gains or withdraw sums from the account. Suppose the proportion of our total

investment in stock i at time t is wi (t) so that the amount invested in stock i is

Pn

wi (t)I(t). Note that since wi (t) are proportions, i=1 wi (t) = 1. What is the

return on this investment over the time interval from t to t + 1? At the end of

this period of time, the value of our investment is

n

X

I(t) wi (t)Si (t + 1).

i=1

If we now subtract the value invested at the beginning of the period and divide

by the value at the beginning, we obtain

P P n

I(t) n wi (t)Si (t + 1) ’ I(t) n wi (t)Si (t) X

i=1 i=1

Pn = wi (t)Ri (t + 1)

I(t) i=1 wi (t)Si (t) i=1

which is just a weighted average of the individual stock returns. Note that it

does not depend on the initial price of the stocks or the total amount that we

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.37

invested at time t. The advantage in using returns instead of stock prices to

assess investments is that the return of a portfolio over a period is a value-

weighted average of the returns of the individual investments.

When time is measured continuously, we might consider de¬ning returns by

using the de¬nition above for a period of length h and then reducing h. In other

words we could de¬ne the instantaneous returns process as

Si (t + h) ’ Si (t)

lim .

Si (t)

h’0

In most cases, the returns over shorter and shorter periods are smaller and

smaller, and approach the limit zero so some renormalization is required above.

It seems more sensible to consider returns per unit time and then take a limit

i.e.

Si (t + h) ’ Si (t)

Ri (t) = lim .

hSi (t)

h’0

Notice that by the de¬nition of the derivative of a logarithm and assuming that

this derivative is well-de¬ned,

d ln(Si (t)) 1d

= Si (t)

dt Si (t) dt

Si (t + h) ’ Si (t)

= lim

hSi (t)

h’0

= Ri (t)

In continuous time, if the stock price process Si (t) is di¬erentiable, the natural

de¬nition of the returns process is the derivative of the logarithm of the stock

price. This de¬nition needs some adjustment later because the most common

continuous time models for asset prices does not result in a di¬erentiable process

Si (t). The solution we will use then will be to adopt a new concept of an integral

and recast the above in terms of this integral.

38 CHAPTER 2. SOME BASIC THEORY OF FINANCE

The Capital Asset Pricing Model (CAPM)

We now consider a simpli¬ed model for building a portfolio based on quite basic

properties of the potential investments. Let us begin by assuming a single period

so that we are planning at time t = 0 investments over a period ending at time

t = 1. We also assume that investors are interested in only two characteristics of

a potential investment, the expected value and the variance of the return over

this period. We have seen that the return of a portfolio is the value-weighted

average of the returns of the individual investments so let us denote the return

on stock i by

Si (1) ’ Si (0)

Ri = ,

Si (0)

and de¬ne µi = E(Ri ) and wi the proportion of my total investment in stock i

at the beginning of the period. For brevity of notation, let R, w and µ denote

the column vectors

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

R1 w1 µ1

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ R2 ⎟ ⎜ ⎟ ⎜ ⎟

w2 µ2

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ .⎟ ⎜ ⎟ ⎜ ⎟

. .

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

R =⎜ ⎟,w =⎜ ⎟ ,µ =⎜ ⎟.

⎜ .⎟ ⎜ ⎟ ⎜ ⎟

. .

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ .⎟ ⎜ ⎟ ⎜ ⎟

. .

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Rn wn µn

P

wi Ri or in matrix notation w0 R. Let us

Then the return on the portfolio is i

suppose that the covariance matrix of returns is the n — n matrix Σ so that

cov(Ri , Rj ) = Σij .

We will frequently use the following properties of expected value and covariance.

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.39

Lemma 3 Suppose ⎛ ⎞

R1

⎜ ⎟

⎜ ⎟

⎜ R2 ⎟

⎜ ⎟

⎜ ⎟

⎜ .⎟

⎜ ⎟

R =⎜ ⎟

⎜ .⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ .⎟

⎝ ⎠

Rn

is a column vector of random variables Ri with E(Ri ) = µi , i = 1, ..., n and

suppose R has covariance matrix Σ. Suppose A is a non-random vector or matrix

with exactly n columns so that AR is a vector of random variables. Then AR

has mean Aµ and covariance matrix AΣA0 .

Then it is easy to see that the expected return from the portfolio with weights

P P

wi is i wi E(Ri ) = i wi µi = w0 µ and the variance is

var(w0 R) = w0 Σw.

We will need to assume that the covariance matrix Σ is non-singular, that

is it has a matrix inverse Σ’1 . This means, at least for the present, that our

model covers only risky stocks for which the variance of returns is positive. If

a risk-free investment is available (for example a secure bond whose return is

known exactly in advance), this will be handled later.

In the Capital Asset Pricing model it is assumed at the outset that investors

concentrate on two measures of return from a portfolio, the expected value and

standard deviation. These expected values and variances are computed under

the real-world probability distribution P not under some risk-neutral Q measure.

Clearly investors prefer high expected return, wherever possible, associated with

small standard deviation of return. As a ¬rst step in this direction suppose we

plot the standard deviation and expected return for the n stocks, i.e. the n

p √

points {(σi , µi ), i = 1, 2, ..., n} where µi = E(Ri ) and σi = var(Ri ) = Σii .

These n points do not consist of the set of all achievable values of mean and

40 CHAPTER 2. SOME BASIC THEORY OF FINANCE

standard of return, since we are able to construct a portfolio with a certain

proportion of our wealth wi invested in stock i.In fact the set of possible points

consists of

X

√

{( w0 Σw, w0 µ) as the vector w ranges over all possible weights such that wi = 1}.

The resulting set has a boundary as in Figure 2.2.

