By Webster Anderson (Auth.)
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Extra info for Financial Dec Making Under Uncertainty
In particular, for all dates and states Ct = a + b4>t + cWt where a(= —A) depends only on social risk-preference and has the opposite sign of social proportional risk aversion, c(= 1 — p) depends only on social time-preference and is between zero and one, and b = —ac. Proof Following an analysis identical to the construction of the optimal decision rules for arbitrary consumer /, for the consensus consumer at date i = 0, C 0 = Α[φ(1 - p) - 1] + (1 - p)W0 (c') 2 where, as a result of his immortality, 1 — δ = (1 + p + p + · · ·)"* = 1 — p and φ = 1 4- 1/(1 + rF) + 1/[(1 + RF)2] + ··· is interpreted as the present value of a perpetual default-free annuity yielding one unit of wealth now and at every date in the future.
C0) = AJ[1 + 2,(1 + rFe)~l] + We for all i, where φλβ Ξ1+1(1 + rFe)~l. Summing this over all i, (*JPe) Σ [Pi(l + Pli)(At + Q )] = | Σ ^ ) + (1 + r F e )-Ç Atkt + Σ ^ By definition and closure, W^ Ξ £ f W*, so that (W^-Cjf)x (1 + r Me ) = Wf. Similarly, by definition, 1 + rPe = ne/Pe9 and 1 + RFe = (1 + KF)2/(1 + r Fe ). Letting K " 1 = Σ,. [ρ\(1 + ΡΪΚ^· + C 0 )] and using these relationships to substitute into the previous equation, (1 + rPe) = (1 + rF) - 1 (1 + rF) (1 + RF) -2 (1 + RFe) + [K(W% - C%)](1 + rMe) Therefore, there exist constants a, b, and c such that in equilibrium (1 + rPe) = a(l + rF) + 6(1 + RFe) + c(l + rMe) so that the risky portfolio, which features in-the-portfolio separation property, may be replaced by holdings of short- and long-term default-free bonds and the market portfolio of all securities.
The implications of risk-preference for initial consumption cannot be inferred without knowing the time-state path of aggregate consumption. The signs in the table presume λ > 0. willing to take in options. Likewise, the more patient a consumer, the less his consumption and the greater his investment in both the market portfolio and options. However, greater patience also leads to less investment in bonds of each maturity when the society exhibits decreasing proportional risk aversion. This also is to be expected since, in the neutral case where all consumers have constant proportional risk aversion, there is no demand for default-free investments.