Chapter 4 Introduction to Portfolio Theory

Eric Zivot

Department of Economics

University of Washington

January 26, 2000

This version: February 20, 2001

1 Introduction to Portfolio Theory

Consider the following investment problem. We can invest in two non-dividend paying

stocks A and B over the next month. Let RA denote monthly return on stock A and

RB denote the monthly return on stock B. These returns are to be treated as random

variables since the returns will not be realized until the end of the month. We assume

that the returns RA and RB are jointly normally distributed and that we have the

following information about the means, variances and covariances of the probability

distribution of the two returns:

µA = E[RA ], σ 2 = V ar(RA ),

A

µB = E[RB ], σ 2 = V ar(RB ),

B

σ AB = Cov(RA , RB ).

We assume that these values are taken as given. We might wonder where such values

come from. One possibility is that they are estimated from historical return data for

the two stocks. Another possibility is that they are subjective guesses.

The expected returns, µA and µB , are our best guesses for the monthly returns on

each of the stocks. However, since the investments are random we must recognize that

the realized returns may be di¬erent from our expectations. The variances, σ 2 and

A

2

σ B , provide measures of the uncertainty associated with these monthly returns. We

can also think of the variances as measuring the risk associated with the investments.

Assets that have returns with high variability (or volatility) are often thought to

be risky and assets with low return volatility are often thought to be safe. The

covariance σ AB gives us information about the direction of any linear dependence

between returns. If σ AB > 0 then the returns on assets A and B tend to move in the

1

same direction; if σ AB < 0 the returns tend to move in opposite directions; if σ AB = 0

then the returns tend to move independently. The strength of the dependence between

the returns is measured by the correlation coe¬cient ρAB = σσAB . If ρAB is close to

A σB

one in absolute value then returns mimic each other extremely closely whereas if ρAB

is close to zero then the returns may show very little relationship.

The portfolio problem is set-up as follows. We have a given amount of wealth and

it is assumed that we will exhaust all of our wealth between investments in the two

stocks. The investor s problem is to decide how much wealth to put in asset A and

how much to put in asset B. Let xA denote the share of wealth invested in stock A

and xB denote the share of wealth invested in stock B. Since all wealth is put into

the two investments it follows that xA + xB = 1. (Aside: What does it mean for xA

or xB to be negative numbers?) The investor must choose the values of xA and xB .

Our investment in the two stocks forms a portfolio and the shares xA and xB are

referred to as portfolio shares or weights. The return on the portfolio over the next

month is a random variable and is given by

(1)

Rp = xA RA + xB RB ,

which is just a simple linear combination or weighted average of the random return

variables RA and RB . Since RA and RB are assumed to be normally distributed, Rp

is also normally distributed.

1.1 Portfolio expected return and variance

The return on a portfolio is a random variable and has a probability distribution

that depends on the distributions of the assets in the portfolio. However, we can

easily deduce some of the properties of this distribution by using the following results

concerning linear combinations of random variables:

(2)

µp = E[Rp ] = xA µA + xB µB

σ 2 = var(Rp ) = x2 σ 2 + x2 σ 2 + 2xA xB σ AB (3)

p AA BB

These results are so important to portfolio theory that it is worthwhile to go

through the derivations. For the & result (2), we have

rst

E[Rp ] = E[xA RA + xB RB ] = xA E[RA ] + xB E[RB ] = xA µA + xB µB

by the linearity of the expectation operator. For the second result (3), we have

var(xA RA + xB RB ) = E[(xA RA + xB RB ) ’ E[xA RA + xB RB ])2 ]

var(Rp ) =

E[(xA (RA ’ µA ) + xB (RB ’ µB ))2 ]

=

E[x2 (RA ’ µA )2 + x2 (RB ’ µB )2 + 2xA xB (RA ’ µA )(RB ’ µB )]

= A B

x2 E[(RA ’ µA )2 ] + x2 E[(RB ’ µB )2 ] + 2xA xB E[(RA ’ µA )(RB ’ µB )],

= A B

2

and the result follows by the de& nitions of var(RA ), var(RB ) and cov(RA , RB )..