0.2

0.18

0.16

0.14

·=mean return

0.12

Efficient Frontier

0.1

0.08

0.06

(σ ,· )

g g

0.04

0.02

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

σ =standard deviation of return

Figure 2.2: The E¬cient Frontier

[FIGURE 2.2 ABOUT HERE]

Exactly what form this ¬gure takes depends in part on the assumptions ap-

plied to the weights. Since they represent the proportion of our total investment

in each of n stocks they must add to one. Negative weights correspond to selling

short one stock so as to be able to invest more in another, and we may assume

no limit on our ability to do so. In this case the only constraint on w is the

P

constraint wi = 1. With this constraint alone, we can determine the bound-

ary of the admissible set by ¬xing the vertical component (the mean return) of

a portfolio at some value say · and then ¬nding the minimum possible standard

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.41

deviation corresponding to that mean. This allows us to determine the leading

edge or left boundary of the region. The optimisation problem is as follows

√

min w0 Σw subject to

subject to the two constraints on the weights

w0 1 = 1

w0 µ = ·.

where 1 is the column vector of n ones. Since we will often make use of the

method of Lagrange multipliers for constrained problems such as this one, we

interject a lemma justifying the method. For details, consult Apostol (1973),

Section 13.7 or any advanced calculus text.

Lemma 4 Consider the optimisation problem

min{f (w); w ∈ Rn } subject to p constraints (2.10)

of the form g1 (w) = 0, g2 (w) = 0, ..., gp (w) = 0.

Then provided the functions f, g1 , ..., gp are continuously di¬erentiable, a nec-

essary solution for a solution to (2.10) is that there is a solution in the n + p

variables (w1 , ...wn , »1 , ..., »p ) of the equations

‚

{f (w) + »1 g1 (w) + ... + »p gp (w)} = 0, i = 1, 2, ..., n

‚wi

‚

{f (w) + »1 g1 (w) + ... + »p gp (w)} = 0, j = 1, 2, ..., p.

‚»j

This constants »i are called the Lagrange multipliers and the function that

is di¬erentiated, {f (w) + »1 g1 (w) + ... + »p gp (w)} is the Lagrangian.

Let us return to our original minimization problem with one small simpli¬-

√

cation. Since minimizing w0 Σw results in the same weight vector w as does

0

minimizing w Σw we choose the latter as our objective function.

42 CHAPTER 2. SOME BASIC THEORY OF FINANCE

We introduce Lagrange multipliers »1 , »2 and we wish to solve

‚ 0

{w Σw + »1 (w0 1 ’ 1) + »2 (w0 µ ’ ·)} = 0, i = 1, 2, ..., n

‚wi

‚ 0

{w Σw + »1 (w0 1 ’ 1) + »2 (w0 µ ’ ·)} = 0, j = 1, 2.

‚»j

The solution is obtained from the simple di¬erentiation rule

‚0 ‚0

w Σw = 2Σw and µw=w

‚w ‚w

and is of the form

w = »1 Σ’1 1+»2 Σ’1 µ

with the Lagrange multipliers »1 , »2 chosen to satisfy the two constraints, i.e.

»1 10 Σ’1 µ + »2 10 Σ’1 1 = 1

»1 µ0 Σ’1 µ + »2 µ0 Σ’1 1 = ·.

Suppose we de¬ne an n — 2 matrix M with columns 1 and µ,

M =[1 µ]

and the 2 — 2 matrix A = (M 0 Σ’1 M )’1 , then the Lagrange multipliers are

given by the vector

⎛ ⎞ ⎡ ¤

»1 1

»=⎝ ⎠ = A⎣ ¦

»2 ·

and the weights by the vector

⎡ ¤

1

w = Σ’1 M A ⎣ ¦. (2.11)

·

We are now in a position to identify the boundary or the curve in Figure 2.2.

√

As the mean of the portfolio · changes, the point takes the form ( w0 Σw, ·)

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.43

with w given by (2.11). Notice that

⎡ ¤

1

0

w Σw = [ 1 · ]A0 M 0 Σ’1 ΣΣ’1 M A ⎣ ¦

·

⎡ ¤

1

= [ 1 · ]A0 M 0 Σ’1 M A ⎣ ¦

·

⎡ ¤

1

= [ 1 · ]A ⎣ ¦

·

= A11 + 2A12 · + A22 · 2 .

√

Therefore a point on the boundary (σ, ·) = ( w0 Σw, ·) satis¬es

σ 2 ’ A22 · 2 ’ 2A12 · ’ A11 = 0

or

σ 2 = A22 · 2 + 2A12 · + A11

= σg + A22 (· ’ ·g )2

2

where

10 Σ’1 µ

A12

·g = ’ (2.12)

= 0 ’1

A22 1Σ 1

2

|A|

A

σg = A11 ’ 12 =

2

A22 A22

1

(2.13)

= 0 ’1 .

1Σ 1

and the point (σg , µg ) represents the point in the region corresponding to the

minimum possible standard deviation over all portfolios. This is the most

conservative investment portfolio available with this class of securities. What

weights to do we need to put on the individual stocks to achieve this conservative

portfolio? It is easy to see that the weight vector is given by

10 Σ’1

0

(2.14)

wg =

10 Σ’1 1

44 CHAPTER 2. SOME BASIC THEORY OF FINANCE

and since the quantity 10 Σ’1 1 in the denominator is just a scale factor to insure

that the weights add to one, the amount invested in stock i is proportional to

the sum of the elements of the i™th row of the inverse covariance matrix Σ’1 .

An equation of the form

σ2 ’ A22 (· ’ ·g )2 = σg

2

represents a hyperbola since A22 > 0. Of course investors are presumed to prefer

higher returns for a given value of the standard deviation of portfolio so it is

only the upper boundary of this curve in Figure 2.2 that is e¬cient in the sense

that there is no portfolio that is strictly better (better in the sense of higher

return combined with standard deviation that is not larger).

