Chapter 2 Exchange Rate Predictability in International Portfolio Selection
2.7. Numerical illustration
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2.7. Numerical illustration
This section employs the constant case proposed in proposition 2.2 to illustrate the learning effect on the exchange rate return and investigate its affect on the cross country’s asset allocation problem. The following analysis shows that the learning effect on asset allocation can significantly outperform portfolio selection without learning in terms of the terminal wealth of the investor. This section also shows the difference between the asset weights with and without adopting the learning process and provides the conditional variance estimates for a steady state investor. The time horizon of the investors is 20 years. The learning process of the exchange rate return is employed to study the optimal portfolio during the observed period. Appendix A.2 lists the parameters used in the following numerical illustration.
10,000 Monte-Carlo simulations were performed on three scenarios of the exchange rate process, i.e., given the mean return is equal to 1%, 3%, and 5%. These simulations reveal the difference between the terminal wealth with and without learning adjustments. The associated utility gain or loss can also be calculated in the same manner. Figures 2.1, 2.2, and 2.3 present the histogram of the difference on the terminal wealth due to learning. These figures show that the difference increases when mean return of the exchange rate process increases. The probability histogram of the simulations shows that the optimal asset allocation employing the learning can significantly improve terminal wealth performance for strategic investors.
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Figure 2.1 Histogram of the terminal wealth given mean return of the exchange rate process 1%.
Figure 2.2 Histogram of the terminal wealth given mean return of the exchange rate process 3%
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Figure 2.3 Histogram of the terminal wealth given mean return of the exchange rate process 5%
Figures 2.1-2.3 illustrates the histogram of difference on the terminal wealth given mean return of the exchange rate process (1%, 3%, and 5%). 10,000 Monte-Carlo simulations were performed to reveal the differences between the terminal wealth with and without learning adjustments. The time horizon is 20 years.
To identify the learning effect, this study defines the outperform benchmark as the ratio of the extra gain in terminal wealth due to learning to terminal wealth without learning adjustment. Table 2.1 summarizes the outperform benchmark based on various mean returns of the exchange rate processes. This table shows that an increase in the mean return of the exchange rate process can increase the outperform benchmark. Thus, learning is a significant factor in strategic asset allocation.
Table 2.1 Outperform benchmark based on the exchange rate processes Return of the exchange
rate process
1% 3% 5%
Outperform benchmark 2.9% 11% 18.5%
10,000 Monte-Carlo simulations were performed. The outperform benchmark is defined as the ratio of
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the extra gain in terminal wealth due to learning to the terminal wealth without learning adjustment.
The time horizon is 20 years.
Table 2.2 presents the mean returns of the exchange rate process through learning.
This table shows that learning adjustment increases the return on the exchange rate by 7%, confirming that the investor can increase yields through learning.
Table 2.2 Mean return of exchange rate with and without learning
Without learning 1% 3% 5%
With learning 1.07% 3.23% 5.39%
Increase percentage 7% 7.33% 7.8%
10,000 Monte-Carlo simulations were performed to determine the mean returns of the exchange rate process through learning. The time horizon is 20 years.
The figures and tables above show that incorporating exchange rate predictability into the investor’s decision can significantly improve returns and achieves higher terminal wealth.
Figures 2.4 and 2.5 present the optimal asset allocation of the asset mix with and without learning adjustments. The general pattern of these two figures shows similar results. The optimal behavior of the investor does not change significantly through observing the proportions in the foreign money market account, the domestic stock index, and the foreign stock index. The optimal proportions in the domestic discount bond and the foreign discount bond increase as the investment date approaches maturity.
Figures 2.4 and 2.5 show the patterns of the asset allocation which have learning adjustments or not. These figures show the optimal weights of the asset mix with and without learning adjustments. The time horizon is 20 years.
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Figure 2.4 Asset allocation with learning adjustments
Figure 2.5 Asset allocation without learning adjustments
Figure 2.6 shows the difference between asset allocation with and without learning. Under the learning effect, the investor will increase investment in foreign
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cash and domestic cash, and simultaneously decrease investment in the foreign stock index, the foreign discount bond, and the domestic stock index.
Figure 2.6 Proportion difference in asset weights with/without learning adjustments.
This figure shows the difference in optimal weights of the asset mix with and without learning adjustments. The time horizon is 20 years.
Following content illustrates compositions of the asset allocation in figures 2.4 to 2.6. In this study, the optimal asset allocation can be decomposed to four mutual funds.
