Downside Risk Hedging Under Comonotonicity

1. Introduction

The use of downside monetary risk measures in actuarial and financial mathematics has been very common in recent years (Mansini et al., 2007, Lejeune and Shen, 2016, Balbás et al., 2022a, or Boonen and Jiang, 2024, to name a few). Many of these uses are related to risk optimization, which serves as a clear bridge between operations research and the financial and/or insurance industry. Nevertheless, with the sole exception of the optimal reinsurance problem, to the best of our knowledge the risk optimization under comonotonicity has been addressed very rarely. Random variables are said to be comonotonic if their joint distribution is generated by the Fréchet-Hoeffding copula (Dhaene et al., 2002), and the intuition behind this definition is that the growth of one variable will never result in a decrease in any of the others. There are a lot of examples of comonotonicity in finance (the price of an European call and the price of its underlying asset, for instance) and insurance (the amount of damage caused by an accident and the indemnity paid by the insurer, for instance). Accordingly, the optimization of comonotonic risks may deserve some attention, and this is the main purpose of this paper. The selected downside risk measures are the value at risk ($V@R$) and the conditional value at risk ($CV@R$), also called expected shortfall. Both measures have very intuitive interpretations, which is why they have been chosen by a large number of researchers (see some of the references cited above, Buch et al., 2023, Nguyen and Nguyen, 2026, etc.).

In order to make the paper easier to read, Section 2 will summarize several already known properties of both risk measures, but there is also a new result with important consequences. Theorem 2 gives a new polynomial upper bound of the type

$$V@R_{1/x}\le CV@R_{1/x}\le \vartheta x+V,$$

(1)

where $\vartheta$ and V are constant parameters and x is the reciprocal of the level of confidence, $i.e.$, $1<x<\infty$ and $1/x$ is the $V@R$ and $CV@R-$confidence level. Since $\vartheta$ is proportional to the difference $CV@R_{1/x_0}-V@R_{1/x_0}$ for a specific (but arbitrary) confidence level $1/x_0$, the $CV@R_{1/x}-$value may be accurately controlled for every x. Thus, select an initial confidence level $1/x_0$ (probably related to the legal framework or some internal risk management model), compute the difference $CV@R_{1/x_0}-V@R_{1/x_0}$ and you will have the constants $\vartheta$ and V such that (1) holds for every $1<x<\infty$.

Section 3 is devoted to presenting the main problem we are going to deal with. A manager will have random earnings in a future planning period and tries to protect its wealth against scenarios that could be adverse. The manager problem is to find the optimal hedging strategy among those that are anti-comonotonic with respect to its future earnings. Optimality is understood in the Pareto sense, since the vector

$$\left(CV@R_{1/x_0}-V@R_{1/x_0},V@R_{1/x_0},expected-wealth\right)$$

(2)

is simultaneously optimized. The couple $\left(CV@R_{1/x_0}-V@R_{1/x_0},V@R_{1/x_0}\right)$ is minimized in order to obtain an upper bound (1) as good as possible, whereas the expected wealth is maximized. It will be pointed out that the optimization of the couples $\left(V@R_{1/x_0},expected-wealth\right)$ and $\left(CV@R_{1/x_0},expected-wealth\right)$ are particular cases of the optimization of (2). Theorem 8 is the most important result of Section 3 because it completely resolves the manager problem. A very important implication of this theorem is that the manager budget to buy the hedging strategy is hardly relevant. There exists an anti-comonotonic optimal solution that remains the same for every budget. If the price of this solution is lower than the budget, the manager must invest the excess capital in a riskless security. If the budget is not enough to buy the optimal strategy, the manager must borrow money.

There are many important applications of the general approach of Section 3 (portfolio insurance, reduction in capital requirements, optimal reinsurance, optimal risk sharing, etc.). Section 4 is devoted to applying Theorem 8 to an important financial problem, namely, the portfolio insurance problem. Portfolio insurance may play a critical role in risk management problems, at least if the financial market is going through turbulent times. The wealth protection when facing market turmoils was always a major question for many traders (Bertrand and Prigent, 2003, Annaert et al., 2009, Zagst and Kraus, 2011, Nangolo et al., 2023, etc.). Classical portfolio insurance strategies are the sale of future contracts and the purchase of European puts (Joossens and Schoutens, 2008), but it will be seen that they are not necessarily optimal. Moreover, when the purchase of puts is optimal, the role of the selected strike is critical. This seems to be an important finding, at least for practitioners. A particular attention is devoted to the Black-Scholes-Merton ($BSM$) model, and some numerical experiments are implemented in order to illustrate the findings of this section.

The focus of Section 5 is a classic in actuarial science, namely, the optimal reinsurance problem. The insurer must buy reinsurance if the probability of large indemnifications is significant. Theorem 8 implies that the optimal contract is frequently a stop-loss reinsurance, at least (Proposition 16) if the reinsurer prices according to the expected value premium principle ($EVPP$). This result is in the line of many former studies (Albrecher et al., 2017), but there is a very important novelty. Indeed, the deductible of the stop-loss contract usually depends on the insurer budget, but as said above, this is not true under our approach. That deductible always remains the same, and this important difference is a consequence of the presence of the financial market. The approach of this paper allows the insurer to trade a riskless asset along with the purchase of reinsurance. The section ends with an example involving the Wang’s premium principle (Wang, 2000). With respect to the $EVPP$; the Wang’s principle makes the contract more expensive if the reinsurer tail risk becomes high enough. This example also illustrates that Theorem 8 applies under very general frameworks.

Section 6 concludes the paper.

2. Mathematical Framework

2.1. Mathematical Background

Let us consider the probability space $\left(\Omega ,\mathcal{F},\mathbb{P}\right)$ composed of the set of “states of the world” $\Omega$, the $\sigma -$algebra $\mathcal{F}$ and the probability measure $\mathbb{P}$. As usual, $L^0\left(\mathbb{P}\right)$ denotes the space of $\mathbb{R}-$valued random variables on $\Omega$, $\mathsf{I}\mathsf{E}\left(.\right)$ denotes “mathematical expectation”, if $1\le p<\infty$ then $L^p\left(\mathbb{P}\right)$ denotes the space of random variables $\mathcal{Y}$ such that $\mathsf{I}\mathsf{E}\left({\left|\mathcal{Y}\right|}^p\right)<\infty$, and $L^{\infty }\left(\mathbb{P}\right)$ denotes the space of $\mathbb{P}-$essentially bounded random variables. The natural norm of $L^p\left(\mathbb{P}\right)$ is given by ${∥\mathcal{Y}∥}_p={\left(\mathsf{I}\mathsf{E}\left({\left|\mathcal{Y}\right|}^p\right)\right)}^{1/p}$ if $1\le p<\infty$ and ${∥\mathcal{Y}∥}_p=esssup\left|\mathcal{Y}\right|$ for $p=\infty$, $esssup$ denoting “essential supremum”. Recall that $L^{p_1}\left(\mathbb{P}\right)\supset L^{p_2}\left(\mathbb{P}\right)$ if $\left\{p_1,p_2\right\}\subset \left\{0\right\}\cup \left[1,\infty \right]$ and $p_1<p_2$ (Kopp, 1984, for further details about all of these topics). Let us also deal with the measure space $\left(\mathbb{R},\mathcal{B},\mathbb{L}\right)$ composed of the set of real numbers $\mathbb{R}$, its Borel $\sigma -$algebra $\mathcal{B}$ and the Lebesgue measure $\mathbb{L}$. If $1\le p<\infty$ then $L^p\left(\mathbb{L}\right)$ denotes the space of real valued measurable functions x such that ${\int }_{-\infty }^{\infty }{\left|x\left(t\right)\right|}^{p}dt<\infty$, and $L^{\infty }\left(\mathbb{L}\right)$ denotes the space of $\mathbb{L}-$essentially bounded real valued measurable functions. The natural norm of $L^p\left(\mathbb{L}\right)$ is given by ${∥x∥}_p={\left({\int }_0^{\infty }{\left|x\left(t\right)\right|}^{p}dt\right)}^{1/p}$ if $1\le p<\infty$ and ${∥x∥}_p=esssup\left|x\right|$ for $p=\infty$. Recall that $L^{p_1}\left(\mathbb{L}\right)\subset L^{p_2}\left(\mathbb{L}\right)$ if $\left\{p_1,p_2\right\}\subset \left[1,\infty \right]$ and $p_1<p_2$.

2.2. Value at Risk and Conditional Value at Risk

If $0<\beta <1$ and $\mathcal{Y}\in L^0\left(\mathbb{P}\right)$ then the value at risk of $\mathcal{Y}$ with the confidence level $1-\beta$ is given by

$$V@R_{1-\beta }\left(\mathcal{Y}\right):=-inf\left\{x\in \mathbb{R};\mathbb{P}\left(\mathcal{Y}\le x\right)>\beta \right\}.$$

We will also need the upper bound of $V@R_{1-\beta }\left(\mathcal{Y}\right)$ given by

$$W@R_{1-\beta }\left(\mathcal{Y}\right):=-inf\left\{x\in \mathbb{R};\mathbb{P}\left(\mathcal{Y}\le x\right)\ge \beta \right\}.$$

For $\mathcal{Y}\in L^1\left(\mathbb{P}\right)$ the conditional value at risk of $\mathcal{Y}$ with the confidence level $1-\beta$ is given by

$$CV@R_{1-\beta }\left(\mathcal{Y}\right):={\displaystyle \frac{1}{\beta }}{\int }_0^{\beta }V@R_{1-t}\left(\mathcal{Y}\right)dt={\displaystyle \frac{1}{\beta }}{\int }_0^{\beta }W@R_{1-t}\left(\mathcal{Y}\right)dt.$$

Rockafellar and Uryasev (2000) proved that $CV@R_{1-\beta }\left(\mathcal{Y}\right)$ can be computed by linear programming methods, and the idea was later extended in Rockafellar et al. (2006) and more recently in Herdegen and Munari (2023). Thus, let us present without proof a “slightly more sophisticated approach” which is widely discussed in Balbás et al. (2017) and (2023a).

Theorem 1.

Consider $0<\beta <1$ and $\mathcal{Y}\in L^1\left(\mathbb{P}\right)$.

$a)$ $CV@R_{1-\beta }\left(\mathcal{Y}\right)$ is the optimal value of the (perhaps infinite-dimensional) solvable linear programming problem

$$max\left\{-\mathsf{I}\mathsf{E}\left(\mathcal{Y}z\right);0\le z\le 1/\beta ,\mathsf{I}\mathsf{E}\left(z\right)=1\right\}$$

(3)

$z\in L^{\infty }\left(\mathbb{P}\right)$ being the decision variable.

$b)$ $CV@R_{1-\beta }\left(\mathcal{Y}\right)$ is the optimal value of the (perhaps infinite-dimensional) solvable linear programming problem

$$max\lambda +{\displaystyle \frac{1}{\beta }}\mathsf{I}\mathsf{E}\left({\lambda }_M\right)\left\{\begin{array}{c}\mathcal{Y}={\lambda }_m-{\lambda }_M-\lambda \hfill \\ {\lambda }_m\ge 0,{\lambda }_M\ge 0\hfill \end{array}\right.$$

(4)

$\left(\lambda ,{\lambda }_m,{\lambda }_M\right)\in \mathbb{R}\times L^1\left(\mathbb{P}\right)\times L^1\left(\mathbb{P}\right)$ being the decision variable. Moreover, if $\mathcal{Y}\in L^p\left(\mathbb{P}\right)$ with $1\le p\le \infty$, then one can take $\left(\lambda ,{\lambda }_m,{\lambda }_M\right)\in \mathbb{R}\times L^p\left(\mathbb{P}\right)\times L^p\left(\mathbb{P}\right)$.

$c)$ (4) is the dual problem of (3). If z is (3)-feasible and $\left(\lambda ,{\lambda }_m,{\lambda }_M\right)$ is (4)-feasible, then they solve the corresponding problem if and only if

$$z{\lambda }_m=\left(1/\beta -z\right){\lambda }_M=0.$$

(5)

$d)$ $\left(\lambda ,{\lambda }_m,{\lambda }_M\right)\in \mathbb{R}\times L^1\left(\mathbb{P}\right)\times L^1\left(\mathbb{P}\right)$ solves (4) if and only if $V@R_{1-\beta }\left(\mathcal{Y}\right)\le \lambda \le W@R_{1-\beta }\left(\mathcal{Y}\right)$, ${\lambda }_m={\left(\mathcal{Y}+\lambda \right)}^{+}$ and ${\lambda }_M=\left(\mathcal{Y}+\lambda \right)$. In particular,

$$\left(\lambda ,{\lambda }_m,{\lambda }_M\right)=\left(V@R_{1-\beta }\left(\mathcal{Y}\right),{\left(\mathcal{Y}+V@R_{1-\beta }\left(\mathcal{Y}\right)\right)}^{+},{\left(\mathcal{Y}+V@R_{1-\beta }\left(\mathcal{Y}\right)\right)}^{-}\right)$$

and

$$\left(\lambda ,{\lambda }_m,{\lambda }_M\right)=\left(W@R_{1-\beta }\left(\mathcal{Y}\right),{\left(\mathcal{Y}+W@R_{1-\beta }\left(\mathcal{Y}\right)\right)}^{+},{\left(\mathcal{Y}+W@R_{1-\beta }\left(\mathcal{Y}\right)\right)}^{-}\right)$$

solve (4).□

According to Theorem 2 below, if one fixes $0<\beta <1$ and computes $V@R_{1-\beta }\left(\mathcal{Y}\right)$ (or $W@R_{1-\beta }\left(\mathcal{Y}\right)$) and $CV@R_{1-\beta }\left(\mathcal{Y}\right)$, then one has an upper bound of $CV@R_{1-s}\left(\mathcal{Y}\right)$ applying for every $0<s<1$, $i.e.$, for every level of confidence.

