2.1. Financial market

The financial market is composed of one risk-free asset and one risky asset, where the former has a risk-free return rate r and no volatility, while the latter has a higher expected return rate $\mu \! > \! r$ and a positive volatility $\sigma \! > \! 0$ . Denote by $\theta \triangleq \frac{\mu-r}{\sigma} \! > \! 0$ the market price of risk. The augmented filtered complete probability space $\big(\Omega, \mathcal{F}_T,\{\mathcal{F}_t\}_{0 \leq t \leq T}, \mathbb{P}\big)$ in the financial market is generated by a Brownian motion $W=\{W_{t}\,:\, 0 \leq t \leq T\}$ on $(\Omega, \mathcal{F}_T, \mathbb{P})$ ; i.e., the equipped filtration $ \mathcal{F} = \{\mathcal{F}_t\}_{0 \leq t \leq T}$ is denoted by $ \mathcal{F}_t = \sigma\{W_{u}\,:\, 0 \leq u\leq t\}$ and satisfies the usual conditions. The interest rate is a constant $r\!>\!0$ , and the price of the risk-free asset $R=\{R_t\,:\, 0 \leq t \leq T\}$ is ${\textrm{d} R_t} = r {R_t} \textrm{d} t$ . The price of the risky asset $S=\{S_t\,:\, 0 \leq t \leq T\}$ is a geometric Brownian motion satisfying ${\textrm{d} S_t} = S_t\big(\mu \textrm{d} t+\sigma \textrm{d} W_t\big) \equiv S_t\big( r \textrm{d} t+\sigma ( \textrm{d} W_t + \theta \textrm{d} t )\big)$ . As a result, the financial market is complete, and we denote the state price density process $\{\xi_t \,:\, 0 \leq t \leq T\}$ by $\xi_t \triangleq \exp \big(\!-\!\big(r+ \frac{1}{2}\theta^2\big)t - \theta W_t\big)$ . The wealth process $X=\{X_t \,:\, 0 \leq t \leq T\}$ is uniquely determined by a portfolio process $\pi=\{\pi_t\,:\, 0\leq t \leq T\}$ and an initial wealth $x_0$ :

(3) \begin{equation}\textrm{d} X_t = r X_t \textrm{d} t +\pi_t \sigma( \textrm{d} W_t + \theta \textrm{d} t).\end{equation}

For any time $t\in[0,T]$ , $\pi_t$ represents the amount of wealth invested in the risky asset at time t. A portfolio $\pi$ belongs to an admissible portfolio set $\mathcal{V}[0,T]$ if and only if

1. (i) $\pi$ is an $\{\mathcal{F}_t\}_{0 \leq t \leq T}$ -progressively measurable ${\mathbb{R}}$ -valued process and $\int_0^T |\pi_t|^2 \textrm{d} t < \infty$ almost surely, and

2. (ii) X is not lower than the liquidation boundary, i.e., $X_t\geq b x_0 e^{-r(T-t)}$ almost surely for all $t \in [0,T]$ .

The first condition guarantees the existence and finiteness of the wealth process X. The second condition comes from the so-called liquidation boundary $bx_0$ , which ensures that the wealth at time t is always above the liquidation level $b x_0 e^{-r(T-t)}$ , where $b\in[0,1)$ . The liquidation boundary is also considered in the studies of [Reference Hodder and Jackwerth17] and [Reference He and Kou16].

2.2. S-shaped utilities and first-loss contracts

In our model, we assume that the investor $(i=0)$ and the manager $(i=1)$ have S-shaped utilities:

(4) \begin{equation}\widehat{U}_i(x) =\left\{\begin{aligned}& x^{p_i}, &x \geq 0,\\& -\lambda_i ({-}x)^{q_i}, &x < 0.\end{aligned}\right.\end{equation}

The functions $\widehat{U}_0({\cdot})$ and $\widehat{U}_1({\cdot})$ reflect respectively the investor’s and the manager’s preferences, which are S-shaped utilities as in cumulative prospect theory. They are illustrated in Figure 1. Note that in Figure 1 we plot $\widehat{U}_0(\cdot - x_0)$ instead of $\widehat{U}_0({\cdot})$ to make the curves clearer. For $i=0,1$ , $p_i\in(0,1)$ and $q_i\in(0,1)$ respectively measure the degrees of risk-aversion and risk-seeking, and $\lambda_i> 1$ measures the degree of loss-aversion. The investor and the manager compare their P&L with a reference point 0.