Now let us return to a portfolio whose standard deviation and mean return

lie on the e¬cient frontier. Let us call these e¬cient portfolios. It turns out

that any portfolio on this e¬cient frontier has the same covariance with the

0

minimum variance portfolio wg R derived above.

1

Proposition 5 Every e¬cient portfolio has the same covariance with

10 Σ’1 1

0

the conservative portfolio wg R.

Proof. We noted before that such a portfolio has mean return · and stan-

dard deviation σ which satisfy the relation

σ 2 ’ A22 · 2 ’ 2A12 · ’ A11 = 0.

Moreover the weights for this portfolio are described by

⎡ ¤

1

w = Σ’1 M A ⎣ ¦. (2.15)

·

so the returns vector from this portfolio can be written as

w0 R = [ 1 · ]AM 0 Σ’1 R.

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.45

It is interesting to observe that the covariance of returns between this e¬cient

0

portfolio and the conservative portfolio wg R is given by

0

cov(wg R, [ 1 · ]AM 0 Σ’1 R)= [ 1 · ]AM 0 Σ’1 Σwg

⎡ ¤

0

1

¦ Σ’1 1 1

= [ 1 · ]A ⎣

10 Σ’1 1

µ0

⎡ ¤

0 ’1

1Σ 1 1

= [ 1 · ]A ⎣ ¦

0 Σ’1 1

µ0 Σ’1 1 1

⎡ ¤

1 1

= [ 1 · ]⎣ ¦

10 Σ’1 1

0

1

=

10 Σ’1 1

where we use the fact that, by the de¬nition of A,

⎡ ¤⎡ ¤

0 ’1 0 ’1

1Σ 1 µΣ 1 10

A⎣ ¦=⎣ ¦.

µ0 Σ’1 1 µ0 Σ’1 µ 01

Now consider two portfolios on the boundary in Figure 2.2. For each the

weights are of the same form, say

⎡ ¤ ⎡ ¤

1 1

wp = Σ’1 M A ⎣ ¦ and wq = Σ’1 M A ⎣ ¦ (2.16)

·p ·q

where the mean returns are ·p and ·q respectively. Consider the covariance

between these two portfolios

0 0 0

cov(wp R, wq R) = wp Σwq

⎡ ¤

1

](M 0 Σ’1 M )’1 ⎣ ¦

=[ 1 ·p

·q

= A11 + A12 (·p + ·q ) + A22 ·p ·q

⎡ ¤

0

= var(wp R) ’ [ 1 ·p ]A ⎣ ¦

0

·p ’ ·q

46 CHAPTER 2. SOME BASIC THEORY OF FINANCE

An interesting special portfolio that is a “zero-beta” portfolio, one that is

0

perfectly uncorrelated with the portfolio with weights wp R. This is obtained by

setting the above covariance equal to 0 and solving we obtain

A11 + A12 ·p

·q = ’

A12 + A22 ·p

µ0 Σ’1 µ ’ (µ0 Σ’1 1)·p

= 0 ’1 .

µ Σ 1 ’ (10 Σ’1 1)·p

There is a simple method for determining the point (, ·q ) graphically indicated

in Figure ??. From the equation relating points on the boundary,

σ2 ’ A22 (· ’ ·g )2 = σg

2

we obtain

‚· σ

=

A22 (· ’ ·g )

‚σ

and so the tangent line at the point (σp , ·p ) strikes the σ = 0 axis at a point

·q which satis¬es

·p ’ ·q σp

=

A22 (·p ’ ·g )

σp

or

2

σp

·q = ·p ’

A22 (·p ’ ·g )

2

A22 ·p + 2A12 ·p + A11

= ·p ’

A22 ·p + A12

A11 + A12 ·p

=’ (2.17)

.

A12 + A22 ·p

Note that this is exactly the same mean return obtained earlier for the portfolio

0

which has zero covariance with wp R. This shows that we can ¬nd the standard

deviation and mean of this uncorrelated portfolio by constructing the tangent

line at the point (σp , ·p ) and then setting ·q to be the y-coordinate of the

point where this tangent line strikes the σ = 0 axis as in Figure 2.3.

[FIGURE 2.3 ABOUT HERE]

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.47

Figure 2.3: The tangent line at the point (σp , ·p )

Now suppose that there is available to all investors a risk-free investment.

Such an investment typically has smaller return than those on the e¬cient

frontier but since there is no risk associated with the investment, its standard

deviation is 0. It may be a government bond or treasury bill yielding interest

rate r so it corresponds to a point in Figure 2.4 at (0, r). Since all investors are

able to include this in their portfolio, the e¬cient frontier changes. In fact if

an investor invests an amount β in this risk-free investment and amount 1 ’ β

(this may be negative) in the risky portfolio with standard deviation and mean

return (σp , ·p ) then the resulting investment has mean return

0

E(βr + (1 ’ β)wp R) = βr+(1 ’ β)· p

and standard deviation of return

q

0

V ar(βr + (1 ’ β)wp R) = (1 ’ β)σp .

This means that every point on a line joining (0, r) to points in the risky portfolio

are now attainable and so the new set of attainable values of (σ, ·) consists of a

cone with vertex at (0, r),the region shaded in Figure 2.4. The e¬cient frontier

48 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Figure 2.4: _____

is now the line L in Figure 2.4. The point m is the point at which this line is

tangent to the e¬cient frontier determined from the risky investments. Under

this theory, this point has great signi¬cance.

[FIGURE 2.4 ABOUT HERE]

Lemma 6 The value-weighted market average corresponds to the point of tan-

gency m of the line to the risky portfolio e¬cient frontier.

