In the real world, however, it is possible that over short periods the marke

equilibrium and profitable opportunities arise. This is more likely to be the ca

agents have divergent expectations, or if they take time to learn about a new en

which affects the returns on stocks of a particular company, or if there are so

who base their investment decisions on what they perceive are ˜trends™ in t

Future chapters will show that, if there is a large enough group of agents w

trends™ then the rational agents (smart money) will have to take account of the m

in stock prices produced by these ˜trend followers™ or noise traders (as they are o

in the academic literature).

It would be useful if we could assess actual investment performance ag

overall index of performance. Such an index would have to measure the act

rank alternative portfolios accordingly.

One area where a performance index would be useful is in ranking the perf

specific mutual funds. A mutual fund allows the managers of that fund to inve

portfolio of securities. For the moment, put to one side the difference in transa

of the individual versus the mutual fund manager (who buys and sells in larg

and may reap economies of scale). What one might wish to know is whether

fund manager provides a ˜better return™ than is provided by a random selectio

by the individual. It may also be the case that whatever performance index i

may itself be useful in predicting the fiture performance of a particular mutu

Any chief executive or group manager of a mutual fund will wish to kno

his subordinates are investing wisely. Again, a performance index ought to be

us whether some investment fund managers are either doing better than othe

or better than a policy of mere random selection of stocks. The indices we w

are known as the Sharpe, Treynor and Jensen performance indices.

3.3.1 Sharpe™s Performance Index (S)

The index suggested by Sharpe is a reward to variability ratio and is define

folio i as:

where ERj = expected return on portfolio i , a = variance of portfolio i and r

j

rate.

Sharpe™s index measures the slope of the transformation line and can be ca

any portfolio using historic data. Over a run of (say) quarterly periods we ca

the average ex-post return and the standard deviation on the portfolio, held by

fund manager. We can also do the same for a portfolio of stocks which has bee

selected. The mutual fund manager will be outperforming the random selecti

provided his value of Si is greater than that given by the randomly selecte

The reason for this is that the mutual fund portfolio will then have a transfor

with a higher slope than that given by the randomly selected portfolio. It follo

investor could end up with a higher level of expected utility if he mixed the

the mutual fund manager with the riskless asset.

Underlying the use of Sharpe™s performance index are the following assum

(i) Investors hold only one risky portfolio (either the mutual fund or a random

portfolio) together with the risk-free asset.

(ii) Investors are risk averse and rates of return are normally distributed

assumption is required if we are to use the mean-variance framework).

3.3.2 Weynor™s Performance Index (T)

When discussing Sharpe™s performance index it was assumed that individua

hold only the riskless asset and a single portfolio of risky assets. Treynor™s p

Pi

It is therefore a measure of excess return per unit of risk but this time the risk i

by the beta of the portfolio. The Treynor index comes directly from the CA

may be written as

Under the CAPM the value of Ti should be the same for all portfolios of secu

the market is in equilibrium. It follows that if the mutual fund manager in

portfolio where the value of Ti exceeds the excess return on the market portfo

be earning an abnormal return relative to that given by the CAPM.

We can calculate the sample value of the excess return on the market portfol

the right-hand side of equation (3.29). We can also estimate the Pi for any give

using a time series regression of (Ri - r)f on (RM- r)t. Given the fund manage

excess return (ERi - r ) we can compute all the elements of (3.29). The fun

outperforms the market if his particular portfolio (denoted i) has a value of

exceeds the expected excess return on the market portfolio. Values of Ti can

rank individual investment manager™s portfolios. There are difficulties in inter

Treynor index if Pi < 0, but this is uncommon in practice.

3.3.3 Jensen™s Performance Index (J)

Jensen™s performance index also assumes that investors can hold either the m

denoted i or a well-diversified portfolio such as the market portfolio. Jensen

given by the intercept J i in the following regression:

To run the regression we need time series data on the expected excess ret

market portfolio and the expected excess return on the portfolio i adopted by

fund manager to obtain estimates for Ji and P i . In practice, one replaces th

return variables by the actual return (by invoking the rational expectations a

(see Chapter 5)). It is immediately apparent from equation (3.30) that if Ji =

have the standard CAPM. Hence the mutual fund denoted i earns a return in

that given by the CAPM if J i is greater than zero. For J i less than zero,

fund manager has underperformed relative to the risk adjusted rate of return gi

CAPM. Hence Jensen™s index actually measures the abnormal return on the p

the mutual fund manager (i.e. a return in excess of that given by the CAPM).

