Heterogeneity of Investors and Asset Pricing in a Risk-Value World

E (è) ≥ W₀R

Varying R parametrically allows us to derive all risk-value efficient port-
folios. In the following the function

will be of special importance. From the properties of the risk function
F{F^f < 0, F^zz > 0, and F^zzz < 0), we immediately get:f > 0, f^z < 0, f^zz > 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 (e_ε) + E[f (è)] = ηπ_ε + 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].

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