Portfolio optimization under Solvency II

Abstract

In the current low interest-rate and highly-regulated environment investing capital efficiently is one of the most important challenges insurance companies face. Certain quantitative parts of regulatory requirements (e.g. Solvency II capital requirements) result in constraints on the investment strategies. This paper mathematically describes the implications of Solvency II constraints on the investment strategies of insurance companies in an expected utility framework with a focus on the market risk module. For this constrained expected utility problem, we define a two-step approach that leads to closed-form approximations for the optimal investment strategies. This proposal circumvents the technical difficulties encountered when applying the convex duality approach or the theory of viscosity solutions. The investment strategies found using the two-step approach can be understood as the optimal investment strategies for constraint problems according to Solvency II. The impact of such constraints on the asset allocation and the performance of these strategies is assessed in a numerical case study.

Appendix

1.1 Proof of Lemma 1

Let us assume that the Solvency II capital requirements for the different assets are positive, i.e.

$$\begin{aligned} SCR_i^{mkt}(t)= & {} \pi _i(t)k_i V_t> 0 \ \ \text{ for } i \in \left\{ \text{ equity, } \text{ real } \text{ estate, } \text{ spread } \right\} ,\\ SCR_{interest}^{mkt}(t)= & {} k_{interest} \left( d_L L - d_{interest} \ \pi _{interest} V_t - d_{spread} \ \pi _{spread} V_t \right) > 0 . \end{aligned}$$

The difference between the constraint sets $\tilde{K}(c,t)$ and K(c, t) is the vector $\tilde{v}$ on the right-hand side of the inequality in the definition of the constraint set. Per definitionem it holds, that

$$\begin{aligned} v_1= & {} \frac{L}{V_t}d_L \le (1-c(t)) d_L := \tilde{v}_1\nonumber \\&\text{ with } \tilde{v}:= \left( (1-c(t)) d_L, 0, 0, 0\right) ^T . \end{aligned}$$

(30)

As the square root function is monotonously increasing and a composition of monotonously increasing functions is also monotonously increasing, let us consider the gradient

$$\begin{aligned}&\nabla g(v) \text{ with } g(v):= \left( B \pi (t) + v \right) ^T W C W \left( B \pi (t) + v \right) \\&\quad \nabla g(v) = 2 WCW (B \pi (t) + v). \end{aligned}$$

Since the matrix WCW is positive-definite $\nabla g(v)$ is non-negative if and only if

$$\begin{aligned} B \pi (t) + v \ge 0 \Leftrightarrow \begin{pmatrix} d_L L - d_{interest} \ \pi _{interest}(t) V_t - d_{spread} \ \pi _{spread}(t) V_t \\ \pi _{equity}(t) \\ \pi _{realestate}(t) \\ \pi _{spread}(t) \\ \end{pmatrix} \ge \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix}\nonumber \\ \end{aligned}$$

(31)

As we assumed that all $SCR_i$ are positive, which is equivalent to inequality (31), the function

$$\begin{aligned} f(v):= \sqrt{\left( B \pi (t) + v \right) ^T W C W \left( B \pi (t) + v \right) } \end{aligned}$$

is monotonously increasing in v as a composition of monotonously increasing functions. Hence, using inequality (30), it holds

$$\begin{aligned} f(v) \le f(\tilde{v}). \end{aligned}$$

We can follow that

$$\begin{aligned} c(t) \ge f(\tilde{v}) \Rightarrow c(t) \ge f(v) , \end{aligned}$$

which concludes the statement

$$\begin{aligned} \pi \in \tilde{K}(c,t) \Rightarrow \pi \in K(c,t). \end{aligned}$$

1.2 Proof of Lemma 2

The support function of a convex set is defined as $\delta (x,c,t):=\sup _{\pi \in K(c,t)}(-\pi ^Tx)$. Let $t \in \left[ 0,T\right]$ and $c(t)>0$ be fixed. Let x be inside the barrier cone $X_{\tilde{K}}=\mathbb {R}^N$. We consider the Lagrangian function

$$\begin{aligned} L(\pi ,x)=-x^T \pi + l \left( c(t)^2- \left( B \pi + \tilde{v} \right) ^T WCW \left( B \pi + \tilde{v} \right) \right) \end{aligned}$$

with the Lagrange multiplier $l \ge 0$. The two conditions are derived as

$$\begin{aligned} \nabla _{\pi }L(\pi ,x)=-x^T-2l \left( B \pi + \tilde{v} \right) ^T WCW B= 0 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial L}{\partial l}(\pi ,l)=c(t)^2- \left( B \pi + \tilde{v} \right) ^T WCW \left( B \pi + \tilde{v} \right) =0. \end{aligned}$$

