2.1. Financial market

We consider a complete financial market without transaction costs on a filtered probability space $\left(\mathbf{\Omega},\mathcal{F},\mathbb{P};\,(\mathcal{F}_t)_{t\in[0,T]}\right)$ , equipped with d-dimensional Brownian motion W, where $(\mathcal{F}_t)_{t\in[0,T]}$ is the natural filtration of W, and $T>0$ is the terminal time. It includes one risk-free asset $S^0$ and $d\ge1$ risky assets $S=(S^{(1)},\cdots, S^{(d)})$ whose price dynamics are $\mathcal{F}$ -adapted semimartingales. Under the no-arbitrage condition, there exists a unique equivalent martingale measure $\mathbb{Q}\sim \mathbb{P}$ so that the discounted risky asset processes $\left(S_t/S^0_t\right)_{t\in[0,T]}$ are $(\mathcal{F},\mathbb{Q})$ -local martingales.

Denote the fractions of wealth invested in the risky assets by a d-dimension $\mathcal{F}$ -predictable process $\pi=(\pi_t)_{t\in[0,T]}=\left(\pi_t^{(1)},\cdots,\pi_t^{(d)}\right)_{t\in[0,T]}$ and assume that it is a self-financing strategy. Then, the corresponding portfolio process $X^{\pi}$ with an initial capital x can be expressed as

(2.1) \begin{equation} X_t^{\pi}=x+\int_0^t\pi_sdS_s,\quad t\in[0,T].\end{equation}

As usual, we only consider investment strategies $\pi_t$ for which the portfolio process (2.1) admits a unique strong solution, and $X_t^{\pi}$ is bounded from below so that arbitrage opportunities are excluded. It is known that the discounted portfolio process $X^{\pi}/S^0$ is a $(\mathcal{F},\mathbb{Q})$ super-martingale. Below, the superscript $\pi$ in $X^{\pi}$ will be omitted when there is no confusion.

We remark that such a complete market setting implies a unique stochastic discount factor or pricing kernel defined by $\xi_T\,:\!=\,\frac{S^0_0}{S_T^0}\frac{d\mathbb{Q}}{d\mathbb{P}}$ , where $\frac{d\mathbb{Q}}{d\mathbb{P}}$ is the Radon-Nikodym derivative of $\mathbb{Q}$ with respect to $\mathbb{P}$ . Economically, $\xi_T$ reflects the status of the economy at time T in the sense that it is low if the economy is booming and is high in a depression. Specifically, for any $\omega\in\mathbf{\Omega}$ , $\xi_T(\omega)$ can be viewed as the Arrow-Debreu value per unit probability P of $1 payoff of an asset in state $\omega$ at time T, and 0 otherwise. Owing to this explanation, we can later construct benchmarks characterizing optimal wealth and strategy as functions of $\xi_T$ to show how they evolve with respect to the state of the economy. Also, we assume that $\xi_T$ is atomless throughout this paper.

By the martingale approach (see, e.g., Karatzas et al. (Reference Karatzas, Lehoczky and Shreve1987); Cox and Huang (Reference Cox and Huang1989)), solving the dynamic utility maximization problem by finding a self-financing optimal investment strategy $\pi$ is equivalent to selecting an optimal terminal wealth $X_T$ financed by the initial capital x. We denote the set of all attainable terminal portfolios $X_T$ by

\begin{equation*} \mathbb{X}\,:\!=\, \left\{X_T\,:\, \mathbb{E}(\xi_TX_T)\leq x,\ X_T\text{ is }\mathcal{F}_T\text{-measurable, and }X_T\geq 0\ a.s.\right\},\end{equation*}

where $\mathbb{E}(\xi_TX_T)\leq x$ is the so-called budget constraint.

Later, we will adopt a one-dimension Black-Scholes model for numerical demonstration. Since we consider a general complete market in the theoretical part, the results can be easily extended to other complete market models like the local volatility model, stochastic interest rate model with a zero-coupon bond, and the Heston stochastic volatility model with an additional financial derivative for completion.

2.2. The agent’s preference

In this paper, the agent’s preference is modeled by a strictly increasing, strictly concave, and continuously differentiable utility function $U\,:\,(0,+\infty)\mapsto({-}\infty,+\infty)$ satisfying the following two conditions:

Assumption 1. (Inada’s conditions)

(2.2) \begin{equation} \lim_{y\rightarrow 0}U^{\prime}(y)=+\infty\quad\text{and}\quad \lim_{y\rightarrow{+\infty}}U^{\prime}(y)=0. \end{equation}

Note that the Inada’s conditions imply that

(2.3) \begin{equation} \lim_{y\rightarrow 0}I(y)=+\infty\quad\text{and}\quad \lim_{y\rightarrow+\infty}I(y)=0,\end{equation}

where $I(\cdot)=(U^{\prime})^{-1}(\cdot)$ , which excludes utility functions with constant absolute relative aversion. However, our approach can be easily extended to such a framework.

Assumption 2. (Integrability Conditions) For all $\lambda>0$ , it holds that

\begin{equation*} (i)\ \mathbb{E}(\xi_TI(\lambda\xi_T))<+\infty\qquad (ii)\ \mathbb{E}(U(I(\lambda\xi_T)))<+\infty\qquad (iii)\ \mathbb{E}\left(\xi_T^2|I^{\prime}(\lambda\xi_T)|\right)<+\infty. \end{equation*}

As shown below, while the first and second integrability conditions are technically needed to ensure the existence of Lagrangian multipliers, the third condition in Assumption 2 allows one to switch the expectation and the differential operators (see Appendix D) when examining the relationship between Lagrange multipliers and other parameters in the optimization problem.

3.1. General benchmark setting

Benchmarking is a universal practice in both active and passive asset management. This subsection provides specifications on the benchmark process $(Y_t)_{t\in[0,T]}$ . Typically, a benchmark represents a general indicator of market sentiment and direction.Footnote 1 A specific choice for the benchmark falls into three usual categories: a portfolio, an index (including the stock indices, e.g., S&P 500 as the most popular example), or any economic indicator. Motivated by various works on portfolio management with benchmarking, for example, Basak et al. (Reference Basak, Shapiro and Teplá2006); Cuoco and Kaniel (Reference Cuoco and Kaniel2011), we focus on a class of benchmark portfolios that are positively correlated with the stock market. These benchmarks are the mostly used benchmark when considering mutual funds and pension funds whose investment management is typically procyclical to the overall state of the economy represented by the market price density $\xi_T$ (see the discussion in Section 2.1). Based on this, it is reasonable to set the benchmark process as an inversely monotone function with respect to $\xi_T$ . More precisely, we assume that $Y_T=f(\xi_T)$ where $f\,:\,[0,+\infty)\mapsto[0,+\infty)$ is a non-increasing and continuous function satisfying the following assumption:

As will be shown in Section 3.2, Assumption 3 can be checked directly for various popular benchmarks. For convenience of presentation, we distinguish the following 3 cases: Cases (1), (2) (when the benchmark curve $f(\xi)$ and the Merton curve $I(\lambda\xi)$ cross at a unique intersection) and Case (3) (when $f(\xi)=I(\lambda_B\xi)$ ), which are summarized in Table 1. Obviously, Case (3) means that the benchmark belongs to the family of Merton portfolios with some multiplier $\lambda_B>0$ . As illustrated in Figure 1, Case (1) (resp. Case (2)) means that the Merton portfolio (resp. the benchmark) is more responsive to the economy change than the benchmark (resp. the Merton curve). Therefore, below, Case (1) and Case (2) will be named as out/underperformance and under/outperformance respectively. Note that by the intermediate value theorem, it is easy to see that Case (1) is fulfilled if for any $\lambda>0$ , the function $\xi\mapsto f(\xi)/I(\lambda\xi)$ is increasing and the following asymptotic results hold

\begin{equation*} \lim_{\xi\nearrow +\infty}\frac{f(\xi)}{I(\lambda\xi)}>1 \quad \mbox{and} \quad\lim_{\xi\searrow 0}\frac{f(\xi)}{I(\lambda\xi)}<1.\end{equation*}

A similar condition can be easily used to verify Case (2).