Theorem 2.

Consider $0<\beta <1$ and $\mathcal{Y}\in L^1\left(\mathbb{P}\right)$. Consider $V\in \mathbb{R}$ such that $V@R_{1-\beta }\left(\mathcal{Y}\right)\le V\le W@R_{1-\beta }\left(\mathcal{Y}\right)$.Then,

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le V+{\displaystyle \frac{\beta \left(CV@R_{1-\beta }\left(\mathcal{Y}\right)-V\right)}{s}}$$

(6)

holds for every $0<s<1$. In particular,

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le V@R_{1-\beta }\left(\mathcal{Y}\right)+{\displaystyle \frac{\beta \left(CV@R_{1-\beta }\left(\mathcal{Y}\right)-V@R_{1-\beta }\left(\mathcal{Y}\right)\right)}{s}}$$

(7)

for every $0<s<1$, and

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le W@R_{1-\beta }\left(\mathcal{Y}\right)+{\displaystyle \frac{\beta \left(CV@R_{1-\beta }\left(\mathcal{Y}\right)-W@R_{1-\beta }\left(\mathcal{Y}\right)\right)}{s}}$$

(8)

for every $0<s<1$. Moreover, if $esssup\left(\mathcal{Y}\right)<\infty$, then

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le {\displaystyle \frac{esssup\left(\mathcal{Y}\right)-\mathsf{I}\mathsf{E}\left(\mathcal{Y}\right)}{s}}-esssup\left(\mathcal{Y}\right)$$

(9)

for every $0<s<1$. Finally, if $essinf\left(\mathcal{Y}\right)>-\infty$, then

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le -essinf\left(\mathcal{Y}\right)$$

(10)

for every $0<s<1$.

Proof. 

Theorem $1d$ implies that

$$\left(V,{\lambda }_m={\left(\mathcal{Y}+V\right)}^{+},{\lambda }_M={\left(\mathcal{Y}+V\right)}^{-}\right)$$

resolves (4). Consider a solution $z_{\beta }$ of (3) and a solution $z_s$ of (3) if $\beta$ is replaced by s. Since (4) is the dual of (3), the results of Luenberger (1969) about convex programming show that

$$\mathsf{I}\mathsf{E}\left(\mathcal{Y}{z}_s\right)\ge \mathsf{I}\mathsf{E}\left(\mathcal{Y}{z}_{\beta }\right)-\mathsf{I}\mathsf{E}\left({\lambda }_M\left({\displaystyle \frac{1}{s}}-{\displaystyle \frac{1}{\beta }}\right)\right),$$

and Theorem $1a$ leads to

$$-CV@R_{1-s}\left(\mathcal{Y}\right)\ge -CV@R_{1-\beta }\left(\mathcal{Y}\right)-\mathsf{I}\mathsf{E}\left({\lambda }_M\left({\displaystyle \frac{1}{s}}-{\displaystyle \frac{1}{\beta }}\right)\right),$$

that is,

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le CV@R_{1-\beta }\left(\mathcal{Y}\right)+\mathsf{I}\mathsf{E}\left({\lambda }_M\left({\displaystyle \frac{1}{s}}-{\displaystyle \frac{1}{\beta }}\right)\right).$$

(11)

Theorem $1b$ implies that

$$CV@R_{1-\beta }\left(\mathcal{Y}\right)=V+{\displaystyle \frac{1}{\beta }}\mathsf{I}\mathsf{E}\left({\lambda }_M\right),$$

(12)

and therefore (11) leads to

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le V+{\displaystyle \frac{1}{\beta }}\mathsf{I}\mathsf{E}\left({\lambda }_M\right)+\mathsf{I}\mathsf{E}\left({\lambda }_M\left({\displaystyle \frac{1}{s}}-{\displaystyle \frac{1}{\beta }}\right)\right)=V+{\displaystyle \frac{\mathsf{I}\mathsf{E}\left({\lambda }_M\right)}{s}}.$$

(13)

(12) leads to $\beta \left(CV@R_{1-\beta }\left(\mathcal{Y}\right)-V\right)=\mathsf{I}\mathsf{E}\left({\lambda }_M\right)$, and (6) follows from (13). In particular, (7) and (8) become trivial if one takes $V=V@R_{1-\beta }\left(\mathcal{Y}\right)$ and $V=W@R_{1-\beta }\left(\mathcal{Y}\right)$, respectively. Furthermore, (9) is a consequence of the well known expressions (Rockafellar et al., 2006) ${lim}_{\beta \to 1}CV@R_{1-\beta }\left(\mathcal{Y}\right)=-\mathsf{I}\mathsf{E}\left(\mathcal{Y}\right)$ and ${lim}_{\beta \to 1}V@R_{1-\beta }\left(\mathcal{Y}\right)=-esssup\left(\mathcal{Y}\right)$, whereas (10) follows from

$$\underset{\beta \to 0}{lim}\beta \left(CV@R_{1-\beta }\left(\mathcal{Y}\right)-V@R_{1-\beta }\left(\mathcal{Y}\right)\right)=0$$

and ${lim}_{\beta \to 0}V@R_{1-\beta }\left(\mathcal{Y}\right)=essinf\left(\mathcal{Y}\right)$.□

Remark 3.

Notice that (6) indicates that $CV@R_{1-s}\left(\mathcal{Y}\right)$ is bounded by a simple hyperbolic function of s. Indeed, by fixing $0<\beta <1$, choosing $V_{\beta }\left(\mathcal{Y}\right)$ such that $V@R_{1-\beta }\left(\mathcal{Y}\right)\le V_{\beta }\left(\mathcal{Y}\right)\le W@R_{1-\beta }\left(\mathcal{Y}\right)$ and taking

$${\vartheta }_{\beta }\left(\mathcal{Y}\right)=\beta \left(CV@R_{1-\beta }\left(\mathcal{Y}\right)-V_{\beta }\left(\mathcal{Y}\right)\right),$$

(14)

(6) leads to

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le {\vartheta }_{\beta }\left(\mathcal{Y}\right)/s+V_{\beta }\left(\mathcal{Y}\right)$$

(15)

for every $0<s<1$. A similar upper bound of $CV@R_{1-s}\left(\mathcal{Y}\right)$ holds if (6) is replaced by (9) and $esssup\left(\mathcal{Y}\right)<\infty$. Indeed, if this case $V_{\beta }\left(\mathcal{Y}\right)$ is replaced by $-esssup\left(\mathcal{Y}\right)$ and ${\vartheta }_{\beta }\left(\mathcal{Y}\right)$ is replaced by $esssup\left(\mathcal{Y}\right)-\mathsf{I}\mathsf{E}\left(\mathcal{Y}\right)$. Lastly, one can fix a finite family $\left\{{\beta }_1,{\beta }_2,...,{\beta }_m\right\}\subset \left(0,1\right)$ and, under the obvious notation,

$$CV@R_{1-s}\left(\mathcal{Y}\right)\le min\left\{{\vartheta }_j/s+V_j;j=1,2,...,m\right\}$$

(16)

for every $0<s<1$.□

3. Optimizing Tail Risk Under Comonotonicity

3.1. Problem Set Up

Consider $\mathcal{U}\in L^2\left(\mathbb{P}\right)$ representing a random monetary income that an agent will receive at a future planning period T. Without loss of generality let us consider that $\left(\mathbb{R},\mathcal{B},\mathbb{P}\right)=\left(\Omega ,\mathcal{F},\mathbb{P}\right)$, where we are still denoting by $\mathbb{P}$ the probability induced by $\mathcal{U}$ on $\mathcal{B}$. Accordingly, $\mathcal{U}:\mathbb{R}⟶\mathbb{R}$ becomes the identity map. Suppose that $\mathcal{U}$ may achieve very small or even very negative values, and therefore the agent may be interested in protecting its wealth by adding at T a (probably random) second income $f\left(\mathcal{U}\right)$, in which case the final wealth at T will become $\mathcal{U}+f\left(\mathcal{U}\right)$. In order to guarantee that $f\left(\mathcal{U}\right)$ is a real protection, the agent may impose $\mathcal{U}$ and $f\left(\mathcal{U}\right)$ to be anti-comonotonic, that is, $\mathcal{U}$ and$-f\left(\mathcal{U}\right)$ must be comonotonic. Although a formal mathematical definition of comonotonicity may be found in Dhaene et al. (2002), let us indicate that under the present framework the idea is that $f\left(\mathcal{U}\right)$ should be a decreasing function of $\mathcal{U}$,1 that is, $f\left(\mathcal{U}\right)$ should decrease as $\mathcal{U}$ increases. In this manner one guarantees that $f\left(\mathcal{U}\right)$ may become high enough if $\mathcal{U}$ becomes low or negative, and the final wealth $\mathcal{U}+f\left(\mathcal{U}\right)$ may (perhaps partially) counteract the potential capital losses associated with $\mathcal{U}$ under “negative scenarios”. Intuitively, the comonotonicity between $-f\left(\mathcal{U}\right)$ and $\mathcal{U}$ may be guaranteed if one deals with the “marginal protection or first order derivative $x\left(\mathcal{U}\right)=d\left(f\left(\mathcal{U}\right)\right)/d\mathcal{U}$” instead of the protection itself $f\left(\mathcal{U}\right)$, because $f\left(\mathcal{U}\right)$ will be decreasing if $x\left(\mathcal{U}\right)\le 0$. In order to deal with this marginal protection and maintain mathematical rigor, recall that the usual bilinear product $L^{\infty }\left(\mathbb{L}\right)\times L^1\left(\mathbb{L}\right)\ni \left(y_1,y_2\right)\to 〈y_1,y_2〉\in \mathbb{R}$ is given by

$$〈y_1,y_2〉:={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}{y}_1\left(\tau \right)y_2\left(\tau \right)d\tau$$

(17)

and consider Proposition 4 below, whose prove is omitted because it may be found in Balbás et al. (2019).

Proposition 4.

Consider

$$J\left(x\right)\left(\omega \right):={\displaystyle \underset{0}{\overset{\omega }{\int }}}x\left(\tau \right)d\tau$$

(18)

for $x\in L^{\infty }\left(\mathbb{L}\right)$ and $\omega \in \mathbb{R}$. Then, $J\left(x\right)\in L^2\left(\mathbb{P}\right)$ for every $x\in L^{\infty }\left(\mathbb{L}\right)$, and $J:L^{\infty }\left(\mathbb{L}\right)\to L^2\left(\mathbb{P}\right)$ is linear and continuous. The adjoint operator $J^{*}$ is $L^1\left(\mathbb{L}\right)$-valued. $J^{*}$:$L^2\left(\mathbb{P}\right)\to L^1\left(\mathbb{L}\right)$ is linear and continuous, and it is given by

$$J^{*}\left(z\right)\left(\tau \right)=\left\{\begin{array}{cc}-\underset{\omega \le \tau }{\int }z\left(\omega \right)\mathbb{P}\left(d\omega \right),\hfill & \tau <0\hfill \\ \underset{\omega \ge \tau }{\int }z\left(\omega \right)\mathbb{P}\left(d\omega \right),\hfill & \tau >0\hfill \end{array}\right.$$

(19)

for every $z\in L^2\left(\mathbb{P}\right)$ and every $\tau \in \mathbb{R}$.2,3□

Lemma 5 below will permit us to give analytically tractable expressions for the risk of $\mathcal{U}+f\left(\mathcal{U}\right)$.

Lemma 5.

Consider $0<\beta <1$, ${\omega }_{\beta }:=V@R_{1-\beta }\left(\mathcal{U}\right)$, ${\mathcal{Z}}_{\beta }\in L^{\infty }\left(\mathbb{P}\right)$ given by

$${\mathcal{Z}}_{\beta }:=\left\{\begin{array}{cccc}1/\beta ,\hfill & \omega <-{\omega }_{\beta }\hfill & \hfill & \\ 1/\beta ,\hfill & \omega =-{\omega }_{\beta }\hfill & and\hfill & \mathbb{P}\left(\mathcal{U}=-{\omega }_{\beta }\right)=0\hfill \\ {\displaystyle \frac{1-\mathbb{P}\left(\mathcal{U}<-{\omega }_{\beta }\right)/\beta }{\mathbb{P}\left(\mathcal{U}=-{\omega }_{\beta }\right)}},\hfill & \omega =-{\omega }_{\beta }\hfill & and\hfill & \mathbb{P}\left(\mathcal{U}=-{\omega }_{\beta }\right)>0\hfill \\ 0,\hfill & \omega >-{\omega }_{\beta },\hfill & \hfill \end{array}\right.$$

(21)

$\xi \in L^{\infty }\left(\mathbb{L}\right)$ with $\xi \ge 0$, and $\left(V_{\beta }\left(J\left(\xi \right)\right),{\lambda }_m^{\left(\beta ,\xi \right)},{\lambda }_M^{\left(\beta ,\xi \right)}\right)\in \mathbb{R}\times L^2\left(\mathbb{P}\right)\times L^2\left(\mathbb{P}\right)$ given by

$$\left\{\begin{array}{c}{V}_{\beta }\left(J\left(\xi \right)\right):=-J\left(\xi \right)\left(-{\omega }_{\beta }\right)\hfill \\ {\lambda }_m^{\left(\beta ,\xi \right)}:={\left(J\left(\xi \right)+V_{\beta }\left(J\left(\xi \right)\right)\right)}^{+}\hfill \\ {\lambda }_M^{\left(\beta ,\xi \right)}:={\left(J\left(\xi \right)+V_{\beta }\left(J\left(\xi \right)\right)\right)}^{-}.\hfill \end{array}\right.$$

If $\mathcal{Y}=J\left(\xi \right)$, then ${\mathcal{Z}}_{\beta }$ solves (3) and $\left(V_{\beta }\left(J\left(\xi \right)\right),{\lambda }_m^{\left(\beta ,\xi \right)},{\lambda }_M^{\left(\beta ,\xi \right)}\right)$ solves (4).

