Download Portfolio Theory Lecture

Returns and Expected
Returns
1. Holding period return
Pt
Rt
CFt
R1
= Price of asset at time t
= % return from time t-1 to t
= cash flow from time t-1 to t
- e.g. dividend
P1  P0  CF1 P1  P0 CF1


P0
P0
P0
=
= (%capital gain) + (%dividend yield)
QUESTION: How can you measure the return you expect
from an asset?
The best guess at the future return i.e., what one should
expect is the mean return:

n
R   Ri / n
i 1
where Ri is an observation of the variable (return) and

R
is the arithmetic sample mean of the variable (return)
EXAMPLE OF
ARITHMETIC MEAN
P0
P1
= 100
= 200
- price now
- price at time 1 period from now
R1
= (200 - 100)/100
= 1.00 or 100%
Suppose P2
= 100 - period 2 price
R2
= ((100 - 200)/200
= -.5 or -50%
Arithmetic mean return
= (R1 + R2 )/2 = .25 or 25%
QUESTION: Is an average return of 25% a good return? Is it
accurate?
Geometric return (also called compound return) especially
when returns vary a lot from period to period. Geometric
mean is always less than or equal to the arithmetic mean.
n
G
=
 (1  Ri ) 1/ n  1
i 1
G
Ri

= is the geometric mean return
= the return in period i
= the product operator
G
= (1 + 1.0)1/2 x (1 + (-.5))1/2 - 1
= [(2)(.5)]1/2 - 1
=0
QUESTION:
What is risk?
One Answer: The likelihood that you will not receive what you
expect - i.e. risk free means that you always get
what you expect
MEASURES - Variance and Standard Deviation

n
sample variance =
 ( Ri  R)2
   i 1
n
sample standard deviation = 
1.
When variables are normally distributed, the mean and
variance are enough to describe the whole distribution
because higher order moments are simply functions of
the mean and variance.
2.
This will allow us to use only means and variances later
in this lecture to define investment opportunities available
to investors.
-3
-2
-1
Mean

