E (è) ≥ W0R
(6)
Varying R parametrically allows us to derive all risk-value efficient port-
folios. In the following the function
f (è) : =
∂F (è)
∂ è
(7)
will be of special importance. From the properties of the risk function
F{Ff < 0, Fzz > 0, and Fzzz < 0), we immediately get:f > 0, fz < 0, fzz > 0.
3.1 Risk-Value Models With an Endogenous Bench-
mark
Using this notation we can write the first order condition for a solution to the
minimization problem (4) - (6) with an endogenous benchmark è = E(è), (η
is the Lagrange-multiplier of the budget constraint (5) and λ the Lagrange-
multiplier of the payoff constraint (6) ) as6 (after dividing by the probability
density)
-f (eε) + E[f (è)] = ηπε + λ; ∀ ε. (8)
As E(π) = 1, taking expectations yields
0 = η + λ (9)
6The solution of the minimization problem exists and is unique if a) for every e satisfying
the constraints (5) and (6) the value of the objective function is finite and b) for every
λ,η and e satisfying the first order condition (8) E[eτr] is finite [Back and Dybυig 1993].
13
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