From (2.17) the point m has standard deviation, mean return ·m which

solves

A11 + A12 ·m

r=’

A12 + A22 ·m

µ Σ µ ’ (µ0 Σ’1 1)·m

0 ’1

= 0 ’1

µ Σ 1 ’ (10 Σ’1 1)·m

and this gives

µ0 Σ’1 µ ’ r(µ0 Σ’1 1)

·m = .

µ0 Σ’1 1 ’ r(10 Σ’1 1)

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.49

The corresponding weights on individual stocks are given by

⎡ ¤

1

wm = Σ’1 M A ⎣ ¦.

·m

⎡ ¤

A11 + A12 ·m

= Σ’1 [1 µ] ⎣ ¦

A12 + A22 ·m

⎡ ¤

’r

= cΣ’1 [1 µ] ⎣ ¦ , where c = A12 + A22 ·m

1

= cΣ’1 (µ’r1).

These market weights depend essentially on two quantities. If R denotes the

correlation matrix

Σij

Rij =

σi σj

√

where σi = Σii is the standard deviation of the returns from stock i, and

µi ’ r

»i =

σi

is the standardized excess return or the price of risk, then the weight wi on

stock i is such that

wi σi ∝ R’1 » (2.18)

with » the column vector of values of »i . For the purpose of comparison, recall

that the conservative portfolio, one minimizing the variance over all portfolios

of risky stocks, has weights

wg ∝ Σ’1 1

which means that the weight on stock i satis¬es a relation exactly like (2.18)

except that the mean returns µi have all been replaced by the same constant.

Let us suppose that stocks, weighed by their total capitalization in the mar-

ket result in some weight vector w 6= wm . When there is a risk-free investment,

m is the only point in the risky stock portfolio that lies in the e¬cient frontier

and so evidently if we are able to trade in a market index (a stock whose value

50 CHAPTER 2. SOME BASIC THEORY OF FINANCE

depends on the total market), we can ¬nd an investment which is a combination

of the risk-free investment with that corresponding to m which has the same

standard deviation as w0 R but higher expected return. By selling short the

market index and buying this new portfolio, an arbitrage is possible. In other

words, the market will not stay in this state for long.

If the market portfolio m has standard deviation σm and mean ·m , then

the line L is described by the relation

·m ’ r

·=r+ σ.

σm

For any investment with mean return · and standard deviation of return σ

to be competitive, it must lie on this e¬cient frontier, i.e. it must satisfy the

relation

σ

· ’ r = β(·m ’ r), where β = or equivalently (2.19)

σm

·’r (·m ’ r)

= .

σ σm

This is the most important result in the capital asset pricing model. The excess

return of a stock · ’ r divided by its standard deviation σ is supposed constant,

and is called the Sharpe ratio or the market price of risk. The constant β called

the beta of the stock or portfolio and represents the change in the expected

portfolio return for each unit change in the market. It is also the ratio of the

standard deviations of return of the stock and the market. Values of β > 1

indicate a stock that is more variable than the market and tends to have higher

positive and negative returns, whereas values of β < 1 are investments that are

more conservative and less volatile than the market as a whole.

We might attempt to use this model to simplify the assumed structure of

the joint distribution of stock returns. One simple model in which (2.19) holds

is one in which all stocks are linearly related to the market index through a

simple linear regression. In particular, suppose the return from stock i, Ri , is

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.51

related to the return from the market portfolio Rm by

σi 2

Ri ’ r = βi (Rm ’ r) + ²i , where βi = , and σi = Σii .

σm

The “errors” ²i are assumed to be random variables, uncorrelated with the

market returns Rm . This model is called the single-index model relating the

returns from the stock Ri and from the market portfolio Rm .It has the merit

that the relationship (2.19) follows immediately.

Taking variance on both sides, we obtain

2 2 2

var(Ri ) = βi var(Rm ) + var(²i ) = σi + var(²) > σi

2

which contradicts the assumption that var(Ri ) = σi . What is the cause of this

contradiction? The relationship (2.19) assumes that the investment lies on the

e¬cient frontier. Is this not a su¬cient condition for investors to choose this

investment? All that is required for rational investors to choose a particular

stock is that it forms part of a portfolio which does lie on the e¬cient frontier.

Is every risk in an e¬cient market rewarded with additional expected return?

We cannot expect the market to compensate us with a higher rate of return for

additional risks that could be diversi¬ed away. Suppose, for example, we have

two stocks with identical values of β. Suppose their returns R1 and R2 both

satisfy a linear regression relation above

Ri ’ r = β(Rm ’ r) + ²i , i = 1, 2,

where cov(²1 , ²2 ) = 0. Consider an investment of equal amounts in both stocks

so that the return is

R1 + R2 ²1 + ²2

= β(Rm ’ r) + .

2 2

For simplicity assume that σ1 · σ2 and notice that the variance of this new

investment is

1

β 2 σm + [var(²1 ) + var(²2 )] < var(R2 ).

2

4

52 CHAPTER 2. SOME BASIC THEORY OF FINANCE

The diversi¬ed investment consisting of the average of the two results in the

same mean return with smaller variance. Investors should not compensated for

the additional risk in stock 2 above the level that we can achieve by sensible

diversi¬cation. In general, by averaging or diversifying, we are able to provide

an investment with the same average return characteristics but smaller variance

than the original stock. We say that the risk (i.e. var(²i )) associated with

stock i which can be diversi¬ed away is the speci¬c risk, and this risk is not

rewarded with increased expected return. Only the so-called systematic risk σi

which cannot by removed by diversi¬cation is rewarded with increased expected

return with a relation like (2.19).

The covariance matrix of stock returns is one of the most di¬cult parameters

to estimate in practice form historical data. If there are n stocks in a market

(and normally n is large), then there are n(n + 1)/2 elements of Σ that need

to be estimated. For example if we assume all stocks in the TSE 300 index

are correlated this results in a total of (300)(301)/2 = 45, 150 parameters

to estimate. We might use historical data to estimate these parameters but

variances and covariances among stocks change over time and it is not clear

over what period of time we can safely use to estimate these parameters. In

spite of its defects, the single index model can be used to provide a simple

approximate form for the covariance matrix Σ of the vector of stock returns.