+1 
+2
+3


For the normal distribution, one, two and three standard deviations about the
mean delineate where observations of a variable should fall 68, 95 and 99
percent of the time, respectively.
Ibbotson Sinquifield Data –
1926-2005
1.
Often, people use these measures to select among
investments
Stocks
Large Cos.
Small Cos.
Geometric
Mean
10.4
12.7
Arithmetic
Mean
12.4
17.5
Standard
Deviation
20.3
33.1
Bonds
Long Corp.
Long Treas.
Med. Treas.
Tbills
5.9
5.4
5.4
3.7
6.2
5.8
5.5
3.8
8.6
9.3
5.7
3.1
QUESTION: Compare the Risk/Return tradeoff of Bonds
to stocks in general.
Compare Long Corporate Bonds to Long Treasury
Bonds. -anything unusual here?
SMALL STOCKS
mean=18%
standard deviation=35%
-87% -52% -17%
-48
-28
-8
18% 53%
12
32
88% 123%
52
72
LARGE STOCKS
mean=12%
standard deviation=20%
LONGTERM U.S.
GOVERNMENT BONDS
mean=5%
standard deviation=9%
-22
-13
-4
5
14
23
32
U.S. TREASURY BILLS
mean=4%
standard deviation=3%
-5 -2 1
4 7
10 13
There are considerable differences among return distributions for these common asset types.
Expectations (means),
Variances and Covariances
1.
In the real world, people typically use sample data,
however, the idea of a sample mean or variance is based
upon probability theory where one assumes each data
observation is equally probable.
2.
If we had all potential outcomes and the probabilities of
each one, then we could calculate expectations and
variances with probabilities. For example,
~
~
N
mean( X )    E ( X )   piXi
i 1
~
~
N
~
Var( X )    E[( Xi  E ( X )) ]   pi ( Xi  E ( X )) 2
2
2
i 1
3. Many finance models involve the derivation of expectations,
variances, and covariances, hence, it is important to
have the ability to derive them for various cases. Some
important cases are listed below with their derivations.
4. Important Property: Expectation is a linear function of
random variables, which means we can take the function
through multiplicative constants and across additive
random variables. In general, we cannot take the
expectation through a product of random variables
unless the variables are independently distributed.
5. These properties imply the following for random variables X
and Y and constants a and b:
A.
E(X + a) = E(X) + a
B.
E(aX) = aE(X)
C.
Var(X + a) = E[((X + a) – E(X + a))2] = Var(X)
D.
Var(aX) = E[(aX – aE(X))2] = a2Var(X)
E.
Cov(X, Y) = Xy = E[(X - E(X))(Y - E(Y))]
= E(XY) - E(X)E(Y)
or restated as
E(XY) = Cov(X, Y) + E(X)E(Y)
6. The two asset case. The return on a portfolio composed of
a% of a stock earning X and b%=(1-a%) of a stock
earning Y we have
•
E(Rp) = E[aX + bY] = aE(X) + bE(Y)
•
VAR(Rp) = E[(aX – aE(X) + bY – bE(Y))2]
= a2Var(X) + b2Var(Y) + 2abE[(X – E(X))(Y – E(Y))]
= a2Var(X) + b2Var(Y) + 2abCov(X, Y)
= a2X2 + b2y2 + 2abXy
Correlationxy = rxy = Xy / X y
Some Properties of
Expectations Useful in
Finance
1. Stein’s Lemma - assume random variables X and Y follow a
bivariate normal distribution, ƒ is a linear or non-linear
function, and ƒy is the partial derivative of ƒ with respect to y,
then
Cov[X, ƒ(Y)] = E[ƒy(Y)]Cov(X,Y)
Example: Maxβ E{U[fee(β)]}
First Order is: 0 = E{U’ * ∂fee/∂β}
Subst. for E[Y*X]
0 = E[U’] * E[∂fee/∂β] + Cov(U’, ∂fee/∂β)
0 = E[U’] * E[∂fee/∂β] + E[U’’]Cov(fee(β), ∂fee/∂β)
2. Multivariate Generalization - assume random variables X, Y
and Z follow a multivariate normal distribution, then
Cov[X, ƒ(Y, Z)] = E[ƒy(Y, Z)]Cov(X,Y) + E[ƒZ(Y, Z)]Cov(X, Z)
3. Unconditional (E(Y)) and Conditional Expectations (E(Y|X) All expectations are based upon some information.
Unconditional expectations are often assumed to involve only
basic information of a distribution, say, past data on a random
variable or a “small” information set.
Conditional expectations involve additional information such
as a correlated variable, X.
4. Law of Iterated Expectations - expectations of expectations
E[E(Y|X)] = E(Y)
(1)
E[E(Y)|X] =E(Y)
(2)
5. Restatement of L.I.E. - when you take expectations of
expectations, the expectations conditional on the least
information of the two (unconditional expectations) prevails.
This is useful in cases where you must forecast the behavior
of others with a different information set or get some average
of a set of conditional expectations.
6. Concrete Example - Suppose you want to predict whether a
risk-neutral investor faced with the choice between buying
stock with return R or bonds with return B will buy stock today.
Assume that everyone knows that 60% of the time R > B. For
the two cases, (1) and (2), either the investor or you get a
signal X that says R < B for sure today :
Equation (1) says, if the investor gets the signal, then you
should still predict the investor will buy stock (the
unconditional expectation). Since you don’t have the
information, you go with the lesser informed prediction. Of
course, you will be wrong in this case but there was no way
for you to know this beforehand.
Equation (2) says, if the you get the signal then you predict
the investor will buy stock (again, the unconditional
expectation). Here, your prediction will be correct even though
you know that had the investor known what you knew, he
would not buy stock.
Example of Iterated Conditional
Expectations For Linearly
Related (or normal) Variables

