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This paper describes optimal investment strategies for kinked utility functions. One example is a CRRA utility function with a kink at a maximum wealth, which leads a covered call “like” strategy and the other is a CRRA utility function with a kink at a minimum wealth, which leads a protective put “like” strategy. This paper introduces analytic mathematical solutions providing a mathematical explanation of a dual utility where Black-Sholes assumption is utilized in the solutions. The intuitive solutions are clear for cases of those kinked utilities but minute mathematical explanation is described. Also a numerical simulation is performed for a covered call like strategy case.

The modelling of problems to maximize the expected utility of end-of-period wealth by allocating wealth between a risky security and a riskless security over some investment horizon is popular among academic circles and investment practitioners. CRRA utility maximization investment strategy problems [1-5] are typical examples.

One of the ways of finding an optimal investment strategy to such a utility maximization problem is to set up the problem with a value function and Hamilton-Jacobi-Bellman (HJB) equation. However, an important condition required to use this method is that the utility function must be twice differentiable.

Recently, kinked utility maximization has been suggested as an important problem to solve. (Basic papers include [

This paper mainly treats the target wealth level case. The optimal investment strategy is to use a covered call option strategy as described in this paper. The other case is well researched and results in a protective put option strategy. Back to 1980’s, there were also discussions about if a protective put option strategy or a portfolio insurance strategy is an optimal strategy or not. [8,9] discussed about one period buying put option strategy with Black-Sholes assumption. [

This paper consists of the following Sections. In Section 2, I set up the problem. In Section 3, I provide the details of the mathematical procedure and present the analytic solutions. In Section 4, I provide a numerical simulation example of one of the strategies presented in Section 3. In Section 5, I address about optimality discussions of option strategies. Session 6 discusses the summary and related discussions.

An investor’s objective is to maximize the expected utility of end-of-period (time t = T) wealth

A constant relative risk aversion utility function was used in [

If

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

・ The portfolio is managed by a strategy process

・ The asset amount,

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

Brownian motion

The decision of the risky asset weight

In this paper, Risky Asset’s characteristics is set as its price S is under geometric Brownian motion with drift

Regarding Risky Asset, P-measure of

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

Mathematical expression is as follows:

Subject to:

P: Market measure. Q: Risk neutral measure.

For the convenience,

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

This implies:

[

The relationship is as follows. (The below, including

We use x and y for general variables. The maximization problem

lows with some simplification for interception (No interception affine transform. Notation of

(For illustrative purpose, see

Note that

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

This leads to the following:

The optimal solution

are supposing so to speak an American type option and any time arriving

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

at any time

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

The optimal solution:

for the left side (For reference, see

for the right side (For reference, see

“Put(K,

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

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

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

Using the solution of the case of a kinked utility at a maximum wealth (a covered call strategy case), I performed a Monte Carlo simulation. The solution strategy means that the investor’s wealth is under the target wealth currently (under-funding), and the strategy aims to achieve the target wealth (full-funding). In the simulation, 10,000 return patters are generated for Risky Asset’s using a geometric Brownian motion. Details of parameters are as follows. Each of the periods means one year and the total number of years is 20. As a benchmark, we denote the Merton model solution as “Standard strategy (STD).” The solution of a covered call strategy is “Covered Call strategy (CC).” A funding target “L” is exogenously given at each time. I understand CC is an optimal strategy under the kinked utility and STD is an optimal strategy under the “normal” utility, so I compare apples and oranges. However, I think it is worthwhile to compare those.

Before moving to Monte Carlo simulation results, we show the case of Risky Asset return is always constant.

In those, “return1” is in amount base, and “return2” is percentage base.

In the following simulations, the strategies’ actions at the beginning of period t = 1 are identical and as follows:

・

・ 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% (This background will be shown below).

The return of Risky Asset t = 1 decides which kind of call option should be sold in the CC Strategy at the beginning of period t = 2 and so forth for the full multi-period case.

