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Unformatted text preview: Eﬃcient Portfolios Lecture 10:
MeanVariance:
Portfolio Choice with a Riskless Asset Valentyn Panchenko Indiﬀerence Curves Meanvariance theory provides a neat separation between
Investor preferences and capital market opportunities. The latter are summarised in the feasible meanvariance
opportunity set and its eﬃcient frontier. The former can be shown with a set of Investor indiﬀerence
curves Meanvariance 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 meanvariance 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 normallydistributed 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 upwardsloping 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 (1x)
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: es diagram Eﬃcient Frontier: ev diagram Optimal Portfolio Choice: ev diagram ...
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This note was uploaded on 07/13/2011 for the course ECON 3107 at University of New South Wales.
 '11
 valentyn
 Economics

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