Suppose Y and X have a bivariate normal distribution with
means µy and µx and variances σy2 and σx2 and correlation
coefficient ρ. Then conditional expectation of Y is linear in
X so that

E[Y|X] = a + bX

Which can be shown to equal something that looks like a
regression equation with β = ρσy/σx

E[Y|X] = µy + β(X - µx)

For iterated expectations, take the expectation again with
respect to X and get

E[E[Y|X]|X] = µy + β(E[X] - µx) = µy

Therefore, the iterated expectation is just the unconditional
expectation
Intermediate steps for
previous slide

E[Y|X] = a + bX

Integrating (taking expectations) over X gives

E[Y] = a + bE[X]
µ y = a + b µx





or
Now consider the expectation of the product
E[Y|X]X = aX + bX2
Integrating over X gives
E[YX] = aE[X] + bE[X2]

Using the definition of Cov[Y,X] = ρσyσx = E[YX] - µy µx and
E[X2] = σx2 + µx2 (see next few slides) then

ρσyσx + µy µx = a µx + b(σx2 + µx2 )

Use this and µy = a + b µx to solve for a and b as
a = µy – µxρσy/σx and b = ρσy/σx




E[Y|X] = µy + ρσy/σx (X - µx)
Rao-Blackwell Theorem –
Variance vs. Conditional
Variance

This theorem simply separates a variable’s unconditional variance
into that explained by a conditioning variable and the residual
variation that is left unexplained (also called the conditional
variance). Like above, assume that Y and X are bivariate normal.

