7.2 Implications of model ambiguity for the optimal stock holdings
In this section, I compare the optimal shares selected using the model combination, with the same quantities
obtained employing the best candidate models. Investors choose different shares to invest in the risky asset
according to their level of risk aversion and investment horizon. I start reporting the results relative to each
single model for three different level of risk aversion and for an investment horizon equal to one period.
However, since returns are i.i.d, this particular choice of the time horizon does not have any impact.
Under the assumption that the Gamma distribution is the optimal model for stock returns, for all three
samples under consideration I obtain the following results:
|
N=7242 |
a* |
|
R.A=2 |
Л |
|
R.A=6 |
0.41 |
|
R.A=10 |
0.27 |
|
E, N=5921 |
a* |
|
R.A=2 |
Л |
|
R.A=6 |
0.77 |
|
R.A=10 |
0.45 |
Table VIII
|
C, N=1321 |
a* |
|
R.A=2 |
0 |
|
R.A=6 |
0 |
|
R.A=10 |
0 |
In contrast, under the assumption that the Gaussian distribution is the optimal model, we have:
|
N=7242 |
a* |
|
R.A=2 |
1 |
|
R.A=6 |
0.99 |
|
R.A=10 |
0.594 |
|
E, N=5921 |
a* |
|
R.A=2 | |
|
R.A=6 | |
|
R.A=10 |
0.88 |
Table VIV
|
C, N=1321 |
a* |
|
R.A=2 |
0 |
|
R.A=6 |
0 |
|
R.A=10 |
-0- |
In the case of the double Gamma distribution, the results relative to the entire sample, are very similar
to those reported by Avramov (2000) for the i.i.d model (Figure 5, p63). The comparison is made even
easier from the fact that I used his same values for the coefficient of the risk aversion. The values reported
in table VIII seem very reasonable also when compared to the most recent evidence from the 2001 Survey
of Consumer Finances21. Among the families holding stocks, corresponding to 21.3% of the interviewed
population, on average the median value of wealth invested in stock is around 32.9% (which is in between
0.41 and 0.27). In contrast, the values obtained for the Gaussian distribution tend to overestimate the actual
share, most likely because this model is not able to account for the fat tails of the distribution which can
strongly affect the results.
In analyzing the results for the case of expansion (E), it can be noticed that for the double Gamma I
obtain values very close to those of Guidolin and Timmermann under: no predictability, bull state probability
equal to one, investment horizon equal one and by the same values for the risk aversion (Figure 5, Guidolin-
Timmermann (2002)). On the contrary, these values are very different from those obtained using the Normal
distribution.
Unfortunately the case of contraction does not provide very interesting results, due to the fact that with
any model the estimated average of stock return is negative, causing the optimal share to be zero for any
value of the risk aversion. Similar results are also reported by Guidolin and Timmermann in the case of
21 All the values reported are obtained from Tab le B, pg 13 of “Recent changes in U.S. Fam ily Finances: Evidence from the
1998 and 2001 Survey of Consumer Finances”. Federal Reserve Bulletin.
20