Since raising R raises the risk of the efficient portfolio, λ > 0 so that, by
(9), η < 0. Substituting λ in equation (8) yields
-E[/(ê)] + / (⅛) = η [1 - π,]; V ε (10)
Defining θε = (1 — πε ) this equation can be rewritten as :
-E[/(ê)] + /(⅛) = η θ.; V ε (11)
The investor’s efficient portfolio is characterized by equation (11) and
constraints (5) and (6). θε is the difference between the forward price of
the risk-free claim, 1, and the probability deflated forward price of a state
ε-contingent claim. θε is negative if the probability deflated forward price of
a state ε-contingent claim exceeds the forward price of the risk-free claim. θ
has zero expectation. The higher θε, the cheaper are the state ε-contingent
claims, and the more state -contingent claims the investor buys because a
higher level of these claims raises — /(eε).
The risk-value efficient frontier can be derived by varying parametrically
the required expected return R*. If R* is not greater than 1, then the La-
grange multiplier λ = 0 and risk is zero. All endowment is invested in the risk-
free asset so that eε is the same for every state. λ grows with R* (R* > 1), be-
cause the objective function is strictly convex. Since λ = dE[F(e')]∕d(WoR*)
for efficient portfolios, it follows that the risk-value efficient frontier is strictly
convex as shown in figure 1.
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