σy2 = E[(Y - µy)2] = E{[(Y – E(Y|X)) + (E(Y|X) - µy)]2}
= E[Y – E(Y|X)]2 + E[E(Y|X) - µy)]2
+ 2E{[Y – E(Y|X)][E(Y|X) - µy]}
= E[Y – E(Y|X)]2 + E[E(Y|X) - µy]2 + 2{0}
= residual (conditional) variance + explained variance
From the previous slide E[Y|X] = µy + ρσy/σx (X - µx) so
For explained variance
E[E(Y|X) - µy]2 = E[ρσy/σx (X - µx)]2
= (ρσy/σx)2E(X - µx)2 = ρ2σy2
For residual (conditional) variance
E[Y – E(Y|X)]2 = σy2 - ρ2σy2 = (1 - ρ2)σy2
Finance Example: Shiller (1981) makes the point that stock price
is the conditional expectation of future dividends (D). He
develops a series of the actual discounted value of dividends
(call this V) over time for the market portfolio. He finds that
the variance of the market price (call this P) of the market
portfolio is 5 times as large as the variance of the series of
discounted dividends he constructs. Therefore, he says the
market is irrational because if the market price is an
expectation of the value of future dividends, V = E[P|(Div.
information)] + e, (e is an error term) then E[P|D] should have
a smaller variance than V. Using terms from above, the
explained variation (ρ2σy2) cannot be larger than the variation
in the variable we wish to explain (σy2) unless we irrationally
make the expectation very volatile by driving the expectation
too high (market bubbles) or too low (market crashes).
The Rao-Blackwell theorem relies on the statistical property of
conditional expectations, which is a forecast of Y based on some
information X. Here, you can think of a regression where the
conditioning variables in X are used to forecast Y, and the fitted values
from the regression are the conditional expectation of Y given the
values of X.
The variance of Y is then split into the part explained by the X
variables, and the residual variance, which is the (conditional)
variance left over after conditioning on the observed X values.
An optimal forecast has the property that the residuals (forecast
errors) and the forecasts are uncorrelated, hence, the third term in the
variance equation above drops out. This forms the basis of Shiller’s
point that the variance in Y is composed of two parts, the part
explained by the forecast (explained variance) and the error variance.
So how can the explained variance, the variance of the stock index
price (X variable) be five times the variance of the value of discounted
dividends (Y variable)? Mathematically, this can only happen if there is
a strong negative correlation between the forecast and the forecast
error so that the third term above is negative, not zero. A negative
covariance implies that investors drive the stock market too high in
good times and too low in bad times. That is, investors forecast very
large (small) dividends in good (bad) times that are systematically in
error. So in good times investors forecast a too large value for Y, and
the error in the forecast (Y – E(Y|X) is a large negative. In bad times,
they forecast a too small value for Y, and then the error is a large
positive, hence, the large negative correlation and covariance.
Because these errors are correlated with the forecasts, they cannot be
optimal and investor behavior is irrational – they overreact.
Properties of Some Useful
Random Variables
1.
Suppose X is a random variable (rv) with mean  and
variance 2 and Z = X – (X - )/, find E(Z) and Var(Z).
E(Z) = E(X) – (E(X) - )/ = E(X) – ( - )/ = E(X)
Var(Z) = 2 - 2Cov[X, (X - )/] + Var[(X - )/]
= 2 - 2Cov[X, (X/ - /)] + Var[X/ - /]
= 2 – 2(2/ ) + 1 =(1 - )2
2. Suppose X is an rv with mean  and b is a constant
different from , show E(X - )2 < E(X – b)2
E(X - b)2 = E[(X - ) + ( - b)]2 = E(X - )2 + ( - b)2
+ 2(u – b)E(X - )
= E(X - )2 + ( - b)2 > E(X - )2
3. Suppose X and Y are two rvs and a and b are constants,
show Var(X + a) = Var(X), Cov[(X+a),(Y+b)] = Cov(X, Y).
Var(X+a) = E[((X+a) – E(X+a))2] = E[(X+a -  -a)2]
= E[(X-)2] = E[(X – E(X))2] = Var(X)
Cov[(X+a),(Y+b)] = E[X+a – E(X)-a] E[Y+b – E(Y)-b]
=E[X - x] E[Y - y] = Cov(X, Y)
Assume the sample of rvs X1, X2, …, Xn are independently
distributed, each with mean  and variance 2. For i = 1,
2, …, n and i  j find:
