Thus, the integral defining the second component of the optimal portfolio is given by
∫
tT ∂∂ + =⎢⎣⎡ − Λ′Γ′ ΓΓ′ − − − + R s−t ⎥⎦⎤Finally, we rewrite the second optimal portfolio component as ) .
4. NUMERICAL ILLUSTRATION
In this numerical illustration, we use the numerical simulation to demonstrate not only the dynamic behaviors of the optimal portfolio strategy but also the optimal separation portfolio
strategy, which were derived in the previous section. Table 1 reports the set of parameters representing the financial market and background risks. Note that some parameters are consistent with the numerical analysis presented by Battocchio and Menoncin (2004).
Table 1. Parameter Values Used in the Numerical Analysis Parameter
Description
Notation Values Parameter Description Notation Values Interest rate Fix-maturity bond
Mean reversion a 0.2 Maturity K 10
Mean rate b 0.05 Market price of risk λr 0.15 Volatility factor σr 0.02 Defined contribution
process
Initial rate r 0 0.03 Labor income growth µL 0.046
Stock Volatility factor σL,r 0.014
Market price of risk λr 0.15 Volatility factor σL,m 0.153 Market price of risk λm 0.31 Volatility factor σL 0.01 Volatility factor σS ,r 0.06 Initial labor income L 0 100 Volatility factor σS ,m 0.17 Contribution rate γ 0.06
Inflation process Time horizon T 10
Mean rate µπ 0.015 Expense rate e 0.01
Volatility factor σπ,r 0.018 Utility parameter β2 -20
Volatility factor σπ,m 0.136
Volatility factor σπ 0.015
To provide the calculations of labor income and CPI effects on the optimal choice, the increase rate of labor income and rate of inflations are simulated in Figure 4. Figure 5 plots the simulated market values of the cash, the stock index and the bond funds over the investment horizon.
Figure 4 Processes of Labor Income and CPI
Figure 5 Dynamic Processes of the Underlying Assets
Figure 6 Proportion Compositions of Optimal Portfolio and Real Wealth
Figure 6 plots the optimal portfolio holdings of cash, stock and nominal bonds as a function of investment of horizon, i.e., 120 months, for an investment. The real fund wealth is also plotted for comparison.
Figure 7 confirms the separation of four fund effects in the optimal portfolio selection and their behaviors in each component over time. The market portfolio has shown a decreasing trend for stock index and bond fund holdings due to the utility maximization principle. In contrast the state variable hedge portfolio shows a steady pattern for the optimal weight for bond fund and stock index holdings. To hedge the risk from the state variables, the investment strategy needs to hold a fixed proportion of bond fund and also reduce the holding of the stock index. In the inflation hedge portfolio, the investors are required to hold a high proportion of the stock index, up to 80%, to hedge the inflation risks, while only a small proportion of bond fund is sufficient in the hedge portfolio. However, in the labor income hedge portfolio, the investor should short sell
his stock index and the bond portfolio in order to preserve the salary uncertainty over his investment horizon.
Figure 7 Percentage Composition of Optimal Separated Portfolio
In Figure 8, the weights of the stock index in the entire optimal portfolio and the weights for the separated mutual funds are shown as illustration. The results indicate that the inflation hedge portfolio constitute the overwhelming proportion (75%) of the optimal portfolios. While, the state variable hedge portfolio, the market myopic portfolio and labor income hedge portfolio play only minor parts in the optimal portfolio selection problem.
The weights of the bond fund in the entire optimal portfolio and the weights for the separated mutual funds are shown in Figure 9. These results indicate that the inflation hedge portfolio (around 35%) and the state variable hedge portfolio (around 20%) constitute the largest proportions of all long-term financial portfolios. However, the market myopic portfolio and labor income hedge portfolio play only minor parts in the optimal portfolio selection problem. Further work is required to asses more precisely the dynamics of the optimal portfolio under several plausible scenarios.
Figure 8 Proportion of Stock in the Optimal Separated Portfolio
Figure 9 Proportion of Bond in the Optimal Separated Portfolio 5. Conclusion
In this paper, we investigate the asset allocation problem for defined contribution pension fund which considers not only the market risk and interest risk but also the uncertainties from labor incomes, the inflation risk and the management charges. We find that if the pension fund is to maximize the expected exponential utility of its terminal wealth, then there should be five components in its optimal asset allocation. Therefore, the optimal investment behaviors of the pension fund managers are characterized through the relative weights among the separated mutual funds according to their preference, financial market and the influential factors.
In this study, we investigate the optimal asset allocation problem incorporating both the financial and background risks. Pension fund managers must consider the short-term fund performance and the hedge requirements simultaneously. Since there exists background risks that cannot be controlled by the fund managers, a comprehensive dynamic framework is formulated to describe the decision-making process. As our results show, the dynamic portfolio that maximizes the expected utility of the plan participant consists of five components: the market portfolio, the state variables hedge portfolio, the inflation hedge portfolio, the salary uncertainty hedge portfolio and the riskless asset. By explicitly solving the optimal portfolio problem, the numerical results indicate that the inflation hedge portfolio constitutes the overwhelming proportion of stock in the optimal portfolios. In addition, the inflation hedge portfolio and the state variable hedge portfolio constitute the over-whelming proportions of bond holdings. This shows that long-term investors should hedge inflation rate risk by holding the stock index. In addition, these investors should respond to the inter-temporal hedging demands in the financial markets by increasing the average allocation to their bond fund.
