions see: William
F. Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium
Conditions of Risk", The Journal of Finance, Vol. XIX, No. 3.
(September, 1964), pp. 425-42. For criticism and expansion on the
theme see: John Lintner,"Security Prices, Risk, and Maximal Gains
from Diversification," Ime_gqurnal of Finance, Vol. XX, No. 4
(December ,1965), pp. 587-615. For a reconciliation of the views
see: Eugene F. Fama, "Risk Return and Equilibrium: Some Clarifying
Comments," The Journal of Finance, Vol. XXIII, No. 1, (March, 1968),
pp. 29-40.

 

64

9See Lintner, Journal of Finance, pp. 587-615; Sharpe,
pp. 425-42; Fama, pp. 290-40; and James C. T. Mao and John F.
Hellewell, "Investment Decision Under Uncertainty; Theory and
Practice," Journal of Finance, Vol. XXIV, No. 2, (May, 1969),
pp. 323-38.

10Note: c.i can take on negative values when the firms
returns are invergely related to the factor.

 

 

11The basic approach used by this study could be modified
and would lead to proper valuation if a higher order utility
function were used. In this situation the optimal portfolio
set, set M of Chapter Two, would have to be in n dimensional space
as Opposed to (e,v) space with the binomial utility function. L
would also be an n dimensional set. REL would be the ordered n
tuple (2 ,2 ,...,2n) where 2- would be the value of the ith
moment 0 tEe distribution of a specific portfolio at a specific
price. The conditions (2-33) through (2—38) would lead to the
definition of the rule, L, where to each element keK there would
be a unique peP called p . Using the n dimensional sets instead
of the two dimensional (E,V) sets we would arrive at the same
basic decision structure (2-40) and the maximization problem
(2-42). Though solution is theoretically possible the method is
currently computationally infeasible for n greater than two. This
occurs due to the inability to determine the optimal decision
set, M, of the n-tuple values. Quadratic programming can be used
for a two dimensional (e,v) value set M. However, a technique
is not currently available for the general, n-tuple case. Thus,
the higher polynomial utility function approach is not accepted
fOr use by this study. The model could not maintain the desired
condition of being able to be practically applied. Explaining
higher moments in terms of the first two moments is one approach
that would be able to give some consideration to the incorporation
of the assumption of higher degree polynomial utility functions
in this study.

128. James Press, "A Compound Events Model for Security
Prices," The Journal of Business, Vol. XL, No. 3, (July, 1967),
pp 0 317-35 0

 

13For a discussion see: William E. Young and Robert H.
Trent, "Geometric Mean Approximations of Individual Security and
Portfolio Performance," Journal of Financial mud Quantitative
Analysis, Vol. iii, No. 3, (June, 1969), p. 179.

14For a discussion and explanation see: James C. T. Mac
6 John F. Hellewell, "Investment Decision Under Uncertainty:
Theory and Practice", The Journal of Finance, Vol. XXIV,
No. 2, (May, 1969), pp. 325-327.

 

65

FOOTNOTES
Chapter 4

1Buff, et al., "The Application of New Decision Analysis
Techniques to Investment Research," Financial Analysts Journal,
November-December, 1968.

 

66

FOOTNOTES
Appendix A

1Equations (2-1) and (2-4) through (2-6) can be used to
generate this information.

2Rather than laboriously reiterate the work previously
presented by another, one is referred to the solution technique
that would be employed in this study. See Harry M. Markowitz,
Portfolio Selection, Monograph 16 of the Cowles Foundation for
Research in Economics at Yale University, John Wiley 8 Sons, Inc.,
(New York, 1959), pp. 170-86, 309-312.

 

3For the more complete explanation see: Markowitz, pp.
170-86.

4This study does not intend to specify the nature of
this unknown risk return trade-off function. If for example
it were a hyperbola segment, category two securities would be
non existant. If the risk return trade-off function were a
perabola segment, category two securities could exist.

67

FOOTNOTES

Appendix B

1With the knowledge of each investors utility functions,

referring to Chapter Two's section on price discrimination and
Figure 2.6, one would be able to specify the portfolio keK held
by investor i and the peP the investor would be willing to pay
for j* in his portfolio.

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Review of Economics and Statistics, Vol. XLVII, No. l,

iFeb bruary, 1965), pp. 13- 37.

 

Lintner, John, "The Valuation of Risk Assets and the Selection of
Risky Investments In Stock Portfolios and Capital Budgets:
A Reply." The Review of Economics and Statistics, Vol. LI,
No. 2, (May, 1969), pp. 222-4.

 

7O

Litzenberger, Robert H., "Equilibrium In the Equity Market Under
Uncertainty." The Journalpf Finance, Vol. XXIV, No. 4,
(September, 1969), pp. 663-71.