1.
E(Xi2) = E[((Xi - ) + )2] = E [(Xi - )2 + 2(Xi - ) + 2]
= 2 + 2 + 0
_
2. E(Xi X) = E(Xi)(1/nXj) = 1/n E(Xj)E(Xj)
= 1/n(2 + 2 + … + 2 + (2 + 2))
last term is for i=j and from (1) above
= 1/n[(n-1)2 + (2 + 2)] = 1/n(2 ) + 2
_
2
_
3. E( X) = E[1/n Xi X ] = 1/n E[Xi X ]
= 1/n [1/n(2 ) + 2] = 1/n(2 ) + 2
_
_
4. E[Xi(Xi - X )] = E(Xi2) - E(Xi X)
= 2 + 2 – (1/n(2 ) + 2) = (1 – 1/n) 2
_
_
_
2
5. E[ X (Xi - X )] = E[Xi X] – E[ X ]
= [1/n(2 ) + 2] - [1/n(2 ) + 2] = 0
_
_
6. E[Xi(Xj - X )] = E(XiXj) - E(Xi X ) = 2 - [1/n(2 ) + 2]
= - 1/n(2 )
_
7.
Cov[Xi, X ] = Cov[ Xi, 1/nXj ] = (1/n)Cov[Xi, Xi]
+ (1/n) Cov[Xi, jiXj] = 1/n(2)
_
_
_
8. Var[Xi - X ] = Var[Xi] - 2 Cov[Xi, X] + Var[ X]
_
= 2 – 2(1/n)(2 ) + E[( X - )2]
= 2 – 2(1/n)(2 ) + E[(1/nXj - )2]
= 2 – 2(1/n)(2 ) + 1/n2E(Xj - )2]
= 2 – 2(1/n)(2 ) + 1/n2(n 2 )]
= 2 – 2(1/n)(2 ) + 1/n(2 ) = 2 (1 – 1/n)
_
_
_
_
_
9. Cov [(Xi - X ), X ] = Cov[Xi, X ] – Cov[ X , X ]
_
_
= Cov[Xi, X ] – Var[ X]
= 1/n(2) - 1/n(2 ) = 0
_
_
_
10. Cov [(Xi - X ), (Xj - X )] = Cov[Xi, Xj] - Cov[Xi, X ]
_
_
- Cov[Xi, X ] + Var[ X ]
= 0 - 1/n(2) - 1/n(2) + 1/n(2) = -1/n(2)
Finding the Minimum
Variance Portfolio
1. Two Asset Case
Var(Rp)
= a2X2 + b2y2 + 2abXy
Correlationxy = rxy = Xy / Xy
Rewrite with b = (1-a) and substitute for correlation
Var(Rp)
= a2X2 + (1-a)2y2 + 2a(1-a)rxy X y
First order condition for minimum w.r.t. portfolio weight a
dVar(Rp)/da = 2aX2 - 2y2 + 2ay2 + 2rxyXy - 4arxyXy = 0
Solve for a* = (y2 - rxyXy )/(X2 + y2 - 2rxyXy )
2. To get an idea about how the minimum variance portfolio
weights are set, consider the following case.
A. rxy = 0, uncorrelated returns imply that the asset weights
depend simply on the relative sizes of the assets’ variances.
a* = (y2 )/(X2 + y2). Here, the larger the variance of asset Y,
the larger the weight on asset X (asset X’s weight is a*).
The weight on Y is just (1 - a*) = (X2 )/(X2 + y2).
B. rxy = -1, perfectly negatively correlated returns imply that
the asset weights depend simply on the relative sizes of the
assets’ standard deviations.
a* = (y2 + Xy )/(X2 + y2 + 2Xy )
a* = y (y + X)/(y + X)2 = y /(y + X)
Here, the larger the standard deviation of asset Y, the larger
the weight on asset X (asset X’s weight is a*).
The weight on Y is just (1 - a*) = x /(y + X).
When two assets are perfectly negatively correlated, one can
buy the two assets in a combination such that the portfolio
variance will be zero.
C. rxy = 1, perfectly positively correlated returns imply that the
asset weights depend simply on the relative sizes of the
assets’ standard deviations.
a* = (y2 - Xy )/(X2 + y2 - 2Xy )
a* = y (y - X)/(y - X)2 = y /(y - X)
Here, one of the weights will be negative (short sales).
Assuming that (y > X) then the weight on X will be positive
and the weight on Y will be negative, (1 - a*) = -x/(y - X).
When two assets are perfectly positively correlated, one can
buy one and short the other in a combination such that the
portfolio variance will be zero.
If we assume that weights can’t be negative, then the
minimum variance portfolio will place all the funds in the
lowest variance asset (a=1 if y > X or 1-a=1 if y < X ).
Finding the Minimum
Variance Opportunity Set
1.
The minimum variance opportunity set is the locus of
combinations of expected return and variance offered by
portfolios that have the minimum variance for each given
expected return.
2.
Use the Lagrange Method and assume Ex and Ey are the
expected returns of assets x and y, respectively. For each
given portfolio expected return , solve for the minimum
portfolio variance as follows:
Min[a 2 x2  (1  a )2 y2  2a(1  a )rxy x y ]
a
s.t aE. x  (1  a ) Ey  
The Lagrangian is
1 2 2
Min L  [a  x  (1  a )2 y2  2a (1  a )rxy x y ]
a ,
2
 [  aE. x  (1  a ) Ey ]
Note: In the case of more than two assets, there will be a
second constraint that requires the sum of the portfolio
asset weights to equal 1, this is implicit for the two asset
case (a, and b=(1-a)). We also multiply by ½ to eliminate
some constants – results are unaffected.
3. Get the first-order conditions.
L
 a  x2   y2  a y2  rxy x y
a
 2arxy x y   ( Ey  Ex )  0
L
   aE  (1  a ) E  0

