## Abstract

We study a portfolio optimization problem involving the loss averse policyholder of a variable annuity with a guaranteed minimum maturity benefit. This financial guarantee is financed via a fee withdrawn directly from the investment account, which impacts the net investment return. A fair pricing constraint is defined in terms of the risk-neutral value of the final contract payout. We propose a new fee structure that adjusts to the investment mix maximizing policyholder’s utility while keeping the contract fairly priced. We seek the investment strategy that maximizes the policyholder’s expected utility of terminal wealth after the application of a financial guarantee and subject to the fair pricing constraint. We assume that the policyholder’s risk attitude is relative to a reference level, risk-seeking towards losses and risk-averse towards gains. We solve the associated constrained stochastic control problem using a martingale approach and analyze the impact of the fee structure on the optimal investment strategies and payoff. Numerical results show that it is possible to find an optimal portfolio for a wide range of fees, while keeping the contract fairly priced.

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## References

Barucci E, Marazzina D, Mastrogiacomo E (2021) Optimal investment strategies with a minimum performance constraint. Ann Oper Res 299(1):215–239. https://doi.org/10.1007/s10479-019-03348-2

Bichuch M, Sturm S (2014) Portfolio optimization under convex incentive schemes. Finance Stochast 18(4):873–915. https://doi.org/10.1007/s00780-014-0236-9

Carpenter JN (2000) Does option compensation increase managerial risk appetite? J Financ 55(5):2311–2331. https://doi.org/10.1111/0022-1082.00288

Chen A, Hieber P, Nguyen T (2019) Constrained non-concave utility maximization: An application to life insurance contracts with guarantees. Euro J Oper Res 273(3):1119–1135. https://doi.org/10.13140/RG.2.2.34779.87849

Chen Z, Li Z, Zeng Y, Sun J (2017) Asset allocation under loss aversion and minimum performance constraint in a dc pension plan with inflation risk. Insurance: Math Econ 75:137–150. https://doi.org/10.1016/j.insmatheco.2017.05.009

Cox JC, Cf Huang (1989) Optimal consumption and portfolio policies when asset prices follow a diffusion process. J Econ Theo 49(1):33–83. https://doi.org/10.1016/0022-0531(89)90067-7

Cui Z, Feng R, MacKay A (2017) Variable annuities with vix-linked fee structure under a heston-type stochastic volatility model. North American Actuarial J 21(3):458–483. https://doi.org/10.1080/10920277.2017.130776

Cvitanić J, Karatzas I (1992) Convex duality in constrained portfolio optimization. The Annals of Applied Probability pp 767–818

Dong Y, Zheng H (2020) Optimal investment with s-shaped utility and trading and value at risk constraints: An application to defined contribution pension plan. Eur J Oper Res 281(2):341–356. https://doi.org/10.1016/j.ejor.2019.08.034

Dong Y, Lv W, Wei S, Gong Y (2020) Optimal investment of dc pension plan under incentive schemes and loss aversion. Mathematical Problems in Engineering. https://doi.org/10.1155/2020/5145848

El Karoui N, Jeanblanc M, Lacoste V (2005) Optimal portfolio management with american capital guarantee. J Econ Dyn Control 29(3):449–468. https://doi.org/10.1016/j.jedc.2003.11.005

Guan G, Liang Z (2016) Optimal management of dc pension plan under loss aversion and value-at-risk constraints. Insurance: Math Econ 69:224–237. https://doi.org/10.1016/j.insmatheco.2016.05.014

He L, Liang Z, Liu Y, Ma M (2019) Optimal control of dc pension plan management under two incentive schemes. North American Actuarial J 23(1):120–141

He XD, Kou S (2018) Profit sharing in hedge funds. Math Financ 28(1):50–81. https://doi.org/10.1111/mafi.12143

Kahneman D, Tversky A (1979) Prospect theory: An analysis of decision under risk. Econometrica 47(2):363–391. https://doi.org/10.2307/1914185

Korn R (1997) Optimal portfolios: stochastic models for optimal investment and risk management in continuous time. World Sci

Kramkov D, Schachermayer W (1999) The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann Appl Prob. pp 904–950