Figure 1. S-shaped utilities: $\widehat{U}_1$ (manager), $\widehat{U}_0(\cdot - x_0)$ (investor).

The S-shaped utility is related to cumulative prospect theory (cf. [Reference Tversky and Kahneman33]), where the individual displays loss-aversion below a certain reference level and risk-aversion above that level. The theoretical results in this area from behavioral economics are supported by empirical evidence (cf. [Reference Barberis, Thaler, Constantinides, Stulz and Elsevier3]) and have been successfully applied in many fields of decision-making, including in the contexts of hedge funds (cf. [Reference Kouwenberg and Ziemba20]) and insurance (cf. [Reference Lin, Saunders and Weng23]). Although the investor is not a decision-maker for the investment as the manager is, the investor also sets a profit goal as his/her own reference level in the S-shaped utility in order to choose an appropriate contract.

Next we consider a recently popular first-loss contract (cf. [Reference He and Kou16]). The mechanism of the contract is as follows. There is a guarantee benchmark and a first-loss capital. The guarantee benchmark is the initial fund wealth, and the first-loss capital is provided by the manager in the total fund wealth. When the fund wealth is just below the guarantee benchmark, the manager has to lose first and the investor loses nothing until the first-loss capital is all lost. As a reward, a higher incentive rate for good performance is shared with the manager. The reference level in the S-shaped utilities and the guarantee benchmark in the first-loss contract naturally fit each other. This is why we use S-shaped utilities and the first-loss contract in our model; this is also reflected in [Reference Berkelaar, Kouwenberg and Post4] and [Reference Kouwenberg and Ziemba20], which study S-shaped utilities, and in [Reference He and Kou16], which studies both of them.

Specifically, the manager’s P&L is given by a function $\Theta(X_T)$ of the terminal wealth $X_T$ , i.e.,

(5) \begin{equation}\Theta(x) =\left\{\begin{aligned}&(\omega + \alpha(1-\omega))(x -x_0), && x \geq x_0,\\&x -x_0, && (1 - \omega)x_0 \leq x < x_0,\\&- \omega x_0, && b x_0 \leq x < (1 - \omega)x_0.\end{aligned}\right.\end{equation}

The investor’s P&L is given by $X_T - \Theta(X_T)- x_0$ . A first-loss contract has two parameters: $\omega \in [0, 1]$ is the proportion of managerial ownership in the fund wealth, and $\alpha \in [0, 1]$ is the incentive rate for good performance above the benchmark $x_0$ . In a first-loss contract, the manager provides the first-loss capital $\omega x_0$ . The investor does not suffer a loss until the terminal wealth is under $(1-\omega) x_0$ . Practically, the manager is incapable of covering the whole loss of funds for the investor, and thus the liquidation boundary is (much) lower than the minimum guarantee (i.e., $0 < bx_0 < (1-\omega) x_0$ ). Figure 2 gives a numerical demonstration of the above details.

Figure 2. P&L: $\Theta(X_T)$ (manager), $X_T-\Theta(X_T)-x_0$ (investor).

We do not want too many contract parameters involved. Thus, we consider a specific first-loss contract and fix it in the optimization problem. Nevertheless, our insights regarding the centrally planned problem (1) and the following results can be applied in other models.

4.1. Three-term decomposition of the optimal portfolio

To further illustrate the patterns in Figure 7 based on Theorem 1, we analyze the structure of the optimal portfolio, which is significantly different from that in the classical theory (cf. [Reference Merton25, Reference Merton26]). To be precise, the optimal portfolio is decomposed into three terms in Theorem 1, and the second and third terms reflect the novelty of the optimal portfolio solved out in our model.

The first term, $\pi_t^{(1)}=\frac{\theta}{(1-p)\sigma}X_t^*$ , is the Merton term, the constant-percentage risky investment based on the CRRA utility. In [Reference Merton25, Reference Merton26], the optimal percentage allocated to the risky asset is a constant $\frac{\theta}{(1-p)\sigma}$ .

