As we described, in series 1 simulations, type 2 agents and CAPM believers with low γ exhibit the strongest survivability; in series 2 simulations, type 1 and 2 agents and CAPM believers with low γ exhibit the strongest survivability.
Therefore, we shall now discuss the survival factors of type 1 and 2 agents and CAPM believers with low γ, respectively.
The survival factor of type 2 agents The reason for their strong survivabil-ity is nothing to do with their beliefs. This is because it is impossible for certain
agents’ average lifetime beliefs to always be the most accurate in several different simulations. In other words, the reason why agents with the squared-root (c) util-ity always survive is not likely to be that their lifetime beliefs always happen to be the most accurate. In fact, we take some typical simulation of series 1 in which type 2 agents are the only survivors except for CAPM believers as an example.
We compute the average cdf for each agent’s lifetime beliefs, and compute the largest distance when compared with the true distribution, the Kolmogorov-Smirnov statistics. Then we average the Kolmogorov-Kolmogorov-Smirnov statistics of 5 agents with the same utility, to obtain the Kolmogorov-Smirnov statistics of 7 utility types which are: (0.01866, 0.01925, 0.01905, 0.01867, 0.01941, 0.01818, 0.01888). Obviously, the Kolmogorov-Smirnov statistics for type 2 agents, 0.01925, is not the smallest one. That is, their lifetime belief is not the most accurate one.
However, they survive in this simulation.
Therefore, the only possible reason is that the investment rule derived from maximizing the expected discounted square-root (c) utility function carries some characteristics that support their survival. The investment rule includes two parts, one being the portfolio rule and another is the saving rate. From several examples we find that the portfolio rule of agents with the squared-root (c) utility is not better then that of others. Taking the same case above as an example, we first compute each agent’s lifetime average portfolio, (AP1i, AP2i, AP3i, AP4i, AP5i).
Then we compute the expected payoff: EPi= AP1i∗ Ew1+ AP2i∗ Ew2+ . . . + AP5i∗ Ew5 for each agent i’s lifetime average portfolio, where Ewm,∀m = 1...5 is the probability of a state m occurrence times the dividend of asset m. Then we average the EP value of 5 agents with the same utility, and we obtain the av-eraged EP for 7 utility types: (0.611386, 0.609724, 0.615229, 0.609119, 0.647657, 0.645967, 0.603597). As we can see, the average EP of agents with the squared-root (c) utility, 0.609724, is not the highest one. Therefore, the portfolio rule for agents with the squared-root (c) utility is not the best in this case. However, they indeed survive.
This forces us to notice the characteristics of saving rates that agents choose.
Furthermore, an obvious characteristic has been observed to support the argu-ment that the saving rate plays a much more important role than the portfolio in determining the survival of agents. The obvious characteristic is that the saving rate chosen by agents with square-root (c) utility is most stable for life. Unlike other agents, they never choose a saving rate that is too low in their life. We believe that this is the main reason for their survival.
Let us take 10 typical simulations of series 1 (5 simulations are conducted under the true model being iid, and 5 under the markov process. All of them are simulated under a length of validation period of 100) in which agents with square-root(c) utility survive with some CAPM believers as an example. For each example, we average the maximum, higher quantile point, medium, lower quantile point and minimum of their saving rates over the 100 periods for 5 agents with the same utilty. Then, we average the values in 10 simulations, and obtain Table 5 in which each column shows the values of each utility type, respectively.
Table 5.
min 0.0027 0.20773 0.0019 0.0011 0.0026 0.05245 0.0016
Q1 0.2935 0.42391 0.23788 0.2387 0.4979 0.11594 0.49093
medium 0.575 0.61456 0.91318 0.44552 0.92016 0.72471 0.57388
Q3 0.72688 0.90163 0.95569 0.84291 0.95553 0.94941 0.88032
max 0.96668 0.96034 0.95968 0.96156 0.9685 0.96157 0.96918
mean 0.246825 0.391177 0.406327 0.248742 0.439013 0.297095 0.286658
As we can see, the mean of the saving rates of agents with the squared-root (c) utility, 0.39118, is not the highest. However, they never choose an extremely low saving rate. Figure 7 is the corresponding Box-Whisker plot. The horizontal axis represents the different utility types, and the vertical axis represents the levels of saving rates.