Notice that the variance of the portfolio is a weighted average of the variances

of the individual assets plus two times the product of the portfolio weights times

the covariance between the assets. If the portfolio weights are both positive then a

positive covariance will tend to increase the portfolio variance, because both returns

tend to move in the same direction, and a negative covariance will tend to reduce the

portfolio variance. Thus & nding negatively correlated returns can be very bene& cial

when forming portfolios. What is surprising is that a positive covariance can also be

bene& cial to diversi&cation.

1.2 E¬cient portfolios with two risky assets

In this section we describe how mean-variance e¬cient portfolios are constructed.

First we make some assumptions:

Assumptions

• Returns are jointly normally distributed. This implies that means, variances

and covariances of returns completely characterize the joint distribution of re-

turns.

• Investors only care about portfolio expected return and portfolio variance. In-

vestors like portfolios with high expected return but dislike portfolios with high

return variance.

Given the above assumptions we set out to characterize the set of portfolios that

have the highest expected return for a given level of risk as measured by portfolio

variance. These portfolios are called e¬cient portfolios and are the portfolios that

investors are most interested in holding.

For illustrative purposes we will show calculations using the data in the table

below.

Table 1: Example Data

σ2σ2

µA µB σA σB σ AB ρAB

A B

0.175 0.055 0.067 0.013 0.258 0.115 -0.004875 -0.164

The collection of all feasible portfolios (the investment possibilities set) in the

case of two assets is simply all possible portfolios that can be formed by varying

the portfolio weights xA and xB such that the weights sum to one (xA + xB = 1).

We summarize the expected return-risk (mean-variance) properties of the feasible

portfolios in a plot with portfolio expected return, µp , on the vertical axis and portfolio

standard-deviation, σ p , on the horizontal axis. The portfolio standard deviation is

used instead of variance because standard deviation is measured in the same units as

the expected value (recall, variance is the average squared deviation from the mean).

3

Portfolio Frontier with 2 Risky Assets

0.250

0.200

Portfolio expected return

0.150

0.100

0.050

0.000

0.000 0.100 0.200 0.300 0.400

Portfolio std. deviation

Figure 1

The investment possibilities set or portfolio frontier for the data in Table 1 is

illustrated in Figure 1. Here the portfolio weight on asset A, xA , is varied from

-0.4 to 1.4 in increments of 0.1 and, since xB = 1 ’ xA , the weight on asset is

then varies from 1.4 to -0.4. This gives us 18 portfolios with weights (xA , xB ) =

(’0.4, 1.4), (’0.3, 1.3), ..., (1.3, ’0.3), (1.4, ’0.4). For each of these portfolios we use

q

the formulas (2) and (3) to compute µp and σ p = σ 2 . We then plot these values1 .

p

Notice that the plot in (µp , σ p ) space looks like a parabola turned on its side (in

fact it is one side of a hyperbola). Since investors desire portfolios with the highest

expected return for a given level of risk, combinations that are in the upper left corner

are the best portfolios and those in the lower right corner are the worst. Notice that

the portfolio at the bottom of the parabola has the property that it has the smallest

variance among all feasible portfolios. Accordingly, this portfolio is called the global

minimum variance portfolio.

It is a simple exercise in calculus to & the global minimum variance portfolio.

nd

We solve the constrained optimization problem

min σ 2 = x2 σ 2 + x2 σ 2 + 2xA xB σ AB

p AA BB

x ,x

A B

s.t. xA + xB = 1.

1

The careful reader may notice that some of the portfolio weights are negative. A negative

portfolio weight indicates that the asset is sold short and the proceeds of the short sale are used to

buy more of the other asset. A short sale occurs when an investor borrows an asset and sells it in

the market. The short sale is closed out when the investor buys back the asset and then returns the

borrowed asset. If the asset price drops then the short sale produces and pro& t.

4

Substituting xB = 1 ’ xA into the formula for σ 2 reduces the problem to

p

min σ 2 = x2 σ 2 + (1 ’ xA )2 σ 2 + 2xA (1 ’ xA )σ AB .

p AA B

xA

The & order conditions for a minimum, via the chain rule, are

rst

dσ 2

p

= 2xmin σ 2 ’ 2(1 ’ xmin )σ 2 + 2σ AB (1 ’ 2xmin )

0= A A A B A

dxA

and straightforward calculations yield

σ 2 ’ σ AB

= 2 B2

xmin , xmin = 1 ’ xmin . (4)

A

σ A + σ B ’ 2σ AB B A

For our example, using the data in table 1, we get xmin = 0.2 and xmin = 0.8.