Mao, James C. T., "Survey of Capital Budgeting: Theory and Practic."
The_gournal of Finance, Vol. XXV, No. 2, (May, 1970), pp.
349-360.

Mao, James C. T., and Hellewell, John F., Investment Decision Under
Uncertainty: Theory and Practice." The Journal of Finance,
Vol. XXIV, No. 2, (May, 1969), pp. 323-38.

Markowitz, Harry M., Portfolio Selection: Efficient Diversification
of Investments, Monograph 16 of the Cowles Foundation for
Research in Economics at Yale University, John Wiley 8 Sons,
Inc., (New York, 1959).

 

Mossin, Jan, "Security Pricing and Investment Criteria in Comparative
Markets." American Economic Review, Vol. LIX, No. 5,
(December, 1969), pp. 749-56.

Myers, Stewart C., "A Time-State-Preference Model of Security
Valuation." Journal of Financial and Quantitatiye_
Analysis, Vol. III, No. 1, (March, 1968), pp. 1-33.

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The Journal of Business, Vol. 40, No. 3, (July, 1967),
pp. 317-35.

Pye, Gordon, "Portfolio Selection and Security Prices." The Review
of Economics and Statistics, Vol. XLIX, No. 1, (February,

1967). pp. 111-15.

Pyle, David H. and Turnovsky, Stephen J., "Safety-First and Expected
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No. 4, (December, 1969), pp. 513-38.

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The Journal of Finance, Vol. XXI, No. 2, (May, 1966),
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Samuelson, Paul A., "General Proof that Diversification Pays."
Journal of Financial andqguantitative Analysis, (March,

1967), pp. 1-131

71

Sharpe, William F., "Capital Asset Prices: A Theory of Market
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Journal of Finance, Vol. XXII,No. 3, (September, 1967),
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Appendix A
THE COMPUTATIONAL PROCEDURE

FOR MODEL SOLUTION

A numerical solution technique very similar to the one
suggested in Chapter Two is now presented in detail. An
operational means of determing security value when multi-
security portfolio membership exists is provided. The numerical
approach used in forming an operational model is represented by
a programming model, Figure A.1. A great deal of the detail was
omitted in the diagram so that the basic approach could be presented
concisely. This allowed a more effective means of examining the
logic of the program. The omissions were mainly in the sub-

routines, one though four, that will be covered in detail.

A workable, but not necessarily highly efficient, program
was developed. Thus, procedures and processes that were used in
parts of the computational algorithm of this Appendix were not
the most efficient available; they were chosen to limit the
complexity and thereby hopefully improve one's ability to under-

stand the algorithm.

The computational procedure is now covered section by
section as it appears in Figure A.1. In the program flow diagram

double capital letters were used to represent capital letters and

A-1

Figure A.l

A Flow Diagram of the Algorithm

CATEGORY=1
P(N+l,G)=
NOMINAL

8

 

 

‘ CATEGORY=2
P(N+l,G)=
NOMINAL

 

 

 

fCATEGORY=3
P(N+l,G)=

PLOW

 

 

CATEGORY=4
P(N+l,G)=
P H+

10

Yes

12

14

INPUT
READ

(:;:) Start

 

 

 

 

 

 

(H

 

 

___M_LL,.
Calculates the op t 1—
mal. frontier G the
e—v values and con-

struction of the

 

 

portfolios.

 

 

1‘ ““ >

 

 

 

Calculates the e 6 v
of the new security

 

.... . .L-.L_..~

 

Calculates tangency
line B B and price

on thii ine, fig.

A.2.

 

 

 

 

 

Calculates the price
of the new security

as an Optimal single
security portfolio

 

 

 

 

 

 

 

 

 

W = W + 1 I I ’15
- v18 AA(G)=ALPH

 

 

ng+11+AArc1 16

 

 

 

 

 

 

 

 

 

N0 w = M?
17
P(N+l, G) -P(
N+l ,G)+INC
21
INCRMNT=INC
No No
Ye RMNT/Z
CATE * —9I
=3§0RY P(N+l,G)=P(N
' 20 +1,C)+INCRMNT
22
Yes
N° G = 2? ,-

v

 

 

 

PRINT OUT
RESULTS

fl“ *

 

 

 

No 25 I

 

 

 

Price range
is P(N+l,G-l)
to PHIGH type

Price range
is PLOW to
PHIGH type

 
    
 

27

 

 

 

 

A-4

single letters to represent small letters. Subscripts were repre-
sented with array notation; thus pnb from text notation would be
P(N,B) in the flow diagram or on program statement cards. Where
program statements were given FORTRAN IV was used. Standard

notation was used in the flow chart symbols.

The application of the program approach presented in this
section would use a sample of the universe of securities that
theoretically go into equilibrium market pricing of securities.
In the following material a sample of n securities will be used.
The underlying model, of Chapter Two, had the assumption of full
information and the inclusion of all marketed securities even
though its use could only be implemented in a much less perfect

situation.

