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Unformatted text preview: 6. PORTFOLIO THEORY
Reading: Luenberger Chapter 6
• Consider a portfolio of n assets (e.g. stocks). A
portfolio is constructed by weighting asset i by weight
wi .Stock picking is accomplished by increasing the
weight of the stock that you like, and setting the weight
to zero for the stock that you do not want to include in
the portfolio, or even shorting it (with negative weight).
• We want to maximize the portfolio return while
minimize the portfolio risk (many definitions) by varying
the weights, wi. 10/17/2011 Many slides are from Prof. Martin’s previous course material 1 Optimization Problems Difficult to optimize two quantities together, namely,
maximize mean portfolio return while minimize
portfolio variance. This may even be illdefined. Do one optimization at a time. Maximize portfolio return, without consideration of
risk? Maximize mean portfolio return for a given portfolio
risk (variance). Minimize portfolio variance for a given portfolio return
(Markowitz Nobel winning work) 2 Markowitz (1952, 1959)* meanvariance optimal portfolio
theory for the idealized case where optimality depends only
on asset return s expected values and covariances, with no
restriction on shortselling and no other constraints on
portfolio weights. In this case analytical formulas for optimal
portfolios on the efficient frontier are obtained. Risk was
defined to be the portfolio variance.
Comment : The Markowitz theory is to this day often called
Modern Portfolio Theory (MPT) in spite of being known for
more than fifty years.
Recent advances in minimizing downside risks not discussed
here.
* Markowitz, H. M. (1952). “Portfolio Selection”, Journal of Finance, 7, no. 1, 7791.
Markowitz, H. M. (1959). Portfolio Selection: Efficient Diversification of Investments, Wiley.
3 6.1 Mean Vector and Covariance Matrix
Let r′ = (r ,..., rn ) be a vector of single period returns. The vector of
1
expected returns µ and the corresponding covariance matrix Σ are: µ ' = E (r!) = ( E ( r1 ), ..., E ( rn )) = (µ1 , ..., µ n )
$
" = cov(r ) = E & r # µ r # µ
% ( )( !'
)
( ) ! is symmetric σ ij = cov(ri , rj ) = E ⎡( ri  µi ) ( rj  µ j )⎤ , 1 ≤ i, j ≤ n,
⎣
⎦
σ ii = Var (ri ) = σ i2 , 1 ≤ i ≤ n,
We assume that ! is nonsingular and so has an inverse !"1 .
4 Portfolio Returns (random)
Portfolio Mean Returns rP = ∑ i =1 wi ri = w ′r
n µ P = E ( rP ) = E (w′r ) = w′E (r ) = w′µ
= ∑ i =1 wi µi
n Portfolio Variance 2
! P = var( rP ) = var( w!r ) (( = E w! r " µ )) 2 ! r " µ r " µ ! w)
= E(w ( )( n ) n = w!#w = $$ wi w j! ij
i =1 j =1 Portfolio Volatility 1/ 2 ! P = ( w!!w ) 5 6.2 Global Minimum Variance Portfolio
min w!"w
w subject to w ′1 = 1, which is just: n ∑w =1
i =1 i 1
L w = w!!w + ! 1 ! w"1
2 () d
L w = !w " ! 1 = 0
dw () ( )
w = !!"11 # Apply the constraint w ' 1 = 1
to get: ! 1 ' !"11 = 1 ( 1 ) ! = 1!" 1 #1 6 w MV !"11
=
!"#11
1 µP = w ′µ ⇒ 2
P ! = w!"w # Knowledge of mean returns is not required,
except as nuisance parameters in Σ ! µ MV 1!"#1µ
=
1!"#11 ! MV = 1
1!"#11 7 6.3 MeanVariance Efficient Frontier
Portfolio mean return:
Portfolio variance:
Minimize variance:
subject to µP = w′µ
2
P ! = w!!w min w!"w
w w′µ = µ P (specified by investor )
w′1 = 1 (fully invested) This is to find the minimum variance portfolio for a given return
unlike the previous global minimum. Difficult!
8 Solution via Lagrange Multipliers
1
L w = w!!w + !1 µ P ! w"µ + !2 1 ! w"1
2
! "#
#$
! "#
#$ () ( ) Constraint 1 Solving ( ) Constraint 2 d
L(w) = 0
dw gives the general form of optimal weight vector: w = !1!"1µ + !2 !"11
Messy algebra left to Appendix A1. Shows hyperbola:
2 a⎞
⎛
2
⎜ µP − ⎟
σP ⎝
c⎠
−
=1
2
1c
dc a = µ ' !"11
c = 1!!"11 b = µ ' !"1µ
d = bc ! a 2
9 Mathematics in Appendix A1 Give the Results Below
The upper part is the Efficient Frontier
µP = a c + d c σ P µP
a
c µMV = a c µP = a c − d c σ P µP = ± σ MV = 1 c d2d
σP − 2
c
c 2
for σ P ≥ 1 c σP A hyperbola called the Portfolio Frontier
10/14/2010 Copyright R. Douglas Martin 10 0.06 EFFICIENT FRONTIER 0.04 GENZ ORCL 0.02 0.03 LLTC
MSFT 0.01 MEAN RETURN 0.05 SHARPE RATIO = 0.294 0.0 HON 0.0 0.05 0.10 0.15 0.20 SIGMA 10/14/2010 Copyright R. Douglas Martin 11 $w.mv:
GENZ HON LLTC MSFT .308 .375 .083 .146 ORCL
.088 $mu.mv:
[1] 0.026
$sigma.mv:
[1] 0.087 10/14/2010 Copyright R. Douglas Martin 12 6.4 Efficient Frontier with Cash
The optimization problem becomes much simpler
when we add cash.
