Introduction to Financial Econometrics
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:

ВµA = E[RA ], Пѓ 2 = V ar(RA ),
A
ВµB = E[RB ], Пѓ 2 = V ar(RB ),
B
Пѓ 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, ВµA and ВµB , 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 diп¬Ђerent from our expectations. The variances, Пѓ 2 and
A
2
Пѓ 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 Пѓ AB gives us information about the direction of any linear dependence
between returns. If Пѓ AB > 0 then the returns on assets A and B tend to move in the

1
same direction; if Пѓ AB < 0 the returns tend to move in opposite directions; if Пѓ AB = 0
then the returns tend to move independently. The strength of the dependence between
the returns is measured by the correlation coeп¬ѓcient ПЃAB = ПѓПѓAB . If ПЃAB is close to
A ПѓB
one in absolute value then returns mimic each other extremely closely whereas if ПЃAB
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)
Вµp = E[Rp ] = xA ВµA + xB ВµB
Пѓ 2 = var(Rp ) = x2 Пѓ 2 + x2 Пѓ 2 + 2xA xB Пѓ 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 ВµA + xB ВµB
by the linearity of the expectation operator. For the second result (3), we have
var(xA RA + xB RB ) = E[(xA RA + xB RB ) в€’ E[xA RA + xB RB ])2 ]
var(Rp ) =
E[(xA (RA в€’ ВµA ) + xB (RB в€’ ВµB ))2 ]
=
E[x2 (RA в€’ ВµA )2 + x2 (RB в€’ ВµB )2 + 2xA xB (RA в€’ ВµA )(RB в€’ ВµB )]
= A B
x2 E[(RA в€’ ВµA )2 ] + x2 E[(RB в€’ ВµB )2 ] + 2xA xB E[(RA в€’ ВµA )(RB в€’ ВµB )],
= 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 Eп¬ѓcient portfolios with two risky assets
In this section we describe how mean-variance eп¬ѓcient portfolios are constructed.
First we make some assumptions:
Assumptions

вЂў Returns are jointly normally distributed. This implies that means, variances
and covariances of returns completely characterize the joint distribution of re-
turns.

вЂў 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 eп¬ѓcient 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
Пѓ2Пѓ2
ВµA ВµB ПѓA ПѓB Пѓ AB ПЃAB
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, Вµp , on the vertical axis and portfolio
standard-deviation, Пѓ 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 в€’ xA , the weight on asset is
then varies from 1.4 to -0.4. This gives us 18 portfolios with weights (xA , xB ) =
(в€’0.4, 1.4), (в€’0.3, 1.3), ..., (1.3, в€’0.3), (1.4, в€’0.4). For each of these portfolios we use
q
the formulas (2) and (3) to compute Вµp and Пѓ p = Пѓ 2 . We then plot these values1 .
p
Notice that the plot in (Вµp , Пѓ 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 Пѓ 2 = x2 Пѓ 2 + x2 Пѓ 2 + 2xA xB Пѓ 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 в€’ xA into the formula for Пѓ 2 reduces the problem to
p

min Пѓ 2 = x2 Пѓ 2 + (1 в€’ xA )2 Пѓ 2 + 2xA (1 в€’ xA )Пѓ AB .
p AA B
xA

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

dПѓ 2
p
= 2xmin Пѓ 2 в€’ 2(1 в€’ xmin )Пѓ 2 + 2Пѓ AB (1 в€’ 2xmin )
0= A A A B A
dxA
and straightforward calculations yield

Пѓ 2 в€’ Пѓ AB
= 2 B2
xmin , xmin = 1 в€’ xmin . (4)
A
Пѓ A + Пѓ B в€’ 2Пѓ AB B A