Proof. 

It is easy to verify that ${\mathcal{Z}}_{\beta }$ and $\left(V_{\beta }\left(J\left(\xi \right)\right),{\lambda }_m^{\left(\beta ,\xi \right)},{\lambda }_M^{\left(\beta ,\xi \right)}\right)$ satisfy the constraints of (3) and (4) respectively. Therefore, according to Theorem $1c$, it is enough to prove that ${\mathcal{Z}}_{\beta }$ and $\left(V_{\beta }\left(J\left(\xi \right)\right),{\lambda }_m^{\left(\beta ,\xi \right)},{\lambda }_M^{\left(\beta ,\xi \right)}\right)$ satisfy (5). (21) implies that ${\mathcal{Z}}_{\beta }{\lambda }_m^{\left(\beta ,\xi \right)}=0$ if $\omega >-{\omega }_{\beta }$, so let us consider $\omega \le -{\omega }_{\beta }$. It is sufficient to prove that ${\lambda }_m^{\left(\beta ,\xi \right)}=0$. Since $\xi \ge 0$, $\omega \to J\left(\xi \right)\left(\omega \right)$ is increasing, and therefore $J\left(\xi \right)\left(\omega \right)\le J\left(\xi \right)\left(-{\omega }_{\beta }\right)=-V_{\beta }\left(J\left(\xi \right)\right)$, that is, $J\left(\xi \right)+V_{\beta }\left(J\left(\xi \right)\right)\le 0$ and ${\lambda }_m^{\left(\beta ,\xi \right)}={\left(J\left(\xi \right)+V_{\beta }\left(J\left(\xi \right)\right)\right)}^{+}=0$. Besides, $\left(1/\beta -{\mathcal{Z}}_{\beta }\right){\lambda }_M^{\left(\beta ,\xi \right)}=0$ is obvious if $\omega <-{\omega }_{\beta }$, so let us consider $\omega \ge -{\omega }_{\beta }$. It is sufficient to prove that ${\lambda }_M^{\left(\beta ,\xi \right)}=0$. Since $\omega \to J\left(\xi \right)\left(\omega \right)$ is increasing, $J\left(\xi \right)\left(\omega \right)\ge J\left(\xi \right)\left(-{\omega }_{\beta }\right)=-V_{\beta }\left(J\left(\xi \right)\right)$, that is, $J\left(\xi \right)+V_{\beta }\left(J\left(\xi \right)\right)\ge 0$ and ${\lambda }_M^{\left(\beta ,\xi \right)}={\left(J\left(\xi \right)+V_{\beta }\left(J\left(\xi \right)\right)\right)}^{-}=0$.□

The additional income will take the form $f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)$ with $a\in \mathbb{R}$, $x\in L^{\infty }\left(\mathbb{L}\right)$ and $x\le 0$. The constraint $x\le 0$ will guarantee that $f\left(\mathcal{U}\right)$ is a decreasing function of $\mathcal{U}$, whereas a will reflect the value of $f\left(\mathcal{U}\right)$ if $\mathcal{U}=0$. Besides, let us also impose $\mathcal{U}$ and $\mathcal{U}+f\left(\mathcal{U}\right)$ to be comonotonic, that is, $\mathcal{U}+f\left(\mathcal{U}\right)$ will be an increasing function of $\mathcal{U}$. Otherwise the agent would not be desiring high realized values of $\mathcal{U}$, and perhaps a strategy against $\mathcal{U}$ would be preferable to a defensive protection. Since the first order derivative of $\mathcal{U}+f\left(\mathcal{U}\right)$ with respect to $\mathcal{U}$ equals $1+x\left(\mathcal{U}\right)$, one has that $1+x\left(\mathcal{U}\right)\ge 0$ must hold, that is $-1\le x\left(\mathcal{U}\right)\le 0$. Nevertheless, for some reasons that will be justified in the actuarial application, let us assume the existence of $H\in L^{\infty }\left(\mathbb{L}\right)$ such that

$$0\le H\le 1,$$

(22)

and let us consider the more general constraints

$$\left\{\left(a,x\right)\in \mathbb{R}\times L^{\infty }\left(\mathbb{L}\right);-H\le x\le 0\right\}.$$

(23)

Obviously, $H=1$ is a particular case.

The pay-off $f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)$ will rarely be achieved for free, so let us assume the existence of a market where it may be bought for a price $\Pi \left(f\left(\mathcal{U}\right)\right)$. Suppose that $\Pi \left(.\right)$ takes the form

$$\Pi \left(a+J\left(x\right)\left(\mathcal{U}\right)\right)=a+k_1\mathsf{I}\mathsf{E}\left(J\left(x\right)\right)+k_2\rho \left(-J\left(x\right)\right),$$

(24)

$k_1\ge 0$, $k_2\ge 0$, $k_1+k_2>0$, and $\rho :L^2\left(\mathbb{P}\right)\to \mathbb{R}$ being a comonotonically additive and coherent (Artzner et al., 1999) risk measure.4 At first glance the pricing rule (24) may seem strange, but it will be shown in Section 4 and Section 5 that $\Pi$ generalizes and integrates the most important financial and actuarial examples.

According to the representation theorem of coherent risk measures given in Artzner et al. (1999) and Rockafellar et al. (2006), the set

$${\partial }_{\rho }=\left\{z\in L^2\left(\mathbb{P}\right);\rho \left(\mathcal{Y}\right)\ge -\mathsf{I}\mathsf{E}\left(\mathcal{Y}z\right))\forall \mathcal{Y}\in L^2\left(\mathbb{P}\right)\right\}$$

(25)

is weakly-compact (Zeidler, 1995) and

$$\rho \left(\mathcal{Y}\right)=max\left\{-\mathsf{I}\mathsf{E}\left(\mathcal{Y}z\right);z\in {\partial }_{\rho }\right\}$$

(26)

holds for every $\mathcal{Y}\in L^2\left(\mathbb{P}\right)$.5 Moreover,

$$\left\{\begin{array}{c}\mathsf{I}\mathsf{E}\left(z\right)=1\hfill \\ z\ge 0\hfill \end{array}\right.$$

(27)

for every $z\in {\partial }_{\rho }$.

The proof of Lemma 6 below is omitted because a similar one may be found in Balbás et al. (2022b).

Lemma 6.

Consider ${\mathcal{Z}}_{\rho }\in {\partial }_{\rho }$ such that $\rho \left(\mathcal{U}\right)=\mathsf{I}\mathsf{E}\left(-{\mathcal{UZ}}_{\rho }\right)$ (recall (26)). Then, $\rho \left(-J\left(\xi \right)\right)=\mathsf{I}\mathsf{E}\left(J\left(\xi \right){\mathcal{Z}}_{\rho }\right)=〈x,J^{*}\left({\mathcal{Z}}_{\rho }\right)〉$ (recall (20)) for every $\xi \in L^{\infty }\left(\mathbb{L}\right)$ such that $-1\le \xi \le 0$.□

Remark 7.

(The agent problem) Henceforth let us fix ${\mathcal{Z}}_{\rho }\in {\partial }_{\rho }$ such that $\rho \left(\mathcal{U}\right)=\mathsf{I}\mathsf{E}\left(-{\mathcal{UZ}}_{\rho }\right)$ and ${\mathcal{Z}}_{\beta }$ given by (21). The agent may select $\beta \in \left(0,1\right)$ and the value of $V@R_{1-\beta }\left(\mathcal{U}+f\left(\mathcal{U}\right)\right)$, $W@R_{1-\beta }\left(\mathcal{U}+f\left(\mathcal{U}\right)\right)$ or $CV@R_{1-\beta }\left(\mathcal{U}+f\left(\mathcal{U}\right)\right)$ in order to measure the risk after purchasing $a+J\left(x\right)\left(\mathcal{U}\right)$. The value of β may be related to the legal setting or to some risk management internal model. Nevertheless, this agent may be also interested in confidence levels beyond the chosen β. In such a case, $V@R_{1-s}\left(\mathcal{U}+f\left(\mathcal{U}\right)\right)$ and $CV@R_{1-s}\left(\mathcal{U}+f\left(\mathcal{U}\right)\right)$ may also become important for other values of s, so (14), (15) and (16) imply that the joint minimization of $CV@R_{1-\beta }\left(J\left(1+x\right)\right)-V_{\beta }\left(J(1+x)\right)$ and $V_{\beta }\left(J(1+x)+a\right)=V_{\beta }\left(J\left(1+x\right)\right)-a$ may be a reasonable and feasible approach.6 Denote by C the budget that the agent can spend in order to buy the protection $f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)$. The agent problem may become (recall (23), (24) and Lemma 6)

$$\left\{\begin{array}{c}min\left(C_{1-\beta }\left(1+x\right)-W_{\beta }\left(1+x\right),W_{\beta }\left(1+x\right)-a,-\mathsf{I}\mathsf{E}\left(a+J\left(1+x\right)\right)\right)\hfill \\ a+k_1\mathsf{I}\mathsf{E}\left(J\left(x\right)\right)+k_2\mathsf{I}\mathsf{E}\left(J\left(x\right){\mathcal{Z}}_{\rho }\right)\le C\hfill \\ -H\le x\le 0,a\in \mathbb{R}\hfill \end{array}\right.$$

(28)

where $C_{1-\beta }\left(1+x\right)$ stands for $CV@R_{1-\beta }\left(J\left(1+x\right)\right)$ and $W_{\beta }\left(1+x\right)$ stands for $V_{\beta }\left(J\left(1+x\right)\right)$ in order to simplify the notation. Theorem 1, (20) and Lemma 5 imply that the problem may be also given by

$$\left\{\begin{array}{c}min-\mathsf{I}\mathsf{E}\left(J\left(1+x\right){\mathcal{Z}}_{\beta }\right)+J\left(1+x\right)\left(-{\omega }_{\beta }\right),-J\left(1+x\right)\left(-{\omega }_{\beta }\right)-a,-\mathsf{I}\mathsf{E}\left(a+J\left(1+x\right)\right)\hfill \\ a+k_1〈x,J^{*}\left(1\right)〉+k_2〈x,J^{*}\left({\mathcal{Z}}_{\rho }\right)〉\le C\hfill \\ -H\le x\le 0,a\in \mathbb{R}\hfill \end{array}\right.$$

(29)

(18) implies that

$$-J\left(1+x\right)\left(-{\omega }_{\beta }\right)={\displaystyle \underset{-{\omega }_{\beta }}{\overset{0}{\int }}}\left(1+x\left(\tau \right)\right)d\tau ={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}\left(\tau \right)\left(1+x\left(\tau \right)\right)d\tau$$

where

$$\mathsf{I}{\mathsf{1}}_A\left(\tau \right)=\left\{\begin{array}{cc}1,\hfill & \tau \in A\hfill \\ 0,\hfill & \tau \notin A\hfill \end{array}\right.$$

is the usual indicator of an arbitrary $A\subset \mathbb{R}$,7 and we have taken

$$\mathsf{I}{\mathsf{1}}_{\left({\omega }_1,{\omega }_2\right)}=-\mathsf{I}{\mathsf{1}}_{\left({\omega }_2{\omega }_1\right)}$$

(30)

for every ${\omega }_i\in \mathbb{R}$, $i=1,2$. In other words (recall (17)), $-J\left(1+x\right)\left(-{\omega }_{\beta }\right)=〈1+x,\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}〉$, and (28) and (29) are the same problem as

$$\left\{\begin{array}{c}min\left(-〈1+x,J^{*}\left({\mathcal{Z}}_{\beta }\right)+\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}〉,〈1+x,\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}〉-a,-a-〈1+x,J^{*}\left(1\right)〉\right)\hfill \\ a+〈x,J^{*}\left(k_1+k_2{\mathcal{Z}}_{\rho }\right)〉\le C\hfill \\ -H\le x\le 0,a\in \mathbb{R}\hfill \end{array}\right.$$

(31)

Since (31) is linear, so are (28) and (29) (they are the same problem), so every Pareto solution may be found by the scalarization method. In particular, taking the weights ${\theta }_c\ge 0$, ${\theta }_v\ge 0$ an ${\theta }_E\ge 0$ such that $\left({\theta }_c,{\theta }_v,{\theta }_E\right)\ne \left(0,0,0\right)$ the agent problem becomes

$$\left\{\begin{array}{c}min〈1+x,\left({\theta }_v-{\theta }_c\right)\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}-{\theta }_c{J}^{*}\left({\mathcal{Z}}_{\beta }\right)-{\theta }_E{J}^{*}\left(1\right)〉-\left({\theta }_v+{\theta }_E\right)a\hfill \\ a+〈x,J^{*}\left(k_1+k_2{\mathcal{Z}}_{\rho }\right)〉\le C\hfill \\ -H\le x\le 0,a\in \mathbb{R}\hfill \end{array}\right.$$

(32)

whose solutions obviously coincide with the solutions of

$$\left\{\begin{array}{c}min〈x,\left({\theta }_v-{\theta }_c\right)\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}-{\theta }_c{J}^{*}\left({\mathcal{Z}}_{\beta }\right)-{\theta }_E{J}^{*}\left(1\right)〉-\left({\theta }_v+{\theta }_E\right)a\hfill \\ a+〈x,J^{*}\left(k_1+k_2{\mathcal{Z}}_{\rho }\right)〉\le C\hfill \\ -H\le x\le 0,a\in \mathbb{R}\hfill \end{array}\right.$$

(33)

Notice that (28) (and therefore (32)) is very general and contains as particular cases both the minimization of $CV@R_{1-\beta }\left(a+J\left(1+x\right)\right)$ and the minimization of $V@R_{1-\beta }\left(a+J\left(1+x\right)\right)$. Indeed, it is enough to take $\left({\theta }_c,{\theta }_v,{\theta }_E\right)=\left(1,1,0\right)$ and $\left({\theta }_c,{\theta }_v,{\theta }_E\right)=\left(0,1,0\right)$, respectively. In other words, the joint minimization of $CV@R_{1-\beta }\left(J\left(1+x\right)\right)-V_{\beta }\left(J(1+x)\right)$ and $V_{\beta }\left(J(1+x)+a\right)$ is more general than the minimization of $CV@R_{1-\beta }\left(J\left(1+x\right)+a\right)$ and the minimization of $V@R_{1-\beta }\left(J\left(1+x\right)+a\right)$. Consequently, the general solution of (28) given in Theorem 8 below will also apply if the agent is only interested in the confidence level β.□

3.2. Problem Solution

Theorem 8 below provides us with the general solution of (32) and (33), and therefore with a general Pareto solution of (28).

