February 8, 2021, Christopher D. Carroll Portfolio-Multi-CRRA
Merton (1969) and Samuelson (1969) study optimal portfolio allocation for a consumer with Constant Relative Risk Aversion utility who can choose among many risky investment options.
Using their framework, here we study a consumer who has wealth at the end of period , and is deciding how much to invest in two risky assets with lognormally distributed return factors , , with covariance matrix
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If the period- consumer invests proportion of in risky asset , (so that and vice-versa), spending all available resources in the last period of life1 will yield:
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where is the portfolio-weighted return factor.
Campbell and Viceira (2002) point out that a good approximation to the portfolio rate of return is obtained by
| (1) |
where
Using this approximation, the expectation as of date of utility at date is:
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where the first term is a negative constant under the usual assumption that relative risk aversion
Our foregoing assumptions imply that
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Substituting from (2) for the log of the expectation in (2), the log of the ‘excess return utility factor’ in (2) is
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The that minimizes this log will also minimize the level; the FOC for minimizing this expression is
| (2) |
So
| (3) |
and note that if the first asset is riskfree so that then this reduces to
| (4) |
but the log of the expected return premium (in levels) on the risky over the safe asset in this case is (recalling that we have assumed ), so (4) becomes
| (5) |
which corresponds to the solution obtained for the case of a single risky asset in Portfolio-CRRA.
Campbell, John Y., and Luis M. Viceira (2002): Appendix to ‘Strategic Asset Allocation: Portfolio Choice for Long-Term Investors’. Oxford University Press, USA, Available on 2011/01/22 at http://kuznets.fas.harvard.edu/campbell/papers/bookapp.pdf.
Merton, Robert C. (1969): “Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case,” Review of Economics and Statistics, 51, 247–257.
Samuelson, Paul A (1963): “Risk and Uncertainty: A Fallacy of Large Numbers,” Scientia, 98(4-5), 108–113.
Samuelson, Paul A. (1969): “Lifetime Portfolio Selection by Dynamic Stochastic Programming,” Review of Economics and Statistics, 51, 239–46.
__________ (1989): “The Judgment of Economic Science on Rational Portfolio Management: Indexing, Timing, and Long-Horizon Effects,” The Journal of Portfolio Management, 16(1), 4–12.