A B

E¬cient portfolios are those with the highest expected return for a given level

of risk. Ine¬cient portfolios are then portfolios such that there is another feasible

portfolio that has the same risk (σ p ) but a higher expected return (µp ). From the

plot it is clear that the ine¬cient portfolios are the feasible portfolios that lie below

the global minimum variance portfolio and the e¬cient portfolios are those that lie

above the global minimum variance portfolio.

The shape of the investment possibilities set is very sensitive to the correlation

between assets A and B. If ρAB is close to 1 then the investment set approaches a

straight line connecting the portfolio with all wealth invested in asset B, (xA , xB ) =

(0, 1), to the portfolio with all wealth invested in asset A, (xA , xB ) = (1, 0). This

case is illustrated in Figure 2. As ρAB approaches zero the set starts to bow toward

the µp axis and the power of diversi& cation starts to kick in. If ρAB = ’1 then

the set actually touches the µp axis. What this means is that if assets A and B

are perfectly negatively correlated then there exists a portfolio of A and B that has

positive expected return and zero variance! To & the portfolio with σ 2 = 0 when

nd p

ρAB = ’1 we use (4) and the fact that σ AB = ρAB σ A σ B to give

σB

xmin = , xmin = 1 ’ xA

A

σA + σB B

The case with ρAB = ’1 is also illustrated in Figure 2.

5

Portfolio Frontier with 2 Risky Assets

0.250

0.200

P ortfolio e x pe cte d re turn 0.150

0.100

0.050

0.000

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450

P ortfolio std. de via tion

correlation=1 correlation=-1

Figure 2

Given the e¬cient set of portfolios, which portfolio will an investor choose? Of

the e¬cient portfolios, investors will choose the one that accords with their risk

preferences. Very risk averse investors will choose a portfolio very close to the global

minimum variance portfolio and very risk tolerant investors will choose portfolios

with large amounts of asset A which may involve short-selling asset B.

1.3 E¬cient portfolios with a risk-free asset

In the preceding section we constructed the e¬cient set of portfolios in the absence of

a risk-free asset. Now we consider what happens when we introduce a risk free asset.

In the present context, a risk free asset is equivalent to default-free pure discount bond

that matures at the end of the assumed investment horizon. The risk-free rate, rf , is

then the return on the bond, assuming no in! ation. For example, if the investment

horizon is one month then the risk-free asset is a 30-day Treasury bill (T-bill) and

the risk free rate is the nominal rate of return on the T-bill. If our holdings of the

risk free asset is positive then we are lending money at the risk-free rate and if our

holdings are negative then we are borrowing at the risk-free rate.

1.3.1 E¬cient portfolios with one risky asset and one risk free asset

Continuing with our example, consider an investment in asset B and the risk free

asset (henceforth referred to as a T-bill) and suppose that rf = 0.03. Since the risk

free rate is &xed over the investment horizon it has some special properties, namely

µf = E[rf ] = rf

6

var(rf ) = 0

cov(RB , rf ) = 0

Let xB denote the share of wealth in asset B and xf = 1 ’ xB denote the share of

wealth in T-bills. The portfolio expected return is

Rp = xB RB + (1 ’ xB )rf

= xB (RB ’ rf ) + rf

The quantity RB ’ rf is called the excess return (over the return on T-bills) on asset

B. The portfolio expected return is then

µp = xB (µB ’ rf ) + rt

where the quantity (µB ’ rf ) is called the expected excess return or risk premium

on asset B. We may express the risk premium on the portfolio in terms of the risk

premium on asset B:

µp ’ rf = xB (µB ’ rf )

The more we invest in asset B the higher the risk premium on the portfolio.

