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X
Standard
Deviation
Figure 2.2 Efficient Frontier and Correlation.
any point in time) and hence only one riskreturn locus and corresponding xi v
riskreturn combination is part of the feasible set or opportunity set for every
More than Two Securities
If we allow the investor to distribute his wealth between any two securities A
and C or A and C we obtain ( p p ,u p )combinations given by curves I, 11, I11 r
in Figure 2.3. If we now allow the agent to invest in all three securities, that
the proportions x1, x2, x3 (with Cixi = 1) then the ( p p ,u p )locus is a cur
This demonstrates that holding more securities reduces portfolio risk for any
of expected return (i.e. the threesecurity portfolio at Y is preferred to portfolio
(securities B and C only) in Figure 2.3 and X also dominates Y and 2). It f
any agent adopting the meanvariance criterion will wish to move from points
Z to points at X.
The slope of IV is a measure of how the agent can tradeoff expected ret
risk by altering the proportions Xi held in the three assets. By altering the c
of his portfolio from M to Q he can obtain an increase in expected return (p
taking on an amount of additional risk ( 0 2  01).
Note that the dashed portion of the curve IV indicates a meanvariance
portfolio. An investor would never choose portfolio L rather than M becau
lower expected return for a given level of portfolio risk than does portfolio M
M is said to dominate portfolio L on the meanvariance criterion.
Limited Investment and Portfolio Variance
In general the agent may reduce op for any given p p by including additiona
his portfolio (particularly those that have negative covariances with the exis
already held). In fact portfolio variance (0;) falls very quickly as one in
Figure 2.3 Efficient Frontier.
5 Numbers of
20
10
Stocks
Figure 2.4 Portfolio Combinations.
number of stocks held from 1 to 10, and thereafter the reduction in portfol
is quite small (Figure 2.4). This, coupled with the brokerage fees and inform
of monitoring a large number of stocks, may explain why individuals tend t
only a relatively small number of stocks. Individuals may also obtain the
diversification by investing in mutual funds (unit trusts) and pension funds
institutions use funds from a large number of individuals to invest in a very
of financial assets and each individual then owns a proportion of this â€˜large p
2.23 The Efficient Frontier
Consider now the case of N assets. When we vary the proportions Xi (i = 1 ,
form portfolios it is obvious that there is potentially a large number of such
We can form 2,3, . . . to N asset portfolios. We can also form portfolios consis
same number of assets but in different proportions. The set of every possible
given by the convex â€˜eggâ€™ of Figure 2.5.
If we apply the meanvariance dominance criterion then all of the points in
of the porlcfolio opportunity set (e.g. PI, Figure 2.5) are dominated by th
P2,
curve AB since the latter has a lower variance for a given expected return. Po
curve AB also dominate those on BC, so the curve AB represents the propo
the eficient set of portfolios and is referred to as the eficient frontier.
We now turn to the problem of how the investor calculates the Xj values th
the efficient frontier. The investor faces a known set of n expected returns an
of and n(n  1)/2 covariances O i j (or correlation coefficients p i j ) and the fo
the expected return and variance of the portfolio are:

Staendafd
Deviation
Figure 2.5
or
We assume our investor wishes to choose the proportions invested in each ass
is concerned about expected return and risk. Risk is measured by the standard d
returns on the portfolio (up). efficientfiontier shows all the combinations
The
which minimises risk u p for a given level of p,.
The investorâ€™s budget constraint is C x i = 1, that is all his wealth is placed
of risky assets. (For the moment he is not allowed to borrow or lend money i
asset.) Short sales X i < 0 are permitted.
A stylised way of representing how the agent seeks to map out the efficien
as follows:
1. Choose an arbitrary â€˜targetâ€™ return on the portfolio pZ, (e.g. p p = 10 perc
2. Arbitrarily choose the proportions of wealth to invest in each asset
1,2, . . . n) such that p p is achieved (using equation (2.15)).
3. Work out the variance or standard deviation of this portfolio (a,)˜ with th
of ( X i ) l using equation (2.17).
4. Repeat (2) and (3) with a new set of ( ˜ i ) 2if ( a p ) 2 < (op)lthen discard t
in favour of ( X i ) 2 (and vice versa).
5 . Repeat (2)(4) until you obtain that set of asset proportions x (with Cx;
;
meets the target rate of return p i and yields the minimum portfolio varian
(up)*. assets held in the proportions x; is an ejfficientportfolio and
The
point in ( p p ,op) space  point A in Figure 2.6.