Table 1. Three situations for benchmarking and portfolio management

Figure 1. Intersection of the Merton curve with the benchmark.

Remark 3.1. We remark that for Case (1), f can even be an increasing function because it does not violate Assumption 3. However, as mentioned above, it is economically reasonable to set $f(\xi_T)$ as a decreasing function of $\xi_T$ . In Case (2), we can deduce that f is monotonically decreasing and satisfying

(3.1) \begin{equation} \lim_{\xi\rightarrow 0}f(\xi)=+\infty \quad\text{and}\quad \lim_{\xi\rightarrow+\infty}f(\xi)=0, \end{equation}

according to (2.3) and Table 1. It is also observed that Case (1) holds true if f is a positive constant (see Example 1) or f is decreasing and bounded below by a positive constant.

Throughout the paper, we make use of the following assumption:

Assumption 4.

\begin{equation*} \mathbb{E}(\xi_Tf(\xi_T))<+\infty \quad\text{and}\quad \mathbb{E}(U(f(\xi_T)))<+\infty.\end{equation*}

Assumption 4 means that the benchmark is replicable and the benchmark expected utility is finite. The above endogenous benchmark structure is quite general, and various widely used benchmark settings in the literature can be included in our setting.

3.2. Benchmark examples

Before giving benchmark examples, we first introduce the Black-Scholes model consisting of one risky asset S with return $\mu$ and volatility $\sigma$ and a risk-free asset $S^0$ with constant interest rate r. In particular, the asset price dynamics are given by

\begin{equation*} dS_t=S_t(\mu dt+\sigma dW_t),\quad dS_t^0=rS_t^0dt,\quad S_0^0=1.\end{equation*}

It is well-known that the risky asset price can be expressed as $S_t=S_0\exp{((\mu-\frac{\sigma^2}{2})t+\sigma W_t)}$ by Itô’s lemma, where $S_0>0$ is the risky asset price at time 0. In such a complete market setting, the unique market price density is defined by

\begin{equation*} d\xi_t=-\xi_t(rdt+\theta dW_t),\quad \xi_0=1,\end{equation*}

where $\theta=\frac{\mu-r}{\sigma}$ is the market price of risk. Similarly to $S_t$ , we have $\xi_t=\exp\left({-}rt-\frac{1}{2}\theta^2t-\theta W_t\right)$ . Let $\pi_t$ be the fraction invested in the risky asset at time t. Then, the corresponding wealth process with an initial capital $x>0$ is given by

(3.2) \begin{equation} dX_t=(r+\pi_t(\mu-r))X_tdt+\pi_t\sigma X_tdW_t,\quad X_0=x.\end{equation}

Below, we consider four benchmark examples that satisfy Assumption 3. Note that under a Black-Scholes market, the counter-monotonicity of f to $\xi_T$ indicates a nonnegative position in the risky asset S.

Example 2. (Stock market benchmarks) Another important benchmark is $f(\xi_T)=S_T$ , as considered in (Basak et al. Reference Basak, Shapiro and Teplá2006, Section 4.3) with a power utility $U(y)=\frac{y^{1-\gamma}}{1-\gamma}$ with $\gamma\neq 1$ and $\gamma >0$ . To verify Assumption 3, it suffices to observe that

\begin{equation*} f(\xi_T)=S_T =S_0e^{\left(\mu-\frac{\sigma^2}{2}\right)T+\sigma W_T}=S_0e^{\left(\mu-\frac{\sigma^2}{2}\right)T-\left(r+\frac{\theta^2}{2}\right)\frac{\sigma T}{\theta}}\times e^{\left(\left({-}r-\frac{\theta^2}{2}\right)T-\theta W_T\right)\times \left( -\frac{\sigma}{\theta}\right)}\,:\!=\,A\xi_T^{-\frac{\sigma}{\theta}},\end{equation*}

which is obviously a monotonically decreasing and differentiable function. To discuss the benchmark classification,we can consider the following equation

\begin{equation*} f(y)-I(\lambda y) =Ay^{-\frac{\sigma}{\theta}}-\lambda^{-\frac{1}{\gamma}}y^{-\frac{1}{\gamma}} =\lambda^{-\frac{1}{\gamma}}y^{-\frac{1}{\gamma}}\left(A\lambda^{\frac{1}{\gamma}}y^{\frac{1}{\gamma}-\frac{\sigma}{\theta}}-1\right) =0.\end{equation*}

If $\frac{\sigma}{\theta}=\frac{1}{\gamma}$ , we have $f(y)=I(\lambda_By)$ for $\lambda_B=A^{-\gamma}$ , which shows that the benchmark belongs to family of Merton curves (i.e., Case (3)). If $\frac{\sigma}{\theta}\neq\frac{1}{\gamma}$ , the root is $A^{\frac{1}{\frac{\sigma}{\theta}-\frac{1}{\gamma}}}\lambda^{\frac{1}{\frac{\gamma\sigma}{\theta}-1}}$ , and we obtain the following classification:

\begin{align*} \text{Case (1): }&f(y)-I(\lambda y) \left \{ \begin{array}{l@{\quad}r} <0 & \text{ if }y<A^{\frac{1}{\frac{\sigma}{\theta}-\frac{1}{\gamma}}}\lambda^{\frac{1}{\frac{\gamma\sigma}{\theta}-1}}\\[4pt] >0 & \text{ if }y>A^{\frac{1}{\frac{\sigma}{\theta}-\frac{1}{\gamma}}}\lambda^{\frac{1}{\frac{\gamma\sigma}{\theta}-1}} \end{array} \right. \text{for}\quad\frac{\sigma}{\theta}<\frac{1}{\gamma};\\[6pt] \text{Case (2): }&f(y)-I(\lambda y) \left \{ \begin{array}{r@{\quad}c@{\quad}l} >0 & \text{ if }y<A^{\frac{1}{\frac{\sigma}{\theta}-\frac{1}{\gamma}}}\lambda^{\frac{1}{\frac{\gamma\sigma}{\theta}-1}}\\[4pt] <0 & \text{ if }y>A^{\frac{1}{\frac{\sigma}{\theta}-\frac{1}{\gamma}}}\lambda^{\frac{1}{\frac{\gamma\sigma}{\theta}-1}} \end{array} \right. \text{for}\quad\frac{\sigma}{\theta}>\frac{1}{\gamma}.\end{align*}

As mentioned after Assumption 3, we remark that the benchmark classification can be checked by considering the function $f(\xi)/I(\lambda \xi)$ .

Example 3. (Hybrid benchmarks) Under the Black-Scholes market consisting of a risk-free asset and only one risky asset, the choice of the benchmark is limited by the set of all possible combinations of these two assets. In terms of combinations, one can form a benchmark whose return rate is a weighted average of the risk-free rate r and the return of stock $R_T^S$ , called the hybrid benchmark (see, e.g., (Basak et al. Reference Basak, Shapiro and Teplá2006, Section 4.2)). More precisely, the return of this benchmark $R_T^Y$ from time 0 to time T is

\begin{equation*} R_T^Y=\frac{1}{T}\ln\frac{Y^H_T}{Y^H_0}=\alpha r+(1-\alpha)R_T^S,\end{equation*}

where $R_T^S=\frac{1}{T}\ln\frac{S_T}{S_0}$ is the return of the stock market from time 0 to time T and the riskless investment proportion $\alpha\in[0,1]$ . Then, the benchmark $Y_T^H=f^H(\xi_T)$ can be expressed as a function of $\xi_T$ , namely $f^H(\xi_T) \,:\!=\,B\xi_T^{-\frac{\sigma}{\theta}(1-\alpha)}$ , where $B=Y_0^He^{\left(\alpha r+\left(\mu-\frac{\sigma^2}{2}\right)(1-\alpha)\right)T-\left(r+\frac{\theta^2}{2}\right)\frac{\sigma }{\theta}(1-\alpha)T}$ . Then, it is clear that $f^H$ is a monotonically decreasing function. To check Assumption 3, we solve the following equation

(3.3) \begin{equation} f^H(y)-I(\lambda y) =By^{-\frac{\sigma}{\theta}(1-\alpha)}-\lambda^{-\frac{1}{\gamma}}y^{-\frac{1}{\gamma}} =\lambda^{-\frac{1}{\gamma}}y^{-\frac{1}{\gamma}}\left(B\lambda^{\frac{1}{\gamma}}y^{\frac{1}{\gamma}-\frac{\sigma}{\theta}(1-\alpha)}-1\right) =0.\end{equation}

If $\frac{\sigma}{\theta}(1-\alpha)=\frac{1}{\gamma}$ , we obtain Case (3) by setting $\lambda_B=B^{-\gamma}$ . For other cases, the root is $B^{\frac{1}{\frac{\sigma}{\theta}(1-\alpha)-\frac{1}{\gamma}}}\lambda^{\frac{1}{\frac{\gamma\sigma}{\theta}(1-\alpha)-1}}$ . Following a similar calculation as in Example 2, it belongs to Case (1) (resp. Case (2)) if $\frac{\sigma}{\theta}(1-\alpha)<\frac{1}{\gamma}$ (resp. $\frac{\sigma}{\theta}(1-\alpha)>\frac{1}{\gamma})$ .