x
y
This gives us two equations in two unknowns, a and .
4. These equations look simple to solve but are extremely
tedious to get in any manageable form for more than two
assets. This is why people use matrix notation to present, and
matrix operators to solve, this problem (see below or Ingersol
p. 83). We get “a” from the second first-order condition
a = ( - Ey)/ (Ex - Ey) and (1 – a) = (Ex - )/ (Ex - Ey)
Then subst. a and (1 - a) into the equation of variance for two
variables to get Var(Rp) = p2
=[X2( - Ey)2 -2rxyXy ( - Ex )( - Ey) + y2( - Ex)2]/(Ex - Ey)2
Suppose Y is a risk-free asset, i.e., rxy = y2 = 0. Then
p2 = X2( - Ey)2 / (Ex - Ey)2
or
 = (Ex - Ey) p/ X + Ey
Here, the opportunity set in (, p) space is linear.
5. The first result with two risky assets is the equation for a
hyperbola (parabola) with its nose toward the Expected
Return axis (y axis) in Expected Return – Standard Deviation
(Variance) Space.
The following figure shows the various shapes given different
correlations ( in the figure, rxy in the equation) between two
stocks.
For most cases where –1 <  < 1, we get a curved surface.
At one the extreme, when  = 1, we get a straight line
because both portfolio mean and variance are just linear
combinations of the two assets.
When  = -1, portfolio mean and variance are also linear
combinations of the two assets but in this case we are able to
drive the variance of the portfolio to zero with the proper
combination of the two assets.
The Case of Many Risky
Assets
1. When there are many risky assets, construction of the
overall minimum variance opportunity can be thought of in the
following way. Each pair of assets forms its own hyperbola
and any point on these hyperbolas can be consider a
separate asset which could be combined with a point
(combination of two other assets) from another hyperbola to
form yet another hyperbola. Considering all possible pairs, we
take the points that offer the lowest variance for a given mean
return.
2. The program to find the minimum variance portfolio for a
given portfolio expected return uses the following definitions
 is the assets’ variance-covariance matrix, assumed to be
positive definite with inverse -1
2p is the portfolio variance
 is a column vector of expected asset returns
p is the portfolio expected return
x is the vector of asset weights
1 is a column vector of 1’s
 and  are Lagrangian multipliers
Minx [½ x’ x] = 2p
s.t.
‘ x = p and 1’x = 1
The Lagrangian is
Min L = [½ x’ x] + [p - ‘ x ] + [1 - 1’x]
The first-order condition w.r.t. x is
x’ - ‘ - 1’ = 0
Solving for the optimal portfolio asset weights
x’ = ‘-1 + 1’ -1
Using f.o.c. again, post-multiply by x and use the constraints
2p = p + 
Now we need to find  and  to get the portfolio frontier.
Post-multiply the optimal portfolio weights by  gives
p = x’ = ‘-1 + 1’ -1 = A + C
Post-multiply the optimal portfolio weights by 1 gives
1 = ‘-11 + 1’ -11 = C + B
Where A = ’-1 > 0, B = 1’-11 > 0, C = 1’-1 = -1’-11,
D=AB – C2 > 0 are all scalars that only depend upon constant
parameters from the return distributions of the assets.
Use the two equations above to solve for  and  and then get
the portfolio frontier and the weights.
 = (B p - C)/D
 = (A - Cp)/D
x’ = ‘-1 + 1’ -1
p2 = [B p2 – 2C p + A]/D
Like in the two asset case, the frontier is a hyperbola in meanstandard deviation space. But now, the points on the frontier
are portfolios of portfolios and any two frontier portfolios can
be used to generate the frontier.
Any two portfolios on the frontier can be used to generate any
of the other portfolios on the frontier. In this sense, none of the
portfolios is unique. Although, when we get to the CAPM we
will use the market portfolio in a unique way. Nevertheless,
Roll used the idea of the non-uniqueness of any frontier
portfolio – including the market portfolio, as a way to critique
the CAPM. To see this, note that
x’ = ‘-1 + 1’-1 = [A1’-1 - C‘-1]/D + [B‘-1 - C1’-1]/D p
= a + bp
So any frontier portfolio weights are a linear function of p
with two scalars (constants) a and b. We can compute a and b
and then plug in the p to get the efficient portfolio weights for
each portfolio expected return along the frontier.
*You can get from any two frontier portfolios m and n with
weights xm and xn to another frontier portfolio p.
All the portfolios on the frontier have different returns, so put α
in xm and (1 - α) in xn to get an expected portfolio return μp.
Expected return is a linear function so
μp = αμm + (1 - α)μn
and the portfolio weights for p are
αxm + (1-α) xn = α[a + bμm] + (1-α)[a + bμn]
= a + b[αμm + (1-α)μn] = a + bμp
8. The global minimum variance portfolios can be obtained by
choosing the p that minimizes p2
2 B  2C


