2.1: Ass umptio ns 2.2: Indi ¤erence 2.3 : Frontier 2.4 : Rf2.5: He dging 2.6: S ummar y
Mean-Variance Analysis
George Pennacchi
Unive rsity of I llinois
George Pen nacch i Univers ity of Illino is
Mean-v ariance a nalysis 1/ 52
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distributed, utility of wealth depends only on portfolio mean and variance. ... which depends only on the mean and variance of the portfolio return.
Typology: Exercises
2021/2022
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Mean-Variance Analysis
George Pennacchi
University of Illinois
George Pennacchi University of Illinois
Introduction
How does one optimally choose among multiple risky assets?
Due to diversiÖcation, which depends on assetsíreturn
covariances, the attractiveness of an asset when held in a
portfolio may di§er from its appeal when it is the sole asset
held by an investor.
Hence, the variance and higher moments of a portfolio need
to be considered.
Portfolios that make the optimal tradeo§ between portfolio
expected return and variance are mean-variance e¢ cient.
George Pennacchi University of Illinois
Taylor Series Approximation of Utility
U(
e R p
) = U
E [
e R p
]
e R p
E [
e R p
]
U
0
E [
e R p
]
1
2
e Rp E [
e Rp ]
2
U
00
E [
e Rp ]
1
n!
e Rp E [
e Rp ]
n
U
(n)
E [
e Rp ]
If the utility function is quadratic, (U
(n) = 0, 8 n 3), then
the individualís expected utility is
E
h
U(
e Rp )
i
= U
E [
e Rp ]
1
2
E
e Rp E [
e Rp ]
2
U
00
E [
e Rp ]
= U
E [
e Rp ]
1
2
V [
e Rp ]U
00
E [
e Rp ]
George Pennacchi University of Illinois
Alternative Assumptions for Mean-Variance
Quadratic utility, such as U (W ) = aW bW
2 , is problematic
due to a ìbliss pointî at W =
a
2 b
after which utility declines
in wealth.
Instead, let utility be a general increasing and concave
function but restrict the risky asset probability distribution.
Claim: If individual assets are multi-variate normally
distributed, utility of wealth depends only on portfolio mean
and variance.
Why? Note that the return on a portfolio is a weighted
average (sum) of the returns on the individual assets.
Then since sums of normals are normal, if the joint
distributions of individual assets are multivariate normal, then
the portfolio return is also normally distributed.
George Pennacchi University of Illinois
Centered Normal Moments
E [
e R p
]
1
=
d exp
1
2
2 t
2
dt
t= 0
E [
e R p
]
2
d
2 exp
1
2
2 t
2
dt
2
t= 0
2
E [
e R p
]
3
d
3 exp
1
2
2 t
2
dt
3
t= 0
E [
e Rp ]
4
=
d
4 exp
1
2
2 t
2
dt
4
t= 0
4
George Pennacchi University of Illinois
Normal Distribution of Returns
So moments are either zero or a function of the variance:
E
h
e R p
E [
e R p
]
n
i
= 0 for n odd, and
E
h
e R p
E [
e R p
]
n
i
n!
(n= 2 )!
1
2
V [
e R p
]
n= 2
for n even.
Therefore, in this case the individualís expected utility equals
E
h
U (
e Rp )
i
= U
E [
e Rp ]
1
2
V [
e Rp ]U
00
E [
e Rp ]
- 0 +
1
8
V [
e Rp ]
2
U
0000
E [
e Rp ]
- 0 + ::: +
1
(n= 2 )!
1
2
V [
e Rp ]
n= 2
U
(n)
E [
e Rp ]
- ::: (6)
which depends only on the mean and variance of the portfolio
return.
George Pennacchi University of Illinois
Preference for Return Mean and Variance
Therefore, assume U is a general utility function and asset
returns are normally distributed. The portfolio return
R
p
has
normal probability density function f (R; Rp ;
2
p
), where we
deÖne
R
p
E [ ~R
p
] and
2
p
V [ ~R
p
].
