6.6.2  Parameter values

In our numerical analysis of the heterogeneous agents model with evolutionary learning we want to choose values for the model parameters which are economically sensible. We assume that there are 250 trading days in one year. The trading interval in our model is 1 day. If we are talking about daily frequencies, then the order of magnitude of percentage price changes is in basis points (1/100 of 1%).

We set the risk-free interest rate to 5% at a yearly basis with daily compounding. Thus rf=0.05/250=0.0002, that is 2 basis points daily. The daily standard deviation of the Dow-Jones Industrial Average during the twentieth century is equal to 1.0830%, which translates to a yearly standard deviation of, if we assume that returns are independently distributed, 1.0830*250 ≈ 17%. We take this number as the standard deviation of the returns, that is σ=0.010830. Dividends are assumed to be iid and the mean dividend is set to 50 yearly, paid daily. Hence the fundamental value of the risky asset under the iid assumption is equal to 50/0.05=1000. The standard deviation of the dividend process is set equal to 10 yearly.

We choose the exponential moving average parameter µ to be equal to 0.18. The maximum fraction of individual wealth a moving average trader can go long or can go short in the risky asset we choose to be equal to γ=1.25 and occurs when the price deviates from the moving average with 7 basis points (λ=0.0007).

The fundamental value expectations parameter v we choose to be equal to 0.99. Because

Etfund(Pt+1)=P* + v (Pt-1-P*) ,
the expected two-day return of the stock price, not corrected for dividends, is equal to
Etfund(Pt+1)-Pt-1
Pt-1
=(1-v)
P*-Pt-1
Pt-1
.
Thus, if the price should decline by 2% to return to the fundamental value P*, then for v=0.99 the fundamental trader expects that the two-day price return is equal to 2 basis points, which corresponds with a one-day price return of 1 basis point.

A broad range of studies, taking into account the full range of available assets, places the degree of risk aversion a for the representative investor in the range of 2 to 4, see for example Friend and Blume (1975), Grossman and Shiller (1981). We set a initially to 4.

Costs for implementing the strategy with fundamental beliefs are higher than the costs for implementing the exponential moving-average strategy. We set the costs of determining the fundamentals to 1 basis point daily (Cfund=0.0001), which is 2.5% yearly. The costs of the moving-average strategy we set to zero.

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