Example 4. (Mixed benchmarks) Another benchmark example is to combine the risk-free asset and the risky asset with different proportions. In particular, assuming that the proportion of capital invested in the risk-free asset is $\beta\in(0,1)$ in the mixed benchmark and $S_0=1$ , the mixed benchmark $Y^M$ is defined by

\begin{equation*} Y_T^M=\beta Y_0^Me^{rT} +(1-\beta)Y_0^Me^{\left(\mu-\frac{\sigma^2}{2}\right)T-\left(r+\frac{\theta^2}{2}\right)\frac{\sigma T}{\theta}}\xi_T^{ -\frac{\sigma}{\theta}}.\end{equation*}

Clearly $Y_T^M$ is bounded from below by $\beta Y_0^Me^{rT}$ and we are in Case (1) if $\frac{1}{\gamma}\geq\frac{\sigma}{\theta}$ , (see Remark 3.1). We remark that if $\frac{1}{\gamma}<\frac{\sigma}{\theta}$ , the Merton curve may intersect the benchmark more than once and Assumption 3 is violated.

Example 5. (CPPI benchmarks) With a portfolio insurance constraint in our setting, it is natural to consider other portfolio insurance strategies as a benchmark. One standard method is the so-called constant proportion portfolio insurance (CPPI, see, e.g., Bertrand and Prigent (Reference Bertrand and Prigent2005)). To construct a CPPI portfolio, the investor sets a floor $L_t=L_0e^{rt}$ as a lower bound of portfolio and dynamically calculates a cushion $C_t$ defined as the difference between the portfolio wealth $X_t^{CPPI}$ and the floor $L_t$ . The amount of capital invested in the risky asset is given by the cushion scaled by a predetermined multiplier m. To embed a CPPI benchmark into our framework, we first infer the dynamic of $C_t$ and terminal benchmark value ${Y}_T^{CPPI}=C_T+L_T$ as a function of $\xi_T$ . By construction, we first have

\begin{equation*} d{Y}_t^{CPPI} =dC_t+rL_tdt=\left({Y}_t^{CPPI}-mC_t\right)\frac{dB_t}{B_t}+mC_t\frac{dS_t}{S_t}\\ =(C_t+L_t-mC_t)rdt+mC_t(\mu dt+\sigma dW_t),\end{equation*}

which implies the cushion dynamics $dC_t=C_t[((1-m)r+m\mu)dt+m\sigma dW_t]$ . Solving this SDE, we obtain

\begin{equation*} C_T=\left({Y}_0^{CPPI}-L_0\right)e^{\left[\left(1-\frac{m}{2}-\frac{m\sigma}{\theta}\right)r+\frac{m\mu}{2}-\frac{m^2\sigma^2}{2}\right]T} \xi_T^{-\frac{m\sigma}{\theta}} \,:\!=\,F\xi_T^{-\frac{m\sigma}{\theta}}.\end{equation*}

The terminal value of CPPI portfolio is then quantified as ${Y}_T^{CPPI}=F\xi_T^{-\frac{m\sigma}{\theta}}+L_0e^{rT}$ .

Similarly to the mixed benchmark, the terminal value of CPPI benchmark is bounded from below, implying only Case (1) is applicable when $\frac{m\sigma}{\theta}\leq\frac{1}{\gamma}$ according to Remark 3.1.

4.1. Problem formulation

The LERL-PI problem is stated as

(4.1) \begin{equation} \max_{{X_T\in\mathbb{X}},\, X_T\geq L}\mathbb{E}\left(U(X_T)\right)\quad \text{ s.t. }\mathbb{E}\left(\xi_T(f(\xi_T)-X_T)\mathbf{1}_{X_T\leq f(\xi_T)}\right)\leq\epsilon,\end{equation}

where $X_T$ is the terminal wealth defined by (2.1), $f(\xi_T)$ is the benchmark at time T as discussed in Section 3, $\epsilon\geq 0$ is the LERL loss bound, and $L\geq 0$ is the minimum guarantee level. The PI constraint guarantees that the terminal wealth is above the level L at maturity.

Before solving the above constrained optimization, it is important to note that the LERL constraint penalizes both the expected shortfall below the benchmark and the probability of being underperformance as

\begin{equation*} \mathbb{E}\left(\xi_T(f(\xi_T)-X_T)\mathbf{1}_{X_T\leq f(\xi_T)}\right) =\mathbb{E}\left(\xi_T(f(\xi_T)-X_T)|X_T\leq f(\xi_T)\right)\mathbb{P}(X_T\leq f(\xi_T)).\end{equation*}

To solve Problem (4.1), it is crucial to set a suitable value of the loss bound $\epsilon$ which ensures that the LERL constraint is active. When the LERL constraint is not active, Problem (4.1) degenerates to a PI problem $\max_{{X_T\in\mathbb{X}}, X_T\geq L}\mathbb{E}\left(U(X_T)\right)$ for which case the corresponding optimal terminal wealth is denoted by $X_T^{PI(L)}$ . This case has been extensively considered in the literature, for instance, in Grossman and Vila (Reference Grossman and Vila1989); Basak (Reference Basak1995); Grossman and Zhou (Reference Grossman and Zhou1996); Jensen and Sørensen (Reference Jensen and Sørensen2001); Gabih et al. (Reference Gabih, Sass and Wunderlich2009); Chen et al. (Reference Chen, Nguyen and Stadje2018). In our setting, the upper bound $\overline{\epsilon}$ of the LERL loss can be obtained by substituting $X_T^{PI(L)}$ into the LERL expression, that is

(4.2) \begin{equation} \overline{\epsilon} \,:\!=\, \mathbb{E}\left(\xi_T\left(f(\xi_T)-X_T^{PI(L)}\right)\mathbf{1}_{X_T^{PI(L)} \leq f(\xi_T)}\right).\end{equation}

We note that the lower bound $ \underline{\epsilon}$ of the LERL loss can be set equal to zero. In this case, Problem (4.1) becomes

(4.3) \begin{equation} \max_{X_T\in\mathbb{X}, \,X_T\geq \max\{f(\xi_T),L\}}\mathbb{E}\left(U(X_T)\right),\end{equation}

which is admissible only if $x\geq \mathbb{E}(\xi_T\max\{f(\xi_T),L\})$ . Remark that this special case is more general than the setting with $L=0$ and $f(\xi_T)=S_T$ , where $S_T$ follows a geometric Brownian motion, that has been solved in Teplá (Reference Teplá2001). For completeness, the optimal solution to Problem (4.3) is given in the following proposition for each of the 3 cases of benchmark summarized in Table 1 (see also the discussion on economic interpretations and mathematical conditions on this benchmark classification before Table 1).

