Lecture10Slidesv-4 - Efficient Portfolios Lecture 10:...

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Unformatted text preview: Efficient Portfolios Lecture 10: Mean-Variance: Portfolio Choice with a Riskless Asset ￿ Valentyn Panchenko ￿ Indifference Curves ￿ Mean-variance theory provides a neat separation between Investor preferences and capital market opportunities. ￿ The latter are summarised in the feasible mean-variance opportunity set and its efficient frontier. ￿ The former can be shown with a set of Investor indifference curves Mean-variance theory assumes that Investors prefer: ￿ ￿ (1) higher expected returns for a given level of standard deviation (2) lower standard deviations for a given a level of expected return. Portfolios that provide the maximum expected return for a given standard deviation and the minimum standard deviation for a given expected return are termed efficient portfolios. All others are inefficient. Indifference Curves Optimal Portfolio Choice Negative Exponential Utility ￿ A particularly useful utility function for mean-variance analysis is the negative exponential: u(w) = 1 − e−cw , where w is a measure of wealth at a future date and c is a positive parameter. ￿ Negative Exponential Utility For purposes of portfolio theory it is desirable to state utility in terms of return (the relative change in wealth over the future period: u(r) = 1 − e−cr , Negative Exponential Utility ￿ The negative exponential utility function is especially convenient in a world of normally-distributed outcomes. ￿ Expected utility is the integral of the utility function using the probability distribution as weights. If the former is negative exponential and the latter is normal, expected utility will be a simple function of the mean and variance of the distribution: eu = e − (v/t) ￿ Here, e is the expected outcome, v is the variance of the outcome, and t equals (2/c), where c is the parameter from the investor’s utility function. Parameter t measures the Investor’s risk tolerance. ￿ The greater an Investor’s risk aversion ,c, the smaller is her risk tolerance, 2/c. Inferring Investor Risk Tolerance Inferring Investor Risk Tolerance Indifference Curves Indifference Curves Portfolio utility depends on both portfolio characteristics and the risk tolerance of the Investor in question: pu(p, k ) = e(p) − v ( p) x = e ( p) − t( k ) e ( p) = x + In a diagram with mean (e) on the vertical axis and variance (v) on the horizontal axis, such an indifference curve will plot as an upward-sloping straight line, the intercept of which will indicate the associated portfolio utility. ￿ The slope of such a line indicates the rate at which the Investor is willing to trade expected value (or return) for variance. ￿ But the slope is 1/t(k). ￿ Thus, t(k), the reciprocal of this slope, is the rate at which the Investor is willing to trade variance for expected return. ￿ We can define t(k) as Investor k’s marginal rate of substitution of variance for expected value. v ( p) , t( k ) where e(p) is the expected value (or return) of portfolio p, v(p) is its variance, t(k) is Investor k’s risk tolerance, and pu(p,k) is the utility of portfolio p for investor k. Consider the set of all portfolios that provide a given level of utility, say x. All such portfolios must satisfy the equation: or: ￿ 1 v ( p) t( k ) Inferring Investor Risk Tolerance Inferring Investor Risk Tolerance Indifference Curves An Example ￿ Assume that the riskless rate of interest is 4% and that by investing in a diversified stock index portfolio it is possible to obtain an expected excess return over the riskless investment of 6% per year, with a standard deviation of 15% per year. An amount x is invested in the stock index and an amount (1-x) in the riskless asset. ￿ Our objective is to find a characterization of the efficient frontier. Inferring Investor Risk Tolerance Inferring Investor Risk Tolerance Some useful formulae An Example (cont’d) Let X and Y be random variables with expectations E (X ) and E (Y ) , variances Var(X ) and Var(Y ) and covariance Cov(X, Y ). Let a and b be real numbers, then: ￿ E (aX + bY ) = aE (X ) + bE (Y ) ￿ Var(aX + bY ) = a2 Var(X ) + b2 Var(Y ) + 2abCov(X, Y ) ￿ Var(aX + bY ) = ￿ ￿ a2 Var(X ) + b2 Var(Y ) + 2ab Var(X ) Var(Y )rXY , where rXY is the correlation coefficient between X and Y . Efficient Frontier: e-s diagram Efficient Frontier: e-v diagram Optimal Portfolio Choice: e-v diagram ...
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This note was uploaded on 07/13/2011 for the course ECON 3107 at University of New South Wales.

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