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empirical work on whether the stock market is efficient centres on trying t
whether one-period returns H r + l are predictable. Later empirical work conce
whether the stock price equalled the DPV of future dividends and the most recen
work brings together these two strands in the empirical literature.
With slight modifications the one-period holding period return can be defin
asset. For a coupon paying bond with initial maturity of n periods and coupo
of C we have



and is referred to as the (one-period) holding period yield (HPY). The first
capital gain on the bond and the second is the coupon (or running) yield. Broadl
stocks
The difficulty with direct application of the DPV concept to stocks is that futur
namely the dividends, are uncertain. Also because the future dividend payment
tain these assets are risky and one therefore might not wish to discount all futur
some constant risk-free interest rate. It can be shown (see Chapter 4) that if t
one-period holding period return E,Hf+I equals qt then the fundamental value
can be viewed as the DPV of expected future dividends E,D,+j deflated by the
discount factors (which are likely to embody a risk premium). The fundamen




+
+
In (1.30) qi is the one-period return between time period t i - 1 and t i(
If there are no systematic profitable opportunities to be made from buying
shares between well-informed rational traders, then the actual market price o
P, must equal fundamental value V , , that is, the DPV of expected future div
example, if P, < V f then investors should purchase the undervalued stock and
a capital gain as P, rises towards V,.In an efficient market such profitable o
should be immediately eliminated.
Clearly one cannot directly calculate V , to see if it does equal P,because ex
dends (and discount rates) are unobservable. However, in Chapters 6 and 16
methods of overcoming this problem and examine whether the stock market is
the sense that P, = V,. Also if we add some simplifying assumptions to the D
(e.g. future dividends are expected to be constant) then it can be used in a rela
manner to calculate V , and assess whether shares are under- or over-valued
to their current market price. Such models are usually referred to as dividen
models (see Elton and Gruber (1987)) and are dealt with in Chapter 4.


1.2 UTILITY AND INDIFFERENCE CURVES
In this section we briefly discuss the concept of utility but only to a level su
reader can follow the subsequent material on portfolio choice.
Economists frequently set up portfolio models where the individual choo
assets in order to maximise either some monetary amount such as profits or
returns on the portfolio or the utility (satisfaction) that such assets yield. For
certain level of wealth will imply a certain level of satisfaction for the indiv
contemplates the goods and services he could purchase with the wealth. If h
doubled his level of satisfaction may not be. Also, for example, if the individua
one bottle of wine per night the additional satisfaction from consuming an
may not be as great as from the first. This is the assumption of diminishin
utility. Utility theory can also be applied to decisions involving uncertain o
fact we can classify investors as ˜risk averters™, ˜risk lovers™ or ˜risk neutral™
the shape of their utility function. Finally, we can also examine how indivi
Suppose W represents the possible outcomes of a football game, namely, win, lo
Suppose an individual attaches probabilities p ( W ) to these outcomes, that
N ( W ) / T where N ( W ) equals the number of wins, losses or draws in the
T = total number of games played. Finally, suppose the individual attaches
levels of satisfaction or utility U to win (= 4 units), lose (= 0 units) and draw
so that U(win) = 4, etc. Then his expected utility from the season™s forthcomin


W


Uncertainty and Risk
The first restriction placed on utility functions is that more is always preferred
that U™( W) > 0 where U ™ ( W )= aU(W)/aW. Now, consider a simple gamble o
$2 for a ˜head™ on the toss of a coin and $0 for tails. Given a fair coin th
monetary value of the risky outcome is $1:

+ (1/2)0 = $1
(1/2)2

Suppose it costs the investor $1 to ˜invest™ in the game. The outcome from n
the game (i.e. not investing) is the $1 which is kept. Risk aversion means t
will reject a fair gamble; $1 for certain is preferred to an equal chance of $2
aversion implies that the second derivative of the utility function is negative U
To see this, note that the utility from not investing U(1) must exceed the expe
from investing
™ +
U(1) (1/2)U(2) (1/2)U(O)

or
U(1) - U ( 0 ) > U(2) - U(1)

so that the utility function has the concave shape given in Figure 1.2 marked ˜ri
It is easy to deduce that for a risk lover the utility function is convex while
neutral investor who is just indifferent to the gamble or the certain outcome,
function is linear (i.e. the equality sign applies to equation (1.33)). Hence we
U ” ( W ) < 0 risk averse
U ” ( W )= 0 risk neutral
U ” ( W ) > 0 risk lover