6 . Choose another arbitrary â€˜targetâ€™ rate of return p y (= 9 percent say) and
above to obtain the new efficient portfolio with proportions xi** and minimu
( , *  point B.
a)*
Figure 2.6
We could repeat this exercise for a wide range of values for alternative expe
rates p , and hence trace out the curve XABCD. However, only the upper po
curve, that is XABC, yields the set of efficient portfolios and this is the eficie
It is worth noting at this stage that the general solution to the above problem
usually does) involve some xf being negative as well as positive. A positive x
stocks that have been purchased and included in the portfolio (i.e. stocks h
Negative xf represent stocks held â€˜shortâ€™, that is stocks that are owned by so
(e.g. a broker) that the investor borrows and then sells in the market. He th
a negative proportion held in these stocks (i.e. he must return the shares to
at some point in the future). He uses the proceeds from these short sales to a
holding of other stocks.
Interim Summary
1. Our investor, given expected returns pi and the variances a, and covarian
p i j ) on all assets, has constructed the efficient frontier. There is only on
frontier for a given set of pi, a;,ajj, ( p i j ) .
2. He has therefore chosen the optimal proportions xi* which satisfy the budge
Cxf = 1 and minimise the risk of the portfolio a for any given level o
,
return on the portfolio p p .
3. He has repeated this procedure and calculated the minimum value of a fo,
of expected return p p and hence mapped out the (p,, a) points which co
,
efficient frontier.
4. Each point on the efficient frontier corresponds to a different set of optim
tions x : , x z , x;, . . . in which the stocks are held.
Points 14 constitute the first â€˜decisionâ€™ the investor makes in applying th
separation theorem  we now turn to the second part of the decision process
2.2.4 Borrowing and Lending: The â€™hansformation Line
Our agent can now be allowed to borrow or lend an asset which has the same
the holding period and yields a â€˜certainâ€™ and hence riskfree rate of interest,
(ii) invest less than his total wealth in the risky assets and use the remainde
the riskfree rate,
(iii) invest more than his total wealth in the risky assets by borrowing the
funds at the riskfree rate. In this case he is said to hold a leveredportf
The transformation line is a relationship between expected return and risk on
portfolio. This specific portfolio consists of (i) a riskless asset and (ii) a portfo
assets.
The transformation line holds for any portfolio consisting of these two as
turns out that the relationship between expected return and risk (measured by th
deviation of the â€˜newâ€™ portfolio) is linear. Suppose we construct a portfolio
consisting of one risky asset with expected return ER1 and variance 0 and :
asset. Then we can show that the relationship between the return on this new
and its standard deviation is
pk = a k buk
where â€˜aâ€™ and â€˜bâ€™ are constants and PI( = expected return on the new portf
standard deviation on the new portfolio. Similarly we can create another new
â€˜Nâ€™ consisting of (i) a set of q risky assets held in proportions xi (i = 1,2, .
together constitute our one risky portfolio and (ii) the riskfree asset. Again w
pN = 60 k 6 I O N
To derive the equation of the transformation line let us assume the individual ha
already chosen a particular combination of proportions (i.e. the x i ) of q r
(stocks) with actual return R, expected return p˜ and variance 0 Note that
.
:
not be optimal proportions but can take any values. Now he is considering what
of his wealth to put in this one portfolio of q assets and how much to borr
at the riskless rate. He is therefore considering a â€˜newâ€™ portfolio, namely co
of the riskfree asset and his â€˜bundleâ€™ of risky assets. If he invests a proportio
own wealth in the riskfree asset, then he invests (1  y ) in the risky â€˜bund
the actual return and expected return on his new portfolio as RN and p ˜resp ,
+
RN = yr (1  y ) R
+
PN = yr (1  y)pR
where (R, p ˜ is) the (actual, expected) return on the risky â€˜bundleâ€™ of his po
in stocks. When y = 1 all wealth is invested in the riskfree asset and p˜ = r
y = 0 all wealth is invested in stocks and p˜ = p ˜For y < 0 the agent borro
.
at the riskfree rate r to invest in the risky portfolio. For example, when y =
initial wealth = $100, the individual borrows $50 (at an interest rate r ) but in
in stocks (i.e. a levered position).