The Input Read

 

The input data, block 1 of Figure A.l, is read in and
stored. This is the input information required by the program.
For every security this includes an identifier, the expected
dollar return in each possible outcome, and the current market
price. The price is not needed for j*, the security to be valued.
The probability of occurrence of each of the outcomes for each
security and the probability of occurrence of each possible joint
outcome between each pair of securities is also required. Four
other variables controlable by the program user, NOMINAL, LIMIT,
INCRMNT, and SSLIMIT are also read. An explanation of their

purpose will be covered in later sections.

A-S

Adjustment of Input Data

 

The expected return, variance, and covariance between the
other securities in the sample is needed for every security except
the new security, j*, whose price is to be determined. The vector,
e, is formed and is the n element column vector whose jth element
is the expected return of security j. The n x n matrix, S, is

formed and is the variance-covariance matrix for the n securities.1

With the e vector and S matrix one has all the necessary
information for the calculation of any attainable portfolio's
expected return and variance. The modification of the input
information and the calculation of the vector and matrix are
accomplished in a subsection of SUBROUTINE 1, block 4 of Figure

A.1.

Portfolio Return and Variance

 

e is used to represent the eXpected return of a specific
w

portfolio, w, where

w = 1,2,...,2.

For a portfolio ew is the weighted sum of the portfolio’s indi-

vidual securities' expected returns, that is,
n
A-l) e = E a. e.

where aj represents the percentage of the portfolio in security

j. The following constraints exist:

A-6

n

A-Z) Z a. = 1
i=1 J

and

A—3) aj 3 O.

In vector form the expected return for a given portfolio is
easily calculated. First, let kw represent the n by one vector

of aj values, that is

A-4) k = [a1 a

A-S) e'k = e .

For a portfolio, the expected variance of returns, vw,
will be used to represent risk. vw is a function of each
security‘s variance, ojz (or Ojj)’ and the covariance between

every security in the portfolio, Oji where i_# j. The variance

of the portfolio is

n
A-6) v = Z

The same constraints: (A—Z), (A-S), and (A—4) apply. In matrix
form vw is solved by premultiplying S by kw' and postmultiplying

by kw’ that is

A-7) k'Sk=v.
W W W

A-7

With the above information any portfolio's return and
variance can be calculated. We now need a method that will enable
a specification of the optimal decision set portfolios without

covering all possible portfolio combinations containing n securities.

The Optimal Portfolios

 

The "critical line method" used by Markowitz would be

2 The

employed to determine the optimal decision set in T space.
technique does not determine the entire attainable set. With the
quadratic programming technique used by Markowitz the single
security portfolio offering the highest return, e, is determined.
Further, the awj weight construction of several of the optimal
portfolios, called turning point portfolios, is determined. The

final optimal portfolio is the one that minimizes variance, v.3

The organization of input data and the calculation of
the Optimal decision set and the portfolios stated above would

be accomplished by block 4, SUBROUTINE l, of Figure A.1.

Program Routing. Blocks 2,3 and S are used to set values

 

at zero and direct control of the program. If this is the first
interation of SUBROUTINE l the Optimal decision set for the
equilibrium priced securities, j = 1,2,...,n, has been determined
and the new security, n + 1 also called j*, whose value is to be
determined has not entered the set of marketed securities. Thus,
the starting price of the new security for multi-security portfolio

entry is first determined. Blocks 2,3, and S are used to direct

control of the program to the determination Of the starting
price, right after the first iteration of SUB l (SUB will be used
as an abbreviation for SUBROUTINE). After initial setting of the
price there is no longer a need to go back to the starting price
section of the program, blocks 6 through 14. After the second
iteration of SUB 1, control is then directed to the section of
the program that tests for the final equilibrium valuation

condition, blocks 15 through 28.

The Value of a Security as a Single Security Portfolio Member

 

In this section the value of a security as a single-
security portfolio member is determined. Some securities will have
no value held by themselves. Other new securities will have a
definable positive value as single security portfolio members.
Blocks 6 through 14 of Figure A.l determine the equilibrium price

Of the new security.

In the program an initial value, NOMINAL, that is close
to zero will be used for the new security. This value is set
by the model user in block 1, input data. For stock valuation
this number will be accepted as the minimum value of the new
security for multi—security portfolios. If this value is set
too high the security might not be in any optimal multi-security
portfolios. For instance, if the security's value as a single
security portfolio is zero, and the security would be in an
Optimal multi-security portfolio at some value between zero and

NOMINAL, the model would not determine this Optimal multi-security

A-9

portfolio containing j*. Further, the initial single security
value, NOMINAL, cannot be set at zero; this would give an undefined

e and v and prevent solution.

The New Security's Return and Variance

 

First the new security's return and variance must be
calculated and its t location in (e,v) space compared with the
optimal decision set, M, to determine if the equilibrium condition

exists, teM.