0 0.2 0.4 0.6 0.8 1 1.2
1 2 3 4 5 6 7
Fig. 7. Box-Whisker plot of saving rates
Now let us observe and discuss these points in more detail. We notice that every time when the asset prices for the last period are particularly high on average, the agents’ saving rates decrease for this period. This is because they use the prices of the last period as expected prices to estimate the return from the investing their J rules as described in Section 3.1; therefore, when the prices of the last period are on average high, they predict that the investing will bring a low return and, therefore, they will tend to choose the investment rule with a low saving rate.
Although all agents lower their saving rates, type 2 agents react mildly com-pared to others. Take a typical simulation as an example. In this simulation, the average prices of periods 10, 12, 18 and 20 are particularly high, and the saving rates of agents, excluding type 2 agents, for periods 11, 13, 19 and 21 decrease dramatically. Table 6 shows the saving rates for different types of agents in the first 25 periods.
Table 6.
type 1 type 2 type 3 type 4 type 5 type 6 type 7 period 1 0.794 0.919 0.959 0.8405 0.953 0.4023 0.9566 period 2 0.267 0.3036 0.006 0.006 0.55 0.32 0.049 period 3 0.828 0.9522 0.954 0.869 0.961 0.3225 0.952 period 4 0.8519 0.679 0.917 0.944 0.961 0.8256 0.9578 period 5 0.8519 0.679 0.9327 0.944 0.96 0.8226 0.9578
period 6 0.017 0.463 0.0347 0.015 0.96 0.2964 0.067
period 7 0.8957 0.855 0.958 0.9457 0.96 0.7101 0.959 period 8 0.051 0.396 0.0142 0.1509 0.6511 0.024 0.013 period 9 0.937 0.8614 0.9626 0.949 0.9638 0.956 0.9503 period 10 0.8097 0.5839 0.7807 0.5977 0.9638 0.6641 0.889 period 11 0.005 0.4423 0.006 0.001 0.004 0.003 0.0729 period 12 0.917 0.6024 0.8795 0.8768 0.9562 0.872 0.6662 period 13 0.001 0.2109 0.001 0.003 0.007 0.001 0.001 period 14 0.8869 0.9556 0.956 0.8906 0.951 0.052 0.9567 period 15 0.1179 0.3861 0.007 0.01 0.004 0.153 0.0355 period 16 0.9532 0.9104 0.951 0.949 0.9482 0.956 0.951 period 17 0.9566 0.926 0.957 0.9561 0.9605 0.956 0.959 period 18 0.9566 0.926 0.9552 0.9561 0.9605 0.956 0.958
period 19 0.006 0.241 0.003 0.001 0.007 0.002 0.006
period 20 0.943 0.9422 0.9653 0.9418 0.9591 0.957 0.956 period 21 0.009 0.2209 0.073 0.008 0.1601 0.059 0.036 period 22 0.963 0.9535 0.959 0.9601 0.954 0.96 0.9375 period 23 0.9184 0.6517 0.9345 0.905 0.9235 0.9444 0.907
period 24 0.037 0.3027 0.07 0.453 0.002 0.001 0.619
period 25 0.9564 0.9486 0.95 0.9643 0.9592 0.959 0.9548
We also observe that, when the period that an agent chooses an extremely low saving rate, his wealth decreases terribly. Furthermore, after this kind of event happens several times, he is driven out of the market. Therefore, we conclude that the stablility of saving rates is the most important factor in terms of the survival of agents with square-root (c) utility functions.