The second term, $\pi_t^{(2)}$ , is called the aggressive term. It increases the allocation on the risky asset, and it arises from the non-concavity of the weighted utility. We observe that $\pi_t^{(2)}$ is related to the linear intervals of the corresponding concave envelope. Quantitatively, the aggressive term is a sum of the product of each linear interval length, the function of the linear interval slope, and the exponential discount factor.

The third term, $\pi_t^{(3)}$ , is called the conservative term. It decreases the allocation to the risky asset, and it arises from certain wealth levels related to the concave envelope: the liquidation boundary $bx_0$ , the first-loss residue capital $(1-\omega)x_0$ , and the benchmark $x_0$ . Quantitatively, the conservative term is a weighted sum of a certain multiple of these wealth levels. The multiple is the product of the Merton constant $\frac{\theta}{(1-p)\sigma}$ and the exponential discount factor. The weight is the function of the slope of the corresponding linear part’s slope; the weights can be interpreted as probabilities, with the sum 1.

4.2. Asymptotic analysis of the optimal portfolio

In this subsection, we conduct an asymptotic analysis of $X^*$ and $ \pi^*$ in Theorem 2 below and discuss the resulting economic insights.

Theorem 2. We have the following statements on $X^*$ and $\pi^*$ :

1. (i) (Monotonic analysis.) $X_t^*$ and $\pi_t^{(1)}$ are decreasing with respect to $\xi_t$ , while $\pi_t^{(3)}$ is increasing with respect to $\xi_t$ . In the first-loss case, $\pi_t^{(2)}$ is increasing if

   $$\xi_t \in \left(0, \frac{\kappa_2}{\nu^*} e^{\big(r - \frac{\theta^2}{2}\big)(T-t)}\right),$$
   and $\pi_t^{(2)}$ is decreasing if
   $$\xi_t \in \left(\frac{\kappa_3}{\nu^*} e^{\big(r - \frac{\theta^2}{2}\big)(T-t)}, +\infty\right).$$
   In the option case, $\pi_t^{(2)}$ is increasing if
   $$\xi_t \in \left(0, \frac{\kappa_1}{\nu^*} e^{\big(r - \frac{\theta^2}{2}\big)(T-t)}\right)$$
   and decreasing otherwise.

2. (ii) (Dynamic analysis.) As $\xi_t \rightarrow 0$ , we have

   \begin{equation*}X_t^* \rightarrow +\infty, \quad\pi_t^{*} \rightarrow +\infty,\quad\frac{\pi_t^{*}}{X_t^*} \rightarrow \frac{\theta}{(1-p)\sigma}, \quad\pi_t^{(2)} \rightarrow 0, \quad\frac{\pi_t^{(3)}}{-\frac{\theta}{(1-p)\sigma} x_0 e^{-r(T-t)}} \rightarrow 1.\end{equation*}

3. (iii) (Dynamic analysis.) As $\xi_t \rightarrow +\infty$ , we have

   \begin{equation*}\frac{X_t^*}{bx_0 e^{-r(T-t)}} \rightarrow 1, \quad\pi_t^{*} \rightarrow 0,\quad\frac{\pi_t^{*}}{X_t^*} \rightarrow 0,\quad\pi_t^{(2)} \rightarrow 0, \quad\frac{\pi_t^{(3)}}{-\frac{b\theta}{(1-p)\sigma} x_0 e^{-r(T-t)}} \rightarrow 1.\end{equation*}
   We list the above results of (ii) and (iii) as follows:

Table 1. Dynamic analysis.

Table 2. Option case: terminal-time analysis.

1. (iv) (Terminal-time analysis.) In the option case, we have the following:

1. (v) (Terminal-time analysis.) In the first-loss case, we have the following:

Table 3. First-loss case: terminal-time analysis.

A basic observation for our analysis is that $\xi_t$ , the state price process at time t, can be interpreted as an indicator showing whether the market is in a good state (small $\xi_t$ ) or a bad state (large $\xi_t$ ).