For our example, using the data in table 1, we get xmin = 0.2 and xmin = 0.8.
A B
Eп¬ѓcient portfolios are those with the highest expected return for a given level
of risk. Ineп¬ѓcient portfolios are then portfolios such that there is another feasible
portfolio that has the same risk (Пѓ p ) but a higher expected return (Вµp ). From the
plot it is clear that the ineп¬ѓcient portfolios are the feasible portfolios that lie below
the global minimum variance portfolio and the eп¬ѓcient 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 ПЃAB 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 ПЃAB approaches zero the set starts to bow toward
the Вµp axis and the power of diversi& cation starts to kick in. If ПЃAB = в€’1 then
the set actually touches the Вµp 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 Пѓ 2 = 0 when
nd p
ПЃAB = в€’1 we use (4) and the fact that Пѓ AB = ПЃAB Пѓ A Пѓ B to give
ПѓB
xmin = , xmin = 1 в€’ xA
A
ПѓA + ПѓB B
The case with ПЃAB = в€’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 eп¬ѓcient set of portfolios, which portfolio will an investor choose? Of
the eп¬ѓcient 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 Eп¬ѓcient portfolios with a risk-free asset
In the preceding section we constructed the eп¬ѓcient 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 Eп¬ѓcient 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
Вµf = E[rf ] = rf

6
var(rf ) = 0
cov(RB , rf ) = 0

Let xB denote the share of wealth in asset B and xf = 1 в€’ xB denote the share of
wealth in T-bills. The portfolio expected return is

Rp = xB RB + (1 в€’ xB )rf
= xB (RB в€’ rf ) + rf

The quantity RB в€’ rf is called the excess return (over the return on T-bills) on asset
B. The portfolio expected return is then

Вµp = xB (ВµB в€’ rf ) + rt

where the quantity (ВµB в€’ 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
Вµp в€’ rf = xB (ВµB в€’ 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

Пѓ 2 = x2 Пѓ 2 .
p BB

The portfolio standard deviation is therefore proportional to the standard deviation
on asset B:
Пѓ p = xB Пѓ B
which can use to solve for xB
Пѓp
xB =
ПѓB
Using the last result, the feasible (and eп¬ѓcient) set of portfolios follows the equation
ВµB в€’ rf
В· Пѓp (5)
Вµp = rf +
ПѓB
Вµ в€’r
which is simply straight line in (Вµp , Пѓ p ) with intercept rf and slope B B f . The slope
Пѓ
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 oп¬Ђ 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:
ВµA в€’ rf 0.175 в€’ 0.03 Вµ в€’ rf 0.055 в€’ 0.03
= 0.562, B
= = = 0.217.
ПѓA 0.258 ПѓB 0.115
Hence, portfolios of asset A and T-bills are eп¬ѓcient relative to portfolios of asset B
and T-bills.

1.3.2 Eп¬ѓcient 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 eп¬ѓcient set in this case will still be a straight line in
(Вµp , Пѓ p )в€’ space with intercept rf . The slope of the eп¬ѓcient set, the maximum Sharpe
ratio, is such that it is tangent to the eп¬ѓcient 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
Вµ в€’r
If we invest in only in asset B and T-bills then the Sharpe ratio is B B f = 0.217
Пѓ
and the CAL intersects the parabola at point B. This is clearly not the eп¬ѓcient set
of portfolios. For example, we could do uniformly better if we instead invest only
Вµ в€’r
in asset A and T-bills. This gives us a Sharpe ratio of A A f = 0.562 and the new
Пѓ
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
Вµp в€’ rf
max s.t.
Пѓp
xA ,xB

Вµp = xA ВµA + xB ВµB
Пѓ 2 = x2 Пѓ 2 + x2 Пѓ 2 + 2xA xB Пѓ AB
p AA BB
1 = xA + xB

After various substitutions, the above problem can be reduced to
xA (ВµA в€’ rf ) + (1 в€’ xA )(ВµB в€’ rf )
max .
1/2
x
(x2 Пѓ 2 + (1 в€’ xA )2 Пѓ 2 + 2xA (1 в€’ xA )Пѓ AB )
A
AA B