Excluding the domestic riskless asset, this study illustrates other mutual funds in figures 2.7 to 2.15. These figures describe three mutual funds and effect of exchange rate predictability. Each figures show the mean value of the weight of five risky assets with 10,000 time’s Monte Carlo simulation. Green line and red line in each pattern illustrates the upper bound and the lower bound of the simulation error (two times of standard deviation).
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Figure 2.7 The international myopic portfolio with learning adjustments.
(𝛒𝐢𝐣 = 𝟎. 𝟕)
Figure 2.8 The domestic interest rate hedge portfolio with learning adjustments.
(𝛒𝐢𝐣 = 𝟎. 𝟕)
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Figure 2.9 The cross-country interest rate differential hedge portfolio with learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕)
Figure 2.10 The international myopic portfolio without learning adjustments.
(𝛒𝐢𝐣 = 𝟎. 𝟕)
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Figure 2.11 The domestic interest rate hedge portfolio without learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕)
Figure 2.12 The cross-country interest rate differential hedge portfolio without learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕)
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Figure 2.13 Proportion difference in international myopic portfolio with/without learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕)
Figure 2.14 Proportion difference in domestic interest rate hedge portfolio with/without learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕)
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Figure 2.15 Proportion difference in cross-country interest rate differential hedge portfolio with/without learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕)
Figures 2.7 to 2.15 compare the portfolio weight in high correlation environment (𝛒𝐢𝐣 = 𝟎. 𝟕; 𝐢, 𝐣 = {𝐞, 𝐫𝐝, 𝐫𝐟, 𝐒𝐝, 𝐒𝐟}). Figures 2.7 to 2.9 illustrate the weight of three mutual funds, considering exchange rate predictability. Figures 2.10 to 2.12 describe the portfolio weight, excluding learning adjustment. Figures 2.13 to 2.15 show the pattern of difference from learning effect. In figures 2.13, the investor will increase the investment in foreign currency and decrease the investment in other foreign assets.
The exchange rate prediction has large effect in the foreign currency, which leads to such result. In figures 2.14, there is no difference in domestic interest rate hedge portfolio with/without learning adjustments because learning effect will not change the return of the domestic asset. Moreover, the exchange rate predictability did not change the domestic interest rate hedge portfolio. Therefore, this study will not discuss the portfolio in the following content. The learning effect of cross-country interest rate differential hedge portfolio will increase the weight of the domestic bond and foreign bond which shows in figure 2.15 evidently.
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Figure 2.16 Asset allocation with learning adjustments (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟐)
Figure 2.17 Asset allocation without learning adjustments (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟐)
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Figure 2.18 Proportion difference in asset weights with/without learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟐)
Figure 2.19 Asset allocation with learning adjustments (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟓)
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Figure 2.20 Asset allocation without learning adjustments (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟓)
Figure 2.21 Proportion difference in asset weights with/without learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟓)
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Figure 2.22 Asset allocation with learning adjustments (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟖)
Figure 2.23 Asset allocation without learning adjustments (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟖)
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Figure 2.24 Proportion difference in asset weights with/without learning adjustments. (𝛒𝐢𝐣 = 𝟎. 𝟕 and 𝛄 = 𝟎. 𝟖)
Figures 2.16 to 2.24 investigate the effect in different risk preference and exchange rate predictability. Figures 2.16, 2.19, and 2.22 describe the asset allocation considering exchange rate learning with different risk preference. These patterns show that the investor increases investment in risky assets as γ increases. Similarly, figures 2.17, 2.20, and 2.23, which do not considering learning effect, have shown the same results. Comparing with figures 2.18, 21, and 2.24, in which they have also shown the similar patterns, where asset allocation changes as γ increases. Our result concludes the risk preference will change portfolio selection.
From figures 2.7 to 2.24, the change in three mutual funds results in change in portfolio selection with learning prediction. In international myopic portfolio, investor usually increases his portfolio in foreign cash and decreases other foreign assets. In cross-country interest rate differential hedge portfolio, investor usually increases bond portfolio. These results finally conclude the pattern in figure 2.6.
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To illustrate the variance of estimate parameters converge. Figure 2.25 shows that the conditional variance converges to a steady state. If the time horizon is 15-20 years, the learning process reduces the variance of b ii, =1, 2. This figure shows that the duration of the time horizon must be longer than 16 years to control the variance and converge to the steady state.
Figure 2.25 The conditional variance estimates over the time horizon.
This figure shows that the conditional variance converges to the steady state. The time horizon is 20 years.