Comparing Treynor ™s and Jensen”s Pegormance

We can rearrange equation (3.30) as follows:

It may be shown, however, that when ranking two mutual funds, say X and

indices they can give different inferences. That is to say a higher value of Ti

over fund X may be consistent with a value of Jj for fund Y which is les

for fund X. Hence the relative performance of two mutual funds depends on

chosen.

In section 6.1 the above indices will be used to rank alternative portfolio

a particular ˜passive™ and a particular ˜active™ investment strategy. This al

ascertain whether the active strategy outperforms a buy-and-hold strategy.

Roll™s Critique and Performance Measures

Roll™s critique, which concerns the estimation of the CAPM using a samp

indicates that in any dataset the following relationship will always hold

= sample mean of the return on portfolio i and

where Ri

Em = sample mean of the return on the market portfolio.

There is an exact linear relationship in any sample of data between the mea

portfolio i and that portfolio™s beta, if the market portfolio is correctly measu

if the CAPM were a correct description of investor behaviour then Treynor™s in

always be equal to the sample excess return on the market portfolio and Jens

would always be zero. It follows that if the measured Treynor or Jensen indice

than suggested by Roll then that simply means that we have incorrectly me

market portfolio.

Faced with Roll™s critique we can therefore only recommend the use of th

mance indices on the basis of a fairly ad-hoc argument. Because of transac

and information costs, investors may not fully diversify their portfolio and h

risky assets in the market portfolio. On the other hand, most investors do n

their holdings to a single security: they hold a diversified, but not the fully

portfolio given by the market portfolio of the CAPM.Thus the appropriate

risk for them is neither the (correctly measured) Bj of the portfolio nor the ow

0; on their own individual portfolio.

It may well be the case that something like the S&P index is a good app

to the portfolio held by the representative investor. Then the Treynor, Sharpe

indices may well provide a useful summary statistic of the relative perform

mutual fund against the S&P index (which one must admit may not be mea

efficient).

3.3.4 Performance of Mutual Funds

There have been a large number of studies of mutual fund performance using

three indices. Most studies have found that fund managers are unable system

beat the market and hence they do not outperform an unmanaged (yet diversified

such as the S&P index. Shawky (1982) examines the performance of 255 mu

different from zero in only 25 out of the 255 mutual funds studied and, o

had negative values of J i and only nine had positive values. Thus, out of mo

mutual funds only nine outperformed an unmanaged diversified portfolio such

index. Using Sharpe™s index, between 15 and 20 percent of mutual funds ou

an unmanaged portfolio, while for Treynor™s index the figure was slightly

around 33 percent outperforming the unmanaged portfolio. Hence, althoug

some mutual funds which outperform an unmanaged portfolio these are not

large in number.

Given the above evidence, there seems to be something of a paradox in

funds are highly popular, and their growth in Western developed nations thro

1970-1990 period has been substantial. While it is true that mutual funds

do not outperform the unmanaged S&P index, nevertheless, they do outperf

any unmanaged portfolio consisting of only a small number of shares. Rela

transaction costs for small investors often imply that investing in the S&P ind

viable alternative and hence they purchase mutual funds. The key results to e

this section are:

Sharpe™s reward-to-variability index is an appropriate performance measur

0

investor holds mutual fund shares plus a riskless asset.

Treynor™s and Jensen™s performance indices are appropriate when the

0

assumed to diversify his portfolio and holds both mutual fund shares, to

many other risky assets and the riskless asset.

3.4 THE ARBITRAGE PRICING THEORY (APT

An alternative to the CAPM in determining the expected rate of return on indivi

and on portfolios of stocks is the arbitrage pricing theory (APT). Broadly sp

APT implies that the return on a security can be broken down into an expe

and an unexpected or surprise component. For any individual stock this surpr

component can be further broken down into ˜general news™ that affects all

specific ˜news™ which affects only this particular stock. For example, news w

all stocks might be an unexpected announcement of an increase in interest r

government. News that affects the stocks of a specific industrial sector, for exam

be the invention of a new radar system which might be thought to influence th

industry but not other industries like chemicals and service industries. The AP

that ˜general news™ will affect the rate of return on all stocks but by differen

For example, a 1 percent unexpected rise in interest rates might affect the retur

of a company that was highly geared, more than that for a company that was

The APT, in one sense, is more general than the CAPM in that it allows a la

of factors to affect the rate of return on a particular security. In the CAPM the

only one factor that influences expected return, namely the covariance between

on the security and the return on the market portfolio.

stock and uit = the unexpected, surprise or news element.