Notice that under the model, assuming uncorrelated random errors ²i with

var(²i ) = δi ,

Ri ’ r = βi (Rm ’ r) + ²i ,

we have

2 22

cov(Ri , Rj ) = βi βj σm , i 6= j, var(Ri ) = βi σm + δi .

Whereas n stocks would otherwise require a total of n(n + 1)/2 parameters in

the covariance matrix Σ of returns, the single index model allows us to reduce

2

this to the n + 1 parameters σm , and δi , i = 1, ..., n. There is the disadvantage

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.53

in this formula however that every pair of stocks in the same market must be

positively correlated, a feature that contradicts some observations of real market

returns.

Suppose we use this form Σ = ββ 0 σm + ∆, to estimate weights on individual

2

stocks, where ∆ is the diagonal matrix with the δi along the diagonal and β

In this case Σ’1 = ∆’1 +

is the column vector of individual stock betas.

c∆’1 ββ 0 ∆’1 where

’1 1

2 P

= ’σm

P

c= ’2 22

2 1+ i βi σm /δi

σm + βi /δi

i

and consequently the conservative investor by (2.14) invests in stock i propor-

tionally to the components of Σ’1 1

X

1

or to + cβi ( βj /δj )

δi j

1

P

or proportional to βi +

cδi ( j βj /δj )

The conditional variance of Ri given the market return Rm is δi . Let us call this

the excess volatility for stock i. Then the weights for the conservative portfolio

are linear in the beta for the stock and the reciprocal of the excess volatility.

The weights in the market portfolio are given by

⎡ ¤ ⎡ ¤

1 1

wm = Σ’1 M A ⎣ ¦ = (∆’1 + c∆’1 ββ 0 ∆’1 )[ 1 µ ](M 0 Σ’1 M )’1 ⎣ ¦

·p ·p

Minimum Variance under Q.

Suppose we wish to ¬nd a portfolios of securities which has the smallest possible

variance under the risk neutral distribution Q. For example for a given set of

weights wi (t) representing the number of shares held in security i at time t,

P

de¬ne the portfolio Π(t) = wi (t)Si (t). Recall from Section 2.1 that under

a risk neutral distribution, all stocks have exactly the same expected return

as the risk-free interest rate so the portfolio Π(t) will have exactly the same

54 CHAPTER 2. SOME BASIC THEORY OF FINANCE

conditional expected rate of return under Q as all the constituent stocks,

X X B(t + 1) B(t + 1)

EQ [Π(t+1)|Ht ] = wi (t)EQ [Si (t+1)|Ht ] = wi (t) Si (t) = Π(t).

B(t) B(t)

i i

Since all portfolios have the same conditional expected return under Q, we

might attempt to minimize the (conditional) variance of the portfolio return of

the portfolio. The natural constraint is that the cost of the portfolio is deter-

mined by the amount c(t) that we presently have to invest. We might assume

a constant investment over time, for example c(t) = 1 for all t. Alternatively,

we might wish to study a self-¬nancing portfolio Π(t), one for which past gains

(or perish the thought, past losses) only are available to pay for the current

portfolio so we neither withdraw from nor add money to the portfolio over its

lifetime. I this case c(t) = Π(t). We wish to minimise

X

varQ [Π(t + 1)|Ht ] subject to the constraint wi (t)Si (t) = c(t).

i

As before, the solution is quite easy to obtain, and in fact the weights are

given by the vector

⎛ ⎞

w (t)

⎜1 ⎟

⎜ ⎟

⎜ w2 (t) ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

. c(t)

⎜ ⎟

Σ’1 S(t).

w(t) = ⎜ ⎟= 0

⎟ S (t)Σt S(t) t

’1

⎜ .

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

.

⎝ ⎠

wn (t)

where Σt = varQ (S(t + 1)|Ht ) is the instantaneous conditional covariance

matrix of S(t) under the measure Q. If my objective were to minimize risk under

the Q measure, then this portfolio is optimal for ¬xed cost. The conditional

variance of this portfolio is given by

c2 (t)

0

varQ (Π(t + 1)|Ht ) = w (t)Σt w(t) = 0 .

S (t)Σ’1 S(t)

t

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.55

Π(t+1)’Π(t)

In terms of the portfolio return RΠ (t + 1) = , if the portfolio is

Π(t)

self-¬nancing so that c(t) = Π(t), the above relation states that the conditional

variance of the return RΠ (t + 1) given the past is simply

1

varQ (RΠ (t + 1)|Ht ) =

S 0 (t)Σ’1 S(t)

t

which is similar to the form of the variance of the conservative portfolio (2.13).

Similarly, covariances between returns for individual stocks and the return

of the portfolio Π are given by exactly the same quantity, namely

1

cov(Ri (t + 1), RΠ (t + 1)|Ht ) = .

S 0 (t)Σ’1 S(t)

t

Let us summarize our ¬ndings so far. We assume that the conditional co-

variance matrix Σt of the vector of stock prices is non-singular. Under the risk

neutral measure, all stocks have exactly the same expected returns equal to the

risk-free rate. There is a unique self-¬nancing minimum-variance portfolio Π(t)

and all stocks have exactly the same conditional covariance β with Π. All stocks

have exactly the same regression coe¬cient β when we regress on the minimum

variance portfolio.

Are other minimum variance portfolios conditionally uncorrelated with the

portfolio we obtained above. Suppose we de¬ne Π2 (t) similarly to minimize the

variance subject to the condition that CovQ (Π2 (t + 1), Π(t + 1)|Ht ) = 0. It is

easy to see that this implies that the cost of such a portfolio at the beginning

of each period is 0. This means that in this new portfolio, there is a perfect

balance between long and short stocks, or that the value of the long and short

stocks are equal.

