An investor’s objective is to maximize the expected utility of end-of-period (time t = T) wealth by allocating his wealth between two assets, a risky security (Risky Asset) and a riskless security (Risk Free Asset), over some investment horizon [0,T], which is called a strategy and expressed by the risky asset weight. Risky Asset’s characteristics are determined by its price S under geometric Brownian motion with drift and volatility. Using utility function, the problem is as follows.

A constant relative risk aversion utility function was used in [1], henceforth “The Merton Model”:

If, we set logarithm function. Figure 1 depicts CRRA utility functions with, , and 1.6.

Figure 2 and Figure 3 depict the utility functions to be used in this paper. As results, Figure 2 leads to a protective put option “like” investment strategy (minimum asset level “M”) and Figure 3 leads to a covered call option “like” investment strategy. The interpretation of the Figure 3 utility function is that once the pension

asset value achieves “L” (indicating its liability level) from under-funding status, the utility will not increase even if the asset value increases. In other words, achieving the full-funding level is the first priority and after that is satisfied, wealth no longer needs to be increased.

In this paper, the mathematical formation of the utility function is described by using both kinks at the minimum wealth level and the target wealth level. This is, however, not saying that the constraints are simultaneously effective. The case of both constraints is presented in the Appendix A.

An objective is set to maximize the expected utility (denoting U) of end-of-period wealth by allocating wealth between two assets, a risky security (Risky Asset) and a riskless security (Risk Free Asset) over some investment horizon [0,T], which is called a strategy and expressed by the risky asset weight. does not become 0 nor negative. Others are as follows.

・ The portfolio is managed by a strategy process and it consists of investing in Risky Asset and Risk Free Asset.

・ The asset amount, , consists of the portfolio assets and derivatives (options) if any.

・ Risky Asset’s characteristic is set as its price S under geometric Brownian motion with drift and volatility.

Brownian motion is on a complete filtered probability space with initial value almost surely. Filtration is all time t available information for the pension fund. Setting a finite time T, satisfies the usual conditions and the augmented sigma-field generated by up to time t. In general expression, the process is a controlled state process valued in and satisfying:

The decision of the risky asset weight is the control. (See Appendix B.) Generally, the control is set as, and it is a progressively measurable process valued in the control set A, a subset of. The Borel function b and on satisfy the usual conditions in order to ensure the existence of a strong solution to the above stochastic process. This is typically satisfied when b and satisfy a Lipschitz condition on (t, x) uniformly in A, and satisfies a square integrability condition.

In this paper, Risky Asset’s characteristics is set as its price S is under geometric Brownian motion with drift and volatility, and Risk Free Asset’s interest rate is supposed to be fixed as.

Regarding Risky Asset, P-measure of and its Equivalent Martingale Measure Q-measure are assumed to exist. We analyse the following stochastic process:

is defined by and. (7)

The utility functions treated in this paper are shown in mathematical form below. Setting the CRRA utility maximization problem as follows, we denote two features of the utility function: There are kinks at the minimum level (M) and at target level (L) of asset wealth. M is for modeling of a minimum solvency level, and L is liability, which should be constant. (See Figure 2 and Figure 3.)

Mathematical expression is as follows:

Subject to:

P: Market measure. Q: Risk neutral measure.

For the convenience, is defined. (Merton model)

Again, the mathematical formation of the utility function is described by using both kinks at the minimum wealth level and the target wealth level. This is, however, not saying that the constraints are simultaneously effective.

Here we define the conjugate value function of as for utility and its dual utility function of such that:

This implies:

[14] shows and also, setting, the below is obtained. (Under. means and so forth.)

is converted to:

The relationship is as follows. (The below, including, reconfirms x is the initial value of wealth.)

We use x and y for general variables. The maximization problem is re-translated as fol-

lows with some simplification for interception (No interception affine transform. Notation of and others are used.);

(For illustrative purpose, see Figure 4 for u(x) and Figure 5 for its dual function.)

Note that is continuous at. To find the optimal, we prepare the following:

H[z] is a hebiside (step) function having value 1 only the area z and others 0.

Using Merton case (See Appendix A.) and setting the Radon-Nikodym derivative as and the Lagrangian coefficient as before, we already know the below:

and

This leads to the following:

The optimal solution which sets at t = T, is as the follows. The solution will be to buy or sell options. The self-financing condition is filled by using as below. is depicted in [13]. To obtain an analytical form of of the case, the stopping time problem of HJB is related to this issue ([13]). We

are supposing so to speak an American type option and any time arriving. It is natural

to invest all money in Risk Free Asset to secure L at t = T. In such a case, is thought to be,

at any time, in Q-measure, to act as satisfy. In case any time arriving

, all money is invested to Risk Free Asset and we omit this description henceforth. De-

rivative investment researches support those above. ([15,16] and etc.)