To understand the roles of these components, it is necessary to explore the economic interpretations by solving the dynamic optimization problems. With respect to the most common
approach used in the literature, the incorporation of the labor income and inflation risks allows us to characterize the general pattern of the optimal strategy.
6. Appendix A
Following the work in Battocchio and Menoncin (2002), the inflation rates can be considered as a background risk affecting only the wealth growth rate, without altering the amount of wealth that can be invested. Actually, fund managers must invest the nominal fund, even though they are interested in maximizing the growth rate of the real fund. Then, we have to consider two different measures for the same fund. In particular, we call F the nominal fund N and F the real fund.
To model how the real fund behaves; a commonly used approximation is the following: the growth rate of the real fund is given by the difference between the nominal fund growth rate and the consumption price growth rate. If we call P the level of consumption prices, then we can write:
P . dP F
dF F dF
N N −
≅
This is the so called Fisher equation but it gives a log-approximation of the exact relation which must hold between F and F . Actually, the true relation comes from an arbitrage N hypothesis. Considering the inflation rate in this framework means considering a possible arbitrage between the financial and the real market. In fact, the nominal interest rate must compensate the opportunity cost of investing in financial assets. The investor who puts his money in the financial market misses the return he could have obtained from a real investment. If the investor buys today a real good and sells it after one period, he gains the inflation rate. If he buys today a financial asset and sells it after one period, he gains a nominal return. Now, we suppose that a particular market, called the real-financial market, exists. If this is the case, then the corresponding “real-financial” return must be such that the investor is indifferent between the two following opportunities:
1. investing one nominal monetary unit in the financial market and missing the return he could have obtained on the real market;
2. investing one nominal monetary unit in the real-financial market.
Accordingly, if we call φ the real-financial return, φN the nominal financial return, and π the inflation rate, then the true equation that must hold between the nominal and the real fund is as follows:
π,
φ = FN ⋅φN −FN ⋅ F
which means that the return on the real wealth must equate the return on the nominal wealth reduced by the loss due to the increase in the price level. By definition it must be true that:
, ,
, P
dP F
dF F
dF
N N
N ≡ ≡
≡ φ π
φ
and so, after substituting in the arbitrage condition, we can write:
dP. F dF
dF = −
7. Appendix B
Here we carry out the original computation for the modified process of L . Similarly, we need to compute the matrix product
[−M′(ΓΓ′)−1ΓΩ+β2(FNΦ′+γLΛ′)(I −Γ′(ΓΓ′)−1Γ)Ω]′. According to what has already been presented above, we can write:
[ ( ) ( )( ( ) ) ] ,
Thus, the modified differential of the state variables z~ can be written as s
( ~) 0 0 .
In particular, for s< , the solution of the interest rate process is t
~( ) ~( ) ( ) 1(1 ( )) ( ).
But since there is no closed-form solution of the labor incomes process, we have to use the numerical method to simulate ~( )
s L .
7. Appendix C
Here we list the Table 2 which is the monthly personal average salary in finance and insurance industries in Taiwan form 1988 to 2004. The data is found from the Directorate General of Budget of Accounting and Statistics of Executive Yuan in Taiwan on 9/28/2005.
(http://win.dgbas.gov.tw/dgbas04/bc5/earning/ht4561.asp)
Year 1988 1989 1990 1991 1992 1993 1994 1995 1996 NTD 32,335 39,230 39,008 40,971 46,311 49,957 56,472 54,658 56,913 Year 13997 1998 1999 2000 2001 2002 2003 2004 NTD 60,641 59,566 60,352 60,871 62,625 65,767 64,693 66,743
Table 2. Personal average salary in finance and insurance industries in Taiwan
Next, we list the Table 3 to show the Consumption Price Index (CPI) from 1995 to 2004 of the major countries worldwide including Taiwan, America, England, Germany, Japan and Korea.
The data is found from the Directorate General of Budget of Accounting and Statistics of
Executive Yuan in Taiwan on 9/28/2005.
(http://www.dgbas.gov.tw/ct.asp?xItem=1633&ctNode=2252)
Taiwan America England Germany Japan Korea
1995 3.7 2.8 3.4 1.7 -0.1 4.4
1996 3.1 2.9 2.4 1.5 0.1 5
1997 0.9 2.3 3.1 1.9 1.7 4.4
1998 1.7 1.6 3.4 0.9 0.7 7.5
1999 0.2 2.2 1.6 0.6 -0.3 0.8
2000 1.3 3.4 2.9 1.4 -0.7 2.2
2001 0 2.8 1.8 2 -0.7 4.1
2002 -0.2 1.6 1.6 1.4 -0.9 2.7
2003 -0.3 2.3 2.9 1.1 -0.3 3.6
2004 1.6 2.7 3 1.7 0 3.6
Table 3. Consumption Price Index of the major countries worldwide (1995-2004) References
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