PORTFOLIO OPTIMIZATION WITH REINSURANCE

PORTFOLIO OPTIMIZATION WITH REINSURANCE INVESTMENT STRATEGY H. C. JIMBO, I. S. NGONGO, AND C. AWONO ONANA Abstract. This paper deals with a portfolio optimization problem in investment - reinsurance set up. Here we seek to minimize the ruin probability over investment horizon. Assuming that we invest both in risk free and risky assets and reinsurance plays the role in total risk reduction, we shall derive the optimal reinsurance - investment strategy and present some analytical results and interesting findings.

1. Introduction In the study of the continuous time portfolio optimization problem of an insurer various investment strategies are often presented. Reinsurance is used for risk spreading at time T < +∞. The insurance risk is modelled a classical risk process and the discounted risk process is given by the dynamical model Z (1.1)

N (t)

t

cs ds −

R(t) = x + 0

X

Xi DTi

i=1

x is the initial reserve, 

N (t)



X

Xi DTi 

 i=1

t∈[0, T ]

denotes the discounted claims process, (Nt )t∈[0, T ] is a counting process and (Xi )i∈N is a sequence of i.d.d. random variables. The variable (Xi )i∈N represents the size of the i-th claim. The discounted factor to time Ti of the i-th claim is denoted by DTi defined to be strictly positive as in Ng et al (2004). If we assume that part of the initial reserve is used for investment in a stock index, without loss of generality we can assume that the price of the riskless asset is constant and equal to one. The discounted stock index is modelled by the diffusion process. (1.2)

dSt = St (bdt + σdWt )

2010 Mathematics Subject Classification. 60G35, 90A09, 90A43. Key words and phrases. Optimal investment, portfolio optimization, investment strategy, optimal strategy, reinsurance, risk reduction, reinsurance, ruin probability. 1

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H. C. JIMBO, I. S. NGONGO, AND C. AWONO ONANA

b ∈ R, σ > 0 are constant and the correlation between stock market and bound market is modelled by assuming that the process w driving the market is given by w = ρw1 + (1 − ρ)w2

(1.3)

Here as usual ρ ∈ (−1, +1), (w1 , w2 ) is a standard two dimensional Brownian motion and w2 is independent of the risk dynamic. As the insurer follows the investment strategy Θ in the risky asset and uses part of the initial reserve z0 for investment in the stock index, Thus obtain a capital gain given by (1.4)

Zt = z 0 +

Z tY 0

Z tY dS = z0 + (bSu dt + σSu dWt ) 0

u

u

and the total wealth of the insurer at time t ∈ [0, T ] discounted to time 0 is (1.5)

Y Y ( , t) = Rt − z0 + Zt

which gives

(1.6)

Z t N (t) X Y Y ( , t) = x − z0 + cs ds + χt − Xi DTi 0

i=1

Next, we introduce the ruin probability to the problem. Assuming that the insurer has the utility function U , the ultimate goal for the insurer is to maximize the expected utility of wealth at the end of the time horizon. (for example at the end of the financial year). Proposition 1.1. During this time period, the insurer can borrow unlimited funds at risk free rate to pay the claims. However, at the end of the time period, the insurer will have to publish the company report and is also concerned with the ruin probability. Proposition 1.2. The ruin probability of the insurer can be incorporated into the optimization problem for a given confidence level . Proposition 1.3. Assuming the total wealth to be collected only at time T, the end of the investment period, we separate the initial wealth at time t = 0 into a part z0 usable for the investment and the remaining part x−z0 ≥ 0 that can be used as a safety cushion for the insurance claims. Such separation is assumed to be given and can have a legal motivation s overall unaffected by the discounting. Q Proposition 1.4. The condition P {Y ( , t) ≤ 0} ≤ α is overall unaffected by the discounting.

PORTFOLIO OPTI. WITH REINSURANCE - INVEST. STRATEGY

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Our paper is organized as follows: In section two we introduce the reinsurance portfolio optimization problem; in section three we develop an approach to solution, we conclude this work in section four with a short conclusion. 2. Reinsurance portfolio optimization Portfolio optimization with constraints for investments in a setting without claims are considered for example in Emmer et al. (2001), Bask and Shapiro (2001), and Gandy (2005) investigated a situation where finding an optimal investment sky asset is driven by a geometric Wiener process, Emmer and Kluppelberg (2004). The problem of finding the optimal investment strategy that maximize and /or minimize the ruin probability has been studied by many authors, among which Paulsen (1998) Browne (1995) and Paulsen (2002) who estimated ruin probability in different settings; Yuen et al. (2004) introduced ruin probability models with stochastic investment return. We have to acknowledge that maximizing the expected utility of insurer over all investment strategies with some constraint on ruin probability is not an easy work. Some authors have considered various subclasses of investment strategies and choose the supposed must likely acceptable such class of investment strategies for solving the optimization problem. 2.1. Problem setup. We investigate the following portfolio-investment strategy with ruin probability and given confidence level α