0

D
2
p
p
p
Solving to get the expected return at the global minimum mp2
 mp = C/B
Substitute into the equation for p2 gives the variance
mp2 = 1/B
For the two-asset case, we have,
 mp = [X2 Ey - rxyXy (Ex + Ey) + y2Ex]/ [X2 - 2rxyXy + y2]
If Y is the risk-free security then y = 0 and
 mp = Ey
meaning the whole portfolio is in the risk-free asset so that
mp2 = 0.
Derivation of the Capital
Market Line
1. When we add a risk-free asset to the previous derivation of
the portfolio frontier, we get the Capital Market Line. The
hyperbola still describes the risk asset portfolios but any
combination of the risk-free asset and any risky portfolio, has
a mean and standard deviation that is linear in the weights.
Define the vector of excess risky returns above the risk-free
rate, rf , as
e =  - rf1
and the portfolio excess return as
ep = e’x
Then we can rewrite the previous Lagrangian as
Min L = [½ x’ x] + [ep - e‘ x ]
The first-order condition is
x’  - e’ = 0
and the optimal portfolio weights are
x’ = e’-1
Then post-multiply by the vector of excess asset returns to get
the excess portfolio return is
ep = e’-1e= F
where F = e’-1e
Then post-multiply the f.o.c. by x to get
p2 = ep
Using these two results to eliminate  gives the CML
ep2 = Fp2
or
ep = F½p
p = ep / F½
So in mean-standard deviation space, the CML defines a
linear efficient frontier, with the minimum variance portfolio
being the risk-free asset.
The CML will be tangent to the risky asset portfolio frontier at
a point T that represents the “market” portfolio.
For any portfolio i on the CML we have
ei = i - rf
so that
i - rf = F½i
i = rf + [(T - rf )/T ] i
The term in square brackets is called the Sharpe Ratio or the
price of risk per unit of the risk of the tangency portfolio.
This is also the slope of the CML.
Portfolio Separation: This result shows that the expected
return of every efficient portfolio can be described by a
combination of the risk-free asset and the tangency portfolio.
Everyone holds an efficient portfolio so that they face the
same price of risk. Therefore, the decisions of firms making
corporate investments can be separated from the decisions of
investors making portfolio allocations. Firm managers do not
have to consider the risk preferences of any particular investor
in a firm’s stock.
9. The constrained portfolio optimization gives us the
minimum variance opportunity set.
The risk averse investor’s constrained utility optimization gives
us his indifference curves in E(R) and (Rp) space.
A unique portfolio choice by the investor is guaranteed by
convexity of the upper-half of the opportunity set (called the
efficient set) and the indifference curves.
At the intersection of the indifference curve and the efficient
set, we have
MRS of investor = MRT for efficiency set of available assets
10. When we assume that a risk-free asset exists with return
Rf, then for the two asset case we will have
E(Rp) = aEx + (1 – a)Rf
Var(Rp) = a2X2
A risk-free asset is assumed to be uncorrelated with other
assets and have zero variance.
The new efficient set becomes a straight line (called the
“Capital Market Line” or CML) that is tangent to the previous
minimum variance opportunity set. Now investors select a
portfolio comprised of the risk-free asset (borrowing or
lending) and a “market” portfolio.
11. Note that points along the CML are linear combinations of
the risk free rate and a portfolio defined by the tangency