Expected utility can then be written as
E
h
U
e Rp
i
Z
1
U(R)f (R; Rp ;
2
p
)dR (7)
Consider an individualís indi§erence curves. DeÖne ex
~ Rp
Rp
p
E
h
U
e Rp
i
Z
1
U( Rp + xp )n(x)dx (8)
where n(x) f (x; 0 ; 1 ). (ex is a standardized normal)
George Pennacchi University of Illinois
Mean vs Variance contíd
Taking the partial derivative with respect to
R
p
@E
h
U
e R p
i
Rp
Z
1
U
0
n(x)dx > 0 (9)
since U
0 is always greater than zero.
Taking the partial derivative of equation (8) with respect to
2
p
and using the chain rule:
@E
h
U
e R p
i
2
p
p
@E
h
U
e R p
i
p
p
Z
1
U
0 xn(x)dx
George Pennacchi University of Illinois
Risk and Utility contíd
Comparing the integrand of equation (10) for equal absolute
realizations of x, we can show
U
0
( Rp + xi p )xi n(xi ) + U
0
( Rp xi p )( xi )n( xi )
= U
0 ( R p
- x i
p
)x i
n(x i
) U
0 ( R p
x i
p
)x i
n(x i
= x i
n(x i
U
0
( R p
- x i
p
) U
0
( R p
x i
p
because
U
0
( R p
- x i
p
) < U
0
( R p
x i
p
due to the assumed concavity of U.
George Pennacchi University of Illinois
Risk and Utility contíd
Thus, comparing U
0 x i
n(x i
) for each positive and negative pair,
we conclude that
@E
h
U
e Rp
i
2
p
p
Z
1
U
0
xn(x)dx < 0 (13)
which is intuitive for risk-averse individuals.
An indi§erence curve is the combinations of
R
p
2
p
that
satisfy the equation E
h
U
e R p
i
= U , a constant. Higher U
denotes greater utility. Taking the derivative
dE
h
U
e Rp
i
@E
h
U
e R p
i
2
p
d
2
p
@E
h
U
e R p
i
Rp
d
Rp = 0
George Pennacchi University of Illinois
Mean and Standard Deviation Indi§erence Curve
As an exercise, show that the indi§erence curve is upward
sloping and convex in
R
p
p
space:
George Pennacchi University of Illinois
Tangency Portfolios
The individualís optimal choice of portfolio mean and variance
is determined by the point where one of these indi§erence
curves is tangent to the set of means and standard deviations
for all feasible portfolios, what we might describe as the ìrisk
versus expected return investment opportunity set.î
This set represents all possible ways of combining various
individual assets to generate alternative combinations of
portfolio mean and variance (or standard deviation).
The set includes ine¢ cient portfolios (those in the interior of
the opportunity set) as well as e¢ cient portfolios (those on
the ìfrontierî of the set).
How can one determine e¢ cient portfolios?
George Pennacchi University of Illinois
Mean/Variance Optimization contíd
The constraint on portfolio weights is!
0 e = 1 where e is
deÖned as an n 1 vector of ones.
A frontier portfolio minimizes the portfolioís variance subject
to the constraints that the portfolioís expected return equals
R
p
and the portfolioís weights sum to one:
min
!
1
2
0 V! +
R
p
0 R
+ [ 1 !
0 e] (18)
The Örst-order conditions with respect to !, , and , are
V!
R e = 0 (19)
Rp !
0 R = 0 (20)
0 e = 0 (21)
George Pennacchi University of Illinois
Mean/Variance Optimization contíd
Solving (19) for!
, the portfolio weights are
= V
1 R + V
1
e (22)
Pre-multiplying equation (22) by
R
0 and e
0 respectively:
R
p
R
0 !
=
R
0 V
1 R +
R
0 V
1 e (23)
1 = e
0
!
= e
0
V
1 R + e
0
V
1
e (24)
Solving equations (23) and (24) for and :
Rp
2
& R
p
2
George Pennacchi University of Illinois