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Portfolio Optimization With a Guaranteed Minimum Maturity Benefit and Risk-Adjusted Fees

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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 This financial guarantee is financed via a fee withdrawn directly from the investment account, which impacts the net investment return.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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Correspondence to Adriana Ocejo.

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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

$$\begin{aligned} \tilde{\xi }_t=1-\int _0^t \tilde{r} \tilde{\xi }_u\,du-\int _0^t \tilde{\beta }\tilde{\xi }_u\,dW_u, \quad \text{ and } \quad F_t=F_0+\int _0^t F_u[\pi _u(\tilde{\mu }-\tilde{r})+\tilde{r}]du+\int _0^t F_u \pi _u \sigma \,dW_u \end{aligned}$$

respectively. By the product rule,

$$\begin{aligned} \tilde{\xi }_t F_t&= F_0 + \int _0^t\tilde{\xi }_u\,dF_u + \int _0^t F_u\,d\tilde{\xi }_u + \langle \tilde{\xi },F\rangle _t \\&= F_0 + \int _0^t \tilde{\xi }_u F_u (\pi _u\sigma -\tilde{\beta })dW_u. \end{aligned}$$

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 \(y=\lambda _1 \tilde{\xi }_T\), \(z=\lambda _2\xi _T\),

$$\begin{aligned} \chi _1(y,z)=[I(y+z)+\theta ]\mathbb {1}_{\left\{ \Delta (y+z)+zG>0 \right\} }, \end{aligned}$$


$$\begin{aligned} \chi _2(y,z)=\chi _1(y,z)\mathbb {1}_{\left\{ 0<y+z<U'_1(\hat{x}-\theta ) \right\} }. \end{aligned}$$

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

$$\begin{aligned} y\mapsto \mathbb {1}_{\{\Delta (y)>0\}}, \qquad z\mapsto \mathbb {1}_{\{\Delta (z)+zG>0\}} \end{aligned}$$

are almost surely non-increasing. First, observe that

$$\begin{aligned} \frac{\partial \Delta (y)}{\partial y}= -[I(y)+\theta ]<0, \end{aligned}$$

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 \(\theta > G\),

$$\begin{aligned} \frac{\partial (\Delta (z)+zG)}{\partial z}= -[I(z)+\theta ]+G<0, \end{aligned}$$

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

$$\begin{aligned} \min _{z > 0}\{ \Delta (z)+zG \}= 0, \end{aligned}$$

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

$$\begin{aligned} \chi _1(y,z)=[I(y+z)+\theta ]\mathbb {1}_{\left\{ \Delta (y+z)+zG>0 \right\} }, \end{aligned}$$


$$\begin{aligned} \chi _2(y,z)=\chi _1(y,z)\mathbb {1}_{\left\{ 0<y+z<U'_1(\hat{x}-\theta ) \right\} }. \end{aligned}$$

First, note that

$$\begin{aligned} \lim _{a \downarrow 0} \Delta (a)&= \lim _{a \downarrow 0} \{U_1(I(a)) - aI(a)\} - \tilde{U}(0)\\&= \lim _{x \rightarrow \infty } \{U_1(x) - xU_1^\prime (x)\} - \tilde{U}(0) = \infty , \end{aligned}$$

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

It follows that

$$\begin{aligned} \lim _{y\downarrow 0} \chi _i(y,0) = \lim _{z\downarrow 0} \chi _i(0,z) = \lim _{c\downarrow 0} I(c) = \infty , \end{aligned}$$

for \(i=1,2\).

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

$$\begin{aligned} \lim _{a \rightarrow \infty } \Delta (a)&= \lim _{x\downarrow 0} \{U_1(x)-xU'_1(x)-\theta U'_1(x)\}-\tilde{U}(0) = -\infty , \end{aligned}$$

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

$$\begin{aligned} \lim _{z \rightarrow \infty } \{\Delta (z) + zG\}&=\lim _{x\downarrow 0} \{U_1(x)-xU'_1(x)-\theta U'_1(x)(\theta -G)\}-\tilde{U}(0) = \infty , \end{aligned}$$

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).

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