The portfolio variance only depends on the variability of asset B and is given by

σ 2 = x2 σ 2 .

p BB

The portfolio standard deviation is therefore proportional to the standard deviation

on asset B:

σ p = xB σ B

which can use to solve for xB

σp

xB =

σB

Using the last result, the feasible (and e¬cient) set of portfolios follows the equation

µB ’ rf

· σp (5)

µp = rf +

σB

µ ’r

which is simply straight line in (µp , σ p ) with intercept rf and slope B B f . The slope

σ

of the combination line between T-bills and a risky asset is called the Sharpe ratio

or Sharpe s slope and it measures the risk premium on the asset per unit of risk (as

measured by the standard deviation of the asset).

The portfolios which are combinations of asset A and T-bills and combinations of

asset B and T-bills using the data in Table 1 with rf = 0.03. is illustrated in Figure

4.

7

Portfolio Frontier with 1 Risky Asset and T-Bill

0.200

0.180

P ortfolio e x pe cte d re turn

0.160

0.140

0.120

0.100

0.080

0.060

0.040

0.020

0.000

0.000 0.050 0.100 0.150 0.200 0.250 0.300

Portfolio std. deviation

Asset B and T-Bill Asset A and T-Bill

Figure 3

Notice that expected return-risk trade o¬ of these portfolios is linear. Also, notice

that the portfolios which are combinations of asset A and T-bills have expected

returns uniformly higher than the portfolios consisting of asset B and T-bills. This

occurs because the Sharpe s slope for asset A is higher than the slope for asset B:

µA ’ rf 0.175 ’ 0.03 µ ’ rf 0.055 ’ 0.03

= 0.562, B

= = = 0.217.

σA 0.258 σB 0.115

Hence, portfolios of asset A and T-bills are e¬cient relative to portfolios of asset B

and T-bills.

1.3.2 E¬cient portfolios with two risky assets and a risk-free asset

Now we expand on the previous results by allowing our investor to form portfolios of

assets A, B and T-bills. The e¬cient set in this case will still be a straight line in

(µp , σ p )’ space with intercept rf . The slope of the e¬cient set, the maximum Sharpe

ratio, is such that it is tangent to the e¬cient set constructed just using the two risky

assets A and B. Figure 5 illustrates why this is so.

8

Portfolio Frontier with 2 Risky Assets and T-Bills

0.350

Portfolio expected return

0.300

0.250

0.200

0.150

0.100

0.050

0.000

0.000 0.100 0.200 0.300 0.400 0.500 0.600

Portfolio std. deviation

Assets A and B Tangency and T-bills Asset B and T-bills

Asset A and t-bills Tangency Asset B

Asset A

Figure 4

µ ’r

If we invest in only in asset B and T-bills then the Sharpe ratio is B B f = 0.217

σ

and the CAL intersects the parabola at point B. This is clearly not the e¬cient set

of portfolios. For example, we could do uniformly better if we instead invest only

µ ’r

in asset A and T-bills. This gives us a Sharpe ratio of A A f = 0.562 and the new

σ

CAL intersects the parabola at point A. However, we could do better still if we invest

in T-bills and some combination of assets A and B. Geometrically, it is easy to see

that the best we can do is obtained for the combination of assets A and B such that

the CAL is just tangent to the parabola. This point is marked T on the graph and

represents the tangency portfolio of assets A and B.

We can determine the proportions of each asset in the tangency portfolio by &nding

the values of xA and xB that maximize the Sharpe ratio of a portfolio that is on the

envelope of the parabola. Formally, we solve

µp ’ rf

max s.t.

σp

xA ,xB

µp = xA µA + xB µB

σ 2 = x2 σ 2 + x2 σ 2 + 2xA xB σ AB

p AA BB

1 = xA + xB

After various substitutions, the above problem can be reduced to

xA (µA ’ rf ) + (1 ’ xA )(µB ’ rf )

max .

1/2

x

(x2 σ 2 + (1 ’ xA )2 σ 2 + 2xA (1 ’ xA )σ AB )

A

AA B

9

This is a straightforward, albeit very tedious, calculus problem and the solution can

be shown to be

(µA ’ rf )σ 2 ’ (µB ’ rf )σ AB

B

T

, xT = 1 ’ xT .

xA =

(µA ’ rf )σ B + (µB ’ rf )σ A ’ (µA ’ rf + µB ’ rf )σ AB B A

2 2

For the example data using rf = 0.03, we get xT = 0.542 and xT = 0.458. The

A B

expected return on the tangency portfolio is

µT = xT µA + xT µB

A B

= (0.542)(0.175) + (0.458)(0.055) = 0.110,

the variance of the tangency portfolio is

³ ´ ³ ´

T2 T2

σ2 σ2 σ 2 + 2xT xT σ AB

= xA + xB

T A B AB

= (0.542)2 (0.067) + (0.458)2 (0.013) + 2(0.542)(0.458) = 0.015,

and the standard deviation of the tangency portfolio is

q √

2

σ T = σ T = 0.015 = 0.124.