A risk averse investor is also said to have diminishing marginal utility of w
additional unit of wealth adds less to utility the higher the initial level of w
U ” ( W ) < 0). The degree of risk aversion is given by the concavity of the utili
in Figure 1.2 and equivalently by the absolute size of U”(W).Two measures of
Figure 1.2 Utility Functions.

of risk aversion are commonly used:




R A ( W ) is the Arrow-Pratt measure of absolute risk aversion, the larger is
greater the degree of risk aversion. R R ( W )is the coefficient of relative risk a
and RR are a measure of how the investor™s risk preferences change with a
wealth. For example, assume an investor with $10000 happens to hold $50
assets. If his wealth were to increase by $10000 and he then put more than $5
into risky assets, he is said to exhibit decreasing absolute risk aversion. (The
of increasing and constant absolute risk aversion are obvioils.)
The natural assumption to make as to whether relative risk aversion is
increasing or constant i s less clear cut. Suppose you have 50 percent of y
(of $lOOQOO) in risky assets. If, when your wealth doubles, you increase the
held in risky assets then you are said to exhibit decreasing relative risk aversi
definitions app!y for constant and increasing relative risk aversion.) Different m
functions give rise to different implications for the form of risk aversion. F
the function
U ( W ) = In W
exhibits diminishing absslute risk aversion and constant relative risk aversio
Certain utility functions allow one to reduce the problem of maximising exp
to a problem involving only the maximisation of a function of expected return
risk of the return (measured by the variance) ch.For example, maximising t
absolute risk aversion utiiity fmction
E[U(W)] = E [ a - bexp(-cW)]
risk aversion. Apart from the unobservable 'c' the maximand (1.38) is in t
mean and variance of the return on the portfolio: hence the term mean-varianc
However, the reader should note that in general maximising E U ( W ) cannot be
a maximisation problem in terms of He and afr only and often portfolio mod
at the outset that investors are concerned with the mean-variance maximan
discard any direct link with a specific utility function(3).

Indifference Curves
Although it is only the case under somewhat restrictive circumstances, let us
the utility function in Figure 1.2 for the risk averter can be represented solely
the expected return and the variance of the return on the portfolio. The link b
of period wealth W and investment in a portfolio of assets yielding an expe
+
l is W = (1 n ) W o where W Oequals initial wealth. However, we assume
l
function can be represented as

U1 > 0, U2 < 0, u11, U22 < 0
U = U(n',a;)

The sign of the first-order partial derivatives ( U l , U2) imply that expected
to utility while more 'risk' reduces utility. The second-order partial derivativ
diminishing marginal utility to additional expected 'returns' and increasing m
utility with respect to additional risk. The indifference curves for the above util
are shown in Figure 1.3.




""r


-
0,
2

Figure 1.3 Indifference Curves.
that at higher levels of risk, say at C, the individual requires a higher expe
(C” to C”™ > A” to A”˜) for each additional increment to risk he undertakes,
at A: the individual is ˜risk averse™.
The indifference curves in risk-return space will be used when analysin
choice in the one-period CAPM in the next chapter and in a simple mean-vari
in Chapter 3.

Intertemporal Utility
A number of economic models of individual behaviour assume that inves
utility solely from consumption goods. At any point in time, utility depends po
consumption and exhibits diminishing marginal utility



The utility function therefore has the same slope as the ˜risk averter™ in Figur
C replacing W). The only other issue is how we deal with consumption which
different points in time. The most general form of such an intertemporal life
function is
U N = u(Cr,C t + l , Cr+2 - . - Cr+N)
However, to make the mathematics tractable some restrictions are usually pla
form of U ,the most common being additive separability with a constant sub
of discount 0 < 6 < 1:



It is usually the case that the functional form of U ( C , ) ,U(Cr+I),etc. are take
same and a specific form often used is




where d < 1. The lifetime utility function can be truncated at a finite valu
if N -+ 09 then the model is said to be an overlapping generations mod
individual™s consumption stream is bequeathed to future generations.
The discount rate used in (1.41) depends on the ˜tastes™ of the individu
+
present and future consumption. If we define S = 1/(1 d) then d is kn
subjective rate of time preference. It is the rate at which the individual will sw
++
+
time t j for utility at time t j 1 and still keep lifetime utility constant. T
separability in (1.41) implies that the extra utility from say an extra consumptio
10 years™ time is independent of the extra utility obtained from an identical c
bundle in any other year (suitably discounted).
L c
C
O

Figure 1.4 Intertemporal Consumption: Indifference Curves.

For the two-period case we can draw the indifference curves that foll
simple utility function of the form U = Czl C y ( 0 < CYI,a < 1) and these
2
in Figure 1.4. Point A is on a higher indifference curve than point B since at
vidual has the same level of consumption in period 1, C1 as at B, but at A, h
of consumption in period zero, CO.At point H if you reduce COby xo units t
individual to maintain a constant level of lifetime utility he must be compens
extra units of consumption in period 1, so he is then indifferent between poin
Diminishing marginal utility arises because at F if you take away xo units of
requires yl(> yo) extra units of C1 to compensate him. This is because at F h
with a lower initial level of CO than at H, so each unit of CO he gives up i
more valuable and requires more compensation in terms of extra C1.
The intertemporal indifference curves in Figure 1.4 will be used in
investment decisions under certainty in the next section and again when
the consumption CAPM model of portfolio choice and equilibrium asset ret
uncertainty.


1.3 PHYSICAL INVESTMENT DECISIONS AND OPTI
CONSUMPTION
Under conditions of certainty about future receipts the investment decisio
section 1.1 indicate that managers should rank physical investment projects
either to their net present value (NPV), or internal rate of return (IRR).
projects should be undertaken until the NPV of the last project undertaken e
or equivalently until IRR = r, the risk-free rate of interest. Under these cir
the marginal (last) investment project undertaken earns just enough net retur
to cover the loan interest and repayment of principal. For the economy a
undertaking real investment requires a sacrifice in terms of lost current co
output. Higher real investment implies that labour skills, man-hours and mach
t = 0, devoted to producing new machines or increased labour skills, which
output and consumption but only in future periods. The consumption profile
prefer, at-the-margin, consumption today rather than tomorrow. How can financ
through facilitating borrowing and lending ensure that entrepreneurs produce
level of physical investment (i.e. which yields high levels of future consump
and also allows individuals to spread their consumption over time accordi
preferences? Do the entrepreneurs have to know the preferences of individual
in order to choose the optimum level of physical investment? How can the
acting as shareholders ensure that the managers of firms undertake the 'correc
investment decisions and can we assume that financial markets (e.g. stock mark
funds are channelled to the most efficient investment projects? These quest
interaction between 'finance' and real investment decisions lie at the heart of
system. The full answer to these questions involves complex issues. Howev
gain some useful insights if we consider a simple two period model of the
decision where all outcomes are certain (i.e. riskless) in real terms (i.e. we a
price inflation). We shall see that under these assumptions a separation princip
If managers ignore the preferences of individuals and simply invest in pr
the NPV = 0 or IRR = r , that is, maximise the value of the firm, then this
given a capital market, allow each consumer to choose his desired consumpt
namely, that which maximises his individual welfare. There is therefore a
process or separation of decisions, yet this still allows consumers to max
welfare by distributing their consumption over time according to their pref
step one, entrepreneurs decide the optimal level of physical investment, disre
preferences of consumers. In step two, consumers borrow or lend in the cap
to rearrange the time profile of their consumption to suit their individual pref
explaining this separation principle we first deal with the production decisio
the consumers' decision before combining these two into the complete model
All output is either consumed or used for physical investment. The entrep
an initial endowment W O at time t = 0. He ranks projects in order of decre
using the risk-free interest rate r as the discount factor. By foregoing consump
obtains resources for his first investment project 10 = W O- Cf'.The physical
in that project which has the highest NPV (or IRR) yields consumption output
Cil) (where C','' > C t ) ,see Figure 1.5). The IRR of this project (in terms of co
goods) is:
1 + IRR") = ˜ 1 1 1) 0 1 )
˜(
(

As he devotes more of his initial endowment W Oto other investment projects
NPVs then the internal rate of return (C1/Co) falls, which gives rise to the
opportunity curve with the shape given in Figure 1.5 (compare the slope at A
The first and most productive investment project has a NPV of



and
c (1) WO Consumption
in Period Zero

Figure 1.5 Production Possibility Curve. Note (A-A" = B-B").