First consider one stock with expected return µ = E[ r ]
and variance σ 2 = var(r )
and some cash earning riskfree return (deterministic) µ f
Portfolio return with given fraction α in cash: µ P = E[α rf + (1 − α )r ] = α rf + (1 − α ) µ . Or µ P ,e = (1 − α ) µe , where µe ≡ µ − rf and µ P ,e = µ P − rf
σ P = (1 − α )σ .
13 Linear relationship between excess
portfolio return and standard deviation
µ P ,e = (1 − α ) µe
σ P = (1 − α )σ . µP
rf σ σP 14
14 What if there are n stocks? LU 6.9: OneFund Theorem:
Treat this collection of stocks as a fund.
The argument in the previous slide still holds
with α in cash and (1 − α ) in this fund. How to allocate (1 − α ) among the n stocks? 15 Efficient Frontier with Cash
Let ∑ n
i =1 wi be invested in risky assets, and 1 − ∑ i =1 wi
n in a riskfree asset with a rate of return
portfolio return and expected return are rP = (1 − w ′1) rf + w ′r So we minimize
subject to 2 . Then, the µP = (1 − w ′1) rf + w ′µ and 1 rf w!"w (1 # w!1) r f + w!µ = µ P
16 Solution via Lagrange Multipliers (only
one constraint, easier)
1
L( w ) = w!"w + ! $ 1 # w!1 rf + w!µ # µ P &
%
'
2
d
w = ! ! "#1 µ ! rf 1 = ! ! "#1µe .
L ( w) = 0 ⇒
dw ( ) ) ( µ P = (1 − w′1) rf + w′µ =γµe ' Σ −1ΣγΣ −1µe ' = rf + w′µe
= rf + γ ⋅ µ 'e Σ µe
−1 µ P ,e = γµe ' Σ µe
−1 σ P 2 = w ' Σ −1w
=γ 2 µe ' Σ −1µe σP = γ µe ' Σ −1µe
17 What is γ ? The fraction allocated to stocks: w ' 1 = γµe ' Σ −11 = (1 − α ) γ = (1 − α )( µe ' Σ −11) −1
So γ ∝ (1 − α )
All cash: α =1, γ =0. 10/17/2011 18 Linear relationship between excess
portfolio return and standard deviation
µ P ,e
σP µP = µe ' Σ −1µe rf σP 10/17/2011 19
19 The straight line looks as follows depending on . µP
rf . γ : T1 γ 1 > 0 T2 γ 2 < 0 σP Main Message : γ will be negative if and only if the global
minimum variance portfolio has a mean return less than that of the
risk free rate. This case can and does sometimes occur in
practice, and no risk averse investor will invest in risky assets in
this case, and will instead invest in a risk–free rate instrument.
20 The AllRiskyAssets Portfolio T ( w! 1 ! 1 " ! = 1!! µe
T In the limit: This portfolio must lie on the hyperbola.
Portfolio T weights: Expected excess returns: Risk "1 ) !1 !"1µe
wT = "1
1#! µe µ 'e !"1µe
µT ,e = µ 'e wT =
1#!"1µe ! T = var( wT ' r ) =  1!"#1µe #1 µ 'e "#1µe
21 Tangency portfolio
T is the tangency portfolio shown below and the efficient frontier with cash
is the straight line tangent to T. µP
T µT . rf σT σP Why can t the line cut the hyperbola?
22 The Sharpe Ratio
For any portfolio P: µP
µP
rf T .. P θ σP 10/17/2011 µ P − rf
SRP =
= Sharpe Ratio
σP
= tan θ
The tangency portfolio
has the maximum Sharpe Ratio σP 23 0.06 EFFICIENT FRONTIER 0.04 GENZ ORCL 0.02 0.03 LLTC
MSFT 0.01 MEAN RETURN 0.05 SHARPE RATIO = 0.294 0.0 HON 0.0 0.05 0.10 0.15 0.20 SIGMA 10/14/2010 Copyright R. Douglas Martin 24 $w.mv:
GENZ HON LLTC MSFT ORCL 0.3081789 0.3753527 0.08328089 0.1456418 0.08754582
$mu.mv:
[1] 0.02577599
$sigma.mv:
[1] 0.08702582
$w.t:
GENZ HON LLTC MSFT ORCL 0.5548272 0.03873531 0.233836 0.1492129 0.1008592
$mu.t:
[1] 0.03648147
$sigma.t:
[,1]
[1,] 0.107126
10/14/2010 Copyright R. Douglas Martin 25 6.5 Risk Penalized Utility Solution (No constraint; User Input for the Risk Aversion
Parameter)
Maximize 1
2
U (w) = µ P (w) ! ! ! ! P (w)
2
Risk aversion parameter
1
= w!µ ! ! ! w"#w
2 With no weight constraints, i.e., no full investment constraint,
the optimal weights are w = ! !1 " #!1µ
The choice ! = 1!"#1µ yields fullinvestment, with possible
short positions. But if this is not the case then 1′w ≠ 1 and
if 1′w < 1 the portfolio contains a positive cash position.