Theorem 8.

Consider $\Lambda \in L^1\left(\mathbb{L}\right)$ given by

$$\left\{\begin{array}{c}\Lambda =\left({\theta }_v-{\theta }_c\right)\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}-{\theta }_c{J}^{*}\left({\mathcal{Z}}_{\beta }\right)\hfill \\ +\left(\left({\theta }_v+{\theta }_E\right)k_1-{\theta }_E\right)J^{*}\left(1\right)+\left({\theta }_v+{\theta }_E\right)k_2{J}^{*}\left({\mathcal{Z}}_{\rho }\right).\hfill \end{array}\right.$$

(34)

Take $\left(a,x\right)$ such that

$$\left\{\begin{array}{c}\Lambda <0⟹x=0\hfill \\ \Lambda >0⟹x=-H\hfill \\ \Lambda =0⟹-H\le x\le 0\hfill \\ a=C-〈x,J^{*}\left(k_1+k_2{\mathcal{Z}}_{\rho }\right)〉.\hfill \end{array}\right.$$

(35)

Then, $\left(a,x\right)$ solves (33), and therefore $f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)$ is an optimal protection.

Proof. (33) is a linear problem whose dual becomes (Anderson and Nash, 1987)

$$\left\{\begin{array}{c}max-C{\Lambda }_0-〈H,{\Lambda }_m〉\hfill \\ \left\{\begin{array}{c}\left({\theta }_v-{\theta }_c\right)\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}-{\theta }_c{J}^{*}\left({\mathcal{Z}}_{\beta }\right)+\left({\Lambda }_0{k}_1-{\theta }_E\right)J^{*}\left(1\right)+{\Lambda }_0{k}_2{J}^{*}\left({\mathcal{Z}}_{\rho }\right)\hfill \\ ={\Lambda }_m-{\Lambda }_M\hfill \end{array}\right.\hfill \\ {\Lambda }_0=\left({\theta }_v+{\theta }_E\right)\hfill \\ {\Lambda }_m\ge 0,{\Lambda }_M\ge 0\hfill \end{array}\right.$$

(36)

$\left({\Lambda }_0,{\Lambda }_m,{\Lambda }_M\right)\in \mathbb{R}\times L^1\left(\mathbb{L}\right)\times L^1\left(\mathbb{L}\right)$ being the decision variable. Moreover, since $\left(a,x\right)$ is obviously (33)-feasible, it will solve the problem if there exists $\left({\Lambda }_0,{\Lambda }_m,{\Lambda }_M\right)$ (36)-feasible such that the complementary slackness conditions

$$\left\{\begin{array}{c}{\Lambda }_0\left(a+〈x,J^{*}\left(k_1+k_2{\mathcal{Z}}_{\rho }\right)〉-C\right)=0\hfill \\ {\Lambda }_m\left(H+x\right)={\Lambda }_{M}x=0\hfill \end{array}\right.$$

(37)

hold. Take ${\Lambda }_0=\left({\theta }_v+{\theta }_E\right)$, ${\Lambda }_m={\Lambda }^{+}$ and ${\Lambda }_M={\Lambda }^{-}$, and the fulfillment of (37) becomes evident.□

Remark 9.

Theorem 8 shows that the budgetary constraint is hardly relevant. Indeed, (34) and (35) imply that x does not depend on the budget C, whereas the value of a only imposes to saturate the constraint. In other words, a budget modification implies that only the purchase/sale of a riskless asset must be modified so as to invest the whole budget. There is a second important consequence of Theorem 8. Indeed, there always exists a solution x such that $x\left(\tau \right)=-H$ or $x\left(\tau \right)=0$ for every $\tau \in \mathbb{R}$. Partial protection ($-H<x<0$) is never better than total protection ($x=-H$ or $x=0$).□

Remark 10.

Fix the solution ${\tilde{x}}_{\beta }$ given by (35). Then, (15) and (16) apply, and one has risk upper bounds holding for every confidence level. In particular, if one deals with (15), and taking into account the ideas of Remark 7,

$$\left\{\begin{array}{c}CV@R_{1-s}\left(\mathcal{U}+f\left(\mathcal{U}\right)\right)=CV@R_{1-s}\left(\mathcal{U}+a+J\left({\tilde{x}}_{\beta }\right)\left(\mathcal{U}\right)\right)\le \hfill \\ 〈1+{\tilde{x}}_{\beta },\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}〉-a-〈1+{\tilde{x}}_{\beta },J^{*}\left({\mathcal{Z}}_{\beta }\right)+\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}〉{\displaystyle \frac{\beta }{s}}\hfill \end{array}\right.$$

(38)

for every $o<s<1$. Similarly, one can consider a set $\left\{{\beta }_1,{\beta }_2,...{\beta }_m\right\}\subset \left(0,1\right)$ and the upper bound (16) holds with (recall (14))

$$\left\{\begin{array}{c}{\vartheta }_j=-{\beta }_j〈1+{\tilde{x}}_{\beta },J^{*}\left({\mathcal{Z}}_{{\beta }_j}\right)+\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{{\beta }_j},0\right)}〉\hfill \\ V_j=〈1+{\tilde{x}}_{\beta },\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{{\beta }_j},0\right)}〉-a\hfill \\ {\omega }_{{\beta }_j}=V@R_{1-{\beta }_j}\left(\mathcal{U}\right)\hfill \end{array}\right.$$

(39)

and ${\mathcal{Z}}_{{\beta }_j}$ given by (21) if ${\beta }_j$ replaces β and ${\omega }_{{\beta }_j}$ replaces ${\omega }_{\beta }$, $j=1,2,...,m$.□

4. Portfolio Insurance

4.1. On the Financial Market

Portfolio insurance may play a critical role in financial risk management problems, at least if the financial market is going through turbulent times. Indeed, the wealth protection when facing market turmoils was always a major question for many managers (Leland and Rubistern, 1976, Bertrand and Prigent, 2003, Annaert et al., 2009, Zagst and Kraus, 2011, Nangolo et al., 2023, etc.). Quite common and very often used by practitioners portfolio insurance strategies are the sale of future contracts or the purchase of European puts (Overhaus et al., 2007, or Joossens and Schoutens, 2008, among many others) but let us deal with the findings of Section 3 in order to verify whether these usual strategies are really optimal. As will be seen, the purchase of an European put is often close to optimal, but the correct computation of the option strike plays a critical role. In contrast, the sale of futures is rarely optimal, though sometimes it might be close to the optimal solution.

Suppose that $\mathcal{U}$ represents the random price (or pay-off) at T of an investor (or manager) portfolio, whereas $f\left(\mathcal{U}\right)$ represents the hedging strategy that allows the investor to (maybe partially) compensate the effects provoked by an unfavorable market trend. According to the framework of Section 3, the investor wealth $\mathcal{U}+f\left(\mathcal{U}\right)$ after adding the hedging strategy $f\left(\mathcal{U}\right)$ must be comonotonic with $\mathcal{U}$, which means that the investor is interested in a favorable evolution of $\mathcal{U}$. Besides, $f\left(\mathcal{U}\right)$ must be anti-comonotonic with $\mathcal{U}$, which means the investor is implementing hedging if the final realized value of $\mathcal{U}$ is not favorable. Both the comonotonicity of $\mathcal{U}+f\left(\mathcal{U}\right)$ and the anti-comonotonicity of $f\left(\mathcal{U}\right)$ are fulfilled if $f\left(\mathcal{U}\right)$ is the pay-off of a sold future or a bought European put, so the usual portfolio insurance strategies satisfy the assumptions of Section 3.

As usual in Financial Mathematics, extend the initial probability space $\left(\Omega ,\mathcal{F},\mathbb{P}\right)$ to the filtered probability space $\left(\Omega ,{\left({\mathcal{F}}_t\right)}_{t\in \mathcal{T}},\mathcal{F},\mathbb{P}\right)$, where $\mathcal{T}$ is a set of trading dates such that $\left\{0,T\right\}\subset \mathcal{T}\subset \left[0,T\right]$, ${\mathcal{F}}_0=\left\{⌀,\Omega \right\}$ and ${\mathcal{F}}_T=\mathcal{F}$. For the sake of mathematical simplicity, let us assume that the model is complete, that is, there exist a riskless asset, whose continuously compounded riskless rate equals r, and n risky assets whose price processes are adapted to ${\left({\mathcal{F}}_t\right)}_{t\in \mathcal{T}}$ and satisfy that every $\mathcal{Y}\in L^2\left(\mathbb{P}\right)$ may be replicated by combining the $n+1$ available securities in a self-financing way. This simplification is not very restrictive, since it is quite usual when dealing with the standard derivative pricing models. For instance, the binomial model and the $BSM$ model are complete. With respect to the usual stochastic volatility models, it is frequently assumed the existence of enough volatility-linked derivatives so as to make the market complete. Otherwise it would not be possible to give a unique price for standard derivatives like, for instance, European options.

The market completeness allows us to suppose that every $\mathcal{Y}\in L^2\left(\mathbb{P}\right)$ is available to the investor. Moreover, the pricing rule $\Pi :L^2\left(\mathbb{P}\right)\to \mathbb{R}$ providing us with the price $\Pi \left(\mathcal{Y}\right)\in \mathbb{R}$ of $\mathcal{Y}\in L^2\left(\mathbb{P}\right)$ is supposed to be linear and continuous, and therefore the Riesz Representation Theorem implies the existence of a unique $z_{\Pi }\in L^2\left(\mathbb{P}\right)$ such that (Duffie, 1988)

$$\begin{array}{cc}\Pi \left(\mathcal{Y}\right)=\mathsf{I}\mathsf{E}\left(z_{\Pi }\mathcal{Y}\right),\hfill & \forall \mathcal{Y}\in L^2\left(\mathbb{P}\right)\hfill \end{array}$$

(40)

and

$$\left\{\begin{array}{c}\mathsf{I}\mathsf{E}\left(z_{\Pi }\right)=1\hfill \\ \mathbb{P}\left(z_{\Pi }>0\right)=1,\hfill \end{array}\right.$$

(41)

$i.e.$, $z_{\Pi }$ satisfies the requirements of the elements of the set ${\partial }_{\rho }$ in (25) (recall (27)). It is known that $z_{\Pi }$ is said to be the stochastic discount factor ($SDF$) of $\Pi$.8 If one considers $\rho \left(\mathcal{Y}\right)=-\mathsf{I}\mathsf{E}\left(z_{\Pi }\mathcal{Y}\right)$, then (41) implies that $\rho$ satisfies the assumptions imposed to $\rho$ in Section 3, whereas (24) with $k_1=0$ and $k_2=1$ leads to

$$\Pi \left(a+J\left(x\right)\right)=a+\mathsf{I}\mathsf{E}\left(z_{\Pi }J\left(x\right)\right)=\mathsf{I}\mathsf{E}\left(z_{\Pi }\left(a+J\left(x\right)\right)\right),$$

that is, the pricing rules of (24) and (40) are exactly the same functional. To sum up, the optimal portfolio insurance problem is particular case of (28), (29), (31), (32) and (33) if one takes

$$\left\{\begin{array}{c}\left(H,k_1,k_2\right)=\left(1,0,1\right)\hfill \\ \rho \left(\mathcal{Y}\right)=-\mathsf{I}\mathsf{E}\left(z_{\Pi }\mathcal{Y}\right)\hfill \\ {\mathcal{Z}}_{\rho }=z_{\Pi }\hfill \end{array}\right.$$

(42)

where the equality $H=1$ is not needed but it is imposed for reasons that will be seen in Section $4.2$.

4.2. Optimal Portfolio Insurance

Notice that the decision variable “$x\left(\mathcal{U}\right)=d\left(f\left(\mathcal{U}\right)\right)/d\mathcal{U}$” may be in this case understood as the Delta Greek of $f\left(\mathcal{U}\right)$ at T, whereas a may be interpreted as the pay-off at T of $f\left(\mathcal{U}\right)$ if the final value of $f\left(\mathcal{U}\right)$ vanishes ($\mathcal{U}=0$). For instance, if $a=1$ and $x\left(\tau \right)=-1$ for every $\tau \in \mathbb{R}$, then (18) implies that

$$f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)=1-{\displaystyle \underset{0}{\overset{\mathcal{U}}{\int }}}dt=1-\mathcal{U}$$

is the pay-off of a sold future contract whose future quotation equals $a=1$. Obviously, the Delta Greek of this position equals $x\left(\tau \right)=-1$. Similarly, if $\mathbb{P}\left(\mathcal{U}\ge 0\right)=1$, $a=2$ and

$$x\left(\tau \right)=\left\{\begin{array}{cc}-1,\hfill & \tau <2\hfill \\ 0,\hfill & \tau >2\hfill \end{array}\right.$$

then

$$f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)=\left\{\begin{array}{cc}2-\mathcal{U},\hfill & 0\le \tau \le 2\hfill \\ 0,\hfill & 2\le \tau \hfill \end{array}\right.$$

is the pay-off of a bought European put with maturity at T, whose underlying asset is $\mathcal{U}$ and whose strike equals 2. Once again, x is the Delta-Greek of this European option.