The above analysis assumes that our objective is minimizing the variance

of the portfolio under the risk-neutral distribution Q. Two objections could be

made. First we argued earlier that the performance of an investment should be

made through the returns , not through the stock prices. Since under the risk

neutral measure Q, the expected return from every stock is the risk-free rate of

56 CHAPTER 2. SOME BASIC THEORY OF FINANCE

return, we are left with the problem of minimizing the variance of the portfolio

return. By our earlier analysis, this is achieved when the proportion of our

total investment at each time period in stock i is chosen as the corresponding

Σ’1 1

component of the vector where now Σt is the conditional covariance

t

10 Σ’1 1

t

matrix of the stock returns. This may appear to be a di¬erent criterion and

hence a di¬erent solution, but because at each time step the stock price is a linear

function of the return Si (t + 1) = Si (t)(1 + Ri (t + 1)) the variance minimizing

portfolios are essentially the same. There is another objection however to an

analysis in the risk-neutral world of Q. This is a distribution which determines

the value of options in order to avoid arbitrage in the system, not the actual

distribution of stock prices. It is not clear what the relationship is between

the covariance matrix of stock prices under the actual historical distribution

and the risk neutral distribution Q, but observations seem to indicate a very

considerable di¬erence. Moreover, if this di¬erence is large, there is very little

information available for estimating the parameters of the covariance matrix

under Q, since historical data on the ¬‚uctuations of stock prices will be of

doubtful relevance.

Entropy: choosing a Q measure

Maximum Entropy

In 1948 in a fundamental paper on the transmission of information, C. E. Shan-

non proposed the following idea of entropy. The entropy of a distribution at-

tempts to measure the expected number of steps required to determine a given

outcome of a random variable with a given distribution when using a simple

binary poll. For example suppose that a random variable X has distribution

ENTROPY: CHOOSING A Q MEASURE 57

given by

x 0 1 2

P [X = x] .25 .25 .5

if we ask ¬rst whether the random variable is ≥ 2 and

In this case,

then, provided the answer is no, if it is ≥ 1, the expected number of queries to

ascertain the value of the random variable is 1+1(1/2) = 1.5. There is no more

e¬cient scheme for designing this binary poll in this case so we will take 1.5 to

be a measure of entropy of the distribution of X. For a discrete distribution,

such that P [X = x] = p(x), the entropy may be de¬ned to be

X

H(p) = E{’ ln(p(X))} = ’ p(x) ln(p(x)).

x

More generally we de¬ne the entropy of an arbitrary distribution through the

form for a discrete distribution. If P is a probability measure (see the appen-

dix),

X

H(P ) = sup{’ P (Ei ) ln(P (Ei ))}

where the supremum is taken over all ¬nite partitions (Ei } of the space.

In the case of the above distribution, if we were to replace the natural log-

arithm by the log base 2, (ln and log2 di¬er only by a scale factor and are

therefore the corresponding measures of entropy are equivalent up a constant

P

multiple) notice that ’ x p(x) log2 (p(x)) = .5(1) + .5(2) = 1.5, so this formula

correctly measures the di¬culty in ascertaining a random variable from a se-

quence of questions with yes-no or binary answers. This is true in general. The

complexity of a distribution may be measured by the expected number of ques-

tions in a binary poll to determine the value of a random variable having that

distribution, and such a measure results in the entropy H(p) of the distribution.

Many statistical distributions have an interpretation in terms of maximizing

entropy and it is often remarkable how well the maximum entropy principle re-

produces observed distributions. For example, suppose we know that a discrete

random variable takes values on a certain set of n points. What distribution p

58 CHAPTER 2. SOME BASIC THEORY OF FINANCE

on this set maximizes the entropy H(p)? First notice that if p is uniform on

P1 1

n points, p(x) = 1/n for all x and so the entropy is ’ x n ln( n ) = ln(n).

Now consider the problem of maximizing the entropy H(p) for any distribution

on n points subject to the constraint that the probabilities add to one. As in

P P

(2.10), the Lagrangian for this problem is ’ x p(x) ln(p(x)) ’ »{ x p(x) ’ 1}

where » is a Lagrange multiplier. Upon di¬erentiating with respect to p(x) for

each x, we obtain ’ ln(p(x)) ’ 1 ’ » = 0 or p(x) = e’(1+») . The probabilities

evidently do not depend on x and the distribution is thus uniform. Applying

the constraint that the sum of the probabilities is one results in p(x) = 1/n

for all x. The discrete distribution on n points which has maximum entropy is

the uniform distribution. What if we repeat this analysis using additional con-

straints, for example on the moments of the distribution? Suppose for example

that we require that the mean of the distribution is some ¬xed constant µ and

the variance ¬xed at σ 2 . The problem is similar to that treated above but with

two more terms in the Lagrangian for each of the additional constraints. The

Lagrangian becomes

X X X X

x2 p(x)’µ2 ’σ 2 }

’ p(x) ln(p(x))’»1 { p(x)’1}’»2 { xp(x)’µ}’»3 {

x x x

whereupon setting the derivative with respect to p(x) equal to zero and ap-

plying the constraints we obtain

p(x) = exp{’»1 ’ »2 x ’ »3 x2 },

with constants »1 , »2 , »3 chosen to satisfy the three constraints. Since the ex-

ponent is a quadratic function of x, this is analogous to the normal distribution

except that we have required that it be supported on a discrete set of points x.