Proposition 4.1. Assume that $x > \mathbb{E}(\xi_T\max\{f(\xi_T),L\})$ . The optimal terminal wealth of Problem (4.3) is

\begin{equation*} X_T^H\,:\!=\,X_T^H(\lambda^H,\xi_T)= \left\{\begin{array}{l@{\quad}r} I(\lambda^H\xi_T)\mathbf{1}_{\xi_T<\underline{\xi}^H} +f(\xi_T)\mathbf{1}_{\underline{\xi}^H\leq \xi_T< \xi_L^H} +L\mathbf{1}_{\xi_T \geq \xi_L^H}, & \text{ Case (1)}\\[5pt] f(\xi_T)\mathbf{1}_{\xi_T<\underline{\xi}^H} +I(\lambda^H\xi_T)\mathbf{1}_{\underline{\xi}^H\leq \xi_T <\underline{\xi}_L^H} +L\mathbf{1}_{\xi_T\geq \underline{\xi}_L^H}, & \text{ Case (2)} \\[5pt] X_T^{PI(L)}=\max\!\left\{I(\lambda^{PI(L)}\xi_T), L\right\}, & \text{ Case (3)} \end{array}\right. \end{equation*}

where $\lambda^H>0$ and $\lambda^{PI(L)}>0$ are calculated by solving the budget constraint with equality, $\underline{\xi}^H$ and $\xi_L^H$ are obtained by solving $f\left(\underline{\xi}^H\right)=I\left(\lambda_H\underline{\xi}^H\right)$ and $f\left(\xi_L^H\right)=L$ , and $\underline{\xi}_L^H=U^{\prime}(L)/\lambda^H$ . If $x = \mathbb{E}(\xi_T\max\{f(\xi_T),L\})$ , then the solution to Problem (4.3) is given by $\max\{f(\xi_T),L\}$ .

When $x < \mathbb{E}(\xi_T\max\{f(\xi_T),L\})$ , the lower bound of $\epsilon$ cannot be zero. In this case, the lower bound of $\epsilon$ can be obtained by solving the following risk minimization problem:

(4.4) \begin{equation} \min_{X_T\in\mathbb{X}, \,X_T \geq L} \mathbb{E}\left(\xi_T(f(\xi_T)-X_T) \mathbf{1}_{X_T \leq f(\xi_T)}\right).\end{equation}

Note that the initial capital x has to be above $e^{-rT}L$ to hedge the minimum guarantee level L. The solution is summarized in the following lemma.

With the above discussion and Lemma 4.1, we can set the lower bound of $\epsilon$ as

(4.5) \begin{equation} \underline{\epsilon} \,:\!=\,\max\{0, \mathbb{E}(\xi_T\max\{f(\xi_T),L\})-x\}.\end{equation}

When the other parameters are fixed, we assume that $\underline{\epsilon}\leq\epsilon<\overline{\epsilon}$ to ensure the bindingness of the LERL constraint in Problem (4.1). Remark that the admissibility of the LERL constraint also depends on other parameters like the initial capital x and the minimum insurance level L. Now, when the other parameters are fixed in Problem (4.1), the LERL constraint is binding if $x_{min}\leq x < x_{max}$ , where $x_{min}$ and $x_{max}$ are defined in (A.3) and (A.2) respectively (See Appendix B for further elaborations).

4.2. Optimal terminal wealth of Problem (4.1)

The optimal terminal wealth of Problem (4.1) for each case of benchmark classification in Table 1 is now summarized in Theorem 4.1 below, assuming that both the LERL and PI constraints are active (see Appendix B for the comprehensive study of the admissibility of Problem (4.1)). We remark that for common utility functions, the corresponding optimal strategies can be derived via the standard argument, see, for example, Chen et al. (Reference Chen, Nguyen and Stadje2018).

Theorem 4.1. Assume that both the LERL and PI constraints are active. Then, the optimal terminal wealth of Problem (4.1) is given by

(4.6) \begin{equation} X_T^{LERL-PI}= \left\{\begin{array}{l@{\quad}r} I(\lambda_1\xi_T)\mathbf{1}_{\xi_T<\underline{\xi}} +f(\xi_T)\mathbf{1}_{\underline{\xi}\leq \xi_T<\overline{\xi}} +I((\lambda_1-\lambda_2)\xi_T)\mathbf{1}_{\overline{\xi}\leq\xi_T<\overline{\xi}_L} +L\mathbf{1}_{\xi_T\geq\overline{\xi}_L}, &\text{ Case (1)}\\[5pt] I((\lambda_1-\lambda_2)\xi_T)\mathbf{1}_{\xi_T<\overline{\xi}} +f(\xi_T)\mathbf{1}_{\overline{\xi}\leq\xi_T<\underline{\xi}} +I(\lambda_1\xi_T)\mathbf{1}_{\underline{\xi}\leq\xi_T<\underline{\xi}_L} +L\mathbf{1}_{\xi_T\geq\underline{\xi}_L}, &\text{ Case (2)} \\[5pt] X_T^{PI(L)}=\max\left\{I(\lambda^{PI(L)}\xi_T), L\right\}, &\text{ Case (3)} \end{array}\right. \end{equation}

where $\lambda_1>\lambda_2>0$ are obtained by solving the budget and LERL constraints with equality simultaneously and ${\lambda^{PI(L)}>0}$ is calculated by solving the budget constraint with equality. Here, $\underline{\xi}$ and $\overline{\xi}$ are obtained by solving $I(\lambda_1\underline{\xi})=f(\underline{\xi})$ and $I((\lambda_1-\lambda_2)\overline{\xi})=f(\overline{\xi})$ respectively, $\overline{\xi}_L=U^{\prime}(L)/(\lambda_1-\lambda_2)$ , and $\underline{\xi}_L=U^{\prime}(L)/\lambda_1$ . In Case (3), the LERL loss bound is equal to $\underline{\epsilon}$ or the LERL constraint is inactive.

From the solution structure of Theorem 4.1, we can observe that both Case (1) and Case (2) share a 4-region solution form. The optimal terminal wealth follows the benchmark in the second region, but it takes the PI level L in the fourth region. Therefore, we name the former the benchmarking region and the latter the PI region. However, in the first and third regions, it may refer to the different performance of the portfolio relative to the benchmark. In particular, for the first (resp. third) region, the portfolio outperforms (resp. underperforms) the benchmark for Case (1) and underperforms (resp. outperforms) the benchmark for Case (2). The above observations are summarized in Table 2, where $\xi_L$ is the crossing point of the benchmark curve f and the PI level L,

Table 2. LERL-PI solution regions for Case (1) and Case (2)

According to Theorem 4.1 and Table 2, the agent in Case (1) is willing to surpass the benchmark (i.e., outperform) in good market scenarios at the cost of being below the benchmark (i.e., underperform) in bad market scenarios due to the budget constraint. On the contrary, it is observed that the agent in Case (2) wants to beat the benchmark in bad market states at the expense of being underperforming in good market states. Therefore, we name the former case as out/underperformance portfolio management (PM) and the latter as under/outperformance PM.

To prove Theorem 4.1, we adopt the static martingale approach. The idea is to find the optimal terminal wealth and derive the corresponding optimal strategy thereon. For $\lambda_1>\lambda_2>0$ , $L>0$ and $\xi>0$ , we first solve the following Lagrangian maximization problem:

(4.7) \begin{equation} \max_{X\geq L}G(\lambda_1,\lambda_2,X) \,:\!=\,{\max_{X\geq L}\left(U(X)-\lambda_1\xi X-\lambda_2\xi(f(\xi)-X)\mathbf{1}_{X\leq f(\xi)}\right)}. \end{equation}

Lemma 4.2. The unique solution to problem (4.7) is given by

\begin{equation*} X^{*}(\lambda_1,\lambda_2,\xi) =\left\{\begin{array}{l@{\quad}r} I(\lambda_1\xi)\mathbf{1}_{\xi<\underline{\xi}} +f(\xi)\mathbf{1}_{\underline{\xi}\leq \xi<\overline{\xi}} +I((\lambda_1-\lambda_2)\xi)\mathbf{1}_{\overline{\xi}\leq\xi<\overline{\xi}_L} +L\mathbf{1}_{\xi\geq\overline{\xi}_L}, & \text{ Case (1)},\\[5pt] I((\lambda_1-\lambda_2)\xi)\mathbf{1}_{\xi<\overline{\xi}} +f(\xi)\mathbf{1}_{\overline{\xi}\leq\xi<\underline{\xi}} +I(\lambda_1\xi)\mathbf{1}_{\underline{\xi}\leq\xi<\underline{\xi}_L} +L\mathbf{1}_{\xi\geq\underline{\xi}_L}, & \text{ Case (2)}{,} \end{array}\right. \end{equation*}

where $\underline{\xi}$ , $\overline{\xi}$ , $\overline{\xi}_L$ and $\underline{\xi}_L$ are defined in Theorem 4.1. For Case (3), the unique solution to Problem (4.7) is stated in Table 3.