In both examples above $H=1$ must hold in order to satisfy (23), and therefore, as anticipated in (42), let us take this value of H throughout the whole Section 4. Otherwise, the common portfolio insurance strategies would not be feasible (recall (22)).

Theorem 8 and (34) provide us with the solution of the optimal portfolio insurance problem. Further properties of this solution may be found if one has an explicit expression of $J^{*}\left(1\right)$ and $J^{*}\left({\mathcal{Z}}_{\beta }\right)$. A general explicit expression of $J^{*}\left({\mathcal{Z}}_{\rho }\right)=J^{*}\left(z_{\Pi }\right)$ cannot be given because it depends on $z_{\Pi }$, that is, on the concrete pricing model one is dealing with. Recall that the value of $J^{*}\left(1\right)$, $J^{*}\left({\mathcal{Z}}_{\beta }\right)$ and $J^{*}\left(z_{\Pi }\right)$ in a finite subset of $\mathbb{R}$ is irrelevant because this set is $\mathbb{L}-$null.

Proposition 11.

$a)$

$$J^{*}\left(1\right)\left(\tau \right)=\left\{\begin{array}{cc}-\mathbb{P}\left(\mathcal{U}\le \tau \right),\hfill & \tau <0\hfill \\ \mathbb{P}\left(\mathcal{U}\ge \tau \right),\hfill & 0<\tau \hfill \end{array}\right.$$

$b)$ If $V@R_{1-\beta }\left(\mathcal{U}\right)>0$, that is, $-{\omega }_{\beta }<0$, then

$$J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)=\left\{\begin{array}{cc}-\mathbb{P}\left(\mathcal{U}\le \tau \right)/\beta ,\hfill & \tau <-{\omega }_{\beta }\hfill \\ -1,\hfill & -{\omega }_{\beta }<\tau <0\hfill \\ 0,\hfill & 0<\tau \hfill \end{array}\right.$$

$c)$ If $V@R_{1-\beta }\left(\mathcal{U}\right)=0$, that is, $-{\omega }_{\beta }=0$, then

$$J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)=\left\{\begin{array}{cc}-\mathbb{P}\left(\mathcal{U}\le \tau \right)/\beta ,\hfill & \tau <0\hfill \\ 0,\hfill & 0<\tau \hfill \end{array}\right.$$

$d)$ If $V@R_{1-\beta }\left(\mathcal{U}\right)<0$, that is, $-{\omega }_{\beta }>0$, then

$$J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)=\left\{\begin{array}{cc}-\mathbb{P}\left(\mathcal{U}\le \tau \right)/\beta ,\hfill & \tau <0\hfill \\ \mathbb{P}\left(\tau \le \mathcal{U}<-{\omega }_{\beta }\right)/\beta ,\hfill & \left\{\begin{array}{c}0<\tau <-{\omega }_{\beta }\hfill \\ \mathbb{P}\left(\mathcal{U}=-{\omega }_{\beta }\right)=0\hfill \end{array}\right.,\hfill \\ \left\{\begin{array}{c}\mathbb{P}\left(\tau \le \mathcal{U}<-{\omega }_{\beta }\right)/\beta \hfill \\ +1-\mathbb{P}\left(\mathcal{U}<-{\omega }_{\beta }\right)/\beta \hfill \end{array}\right.,\hfill & \left\{\begin{array}{c}0<\tau <-{\omega }_{\beta }\hfill \\ \mathbb{P}\left(\mathcal{U}=-{\omega }_{\beta }\right)>0\hfill \end{array}\right.\hfill \\ 0,\hfill & -{\omega }_{\beta }<\tau \hfill \end{array}\right.$$

Proof. 

$a)$ is an evident consequence of (19), whereas $b)$, $c)$ and $d)$ are evident consequences of (19), (21) and $\mathsf{I}\mathsf{E}\left({\mathcal{Z}}_{\beta }\right)=1$.□

Remark 12.

(42) implies that (34) becomes

$$\Lambda =\left\{\begin{array}{c}\left({\theta }_v-{\theta }_c\right)\mathsf{I}{\mathsf{1}}_{\left(-{\omega }_{\beta },0\right)}-{\theta }_c{J}^{*}\left({\mathcal{Z}}_{\beta }\right)\hfill \\ -{\theta }_E{J}^{*}\left(1\right)+\left({\theta }_v+{\theta }_E\right)J^{*}\left(z_{\Pi }\right).\hfill \end{array}\right.$$

(43)

Since $k_1+k_2{\mathcal{Z}}_{\rho }=z_{\Pi }$ (recall (42)), take $\left(a,x\right)\in \mathbb{R}\times L^{\infty }\left(\mathbb{L}\right)$ given by

$$\left\{\begin{array}{c}\Lambda <0⟹x=0\hfill \\ \Lambda >0⟹x=-1\hfill \\ \Lambda =0⟹-1\le x\le 0\hfill \\ a=C-〈x,J^{*}\left(z_{\Pi }\right)〉\hfill \end{array}\right.$$

and $f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)$ is an optimal portfolio insurance (Theorem 8).

Suppose for instance that $\mathbb{P}\left(\mathcal{U}>0\right)=1$ and $\mathbb{P}\left(\mathcal{U}=u\right)=0$ for every $u\in \mathbb{R}$. Proposition $11d$ implies that

$$J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)=\left\{\begin{array}{cc}0,\hfill & \tau <0\hfill \\ \mathbb{P}\left(\tau \le \mathcal{U}\le -{\omega }_{\beta }\right)/\beta ,\hfill & 0\le \tau \le -{\omega }_{\beta },\hfill \\ 0,\hfill & -{\omega }_{\beta }<\tau ,\hfill \end{array}\right.$$

(19) implies that $J^{*}\left(z_{\Pi }\right)\left(\tau \right)=0$ if $\tau <0$, and (43) implies that

$$\Lambda ={\theta }_v{J}^{*}\left(z_{\Pi }\right)+{\theta }_E{J}^{*}\left(z_{\Pi }-1\right)$$

if $\tau >-{\omega }_{\beta }=-V@R_{1-\beta }\left(\mathcal{U}\right)>0$ and $\Lambda =0$ if $\tau <0$. If the market is risk neutral ($\mathbb{P}\left(z_{\Pi }=1\right)=1$, Duffie, 1988), then $\Lambda ={\theta }_v{J}^{*}\left(z_{\Pi }\right)$ implies that $\Lambda \ge 0$ for $\tau >-{\omega }_{\beta }$, and Theorem 8 implies that one can take a solution of (33) such that $x=-1$ for $\tau >-{\omega }_{\beta }$, $i.e.$, values of $\mathcal{U}$ higher than $-{\omega }_{\beta }$ have to be fully protected. This finding might partially justify why the sale of future contracts is often used by traders interested in portfolio insurance. In contrast, as will be seen in Section $4.3$, non-risk neutral models frequently imply that $\mathbb{R}\ni \mathcal{U}⟶z_{\Pi }\left(\mathcal{U}\right)\in \mathbb{R}$ is strictly increasing or strictly decreasing in $\mathcal{U}\ge 0$. If $z_{\Pi }$ is decreasing then $z_{\Pi }\left(\mathcal{U}\right)-1<0$ for $\mathcal{U}$ higher than some threshold (recall (41)). Hence, if the expected wealth were important enough for the investor (or “${\theta }_E$ were large enough”), then $\Lambda \left(\mathcal{U}\right)<0$ might hold, and then $x\left(\mathcal{U}\right)=0$ should hold. High values of $\mathcal{U}$ will not be protected under the optimal solution of (33). This finding might partially justify why the empirical evidence also shows that the purchase of European puts is also often used by practitioners. At any rate, notice that all of these comments only indicate that the classical portfolio insurance strategies might be close to optimal one under some conditions. In Section $4.3$ we will find exact solutions under the assumptions of the $BSM$ model.

Notice that the assumptions above $\mathbb{P}\left(\mathcal{U}>0\right)=1$ and $\mathbb{P}\left(\mathcal{U}=u\right)=0$ may be relaxed and Theorem 8 still yields the optimal portfolio insurance. Under different assumptions about $\mathcal{U}$ the role of Proposition $11d$ may be replaced by other statements of Proposition 11, and then one can check whether $\Lambda \left(\mathcal{U}\right)>0$ or $\Lambda \left(\mathcal{U}\right)<0$ holds in order to take $x\left(\mathcal{U}\right)=-1$ or $x\left(\mathcal{U}\right)=0$. We will not do that in order to prevent a tedious casuistry and shorten the exposition.□

4.3. Optimality Under the Black-Scholes-Merton Model

Let us particularize the findings above under the assumptions of the $BSM$ model. Accordingly, consider an international stock index ($i.e.$, an international benchmark) whose evolution is given by a geometric Brownian motion ($GBM$). For the sake of mathematical simplicity, let us consider derivatives whose underlying asset is the future contract rather than the index itself. If so, the future quotation F is also a $GBM$ given by the stochastic differential equation

$$dF=F\left(\mu dt+\sigma dz\right),$$

(44)

where $\mu$ is the index excess return (that is, the difference between the index drift and the riskless rate) and $\sigma$ is the index volatility. The simplification is provoked by the absence of the index dividend yield in (44). Moreover, recall that the purchase of the future contract plus the investment of the index quotation in the riskless asset replicates the purchase of the index along with the investment of every generated dividend in the riskless asset (Hull, 2021). In other words, to deal with the future contract does not restrict the analysis.

Given the planning period T, denote by $F_T$ the future quotation at T, take $\mathcal{U}=F_T$, and recall that (Wang, 2000, or Hamada and Sherris, 2003)

$$z_{\Pi }=M{\left(F_T\right)}^{-\mu /{\sigma }^2},$$

(45)

where $M=1/\mathsf{I}\mathsf{E}\left(F_T^{-\mu /{\sigma }^2}\right)>0$. Since $F_T$ has a continuous distribution and $F_T>0$, Proposition 11 and (19) lead to

$$J^{*}\left(1\right)\left(\tau \right)=J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)=J^{*}\left(z_{\Pi }\right)\left(\tau \right)=0$$

if $\tau <0$ and

$$\left\{\begin{array}{c}{J}^{*}\left(1\right)\left(\tau \right)=\mathbb{P}\left(F_T\ge \tau \right)\hfill \\ J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)=\left\{\begin{array}{cc}\mathbb{P}\left(\tau \le F_T<-{\omega }_{\beta }\right)/\beta ,\hfill & 0<\tau <-{\omega }_{\beta },\hfill \\ 0,\hfill & -{\omega }_{\beta }<\tau \hfill \end{array}\right.\hfill \\ J^{*}\left(z_{\Pi }\right)\left(\tau \right)=M\underset{\omega \ge \tau }{\int }{\omega }^{-\mu /{\sigma }^2}\mathbb{P}\left(d\omega \right)\hfill \end{array}\right.$$

(46)

for $\tau >0$. The solution of (44) leads to (Hull, 2021)

$$F_T=F_{0}exp\left(\left(\mu -{\displaystyle \frac{{\sigma }^2}{2}}\right)T+\sigma \sqrt{T}W\right),$$

(47)

$F_0$ being the current quotation of the future contract and W being a random variable with the standard normal distribution. Thus, if $\Phi$ is the cumulative distribution function of W, it is known that

$$-{\omega }_{\beta }=-V@R_{1-\beta }\left(F_T\right)=F_{0}exp\left(\left(\mu -{\displaystyle \frac{{\sigma }^2}{2}}\right)T+\sigma \sqrt{T}{\Phi }^{-1}\left(\beta \right)\right)$$

(48)

and

$$\left\{\begin{array}{c}\mathbb{P}\left(\tau \le F_T<-{\omega }_{\beta }\right)=\hfill \\ \Phi \left({\displaystyle \frac{log\left(-{\omega }_{\beta }/F_0\right)-\left(\mu -{\sigma }^2/2\right)T}{\sigma \sqrt{T}}}\right)-\Phi \left({\displaystyle \frac{log\left(\tau /F_0\right)-\left(\mu -{\sigma }^2/2\right)T}{\sigma \sqrt{T}}}\right),\hfill \end{array}\right.$$

which complements the expression of $J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)$ given in (46). Moreover, (48) trivially implies that

$$\Phi \left({\displaystyle \frac{log\left(-{\omega }_{\beta }/F_0\right)-\left(\mu -{\sigma }^2/2\right)T}{\sigma \sqrt{T}}}\right)=\beta ,$$

and consequently

$$\mathbb{P}\left(\tau \le F_T<-{\omega }_{\beta }\right)=\beta -\Phi \left({\displaystyle \frac{log\left(\tau /F_0\right)-\left(\mu -{\sigma }^2/2\right)T}{\sigma \sqrt{T}}}\right).$$