From the first equation we obtain

$$\begin{aligned} \pi = B^{-1} \left( -\frac{1}{2l} (WCW)^{-1} (B^T)^{-1} x - \tilde{v} \right) , \end{aligned}$$

where we implicitly used $l>0$. The second condition then becomes

$$\begin{aligned} c(t)^2- \left( -\frac{1}{2l} (WCW)^{-1} (B^T)^{-1} x \right) ^T WCW \left( -\frac{1}{2l} (WCW)^{-1} (B^T)^{-1} x \right) = 0 \end{aligned}$$

and thus

$$\begin{aligned} l=\frac{1}{2c(t)}\left( x^T B^{-1} (WCW)^{-1} (B^T)^{-1} x\right) ^{\frac{1}{2}} \end{aligned}$$

Now we can conclude again using the symmetry of $\left( WCW \right) ^{-1}$ that

$$\begin{aligned} \delta (x,c,t)=\sup _{\pi \in K(c,t)}(-\pi ^Tx)= c(t) \sqrt{x^T B^{-1} (WCW)^{-1} (B^T)^{-1}x} + x^T B^{-1}\tilde{v}. \end{aligned}$$

1.3 Proof of Example 1

First we compute the support function of the set $K(c,t)=\left\{ \pi (t): c(t) \ge |\pi (t) k| \right\}$ by considering two cases.

$$\begin{aligned} \delta (c,x,t):=\sup _{\pi \in K(c,t)}(-\pi x) \end{aligned}$$

Case 1: $x \ge 0$

$$\begin{aligned} \sup _{\pi \in K(c,t)}(-\pi x)=\frac{c(t)}{k}x \end{aligned}$$

Case 2: $x \le 0$

$$\begin{aligned} \sup _{\pi \in K(c,t)}(-\pi x)=-\frac{c(t)}{k}x \end{aligned}$$

Combining the two cases we see

$$\begin{aligned} \delta (c,x,t)=\frac{c(t)}{k} | x | \end{aligned}$$

The optimal dual is calculated by considering the same two cases.

Case 1: $x \ge 0$

$$\begin{aligned} \lambda ^*(c,t)= \text {arg}\inf \limits _{x \in X_{K}} \left\{ c(t) \frac{x}{k}+\frac{1}{2}\left( \tilde{\gamma }+\frac{x}{\sigma }\right) ^2(\rho +1)\right\} \end{aligned}$$

In order to find the infimum, the derivative of $f(x):= c(t) \frac{x}{k}+\frac{1}{2}(\tilde{\gamma }+\frac{x}{\sigma })^2(\rho +1)$ is set equal to 0

$$\begin{aligned}&\frac{c(t)}{k}+ \left( \tilde{\gamma }+\sigma ^{-1}x\right) \sigma ^{-1} (\rho +1) = 0\\&\quad \Rightarrow x= - \frac{c(t) \sigma ^2}{k (\rho +1)}-(\mu -r) \end{aligned}$$

As we defined $c(t) \ge 0$ and $\mu \ge r$ it holds $x<0$, which violates the assumption of Case 1. As the value function $c(t)\frac{x}{k}+\frac{1}{2}( \tilde{\gamma }+\frac{x}{\sigma })^2(\rho +1)$ is strictly increasing for $x>- \frac{c(t) \sigma ^2}{k (\rho +1)}-(\mu -r)$ the solution satisfying the condition of Case 1 is found to be $x=0$.

Case 2: $x \le 0$

$$\begin{aligned} \lambda ^*(c,t)= \text {arg} \inf \limits _{x \in X_{K}} \left\{ -c(t) \frac{x}{k}+\frac{1}{2}\left( \tilde{\gamma }+\frac{x}{\sigma }\right) ^2(\rho +1)\right\} \end{aligned}$$

In order to find the infimum, the derivative of $f(x):= -c(t) \frac{x}{k}+\frac{1}{2}(\tilde{\gamma }+\frac{x}{\sigma })^2(\rho +1)$ is set equal to 0

$$\begin{aligned}&-\frac{c(t)}{k}+\left( \tilde{\gamma }+\sigma ^{-1}x\right) \sigma ^{-1} (\rho +1) = 0\\&\quad \Rightarrow x= \frac{c(t) \sigma ^2}{k (\rho +1)}-(\mu -r) \end{aligned}$$

This solution fulfills the condition of Case 2, if

$$\begin{aligned} \frac{c(t) \sigma ^2}{k (\rho +1)} \le (\mu -r) \end{aligned}$$

If this does not hold true the optimal solution is again $x=0$ as the value function is strictly decreasing for $0 \le \frac{c(t) \sigma ^2}{k (\rho +1)}-(\mu -r)$.