Theorem 2(i) provides a monotonic analysis and shows that the wealth process $X_t^*$ is decreasing with respect to $\xi_t$ . This statement additionally explains why a small $\xi_t$ means a good market state (leading to a large wealth $X_t^*$ ) and a large $\xi_t$ means a bad market state (leading to a small wealth $X_t^*$ ). More importantly, our three-term decomposition provides an approach for the monotonic analysis, because the monotonicity of $\pi_t^*$ with respect to $\xi_t$ is unknown in general. While the first term $\pi_t^{(1)}$ , as a multiple of $X_t^*$ , is decreasing in $\xi_t$ , the conservative term $\pi_t^{(3)}$ is increasing in $\xi_t$ . The combined effect of the three terms is that the optimal strategy $\pi_t^*$ is not monotone in $\xi_t$ . Yet we can use the decomposition to show the monotonicity of the separate terms. If the market improves ( $\xi_t$ decreases), the first term $\pi_t^{(1)}$ contributes to an increase on the risky investment, although the third term $\pi_t^{(3)}$ decreases. If the market gets worse ( $\xi_t$ increases), the first term $\pi_t^{(1)}$ contributes to a decrease on the risky investment, although the third term $\pi_t^{(3)}$ increases. Two further scenarios ( $\xi_t \rightarrow 0$ and $\xi_t \rightarrow +\infty$ ) are respectively analyzed in Theorem 2(ii)–(iii), where $\pi_t^{(1)}$ is a dominating term. In the middle range of $\xi_t$ , the monotonicity of $\pi_t^*$ is highly affected by the aggressive term $\pi_t^{(2)}$ and the conservative term $\pi_t^{(3)}$ .

Theorem 2(ii) provides a dynamic analysis and shows that when the market is in a good state ( $\xi_t \rightarrow 0$ ), the wealth is well accumulated and the optimal percentage allocated to the risky asset is asymptotically the Merton constant $\frac{\theta}{(1-p)\sigma}$ . The aggressive term $\pi_t^{(2)}$ tends to zero, showing that there is no need to take on more risk in the good state. The conservative term $\pi_t^{(3)}$ tends to $\frac{\theta}{(1-p)\sigma} x_0$ , the Merton constant multiplied by the initial wealth $x_0$ . This shows that in the good state, the weighted sum $\pi_t^{(3)}$ is tracking one of the wealth levels, the benchmark $x_0$ .

Theorem 2(iii) shows that when the market is in a bad state ( $\xi_t \rightarrow +\infty$ ), the investment performs poorly and tends to be liquidated, and the optimal strategy is to invest everything in the risk-free asset. The aggressive term $\pi_t^{(2)}$ tends to zero, showing that it is impossible to take on any risk in the asymptotic liquidation state. The conservative term $\pi_t^{(3)}$ tends to $\frac{\theta}{(1-p)\sigma} b x_0$ , the Merton constant multiplied by the liquidation boundary $b x_0$ . This shows that in the bad state, the weighted sum $\pi_t^{(3)}$ is tracking one of the wealth levels, the liquidation boundary $b x_0$ . As a result, the amount allocated to the risky asset tends to 0 when the optimal wealth tends to be liquidated, and the wealth is never below the liquidation level $bx_0 e^{-r(T-t)}$ (which coincides with [Reference Bielecki, Jin, Pliska and Zhou5, Reference He, Liang, Liu and Ma14, Reference He and Kou16]). In addition, (ii) and (iii) show that if the wealth $X_t^*$ increases, the higher wealth level which $\pi_t^{(3)}$ is tracking will increase, and thus the conservative term $\pi_t^{(3)}$ will increase.

Theorem 2(iv) provides a terminal-time analysis and shows the asymptotic behavior as one approaches the terminal evaluation time T in the option case. We compare the state price $\xi_T$ with a threshold value $\frac{\kappa_1}{\nu^*}$ . When the market is in a good state ( $\xi_T < \frac{\kappa_1}{\nu^*}$ ), the optimal final wealth is higher than the benchmark $x_0$ and results in a profit. The term $\pi_t^{(2)}$ tends to zero, and $\pi_t^{(3)}$ tends to $-\frac{\theta}{(1-p)\sigma} x_0$ , which is coincident with (ii). However, the optimal percentage $\frac{\theta}{(1-p)\sigma}\frac{X_T^*-x_0}{X_T^*}$ is smaller than the Merton constant $\frac{\theta}{(1-p)\sigma}$ , implying that the optimal investment becomes more conservative when the time is limited and the market is good (which coincides with [Reference Carpenter9]). The bad state is coincident with (iii).