Table 3. The unique solution to Problem (4.7) for Case (3)

By making use of Lemma 4.2, the optimal terminal wealth of Problem (4.1) is given by $X_T^{LERL-PI}=X^{*}(\lambda_1,\lambda_2,\xi_T)$ . To finish, we need to show the existence of the Lagrange multipliers. To this end, for each Case (i), $i=1,2$ , let us define two auxiliary functions:

(4.8) \begin{equation} H_i(\lambda_1,\lambda_2) = \mathbb{E}\left(\xi_TX_T^{LERL-PI}\right)\text{ and }K_i(\lambda_1,\lambda_2) = \mathbb{E}\left(\xi_T\left(f(\xi_T)-X_T^{LERL-PI}\right)\mathbf{1}_{X_T^{LERL-PI}\leq f(\xi_T)}\right).\end{equation}

With the aid of these auxiliary functions, the following proposition asserts the existence of the Lagrange multipliers.

Proposition 4.2. Assume that both the LERL and PI constraints are active. Then, there exists a unique solution $(\lambda_1,\lambda_2)\in \mathbf{D}$ to the system of equations

(4.9) \begin{equation} \left\{\begin{array}{l@{\quad}r} H_i(\lambda_1,\lambda_2)=x,&\\[4pt] K_i(\lambda_1,\lambda_2)=\epsilon,& \end{array}\right. \end{equation}

for each Case (i), $i=1,2$ . The domain $\mathbf{D}$ is defined as

(4.10) \begin{equation} \mathbf{D} =\left\{(\lambda_1,\lambda_2)\in(0,+\infty)\times(0,+\infty)\,:\, \lambda_1\in[\lambda^{PI(L)},\overline{\lambda}_1),\ \lambda_2\in[0,\lambda_1{)}\right\}, \end{equation}

where

\begin{equation*} \overline{\lambda}_1= \left\{\begin{array}{l@{\quad}r} \lambda^H, &\text{if }\mathbb{E}(\xi_T\max\{f(\xi_T),L\})< x,\\[4pt] +\infty, &\text{if } \mathbb{E}(\xi_T\max\{f(\xi_T),L\})\geq x\text{ and Case (1)},\\[4pt] \lambda_a, &\text{if } \mathbb{E}(\xi_T\max\{f(\xi_T),L\})\geq x\text{ and Case (2)}.\\ \end{array}\right. \end{equation*}

The multiplier $\lambda^H$ is defined in Proposition 4.1, and $\lambda_a$ is defined as a limit of $\lambda_1$ such that $\lim_{\lambda_1\rightarrow\lambda_a}\underline{\xi}=\underline{\xi}_L=\xi_L$ .

With the above preparation, we can now complete the proof of Theorem 4.1.

Proof [Proof of Theorem 4.1]. By Assumptions 2,4 and the following observation

\begin{equation*} U\left(X_T^{LERL-PI}\right)\leq U(f(\xi_T))+U(L)+U\left(I\left(\left(\lambda_1-\lambda_2\right)\xi_T\right)\right), \end{equation*}

we have $ \mathbb{E}(U\left(X_T^{LERL-PI}\right))<+\infty$ . Let $X_T$ be any admissible terminal wealth of Problem (4.1), we obtain

\begin{align*} \mathbb{E}(U(X_T)) & \leq \mathbb{E}\left(U(X_T)+\lambda_1(x-\xi_TX_T)+\lambda_2\left(\epsilon-\xi_T(f(\xi_T)-X_T)\mathbf{1}_{X_T\leq f(\xi_T)}\right)\right)\\ & \leq \mathbb{E}\left(\sup_{X_T\geq L}\left(U(X_T)-\lambda_1\xi_TX_T-\lambda_2\xi_T(f(\xi_T)-X_T)\mathbf{1}_{X_T\leq f(\xi_T)}\right)\right)+x\lambda_1+\epsilon\lambda_2\\ & = \mathbb{E}\left(U\left(X_T^{LERL-PI}\right)\right)+\lambda_1\left(x- \mathbb{E}\left(\xi_TX_T^{LERL-PI}\right)\right)\\ & \quad +\lambda_2\left(\epsilon- \mathbb{E}\left(\xi_T\left(f(\xi_T)-{X_T^{LERL-PI}}\right)\mathbf{1}_{X_T^{LERL-PI}\leq f(\xi_T)}\right)\right)\\ & = \mathbb{E}\left(U\left(X_T^{LERL-PI}\right)\right), \end{align*}

where the first inequality follows from the budget and LERL constraints, the first equality follows from Lemma 4.2, and the last equality follows from the bindingness of both constraints:

(4.11) \begin{equation} \mathbb{E}\left(\xi_TX_T^{LERL-PI}\right)=x\quad\text{and}\quad \mathbb{E}\left(\xi_T\left(f(\xi_T)-X_T^{LERL-PI}\right)\mathbf{1}_{X_T^{LERL-PI}\leq f(\xi_T)}\right)=\epsilon. \end{equation}

In the expression of $X_T^{LERL-PI}$ , the Lagrangian multipliers $(\lambda_1, \lambda_2)$ can be obtained by solving (4.11). The existence is proved in Proposition 4.2. Hence, $X_T^{LERL-PI}$ is optimal. To finish, we remark that in Case (3), either the LERL constraint is inactive or its loss bound is equal to $\underline{\epsilon}$ . The former corresponds to cases (3a) and (3b) in Lemma 4.2, whose LERL loss is obviously equal to 0. The latter corresponds to the case (3c) in Lemma 4.2, and its LERL loss can be computed by substituting $X_T^{PI(L)}$ into (4.11).

6.1. Portfolio performance with benchmark

A performance measurement for a risky portfolio typically means a score attached to the portfolio. The goal of any investor who uses a particular performance measure is to select the portfolio for which this measure is the greatest. The literature on portfolio performance evaluation is vast, starting with the Sharpe ratio (see Sharpe (Reference Sharpe1966)). Alternative performance measures are reward-to-risk ratios representing a fraction where a measure of reward is divided by a measure of risk. Examples of such reward-to-risk ratios include the Sortino ratio (see, e.g., Sortino and Price (Reference Sortino and Price1994); Sortino et al. (Reference Sortino, Van Der Meer and Plantinga1999)) and the Omega ratio (see, e.g., Keating and Shadwick (Reference Keating and Shadwick2002)). For further expositions, we refer to Caporin et al. (Reference Caporin, Jannin, Lisi and Maillet2014).

In our framework, we adopt the Omega ratio of the portfolio $X_T$ with respect to the benchmark $Y_T$ defined by

(6.1) \begin{equation} \Omega_{{Y_T}}\left(X_T\right) \,:\!=\,\frac{\mathbb{E}\left(\max\{X_T-{Y_T},0\}\right)} {\mathbb{E}\left(\max\{Y_T-X_T,0\}\right)},\end{equation}

which quantifies the expected outperformance over the expected underperformance of a portfolio $X_T$ with respect to the benchmark ${Y_T}$ at maturity. We remark that the Omega ratio is also known as a gain-loss ratio, see, for example, Bernardo and Ledoit (Reference Bernardo and Ledoit2000); Cochrane and Saa-Requejo (Reference Cochrane and Saa-Requejo2000); Cherny and Madan (Reference Cherny and Madan2009). Furthermore, it is well-documented that the Omega ratio is deemed a better portfolio performance measurement than the Sharpe ratio and the Sortino ratio as its calculation captures the whole portfolio distribution (see also Bernard et al. (Reference Bernard, Vanduffel and Ye2019); Lin et al. (Reference Lin, Saunders and Weng2019); Guan et al. (Reference Guan and Liang2021) for related elaborations).