9
This is a straightforward, albeit very tedious, calculus problem and the solution can
be shown to be
(ВµA в€’ rf )Пѓ 2 в€’ (ВµB в€’ rf )Пѓ AB
B
T
, xT = 1 в€’ xT .
xA =
(ВµA в€’ rf )Пѓ B + (ВµB в€’ rf )Пѓ A в€’ (ВµA в€’ rf + ВµB в€’ rf )Пѓ 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
ВµT = xT ВµA + xT ВµB
A B
= (0.542)(0.175) + (0.458)(0.055) = 0.110,
the variance of the tangency portfolio is
Ві Вґ Ві Вґ
T2 T2
Пѓ2 Пѓ2 Пѓ 2 + 2xT xT Пѓ 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 в€љ
2
Пѓ T = Пѓ T = 0.015 = 0.124.
The eп¬ѓcient 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-oп¬Ђ of these portfolios is given by the line
connecting the risk-free rate to the tangency point on the eп¬ѓcient 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%) Г— (54.2%) =
5.42% of her wealth in asset A, (10%) Г— (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
Вµp = rf + 0.10(ВµT в€’ rf )
= 0.03 + 0.10(0.110 в€’ 0.03)
= 0.038.
and the standard deviation is
Пѓ p = 0.10Пѓ 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%)Г—(54.2%) = 59.62% of her wealth in asset A, (110%)Г—(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

Вµp = 0.03 + 1.1(0.110 в€’ 0.03) = 0.118
Пѓ p = 1.1(0.124) = 0.136.

2 Eп¬ѓcient Portfolios and Value-at-Risk
As we have seen, eп¬ѓcient 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, eп¬ѓcient 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 eп¬ѓcient 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 ВµB = 0.055 and the standard deviation (risk) is Пѓ B = 0.115. An eп¬ѓcient
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 eп¬ѓcient frontier
that is directly above Пѓ 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 в€’ xT invested in T-bills is Пѓ p = xT Пѓ T .
Since we want to & the eп¬ѓcient portfolio with Пѓ p = Пѓ B = 0.115, we solve
nd

ПѓB 0.115
= 0.917, xf = 1 в€’ xT = 0.083.
xT = =
ПѓT 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
eп¬ѓcient portfolio, the expected return should be higher than the expected return on

12
asset B. Indeed it is since

Вµp = rf + xT (ВµT в€’ rf )
= 0.03 + 0.917(0.110 в€’ 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 eп¬ѓcient 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 ВµB = 0.055 and the line representing eп¬ѓcient 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 в€’ xT invested in T-bills has expected return
equal to Вµp = rf + xT (ВµT в€’ rf ). Since we want to & the eп¬ѓcient portfolio with
nd
Вµp = ВµB = 0.055 we use the relation

Вµp в€’ rf = xT (ВµT в€’ rF )

and solve for xT and xf = 1 в€’ xT
Вµp в€’ rf 0.055 в€’ 0.03
= 0.313, xf = 1 в€’ xT = 0.687.
xT = =
ВµT в€’ rf 0.110 в€’ 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 eп¬ѓcient portfolio, the standard deviation (risk) of this portfolio should be
lower than the standard deviation on asset B. Indeed it is since

Пѓ p = xT Пѓ T
= 0.313(0.124)
= 0.039.

Notice how large the risk reduction is by forming an eп¬ѓcient portfolio. The standard
deviation on the eп¬ѓcient 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
eп¬ѓcient 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 eп¬ѓcient portfolio abstracts from the costs asso-
ciated with selling the benchmark portfolio and buying the eп¬ѓcient 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 в‰¤ 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 = в€’0.133.That is, with 5% probability the
return on asset B will be в€’13.3% or less. If RB = в€’0.133 then the loss in portfolio
value3 , which is the 5% VaR, is

loss in portfolio value = V aR = |W0 В· (eq0.05 в€’ 1)| = |\$100, 000(eв€’0.133 в€’ 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 eп¬ѓcient 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 = в€’0.009.
That is, with 5% probability the return on the eп¬ѓcient portfolio will be в€’0.9% or
less. This is considerably smaller than the 5% quantile of the distribution of asset B.
If Rp = в€’0.009 the loss in portfolio value (5% VaR) is

loss in portfolio value = V aR = |W0 В· (eq0.05 в€’ 1)| = |\$100, 000(eв€’0.009 в€’ 1)| = \$892.

