λi as exogenous. Later on, sensitivity of the results with respect to λi will
be discussed.
As discussed before, empirical results on investors’ risk aversion are mixed.
To determine the values for the risk aversion parameters we refer to recent
empirical estimates of aggregate RRA implied by option prices, but stick to
relatively conservative specifications.
-insert Figures 1 to 3 here-
Figures 1 to 3 illustrate the simulation results. The upper graph shows the
shares of claims bought by the three investors as a function of the aggregate
supply of claims. These shares always add up to 1. The lower graph shows
the implied aggregate RRA as a function of the aggregate supply of claims
(fat curve) and an approximation of the fat curve (thin curve) which is used
later. Note that the scale of X , the aggregate supply of claims, is irrelevant
since all investors have constant RRA.
Figure 1 may be viewed as the ”normal” case. The three investors have
RRA levels (5; 3; 1). The weights (1 /λ) are given by the vector (1; 3/5; 1).
As indicated in the upper graph, given a very low aggregate supply of claims,
the most risk averse investor 1 buys almost all available claims, but her share
declines quickly since, first, investor 2 with RRA 3 quickly raises her share
and, second, the least risk averse investor also increases her share gradually.
The RRA-vector (5; 3; 1) violates the condition for a risk aversion regime
shift given in Lemma 2. Therefore the slope of the aggregate RRA curve
does not approach zero for low levels of supply. The lower graph shows that
aggregate RRA is basically a smoothly declining convex curve. Hence, in
this setting there is no room for a crash.
In Figure 2 we raise investor 1’s RRA from 5 to 20 so that (γ) = (20; 3; 1).
(1 /λ) = (10-3; 20; 1) so that the expected marginal utility of investor 1 is
very high indicating a small amount of wealth allocated to claims on the
horizon market portfolio. Yet the upper graph in Figure 2 shows that she
buys almost all claims as long as the supply of claims stays below 1. The
second and third investor come into play at higher supply levels. Therefore
aggregate RRA stays almost constant at a level of 20 for the entire (0; 1)
range of claims supply. Then it declines sharply in the range (1; 1.5) and
thereafter slowly approaches the level 1, the RRA of the least risk averse
investor. Hence this situation paves the ground for a crash. Important is
the sharp decline in aggregate RRA. This reflects the high pseudo-variance
of the investors’ inverse RRA, i.e. the second term in equation (2).
Consider the third example in Figure 3. Now there exist two investors with
high RRA 20 resp. 18 and one investor with RRA 1; (γ) = (20; 18; 1) and
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