Combining the two cases, it holds

$$\begin{aligned} \lambda ^*(c,t)= min \left( \frac{c(t)(1-\alpha ) \sigma ^2}{k}- (\mu - r),0 \right) \end{aligned}$$

Using the result from Chapter 2, the optimal investment strategy is found as

$$\begin{aligned} \pi ^*_{\lambda ^*}(c,t)= & {} \frac{1}{1-\alpha }(\sigma )^{-1} \tilde{\gamma }_{\lambda ^*}= \frac{1}{1-\alpha }(\sigma )^{-1} ( \tilde{\gamma }(t)+\sigma ^{-1}\lambda ^*(c,t))\\= & {} \frac{1}{1-\alpha } (\sigma ^2)^{-1}( \mu - r + \lambda ^*(c,t) )\\= & {} \frac{1}{1-\alpha } (\sigma ^2)^{-1}\left( \mu - r + min \left( \frac{c(1-\alpha ) \sigma ^2}{k}- (\mu - r),0 \right) \right) \\= & {} min \left( \frac{c(t)}{k}, \frac{1}{1-\alpha } \frac{\mu -r}{\sigma ^2} \right) . \end{aligned}$$

1.4 Proof of Proposition 3

In Sect. 3 it has been shown, that for a constant c the wealth process $V^*_t(c)$ behaves like a Geometric Brownian motion with constant drift and diffusion

$$\begin{aligned} dV_t^*(c)= & {} V_t^*(c) \left[ ((\pi ^*(c))^T (\mu -r)+r) dt + (\pi ^*(c))^T \sigma dW_t \right] . \end{aligned}$$

Due to the exponential structure of the solution of the SDE, the minimum of such a Geometric Brownian motion can be written as

$$\begin{aligned} min V^*_{t}(c) = V^*_{0}(c) e^{min R_t} = V_0 e^{min R_t} , \end{aligned}$$

where $R_t= \tilde{\mu }t + \tilde{\sigma }W_t$ is a Wiener process with drift $\tilde{\mu }$ and diffusion $\tilde{\sigma }$

$$\begin{aligned} \tilde{\mu }= ((\pi ^*(c))^T (\mu -r)+r) - \frac{(\Vert (\pi ^*(c))^T \sigma \Vert )^2}{2} \quad \text{ and } \quad \tilde{\sigma }= \Vert (\pi ^*(c))^T \sigma \Vert . \end{aligned}$$

Analyzing the distribution of the RALC process, we find

$$\begin{aligned}&\mathbb {P} \left[ \underset{0 \le t \le T}{min} \frac{V^*_{t}(c)-L}{V^*_{t}(c)} \ge c \right] = \mathbb {P} \left[ \underset{0 \le t \le T}{min} \left( 1-L \frac{1}{V^*_{t}(c)}\right) \ge c \right] \\&\quad = \mathbb {P} \left[ 1- L \underset{0 \le t \le T}{max} \left( \frac{1}{V^*_{t}(c)}\right) \ge c \right] \\&\quad = \mathbb {P} \left[ \underset{0 \le t \le T}{max} \left( \frac{1}{V^*_{t}(c)}\right) \le \frac{1-c}{L} \right] = \mathbb {P} \left[ \frac{1}{\underset{0 \le t \le T}{min} \left( V^*_{t}(c)\right) } \le \frac{1-c}{L} \right] \\&\quad =\mathbb {P} \left[ \frac{L}{1-c} \le \underset{0 \le t \le T}{min} V^*_{t}(c) \right] \\&\quad =1-\mathbb {P} \left[ \underset{0 \le t \le T}{min} V^*_{t}(c) \le \frac{L}{1-c} \right] =1-\mathbb {P} \left[ V_0 e^{min R_t} \le \frac{L}{1-c} \right] \\&\quad =1-\mathbb {P} \left[ \underset{0 \le t \le T}{min} R_t \le \log \left( \frac{L}{1-c} \frac{1}{V_0}\right) \right] \\&\quad = 1-N \left( \frac{\log \left( \frac{L}{(1-c)V_{0}} \right) - \tilde{\mu }T}{\tilde{\sigma } \sqrt{T}}\right) - e^{2\frac{\log \left( \frac{L}{(1-c)V_{0}}\right) \tilde{\mu }}{\tilde{\sigma }^2}} N \left( \frac{\log \left( \frac{L}{(1-c)V_{0}}\right) +\tilde{\mu }T}{\tilde{\sigma } \sqrt{T}}\right) , \end{aligned}$$

where the distribution function of the running minimum of $R_t$ is taken from Björk (2009), p. 209.