For our expected utility maximization under a joint LERL-PI framework, it is reasonable to look at the following portfolio performance measurement, which we call the “utility-transformed” Omega ratio,

(6.2) \begin{equation} U\Omega_{{Y_T}}\left(X_T\right) \,:\!=\,\Omega_{U(Y_T)}\left(U(X_T)\right) =\frac{\mathbb{E}\left((U(X_T)-U({Y_T}))\mathbf{1}_{X_T>{Y_T}}\right)} {\mathbb{E}\left((U({Y_T})-U(X_T))\mathbf{1}_{X_T<{Y_T}}\right)}.\end{equation}

Observe that $U\Omega_{{Y_T}}$ is the Omega ratio of the utility of the portfolio $U(X_T)$ with respect to the utility of the benchmark $U(Y_T)$ , and it quantifies utility gain over utility loss of the portfolio performance relative to the benchmark ${Y_T}$ . Compared to the Omega ratio $\Omega_{{Y_T}}$ , the utility-transformed ratio $U\Omega_{{Y_T}}$ aims at measuring the relative utility performance to the benchmark. Below, these measurements are numerically compared with the new BRBU performance ratio $\kappa/\eta$ defined by (5.6). It turns out that the BRBU ratio is aligned with the certainty equivalent defined by

\begin{equation*} CE\,:\!=\,U^{-1}\left(\mathbb{E}\left(U\left(X_T^{LERL-PI}\right)\right)\right),\end{equation*}

a certain terminal wealth level that generates the same expected utility. As shown below, while the Omega ratio and the utility-transformed Omega ratio do not vary consistently with the CE, the utility gain-loss ratio of the BRBU problem which positively moves with the CE in all the cases of benchmark.

We conclude this subsection by commenting on related literature on performance measurement. Rational performance measures can be constructed in an axiomatization framework (see, e.g., De Giorgi (Reference De Giorgi2005); Cherny and Madan (Reference Cherny and Madan2009)) or in a utility-based approach (see, e.g., Pedersen and Satchell (Reference Pedersen and Satchell2002); Zakamouline and Koekebakker (Reference Zakamouline and Koekebakker2009a,Reference Zakamouline and Koekebakkerb)). In Zakamouline (Reference Zakamouline2014), the author demonstrates that loss aversion plays a vital role in performance measurement. Performance measurement in a portfolio insurance context is also studied in Bertrand and Prigent (Reference Bertrand and Prigent2011); Ameur and Prigent (Reference Ameur and Prigent2018).

6.2. Money benchmark

In this subsection, we set the benchmark as a positive constant $\overline{x}$ (interpreted as a money market benchmark). Note that the condition $\overline{x}>L$ is needed; otherwise, the LEL constraint is not active. As discussed in Example 1, this constant benchmark satisfies Assumption 3 and corresponds to Case (1), that is, out/underperformance PM. Problem (4.1) now becomes an expected utility maximization problem under a joint LEL and PI constraint, named LEL-PI problem. We set the constant benchmark $\overline{x}=100e^{rT}$ , unless specified otherwise. We remark that if the PI level L is set to zero, the LEL-PI problem becomes a limited expected loss (LEL) risk management which has been studied in Basak and Shapiro (Reference Basak and Shapiro2001). We start with a numerical demonstration for the properties of the LEL-PI solution stated in Lemmas A.4–A.9 and Remark A.1. In particular, Figure 2a shows that while keeping the outperformance region and the benchmarking region unchanged, the PI constraint only influences the underperformance region, which demonstrates the discussion in Remark A.1. In addition, Figure 2b demonstrates that the initial capital x has no impact on the underperformance region. When additional capital is provided, the outperformance region enlarges with a shrinking benchmarking region, which aligns with the results from Lemma A.4. Economically, when the LERL loss bound $\epsilon$ is fixed (so is the underperformance), the risk manager in Case (1) would devote the additional capital to improve the portfolio performance in good states so as to maximize her utility.

Table 5. Impact of $\overline{x}$ on the certainty equivalent and the portfolio performance ratios

Figure 2. Impact of the PI level L, the initial capital level x, the LEL loss bound $\epsilon$ , and the constant benchmark level $\overline{x}$ on the optimal terminal wealth for the LEL-PI model in out/underperformance PM.

The impact of the loss bound $\epsilon$ is shown in Figure 2c. It can be observed that the benchmarking region shrinks for a larger loss bound. Specifically, when $\epsilon\geq\overline{\epsilon}=22.8091$ (see (4.2)), the LEL constraint is no longer binding, and the problem turns into a PI problem with minimum capital level L. Moreover, $\epsilon=0$ means that the agent has no tolerance for the underperformance relative to $\overline{x}$ , which corresponds to the PI problem with minimum capital level $\overline{x}$ .

In Figure 2d, the impact of constant benchmark $\overline{x}$ on the optimal terminal wealth $X_T^{LEL-PI}$ is numerically depicted. Notably, the larger $\overline{x}$ , the larger the benchmarking region, but the smaller the outperformance and underperformance regions. Interestingly, as observed from Table 5, all the performance ratios decrease with the benchmark level $\overline{x}$ , hand in hand with a lower certainty equivalent. This means that setting a higher benchmark level (e.g., higher guaranteed payment requirement or higher standard) would induce the agent to have a higher loss aversion, hence suffering a more significant utility loss from underperformance. Consequently, her optimal strategy is to enlarge the benchmarking region as much as possible. Due to the budget restriction, this will reduce the trade-off between the outperformance and the underperformance. In addition, due to the LERL constraint, adjusting the deterministic benchmark would mainly influence the outperformance region, indicating that a LERL-PI-RM with a higher benchmark (i.e., the agent is bearing a greater loss aversion relative to the benchmark) leads to smaller values for all the performance ratios.

The above sensitivity analysis suggests that all the performance ratios depict the portfolio performance quite well for the money benchmark case. Note that when $\overline{x}$ increases to $104.86$ , the initial capital x is no longer enough to hedge such a high level of $\overline{x}$ (see Lemma A.6). In this case, all the ratios tend to zero since $\lambda_1=+\infty$ and $\underline{\xi}=0$ , according to Lemma A.7. We also remark that with a smaller constant benchmark value $\overline{x}$ , it is easier for the agent to achieve a higher relative performance to the benchmark, hence implying a greater expected utility level. However, due to various considerations like solvency requirements and risk management perspectives, the benchmark $\overline{x}$ might be set above a certain value.

Below we will see that for various random benchmarks, while the Omega ratio (as well as the utility-transformed Omega ratio) may no longer well capture the portfolio performance, the BRBU ratio $\kappa/\eta$ still does quite well, however.

6.3. Stock market benchmark

This subsection presents a numerical analysis of the stock benchmark for under/outperformance PM, that is, Case (2). It can be observed from Figure 3b that in contrast to the case with money benchmark, an adjustment of initial capital x has no influence on the underperformance region but on the benchmarking and outperformance regions, substantiating again the result reported in Lemma A.4. It economically implies that the agent using a benchmark that asymptotically dominates the unconstrained Merton curve in good market scenarios (i.e., under/outperformance) is more concerned about the outperformance region and would use any additional capital to improve the portfolio performance in the relatively unfavorable states. From Figure 3a, in contrast to the money benchmark case (out/underperformance), adjusting the PI level L does not change the underperformance region but only influences the outperformance region (see a mathematical justification in Lemma A.8).

Figure 3. Impact of the initial capital x, the PI level L, and the LERL loss bound $\epsilon$ on the optimal terminal wealth for under/outperformance PM with stock market benchmarks.

Figure 3c depicts the impact of the loss bound $\epsilon$ on the optimal terminal portfolio. Compared with the initial capital and the PI level, the upper bound of relative expected loss $\epsilon$ has a notable impact on all regions of market states. As the loss bound $\epsilon$ directly affects the underperformance region (loss region) via the LERL constraint, there is a trade-off between the loss region and the other regions due to the budget constraint. In addition, the higher (resp. lower) the LERL loss bound $\epsilon$ , the lower the optimal terminal wealth in the underperformance (resp. outperformance) region.

