Risk

Fig.l The risk-value efficient frontier depicts the minimal portfolio risk as a func-
tion of the required expected portfolio return R*. The thin curve represents
the frontier for an endogenous benchmark, the thick curve for an exogenous
benchmark.
3.2 Risk-Value Models With an Exogenous Benchmark
Now consider risk functions with an exogenous benchmark e; e ≥0. Then in
the first order condition (8) for a risk-value efficient portfolio the second term
disappears since portfolio choice has no effect on the benchmark. Hence the
first order condition reads:
-f (⅛) = ηπ, + A; Vε.
(l2)
Again, A > 0 so that η < 0 follows.
Taking expectations yields
-E [f (e)] = η + A (l3)
so that subtraction of (l2) from (l3) leads to
-E[f (e)] + f (eε) = ηθε; V ε. (l4)
l5
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3. References
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7. A MARKOVIAN APPROXIMATED SOLUTION TO A PORTFOLIO MANAGEMENT PROBLEM
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