The e¬cient portfolios now are combinations of the tangency portfolio and the

T-bill. This important result is known as the mutual fund separation theorem. The

tangency portfolio can be considered as a mutual fund of the two risky assets, where

the shares of the two assets in the mutual fund are determined by the tangency

portfolio weights, and the T-bill can be considered as a mutual fund of risk free

assets. The expected return-risk trade-o¬ of these portfolios is given by the line

connecting the risk-free rate to the tangency point on the e¬cient frontier of risky

asset only portfolios. Which combination of the tangency portfolio and the T-bill

an investor will choose depends on the investor s risk preferences. If the investor is

very risk averse, then she will choose a combination with very little weight in the

tangency portfolio and a lot of weight in the T-bill. This will produce a portfolio

with an expected return close to the risk free rate and a variance that is close to zero.

For example, a highly risk averse investor may choose to put 10% of her wealth in

the tangency portfolio and 90% in the T-bill. Then she will hold (10%) — (54.2%) =

5.42% of her wealth in asset A, (10%) — (45.8%) = 4.58% of her wealth in asset B

and 90% of her wealth in the T-bill. The expected return on this portfolio is

µp = rf + 0.10(µT ’ rf )

= 0.03 + 0.10(0.110 ’ 0.03)

= 0.038.

and the standard deviation is

σ p = 0.10σ T

= 0.10(0.124)

= 0.012.

10

A very risk tolerant investor may actually borrow at the risk free rate and use these

funds to leverage her investment in the tangency portfolio. For example, suppose the

risk tolerant investor borrows 10% of her wealth at the risk free rate and uses the

proceed to purchase 110% of her wealth in the tangency portfolio. Then she would

hold (110%)—(54.2%) = 59.62% of her wealth in asset A, (110%)—(45.8%) = 50.38%

in asset B and she would owe 10% of her wealth to her lender. The expected return

and standard deviation on this portfolio is

µp = 0.03 + 1.1(0.110 ’ 0.03) = 0.118

σ p = 1.1(0.124) = 0.136.

2 E¬cient Portfolios and Value-at-Risk

As we have seen, e¬cient portfolios are those portfolios that have the highest expected

return for a given level of risk as measured by portfolio standard deviation. For

portfolios with expected returns above the T-bill rate, e¬cient portfolios can also be

characterized as those portfolios that have minimum risk (as measured by portfolio

standard deviation) for a given target expected return.

11

Efficient Portfolios

0.250

Efficient portfolios of T-

bills and assets A and B

0.200

Asset A

0.150

Tangency

Portfolio ER

Portfolio

0.103 Combinations of tangency portfolio

and T-bills that has the same SD as

0.100

asset B

0.055 Asset B

0.050

Combinations of tangency

rf portfolio and T-bills that has

same ER as asset B

0.000 0.039 0.114

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

Portfolio SD

Figure 5

To illustrate, consider & gure 5 which shows the portfolio frontier for two risky

assets and the e¬cient frontier for two risky assets plus a risk-free asset. Suppose

an investor initially holds all of his wealth in asset A. The expected return on this

portfolio is µB = 0.055 and the standard deviation (risk) is σ B = 0.115. An e¬cient

portfolio (combinations of the tangency portfolio and T-bills) that has the same

standard deviation (risk) as asset B is given by the portfolio on the e¬cient frontier

that is directly above σ B = 0.115. To & the shares in the tangency portfolio and

nd

T-bills in this portfolio recall from (xx) that the standard deviation of a portfolio with

xT invested in the tangency portfolio and 1 ’ xT invested in T-bills is σ p = xT σ T .