As indicated in Table 6, the higher the present value $S_0$ of the stock benchmark, the lower the certainty equivalent and the performance ratios. Indeed, for the under/outperformance benchmark portfolio management, the outperformance (resp. underperformance) region corresponds to bad (resp. good) market states. Therefore, in this case, a higher initial benchmark value will induce the agent to have a higher loss aversion. According to Figure 3d, the agent chooses to enlarge the benchmarking region to take care of this concern, similar to the situation in which she faces a higher $\overline{x}$ in the money benchmark case (see Figure 2d). At the same time, both the underperformance and outperformance regions diminish. As a result, a smaller outperformance region causes a decrease in expected utility. As seen above, all performance ratios capture well the portfolio performance in this situation.

Table 6. Impact of $S_0$ on the certainty equivalent and the portfolio performance ratios

Figure 4. Comparison of different optimal strategies

To further investigate the LERL-PI agent’s investment behavior, we plot the optimal strategies for both money market and stock market benchmarks in Figure 4 and compare them with the unconstrained (Merton) optimal strategy. As shown in Figure 4a, compared to the LEL strategy, the LEL-PI strategy is more conservative in the very bad market states due to the additional PI constraint. As pointed out in (Basak and Shapiro, Reference Basak and Shapiro2001, Proposition 5 (ii)), the LEL strategy is bounded from above by the Merton strategy, which also applies to the LEL-PI strategy.

In Figure 4b, we compare the LERL-PI, LERL, and Merton optimal strategies for the stock market benchmark. It is observed that with an extra PI constraint, the LERL-PI strategy falls to zero in the very bad market states. On the contrary, without a PI constraint, the LERL strategy is bounded below by the Merton strategy and asymptotically converges to the Merton ratio when the market state worsens. Aligned with what is observed in Figure 3, the agent is more concerned about the underperformance relative to the benchmark (i.e., good market states).

In the remaining subsections, we take a deeper look at the LERL-PI solution for hybrid, mixed, and CPPI benchmarks.

6.4. Hybrid benchmark

In this subsection, we study the impact of hybrid benchmarks with respect to the riskless investment proportion $\alpha$ . As mentioned in Example 3, out/underperformance (i.e., Case (1)) or under/outperformance (i.e., Case (2)) PM can happen depending on whether $\frac{\sigma}{\theta}(1-\alpha)$ is greater or smaller than $\frac{1}{\gamma}$ .

The result for out/underperformance (resp. under/outperformance) PM is presented in Table 7 (resp. Table 8). We can observe that for out/underperformance PM, the higher the riskless investment proportion $\alpha$ , the lower the certainty equivalent. In addition, we can also observe from Figure 5a and Table 7 that a lower riskless investment proportion $\alpha$ indicates a shrinking benchmarking region. Eventually, it would become the PI solution when $\underline{\xi}=\overline{\xi}$ . It happens when $\alpha\approx 0.083$ , which is obtained by solving

(6.3) \begin{equation} \mathbb{E}\left(\xi_T\left(f^H(\xi_T,\alpha)-X_T^{PI}\right)\mathbf{1}_{X_T^{PI}\leq f(\xi_T)}\right)=\epsilon.\end{equation}

On the other hand, if we keep increasing the riskless investment proportion $\alpha$ , it would eventually become the LEL-PI solution when $\alpha=1$ since $f^H(\xi_T)=Y_0^He^{rT}$ in this situation.

Table 7. Effect of $\alpha$ in the hybrid benchmark: out/underperformance PM (Case (1))

Table 8. Summary of the effect of changing $\alpha$ in under/outperformance PM (Case (2)) with hybrid benchmarks

Figure 5. Out/underperformance PM with hybrid benchmark: impact of the riskless investment proportion $\alpha$ on the optimal terminal wealth $X_T^{LERL-PI}$ .

Interestingly, for out/underperformance hybrid benchmark PM (Case (1)), both the Omega and utility-transformed Omega ratios do not increase with the certainty equivalent. Indeed, Table 7 shows that the Omega ratio reaches the highest value when $\alpha\approx 0.6$ . The utility-transformed Omega ratio shares a very similar trend with the Omega ratio when changing $\alpha$ . In particular, the measurement $U\Omega_{f^H(\xi_T)}$ increases in $\alpha\in(0,0.8)$ but decreases in $\alpha\in(0.8,1)$ and thus attains the maximum value at $\alpha\approx 0.8$ . Apparently, none of these two values of $\alpha$ corresponds to the highest certainty equivalent, which is attained for $\alpha\in[0,0.083]$ (PI case). This suggests that the portfolio performance in such a random benchmark LERL-RM situation may not be well reflected by using the Omega ratio or its utility-transformed version. Nevertheless, the BRBU indicator still captures well the portfolio performance relative to the benchmark in this case. Remarkably, as shown in both Tables 7 and 8, $\kappa/\eta$ is always positively proportional to the certainty equivalent, which is not always the case for the classical and the utility-transformed Omega ratio. Let us take a closer inspection. For Case (1), recall that for a lower value of $\kappa/\eta$ means that the agent is more concerned about underperformance (unfavorable market states) than outperformance (favorable market states). From Table 7, it can be observed that being more concerned about the underperformance region, the agent would prefer a higher risk-free investment proportion benchmark, which explains why $\kappa/\eta$ is negatively proportional to $\alpha$ . In addition, to deal with this concern, the agent chooses to enlarge the benchmarking region so that the underperformance region diminishes. However, due to the budget constraint, this results in smaller upside potentials and so is the outperformance region, according to Figure 5a and Table 7; hence, a decrease in the expected utility.

To gain further insight related to the optimal terminal wealth, we plot in Figure 5b the estimated distribution of $X_T^{LERL-PI}$ by simulating $10^7$ paths of the market price density $\xi_T$ for different values of $\alpha$ in Case (1) of the hybrid benchmark. In this density plot, the left tail (resp. right tail) corresponds to the underperformance region (resp. outperformance region), and the central area reflects the benchmarking region, which can be seen by comparing Figure 5b and 5a (the plot of $X_T^{LERL}$ against $\xi_T$ ). As pointed out in Figure 5b, as the riskless investment proportion $\alpha$ increases, a less dispersed distribution with thinner tails is displayed. The less dispersed shape corresponds to the larger benchmarking region depicted in Figure 5a and indicates a smaller variance; thus, a less risky investment behavior, which matches the less risky benchmark (higher $\alpha$ ). The thinner left tail (resp. right tail) coincides with the shrinking PI region (resp. outperformance region), as observed in Figure 5a. Moreover, the size change of both the left and right tails with respect to the change of $\alpha$ confirms the trade-off between the underperformance and the outperformance. For instance, a decrease of $\alpha$ induces a fatter left tail (higher probability of the underperformance) associated with a fatter right tail (higher probability of the outperformance).

Table 8 shows that when the riskless investment proportion $\alpha$ increases, all the measurements, except for the utility-transformed Omega ratio, rise, capturing well the augmentation in the certainty equivalent. This observation seems surprising as a more prudent investment (i.e., higher $\alpha$ ) leads to greater expected utility. However, this can be explained by recalling that for under/outperformance PM (Case (2)), the outperformance region corresponds to the bad market states, and an increase in $\kappa/\eta$ means that the agent gets more concerned about this region and will hence choose a more conservative benchmark (i.e., higher $\alpha$ ). In this situation, the agent chooses to reduce the benchmarking region, as depicted in Figure 6a and Table 8, while simultaneously expanding the outperformance and underperformance regions. As a result, this leads to a higher expected utility.

Figure 6. Under/outperformance PM with the hybrid benchmark: Impact of the riskless investment proportion $\alpha$ on the optimal terminal wealth $X_T^{LERL-PI}$ .