The new security's return and variance are calculated
in SUB 2, block 6, of Figure A-l. The expected return of the

single security portfolio is

 

 

b.*
A-8) e.* = 1 :3 °a ya
J p]* 3:1 j* j*
j*
and its variance is
bj*
A'g) V.* = 1 2 (Ca - e 0* pug )2 ya
3 p?* a=1 j* J J j*
J 3*

In this equation pj* = NOMINAL. Even though it is shown that the
variance and expected return of the new security are calculated

in SUB 2 in Figure A-l, this information already could have been
readily calculated in SUB l and the information on the new security
made available for this later part of the program. By allowing

the new security to enter in the calculation of vector e and matrix

S the expected return would be found in element e of e and

(n+1)

A-lO

the variance in element 0 S. The security, n+1 or j*,

(n+1)(n+1) °f
would not have entered into the optimal decision set calculations
of the later part of SUB 1, only the original equilibrium priced
n securities would have entered.

Securities With Negative Expected Returns:
Category 1 Securities

 

 

The securities fall into three distinct classes with respect
to their expected return and variance. The first category will
include securities that have a negative expected return. As single
security portfolios the securities in this category have no value.
Thus, if the e of the new security portfolio, block 7 of Figure
A.1, is less than or equal to zero the security is defined to
be in category 1 and its price as a single security portfolio
member will be left at its NOMINAL value, block 8 of Figure A.1.

If it is greater than zero it is in one of the other categories
and may have a single security portfolio value greater than
NOMINAL.

Members Not in the Current Optimal Decision Set:
Category Two and Three Securities

 

 

Securities in these two categories have a positive e
value; category two and three securities are those whose UNW = O.
BZBT; of Figure A.2 represents the locus of points tcL for one
category two single security portfolio while BEBTE' represents
the same for a category three security. When LnM = O for a
single security portfolio the locus Of pointSteL for that portfolio

do not intersect the locus of points teM of the optimal decision

A-ll

set, represented by 212m in Figure A.2.

Category three securities are defined as those whose locus
Of points teL intersects the unknown market required risk-return

trade-Off line,represented by z ’ of the Figure A.2, that

1‘1
extends from the upper region of the currently available optimal

decision set, represented by line 2 2 Category two securities

m 1'

are those whose single security portfolios do not intersect

 

 

 

-———-4
I
zltl .
Figure A.2
The Locus of Points Of A 8'
Category 2 and 3 Security t' 1
B!
V ‘ B5
t .
l
21
B4
8'4
——-—J ___ m . . e
B' 1 3

S

The single security portfolio equilibrium price for one possible
category three security locates the security at location t1

in (e,v,) space. The EEEEH line is not determinable in an
information constrained situation. Thus a range would be used
to specify the approximate single security portfolio equilibrium

value for securities in this category. The category two security

has a single security portfolio equilibrium value of zero.

An analytical approach is used in determining the value
range Of the category three securities. The tangency line,
BlBl' to the upper part of the attainable market risk return

trade-off line, can be represented by the equation

e 2: B1 + B2 v
where B1 is the e axis intercept and B2 is the slope, with respect
to v, of the line 8181' . With e1 and v1, the return and

variance of the first efficient portfolio's 2 value, and with e2

and v2, from the next Optimal portfolio's value 22 generated
in the Markowitz program, an estimate of the values B1 and B2

can be made.

The equation of the line can be determined by the "two-

point" form of the equation of a straight line.

2
e=e+ (v-v)
1 v2 1 1
which solved will yield:
_ = * 1:
A 10) e B1 + 82 v

The locus of points meM between the first two turning
point optimal portfolio’s 2 values form a curvi—linear line
segment. (A-lO) was derived using these two turning points. The
true line of tangency is tangent to the curvi-linear segment. Thus,
(A910) only gives an approximation Of the values B1 and B . The

2

estimated values 81* and 82* will now be represented by B1 and B2

in the following equations.

A-13

It is known that

b.
1 J
A-ll) ej = -—- 2 Ca Ya
pi a =1 J J
J
and
b1
1 - 2
v = -——Q X (c - c ) y
. a. .
J p) a.=1 j J a]
b. 3
_ J
where c. 15 2 c y
a.=1 aj a)
J
Substituting into equation (A-lO),
bJ
-l - -2 - 2
p. = B1 + B2 p. 2 (ca - c.) ya
J J J aj=1 j J J

0"

 

p 2 B - p E + B Zj (c - E )2 y = 0
J 1 J J 2 a.=1 aj J aJ
3

Solving this quadratic equation erbDj. we have

.- + - 2 3 - 2 1/2

cj - [cj - 4 B1B2 E (Ca. - cj) ya.]

a.-l J j
A-lZ) p. = J
3 281

The locus of points tcL Of any given security forms a

parabola which approaches a minimum v at t = (0,0) when pj
B1 1

linear. A system consisting of one quadratic and one linear

approaches infinity. The tangency line ' of Figure A.2 is

equation has (at the most) two real solutions. This accounts for

the possibility of obtaining two pj values in equation (A-12).