between the CML and the investment opportunity set.
If all investors have the same beliefs about the return
distributions of the assets, then this tangency portfolio is the
“market” portfolio and the CML can be defined from the fact
that the slopes of the CML and opportunity set must be equal
at the tangency so
Slope of CML = Slope of Opportunity set
[E(Rp) - Rf ] / [E(RM) – Rf] = p / M
Rearranging gives
E(Rp) = Rf + [E(RM) – Rf]p / M
The second term on the RHS is the product of the units of risk
(p ) and the price of risk [E(RM) – Rf]/ M .
Two-Fund Separation: As a result, all portfolios are defined by
the risk-free security and the market portfolio.
This implies that for all investors
MRS of any investor = MRT from the CML = [E(RM) – Rf]/ M
All investors agree on the price of risk.This implies that
investment decisions can be “separated” from the particular
risk preferences of a firm’s investors. Managers of firms can
use the market equilibrium price of risk to evaluate potential
investments rather than define the particular risk preferences
of their investors.
How Portfolio Diversification
Affects Risk
12. Consider the definition of portfolio variance
N
N
 p2   wiwj ij
i 1 j 1
As we add more securities to the portfolio, we are reducing
the average amount put in each asset. How does the portfolio
variance change as we do this? Take the derivative to get,
 p2
wi
N
 2 wi  2 wj ij
2
i
j 1
As the w’s are getting small, the first term goes to zero. The
second term, however, does not because the number of
covariances increases even as the weight applied to each
decreases. Portfolio variance is dominated by covariance. If
we chose an equal weight for all assets, then w = 1/N, and
take the average σij over all i and j, then for large N we get
 p2
N
1 2
1
1
 2  i  2  ij  0  2 N  ij  2 ij
wi
N
N
j 1 N
Portfolio
Risk
Diversifiable Risk
Nondiversifiable Risk
Number of securities in the portfolio
Diversifiable risk drops as more securities
are added to a portfolio.
Calculating Expected
Returns Variances and
Covariances of Portfolios
1.
A quick way to calculate expected returns,variances and
covariances of a portfolio is to use matrices and matrix
operations.
2.
E(Rp) = R’W
3.
Var(Rp) = W’W
Where W is a vector of weights and is the variance covariance
matrix of the individual assets. This is very convenient
once the number of assets is large. For three assets
 w1
E ( Rp )  E ( R1) E ( R 2) E ( R3) w2
 
 w3
 p2  w1 w2
 12  12  13  w1 

 
2
w3 21  2  23 w2
 
2
 31  32  3   w3


You can perform matrix operations using programs like Excel.
Covariance of Two
Portfolios
4. When you want to calculate the covariance between two
portfolios that have different weights on the same set of
assets you can use
Cov( RA, RB)  WA' WB  w1a
 12  12  w1b 
w2 a 

2 
w
2
b

21


2 

5. One can look at the value of a firm’s equity as long
positions (positive portfolio weights) in its assets and short
positions (negative portfolio weights) in its liabilities.
Return to shareholders is just the net return on the portfolio of
long and short positions.
The variance of shareholder return is just the variance of the
portfolio of long and short positions.
Problems involving a large number of positions, such as
problem 5.14 in CWS can be solved quickly using matrices as
shown above. Use the mmult(x,y) function in Excel and note
that you must highlight the cells that will hold the results, enter
the function as usual and then press control-shift-enter
together.