[αi(Dτ)ηM(Dτ)/ηi(xi)]2. Note that ]Qizi = 1, by equation (1). Hence if
there are many investors, each having a positive consumption share, then
the average zi will be very small. This holds, a fortiori, for zi2 . Therefore if
there are investors with increasing, with constant and with declining RRA,
then the first term on the right hand side is likely to be close to zero while
the positive second term is subtracted. The second term tends to be higher,
the more heterogeneous investor preferences are.5 Since this heterogeneity
appears to be strong in reality, we conclude that aggregate RRA is likely to
decline.
The intuition for this result can be obtained from the following reasoning.
Given an optimal allocation of claims, a highly risk averse investor i tends
to buy claims xi(Dτ) which increase only little with aggregate dividend
Dτ . Her demand curve xi (Dτ ) is rather flat. Hence her share αi (Dτ )=
xi(Dτ)/Dτ tends to be high [low] when Dτ is low [high]. The opposite is
true of an investor with low RRA. Therefore in the low dividend states the
highly risk averse investors dominate the market so that aggregate RRA
turns out to be high. In the high dividend states low risk averse investors
dominate the market so that aggregate RRA turns out to be low. Thus,
aggregate RRA tends to decline with increasing dividend. Hence, we regard
declining aggregate RRA as the normal case and will analyze asset pricing
for this case.
1.2 The Pricing of the Market Portfolio
We investigate the pricing of the market portfolio in a perfect and complete
capital market. We consider a continuous time economy with an infinite
horizon. Since we are interested in the pricing impact of declining aggregate
RRA, we take the instantaneous risk-free rate rf as exogenously given and
non-random.6 The market portfolio pays an exogenously given dividend
stream which is governed by a geometric Brownian motion
dDt = μDDtdt + σDDtdWt , 0 ≤ t < ∞, (3)
where the instantaneous drift μD and the instantaneous volatility σD are
assumed constant. Wt is a one-dimensional standard Brownian motion and
the initial dividend D0 is positive. This represents a simple setting with the
5 The second term approaches zero if one investor buys a very large fraction of the ag-
gregate dividend and the other investors buy very little. This can happen if the aggregate
dividend is very low or very high and marginal utility of consumption of the first investor
relative to that of every other investor goes to infinity for very low resp. very high levels
of consumption.
6The impact of heterogeneous time-preferences of investors on the term structure of
interest rates is analyzed in Lengwiler (2004), for example.