A-14

The two values can be both complex numbers, one complex and one

real ,Or both real.

Category two securities. Category two securities have

 

a positive (e,v) value when pj is positive. With this category

security the locus of points tcL does not intersect the tangency
. “T—

line, 818 1

in this category would be complex numbers. This condition in

of Figure A.2. Both Of the pj values of a security

equation (A-12) can be used to distinguish this category of

securities from category three securities.

At least theoretically, there is a possibility that a
security's locus of points teL could fail to meet or intersect
the tangency line, BIBTI' and still be efficient by intersecting
or meeting the unknown market required risk-return trade-Off line.
The only way one could limit the possibility of overlooking this
specific condition would be by making an estimate of the trade-off
line and by solving an equation based on either equation (A-lO) or
a curvilinear equation that represented the trade—Off function.
Since the specific condition that causes this valuation error would
be rare and the techniques used to test it would be fairly arbi-

trary, with respect to the specification of the trade-Off function,

this technique will not be used.

The category two security as a single security portfolio
member has a value of zero. The value NOMINAL would be used

for the category two securities, block 10 of Figure A.1.

A-15

Category three securities. The higher of the two real

 

pj values generated or the one real number generated from equation
(A-lZ) is the value of the new security in the single security
portfolio located at the intersection of the locus of points teL,

represented by 858' in Figure A.2, and the tangency line,

5

represented by BlB'1 . The value of the security at the other
extreme position would be determined at the intersection of BSB'3
and BSB'5 . Substituting B3 for e in (A-ll)

b.
A 13 ‘1 J

- ) pj _ 83 2 Ca. ya
a.=1 j j

J

where B3 is the maximum attainable e value in an Optimal portfolio.

This equation gives the maximum value of security j. Thus,
referring to the higher real pj of equation (A-12) as pjlow and
the pj of equation (A-lS) as pjhigh (PHIGH in Figure A.l), we

have an estimate of the range of the value of security j held as

an Optimal single security portfolio in category three.

pjhigh > pj > pj low
This is accurate except for possible error resulting from an

incorrect tangency line.

The starting price for multi-security portfolios for
category three securities is the price at the intersection
of the security's locus of points tcL and the tangency line
BIBTI- of Figure A.2. This is the lowest possible price that

a category three security could have in equilibrium if the

A-16

equilibrium were solely based on single security portfolios. More
will be covered on the range of value of category three securities
in single security portfolios and their determined equilibrium
value for multi-security portfolio entry later in the model. Blocks
23 through 27 of Figure A.l are concerned with the specification

of the range for category three securities.

The assumption that a security offering a certainty stream

 

of returns exists can be eliminated. In the program, and not the

 

model Of Chapter Two, the initial assumption was made that a
certainty return portfolio existed or could be constructed. The

effect of dropping this assumption is now examined. The unavailability
Of a certainty return security would cause the attainable 2 set to

be above the e axis in (e,v) space, Figure A.3. A security whose

set L, the locus of points represented by OB of the figure, is

below and does not intersect the optimal decision set cannot

be valued by the current model; p = ¢ for this particular

portfolio k. An estimated single security portfolio value range,

the prices, pcP, corresponding to the two space values from B

1

to B for a specific portfolio k, could be determined for a

2,
security of this type. Like the category three securities of

this Appendix, an undefined market risk-return trade-Off line,
EEIFEI‘, exists. It has a locus of points from the optimal
decision set to the e axis. The true equilibrium value of the
single security portfolio would be at the unidentifiable point t*1.

The rest of the valuation procedure is similar to that for category

three securities, including multi—security portfolio entry and

A-17

Figure A.3

The Risk-Return Trade-Off Curve When
A Certainty Return Portfolio Does Not Exist

 

 

 

the determination of the final multi-security or single security
portfolio equilibrium value. Thus, the certainty return assumption
can be eliminated and with modification, the computational algorithm

still used for valuation

SUBROUTINE 3. The calculation of the two pj values

 

in equation (A-12) and the supplementary information covered

in the past section would be accomplished in SUB 3. If both pj
values were complex numbers control would be directed to block

10 which both defines the new security as a category two security
and initializes its multi-security portfolio starting price at
NOMINAL. If at least one Of the pj values were a real positive
number the security would be either a category three or four
security. If both were positive, the higher would be the value

of p.

J low' Once it is known that a positive real pj and t value

exist, equation (A-l3) could be solved to determine the pj high of the

security. Control then would"gO'to block ll. If p. (PLOW

3 high>Pj low
in Figure A.l) the security would be in category three. If’this Condi-
tion does not hold it would be-a category four security; Control is

paSsed to block 13 if it were a category three security and the price

for multi-security portfolio entry would be initialized at pj low'

Single Security Portfolios That Are
in the Optimal Decision Set.