With more points, positioned more closely together, the distribution becomes

closer to the normal. Let us call such a distribution the discrete normal dis-

tribution. For a simple example, suppose that we wish to use the maximum

entropy principle to approximate the distribution of the sum of the values on

ENTROPY: CHOOSING A Q MEASURE 59

0.18

0.16

0.14

0.12

0.1

probability

0.08

0.06

0.04

0.02

0

2 3 4 5 6 7 8 9 10 11 12

value

Figure 2.5: A discrete analogue of the normal distribution compared with the

distribution of the sum of the values on two dice.

two dice. In this case the actual distribution is known to us as well as the mean

and variance E(X) = 7, var(X) = 35/6;

2 3 4 5 6 7 8 9 10 11 12

x

1 2 3 4 5 6 5 4 3 2 1

P (X = x) 36 36 36 36 36 36 36 36 36 36 36

The maximum entropy distribution on these same points constrained to have

the same mean and variance is very similar to this, the actual distribution. This

can been seen in Figure 2.5.

[FIGURE 2.5 ABOUT HERE]

In fact if we drop the requirement that the distribution is discrete, or equiv-

alently take a limit with an increasing number of discrete points closer and

closer together, the same kind of argument shows that the maximum entropy

distribution subject to a constraint on the mean and the variance is the normal

distribution. So at least two well-known distributions arise out of maximum

60 CHAPTER 2. SOME BASIC THEORY OF FINANCE

entropy considerations. The maximum entropy distribution on a discrete set

of points is the uniform distribution. The maximum entropy subject to a con-

straint on the mean and the variance is a (discrete) normal distribution. There

are many other examples as well. In fact most common distributions in statis-

tics have an interpretation as a maximum entropy distribution subject to some

constraints.

Entropy has a number of properties that one would expect of a measure of

the information content in a random variable. It is non-negative, and can in

usual circumstances be in¬nite. We expect that the information in a function

of X , say g(X), is less than or equal to the information in X itself, equal if

the function is one to one (which means in e¬ect we can determine X from

the value of g(X)). Entropy is a property of a distribution, not of a random

variable. Nevertheless it is useful to be able to abuse the notation used earlier

by referring to H(X) as the entropy of the distribution of X. Then we have the

following properties

Proposition 7 H(X) ≥ 0

Proposition 8 H(g(X)) · H(X) for any function g(x)..

The information or uncertainty in two random variables is clearly greater

than that in one. The de¬nition of entropy is de¬ned in the same fashion as

before, for discrete random variables (X, Y ),

H(X, Y ) = ’E(ln p(X, Y ))

where p(x, y) is the joint probability function

p(x, y) = P [X = x, Y = y].

If the two random variables are independent, then we expect that the uncer-

tainty should add. If they are dependent, then the entropy of the pair (X, Y )

is less than the sum of the individual entropies.

ENTROPY: CHOOSING A Q MEASURE 61

Proposition 9 H(X, Y ) · H(X) + H(Y ) with equality if and only if X and Y

are independent.

Let us now use the principle of maximum entropy to address an eminently

practical problem, one of altering a distribution to accommodate a known mean

value. Suppose we are interested in determining a risk-neutral distribution

for pricing options at maturity T. Theorem 1 tells us that if there is to be no

arbitrage, our distribution or measure Q must satisfy a relation of the form

EQ (e’rT ST ) = S0

where r is the continuously compounded interest rate, S0 is the initial (present)

value of the underlying stock, and ST is its value at maturity. Let us also

suppose that we constraint the variance of the future stock price under the

measure Q so that

varQ (ST ) = σ 2 T.

Then from our earlier discussion, the maximum entropy distribution under

constraints on the mean and variance is the normal distribution so that the

probability density function of ST is

(s ’ erT S0 )2

1

f (s) = √ }.

exp{’

2σ2 T

σ 2πT

If we wished a maximum entropy distribution which is compatible with a

number of option prices, then we should impose these option prices as additional

constraints. Again suppose the current time t = 0 and we know the prices

Pi , i = 1, ..., n of n di¬erent call options available on the market, all on the same

security and with the same maturity T but with di¬erent strike prices Ki . The

distribution Q we assign to ST must satisfy the constraints

E(e’rT (ST ’ Ki )+ ) = Pi , i = 1, ..., n (2.20)

as well as the martingale constraint

E(e’rT ST ) = S0 . (2.21)

62 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Once again introducing Lagrange multipliers, the probability density function

of ST will take the form

n

X

’rT

»i (s ’ Ki )+ + »0 s}

f (s) = k exp{e

i=1

where the parameters »0 , ..., »n are chosen to satisfy the constraints (2.20) and

(2.21) and k so that the function integrates to 1. When ¬t to real option price

data, these distributions typically resemble a normal density, usually however

with some negative skewness and excess kurtosis. See for example Figure XXX.

There are also“sawtooth” like appendages with teeth corresponding to each of

the n options. Note too this density is strictly positive at the value s = 0,

a feature that we may or may not wish to have. Because of the ”teeth”, a

smoother version of the density is often used, one which may not perfectly

reproduce option prices but is nevertheless appears to be more natural.

Minimum Cross-Entropy

Normally market information does not completely determine the risk-neutral

measure Q . We will argue that while market data on derivative prices rather

than historical data should determine the Q measure, historical asset prices

can be used to ¬ll in the information that is not dictated by no-arbitrage con-

siderations. In order to relate the real world to the risk-free world, we need

either su¬cient market data to completely describe a risk-neutral measure Q

(such a model is called a complete market) or we need to limit our candidate

class of Q measures somewhat. We may either de¬ne the joint distributions of

the stock prices or their returns, since from one we can pass to the other. For

convenience, suppose we describe the joint distribution of the returns process.

The conditions we impose on the martingale measure are the following;

1. Under Q, each normalized stock price Sj (t)/Bt and derivative price

Vt /Bt forms a martingale. Equivalently, EQ [Si (t+1)|Ht ] = Si (t)(1+r(t))

ENTROPY: CHOOSING A Q MEASURE 63

where r(t) is the risk free interest rate over the interval (t, t + 1). (Recall

that this risk-free interest rate r(t) is de¬ned by the equation B(t + 1) =

(1 + r(t))B(t).)