Larsen K (2005) Optimal portfolio delegation when parties have different coefficients of risk aversion. Quantitative Finance 5(5):503–512. https://doi.org/10.1080/14697680500305204

Lin H, Saunders D, Weng C (2017) Optimal investment strategies for participating contracts. Insurance: Math Econ 73:137–155. https://doi.org/10.1016/j.insmatheco.2017.02.001

Liu Y (2010) Pricing and hedging the guaranteed minimum withdrawal benefits in variable annuities

MacKay A, Augustyniak M, Bernard C, Hardy MR (2017) Risk management of policyholder behavior in equity-linked life insurance. J Risk Insurance 84(2):661–690. https://doi.org/10.1111/jori.12094

Merton RC (1975) Optimum consumption and portfolio rules in a continuoustime model. Stochastic optimization models in Finance pp 621-661. https://doi.org/10.1016/0022-0531(71)90038-X

Milevsky MA, Salisbury TS (2006) Financial valuation of guaranteed minimum withdrawal benefits. Insurance: Mathematics and Economics 38(1):21–38. https://doi.org/10.1016/j.insmatheco.2005.06.012

Nguyen T, Stadje M (2020) Nonconcave optimal investment with value-at-risk constraint: An application to life insurance contracts. SIAM J Control Optim 58(2):895–936. https://doi.org/10.1137/18M1217322

Reichlin C (2013) Utility maximization with a given pricing measure when the utility is not necessarily concave. Math Financ Econ 7(4):531–556. https://doi.org/10.1007/s11579-013-0093-x

Shen Y, Sherris M, Ziveyi J (2016) Valuation of guaranteed minimum maturity benefits in variable annuities with surrender options. Insurance: Math Econ 69:127-137. https://doi.org/10.1016/j.insmatheco.2016.04.006

Sun J, Shevchenko PV, Fung MC (2018) The impact of management fees on the pricing of variable annuity guarantees. Risks 6(3):103. https://doi.org/10.3390/risks6030103

Tversky A, Kahneman D (1986) Rational choice and the framing of decisions. J Business 59:S251–S278

Wu S, Dong Y, Lv W, Wang G (2020) Optimal asset allocation for participating contracts with mortality risk under minimum guarantee. Communications in Statistics-Theory and Methods 49(14):3481–3497. https://doi.org/10.1080/03610926.2019.1589518

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## Proofs of Auxiliary Results

### Proofs of Auxiliary Results

### 1.1 Proof of Lemma 2.1:

Under \(\mathbb {P}\), we have that \(\tilde{\xi }\) and *F* solve

respectively. By the product rule,

The left-hand side is non-negative while the right-hand side is a local martingale. Hence the supermartingale property is implied.

### 1.2 Proof of Lemma 3.1:

Let \(f(x):=\tilde{U}(x)-x\tilde{U}'(x)-\tilde{U}(0)\) for \(x>M\). Note that for \(x>M\), \(\tilde{U}(x)=U_1(x-\theta )\). Due to the strictly increasing and strictly concave properties of \(U_1\), the function *f*(*x*) is strictly increasing. Also, the continuous differentiability of \(U_1\) implies that *f*(*x*) is continuous.

By the conditions in Eq. (11), \(\lim _{x\uparrow \infty }f(x)=\infty\). Moreover, \(\lim _{x\downarrow M}f(x)<0\). To see this:

Case 1) if \(M=G\) then \(\lim _{x\downarrow G}\tilde{U}(x)=U_1(G-\theta )\equiv \tilde{U}(0)\) (see Eq. (15)). Since \(G\ge \theta\), \(\lim _{x\downarrow G}U'_1(x-\theta )>0\), possibly infinity. Thus, \(\lim _{x\downarrow G}f(x)=-G\lim _{x\downarrow G}U'_1(x-\theta )<0\).

Case 2) if \(M=\theta\) then \(\lim _{x\downarrow \theta }\tilde{U}(x)=U_1(0)=-U_2(0)\) and \(\tilde{U}(0)=-U_2(\theta -G)\). Also, by the Inada condition at \(x=0\) we have that \(\lim _{x\downarrow \theta }\tilde{U}'(x)=\lim _{x\downarrow \theta }U'_1(x-\theta )=\infty\) (see Eq. (16)). So \(\lim _{x\downarrow \theta }f(x)<0\).