(2.1)

h n Y oi max E U Y ,t Q ∈Θ Q Subject to P (Y ( , t) ≤ 0) ≤ α

The problem of finding the optimal investment strategy that maximizes the expected utility and/or minimizes the ruin probability has been investigated by different authors, among which Browne (1995), Hipp and Plum (2000) and Gaier et al. (2003). It was only in 2005 that Yuen et al investigate ruin probability in model with stochastic investment return. We shall always assume the square integrability of predictability of the investment process although it may not be easy especially when we have some constraint on ruin probability. We shall also assume that such suitable class exists and limit ourselves to a class of investment strategies, where a constant proportion of capital is been invested in risky asset in a continuous manner. We shall finally notice that the term proportion means that over time the investment activities separate itself from the insurance cash flows. Such class of investment strategy is shown to be optimal in different settings and under different optimality criteria (see for example Browne (1995, 1997) and Gaier et al (2003)). Since the total wealth is measured only at the end of each investment period (time horizon) we shall separate the initial wealth z0 and the remaining x − z0 used as safety buffer for insurance claim as per Solvency II.

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(2.2)


h n Y oi E U Y ,t max Q ∈Θ Q Subject to sup P (Y ( , t) ≤ 0) ≤ α

Several authors have solved problem (2.2) under various situations. For example (Emmer et al (2001), Basak and Shapiro and Gandy (2005) investigated when the risky asset follows a geometric Wiener process; Emmer and Kupperberg (2001 and 2004) used Gaussian approximation. The problem of finding the optimal investment that minimizes the ruin probability or maximizer the expected utility in various setup has been studied by Hipp and Plumb (2003), Gaier et al. (2003), Paulsen (1998 and -2000) and Yuen et al. (2004). These authors investigated situations with ruin probability models and stochastic return assuming that all investment strategies where predictable thus square-integrable. In this paper we will only consider the specific class of investment strategy with constant proportion of capital gain which is invested into risky asset. 2.2. Insurance vs Reinsurance. The reinsurer must pay a compensation to insurer when loss occurs. Assuming that the insurer has some preference in the strategy, we will formulate the optimization problem taking into account the ruin probability. Part of the initial reserve is invested in stock index and another part of the initial reserve is invested in the stock market with appropriate investment strategy in risky asset.

Figure 1. Chart flow

3. Approach to solution The constant proportion is linked to the wealth process Z starting with initial wealth of class Θp of investment strategies with a constant proportion of capital gain is continuously invested in risky assets z0 ≤ x. As the

PORTFOLIO OPTI. WITH REINSURANCE - INVEST. STRATEGY

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investment activities are separated from the insurance cash flows, this will have some consequences on capital gain. Although the constant proportion strategy is ubiquitous in portfolio optimization (see Merton 1971) but still there are optimal for various types of investment problems. The constant proportion strategy for investment problems with constraints are can be optimal under various optimality criteria (see for example Giaer et al 2003 and Browne 1997). There are many other existing strategies but we will not mention them in this note. The total wealth is considered to be collected at the time horizon representing the end of the investment period. We will separate the initial wealth into two parts: a part z0 ≤ x which is used for the investment and a remaining part x − z0 ≥ 0 which is used as buffer for insurance claims. In such condition the solution to problem (2.2) exists and is finite. For a class of proportion investment strategy and concave utility function, the solution exists if and only if the constraints are fulfilled. The ruin probability is continuous as a function of investment strategy, the ruin constraint α is chosen appropriately and the optimization is done on nonempty and compact set. Assuming that the ruin probability is monitored continuously for a given confidence level α, problem (2.2) is reformulate using the extended argument as follows: Y Y sup P Y ,t ≤0 ≤P inf Y ,t ≤0 t∈[0, T ]

t∈[0, T ]

This does not imply that P

Y inf Y , t ≤ 0 is bounded, but in-

t∈[0, T ]

sured that the ruin probability is controlled pointwise over the entire time period [0, T ]. Lemma 3.1. Assume that for a given Q confidence level α < Bi there exist a strategy π ∈ Θi such that P {(Y ( , t) ≤ 0)} < α. Let U denote the utility function of insurer. If the expected utility of the terminal wealth E [U {Y (π, T )}] is continuous in π ∈ R or if E [U {Y (π, T )}] is monotone increasing in π, then there exists a finite solution π ∗ of optimization problem (2.2) in the class of investment strategies. Proof. We shall prove this fact in five steps. Q Step 1. We shall realize that the ruin probability P (Y ( , t) ≤ 0) is continuous in π ∈ R and α < Bi . Q Step 2. Realize that the set S = {π ∈ R : P (Y ( , t) ≤ 0) < α} is a compact subset of R. Step 3. Assume that the S is non-empty. Step 4. Assume that E[U {Y (π, T )}] is monotone increasing in π. Step 5. Thus an optimal solution to (2.2)Qexists and is finite. Step 6. Finally π ∗ = sup {π ∈ Θ : P (Y ( , t) ≤ 0) < α}.

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Lemma 3.2. Assuming that the utility function U is such that E[U {Y (π, T )}] is continuous in π ∈ R or monotone increasing in π. For the ruin probability α ∈ [P {RT ≤ 0}, P {RT ≤ z0 }] there exists a finite solution π ∗ ∈ Θp to the Q optimization problem (2.2): π ∗ = sup {π ∈ Θ : P (Y ( , t) ≤ 0) < α}. Proof. There exists an admissible finite strategy π ∈ R such that n Y o P Y ,t ≤0