In case all returns are −10%, leveraging and investing into Risky Asset (Standard portfolio

Generally speaking, a worse funding level makes

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

・ Pension fund total asset value

・ Pension Liability

・ Total periods: 20 years.

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

At the start pf the period | At the end of the period | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ζ = 1 + η | Xt | Market | CC Strategy | Market | return | return | STD Strategy | return | return | ||||||

T-t | Lt | wt | η | ζ | ζwt | Call | Value | Xt | Call | Value | (money) | (%) | MV | (money) | (%) |

20 | 100.0 | 70.0 | 0.60 | 1.596 | 111.7 | 41.7 | 70.0 | 100.5 | 32.2 | 68.3 | −1.7 | −2.4% | 63.0 | −7.0 | −10.0% |

19 | 102.0 | 68.3 | 0.43 | 1.43 | 97.7 | 29.4 | 68.3 | 87.9 | 21.9 | 66.1 | −2.3 | −3.3% | 56.7 | −6.3 | −10.0% |

18 | 104.0 | 66.1 | 0.30 | 1.304 | 86.1 | 20.1 | 66.1 | 77.5 | 14.3 | 63.2 | −2.8 | −4.3% | 51.0 | −5.7 | −10.0% |

17 | 106.1 | 63.2 | 0.21 | 1.209 | 76.4 | 13.2 | 63.2 | 68.8 | 8.9 | 59.9 | −3.4 | −5.3% | 45.9 | −5.1 | −10.0% |

16 | 108.2 | 59.9 | 0.14 | 1.138 | 68.1 | 8.3 | 59.9 | 61.3 | 5.2 | 56.1 | −3.8 | −6.3% | 41.3 | −4.6 | −10.0% |

15 | 110.4 | 56.1 | 0.09 | 1.087 | 61.0 | 4.8 | 56.1 | 54.9 | 2.8 | 52.0 | −4.1 | −7.3% | 37.2 | −4.1 | −10.0% |

14 | 112.6 | 52.0 | 0.05 | 1.05 | 54.6 | 2.6 | 52.0 | 49.2 | 1.4 | 47.8 | −4.2 | −8.1% | 33.5 | −3.7 | −10.0% |

13 | 114.9 | 47.8 | 0.03 | 1.027 | 49.1 | 1.3 | 47.8 | 44.2 | 0.6 | 43.6 | −4.2 | −8.9% | 30.1 | −3.3 | −10.0% |

12 | 117.2 | 43.6 | 0.01 | 1.013 | 44.1 | 0.5 | 43.6 | 39.7 | 0.2 | 39.5 | −4.1 | −9.4% | 27.1 | −3.0 | −10.0% |

11 | 119.5 | 39.5 | 0.01 | 1.005 | 39.7 | 0.2 | 39.5 | 35.7 | 0.1 | 35.6 | −3.8 | −9.7% | 24.4 | −2.7 | −10.0% |

10 | 121.9 | 35.6 | 0.00 | 1.002 | 35.7 | 0.1 | 35.6 | 32.1 | 0.0 | 32.1 | −3.5 | −9.9% | 22.0 | −2.4 | −10.0% |

9 | 124.3 | 32.1 | 0.00 | 1.001 | 32.1 | 0.0 | 32.1 | 28.9 | 0.0 | 28.9 | −3.2 | −10.0% | 19.8 | −2.2 | −10.0% |

8 | 126.8 | 28.9 | 0.00 | 1.001 | 28.9 | 0.0 | 28.9 | 26.0 | 0.0 | 26.0 | −2.9 | −10.0% | 17.8 | −2.0 | −10.0% |

7 | 129.4 | 26.0 | 0.00 | 1.001 | 26.0 | 0.0 | 26.0 | 23.4 | 0.0 | 23.4 | −2.6 | −10.0% | 16.0 | −1.8 | −10.0% |

6 | 131.9 | 23.4 | 0.00 | 1.001 | 23.4 | 0.0 | 23.4 | 21.1 | 0.0 | 21.1 | −2.3 | −10.0% | 14.4 | −1.6 | −10.0% |