 

 

The previous stages of the computational algorithm have
led to the Specification of a new security in category four.
The value of the security pj, where the locus of points teL
intersects the tangency line, BEBTE Of Figure A.4, has been
determined. With this value as a starting price a numerical
approach would be used to determine the equilibrium value of

the category four security as a single-security portfolio member.

First the Optimal decision set line segment that the new security's

Figure A.4

Turning PointCe,v) LOcations and
The Locus of Points of a Portfolio

v I B

 

 

 

locus of points teL intersected would be determined- The t values
of all turning point portfolios, numbered 1 through 6 in Figure
A.4, would have been generated in the first iteration of SUB 1,
covered previously. With Ej used to represent 2] c y

a a.
.=]_ °
83 J J

and solving for pj in equation (A-ll) we have

A-l4) p. = E. e.

cj is a constant for any given security j including j*, the
security we are examining. Now by substituting the value of ew
for the Optimal turning point portfolio, w = l,2,...,m into

equation (A-l4) we can derive m distinct pwj values.

A-lS) pw. = E. e

Each of these pwj values is the equilibrium value a Ej return would
be worth if it had a variance of returns equal to vw. A check
is made to see what the ij value for security j would be if

the security had a value pwj

 

b.
A-l6) v = 1 z] (c - E )2 y
wj 2 a =1 aj j aj
pwj 3'

we can compare to see if
i) v . < v

wj w
ii) v . = v

wj w
iii) v . > v

w] w

If (i) applies, the security j has a t whose e is greater than the
e of the Optimal portfolio having the same e, and it is under-
valued as an Optimal single-security portfolio member. If (ii)
applies the single security t value is an element Of the optimal
decision set at a turning point where the wth optimal portfolio

is located. If (iii) applies the security's teL has an e less
than an e Of an optimal portfolio having the same variance; the

security is overpriced.

There is no need to start at w = l and go through all m
turning point portfolios. The pj at j's t value on the tangency
line was calculated in SUB 3. This pj value can be compared
with the pwj values generated in equation (A-lS). Since security
j's intersection with the Optimal set will be at an e equal to
or less than the e of security j's intersection with the tangency
line, the optimal turning point portfolios with an e greater than
the e of j at the tangency intersection point, Optimal turning
point one of Figure A.3, need not be considered. The locus of
points teL in T space could not intersect line segments above

these w locations. Thus the procedure is to :

a) compare to see if

- -1
. < . = C . e
p1 pWJ J w

for w = l,2,...,k. k is reached where the above condition does

not hold.

A-21

b) now having solved

then solve for vj in equation (A—16), using the pwj just derived
and check to see if (i), (ii), or (iii) apply. This is continued
for w = k, k + l, k + 2,...,h + 1. h + l is reached when

condition (iii) holds.

c) security j intersects the optimal decision set at w = h or

on the optimal line segment between turning point h and h + l.

The approximate location Of this line segment in T space
and the price of the security at this location will now be
determined though a numerical approach. The process is quite

similar to that used previously.

First, one can construct an Optimal portfolio between
any two adjacent turning point optimal portfolios. Using k to
now represent the percentage weight Of this new portfolio in
h + l and (l - k) the weight in h, any portfolio between h and

h + 1 can be constructed. In this new portfolio

A-l7 + l-k a .
1 c 1 h]

“(h+k)j . k“(huh
for j = 1,2,...,n.

The expected portfolio return, e(h+k ), can be determined for
i

this new portfolio

M3

_ ' =
A 1 ) e(h+ki) j 1 a(h+ki)jej

A-22

where i is used to count how many times we have iterated the
process which includes equation (A-l7) above through (iii')
below. On the first iteration i = l and k is arbitrarily set at

0.5. This can also be solved in matrix form

A-S') = e

e ' a .
(h+ki) (h+ki)

the variance of this portfolio is

m m

A-6') V = Z Z a . a a.
(h+ki) j=1 £21 (h+ki) J (h+ki)2 11

or in matrix form

A-7') s a

a' = v
(h+ki) (h+ki) (h+ki)

As in the previous equation (A-lS), we can solve for

- -1
C e

A‘ls') P(h+ki)j = j (“*ki)-

This . value is the e uilibrium value a 5. return would be
p(h+ki)j q J
worth if it had a variance of returns equal to v(h+k ). A check
1

is made to see what the V(h+k )j value for security j would be
i

at p(h+ki)j'

1
_ v
A 16 ) V(h+ki)j

2
P (h+ki)j 1

now we can compare to see if

1') Vch+ki)j ‘ Vch+k11

A-23

ii') v . = v
. (h+ki)J (h+ki)

...,
111 ) V(h+ki)j > v(h+ki)

a) if i', then increment i by one and now let

A-18) k. = 2

Control is returned to equation (A-l7).
b) if ii' then shift control to the next section of the model.

c) if iii' then a check is made to see if
A-19) SSLIMIT > ki'

SSLIMIT is set by the model user and is used to terminate the
iterative price change process when the single security portfolio
teL is sufficiently close to a teM. Thus, if equation (A-19) holds,
control is shifted to the next section of the algorithm. If this

condition does not hold then increment i by one and let

A-ZO) k. = k. - 2

Control is returned to equation (A-l7).