Since we want to & the e¬cient portfolio with σ p = σ B = 0.115, we solve

nd

σB 0.115

= 0.917, xf = 1 ’ xT = 0.083.

xT = =

σT 0.124

That is, if we invest 91.7% of our wealth in the tangency portfolio and 8.3% in T-bills

we will have a portfolio with the same standard deviation as asset B. Since this is an

e¬cient portfolio, the expected return should be higher than the expected return on

12

asset B. Indeed it is since

µp = rf + xT (µT ’ rf )

= 0.03 + 0.917(0.110 ’ 0.03)

= 0.103

Notice that by diversifying our holding into assets A, B and T-bills we can obtain a

portfolio with the same risk as asset B but with almost twice the expected return!

Next, consider & nding an e¬cient portfolio that has the same expected return

as asset B. Visually, this involves &nding the combination of the tangency portfo-

lio and T-bills that corresponds with the intersection of a horizontal line with in-

tercept µB = 0.055 and the line representing e¬cient combinations of T-bills and

the tangency portfolio. To & the shares in the tangency portfolio and T-bills in

nd

this portfolio recall from (xx) that the expected return of a portfolio with xT in-

vested in the tangency portfolio and 1 ’ xT invested in T-bills has expected return

equal to µp = rf + xT (µT ’ rf ). Since we want to & the e¬cient portfolio with

nd

µp = µB = 0.055 we use the relation

µp ’ rf = xT (µT ’ rF )

and solve for xT and xf = 1 ’ xT

µp ’ rf 0.055 ’ 0.03

= 0.313, xf = 1 ’ xT = 0.687.

xT = =

µT ’ rf 0.110 ’ 0.03

That is, if we invest 31.3% of wealth in the tangency portfolio and 68.7% of our

wealth in T-bills we have a portfolio with the same expected return as asset B. Since

this is an e¬cient portfolio, the standard deviation (risk) of this portfolio should be

lower than the standard deviation on asset B. Indeed it is since

σ p = xT σ T

= 0.313(0.124)

= 0.039.

Notice how large the risk reduction is by forming an e¬cient portfolio. The standard

deviation on the e¬cient portfolio is almost three times smaller than the standard

deviation of asset B!

The above example illustrates two ways to interpret the bene& from forming

ts

e¬cient portfolios. Starting from some benchmark portfolio, we can & standard de-

x

viation (risk) at the value for the benchmark and then determine the gain in expected

return from forming a diversi& portfolio2 . The gain in expected return has concrete

ed

2

The gain in expected return by investing in an e¬cient portfolio abstracts from the costs asso-

ciated with selling the benchmark portfolio and buying the e¬cient portfolio.

13

meaning. Alternatively, we can & expected return at the value for the benchmark

x

and then determine the reduction in standard deviation (risk) from forming a diver-

si& portfolio. The meaning to an investor of the reduction in standard deviation

ed

is not as clear as the meaning to an investor of the increase in expected return. It

would be helpful if the risk reduction bene& can be translated into a number that is

t

more interpretable than the standard deviation. The concept of Value-at-Risk (VaR)

provides such a translation.

Recall, the VaR of an investment is the expected loss in investment value over a

given horizon with a stated probability. For example, consider an investor who invests

W0 = $100, 000 in asset B over the next year. Assume that RB represents the annual

(continuously compounded) return on asset B and that RB ˜N(0.055, (0.114)2 ). The

5% annual VaR of this investment is the loss that would occur if return on asset B is

equal to the 5% left tail quantile of the normal distribution of RB . The 5% quantile,

q0.05 is determined by solving

Pr(RB ¤ q0.05 ) = 0.05.

Using the inverse cdf for a normal random variable with mean 0.055 and standard

deviation 0.114 it can be shown that q0.05 = ’0.133.That is, with 5% probability the

return on asset B will be ’13.3% or less. If RB = ’0.133 then the loss in portfolio

value3 , which is the 5% VaR, is

loss in portfolio value = V aR = |W0 · (eq0.05 ’ 1)| = |$100, 000(e’0.133 ’ 1)| = $12, 413.

To reiterate, if the investor hold $100,000 in asset B over the next year then the 5%

VaR on the portfolio is $12, 413. This is the loss that would occur with 5% probability.

