(1 /λ) = (10; 6; 1). In this case the expected marginal utilities of the highly
risk averse investors are relatively low indicating relatively high amounts of
wealth allocated to claims on the horizon market portfolio. The lower graph
shows that aggregate RRA is almost constant at a level of 19 for a wide
range of claims supply (0; 1.3). Then it drops sharply to a level of 7 in the
range (1.3; 1.8) and then gradually approaches the level 1. The interesting
result is that even though the condition stated in Lemma 2 does not hold,
aggregate RRA basically stays constant in a wide range of low claims supply.
The reason is that the two highly risk averse investors dominate the market
and change their shares of claims relatively little implying aggregate RRA
to be roughly equal to the average of their risk aversion levels. Again, we
may observe a crash.
How robust are the results shown in Figures 2 and 3? Additional simulations
indicate several properties. First, if as in Figure 3, there are 2 investors such
that their levels of RRA are higher than twice the level of the third investor,
then the aggregate RRA curve is similar to that in Figure 3. Second, if
the weights (1 /λ) for the three investors are changed, then the shape of the
aggregate RRA curve remains similar, but the low supply range with almost
constant aggregate RRA will be shorter or longer depending on the wealth
of the most risk averse investors. Third, if there are many investors instead
of one with the same constant RRA γ , this has no effect on aggregate RRA
as long as the sum of the λ1 /γ across these investors stays the same. The
intuition is that all investors with the same RRA buy the same portfolio
of claims up to multiplicative factors reflecting the levels of their initial
endowments.
Therefore, the shape of aggregate RRA shown in Figures 2 and 3 appears
to be robust to a wide set of parameter changes. The crucial condition for
a sharp decline of aggregate RRA in some range of the supply of claims
appears to be that there is a group of investors with high levels of RRA and
another group of investors with low levels such that the high levels exceed
twice the low levels.
4.2 Simulation Procedure
For the simulation of equilibrium price processes we approximate aggregate
RRA by approximating the equilibrium stochastic discount factor through
a sum of power functions. Let St+h be the value of the market portfolio
at some horizon date t + h which defines the aggregate supply of claims at
that date. The random part of the stochastic discount factor, Φt,t+h (St+h),
is approximated by the generalized polynomial
∑E1 α S-+h
Φt,t+h =
EEJ=ι αi S-+δh∣Dt] '
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