Mean Variance Theory

Mean Variance Theory

September 24th, 2012


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

.


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 .


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.


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 .


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 .


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.


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.


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


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.


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).


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).


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.


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).


Portfolio Diagram (With Shortselling)

Figure: With shortselling. Solid line is |(1 )1 2| and dotted is|(1 )1 + 2|.


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.


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.


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).


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).


Solving with Matlab

A good way to find optimal mean-variance allocation is usingMatlabs fmincon (see Matlab code on Blackboard).


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Mean Variance Theory