Mean Variance Theory
September 24th, 2012
Mean Variance Theory
Mean & Variance of a Portfolio
Let r1, . . . , rN be R.V.s representing random future return rates ofN assets. The portfolio return rate is
r =Nn=1
wnrn
and is also random. Denote rn = Ern and r = Er ,
r = ENn=1
wnrn =Nn=1
wn rn ,
2(r) = var(r) = cov
(Nn=1
wnrn,Nn=1
wnrn
)
=N
m,n=1
wnwm cov(rn, rm) =cnm
.
Mean Variance Theory
Covariance Matrix
C is the covariance matrix of (r1, . . . , rN),
C =
c11 c12 c13 . . . c1Nc21 c22 c23 . . . c2Nc31 c32 c33 . . . c3N
.... . .
cN1 cN2 cN3 . . . cNN
C is a symmetric matrix (CT = C ) and is positive semi-definite
Nm,n=1
xmxncnm 0 for all x RN .
Mean Variance Theory
Example: Two Assets
r1 = .12, 1 = .18, w1 = .25r1 = .15, 2 = .20, w2 = .75 and cov(r1, r2) = .01.
r = .25(.12) + .75(.15) = .1425
2 = var(w1r1 + w2r2) = cov(w1r1 + w2r2,w1r1 + w2r2)
= w21 var(r1) + w22 var(r2) + 2w1w2cov(r1, r2)
= (.25)2(.18)2 + (.75)2(.2)2 + 2(.25)(.75)(.01)
= .028275 ,
and so = .1681. Have compromised on the expected return, buthave lowered the overall variation of the outcome.
Mean Variance Theory
Diversification (Uncorrelated Assets)
For N mutually uncorrelated assets with mean return m andvariance 3, portfolio with equal weights wi =
1N is less risky:
r =1
N
Nn=1
ri ,
r =1
N
Nn=1
Eri =1
N
Nn=1
m = m ,
var(r) =1
N2
Nn=1
2 =2
N 0 as N .
Mean Variance Theory
Diversification (Correlated Assets)
Suppose now that cov(ri , rj) = .32. Then
var(r) =1
N2E
( Ni=1
(ri r)) N
j=1
(rj r)
=1
N2
Ni ,j=1
cov(ri , rj) =1
N2
i=j
2 +i 6=j
cov(ri , rj)
=
1
N2(N2 + N(N 1).32) = .72
N+ .32 .32 .
Mean Variance Theory
Diversification in General
Assets all with equal expected returns is unrealistic.
In general: diversification may reduce overall expected returnwhile reducing the variance.
Mean-Variance approach developed by Markowitz makeexplicit the trade-off between mean and variance.
Mean Variance Theory
Lets Develop a Finanical Model for Measuring Risk
From N-many assets with returns r1, r2, . . . , rN , well constructportfolios based on our
concerns regarding volatility,
and our natural liking for higher returns.
It turns out that a simple 2-D relationship can be formed.
Mean Variance Theory
Simple Example
Two Assets: with expected return rates r1, r2, variance 1,2, and covariance c12 (each asset is a point in themean-standard deviation diagram)
form a portfolio of this two assets with weights w1 = andw2 = (1 ),
r = r1 + (1 )r22 = 221 + 2(1 )c12 + (1 )222
is a new point in the diagram (, r) (is a new asset)different portfolio for different
Mean Variance Theory
Portfolio Diagram (No Shortselling)
Figure: For no short selling: the lines labeled = 1 are the lowerbounds on . The upper bound is the line labeled = 1. The set ofpoints (, r) for [0, 1] are the curved line.
Mean Variance Theory
Variance Bounds
For each define
().
=
(1 )221 + 2(1 )12 + 222 .
For [0, 1] (no short selling), the most variance occurs when = 1,
()
(1 )221 + 2(1 )12 + 222
=
((1 )1 + 2)2 = (1 )1 + 2(this is the dotted line in Figure 1).
Mean Variance Theory
Variance Bounds
Similarly for [0, 1] (no short selling), the minimum varianceoccurs when = 1,
()
(1 )221 2(1 )12 + 222
=
((1 )1 2)2 = |(1 )1 2|(this makes up the two straight lines originating on left in Figure1).
Mean Variance Theory
Variance Bounds
In Figure 1, the point where the two lines meet on the y-axis isr(0) where 0 is s.t. (0) = 0 when = 1, i.e.
(1 )1 02 = 0
0 = 11 + 2
,
and so r(0) =1
1+2r1 +
(1 11+2
)r2.
Mean Variance Theory
Variance Bounds (with short selling)
Same bounds for [0, 1], but for / [0, 1] we have
() |(1 )1 2| case = 1() |(1 )1 + 2| case = 1
(See Figure 2).
Mean Variance Theory
Portfolio Diagram (With Shortselling)
Figure: With shortselling. Solid line is |(1 )1 2| and dotted is|(1 )1 + 2|.
Mean Variance Theory
For 3 Assets
Add a third asset with expected return r3 and std. dev. 3. Let1 equal the total allocation in assets 1 and 2, then repeatanalysis from before.
Results in more options for allocation (hyper place of R3instead of the lower dimensional hyper lane of R2
There is a region of possible (, r) points rather than just acurve (See Figure 3).
In general, can find feasible sets of points (, r) for N-manyassets, and it gives us a good idea of a portfoliosmean-variance trade-off.
Mean Variance Theory
Feasible Region
Figure: Assets 1 and 2 are the same as from slide 4, and for the newasset we have r3 = .11 and 3 = .1.
Mean Variance Theory
Minimum Variance and the Efficient Frontier
For a portfolio allocation w RN to be optimal, we would like itto minimize variance will still maintaining a certain level ofexpected return. This optimization problem is formulate as
minwRN
var
(Nn=1
wnrn
)=
Nn,m=1
wnwmcnm = wTCw
subject to the constraintsn
n=1 wn = 1 andN
n=1 wn rn = r ,where r is our desired level of return.We say portfolio r =
Nn=1 wnrn is efficient if there exists no other
portfolio r such that Er Er and (r) < (r).
Mean Variance Theory
Minimum Variance Set
Figure: The side-ways parabola shows the minimum variance set,minw w
TCw s.t. w1 + w2 = 1 and w1r1 + w2r2 = r . This is the samenumber from slide 4, and the dot on the frontier is the allocation fromslide 4 (its in fact efficient).
Mean Variance Theory
Solving with Matlab
A good way to find optimal mean-variance allocation is usingMatlabs fmincon (see Matlab code on Blackboard).
Mean Variance Theory