Since r is known and fixed over the holding period then the standard devia
â€˜newâ€™ portfolio depends only on the standard deviation of the risky portfolio
OR. From (2.19) and (2.20) we have
2
 y)2E(R  pR)2
= E(&  pN)2 = (1
ON
and (2.22) are both definitional but it is useful to rearrange them into a sing
in terms of mean and standard deviation ( p ˜ N ) of the â€˜newâ€™ portfolio. From
O,
(1  Y ) = O N / O R
Y = 1 (ON/OR)
Substituting for y and (1  y) from (2.23) and (2.24) in (2.20) gives the iden
where SO = r and 61 = ( p  OR. Thus for any portfolio consisting of two
˜
of which is a risky asset (portfolio) and the other is a riskfree asset, the r
between the expected return on this new portfolio p˜ and its standard error O
with slope given by 81 and intercept = r . Equation (2.25) is, of course, an ide
is no behaviour involved. ( p  r ) is always positive since otherwise no one w
˜
the set of risky assets.
When a portfolio consists only of n risky assets, then as we have seen th
opportunity set in returnstandard deviation space is curved (see Figure 2.6)
the opportunity set for a twoasset portfolio consisting of a riskfree asset and
risky portfolio is a positive straight line. This should not be unduly confusin
portfolios considered in the two cases are different and in the case of the â€˜effic
curve is derived under an optimising condition and is not just a rearrangeme
identities.
Equation (2.25) says that p˜ increases with ( O N / G R ) . This arises because f
an increase in O N / O R simply implies an increase in the proportion of wea
the risky asset (i.e. 1  y) and since ER > r this raises the expected return o
portfolio p ˜Similarly for a given ( O N / O R )= (1  y) (see equation (2.23)), a
.
in the expected excess return on the risky asset ( p  r ) increases the overa
˜
return p ˜This is simply because here, the investor holds a fixed proportion
.
the risky asset but the excess return on the latter is higher.
We can see from (2.25) that when all wealth is held in the set of risky
0 and hence ON = OR and this is designated the 100 percent equity portfolio
Figure 2.7). When all wealth is invested in the riskfree asset y = 1 and ,UN
O N / O R = 0 from (2.33)). At points between r and X, the individual holds so
initial wealth in the riskfree asset and some in the equity portfolio. At points
individual holds a levered portfolio (i.e. he borrows some funds at a rate r and
all his own wealth to invest in equities).
DERIVATION OF THE CAPM
23
.
231 The Optimal Portfolio
..
The transformation line gives us the riskreturn relationship for any portfolio
of a combination of investment in the riskfree asset and any â€˜bundleâ€™ of sto
All wealth in risky
assets
I
1
I
risk free asset I
I
I
I
I
I
+
bN
OR
Standard
Deviation
Figure 2.7 The Transformation Line.
is no behavioural or optimisation by agents behind the derivation of the tran
line: it is an identity. At each point on a given transformation line the agen
risky assets in the same fixed proportions x i . Suppose point X (Figure 2.7) r
combination of x; = 20 percent, 25 percent and 55 percent in the three risk
of firms â€˜alphaâ€™, â€˜betaâ€™ and â€˜gammaâ€™. Then points Q, L and Z also represen
proportions of the risky assets. The only â€˜quantityâ€™ that varies along the tran
line is the proportion held in the one risky bundle of assets relative to that
riskfree asset.
The investor can borrow or lend and be anywhere along the transformati
(Exactly where he ends up along rZ depends on his preferences for risk versus r
we shall see this consideration does not enter the analysis until much later.) Fo
point Q in Figure (2.8) might represent 40 percent in the riskless asset and 60
the bundle of risky securities. Hence an investor with $100 would at point Q
Expected
Return
r
Standard
Deviation
Figure 2.8 Portfolio Choice.
because at any point on rZâ€™ the investor has a greater expected return for any
of risk compared with points on rZ. In fact because rZâ€™ is tangent to the effici
it provides the investor with the best possible set of opportunities. Point M
a â€˜bundleâ€™ of stocks held in certain fixed proportions. As M is on the effici
the proportions X i held in risky assets are optimal (i.e. the x: referred to e
investor can be anywhere along rZâ€™, but M is always a fixed bundle of stock
fixed proportions of stocks) held by all investors. Hence point M is known as
portfolio and rZâ€˜ is known as the capital market line (CML). The CML is the
transformation line which is tangential to the efficient frontier.
Investorâ€™s preferences only determine at which point along the CML, rZâ€™,
vidual investor ends up. For example, an investor with little or no risk aver
end up at a point like K where he borrows money (at r) to augment his own
he then invests all of these funds in the bundle of securities represented by
still holds all his risky stocks in the fixed proportions xi*).
