4. Numerical Illustrations
In this section, we use the numerical illustrations to explain how fund managers will respond to different environments of the financial markets in order to fulfill the criterions of the delegated funds. We suppose that the initial wealth is 1 billion dollars.
The parameters for interest rate and stock index in our models are given as follows.
According to the research of Chan, Karolyi, Longstaff, and Sanders (1992), the suppose that the life of the delegated fund is 2 years, i.e. T 2, and separate the time interval into two periods for which time 0 to 1 is the first period and time1 to 2 is the second one. Revising occurs at the point of time 1 which is just the half of the life of the delegated fund, and the proportion of wealth of each investment instrument adjusts every 5 days.
First, we focus on the effect of changing the parameters, K and 1 K , to the 2 position of stock index. We are given that 0.5 and 0.2, and draw the
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weight of stock keeps decreasing until the time 1 which is also the time for revising, and the weight of stock for the next period is very similar to the previous one for which the weight keeps decreasing until the end of this interval. Why is the position of stock at the begging higher than the end of this period and then keep decreasing?Because the fund managers will investment some assets with higher volatilities at the begging in order to fulfill the criterions of minimum guaranteed returns, and with time near the end of the period the fund managers have to reduce holding these assets to prevent the wealth influenced by the downside risks.
Table 1. Parameter values used in the numerical analysis
Parameter Descriptions Notation Parameter Values Initial value for short-term interest rate 0.02
Risk premium for interest rate risk 0.00075 Parameters describing interest dynamics 0.06 Parameters describing interest dynamics 0.25 Volatility (standard deviation) of interest rates 0.02 Correlation between stock index and interest rates -0.25
Risk premium for market risk 0.03
Volatility (standard deviation) of stock index 0.2
Coefficient of risk aversion 0.2
Utility weight of first period 0.5
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difficulty to get the target for fund managers. Without surprisingly, the fund managers must raise the proportion of wealth of the stock at the beginning, because only these assets within high risks can supply enough returns to match our target. Moreover, from figure 9 to 12, we can find that fund managers even need to short bond portfolios to increase the holding of stock index. However, we can also find a weird phenomenon where increasing the weight of the stock accompanies with increasing the weight of the cash. This is just because holding too much high risk assets will make fund managers facing too much downside risks. Therefore, the fund managers need to hold certain proportion of cash to stabilize their fund level process.In addition to that, if we let the difference between K and 1 K be a constant, 2 e.g. 0.06, 0.12, 0.18 and 0.24, we find that the weights of stock, bond, and cash in the second period are very like the same as long as the difference between K and 1 K 2 is the same, even under different scenario of K . In other words, when fund managers 1 are deal with the problem of asset allocation for the next period, what they most take into consideration is the relative mandates to the previous period but not the absolute mandates they are really given. This result can be concluded from comparing figure 1, 5, and 9, figure 2, 6, and 10, figure 3, 7, and 11, or figure 4, 8, and 12.
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Figure 1. Optimal portfolio proportions given K = 1.06, K = 1.12 1 2
Figure 2. Optimal portfolio proportions given K = 1.06, K = 1.18 1 2
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Figure 3. Optimal portfolio proportions given K = 1.06, K = 1.24 1 2
Figure 4. Optimal portfolio proportions given K = 1.06, K = 1.30 1 2
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Figure 5. Optimal portfolio proportions given K = 1.18, K = 1.24 1 2
Figure 6. Optimal portfolio proportions given K = 1.18, K = 1.30 1 2
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Figure 7. Optimal portfolio proportions given K = 1.18, K = 1.36 1 2
Figure 8. Optimal portfolio proportions given K = 1.18, K = 1.42 1 2
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Figure 9. Optimal portfolio proportions given K = 1.30, K = 1.36 1 2
Figure 10. Optimal portfolio proportions given K = 1.30, K = 1.42 1 2
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Figure 11. Optimal portfolio proportions given K = 1.30, K = 1.48 1 2
Figure 12. Optimal portfolio proportions given K = 1.30, K = 1.54 1 2
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Second, if we let parameter K equal to parameter 1 K , we can find that the 2 changing processes of weight of each investment instrument in the first period will depend on the selection of parameter K from figure 13 to 16, and the changing 1 processes of weights in the second period are similar to each other where fund managers will hold almost only bond portfolios. This is because when fund managers have reached the target of given criterions at the end of the first period, they have fulfilled the criterions for the second period at the meantime. Hence, fund managers will lose incentive to pursue higher returns, and they will be more willing to hold assets like bonds to get fixed incomes. This phenomenon also tells us the important of taking multi-period downside risks as a whole risk to do consideration. If we do not have any chance to review the performance of fund managers, i.e. there is only one period in the life of the delegated fund, we may lose the opportunity to pursue higher returns. Furthermore, it will produce principle-agent problem between the investors and portfolio managers and influence fund managers doing asset allocation.