We can further subdivide the surprise or news element Ujr into systematic

risk mt,that is risk that affects a large number of stocks each to a greater or le

and unsystematic (idiosyncratic or specific) risk & i f , which specifically affec

firm or a small group of firms:

+

Uit = mt Ejr

As in the case of the CAPM, the systematic risk cannot be diversified aw

this element of news or new information affects all companies. However, as

unsystematic or specific risk may be diversified away.

In order to make the APT operational we need some idea of what causes

risk. News about economy-wide variables are, for example, a government ann

that GDP is higher than expected or a sudden increase in interest rates by the Ce

These economy-wide factors F (indexed by j ) , may have different effects o

securities and this is reflected in the different values for the coefficients bij

given below:

+ bi2(F2r - EF2,) +

m, = C b i j ( F j - E F j ) , = b i l ( F 1 , - EF1,) ***

j

where the expectations operator E applies to information at time t - 1 or e

example, if for a particular fr the beta attached to the surprise in interest ra

im

to 0.5 then for every 1 percent that the interest rate rises above its expected

would increase the return on security i by half a percent (above its expected v

that the ˜betas™ here are not the same as the CAPM betas and hence are denot

than /?.(Later it will be shown that the betas of the APT may be reconciled wit

of the CAPM under certain restrictive assumptions.)

A crucial assumption of the APT is that the idiosyncratic or specific risk E

related across different securities:

COV(Ei, E j ) = 0

In fact, as we shall see below, specific risk can be diversified away by hold

number of securities.

Return on the Portfolio

For simplicity, suppose there is only one systematic risk factor, F,, and thre

in the portfolio. The return on a portfolio RP of three securities held in propo

I

by definition

3 3

i=l i=l

3 /3 \ 3

to be close to zero. In fact, as the number of securities increases the last te

right-hand side of (3.37) will approach zero and the specific risk will have be

fied away. Hence the return on the portfolio is made up of the expected retu

individual securities and the systematic risk as represented by the single eco

news term F , .

A More Formal Approach

The beauty of the APT is that it does not require any assumptions about uti

or that the mean and variance of a portfolio are the only two elements in the

objective function. The model is really a mechanism (an algorithm almost) that

to derive an expression for the expected return on a security (or a portfolio of

based on the idea that riskless arbitrage opportunities will be instantaneously

Not surprisingly, the APT does, however, require some (arbitrary) assumpt

assume that agents have homogeneous expectations and that the return Rit on

is linearly related to a set of k factors F j t :

where the bij are known as factor weights. Taking expectations of (3.38) and

EEit = 0, and subtracting it from itself

k

+ +

- EF j t )

= ERit bij ( Fj t

Rit &it

i

Equation (3.39) shows that although each security is affected by all the factors,

of any particular Fk depends on the value of bik and this is different for eac

This is the source of the covariance between returns Rit on different securities

Assume that we can continue adding factors to (3.39) until the unexplained

return E; is such that

= 0 for all i # j and all time periods

E(EiEj)

- E F j ) ] = 0 for all stocks and factors (and all t )

E[Ei(Fj

Respectively, (3.40) and (3.41) state that the unsysfematic (or specific) risk is un

across securities and is independent of the factors F, that is of systematic risk

the factors F are common across all securities. Now we perform an ˜experim

investors form a zero-beta portfolio with zero net investment. The zero-beta port

satisfy

n

C x i b i j = 0 for all j = 1 , 2 , .. . k.

i= 1

and the assumption of zero investment implies

n

CX;=O

i=l

can be earned by arbitrage. This arbitrage condition places a restriction on th

return of the portfolio, so using (3.34) we have:

k

n

Using (3.42) and the assumption that for a large well-diversified portfolio th

on the RHS approaches zero gives:

where the second equality holds by definition. Since this artificially constructe

has an actual rate of return RP equal to the expected return ER:, there is a zero

in its return and it is therefore riskless. Arbitrage arguments then suggest that t

return must be zero: n

ER; = C x i E R i , = 0

i=l

We now have to invoke a proof based on linear algebra. Given the conditi

(3.42), (3.43) and (3.46) which are known as orthogonality conditions then

shown that the expected return on any security i may be written as a linear c

of the factor weightings bij. For example, for a two-factor model:

+ hlbjl + h2bj2

ER, = ho

It was noted above that bjl and bi2 in (3.39) are specific to security i . The expe

on security i weights these security-specific betas by a weight hj that is the s

securities. Hence hj may be interpreted as the extra expected return required

a securities sensitivity to the jth

attribute (e.g. GNP or interest rates) of the s

Interpretation of the A j

Assume for the moment that the values of bil and bi2 are known. We can inte

as follows. Consider a zero-beta portfolio (i.e. bil and bi2 = 0 ) which has a

return ERZ.If riskless borrowing and lending exist then ERZ = r , the risk-free r

bjl = bi2 = 0 in (3.47) gives

A0 = ERZ (or r )

Next consider a portfolio having bil = 1 and bi2 = 0 with an expected return E

tuting in (3.47) we obtain

hi = ER1 - A0 = E(R1 - R Z )

+ bilE(R1 - R Z )+ bi2E(R2 - RI)

ERj = E R

Thus one interpretation of the APT is that the expected return on a security i

the sensitivity of the actual return to the factor loadings (i.e. the bij). In add

factor loading (e.g. bil) is ˜weighted™ by the expected excess return E(R1 -

the (excess) return on a portfolio whose beta with respect to the first factor

with respect to all other factors is zero. This portfolio with a ˜beta of 1™ theref

the unexpected movements in the factor F1.

3.4.1 Implementation of the APT

The APT may be summed up in two equations:

k

+

ERi, = ho bijhj

j= 1

where ho = r, or ER“. The APT may be implemented in the following (sty

A ˜first-pass™ time series regression of Ri, on a set of factors F j f (e.g. infla

growth, interest rates) will yield estimates of ai and the bil, bi2, etc. This can b

for i = 1 , 2 , . . .rn securities so that we have rn values for each of the betas, on

of the different securities. In the ˜second-pass™ regression the bi vary over the r

and are therefore the RHS variables in (3.53). Hence in equation (3.53) the

variables which are different across the rn securities. The hj are the same for al

and hence these can be estimated from the cross-section regression (3.53) of

bij (for i = 1,2, . . . rn). The risk-free rate is constant across securities and h

constant term in the cross-section regression.

The above estimation is a two-step procedure. There exists a superior pro

principle at least) whereby both equations (3.52) and (3.53) are estimated simu

This is known as factor analysis. Factor analysis chooses a subset of all the fac

that the covariance between each equation™s residuals is (close to) zero (i.e. E (

which is consistent with the theoretical assumption that the portfolio is fully

One stops adding factors Fj when the next factor adds ˜little™ additional e

Thus we simultaneously estimate the appropriate number of F j S and their cor

bijs. The hj are then estimated from the cross-section regression (3.53).

There are, however, problems in interpreting the results from factor anal

the signs on the bij and h j s are arbitrary and could be reversed (e.g. a positi

negative hj is statistically undistinguishable from a negative bi j and positive hj

there is a scaling problem in that the results still hold if the p i j are doubled

halved. Finally, if the regressions are repeated on different samples of data t

guarantee that the same factors will appear in the same order of importance.

arguments plus a few other minimal restrictive assumptions) nevertheless it is

implement and make operational. As well as the problems of interpretation of

h j which we cannot 'sign' apriori (i.e. either could be positive or negative)

also have problems in that the bij or hj may not be constant over time. In gen

applied work has concentrated on regressions of equation (3.52) in an effort

few factors that explain actual returns.

3.4.2 The CAPM and the APT

It must by now be clear to the reader that these two models of equilibrium

returns are based on rather different (behavioural) assumptions. The APT is oft

to as a multi-factor (or multi-index) model. The standard CAPM in this termin

be shown to be a very special case of the APT, namely a single-factor ver

APT, where the single factor is the expected return on the market portfolio E

return generating equation for security i is hypothesised to depend on only one

this factor is taken to be the return on the market portfolio, then the APT giv

+ bjR7 +

= aj

Rjt &it

This single index APT equation (3.54) can be shown to imply that the expecte

given by:

+

ERi, = rt b j ( E q - r t )

which conforms with the equilibrium return equation for the CAPM.

The standard CAPM may also be shown to be consistent with a multi-inde

see this consider the two-factor APT:

and

+ bjlh1 + bj2h2

ERj = r

Now hj (as we have seen) is the excess return on a portfolio with a bij of un

factor and zero on all other factors. Since h j is an excess return, then ifthe CA

= pl(ER"' - r )

)cl

= p2(ERm - r )

A2

where the pi are the CAPM betas. (Hence pi = cov(Ri, Rm)/var(Rm)- note