Separation Principle
Thus the investor makes two separate decisions
( 9 He uses his knowledge of expected returns, variances and covariances t
the efficient set of stocks represented by the efficient frontier YML (F
He then determines point M as the point of tangency of the line from
efficient frontier. All this is accomplished without any recourse to the i
preferences. All investors, regardless of preferences (but with the same
expected returns, etc.) will â€˜home inâ€™ on the portfolio proportions (xi*) o
securities represented by M. All investors hold the marketportfolio or mor
all investors hold their risky assets in the same proportions as their rel
in the market. Thus if the value of ICI shares constitutes 10 percent of
market valuation then each investor holds 10 percent of his own risky p
ICI shares.
(ii) The investor now determines how he will combine the market portfol
assets with the riskless asset. This decision does depend on his subje
return preferences. At a point to the left of M the individual investor is
risk averse and holds a percentage of his wealth in the market portfolio (i
optimal proportions xi*) and a percentage in the riskfree asset. If the
investor is less risk averse then he ends up to the right of M such as
levered portfolio (i.e. he borrows to increase his holdings of the market p
excess of his own initial wealth). At M the individual puts all his own w
the market portfolio and neither borrows nor lends at the riskfree rate.
The CML, rZâ€™, which is tangential at M, the market portfolio, must have the
by (2.25), that is:
in the market portfolio (point A, Figure 2.8). A less risk averse investor w
borrowing in order to invest more in the risky assets than allowed by his in
(point K, Figure 2.8). However, one thing all investors have in common is that
portfolio of risky assets for all investors lies on the CML and for each invest
slope of CML = ( p m  r)/Om
= slope of the indifference curve
The slope of the CML is often referred to as the â€˜market price of riskâ€™. The s
indifference curve is referred to as the marginal rate of substitution, MRS, sin
rate at which the individual will â€˜trade off more return for more risk.
All investors portfolios lie on the CML and therefore they all face the sa
price of risk. From (2.27) and Figure (2.8) it is clear that for both investors a
the market price of risk equals the MRS. The latter measures the individlclalâ€™
trade off or â€˜tasteâ€™ between risk and return. Hence in equilibrium, all individ
the same tradeoff between risk and return.
The derivation of the efficient frontier and the market portfolio have been
in terms of the standard deviation being used as a measure of risk. When risk i
in terms of the variance of the portfolio then
2
km = ( P m  r)/gm
is also frequently referred to as â€˜the market price of riskâ€™. Since a m and U: are co
very similar, this need not cause undue confusion. (See Roll (1977) for a di
the differences in the representation of the CAPM when risk is measured in
different ways.)
Market Equilibrium
In order that the efficient frontier be the same for all investors they must h
= ERi, U,? a
geneous expectations about the underlying market variables
course, this does not mean that they have the same degree of risk aversion.) H
homogeneous expectations all investors hold all the risky assets in the propor
by point M, the market portfolio. The assumption of homogeneous expectation
in producing a market equilibrium where all risky assets are willingly held in
proportions x: given by M or in other words, in producing market clearing. Fo
if the price of shares of â€˜alphaâ€™ is temporarily low and those of delta shares t
high, then all investors will wish to hold more alpha shares and less delta s
price of the former rises and the latter falls as investors with homogeneous e
buy alpha and sell delta shares.
In What Proportions are the Assets Held?
When we allow borrowing and lending we know that the individual will hold
risky assets in the optimal proportions represented by the point M. He holds
each individual to be on his own highest indifference curve.)
But a problem remains. How can we calculate the risky asset proportions xf
by point M? So far we have only shown how to calculate each set of xf for ea
the efficient frontier. We have not demonstrated how the proportions xf for th
point M are derived. This can, in fact, be quite a technically complicated p
illustrate, note from Figure 2.8 that for any transformation line:
[+I
tan@= ER, 
where â€˜ p â€™ represents any risky portfolio, and as we have seen ER, and opdepen
well as the known values of Pj and Ojj for the risky assets). Hence to achiev
equation (2.29) can be maximised with respect to xi, subject to the budget
Cxi = 1 and this yields the optimum proportions xf. Some of the x; may b
zero, indicating short selling of assets. If short sales are not allowed then the
constraint, xf 3 0 for all i, is required and to find the optimal xf requires a sol
quadratic programming techniques.
23.2 Determining Equilibrium Returns
Let us now use the ideas developed above to derive the CAPM equation re
the equilibrium return on each security. We do so using a mixture of graphic
(Figure 2.9) and simple algebra (see Appendix 2.1 for a more formal deriva
slope of the CML is constant and represents the market price of risk which i
for all investors.
Pm r
slope of CML = 
am
We now undertake a thought experiment whereby we â€˜moveâ€™ from M (which c
assets in fixed proportions) and create an artificial portfolio by investing some o
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