From the above properties satisfied by *f*, there exists a unique root \(\hat{x}>M\) to the equation \(f(x)=0\).

Finally, \(U_c\) as defined above is indeed the smallest concave function dominating \(\tilde{U}\) since \(U_c(x)=\tilde{U}(x)\) for \(x\ge \hat{x}\) and \(U_c(x)\) is linear for \(0\le x< \hat{x}\).

### 1.3 Proof of Proposition 3.3:

From Proposition 3.2, we have that for \(\theta \le G\), \(\chi (y,z)=\chi _1(y,z)\), while or \(\theta >G\), \(\chi (y,z)=\chi _2(y,z)\), where \(y=\lambda _1 \tilde{\xi }_T\), \(z=\lambda _2\xi _T\),

and

Since \(I(\cdot )\) is strictly decreasing, it is sufficient to show that the mappings

are almost surely non-increasing. First, observe that

and thus \(\mathbb {1}_{\{\Delta (y)>0\}}\) is non-increasing.

To study the behavior of \(\mathbb {1}_{\{\Delta (z)+zG>0\}}\), we consider two cases: \(\theta > G\) and \(\theta \le G\). For \(\theta > G\),

and thus \(\mathbb {1}_{\{\Delta (z)+zG>0\}}\) is non-increasing.

For \(\theta \le G\), it is possible to show that \({\mathbb {1}_{\{\Delta (z)+zG>0\}} = 1}\) \(\mathbb P\)-a.s. Indeed, one can show that

with \(\arg \min _{z > 0} \{\Delta (z)+zG \}= U^\prime (G - \theta )\). Thus, \(\mathbb {1}_{\{\Delta (z)+zG>0\}} = 1\) for \(z \in \mathbb R^+ \setminus U^\prime (G - \theta )\), and since \(\xi _T\) has a continuous distribution, \(P(\lambda _2 \xi _T = U^\prime (G - \theta )) = P(\mathbb {1}_{\{\Delta (z)+zG>0\}} \ne 1)=0\).

### 1.4 Proof of Proposition 3.4:

Recall that for \(\theta \le G\), \(\chi (y,z)=\chi _1(y,z)\), while for \(\theta >G\), \(\chi (y,z)=\chi _2(y,z)\), where

and

First, note that

where the result follows from the asymptotic elasticity assumption in Eq. (11).

It follows that

for \(i=1,2\).

We also observe that, from the Inada and asymptotic elasticity conditions,

from which get that \(\lim _{y \rightarrow \infty } \chi (y,0) = 0\), since \({\lim _{y \rightarrow \infty } \mathbb {1}_{\left\{ \Delta (y)>0 \right\} } = 0}\) and \(\lim _{y \rightarrow \infty } I(y) = 0\).

Finally, to study the asymptotic behavior of \(\chi (0,z)\) as \(z \rightarrow \infty\), we consider the cases \(\theta > G\) and \(\theta \le G\) separately. For \(\theta > G\), we note that \(\mathbb {1}_{\{0< z < U^\prime _1(\hat{x} - \theta )\}}\) is equal to 0 for large enough *z*, and so \({\lim _{z \rightarrow \infty } \chi _2(0,z) = 0}\).

For \(\theta < G\), from the Inada and asymptotic elasticity conditions, we have that

from which we obtain that \(\lim _{z \rightarrow \infty } \chi _1(0,z) = \theta\).

Finally, for \(\theta = G\), \(\lim _{z \rightarrow \infty } \chi _1(0,z)\) can be 0 or \(\theta\), depending on the asymptotic behavior of *zI*(*z*).

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MacKay, A., Ocejo, A. Portfolio Optimization With a Guaranteed Minimum Maturity Benefit and Risk-Adjusted Fees.
*Methodol Comput Appl Probab* **24**, 1021–1049 (2022). https://doi.org/10.1007/s11009-022-09942-5

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DOI: https://doi.org/10.1007/s11009-022-09942-5

### Keywords

- Actuarial mathematics
- Variable annuity
- Portfolio optimization
- Non-concave utility
- Stochastic control
- Loss aversion