Besides, the first equality in (46) and (47) imply that

$$J^{*}\left(1\right)\left(t\right)=1-\Phi \left({\displaystyle \frac{log\left(\tau /F_0\right)-\left(\mu -{\sigma }^2/2\right)T}{\sigma \sqrt{T}}}\right)$$

and

$$J^{*}\left(z_{\Pi }\right)\left(\tau \right)=M\underset{\omega \ge 0}{\int }\mathsf{I}{\mathsf{1}}_{\left[\tau ,\infty \right)}\left(\omega \right){\omega }^{-\mu /{\sigma }^2}\mathbb{P}\left(d\omega \right)=M\mathsf{I}\mathsf{E}\left(\mathsf{I}{\mathsf{1}}_{\left[0,\tau \right)}\left(\omega \right){\omega }^{-\mu /{\sigma }^2}\right).$$

To sum up, if

$$\Gamma \left(\tau \right)={\displaystyle \frac{log\left(\tau /F_0\right)-\left(\mu -{\sigma }^2/2\right)T}{\sigma \sqrt{T}}}$$

for $\tau >0$, then (46) becomes

$$\left\{\begin{array}{c}{J}^{*}\left(1\right)\left(\tau \right)=1-\Phi \left(\Gamma \left(\tau \right)\right)\hfill \\ J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)=\left\{\begin{array}{cc}1-\Phi \left(\Gamma \left(\tau \right)\right)/\beta ,\hfill & 0<\tau <-{\omega }_{\beta },\hfill \\ 0,\hfill & -{\omega }_{\beta }<\tau \hfill \end{array}\right.\hfill \\ J^{*}\left(z_{\Pi }\right)\left(\tau \right)=M\mathsf{I}\mathsf{E}\left(\mathsf{I}{\mathsf{1}}_{\left[\tau ,\infty \right)}\left(\omega \right){\omega }^{-\mu /{\sigma }^2}\right).\hfill \end{array}\right.$$

(49)

Consider Problem (33) and suppose henceforth that ${\theta }_c=1$ and ${\theta }_v<1$. Actually, ${\theta }_c=1$ may be easily reached if ${\theta }_c\ne 0$, because the solution of (33) remains the same if $\left({\theta }_c,{\theta }_v,{\theta }_E\right)$ is replaced by $\left(1,,{\theta }_v/{\theta }_c,{\theta }_E/{\theta }_c\right)$, and ${\theta }_c\ne 0$ just means that the investor is interested in the upper bound (15), that is, it is interested in levels of confidence beyond the initially selected one $1-\beta$. Besides, if ${\theta }_c=1$ and ${\theta }_v\ge 1$ held, then (33) would become a scalarization of the objectives

$$\left(CV@R_{1-\beta }\left(J\left(1+x\right)\right),V@R_{1-\beta }\left(J\left(1+x\right)\right),-\mathsf{I}\mathsf{E}\left(a+J\left(x\right)\right)\right),$$

and the interest of (15) would be once again diluted. Obviously, (34) leads to (recall (30))

$$\Lambda =\left(1-{\theta }_v\right)\mathsf{I}{\mathsf{1}}_{\left(0,-{\omega }_{\beta }\right)}-J^{*}\left({\mathcal{Z}}_{\beta }\right)-{\theta }_E{J}^{*}\left(1\right)+\left({\theta }_v+{\theta }_E\right)J^{*}\left(z_{\Pi }\right),$$

(50)

where ${\omega }_{\beta }$ is given in (48) and $J^{*}\left(1\right)$, $J^{*}\left({\mathcal{Z}}_{\beta }\right)$ and $J^{*}\left(z_{\Pi }\right)$ are given in (49).

Next let us show that, in some particular cases, the sign of (50) beyond some threshold may be known without computations. As a consequence, the intuitions of Remark 12 about the optimality of the sale of futures or the purchase of puts might be partially reinforced.

Proposition 13.

$a)$ Suppose that ${\theta }_c=1$, ${\theta }_v=0$ and $\mu >0$.9 Then, there exist ${\tau }_0>0$ and an optimal portfolio insurance x such that $x\left(\mathcal{U}\right)=0$ for $\mathcal{U}\ge {\tau }_0$.

$b)$ Suppose that ${\theta }_c=1$, ${\theta }_v=0$ and $\mu <0$. Then, there exist ${\tau }_0>0$ and an optimal portfolio insurance $x\left(\mathcal{U}\right)$ such that $x\left(\mathcal{U}\right)=-1$ for $\mathcal{U}\ge {\tau }_0$.

$c)$ Suppose that ${\theta }_c=1$ and ${\theta }_E=0$.10 Then, there exist ${\tau }_0>0$ and an optimal portfolio insurance x such that $x\left(\mathcal{U}\right)=-1$ for $\mathcal{U}\ge {\tau }_0$.

Proof. 

$a)$ and $b)$ (49) and (50) imply that $\Lambda ={\theta }_E{J}^{*}\left(z_{\Pi }-1\right)$ if $\tau >-{\omega }_{\beta }$. If $\mu >0$ ($\mu <0$), then (45) implies that $z_{\Pi }\left(\omega \right)<1$ ($z_{\Pi }\left(\omega \right)>1$) if $\omega$ is large enough, and (19) implies that $\Lambda \left(\tau \right)<0$ ($\Lambda \left(\tau \right)>0$) if $\tau$ is large enough. The result follows from Theorem 8 and Proposition $11d$.

$c)$ One has $\Lambda \left(\tau \right)={\theta }_v{J}^{*}\left(z_{\Pi }\right)\left(\tau \right)\ge 0$ if $\tau$ is large enough. The result follows from Theorem 8 and Proposition $11d$.□

4.4. Numerical Experiments

Under the framework of Section $4.3$, numerical experiments have been implemented for $1-\beta =95\%$, $F_0=1000$, $T=1$ and $r=1\%$. The extreme situations ${\theta }_v=0$ or ${\theta }_E=0$ of Proposition 13 have been excluded, and therefore all the experiments have been implemented with ${\theta }_c=1$, $1>{\theta }_v>0$, and ${\theta }_E>0$. During periods of calm in the financial markets such that $\mu >0$ holds and $\sigma$ is not very high, the optimal portfolio insurance strategy has often been close to the purchase of an European put (a classical one), but not necessarily the purchase of that put itself. For instance, if $\mu =1\%$, $\sigma =20\%$ and the budget C equals zero,11 then the optimal portfolio insurance has become the one given in Table I below (the expectations of (49) have been estimated with ${10}^4$ Monte Carlo simulations):

$$\left|\begin{array}{c}\begin{array}{c}\begin{array}{cccccc}\hfill & \hfill & & TableI\hfill & \hfill & \end{array}\hfill \\ \begin{array}{c}Theoptimalportfolioinsurance\hfill \\ Onemustborrowthestrategyprice\hfill \end{array}\hfill \end{array}\hfill \\ \left(\mu ;\sigma \right)=\left(1\%;20\%\right)\hfill \\ \left({\theta }_c;{\theta }_v;{\theta }_E\right)=\left(1;0.9;0.1\right)⟹{\left(1015-F_T\right)}^{+}-{\left(189-F_T\right)}^{+}\hfill \\ \left({\theta }_c;{\theta }_v;{\theta }_E\right)=\left(1;0.5;0.5\right)⟹{\left(1049-F_T\right)}^{+}-{\left(195-F_T\right)}^{+}\hfill \\ \left({\theta }_c;{\theta }_v;{\theta }_E\right)=\left(1;0.1;0.9\right)⟹{\left(1016-F_T\right)}^{+}\hfill \end{array}\right|$$

Remark 10 applies, and beyond the level of confidence $1-\beta =95\%$ one can choose other values $1-{\beta }_j$ in order to apply (16). Tables $II$ and $III$ below present the coefficients of several upper bounds.

$$\left|\begin{array}{c}TableII\\ Valuesof{\vartheta }_{j}and{V}_{j}in\left(16\right)\hfill \\ \left(\mu ;\sigma \right)=\left(1\%;20\%\right),\left({\theta }_c;{\theta }_v;{\theta }_E\right)=\left(1;0.9;0.1\right)\hfill \\ 1-{\beta }_1=99,9\%⟹\left({\vartheta }_1;V_1\right)=\left(1885\times {10}^{-4};-188.5\right)\hfill \\ 1-\beta =95\%⟹\left(\vartheta ;V\right)=\left(1885\times {10}^{-4};-188.5\right)\hfill \\ 1-{\beta }_2=29,03\%⟹\left({\vartheta }_2;V_2\right)=\left(0.709898;-189.5\right)\hfill \\ 1-{\beta }_3=20.03\%⟹\left({\vartheta }_3;V_3\right)=\left(49.923048;-254.5\right)\hfill \\ 1-{\beta }_4=10.03\%⟹\left({\vartheta }_4;V_4\right)=\left(142.222728;-362.5\right)\hfill \end{array}\right|$$

$$\left|\begin{array}{c}TableIII\\ Valuesof{\vartheta }_{j}and{V}_{j}in\left(16\right)\hfill \\ \left(\mu ;\sigma \right)=\left(1\%;20\%\right),\left({\theta }_c;{\theta }_v;{\theta }_E\right)=\left(1;0.1;0.9\right)\hfill \\ 1-{\beta }_1=98\%⟹\left({\vartheta }_1;V_1\right)=\left(0;0\right)\hfill \\ 1-\beta =95\%⟹\left(\vartheta ;V\right)=\left(0;0\right)\hfill \\ 1-{\beta }_2=43,93\%⟹\left({\vartheta }_2;V_2\right)=\left(2.23084;-0.4\right)\hfill \\ 1-{\beta }_3=40,83\%⟹\left({\vartheta }_3,V_3\right)=\left(11.4632;-20\right)\hfill \\ 1-{\beta }_4=10,03\%⟹\left({\vartheta }_4;V_4\right)=\left(198.04802;-262\right)\hfill \end{array}\right|$$

Things are somewhat distinct when facing market turmoils, because the strikes of the involved puts are quite different. Contrary to what many managers do, the sale of futures rarely becomes an optimal solution. In contrast, the purchase of puts, or combinations of long and short puts very often solves (33). For instance, if $\left({\theta }_c;{\theta }_v;{\theta }_E\right)=\left(1;0.5;0.5\right)$, $\mu =-6\%$ and $\sigma =30\%$, the optimal portfolio insurance strategy equals ${\left(1379-F_T\right)}^{+}-{\left(75-F_T\right)}^{+}$. Table $IV$ below gives a sample of values of the coefficients of the upper bound (16) if the strategy implementation is financed through borrowing.

$$\left|\begin{array}{c}TableIV\\ Valuesof{\vartheta }_{j}and{V}_{j}in\left(16\right)\hfill \\ \left(\mu ;\sigma \right)=\left(-6\%;30\%\right),\left({\theta }_c;{\theta }_v;{\theta }_E\right)=\left(1;0.5;0.5\right)\hfill \\ 1-{\beta }_1=99\%⟹\left({\vartheta }_1,V_1\right)=\left(7.55\times {10}^{-5};-75.5\right)\hfill \\ 1-\beta =95\%⟹\left(\vartheta ,V\right)=\left(7.55\times {10}^{-5};-75.5\right)\hfill \\ 1-{\beta }_2=50\%⟹\left({\vartheta }_2,V_2\right)=\left(7.55\times {10}^{-5};-75.5\right)\hfill \\ 1-{\beta }_3=7.73\%⟹\left({\vartheta }_3,V_3\right)=\left(0.922786;-76\right)\hfill \\ 1-{\beta }_4=1\%⟹\left({\vartheta }_4,V_4\right)=\left(409.885115;-499.5\right)\hfill \end{array}\right|$$

Things are essentially similar if the excess return $\mu$ decreases or the volatility $\sigma$ increases. The optimal portfolio insurance almost always equals the purchase of puts or a combination of puts, but never equals the sale of a future contract.12 Needless to say, the manager may replace $BSM$ with many alternative models (see, for instance, Allaj et al., 2025, for a recent revision of many pricing models), but Theorem 8 still applies, (35) solves the problem, and there always exist an optimal portfolio insurance x such that none of the elements $\tau \in \mathbb{R}$ will satisfy $-1<x\left(\tau \right)<0$. Partial hedging will be always beaten by total hedging.

5. Optimal Reinsurance

5.1. The Insurer Problem

Insurers frequently buy reinsurance contracts if the probability of high indemnifications to policyholders is significant enough. Accordingly, the “optimal reinsurance problem” has become a classic in actuarial mathematics, and a relatively recent “state of the art” may be found in Albrecher et al. (2017).

Denote by $\mathcal{U}$ the insurer random wealth at the maturity date T of the policies, that is, suppose that $-\mathcal{U}$ is the global indemnification. Obviously, $\mathbb{P}\left(-\mathcal{U}\ge 0\right)=1$, that is, $\mathbb{P}\left(\mathcal{U}\le 0\right)=1$.13 The insurer can buy reinsurance in order to receive at T a second indemnification $f_0\left(\mathcal{U}\right)\ge 0$ to be paid by the reinsurer. According to the terminology used in the relevant literature, $f_0\left(\mathcal{U}\right)$ is the ceded risk, and $f_0$ must be a decreasing function in order to prevent moral hazard for the insurer. Indeed, if ${\mathcal{U}}_1<{\mathcal{U}}_2$ and $f_0\left({\mathcal{U}}_1\right)<f_0\left({\mathcal{U}}_2\right)$ held, then the reinsurer would pay less if the indemnity increased from $-{\mathcal{U}}_2$ to $-{\mathcal{U}}_1$, $i.e.$, the reinsurer might try to get the insurer to cover more claims. Besides, the so-called retained risk $\mathcal{U}+f_0\left(\mathcal{U}\right)$ must be an increasing function to prevent moral hazard for the reinsurer. Indeed, if ${\mathcal{U}}_1<{\mathcal{U}}_2$ and ${\mathcal{U}}_1+f_0\left({\mathcal{U}}_1\right)>{\mathcal{U}}_2+f_0\left({\mathcal{U}}_2\right)$ held, then the insurer would become richer if the indemnity $-{\mathcal{U}}_1$ were higher than the indemnity $-{\mathcal{U}}_2$, $i.e.$, the insurer might have incentives to recognize more claims than actually occurred. The insurer can also invest money in a riskless asset, and therefore $f\left(\mathcal{U}\right)=a+f_0\left(\mathcal{U}\right)$ is the insurer additional income at T, where a stands for the (recall, final value) amount invested in the riskless security.14 Evidently, for every $a\in \mathbb{R}$, $f\left(\mathcal{U}\right)$ and $\mathcal{U}+f\left(\mathcal{U}\right)$ remain decreasing and increasing, respectively. As in Section 3, the ceded risk $f_0\left(\mathcal{U}\right)$ is going to be replaced by the marginal ceded risk $x\left(\mathcal{U}\right)$, that is, $f_0\left(\mathcal{U}\right)=J\left(x\right)\left(\mathcal{U}\right)$ and $f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)$. Finally, the marginal ceded risk H of (22) represents the maximum amount the reinsurer agrees to pay for every monetary unit that the insurer’s indemnity payments increase.