Now suppose the investor chooses to hold an e¬cient portfolio with the same

expected return as asset B. This portfolio consists of 31.3% in the tangency portfolio

and 68.7% in T-bills and has a standard deviation equal to 0.039. Let Rp denote the

annual return on this portfolio and assume that Rp ˜N (0.055, 0.039). Using the inverse

cdf for this normal distribution, the 5% quantile can be shown to be q0.05 = ’0.009.

That is, with 5% probability the return on the e¬cient portfolio will be ’0.9% or

less. This is considerably smaller than the 5% quantile of the distribution of asset B.

If Rp = ’0.009 the loss in portfolio value (5% VaR) is

loss in portfolio value = V aR = |W0 · (eq0.05 ’ 1)| = |$100, 000(e’0.009 ’ 1)| = $892.

Notice that the 5% VaR for the e¬cient portfolio is almost &fteen times smaller than

the 5% VaR of the investment in asset B. Since VaR translates risk into a dollar &gure

it is more interpretable than standard deviation.

3

To compute the VaR we need to convert the continuous compounded return (quantile) to a

simple return (quantile). Recall, if Rc is a continuously compounded return and Rt is a somple

tc

return then Rc = ln(1 + Rt ) and Rt = eRt ’ 1.

t

14

3 Further Reading

The classic text on portfolio optimization is Markowitz (1954). Good intermediate

level treatments are given in Benninga (2000), Bodie, Kane and Marcus (1999) and

Elton and Gruber (1995). An interesting recent treatment with an emphasis on

statistical properties is Michaud (1998). Many practical results can be found in the

Financial Analysts Journal and the Journal of Portfolio Management. An excellent

overview of value at risk is given in Jorian (1997).

4 Appendix Review of Optimization and Con-

strained Optimization

Consider the function of a single variable

y = f (x) = x2

which is illustrated in Figure xxx. Clearly the minimum of this function occurs at

the point x = 0. Using calculus, we & the minimum by solving

nd

min y = x2 .

x

The & order (necessary) condition for a minimum is

rst

d d2

0= f (x) = x = 2x

dx dx

and solving for x gives x = 0. The second order condition for a minimum is

d2

0< f (x)

dx

and this condition is clearly satis& for f (x) = x2 .

ed

Next, consider the function of two variables

y = f(x, z) = x2 + z 2 (6)

which is illustrated in Figure xxx.

15

y = x^2 + z^2

8

7

6

5

y4

3

2

1.75

1 1

0.25

z

0 -0.5

-2

-1.75

-1.5

-1.25

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

-2

0.75

1

1.25

1.5

1.75

2

x

Figure 6

This function looks like a salad bowl whose bottom is at x = 0 and z = 0. To &nd

the minimum of (6), we solve

min y = x2 + z 2

x,z

and the & order necessary conditions are

rst

‚y

0= = 2x

‚x

and

‚y

0= = 2z.

‚z

Solving these two equations gives x = 0 and z = 0.

Now suppose we want to minimize (6) subject to the linear constraint

(7)

x + z = 1.

The minimization problem is now a constrained minimization

min y = x2 + z 2 subject to (s.t.)

x,z

x+z = 1

16

and is illustrated in Figure xxx. Given the constraint x + z = 1, the function (6) is

no longer minimized at the point (x, z) = (0, 0) because this point does not satisfy

x + z = 1. The One simple way to solve this problem is to substitute the restriction

(7) into the function (6) and reduce the problem to a minimization over one variable.

To illustrate, use the restriction (7) to solve for z as

z = 1 ’ x. (8)

Now substitute (7) into (6) giving

y = f(x, z) = f (x, 1 ’ x) = x2 + (1 ’ x)2 . (9)

The function (9) satis& the restriction (7) by construction. The constrained mini-

es

mization problem now becomes

min y = x2 + (1 ’ x)2 .

x

The & order conditions for a minimum are

rst

d2

(x + (1 ’ x)2 ) = 2x ’ 2(1 ’ x) = 4x ’ 2

0=

dx

and solving for x gives x = 1/2. To solve for z, use (8) to give z = 1 ’ (1/2) = 1/2.

Hence, the solution to the constrained minimization problem is (x, z) = (1/2, 1/2).