2. Q is a probability measure.

A slight revision of notation is necessary here. We will build our joint distri-

butions conditionally on the past and if P denotes the joint distribution stock

prices S(1), S(2), ...S(T ) over the whole period of observation 0 < t < T then

Pt+1 denotes the conditional distribution of S(t + 1) given Ht . Let us denote

the conditional moment generating function of the vector S(t + 1) under the

measure Pt+1 by

X

mt (u) = EP [exp(u0 S(t + 1)|Ht ] = EP [exp( ui Si (t + 1))|Ht ]

i

We implicitly assume, of course, that this moment generating function exists.

Suppose, for some vector of parameters · we choose Qt+1 to be the exponential

tilt of Pt+1 , i.e.

exp(· 0 s)

dQt+1 (s) = dPt+1 (s)

mt (·)

The division by mt (·) is necessary to ensure that Qt+1 is a probability measure.

Why transform a density by multiplying by an exponential in this way?

There are many reasons for such a transformation. Exponential families of dis-

tributions are built in exactly this fashion and enjoy properties of su¬ciency,

completeness and ease of estimation. This exponential tilt resulted from maxi-

mizing entropy subject to certain constraints on the distribution. But we also

argue that the measure Q is the probability measure which is closest to P in

a certain sense while still satisfying the required moment constraint. We ¬rst

introduce cross-entropy which underlies considerable theory in Statistics and

elsewhere in Science.

64 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Cross Entropy

Consider two probability measures P and Q on the same space. Then the

cross entropy or Kullbach-Leibler “distance” between the two measures is given

by

X Q(Ei )

H(Q, P ) = sup Q(Ei ) log

P (Ei )

{Ei }

where the supremum is over all ¬nite partitions {Ei } of the probability space.

Various properties are immediate.

Proposition 10 H(Q, P ) ≥ 0 with equality if and only if P and Q are iden-

tical.

If Q is absolutely continuous with respect to P , that is if there is some

density function f (x) such that

Z

f (x)dP for all E

Q(E) =

E

then provided that f is smooth, we can also write

dQ

H(Q, P ) = EQ log( ).

dP

If Q is not absolutely continuous with respect to P then the cross entropy

H(Q, P ) is in¬nite. We should also remark that the cross entropy is not really

a distance in the usual sense (although we used the term “distance” in reference

to it) because in general H(Q, P ) 6= H(P |Q). For a ¬nite probability space,

there is an easy relationship between entropy and cross entropy given by the

following proposition. In e¬ect the result tells us that maximizing entropy H(Q)

is equivalent to minimizing the cross-entropy H(Q, P ) where P is the uniform

distribution.

Proposition 11 If the probability space has a ¬nite number n points, and P

denotes the uniform distribution on these n points, then for any other probability

measure Q,

H(Q, P ) = n ’ H(Q)

ENTROPY: CHOOSING A Q MEASURE 65

Now the following result asserts that the probability measure Q which is

closest to P in the sense of cross-entropy but satis¬es a constraint on its mean

is generated by a so-called “exponential tilt” of the distribution of P.

Theorem 12 : Minimizing cross-entropy.

Let f (X) be a vector valued function f (X) = (f1 (X), f2 (X), ..., fn (X)) and

µ = (µ1 , ..., µn ). Consider the problem

min H(Q, P )

Q

subject to the constraint EQ (fi (X)) = µi , i = 1, ..., n. Then the solution, if it

exists, is given by

Pn

exp(· 0 f (X)) exp( i=1 ·i fi (X))

dQ = dP =

m(·) m(·)

Pn ‚m

where m(·) = EP [exp( i=1 ·i fi (X))] and · is chosen so that = µm(·).

‚·i

The proof of this result, in the case of a discrete distribution P is a straight-

forward use of Lagrange multipliers (see Lemma 3). We leave it as a problem

at the end of the chapter.

Now let us return to the constraints on the vector of stock prices. In order

that the discounted stock price forms a martingale under the Q measure, we

require that EQ [S(t + 1)|Ht ] = (1 + r(t))S(t). This is achieved if we de¬ne Q

such that for any event A ∈ Ht ,

Z

Zt dP where

Q(A) =

A

s

X

0

·t (St+1 ’ St )) (2.22)

Zs = kt exp(

t=1

where kt are Ht measurable random variables chosen so that Zt forms a mar-

tingale

E(Zt+1 |Ht ) = Zt .

66 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Theorem 9 shows that this exponentially tilted distribution has the property

of being the closest to the original measure P while satisfying the condition

that the normalized sequence of stock prices forms a martingale.

There is a considerable literature exploring the links between entropy and

risk-neutral valuation of derivatives. See for example Gerber and Shiu (1994),

Avellaneda et. al (1997), Gulko(1998), Samperi (1998). In a complete or

incomplete market, risk-neutral valuation may be carried out using a martingale

measure which maximizes entropy or minimizes cross-entropy subject to some

natural constraints including the martingale constraint. For example it is easy

to show that when interest rates r are constant, Q is the risk-neutral measure

for pricing derivatives on a stock with stock price process St , t = 0, 1, ... if

and only if it is the probability measure minimizing H(Q, P ) subject to the

martingale constraint

1

(2.23)

St = EQ [ ¯ St+1 ].

1+r

There is a continuous time analogue of (2.22) as well which we can anticipate

by inspecting the form of the solution. Suppose that St denotes the stock price

at time t where we now allow t to vary continuously in time. which we will

discuss later but (2.22) can be used to anticipate it. Then an analogue of (2.22)

could be written formally as

Z t

0

·t dSt ’ gt )

Zs = exp(