The survival factor of type 1 agents In series 2 simulations in which we set α1 = 0, β1 = 1, we found that type 1 agents show strong survivability, too.
We observed their saving rates, and found that, over the 100 periods, the saving rates became nearly fixed at their discount rate, the β value, along the time path.
[Blume and Easley (1992)] provide the analytical solution for the investment rule maximizing expected discounted utility in Theorem 5.1 that states “Suppose
trader i’s objective function is Ei[∞
t=1log(cit)]. If beliefs over states at date t are qit and the value above is finite for any investment rule, then the optimal investment rule is the simple rule δit= βi, αit= qti at each date.”
Our Low-Level GA does the optimality job quite well. Take some type 1 agent in a simulation as an example. His lifetime average investment rule ( ¯δi, ¯αi1, ¯αi2, ¯αi3, ¯αi4, α¯i5)=(0.5897, 0.0589, 0.1847, 0.3006, 0.2484, 0.2074) approximates the theo-retically optimal rule (β, ¯qi1, ¯qi2, ¯q3i, ¯q4i, ¯q5i)=(0.59, 0.0476, 0.1898, 0.3002, 0.2502, 0.2107).
Similarly, we could find some examples in which they survive while their portfolio rule and the accuracy of their lifetime beliefs are not the best among agents. Again, the stablility of the saving rates is the key factor in their survival.
The reason for type 1 and type 2 agents choosing stable saving rates From the Euler equation, the basic condition for choosing consumption over time is known: r = β− [uu”(c)·c(c) ](c˙c).
where r is the rate of return on investment.
Because the coefficient of RRA (Relative Risk Aversion) is defined as−[uu”(c)·c(c) ], the Euler equation described above can be rewritten as follows:
r = β + RRA· (˙c
c) (31)
From Equation (31), we know that when r decreases, if someone’s coefficient of RRA approaches zero, his c˙c must drop dramatically. This will drive agents with the coefficients of RRA approaching zero to choose an extremely low saving rate when the rate of return on investment clearly decreases.
The following table summarizes the coefficient of RRA for each type of agent in series 1 and 2.
Obviously, in series 1, the only agents with constant RRA are type 2 agents (RRA=0.5). This CRRA characteristics free them from the crisis of zero RRA approaching and then win them a stable saving rate path. The reason for the sta-ble saving rate path of type 1 agents in series 4 is similar. Their utility functions are CRRA (RRA=1), too.
The survival factor of CAPM believers with low γ values CAPM be-lievers do not have to guess the true distribution, but instead follow the formula described earlier.
Notice that the market portfolio part (and the corresponding part in the savings rule) reflects the weighted average of other agents’ investment rules (in this period if they have perfect foresight). In particular, it assigns a larger weight to the dominant agents’ portfolio. If his γ value is 0, then the CAPM believer behaves just as a dominant agents’ imitator. Hence, the market portfolio part is the superior position of their rules.
We further analyze the reason why being endowed with a high γ value is not good for survival as follows. In the risk-free portfolio part, the original idea
Table 7.
utility type RRA in series 1 in series 2
approaches 0 1
The reason why some types of agents’ RRA approaches zero is that the wealth on average is quite small, not to mention the consumption. When c approaches zero, those RRA values also approach zero.
of dividing the asset price by its dividend is that the risk of the asset with a higher dividend is always higher, so investing assets with high dividends are not suggested from the risk-free point of view. However, in our economy, the worst return from investing in any asset is just zero. This risk-free portfolio part makes CAPM believers invest less in assets that give higher dividends than the optimal portfolio. Hence, we think that the risk-free portfolio part is an inferior factor for CAPM believers to survive in our economy.
Besides, agents in the economy are assumed to be boundedly-rational. We further assume that they have static expectations, i.e.ρm,tˆ = ρm,t−1. Therefore, what CAPM believers mimic is other agents’ portfolios in the previous period.
However, this only misleads CAPM believers in one kind of situation. Let us explain this step by step.