The failure of the utility-transformed Omega ratio raises an alert about using performance measures based on merely computing the portfolio performance relative to the benchmark. Unlike the classical Omega ratio, the utility-transformed Omega ratio is inversely proportional to the expected utility. In particular, while an increase in the riskless investment proportion $\alpha$ in the hybrid benchmark (Case (2)) induces a smaller the utility-transformed Omega ratio, it creates a higher $\kappa/\eta$ (i.e., the agent is less loss-averse), resulting in a higher certainty equivalent. Furthermore, as illustrated in Table 8, the BRBU ratio moves in tandem with the certainty equivalent, indicating a potentially suitable candidate for the utility-based performance measure. This aspect will be further investigated in the remaining numerical analysis.

Remark that $\alpha=0$ corresponds to the stock market benchmark studied in Section 6.3, whereas the problem becomes a pure PI problem when the riskless investment proportion $\alpha$ exceeds a specific value ( $\approx$ 0.76) obtained by solving (6.3) with the given parameters of Case (2) (see Table 4).

Similarly to Case (1), it is shown in Figure 6b that an increase in the riskless investment proportion $\alpha$ (less risky benchmark) also induces a less dispersed distribution as observed in Figure 5b. Another similarity between Case (1) and Case (2) lies in the trade-off between the outperformance region and the underperformance region. However, the difference between Case (1) and Case (2) is notable. In particular, for a hybrid benchmark, the LERL-PI terminal wealth in Case (1) displays a single peak distribution (see Figure 5b), whereas it exhibits a multi-peak distribution structure in Case (2), as reported in Figure 6b. Unlike Case (1), the left tail (resp. right tail) of the density plot in Case (2) corresponds to the outperformance region (resp. underperformance region), but the central area (between the first peak and the last peak) also reflects the benchmarking region.

6.5. Mixed benchmark

This subsection considers the mixed benchmark in Example 4 with various proportions of capital invested in the risk-free asset $\beta$ . Recall first from Remark 3.1 that the mixed benchmark can only be fitted in Case (1), that is, out/underperformance PM.

Our numerical result in Table 9 shows that the higher the ratio of capital $\beta$ invested in the money market in the mixed benchmark construction, the higher the certainty equivalent. Surprisingly, both the Omega and utility-transformed Omega ratios decrease with the certainty equivalent, indicating that using these ratios in out/underperformance PM with mixed benchmarks would lead to a largely inaccurate portfolio performance measurement. Interestingly, like Case (1) of the hybrid benchmark, the BRBU indicator well captures the portfolio performance relative to mixed benchmarks. In particular, both the BRBU ratio $\kappa/\eta$ and the certainty equivalent decrease with $\beta$ . This can be explained as in Case (1) of the hybrid benchmark that both metrics decrease in $\alpha$ . In addition, as seen in Table 9, the lower the ratio of capital $\beta$ invested in the money market, the higher the certainty equivalent. This confirms the intuition that a higher risky investment proportion corresponds to a smaller loss aversion, consequently resulting in an increase in the expected utility. According to Figure 7a and Table 9, a lower $\beta$ is also accompanied with expanding outperformance and underperformance regions, whereas the benchmarking region is contracting. Moreover, similarly to Case (1) of the hybrid benchmark, we can see that if $\beta$ decreases to a certain point ( $\approx 0.06$ ), the benchmarking region disappears, and the LERL-PI problem becomes the PI case. Likewise, the PI solution, which corresponds to $\kappa=\eta$ , indicates that the agent is indifferent between the outperformance and underperformance effects.

Table 9. Summary of the effect of changing $\beta$ in the mixed benchmark

Figure 7. Impact of the ration of capital invested in the money market $\beta$ on the optimal terminal wealth for out/underperformance PM with the mixed benchmark.

6.6. CPPI benchmark

In this subsection, we analyze the same metrics as in the previous subsections for CPPI benchmarks presented in Example 5, where it is assumed that the multiplier m is bounded from above by the Merton strategy $\theta/(\sigma\gamma)$ .Footnote 2 We remark that for m in the range of $[0,\theta/(\sigma\gamma)]$ , the LERL constraint is always binding and never turns into the PI case, unlike the other benchmarks considered before. Note also that with such a CPPI benchmark, we are in a out/underperformance PM (Case (1)) because the CPPI benchmark entails a lower bound, see Remark 3.1. In the following numerical demonstration, we assume that the PI level is the same as the terminal floor of CPPI benchmarks ( $L_T=L$ ).

The numerical results are reported in Figure 8 and Table 10 with different values of $m\in[0,0.9]$ . According to Table 10, the higher the predetermined multiplier m, the higher the certainty equivalent and the BRBU ratio $\kappa/\eta$ . The reason is that by adopting a utility function with smaller loss aversion, the agent can gain higher upside potentials in good market states, which eventually leads to an increase in certainty equivalent. As demonstrated for hybrid benchmarks (Case (1)) and mixed benchmarks, neither the Omega ratio nor the utility-transformed Omega ratio fully captures the portfolio performance relative to CPPI benchmarks.

Table 10. Summary of the effect of changing predetermined multiplier m in the CPPI benchmark

Figure 8. Impact of the predetermined multiplier m on the optimal terminal wealth $X_T^{LERL-PI}$ with the CPPI benchmark.

To gain more insights into the investment choice, we plot in Figures 9 and 10 the optimal LERL-PI investment strategies in the out/underperformance PM (Case (1)) for hybrid, mixed, and CPPI benchmarks in comparison with the limiting LEL-PI, PI, Merton, and the LERL strategy. It is interesting to observe that these three benchmark settings share a very similar investment pattern. In all cases, the LERL-PI risky investment ratio is mostly bounded in the range limited by the LEL-PI and PI strategies, whereas the LERL strategy is limited by the LERL-PI and Merton investment ratio. In particular, the LERL-PI agent in very good market scenarios will adopt an investment strategy close to the Merton ratio to beat the benchmark. When the market is no longer extremely good, the agent, due to the loss constraint, is sensible to reduce the risky asset holding significantly but always keep it smaller than that of the PI strategy and simultaneously greater than the LEL-PI one. In intermediate market states, she tries to replicate the benchmark as much as possible, namely enlarging the benchmarking region, by taking higher risky exposures. However, in bad market states, because of dealing with the underperformance and the LERL constraint, the agent adopts a riskier investment behavior by increasing her risky investment exposures in this region. In addition, without PI constraint, the LERL agent takes even riskier investment behavior than the LERL-PI agent in the intermediate and bad market states. When the market states are extremely bad (i.e., with a very high value of the pricing density $\xi_t$ ), the PI constraint now forces her to reduce the risky investment rapidly, whereas the LERL strategy reverts to the Merton ratio. Moreover, for each specific benchmark, a more prudent benchmark (i.e., with higher risk-free investment initiation $\alpha$ (resp. $\beta$ ) in case of the hybrid benchmark (resp. mixed benchmark) or with a lower value of the multiplier m in case of the CPPI benchmark) the agent choose, the closer the LERL-PI strategy reaches the limiting LEL-PI strategy.

Figure 9. LERL-PI optimal investment strategies for out/underperformance PM.

Figure 10. Comparison of optimal investment strategies for out/underperformance PM.

6.7. Discussion

We close this section by commenting on the portfolio performance measurement in a LERL-PI-RM framework, where the agent has to control the portfolio loss relative to a random benchmark. From the above analysis, it is observed that increasing certainty equivalent not only results from a contraction of the benchmarking region but also forms an enlargement of the outperformance region. However, when the loss constraint relative to the benchmark is active, the regulatory effects on the outperformance and underperformance regions may offset each other; hence, the investment performance cannot be fully captured by only looking at the well-known Omega ratio or its utility-transformed version.

Our numerical analysis with various benchmarks shows that the Omega ratio and its utility-transformed version can only reflect the expected utility gain or loss due to the change of loss aversion caused by the different reference levels, but fail to reflect the expected utility performance caused by changing risk profile in the benchmark. Nevertheless, compared with the classical and utility-transformed Omega ratios, the BRBU ratio derived from the benchmark-reference-based utility problem does well for all benchmark types. To account for this result, it is worth recalling that the BRBU ratio takes the agent’s loss aversion into account, whereas the (utility-transformed) Omega ratio does not, as explained in Section 5. Hence, our results suggest a more proper portfolio performance indicator which incorporates the agent’s loss aversion in the LERL benchmarking risk management.