Another way to solve the constrained minimization is to use the method of La-

grange multipliers. This method augments the function to be minimized with a linear

function of the constraint in homogeneous form. The constraint (7) in homogenous

form is

x+z’1=0

The augmented function to be minimized is called the Lagrangian and is given by

L(x, z, ») = x2 + z 2 ’ »(x + z ’ 1).

The coe¬cient on the constraint in homogeneous form, », is called the Lagrange

multiplier. It measures the cost, or shadow price, of imposing the constraint relative

to the unconstrained problem. The constrained minimization problem to be solved

is now

min L(x, z, ») = x2 + z 2 + »(x + z ’ 1).

x,z,»

The & order conditions for a minimum are

rst

‚L(x, z, »)

0= = 2x + »

‚x

‚L(x, z, »)

0= = 2z + »

‚z

‚L(x, z, »)

=x+z’1

0=

‚»

17

The & order conditions give three linear equations in three unknowns. Notice that

rst

the & order condition with respect to » imposes the constraint. The & two

rst rst

conditions give

2x = 2z = ’»

or

x = z.

Substituting x = z into the third condition gives

2z ’ 1 = 0

or

z = 1/2.

The & solution is (x, y, ») = (1/2, 1/2, ’1).

nal

The Lagrange multiplier, », measures the marginal cost, in terms of the value of

the objective function, of imposing the constraint. Here, » = ’1 which indicates

that imposing the constraint x + z = 1 reduces the objective function. To understand

the roll of the Lagrange multiplier better, consider imposing the constraint x + z =

0. Notice that the unconstrained minimum achieved at x = 0, z = 0 satis& this es

constraint. Hence, imposing x + z = 0 does not cost anything and so the Lagrange

multiplier associated with this constraint should be zero. To con& this, the we

rm

solve the problem

min L(x, z, ») = x2 + z 2 + »(x + z ’ 0).

x,z,»

The & order conditions for a minimum are

rst

‚L(x, z, »)

= 2x ’ »

0=

‚x

‚L(x, z, »)

= 2z ’ »

0=

‚z

‚L(x, z, »)

0= = x+z

‚»

The & two conditions give

rst

2x = 2z = ’»

or

x = z.

Substituting x = z into the third condition gives

2z = 0

or

z = 0.

The & solution is (x, y, ») = (0, 0, 0). Notice that the Lagrange multiplier, », is

nal

equal to zero in this case.

18

5 Problems

Exercise 1 Consider the problem of investing in two risky assets A and B and a

risk-free asset (T-bill). The optimization problem to & the tangency portfolio may

nd

be reduced to

xA (µA ’ rf ) + (1 ’ xA )(µB ’ rf )

max 1/2

xA

(x2 σ 2 + (1 ’ xA )2 σ 2 + 2xA (1 ’ xA )σ AB )

AA B

where xA is the share of wealth in asset A in the tangency portfolio and xB = 1 ’ xA

is the share of wealth in asset B in the tangency portfolio. Using simple calculus,

show that

(µA ’ rf )σ 2 ’ (µB ’ rf )σ AB

B

xA = .

(µA ’ rf )σ 2 + (µB ’ rf )σ 2 ’ (µA ’ rf + µB ’ rf )σ AB

B A

References

[1] Benninga, S. (2000), Financial Modeling, Second Edition. Cambridge, MA: MIT

Press.

[2] Bodie, Kane and Marcus (199x), Investments, xxx Edition.

[3] Elton, E. and G. Gruber (1995). Modern Portfolio Theory and Investment Anal-

ysis, Fifth Edition. New York: Wiley.

[4] Jorian, P. (1997). Value at Risk. New York: McGraw-Hill.

[5] Markowitz, H. (1987). Mean-Variance Analysis in Portfolio Choice and Capital

Markets. Cambridge, MA: Basil Blackwell.

[6] Markowitz, H. (1991). Portfolio Selection: E¬cient Diversi& cation of Invest-

ments. New York: Wiley, 1959; 2nd ed., Cambridge, MA: Basil Blackwell.

[7] Michaud, R.O. (1998). E¬cient Asset Management: A Practical Guide to

Stock Portfolio Optimization and Asset Allocation. Boston, MA:Harvard Business

School Press.

19