Let us now turn to the financing problem. In the capital market, any two co
streams COand C1 have a present value (PV) given by:



hence
+ r)Co
V ( l + r ) - (1
c =P
1


For a given value of PV, this gives a straight line in Figure 1.6 with a slop
+ r). Equation (1.45) is referred to as the money market line since it rep
-(1
rate of return on lending and borrowing money in the financial market place.
+
an amount COtoday you will receive C1 = (1 r)Co tomorrow.
Our entrepreneur, with an initial endowment of WO, will continue to invest
assets until the IRR on the nth project just equals the risk-free market interest

IRR'") = r

A
Consumption
in Period One




*
Consumption
in Period Zero

Figure 1.6 Money Market Line.
current consumption of C and consumption at t = 1 of Cl; (Figure 1.6). At a
G
the right of X the slope of the investment opportunity curve (= IRR) exceeds
interest rate (= r ) and at points to the left of X, the opposite applies.
However, the optimal levels of consumption (Cg, C ; ) from the production de
not conform to those preferred by individual consumers. We now leave the
decision and turn exclusively to the consumer™s decision.
Suppose the consumer has income accruing in both periods and this inco
has present value of PV. The consumption possibilities which fully exhaust t
(after two periods) are represented by:

+- C1
PV = CO
(1 +
We now assume that lifetime utility (satisfaction) of the consumer depends on


and there is diminishing marginal utility in both COand C1 (i.e. aU/aCi > 0, a
0, for i = 0, 1). The indifference curves are shown in Figure 1.7. To give up
C the consumer must be compensated with additional units of C1, if he is to m
O
initial level of utility. The consumer wishes to choose CO and C1 to maximi
utility subject to his budget constraint (1.48). Given his endowment PV, h
consumption in the two periods is (C;t*,C;*). general, the optimal production
In
investment plan which yields consumption (C8,C;) will not equal the consume
consumption profile (C;*,C;*).However, the existence of a capital market e
the consumer™s optimal point can be attained. To see this consider Figure 1.8
The entrepreneur has produced a consumption profile (CC;,C;)which max
value of the firm. We can envisage this consumption profile as being paid


t
Consumption
in period One



c;




C; Consumption
in Period Zero

Figure 1.7 Consumers™ Maximisation Problem.
e
C^
O Consumption
in Period Zero

Figure 1.8 Investing 10 and Lending ˜L™ in the Capital Market.

owners of the firm in the form of (dividend) income. The present value of
flow™ is PV* where
+
PV* = c;; CT/(l+ r )
This is, of course, the ˜income™ given to our individual consumer as owner o
But, under conditions of certainty, the consumer can ˜swap™ this amount PV
combination of consumption that satisfies



Given PV* and his indifference curves (i.e. tastes or preferences) in Figure
then borrow or lend in the capital market at the riskless rate r to achieve that co
C:*,Cy* which maximises his own utility function.
Thus, there is a separation of investment and financing (borrowing and len
sions. Optimal borrowing and lending takes place independently of the physical
decision. If the entrepreneur and consumer are the same person(s), the separ
ciple still applies. The investor (as we now call him) first decides how much
initial endowment W Oto invest in physical assets and this decision is indepen
own (subjective) preferences and tastes. This first stage decision is an objecti
tion based on comparing the internal rate of return of his investment project
risk-free interest rate. His second stage decision involves how much to borrow
the capital market to ˜smooth out™ his desired consumption pattern over time.
decision is based on his preferences or tastes, at the margin, for consumption to
(more) consumption tomorrow.
Much of the rest of this book is concerned with how financing decisions

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