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Figure 13. Optimal portfolio proportions given K = 1.06, K = 1.06 1 2
Figure 14. Optimal portfolio proportions given K = 1.12, K = 1.12 1 2
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Figure 15. Optimal portfolio proportions given K = 1.18, K = 1.18 1 2
Figure 16. Optimal portfolio proportions given K = 1.24, K = 1.24 1 2
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Last, we will do the sensitivity analysis to identify how the changing of and
will affect fund managers to do asset allocation. We are given that K11.12 and
2 1.18
K . If is 0.4, 0.2, -0.2, and -0.6, respectively, then we will find that the weight of the stock at the beginning of each period is lower and lower and the weight of the bond is higher and higher from figure 17 to 20. The decreasing of
represents for the increasing of degree of risk aversion, so fund managers will reduce holding high risk assets like stocks. Besides that, fund managers will more prefer to raise the proportion of wealth of the stock when time is near the end of the first period if they are more risk aversion. This is just because the weight of the stock at the beginning is not enough to support fund managers to fulfill the criterions, they need to keep increasing the holdings of assets with high volatilities. However, why they do not hold more assets like stocks at the beginning, because holding too much high risk assets may make fund managers get stuck into the risks of losing principles. Hence, to restrict the holdings of the weight of the stock at the beginning is necessary, and we can expect that increasing the proportion of the stock within time is a certain result.
Comparing the proportion of the wealth of each asset under different degree of , and we can conclude that the changing of will not take significant impact to the asset allocation for fund managers. This result can be derived from comparing figure 17, 21, and 25, figure 18, 22, and 26, figure 19, 23, and 27, or figure 20, 24, and 28.
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Figure 17. Optimal portfolio proportions given β = 0.25, γ = 0.4
Figure 18. Optimal portfolio proportions given β = 0.25, γ = 0.2
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Figure 19. Optimal portfolio proportions given β = 0.25, γ = -0.2
Figure 20. Optimal portfolio proportions given β = 0.25, γ = -0.6
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Figure 21. Optimal portfolio proportions given β = 0.50, γ = 0.4
Figure 22. Optimal portfolio proportions given β = 0.50, γ = 0.2
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Figure 23. Optimal portfolio proportions given β = 0.50, γ = -0.2
Figure 24. Optimal portfolio proportions given β = 0.50, γ = -0.6
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Figure 25. Optimal portfolio proportions given β = 0.75, γ = 0.4
Figure 26. Optimal portfolio proportions given β = 0.75, γ = 0.2
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Figure 27. Optimal portfolio proportions given β = 0.75, γ = -0.2
Figure 28. Optimal portfolio proportions given β = 0.75, γ = -0.6
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Because the financial wealth is not directly managed by the investors in the most developed countries but by the financial intermediaries like portfolio managers, the principal-agency relationships become more and more important. Generally, the long-term funds which include insurance funds, pension funds, or endowment funds, etc. are predominantly managed by corporate treasures who often delegate the asset management to a third party, thus creating an additional layer of agency. In order to reduce the principal-agent cost in delegated fund management, it is not strange to have some mandates like minimum guaranteed returns and protective mechanism in the contract. Thus, how fund managers do asset allocation within multi-period downside risks on the premises of following these criterions is the issue which we focus on.
Effective and efficient downside control approaches are especially important for the delegated fund managers to deal with long-term funds. In order to control these downside risks and maximize the projected investment yield, we assume that the objective of the delegated fund managers is to maximize the expected utility of wealth of the long-term fund at the end of each period. Furthermore, we also ask the fund managers that the wealth at the end of each period must at least be equal to the criterions which have been promised at the beginning or even be higher than them.