5 | 134.6 | 21.1 | 0.00 | 1.001 | 21.1 | 0.0 | 21.1 | 19.0 | 0.0 | 18.9 | −2.1 | −10.0% | 13.0 | −1.4 | −10.0% |

4 | 137.3 | 18.9 | 0.00 | 1.001 | 19.0 | 0.0 | 19.0 | 17.1 | 0.0 | 17.1 | −1.9 | −10.0% | 11.7 | −1.3 | −10.0% |

3 | 140.0 | 17.1 | 0.00 | 1.001 | 17.1 | 0.0 | 17.1 | 15.4 | 0.0 | 15.3 | −1.7 | −10.0% | 10.5 | −1.2 | −10.0% |

2 | 142.8 | 15.3 | 0.00 | 1.001 | 15.4 | 0.0 | 15.4 | 13.8 | 0.0 | 13.8 | −1.5 | −10.0% | 9.5 | −1.1 | −10.0% |

1 | 145.7 | 13.8 | 0.00 | 1.001 | 13.8 | 0.0 | 13.8 | 12.4 | 0.0 | 12.4 | −1.4 | −10.0% | 8.5 | −0.9 | −10.0% |

0 | 148.6 |

The case of always −10% return for Risky Asset.

(100*(1 + 2%)^20) in 20 years. The expected return of STD should be 3.835%. This leads to

In CC Strategy, the underlying asset of call option is

At the start pf the period | At the end of the period | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ζ = 1 + η | Xt | Market | CC Strategy | Market | return | return | STD Strategy | return | return | ||||||

T-t | Lt | wt | η | ζ | ζwt | Call | Value | Xt | Call | Value | (money) | (%) | MV | (money) | (%) |

20 | 100.0 | 70.0 | 0.60 | 1.596 | 111.7 | 41.7 | 70.0 | 122.9 | 49.8 | 73.1 | 3.1 | 4.5% | 77.0 | 7.0 | 10.0% |

19 | 102.0 | 73.1 | 0.61 | 1.606 | 117.4 | 44.3 | 73.1 | 129.2 | 52.9 | 76.3 | 3.2 | 4.3% | 84.7 | 7.7 | 10.0% |

18 | 104.0 | 76.3 | 0.61 | 1.613 | 123.0 | 46.8 | 76.3 | 135.4 | 55.9 | 79.5 | 3.2 | 4.2% | 93.2 | 8.5 | 10.0% |

17 | 106.1 | 79.5 | 0.62 | 1.616 | 128.4 | 49.0 | 79.5 | 141.3 | 58.6 | 82.7 | 3.2 | 4.1% | 102.5 | 9.3 | 10.0% |

16 | 108.2 | 82.7 | 0.62 | 1.615 | 133.6 | 50.9 | 82.7 | 147.0 | 61.0 | 86.0 | 3.3 | 4.0% | 112.7 | 10.2 | 10.0% |

15 | 110.4 | 86.0 | 0.61 | 1.611 | 138.6 | 52.5 | 86.0 | 152.4 | 63.1 | 89.3 | 3.3 | 3.9% | 124.0 | 11.3 | 10.0% |

14 | 112.6 | 89.3 | 0.60 | 1.602 | 143.1 | 53.8 | 89.3 | 157.4 | 64.7 | 92.7 | 3.4 | 3.8% | 136.4 | 12.4 | 10.0% |

13 | 114.9 | 92.7 | 0.59 | 1.589 | 147.3 | 54.6 | 92.7 | 162.0 | 65.9 | 96.1 | 3.4 | 3.7% | 150.1 | 13.6 | 10.0% |

12 | 117.2 | 96.1 | 0.57 | 1.571 | 151.0 | 54.9 | 96.1 | 166.2 | 66.5 | 99.7 | 3.5 | 3.6% | 165.1 | 15.0 | 10.0% |