There are two factors that determine non-formula agents’ portfolios. One is the assets’ dividends, but they are time-invariant. Another is the belief regarding the probability that each asset will give a dividend, and the belief is time-variant.
When the true model follows the iid process, the probability that each state will occur will never change from the beginning to the end. If agents believe that the true model follows the iid process, they will try to fit the same model in each period. Therefore, it does not make much difference if the CAPM believers mimic non-formula agents’ beliefs in the last or current period. Even if they be-lieve that true model follows a Markov process, each row in their Markov table approximates the same time-invariant true iid probability vector. Therefore, the CAPM believers’ mimicking the non-formula traders’ beliefs in different periods is the same as mimicking similar probability functions in different rows. There-fore, when the iid is the true model, the effect of boundedly-rational CAPM
believers mimicking other agents’ investment rules in the last period is similar to the effect where perfect foresight is assumed.
When the true model follows the Markov process, if other agents’ beliefs are trying to fit the stationary distribution of the true model, the effect of boundedly-rational CAPM believers mimicking other agents’ last-period portfolios is the same as the effect where perfect foresight is assumed, because the stationary distribution will also never change from beginning to the end. However, if agents believe that the Markov process is the true model, boundedly-rational CAPM believers will be misled. Take two of our simulations for a comparison. We assume that the true model follows the Markov process, that the lenghth of the validation period is 100, and that non-formula agents’ saving rates are equal to 0.59 in both cases. The γ values assigned to each set of CAPM believers are 0.1, 0.2, 0.3, 0.4, 0.5 in both cases. The only difference is that, in the first case, most of the time agents believe that the true model follows the iid process and, in the second case, the markov process. The CAPM believer with the γ value =0.1 survives in both cases. However, the agent with γ value=0.2 can only survive in the first case. Therefore, we believe that CAPM believers are more or less misled in the second situation, and hence, only the CAPM believers with the lowest γ value can survive. This finding tends to support our hypothesis.
6 Conclusion
Our agent-based simulation results are largely consistent with [Blume and Easley (1992)]. First, we also find that rational log-utility traders survive in series 2 simulations.4Secondly, rational CRRA agents with moderately high RRA coef-ficients such as type 2 agents also survive. In fact, we also try u(c) = cα2, and we explore several levels of parameter α2values. We find that when α2increases (meaning that the agent’s coefficient of RRA decreases), his survivability de-creases. Thirdly, forecasting accuracy does not guarantee survival. The example mentioned in Section 5.2 shows that, despite their lowest lifetime Kolmogorov-Smirnov statistics, 0.01818, the type 6 agents are driven out in that simulation.
Furthermore, to enlarge the differences in the accuracy of lifetime beliefs among agents, in series 3 and 4, we set that the same types of agents have different lengths of validation, say 150, 100, 70, 50 and 30, and that each type of agents shares these levels of length. We find that the survivors are always type 2 agents with lengths of validation equal to 150 and 100 (70 also survive sometimes) in series 3, and type 1 and 2 agents with length of validation equal to 150 and 100 (70 also survive sometimes) in series 4. The survivability of other types of agents with a length of validation equal to 150 is worse than that of type 2 agents with a length of validation equal to 70 in series 3. Hence, we conclude that preference plays a more important role in survivability, although accuracy of belief does still matter.
4 The type 1 agents in series 1 have a different coefficient of RRA from that of the log utility function discussed in [Blume and Easley (1992)].
However, in using the agent-based model, we also find something which was not shown in [Blume and Easley (1992)]. First, there are other types of traders that may survive in the market as well, e.g., the CAPM traders. Secondly, the reason why CRRA traders can survive in the market is because of their implied stable saving behavior. This may help us understand locked-up saving contracts, such as the national annuity program. Finally, if we assume that agents have the same discount factor, the result of [Blume and Easley (1992)] remains robust even if saving rates are determined endogeously. Furthermore, their savings rule also exhibits superiority in determining survival.
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