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) , n = 1, 2, · · · , N. Sn(0)

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

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

RP = n=1 wnRn .

Assumptions 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 1 · · · 1) are linearly independent, otherwise 2. µ = (R1 R2 · · · RN ) and RP is a constant irrespective of any choice of portfolio weights.

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The first two moments of RP are
N N

µP = E[RP ] = n=1 E[wn Rn] = n=1 N

wnµn, where µn = Rn,

and
2 σP = var(RP ) = N N N

wiwj cov(Ri, Rj ) = i=1 j=1 i=1 j=1

wiσij wj .

Let Ω denote the covariance matrix so that
2 σP = wT Ωw.

For example when n = 2, we have (w1 w2 ) σ11 σ12 σ21 σ22 w1 w2
2 2 2 2 = w1 σ1 + w1w2(σ12 + σ21) + w2 σ2 .

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

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 µi, σi and σij are all known.

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Two-asset portfolio Consider two risky assets with known means R1 and R2, variances 2 2 σ1 and σ2 , of the expected rates of returns R1 and R2, together with the correlation coefficient ρ. Let 1 − α and α be the weights of assets 1 and 2 in this two-asset portfolio. Portfolio mean: RP = (1 − α)R1 +