11 | 119.5 | 99.7 | 0.55 | 1.551 | 154.6 | 54.9 | 99.7 | 170.0 | 66.8 | 103.2 | 3.6 | 3.6% | 181.6 | 16.5 | 10.0% |

10 | 121.9 | 103.2 | 0.53 | 1.526 | 157.5 | 54.3 | 103.2 | 173.3 | 66.4 | 106.9 | 3.7 | 3.5% | 199.7 | 18.2 | 10.0% |

9 | 124.3 | 106.9 | 0.50 | 1.497 | 160.0 | 53.1 | 106.9 | 176.0 | 65.4 | 110.7 | 3.8 | 3.5% | 219.7 | 20.0 | 10.0% |

8 | 126.8 | 110.7 | 0.47 | 1.466 | 162.2 | 51.6 | 110.7 | 178.4 | 63.9 | 114.5 | 3.9 | 3.5% | 241.7 | 22.0 | 10.0% |

7 | 129.4 | 114.5 | 0.43 | 1.431 | 163.9 | 49.3 | 114.5 | 180.3 | 61.7 | 118.5 | 4.0 | 3.5% | 265.8 | 24.2 | 10.0% |

6 | 131.9 | 118.5 | 0.39 | 1.393 | 165.1 | 46.6 | 118.5 | 181.6 | 58.9 | 122.6 | 4.1 | 3.5% | 292.4 | 26.6 | 10.0% |

5 | 134.6 | 122.6 | 0.35 | 1.351 | 165.7 | 43.0 | 122.6 | 182.3 | 55.3 | 126.9 | 4.3 | 3.5% | 321.6 | 29.2 | 10.0% |

4 | 137.3 | 126.9 | 0.31 | 1.306 | 165.8 | 38.8 | 126.9 | 182.4 | 50.9 | 131.4 | 4.5 | 3.5% | 353.8 | 32.2 | 10.0% |

3 | 140.0 | 131.4 | 0.26 | 1.256 | 165.0 | 33.6 | 131.4 | 181.5 | 45.5 | 136.1 | 4.7 | 3.5% | 389.2 | 35.4 | 10.0% |

2 | 142.8 | 136.1 | 0.20 | 1.197 | 162.9 | 26.8 | 136.1 | 179.2 | 38.2 | 141.0 | 4.9 | 3.6% | 428.1 | 38.9 | 10.0% |

1 | 145.7 | 141.0 | 0.12 | 1.124 | 158.4 | 17.5 | 141.0 | 174.3 | 28.6 | 145.7 | 4.7 | 3.3% | 470.9 | 42.8 | 10.0% |

0 | 148.6 |

The case of always +10% return for Risky Asset.

CC Strategy shows that its return has a 3.856% cap (upper limit). This means that the strategy’s aim is achieve a funding level of 100%, and the 3.835% return is enough. (As described before, the 3.835% return makes the funding level from 70% to 100% based on 2% p.a. liability increase for 20 years.) The possibility of a final return of CC is better than STD is almost every case. But, there is an effect of additional return especially if STD’s return is negative.

In addition to the full 20-year results,

The shape tends to become covered call type payoff.

CC Strategy return distribution shows that both fewer big positive return and fewer big negative return arise.

CC has a smaller and more narrowly distributed volatility as shown in

The following summarize the CC Strategy characteristics:

・ Full period total return of the CC Strategy and STD Strategy look like the same, but the CC Strategy has downside resistance, meaning superior returns especially when STD Strategy has negative returns.

・ According to the Monte Carlo simulation, the CC Strategy has fewer negative and more positive return opportunities.

・ Volatility of the CC Strategy is very small on average.

Setting parameters differently versus Case 1, I checked the sensitivity of the volatility increase and expected return improvement.

This sensitivity analysis shows the following:

・ In

・ 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.