Notice that the 5% VaR for the eп¬ѓcient 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 в€’ 1.
t

14
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
в€‚y
0= = 2x
в€‚x
and
в€‚y
0= = 2z.
в€‚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 в€’ x. (8)

Now substitute (7) into (6) giving

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

The function (9) satis& the restriction (7) by construction. The constrained mini-
es
mization problem now becomes

min y = x2 + (1 в€’ x)2 .
x

The & order conditions for a minimum are
rst
d2
(x + (1 в€’ x)2 ) = 2x в€’ 2(1 в€’ x) = 4x в€’ 2
0=
dx
and solving for x gives x = 1/2. To solve for z, use (8) to give z = 1 в€’ (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+zв€’1=0
The augmented function to be minimized is called the Lagrangian and is given by

L(x, z, О») = x2 + z 2 в€’ О»(x + z в€’ 1).

The coeп¬ѓcient on the constraint in homogeneous form, О», 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, О») = x2 + z 2 + О»(x + z в€’ 1).
x,z,О»

The & order conditions for a minimum are
rst
в€‚L(x, z, О»)
0= = 2x + О»
в€‚x
в€‚L(x, z, О»)
0= = 2z + О»
в€‚z
в€‚L(x, z, О»)
=x+zв€’1
0=
в€‚О»
17
The & order conditions give three linear equations in three unknowns. Notice that
rst
the & order condition with respect to О» imposes the constraint. The & two
rst rst
conditions give
2x = 2z = в€’О»
or
x = z.
Substituting x = z into the third condition gives
2z в€’ 1 = 0
or
z = 1/2.
The & solution is (x, y, О») = (1/2, 1/2, в€’1).
nal
The Lagrange multiplier, О», measures the marginal cost, in terms of the value of
the objective function, of imposing the constraint. Here, О» = в€’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, О») = x2 + z 2 + О»(x + z в€’ 0).
x,z,О»

The & order conditions for a minimum are
rst
в€‚L(x, z, О»)
= 2x в€’ О»
0=
в€‚x
в€‚L(x, z, О»)
= 2z в€’ О»
0=
в€‚z
в€‚L(x, z, О»)
0= = x+z
в€‚О»
The & two conditions give
rst
2x = 2z = в€’О»
or
x = z.
Substituting x = z into the third condition gives
2z = 0
or
z = 0.
The & solution is (x, y, О») = (0, 0, 0). Notice that the Lagrange multiplier, О», 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 (ВµA в€’ rf ) + (1 в€’ xA )(ВµB в€’ rf )
max 1/2
xA
(x2 Пѓ 2 + (1 в€’ xA )2 Пѓ 2 + 2xA (1 в€’ xA )Пѓ AB )
AA B

where xA is the share of wealth in asset A in the tangency portfolio and xB = 1 в€’ xA
is the share of wealth in asset B in the tangency portfolio. Using simple calculus,
show that
(ВµA в€’ rf )Пѓ 2 в€’ (ВµB в€’ rf )Пѓ AB
B
xA = .
(ВµA в€’ rf )Пѓ 2 + (ВµB в€’ rf )Пѓ 2 в€’ (ВµA в€’ rf + ВµB в€’ rf )Пѓ AB
B A

References
 Benninga, S. (2000), Financial Modeling, Second Edition. Cambridge, MA: MIT
Press.

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

 Elton, E. and G. Gruber (1995). Modern Portfolio Theory and Investment Anal-
ysis, Fifth Edition. New York: Wiley.

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

 Markowitz, H. (1987). Mean-Variance Analysis in Portfolio Choice and Capital
Markets. Cambridge, MA: Basil Blackwell.

 Markowitz, H. (1991). Portfolio Selection: Eп¬ѓcient Diversi& cation of Invest-
ments. New York: Wiley, 1959; 2nd ed., Cambridge, MA: Basil Blackwell.

 Michaud, R.O. (1998). Eп¬ѓcient Asset Management: A Practical Guide to
Stock Portfolio Optimization and Asset Allocation. Boston, MA:Harvard Business
School Press.

19