Q. What if 1′w > 1 ?
26 An interesting solution is obtained by optimizing with
the fullinvestment constraint, i.e., maximize subject to Lagrangian: 1
U ( w ) = w!µe ! ! ! w"#w
2
1′ w = 1 (full investment) 1
L( w ) = w!µe ! ! ! w"!w ! ! ! (1"w ! 1)
2
dL( w )
= µe ! ! ! "w # ! ! 1 = 0
dw ! w = ! !1 ! "#1 ! µe ! ! ! 1 ( )
27 Use the constraint to solve for the Lagrange multiplier and
obtain the following result.
TwoFund Separation. The optimal weight vector has the
following representation:
#!1µe
#!11
w = ! !11"#!1µe
+ 1 ! ! !11"#!1µe $ !1
1"#!1µe
1"# 1 (
= (! !1 )
(
µ ) $ w + (1 ! ! 1"#!1 e T Tangency portfolio
"1
It is achieved at ! = 1!! µ e . 10/14/2010 ) !1 ) 1"#!1µe $ w MV Global minimum variance portfolio
It is achieved when λ → ∞ . Copyright R. Douglas Martin 28 A1: Frontier Mathematics
Now apply the two constraints w ′µ = µ P and w ′1 = 1 to the
optimal weight vector w = !1!"1µ + !2 !"11
and solve for the two Lagrange multipliers:
c ⋅ µP − a
λ1 =
d where b − a ⋅ µP
λ2 =
d a = µ ' !"11 b = µ ' !"1µ c = 1!!"11
10/14/2010 d = bc ! a 2 Copyright R. Douglas Martin 29 Proposition A1: The optimal weight vector may written as w = g1 + g 2 µ P
where g1 and g 2 are fixed vectors determined by µ and ! : g1 = 1
b ! "#11 ! a ! "#1µ
d ( ) g2 = 1
c ! "#1µ ! a ! "#11
d ( ) Proof: Exercise.
Note that as you vary µ P arbitrarily you will typically get
some wi < 0 that correspond to short selling. 10/14/2010 Copyright R. Douglas Martin 30 Proposition A2: Let P , P2 be any two efficient frontier
1
portfolios. Then ( ) cov rP , rP = w!P "w P
1 2 1 2 1 c$
a '$
a'
= + & µ P # )& µ P # )
c d % 1 c (% 2 c (
Proof: Exercise (Hint: use Proposition 1) The special case of the above with P = P2 gives the
1
Portfolio Frontier formula: 1 c⎛
a⎞
σ = cov ( rP , rP ) = + ⎜ µP − ⎟
c d⎝
c⎠ 2 2
P 10/14/2010 Copyright R. Douglas Martin 31 The Portfolio Frontier is a Hyperbola
Rearranging gives 2 a⎞
⎛
2
⎜ µP − ⎟
σP ⎝
c⎠
−
=1
2
1c
dc Recall that the equation of a hyperbola centered at (0, 0) is x 2 y 2 ⎛ x y ⎞⎛ x y ⎞
B
− 2 = ⎜ − ⎟⎜ + ⎟ = 1, with asymptotes y = ± x
2
A B ⎝ A B ⎠⎝ A B ⎠
A
Thus µ P versus
with asymptotes 10/14/2010 σ P is a hyperbola centered at (0, a / c),
a
d
µP = ± σ P
c
c
Copyright R. Douglas Martin 32 A2. Positive Definite Covariance Matrices
Variances are nonnegative, which gives:
2
! P = w!!w ! 0 Thus ! is always nonnegative definite. A covariance
matrix ! is said to be positive definite if
2
! P = w!"w > 0 !w " 0 It can be shown that if w!"w = 0 for some w ≠ 0 then
there is a linear dependency among the returns r of the
following form: there exist constants a1 , a2 ,L , an not all
of which are zero and a constant c such that Prob( a1r1 + a2 r2 + L + an rn = c ) = 1
10/14/2010 Copyright R. Douglas Martin 33 Some Matrix Algebra Results
See for example the Appendix of Matrix Analysis (1979) by
Mardia, Kent and Bibby, Academic Press 1) Definition: A square matrix A is nonsingular if and
only if its determinant A is nonzero.
2) The inverse A −1 of a square matrix A exists if and
only if A is nonsingular.
3) A positive definite matrix is nonsingular and hence its
inverse exists
4) The inverse of a positive definite matrix is positive
definite. 10/14/2010 Copyright R. Douglas Martin 34 ...
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This note was uploaded on 03/23/2012 for the course AMATH 541 taught by Professor Kk.t during the Winter '11 term at University of Washington.
 Winter '11
 KK.T

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