The price of the new category four security as an optimal single

security portfolio member is the last price, p(h+k )j’ calculated
i

A-24

before existing the above iterative process. This also is the

starting price for multi-security portfolio entry.

The above process for determing a category four security's
single portfolio value would be accomplished in SUB 4, block 13,
of Figure A.1. Block 14 would be used to identify the category
of the security and set the starting price for multi-security
portfolio entry.

Single Security Portfolio Equilibrium Value:
A Conclusion

 

 

The initial price for membership in multi-security optimal
decision set portfolios has been determined for any type of new
security. The program could identify both the category of this
new security and its single security portfolio equilibrium value.

Multi—Security Portfolio Entry and
Final Equilibrium Value

 

 

This section determines if the new security has a final
equilibrium value equal to or greater than its initial value
determined when held only by itself. Its final value can only
be greater if it enters Optimal multi-security portfolios. Thus,
the first test determines if the new security at its initial
equilibrium single security value is also a member of optimal

multi-security portfolios.

After both the category and initial multi-security portfolio

membership value are determined, block 8, 10, 12, or 14 of Figure

A-25

A.1, control reverts to SUB 1 after first incrementing the iteration
counter, g in block 3. For all passes, other than the initial pass
through SUB l, the Optimal decision set is determined for the n + l
securities which include the new market member, j*, or for n
securities if one of the original securities cash returns distri-
bution is changed. Information is generated on the t values of

the turning point portfolios and the awj weights for j = l,2,...,n + l

for each w = l,2,...,m.

Next, the test is made to see if the new security is a
member of Optimal multi-security portfolios. Blocks 15 through

20 perform this test. Blocks 15 through 18 first determine

m
Aer) A = X

where Ag represents the sum of the Owj* weights Of the m turning
point portfolios only. It should be noted that if awj* = 0

for w = l,2,...,m, this security will not appear in any optimal
portfolios, including all locations not at turning points. Thus,

block 19 determines if the new security has membership in any

Optimal portfolios. If

A > 0
g

optimal multi-portfolio membership exists. If this condition
holds the equilibrium value for the new security has not yet
been determined. The price of the new security is incremented,

block 21, and control is returned to the calculation of the

A-26

Optimal decision set with the higher value for the new security.
This process of iterating from block 21 to 3,4,5 and 15 through 19

is continued until

Control then shifts to block 20 to see if the Ag-l value is within
the user specified proximity to zero that must be attained before

termination of price changes. Thus, if

A > LIMIT
g-l

holds, the increment is reduced in size and added to pj*(g-l)
which was the last determined price for security j* where

Optimal multi-portfolio membership occurred. Control returns

to block 3. If the condition Ag_1 > LIMIT does not hold, the
equilibrium pricing of the security is completed, except possibly

for category three securities. Control is then shifted to the

printing Of results.

The Special Case of Category Three Securities

 

Category three securities were the only type having an
indeterminate price; multiple possible prices could exist. Thus,
blocks 23 through 25 are used to determine the specific price or
price range with a category three security in final equilibrium.
Block 23 is used to direct control for non—category three securities
to either the final printing Of results in block 28 or to block
24 for category three securities. Block 24 checks to see if the

final value of the security pj exceeds pj*high from equation

*(g-l)

A-27

(A-l3). If so, the final value for this category three security
is pj*(g-l) and control is switched to the print out of results,
block 28. The final value of this security comes from membership
in Optimal multi-security portfolios. Since this value exceeds
the range of values derived from single-security portfolio member-
ship this security would only be held in multi-security portfolios

at the specific value derived. If the condition

burnipj*mm

holds, control is directed to block 25 where the iteration counter,
g , is checked. If -both g < 2 or A8 = 0 the security is priced
as a single security portfolio member. The single security value

range was already established as

Pj* high > pj* > Pj* low'

This information would be printed out in block 27. If the iteration
counter, g, is greater than two the category three security has
membership in optimal multi-security portfolios at a pj*(g-l)

greater than pj* , and the price range for this security would

low
be

%*mm>pr’Prm41

This information would be printed out in block 26.