Interestingly, when the market is in a specific state ( $\xi_T = \frac{\kappa_1}{\nu^*}$ ), the optimal portfolio $\pi^*$ tends to infinity while the optimal wealth process $X^*$ tends to liquidation, implying that the best strategy is to gamble as much as possible, but that the resulting outcome is either liquidation or profit for both parties. In this case, $\pi_t^{(2)}$ and $\pi_t^{(3)}$ behave strangely. The aggressive term tends to infinity, reflecting the strategy of gambling for a profit, while the conservative term tends to $\frac{\theta}{(1-p)\sigma} \frac{b+1}{2} x_0$ , the exact midpoint between the conservative terms in (ii) and (iii). Theorem 2(v) shows similar asymptotic behavior as one approaches the terminal evaluation time T in the first-loss case.

To sum up, because of $\pi^{(2)}$ , the portfolio gambles on the risky asset whenever the utility is not strictly concave, resulting in two peaks in the first-loss case and one peak in the option case; because of $\pi^{(3)}$ , the portfolio becomes conservative at some wealth levels, resulting in three valleys in the first-loss case and two valleys in the option case. The term $\pi^{(3)}$ further explains why the portfolio becomes more conservative in a better market state (cf. [Reference Carpenter9]) and the total wealth is always above the liquidation boundary $b x_0$ (cf. [Reference Bielecki, Jin, Pliska and Zhou5]).

5.1. Pareto frontier

In this subsection, we establish the collection of Pareto points (2) under the optimal portfolio $ \pi^*$ given in Theorem 1. Based on Proposition 1, we can equivalently establish $\text{PF} \equiv \big\{\big(\mathbb{E}\big[U_1\big(\hat{X}_\gamma\big)\big], \mathbb{E}\big[U_0\big(\hat{X}_\gamma\big)\big]\big)\,:\,{\gamma \in [0,1]} \big\} \subset \mathbb{R}^2,$ where, with a little abuse of notation, $\hat{X}_\gamma \triangleq X_T^{\pi^{*,\gamma}}$ denotes the terminal wealth of the optimal portfolio $\pi^{*,\gamma}$ in Problem (6) with weight $\gamma \in [0,1]$ . The point $\big(\mathbb{E}\big[U_1\big(\hat{X}_\gamma\big)\big], \mathbb{E}\big[U_0\big(\hat{X}_\gamma\big)\big]\big)$ is a manager–investor expected utility (EU) pair.

Our first result is Theorem 3 below, showing that the collection (2) is a decreasing and concave frontier, which is referred to as the Pareto frontier (PF). Above all, we need to introduce the manager’s single utility maximization problem,

(21) \begin{eqnarray}\sup_{X \in D} \mathbb{E}\big[\widehat{U}_1\big(\Theta(X_T)\big)\big],\end{eqnarray}

and the investor’s single utility maximization problem,

(22) \begin{eqnarray}\sup_{X \in D} \mathbb{E}\big[\widehat{U}_0\big(X_T - \Theta(X_T) - x_0\big)\big].\end{eqnarray}

Similarly to Theorem 1, we show in Proposition 2 that there exist (distribution-wise unique) solutions to Problems (21) and (22).

We now characterize the PF, in Theorem 3.

Theorem 3 states that the collection (2) is a strictly decreasing and strictly concave frontier if it is not reduced to a single point. A decreasing PF shows that the total welfare is shared between the investor and the manager. Nevertheless, in some multi-objective problems (e.g., when $U_1$ is a multiple of $U_0$ ), if Problem (21) and Problem (22) have the same solution, the PF will reduce to a single point (i.e., $\big(\mathbb{E}\big[U_1\big(\hat{X}_0\big)\big], \mathbb{E}\big[U_0\big(\hat{X}_0\big)\big]\big) = \big(\mathbb{E}\big[U_1\big(\hat{X}_1\big)\big], \mathbb{E}\big[U_0\big(\hat{X}_1\big)\big]\big)\big)$ .