The price of $f\left(\mathcal{U}\right)=a+J\left(x\right)\left(\mathcal{U}\right)$ will be given by

$$a+k_1\mathsf{I}\mathsf{E}\left(J\left(x\right)\right)+k_2\rho \left(-J\left(x\right)\right)=a+k_1\mathsf{I}\mathsf{E}\left(J\left(x\right)\right)+k_2\mathsf{I}\mathsf{E}\left(J\left(x\right){\mathcal{Z}}_{\rho }\right)$$

(recall (24) and Lemma 6). It means that the financial market is pricing according to the ideas presented in Section 4, whereas the reinsurer prices according to the usual premium principle

$$\Pi \left(f_0\left(\mathcal{U}\right)\right)=\Pi \left(J\left(x\right)\left(\mathcal{U}\right)\right)=k_1\mathsf{I}\mathsf{E}\left(J\left(x\right)\right)+k_2\rho \left(-J\left(x\right)\right)$$

(Albrecher et al., 2017).

5.2. The Optimal Contract

$\mathcal{U}\le 0$ and (19) imply that $J^{*}\left(z\right)\left(\tau \right)=0$ for every $z\in L^2\left(\mathbb{P}\right)$ if $\tau >0$. Besides, it $\tau <0$,

$$J^{*}\left(z\right)\left(\tau \right)=-\underset{\omega \le \tau }{\int }z\left(\omega \right)\mathbb{P}\left(d\omega \right).$$

Furthermore, $\tau <0$, $\mathcal{U}\le 0$ and Proposition $11a$ lead to

$$J^{*}\left(1\right)\left(\tau \right)=-\mathbb{P}\left(\mathcal{U}\le \tau \right),$$

while $\tau <0$, $\mathcal{U}\le 0$ and Proposition $11b$ lead to

$$J^{*}\left({\mathcal{Z}}_{\beta }\right)\left(\tau \right)=\left\{\begin{array}{cc}-\mathbb{P}\left(\mathcal{U}\le \tau \right)/\beta ,\hfill & \tau <-{\omega }_{\beta }\hfill \\ -1,\hfill & -{\omega }_{\beta }<\tau <0.\hfill \end{array}\right.$$

Remark 14.

Bearing in mind the latter equalities, Theorem 8, and specifically (34) and (35), lead to optimal reinsurance

$$\left\{\begin{array}{c}\left\{\begin{array}{c}\left({\theta }_c/\beta -\left(\left({\theta }_v+{\theta }_E\right)k_1-{\theta }_E\right)\right)\mathbb{P}\left(\mathcal{U}\le \tau \right)+\left({\theta }_v+{\theta }_E\right)k_2{J}^{*}\left({\mathcal{Z}}_{\rho }\right)<0\hfill \\ ⟹x=0,\tau <-{\omega }_{\beta }\hfill \end{array}\right.\hfill \\ \left\{\begin{array}{c}\left({\theta }_c/\beta -\left(\left({\theta }_v+{\theta }_E\right)k_1-{\theta }_E\right)\right)\mathbb{P}\left(\mathcal{U}\le \tau \right)+\left({\theta }_v+{\theta }_E\right)k_2{J}^{*}\left({\mathcal{Z}}_{\rho }\right)>0\hfill \\ ⟹x=-H,\tau <-{\omega }_{\beta }\hfill \end{array}\right.\hfill \\ \left\{\begin{array}{c}{\theta }_v-\left(\left({\theta }_v+{\theta }_E\right)k_1-{\theta }_E\right)\mathbb{P}\left(\mathcal{U}\le \tau \right)+\left({\theta }_v+{\theta }_E\right)k_2{J}^{*}\left({\mathcal{Z}}_{\rho }\right)<0\hfill \\ ⟹x=0,-{\omega }_{\beta }<\tau \hfill \end{array}\right.\hfill \\ \left\{\begin{array}{c}{\theta }_v-\left(\left({\theta }_v+{\theta }_E\right)k_1-{\theta }_E\right)\mathbb{P}\left(\mathcal{U}\le \tau \right)+\left({\theta }_v+{\theta }_E\right)k_2{J}^{*}\left({\mathcal{Z}}_{\rho }\right)>0\hfill \\ ⟹x=-H,-{\omega }_{\beta }<\tau \hfill \end{array}\right.\hfill \\ a=C-〈x,J^{*}\left(k_1+k_2{\mathcal{Z}}_{\rho }\right)〉\hfill \end{array}\right.$$

(51)

along with $x\left(\tau \right)=0$ if $\tau >0$. As indicated in Remark 9, the insurer budget C is not at all relevant, because it only affects the amount a to be invested in the riskless security. The reinsurance contract $J\left(x\right)\left(\mathcal{U}\right)$ remains the same if C is modified. This is a significant difference with respect to previous studies (Albrecher et al., 2017), and this difference is provoked by the incorporation of the riskless asset in the insurer problem. If one only looks for optimal reinsurance contracts, the insurer budget matters and affects the selected contract. If the insurer may combine reinsurance with a riskless asset, the insurer budget becomes irrelevant, in the sense that it has no influence on the selected reinsurance contract.

(51) is totally general, and it shows the existence of an optimal contract $J\left(x\right)\left(\mathcal{U}\right)$ saturating the constraint (23) on the marginal ceded risk. In particular, if $H=1$, $i.e.$, if the reinsurer accepts to pay one more monetary unit for every monetary unit that the insurer’s indemnity payments increase, then there is an optimal contract such that $x\left(\mathcal{U}\right)=-1$ or $x\left(\mathcal{U}\right)=0$ for every potentially realized value of $\mathcal{U}$.

Remark 10 also applies now, $i.e.$, denoting by ${\tilde{x}}_{\beta }$ the solution given in (51), one has that (38) and (39) hold and provide us with upper bounds of $CV@R_{1-s}\left(\mathcal{U}+a+J\left({\tilde{x}}_{\beta }\right)\left(\mathcal{U}\right)\right)$ for every level of confidence (for every $0<s<1$).□

(51) shows that the reinsurer premium principle (and specifically ${\mathcal{Z}}_{\rho }$) has an important role to determine the optimal contract. In a similar way to how it was with the portfolio insurance problem, the particularization of (51) to concrete cases would imply to deal with a tedious and probably useless casuistry that would significantly enlarge the paper. For that reason, let us focus on the most famous premium principle, that is, the $EVPP$.

Remark 15.

Suppose that $\mathbb{L}$ and $\mathbb{P}$ are equivalent on $\left(-\infty ,0\right]$. Then, the cumulative distribution function $F_{\mathcal{U}}:\left(-\infty ,0\right]⟶\left(0,1\right]$ of $\mathcal{U}$ is a continuous and strictly increasing bijection whose inverse $F_{\mathcal{U}}^{-1}:\left(0,1\right]⟶\left(-\infty ,0\right]$ is also continuous and strictly increasing. Furthermore, it is easy to see that

$$F_{\mathcal{U}}^{-1}\left(v\right)=-V@R_{1-v}\left(\mathcal{U}\right)$$

(52)

holds for every $\nu \in \left(0,1\right)$ and $F_{\mathcal{U}}^{-1}\left(1\right)=0$.□

Proposition 16.

Suppose that the reinsurer prices according to the $EVPP$ ($k_1>1$ and $k_2=0$). Suppose also that $\mathbb{P}$ and $\mathbb{L}$ are equivalent in $\left(-\infty ,0\right]$. Then:

$a)$ Suppose that ${\theta }_V=0$. If

$${\theta }_c/\beta >\left(k_1-1\right){\theta }_E$$

then $x=-\mathsf{I}{\mathsf{1}}_{\left(-\infty ,-{\omega }_{\beta }\right)}H$ solves (33), and $x=0$ solves (33) otherwise.

$b)$ Suppose that ${\theta }_V>0$. If

$${\theta }_c/\beta >k_1{\theta }_v+\left(k_1-1\right){\theta }_E$$

(53)

then $x=\mathsf{I}{\mathsf{1}}_{\left(-\infty ,-\gamma \right)}H$ solves (33), where

$$-\gamma =F_{\mathcal{U}}^{-1}\left({\displaystyle \frac{{\theta }_v}{k_1{\theta }_v+\left(k_1-1\right){\theta }_E}}\right)$$

satisfies -$\gamma >-{\omega }_{\beta }$. If (53) fails, then $x=0$ solves (33).

Proof. 

Notice that (51) becomes

$$\left\{\begin{array}{c}\left({\theta }_c/\beta -\left(\left(k_1{\theta }_v+\left(k_1-1\right){\theta }_E\right)\right)\right)F_{\mathcal{U}}\left(\tau \right)<0⟹x=0,\tau <-{\omega }_{\beta }\hfill \\ \left({\theta }_c/\beta -\left(\left(k_1{\theta }_v+\left(k_1-1\right){\theta }_E\right)\right)\right)F_{\mathcal{U}}\left(\tau \right)>0⟹x=-H,\tau <-{\omega }_{\beta },\hfill \\ {\theta }_v-\left(\left(k_1{\theta }_v+\left(k_1-1\right){\theta }_E\right)\right)F_{\mathcal{U}}\left(\tau \right)<0⟹x=0,-{\omega }_{\beta }<\tau ,\hfill \\ {\theta }_v-\left(\left(k_1{\theta }_v+\left(k_1-1\right){\theta }_E\right)\right)F_{\mathcal{U}}\left(\tau \right)>0⟹x=-H,-{\omega }_{\beta }<\tau ,\hfill \end{array}\right.$$

Accordingly, for $\tau \in \left(-\infty ,-{\omega }_{\beta }\right)$ one can take $x=-H$ if (53) holds, and and $x=0$ otherwise, while for $\tau \in \left(-{\omega }_{\beta },0\right)$ one can take $x=-H$ if

$${\theta }_v>\left(\left(k_1{\theta }_v+\left(k_1-1\right){\theta }_E\right)\right)F_{\mathcal{U}}\left(\tau \right)$$

(54)

and $x=0$ otherwise. (54) cannot hold if ${\theta }_V=0$, so $x=0$ if ${\theta }_V=0$ and $\tau >-{\omega }_{\beta }$ Statement $a)$ has been proved. Suppose that ${\theta }_V>0$. If (53) fails one has

$${\displaystyle \frac{{\theta }_v}{k_1{\theta }_v+\left(k_1-1\right){\theta }_E}}\le \beta$$

and therefore (recall (52))

$$-\gamma \le F_{\mathcal{U}}^{-1}\left(\beta \right)=-V@R_{1-\beta }\left(\mathcal{U}\right)=-{\omega }_{\beta }.$$

Besides, (54) would lead to

$${\displaystyle \frac{{\theta }_v}{k_1{\theta }_v+\left(k_1-1\right){\theta }_E}}>F_{\mathcal{U}}\left(\tau \right)$$

and $-\gamma >\tau$. Thus, $\tau <-{\omega }_{\beta }$ and $\tau$ does not belong to $\left(-{\omega }_{\beta },0\right)$. In other words, if (53) fails then (54) fails too for $\tau \in \left(-{\omega }_{\beta },0\right)$ and $x=0$ solves (33). Suppose that (53) holds. Proceeding as above -$\gamma >-{\omega }_{\beta }$, and (54) holds for $\tau \in \left(\mathsf{I}{\mathsf{1}}_{\left(-\infty ,--\gamma \right)}-{\omega }_{\beta },-\gamma \right)$. Hence, $b)$ has been proved.□

Remark 17.