Case 2 | Case 1 | Case 3 | Case 4 | Case 1 | Case 5 | ||
---|---|---|---|---|---|---|---|

Parameters | |||||||

Initial Liability: L0 | 100 | 100 | 100 | 100 | 100 | 100 | |

Initial Asset: w0 | 70 | 70 | 70 | 70 | 70 | 70 | |

Liabiilty growth ratio | 2% | 2% | 2% | 2% | 2% | 2% | |

risk free interest rate | 1% | 1% | 1% | 1% | 1% | 1% | |

γ of Utility | 5.650 | 1.411 | 0.627 | 0.353 | 1.6 | 3.170 | |

Risky Asset’s return: μ | 5% | 5% | 5% | 3% | 5% | 7% | |

Risky Asset’s volatility: σ | 10% | 20% | 30% | 20% | 20% | 20% | |

Risky Asset’s Sharpe Ratio | 0.40 | 0.20 | 0.13 | 0.10 | 0.20 | 0.30 | |

periods (years) | 20 | 20 | 20 | 20 | 20 | 20 | |

Characteristics | |||||||

Risky Asset weight | 71% | 70.9% | 71% | 142% | 63% | 47% | |

Portfolio return | 3.8% | 3.8% | 3.8% | 3.8% | 3.5% | 3.8% | |

Portfolio risk | 7.1% | 14.2% | 21.3% | 28.3% | 12.5% | 9.5% | |

Portfolio Sharpe Ratio | 0.40 | 0.20 | 0.13 | 0.10 | 0.20 | 0.30 | |

Initial Funding Ratio | 70% | 70% | 70% | 70% | 70% | 70% | |

Liability Value at the end of the period | 149 | 149 | 149 | 149 | 149 | 149 | |

Necessary returns calculated from initial Funding Ratio | 3.8% | 3.8% | 3.8% | 3.8% | 3.8% | 3.8% | |

Simulation Results | |||||||

Ratio of CC is superior to STD | 19.3% | 46.5% | 58.9% | 67.5% | 46.5% | 30.3% |

This shows 10,000 times Monte Carlo simulation results of total 20 years returns like

In Section 1, I introduced previous works about portfolio insurance and put option optimality discussions. Although works of [11-13] and this paper treat dynamic strategy meaning not buying and holding strategies nor one period models, this Section starts one period model discussion of [8,10].

Actually, regarding [

They discussed that above (35) indicates that risk premium should be zero and this turns out with (36) that the utility function U is a linear function.

[

In their case, they set

With our model, [

For an investment strategy with a smooth utility function, the Merton model solution is obtained under strict conditions that the utility function is differentiable and strictly concave. When [

utility function, such as once the pension asset value achieves “L (indicating its liability level)” from under-funding status, the utility will not increase as even though the asset value increases, and the Merton conditions are not satisfied.

This paper described optimal investment strategies for kinked utility functions like CRRA utility function with a kink at a maximum wealth. The solutions are analytic mathematical solutions, one of which expresses a covered call option “like” strategy with dynamically managed and the option parameter varies. The other is a protective put option “like” strategy with dynamically managed and the option parameter varies. Dual utility with a Black-Sholes assumption is utilized in the solutions.

Some related discussions are described. For generally, non-smooth and/or non-strictly-concave utility functions, it is not clear if there exist smooth solutions to the HJB. To deal with the lack of a priori knowledge of the differentiability of the value function one may use a weak solution concept and characterize the value function as a unique viscosity solution to the HJB ([17-21] and etc.). It is in general difficult to show the differentiability of the value function even it is known to be a viscosity solution to the HJB. The lack of the differentiability of the value function makes it impossible to apply the verification theorem to find the optimal control. Another related issue is the Backward Stochastic Differential equations ([

Finally, the following summarize the numerical simulations for the Merton model strategy (STD) and the covered call strategy (CC) described above.

・ Full period total return of the CC Strategy and STD Strategy look the same, but the CC Strategy has downside resistance, meaning superior returns especially when the STD Strategy has negative returns.

・ According to the Monte Carlo simulation, the CC Strategy has fewer negative and more positive return opportunities.

・ Volatility of CC Strategy is very small on average.

・ The CC Strategy merit will increase if volatility increases but the merit decreases if expected return increases.