A-28

Printing of Results

 

Information that would be generated in the calculations
include: 1) the starting category Of the new security; 2) the
equilibrium single-security portfolio value; 3) whether Optimal
multi-security portfolio membership exists; 4) and if so, the t
and awj construction of the turning point portfolios it enters;
and finally 5) the final equilibrium price for the security. All
or part of this information could be made available to the program
user.

Further Remarks on the Computational
Procedure for Model Solution

 

 

The basic program presented omitted some of the smaller
steps not critical to the explanation of the operational means of

Obtaining security valuation.

Some of the subroutines could have used more efficient
optimal seeking techniques in the iterative sections rather than
using externally supplied increments and limits. It should be
noted that none of the four variables, NOMINAL, LIMIT, INCRMNT, of
SSLIMIT could be equal to or less than zero. Further, computational
efficiency would be lost with no gains in final solution if INCRMNT
were set very low. The aim was in providing a model that could be

used with Observed or observable data.

Appendix B
PRICING LARGE SHARE OFFERINGS

This section investigates the relationship that exists
between the supply of a new security and the market pricing of
the supply. If the constraining assumption is still held that the
offering of a new security and its supply does not affect the
valuation Of other securities, set M of Chapter Two remains

unchanged when a new security, j*, is traded.

From Chapter Two, the nominal effect assumption of a new
security led to the definitions that the j* security would be
held: 1) by the investors that achieve the greatest value for

j* in their particular portfolio,

A

2—38) pk = max pk

A

pkePk
and 2) in that portfolio where value is maximized

2-39) p* = max _ pk.
p eP
k k

This definition might be faulty if the share Offering were large.

There are a number Of investors, i = 1,2,...,n1, each
holding a specific portfolio, keK. Each investor has a given
wealth, wi, invested in his Optimal portfolio. The percentage
Of the given security, j*, held in an investor's portfolio, k,

B-l

B-2

is 6. There are x shares of security j*. From equation (2-39)

J*i'

p* was the equilibrium price. If

n
B-l) p*x > Z w. a. .,

then there are excess or unsold shares, and the supply of security
j* at p* exceeds the demand. This condition contradicts the
nominal effect assumption of the model in Chapter Two and by
definition could not occur at that time. Since the nominal effect
assumption was not made in this Appendix, condition (B-l) can

occur and j* is not equilibrium priced at p*.

The assumption of no price discrimination. If no price
discrimination is allowed, the equilibrium price would be the
marginal value of j* to the portfolio it entered at its lowest
price. To describe this condition, first the set P is formed.
It is the set of all possible Pk sets. Going from the

max pcP called p1 to the next highest p2, and so on, the check

is made to see if

 

B-2) x s X

3 represents the n2 where condition

(B-2) is satisfied. “j*i is the percentage of a portfolio that

In (B-2) n2 = l,2,...,n3, and n

would be held by the ith investor in security j* at the specific

price 62.1 When the condition is reached that (B-2) is satisfied

B-3

n

A

p will be the final equilibrium price of j*. This will be the

marginal value of j* to the portfolio it entered at its lowest price.
n
For all investors, 1, whose k portfolio has j* at a value p£> p 2 a
n
surplus value equivalent to the spread (pl - p 2) will be gained

if there is no price discrimination.

An Estimate of the Price of
Largp Share Offerings

 

 

An approximation of the equilibrium pricing of a large
share offering could be made using the computational algorithm
of Appendix A. One would need added input information or estimates
of the distribution of wealth over the meM, before estimating
prices for j* given different issue sizes. With the algorithm
of Appendix A the ch containing j* for a sample of11 of the
elementSOf M can be determined for each pj* iterated. The size
of the sample, n, would be determined by the model user. With
wk standing for the estimate of wealth in the portfolio k
corresponding to meM, an estimate Of the aggregate shares demanded,

xd, at a price pj* would be

G.

n
xd = p.* 2 wk

J k 1 j*k'
The reliability of the estimated demand, xd, would be a function
of both the sample points selected and the size, n, of the sample.

One could construct a schedule showing the relationship between

estimated pj* values and xd.

B-4

Usefulness of Security Demand Information

 

Information on the absorption of shares at different prices
would be useful when the issue is very large and when it must enter
a large number Of investor portfolios before it is fully absorbed.
A very small portfolio weight, “j*k’ for the security in an optimal
portfolio would clear the market Of a very small issue of a
security. This would not be the situation for a large new issue.
For example, AT&T and its affiliates accounted for about 23% Of
all new common stock issuance between 1946 and 1959. A small

a in a few Optimal portfolios would not have cleared the market

j*k
of ATGT's and its affiliates' shares. Errors in valuation could
result if it were assumed that a large new issue of AT&T shares
would leave the original equilibrium price Of the already out-
standing shares unchanged. The price would have to adjust on all
shares, (new and outstanding) until there were a large enough
addition in both the number of optimal portfolios and the

0j*k for the ATGT securities in those portfolios, to absorb the

new issue.

"71111111111111“