5.2. Comparison of fixed-weight contracts

We first compare different contracts with the weight $\gamma$ fixed ( $\gamma = 0, 0.5, 1$ ); see Figures 8–9. Each contract contains two parameters $(\omega, \alpha)$ . Recall in Equation (5) that $\omega$ is the manager’s proportion in the total asset and $\alpha$ is the incentive rate for the wealth above the benchmark $x_0$ .

Figure 8. Fixed-weight contract comparison: each circle point corresponds to a manager–investor EU pair under a contract with $(\omega, \alpha)$ . For all panels, $T = 0.01$ ; left panel, $\gamma = 0$ ; middle panel, $\gamma = 0.5$ ; right panel, $\gamma = 1$ .

Figure 9. Fixed-weight contract comparison: each circle point corresponds to a manager–investor EU pair under a contract with $(\omega, \alpha)$ . For all panels, $T = 20$ ; left panel, $\gamma = 0$ ; middle panel, $\gamma = 0.5$ ; right panel, $\gamma = 1$ .

Figure 10. PFs of different contracts (left panel, $T=20$ ; right panel, $T=0.01$ ). Each circle point corresponds to a manager–investor EU pair (a Pareto point) under a contract with $(\omega, \alpha, \gamma)$ . A PF is plotted by a three-point curve, consisting of three Pareto points, for $\gamma = 0, 0.5, 1$ , under a specific contract setting $(\omega, \alpha)$ . In the right panel, each three-point curve represents a PF which is decreasing and concave. In the left panel, each three-point curve is almost reduced to a single point.

We find that there is no Pareto improvement, in terms of the utilities of both the investor and the manager, between any two fixed-weight contracts. In any subfigure, for any two contracts, there is no contract outperforming the other one in both objectives, no matter whether the evaluation time is long or short. This motivates us to incorporate the weight parameter $\gamma$ as one of the contract parameters in order to seek possible Pareto improvement.

In addition, when the evaluation time is long, we can see that the EUs of the investor (and of the manager) are similar for different weight parameters. The investor gets a higher EU for a smaller incentive rate $\alpha$ and a smaller managerial ownership proportion $\omega$ . This finding will be revisited below.

5.3. A contract providing a Pareto improvement

In this subsection, we focus on PFs of different contracts. We propose a novel contract, simultaneously changing the weight and the parameter to improve the utilities of both parties, as illustrated in Figure 10, which depicts several PFs under different sets of contract parameters $(\omega, \alpha)$ .

In the left panel of Figure 10, when the evaluation time is long ( $T=20$ ), the Pareto-optimal portfolios are quite similar and do not involve gambling (see the fourth subfigure in Figure 7). As a result, the PF of each contract is almost reduced to a single point, as if the Pareto points of each PF are clustered around the single point. Meanwhile, the PFs of different contracts are separated. Graphically, a new frontier of these PFs appears and is again decreasing, which shows that the total welfare is shared among the different contracts when the evaluation time is long. We find that among first-loss contracts with long evaluation time, the investor benefits from contracts with smaller incentive rate $\alpha$ or smaller managerial ownership proportion $\omega$ .

Significantly, we can find a contract providing a Pareto improvement from the right panel of Figure 10. When the evaluation time is short ( $T=0.01$ ), the performance of Pareto-optimal portfolios under each contract is much different (see the first subplot in Figure 7). As a result, the PFs of different contracts are clustered. If we treat the objective weight $\gamma$ as a parameter of a contract, there exists some contract dominating another contract with a different weight. For instance, the contract with $\omega=0.05$ , $\alpha=0.4$ and objective weight $\gamma=0.5$ dominates another contract with $\omega=0.05$ , $\alpha=0.2$ and objective weight $\gamma=1$ on the PF, which indicates that one contract has been found to be strictly better than others.