Proposition 16 shows that $x=0$ or $x=-\mathsf{I}{\mathsf{1}}_{\left(-\infty ,-\gamma \right)}H$ for some $0\ge -\gamma \ge -V@R_{1-\beta }\left(\mathcal{U}\right)$ solve (33), depending on the selected weights $\left({\theta }_c,{\theta }_v,{\theta }_E\right)$. The higher the value of ${\theta }_c$, the higher the probability of obtaining a solution of the type $\mathsf{I}{\mathsf{1}}_{\left(-\infty ,-\gamma \right)}H$. If $H=1$, that is, if the reinsurer agrees to pay one more unitary unit per new unitary unit of claim,15 then this solution becomes $x=-\mathsf{I}{\mathsf{1}}_{\left(-\infty ,-\gamma \right)}$, that is, (recall (18))

$$J\left(-\mathsf{I}{\mathsf{1}}_{\left(-\infty ,-\gamma \right)}\right)\left(\mathcal{U}\right)={\displaystyle \underset{\mathcal{U}}{\overset{0}{\int }}}\mathsf{I}{\mathsf{1}}_{\left(-\infty ,-\gamma \right)}\left(\tau \right)d\tau ={\left(-\mathcal{U}-\gamma \right)}^{+}.$$

This is the well-known stop-loss contract with deductible $\gamma \ge 0$. If the global indemnification $-\mathcal{U}$ becomes higher than the deductible γ, the reinsurer will be required to pay the entire excess. Actually, stop-loss contracts are very often optimal in the actuarial literature (Albrecher et al., 2017, Román et al., 2018, Boonen and Jiang, 2024,Aboagye et al., 2025, etc.), so, in this sense, (28) has a natural solution. Nevertheless, there is an interesting difference with respect to previous approaches. The deductible and the insurer budget are independent. If the budget were not enough to buy the optimal contract, the insurer should borrow money. If the budget were higher than the optimal contract premium, the excess should be lent in the financial market.□

5.3. Illustrative Example

Let us deal with an illustrative numerical example beyond the $EVPP$. Suppose that $\mathbb{P}$ and $\mathbb{L}$ are equivalent in $\left(-\infty ,0\right]$, $F_{\mathcal{U}}$ denotes the cumulative distribution function of $\mathcal{U}$ and $f_{\mathcal{U}}$ denotes the density function of $\mathcal{U}$. Notice that $f_{\mathcal{U}}$ is strictly positive (outside a null set) and, as in Remark 15, $F_{\mathcal{U}}^{-1}:\left[0,1\right)\to \left[0,\infty \right)$ is a one to one continuous and strictly increasing bijection satisfying (52). Moreover, under the obvious notation,

$$F_{-\mathcal{U}}\left(\tau \right)=1-F_{\mathcal{U}}\left(-\tau \right).$$

(55)

Suppose that $\rho$ is the Wang’s risk measure (Wang, 2000, or Hamada and Sherris, 2003) given by the distortion function

$$g\left(\tau \right)=\Phi \left(b+{\Phi }^{-1}\left(\tau \right)\right),$$

$0<\tau <1$, $\Phi$ denoting the cumulative distribution function of the standard normal distribution, and $b>0$ being a parameter. Evidently, $g^{\prime }\left(\tau \right)=exp\left(-b^2/2-b{\Phi }^{-1}\left(\tau \right)\right)$ and (recall (52))

$$\rho \left(\mathcal{U}\right)={\displaystyle \underset{0}{\overset{1}{\int }}}V@R_{1-s}\left(\mathcal{U}\right)g^{\prime }\left(s\right)d\tau =-{\displaystyle \underset{0}{\overset{1}{\int }}}{F}_{\mathcal{U}}^{-1}\left(s\right)e^{-b^2/2-b{\Phi }^{-1}\left(s\right)}ds.$$

(56)

Lemma 18.

If ρ is the Wang’s risk measure of parameter $b>0$, then ${\mathcal{Z}}_{\rho }\left(\omega \right)=exp\left(-b^2/2-b{\Phi }^{-1}\left(F_{\mathcal{U}}\left(\omega \right)\right)\right)$ for $\omega <0$.

Proof. 

The change of variable $s=F_{\mathcal{U}}\left(\omega \right)$ in (56) leads to

$$\begin{array}{c}\rho \left(\mathcal{U}\right)=-{\displaystyle \underset{-\infty }{\overset{0}{\int }}}{F}_{\mathcal{U}}^{-1}\left(F_{\mathcal{U}}\left(\omega \right)\right)e^{-b^2/2-b{\Phi }^{-1}\left(F_{\mathcal{U}}\left(\omega \right)\right)}{f}_{\mathcal{U}}\left(\omega \right)d\omega \hfill \\ =-{\displaystyle \underset{0}{\overset{\infty }{\int }}}\omega e^{-b^2/2-b{\Phi }^{-1}\left(F_{\mathcal{U}}\left(\omega \right)\right)}{f}_{\mathcal{U}}\left(\omega \right)d\omega ={\displaystyle \underset{0}{\overset{\infty }{\int }}}\omega \left(-e^{-b^2/2-b{\Phi }^{-1}\left(F_{\mathcal{U}}\left(\omega \right)\right)}\right)\mathbb{P}\left(d\omega \right),\hfill \end{array}$$

where the last equality follows from $f_{\mathcal{U}}=d\mathbb{P}/d\mathbb{L}$. Thus, since $\mathcal{U}$ is the identity map, $\rho \left(\mathcal{U}\right)=\mathsf{I}\mathsf{E}\left(\mathcal{U}\left(\omega \right)e^{-b^2/2-b{\Phi }^{-1}\left(F_{\mathcal{U}}\left(\omega \right)\right)}\right)$.□

Without lost of generality one can accept the equality $\mathsf{I}\mathsf{E}\left(\mathcal{U}\right)=-1$ because the monetary unit may be normalized. Besides, let us assume that $H=1$ and that $-\mathcal{U}$ has a log-normal distribution, $i.e.$, $log\left(-\mathcal{U}\right)$ is normal. Denote $\mathsf{I}\mathsf{E}\left(log\left(-\mathcal{U}\right)\right)=m$ and ${\sigma }^2\left(log\left(-\mathcal{U}\right)\right)={\delta }^2$, where ${\sigma }^2\left(.\right)$ represents “variance”. $\mathsf{I}\mathsf{E}\left(-\mathcal{U}\right)=1$ trivially implies that $m=-{\delta }^2/2$.16 Moreover, $F_{-\mathcal{U}}\left(\tau \right)=\Phi \left(\left(log\left(\tau \right)+{\delta }^2/2\right)/\delta \right)$ and (55) lead to

$$F_{\mathcal{U}}\left(\tau \right)=1-\Phi \left({\displaystyle \frac{log\left(-\tau \right)+{\delta }^2/2}{\delta }}\right)$$

(58)

for $\tau <0$. Let us take $\delta =\sqrt{log\left(5\right)}$, which implies that ${\sigma }^2\left(\mathcal{U}\right)=4$, as shown by (57).

Suppose that the reinsurer prices according to the Wang’s premium principle, $i.e.$, $k_1=0$, $k_2=1$ and $\rho$ is the Wang’s risk measure of (56). Bearing in mind the equality ${\Phi }^{-1}\left(1-\Phi \left(\tau \right)\right)={\Phi }^{-1}\left(\Phi \left(-\tau \right)\right)=-\tau$ for every $\tau \in \mathbb{R}$, Lemma 18 and (58) lead to

$$\left\{\begin{array}{c}{\mathcal{Z}}_{\rho }\left(\tau \right)=exp\left(-b^2/2-b{\Phi }^{-1}\left(F_{\mathcal{U}}\left(\tau \right)\right)\right)=\hfill \\ exp\left(-{\displaystyle \frac{b^2}{2}}-b{\Phi }^{-1}\left(1-\Phi \left({\displaystyle \frac{log\left(-\tau \right)+{\delta }^2/2}{\delta }}\right)\right)\right)\hfill \\ =exp\left(-{\displaystyle \frac{b^2}{2}}+b\left({\displaystyle \frac{log\left(-\tau \right)+{\delta }^2/2}{\delta }}\right)\right)\hfill \\ ={\left(-\tau \right)}^{b/s}exp\left({\displaystyle \frac{b\left(\delta -b\right)}{2}}\right).\hfill \end{array}\right.$$

(59)

if $\tau <0$. Take (for instance) $\beta =0.05$, ${\theta }_C=1$, $0\le {\theta }_V\le 1$ and ${\theta }_E=0.47$. The selected $\beta$ implies that ${\omega }_{\beta }\approx 3.631723227$. Besides, Theorem 8 and (35) show that the solutions of (33) are characterized by the sign of $\Lambda$ in (34). The estimation of $\Lambda$ requires the previous estimation of $J^{*}\left({\mathcal{Z}}_{\rho }\right)$. (19) leads to

$$J^{*}\left({\mathcal{Z}}_{\rho }\right)\left(t\right)=-{\displaystyle \underset{-\infty }{\overset{\tau }{\int }}}{\mathcal{Z}}_{\rho }\left(\omega \right)\mathbb{P}\left(d\omega \right)=-{\displaystyle \underset{-\infty }{\overset{\tau }{\int }}}{\mathcal{Z}}_{\rho }\left(\omega \right)f_{\mathcal{U}}\left(\omega \right)d\omega .$$

Taking derivatives in (58),

$$f_{\mathcal{U}}\left(\tau \right)={\displaystyle \frac{1}{\delta \tau }}\varphi \left({\displaystyle \frac{log\left(-\tau \right)+{\delta }^2/2}{\delta }}\right),$$

$\varphi \left(\omega \right)=e^{-{\omega }^2/2}/\sqrt{2\pi }$ denoting the standard normal density, so (59) leads to

$$J^{*}\left({\mathcal{Z}}_{\rho }\right)\left(\tau \right)={\displaystyle \frac{1}{\delta }}exp\left({\displaystyle \frac{b\left(\delta -b\right)}{2}}\right)\left({\displaystyle \underset{-\infty }{\overset{\tau }{\int }}}{\left(-\omega \right)}^{\left(b-\delta \right)/\delta }\varphi \left({\displaystyle \frac{log\left(-\omega \right)+{\delta }^2/2}{\delta }}\right)d\omega \right),$$

$\tau <0$. In order to simplify some tedious computations, suppose that $b=\delta =\sqrt{log\left(5\right)}$. One has that

$$J^{*}\left({\mathcal{Z}}_{\rho }\right)\left(\tau \right)={\displaystyle \frac{1}{\delta }}{\displaystyle \underset{-\tau }{\overset{\infty }{\int }}}\varphi \left({\displaystyle \frac{log\left(\omega \right)+{\delta }^2/2}{\delta }}\right)d\omega$$

for $\tau <0$, and this integral may be estimated by Monte Carlo simulation. Since Propositions $11a$ and $11d$ enable us to compute $J^{*}\left(1\right)$ and $J^{*}\left({\mathcal{Z}}_{\beta }\right)$, one has that $\Lambda$ in (34) is easily estimated by Monte Carlo simulation too, and the solution (35) may be given explicitly.

Under the given parameters, the solution of (33) is always a stop-loss contract, and Table V below shows some selected numerical examples. Every numerical value has been rounded to the third decimal place and every integral has been estimated with ${10}^5/2$ Monte-Carlo simulations.

$$\left|\begin{array}{c}TableV:Riskandexpectedwealthdependon{\theta }_V\left(Wan{g}^{\prime }sPP\right)\hfill \\ \hfill \begin{array}{cccccc}{\theta }_V\hfill & 1;\hfill & 0.9;\hfill & 0,6;\hfill & 0.3;\hfill & 0\hfill \\ Deductible\hfill & 0.835;\hfill & 0.93;\hfill & 1.405;\hfill & 2.72;\hfill & 3.604\hfill \\ \mathsf{I}\mathsf{E}\hfill & a-1.259;\hfill & a-1.261;\hfill & a-1.257;\hfill & a-1.21;\hfill & a-1.18\hfill \\ V@R_{95\%}\hfill & 1.614;\hfill & 1.683;\hfill & 2.046;\hfill & 3.157;\hfill & 3.956\hfill \\ CV@R_{95\%}\hfill & 1.614;\hfill & 1.683;\hfill & 2.046;\hfill & 3.157;\hfill & 3.956\hfill \end{array}\end{array}\right|$$

For ${\theta }_V=1$ and ${\theta }_V=0.9$ the optimal solutions become the stop-loss reinsurance contracts with deductible $0.835$ and $0.93$, respectively. The risk increment equals $1.683-1,614=0.069$, while the expected wealth increment only equals $1.261-1.259=0.0002$. Consequently, the insurer will probably prefer to chose ${\theta }_V=1$, which is equivalent to the optimization of the couple $\left(CV@R,expected\_wealth\right)$. Actually, the selected sample of Table V illustrates that the risk increment is never compensated by the expected wealth growth. Besides, (7) and (9) lead, respectively, to (recall also Remark 10)

$$CV@R_{1-s}\left(J\left(1+x\right)\right)\le 1.614$$

and

$$CV@R_{1-s}\left(J\left(1+x\right)\right)\le {\displaystyle \frac{0.48}{s}}+0.779$$

for every $0<s<1$. Better upper bounds may be obtained by considering alternative levels of confidence. For instance, by adding ${\beta }_1=0.8$ one obtains (recall (16) and Remark 10)

$$\left\{\begin{array}{c}CV@R_{1-s}\left(J\left(1+x\right)\right)\le \hfill \\ \\ min\left\{1.614;{\displaystyle \frac{0.339}{s}}+0.933;{\displaystyle \frac{0.48}{s}}+0.779\right\}\hfill \end{array}\right.$$

6. Conclusion

The simultaneous optimization of the tail risk and the expected wealth has been studied under comonotononicity, and two important novelties may have been found. On the one hand, the new proposal may permit us to control the downside risk beyond and initially chosen confidence level. On the other hand, the problem has been completely solved and the solution does not depend on the decision maker budget. Since the involved pricing method is quite general and compatible with financial pricing models and actuarial premium principles, the two selected applications involve a financial mathematics problem and an actuarial mathematics one. The financial problem has been the portfolio insurance, and it has been pointed out that usual strategies are not necessarily optimal. Furthermore, if the classic purchase of puts is optimal, the chosen strikes are quite important. This seems to be relevant, at least for practitioners facing a market turmoil. The actuarial example has been the optimal reinsurance problem. The independence between the optimal contract and the insurer budget seems to be quite relevant too, as it differs from the results of previous studies. This difference is caused by the incorporation of a financial riskless asset the can by traded by the insurer at the same time that it purchases a reinsurance contract. Illustrative numerical experiments have been presented.