The optimal solution:

for the left side (For reference, see Figure 2), and;

for the right side (For reference, see Figure 3).

“Put(K,)” and “Call(K,)” in equations above are meaning Black-Scholes put option and call option prices respectively with the parameters of strike price K and maturity with underlying asset. Other parameters are as follows. Again, is decided by a self-financing condition with option premium taken into account.

a scalar and it varies if time varies.

The following presents the intuition of the solution above.

The solution (31) is a “protective put option” “like” strategy. Because the utility suddenly has negative infinite value if the pension asset value becomes below “M.” In order to avoid investor’s wealth value’s being below “M,” the solution will be to buy a put option of asset with strike price M. In reality, the strike price will be adjusted by some fixed numbers because to buy the put option, the option premium amount should be paid.

The solution (32) is a “covered call option” “like” strategy. Because there is no incentive to let the wealth increase once the wealth achieves the target amount “L.” The asset consists of and a short position of the call option. To sell a call option (covered call) meaning that the investor wants to achieve the target amount but there is no need to be above level “L.” As is the same of the protective put option case, the strike price will be adjusted by a fixed figure because selling call options makes money, and the money should also invested as the same way of.

The solution of the case of both constraints (lower and upper bounds) is in Appendix A.

The following are assumptions are made:

・ Risky Asset: Return 5% p.a., Volatility 20% p.a., Brownian motion.

・ Cash: Return (risk free return) 1% p.a., Volatility 0%.

・ CC Strategy: Utility function is used..

・ Pension fund total asset value: Initially 70 at t = 0.

・ Pension Liability: Initially 100 at t = 0. Will increase at 2% p.a.

・ Total periods: 20 years.

As a benchmark strategy, STD Strategy is targeting making asset from under-funding (70) to full-funding

The case of always −10% return for Risky Asset. is decided by a self-financing condition with option premium taken into account. is defined as and is the option premium ratio versus the notional amount. In the simulation, the strategies’ actions at the beginning of period t = 1 is set as; is 1.596. Call option strike price 63 (L0/1.596), current price 70, tenor 20, Premium 38%, Principal amount 112 (70 × 1.596). Initial money available = “70 + option premium 41.7” = 111.7. Initial money is invested into Risky Asset 70% and Cash 30%. The return of Risky Asset t = 1 decides which kind of call option should be sold in the Covered Call (CC) Strategy at the beginning of period t = 2 and so forth for the full multi-period case. Here all returns are −10%, so, CC Strategy’s leveraging and investing into Risky Asset (Standard portfolio) makes the difference between pension asset and liability worse as a result.

(100*(1 + 2%)^20) in 20 years. The expected return of STD should be 3.835%. This leads to’s composition of Risky Asset around 70% and Cash around 30%.

In CC Strategy, the underlying asset of call option is, and the notional amount is.

Setting parameters differently versus Case 1, I checked the sensitivity of the volatility increase and expected return improvement. Table 3 shows those new parameters. The return distribution is shown in Figure 12.

This sensitivity analysis shows the following:

・ In Table 3, you see the ratio of CC to STD is superior by 46.5% (almost half and half) in Case 1. Case 3 is higher volatility of Risky Asset and the ratio of Case 3 is as large as 58.9%. Instead, in Case 2, which is the smaller volatility case, the ratio is only 19.3%.

・ In case of a higher expected return of Risky Asset (Case 5), we see the ratio is 30.3% and in case of a smaller expected return, we see the ratio is 67.5%.

Therefore, the following observations can be made:

・ Reducing volatility makes less risk averse and less cash position. As a result, portfolio volatility increases and the premium earned by covered call increases.

・ Increasing volatility makes more risk averse and more cash position. As a result, portfolio volatility decreases and the Covered Call effect decreases.

・ In case expected return decreases, more Risky Asset ratio makes more Covered Call merit.

This shows 10,000 times Monte Carlo simulation results of total 20 years returns like Figure 8. Several cases of the parameters of drift and volatility are tested. The upper part of the table shows parameters. The middle part of the table shows characteristics. The bottom part of the table shows the results of the comparison for several cases. The ratios of simulation results, which shows Covered Call Strategy (CC) is superior to Standard Strategy (STD) regarding the end of returns (the end of asset values), are shown.