Therefore, if we simultaneously consider the investor’s utility in the investment objective and increase the incentive rate, we can obtain a Pareto improvement on some other contract ( $\omega=0.05$ , $\alpha=0.2$ , $\gamma=1$ ) in the traditional maximization of the single utility of the manager. This may be beneficial for investors and also helpful in enabling fund managers to design competitive contracts.

Figure 11. PFCEs of different contracts (Left panel: $T=20$ , Right panel: $T=0.01$ ). Each circle points means an Manager-Investor CE pair (a Pareto point) under a contract with $(\omega, \alpha, \gamma)$ . A PFCE is plotted by a three-point curve, consisting of three Pareto points of $\gamma = 0, 0.5, 1$ under a specific contract setting $(\omega, \alpha)$ . In the right panel, each three-point curve represents a PF, which is decreasing and concave. In the left panel, each three-point curve is almost reduced to a single point.

5.4. Certainty equivalent

To enlarge the scope of our results above, we investigate the certainty equivalent in addition to the expected utility. Denote by $\widehat{U}_i^{-1}$ the inverse function of $\widehat{U}_i$ , $i = 0, 1$ . For $\gamma \in [0, 1]$ , we define the P&L certainty equivalents of the manager and the investor under the optimal wealth $\hat{X}_\gamma$ (note that $\hat{X}_\gamma \equiv X_T^{\pi^*, \gamma}$ ) as follows:

\begin{equation*}\text{CE}_1\big(\hat{X}_\gamma\big) \triangleq \widehat{U}_1^{-1} \left( \mathbb{E}\big[\widehat{U}_1\big(\Theta\big(\hat{X}_\gamma\big)\big)\big] \right),\qquad\text{CE}_0 \triangleq \widehat{U}_0^{-1} \left(\mathbb{E}\left[\widehat{U}_0\big(\hat{X}_\gamma - \Theta\big(\hat{X}_\gamma\big) - x_0\big)\right]\right).\end{equation*}

In our context, the certainty equivalent means the guaranteed amount of cash allocated to the corresponding agent, who would consider this to be the same amount as the optimal terminal wealth. That is,

$$\widehat{U}_1 \big(\text{CE}_1\big(\hat{X}_\gamma\big)\big) = \mathbb{E}\big[\widehat{U}_1\big(\Theta\big(\hat{X}_\gamma\big)\big)\big],\qquad \widehat{U}_0 \big(\text{CE}_0\big(\hat{X}_\gamma\big)\big) = \mathbb{E}\big[\widehat{U}_0\big(\hat{X}_\gamma - \Theta\big(\hat{X}_\gamma\big) - x_0\big)\big].$$

Indeed, as $\widehat{U}_i$ , $i = 0, 1$ , are increasing, this is equivalent to maximizing the manager’s (or the investor’s) expected utility and certainty equivalent. The advantage of using the certainty equivalent is that we can compare the numerical values for the manager and the investor on the same scale. We define the Pareto frontier of the certainty equivalents (PFCE) as follows:

$$\text{PFCE} = \left\{ \left(\text{CE}_1\big(\hat{X}_\gamma\big), \text{CE}_0\big(\hat{X}_\gamma\big)\right): \gamma \in [0, 1] \right\} \subset {\mathbb{R}}^2.$$

Figure 11 shows the PFCEs under the different contract settings. The curves in Figure 11 show a similar pattern to those in Figure 10, and the behaviors in terms of CE and EU are similar.

From the left panel of Figure 11, we find that if the evaluation time is long (i.e., T is large), the investor has a higher certainty equivalent if the contract has a smaller incentive rate $\alpha$ or a smaller managerial ownership proportion $\omega$ . In the right panel of Figure 11, if the evaluation time is short (i.e., T is small), we can find a similar means of Pareto improvement on CE. If we simultaneously consider the investor’s utility in the investment objective and increase the incentive rate ( $\omega = 0.05$ , $\alpha = 0.4$ , $\gamma = 0.5$ ), we can obtain a Pareto improvement over some other contract ( $\omega=0.05$ , $\alpha=0.2$ , $\gamma=1$ ) in the traditional problem of maximization of the single utility of the manager. This indicates that the Pareto improvement results hold in wide generality, as the numerical values for the manager and the investor are compared on the same scale of CE.