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Unformatted text preview: ACTL3004: Week 3 Financial Economics for Insurance and Superannuation: Week 3 The Mean Variance Portfolio Theory 1 / 32 ACTL3004: Week 3 The MeanVariance Portfolio Theory Motivations Motivation 1 Consider an individual with utility function u ( · ) . We can use a Taylor series expansion around the expected end of period wealth W : u ( W ) = u ( E [ W ]) + u ( E [ W ])( W E [ W ]) + 1 2 u 00 ( E [ W ])( W E [ W ]) 2 + ∞ X n = 3 1 n ! u ( n ) ( E [ W ])( W E [ W ]) n Supposing the technical conditions are satisfied, his expected utility can be seen to be E [ u ( W )] = u ( E [ W ]) + 1 2 u 00 ( E [ W ]) σ 2 W + E " ∞ X n = 3 1 n ! u ( n ) ( E [ W ])( W E [ W ]) n # so this indicates a preference for u ( E [ W ]) and dislike for σ 2 W (recall the properties for u ( · ) ) 2 / 32 ACTL3004: Week 3 The MeanVariance Portfolio Theory Motivations Motivation 2 Suppose the utility function of a riskaverse investor has the quadratic form u ( w ) = w dw 2 , where w < 1 2 d and d ≥ 0. Suppose an investor can choose from one or any combination of two available securities with returns e r 1 and e r 2 with respective means and variances E ( e r 1 ) = z 1 , Var ( e r 1 ) = σ 2 1 and E ( e r 2 ) = z 2 , Var ( e r 2 ) = σ 2 2 , and covariance Cov ( e r 1 , e r 2 ) = σ 12 . Let w 1 and w 2 be the proportion of assets invested in securities 1 and 2, respectively. Thus, w 1 + w 2 = 1 and the portfolio return will be e R p = w 1 e r 1 + ( 1 w 1 ) e r 2 . 3 / 32 ACTL3004: Week 3 The MeanVariance Portfolio Theory Motivations Motivation 2 Assume that the initial wealth is 1. For arbitrary distributions of returns, the wealth after one period becomes e W = 1 + e Rp Therefore, the expected utility becomes E h u e W i = E h u 1 + e Rp i = E 1 + e Rp d 1 + e Rp 2 = ( 1 d ) + ( 1 2 d ) E e Rp dE e R 2 p = ( 1 d ) + ( 1 2 d ) μ p d μ 2 p + σ 2 p = ( 1 d ) + h ( 1 2 d ) μ p d μ 2 p i d σ 2 p . So, for a fixed and known E e Rp = μ p , maximizing this expected utility is equivalent to minimizing the variance of the portfolio Var e Rp = σ 2 p . The variance of the portfolio can be expressed as σ 2 p = Var [ w 1 e r 1 + ( 1 w 1 ) e r 2 ] = w 2 1 σ 2 1 + ( 1 w 1 ) 2 σ 2 2 + 2 w 1 ( 1 w 1 ) σ 12 . 4 / 32 ACTL3004: Week 3 The MeanVariance Portfolio Theory Motivations Motivation 3 For an arbitrary utility function we have found that E [ u ( W )] = u ( E [ W ]) + 1 2 u 00 ( E [ W ]) σ 2 W + E " ∞ X n = 3 1 n ! u ( n ) ( E [ W ])( W E [ W ]) n # = u ( E [ W ]) + 1 2 u 00 ( E [ W ]) σ 2 W + ∞ X n = 3 1 n ! u ( n ) ( E [ W ]) E ( W E [ W ]) n assume now that asset returns are multivariate normally distributed. With wealth a linear function of asset prices, wealth is also normally distributed....
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