(2.1) Markowitz’s mean-variance formulation (2.2) Two-fund ...

2. Mean-variance portfolio theory

(2.1) Markowitz's mean-variance formulation (2.2) Two-fund theorem (2.3) Inclusion of the riskfree asset

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2.1 Markowitz mean-variance formulation

Suppose there are N risky assets, whose rates of returns are given by the random variables R1, ? ? ? , RN , where

Rn

=

Sn(1) - Sn(0), Sn(0)

n

=

1,

2,

?

?

?

,

N.

Let w = (w1 ? ? ? wN )T , wn denotes the proportion of wealth invested in asset n,

N

with wn = 1. The rate of return of the portfolio is

n=1

Assumptions

N

RP = wnRn.

n=1

1. There does not exist any asset that is a combination of other assets in the

portfolio, that is, non-existence of redundant security.

1 2. ? = (R1 R2 ? ? ? RN ) and = (1 1 ? ? ? 1) are linearly independent, otherwise

RP is a constant irrespective of any choice of portfolio weights.

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The first two moments of RP are

N

N

?P = E[RP ] =

E[wnRn] =

wn?n, where ?n = Rn,

n=1

n=1

and

NN

NN

P2 = var(RP ) =

wiwjcov(Ri, Rj) =

wiij wj .

i=1 j=1

i=1 j=1

Let denote the covariance matrix so that

P2 = wT w. For example when n = 2, we have

(w1

w2)

11 12 21 22

w1 w2

= w1212 + w1w2(12 + 21) + w2222.

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Remark

1. The portfolio risk of return is quantified by P2 . In mean-variance analysis, only the first two moments are considered in the portfolio model. Investment theory prior to Markowitz considered the maximization of ?P but without P .

2. The measure of risk by variance would place equal weight on the upside deviations and downside deviations.

3. In the mean-variance model, it is assumed that ?i, i and ij are all known.

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Two-asset portfolio Consider two risky assets with known means R1 and R2, variances 12 and 22, of the expected rates of returns R1 and R2, together with the correlation coefficient . Let 1 - and be the weights of assets 1 and 2 in this two-asset portfolio. Portfolio mean: RP = (1 - )R1 + R2, 0 1 Portfolio variance: P2 = (1 - )212 + 2(1 - )12 + 222.

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