a hybrid estimate for the finite-time ruin probability in...

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The North American Actuarial Journal (NAAJ) 2012 – Volume 16 Issue 3

378

A HYBRID ESTIMATE FOR THE FINITE-TIME RUINPROBABILITY IN A BIVARIATE AUTOREGRESSIVE

RISK MODEL WITH APPLICATION TO PORTFOLIOOPTIMIZATION

Qihe Tang* and Zhongyi Yuan†

ABSTRACT

Consider a discrete-time risk model in which the insurer is allowed to invest a proportion of itswealth in a risky stock and keep the rest in a risk-free bond. Assume that the claim amounts withinindividual periods follow an autoregressive process with heavy-tailed innovations and that the log-returns of the stock follow another autoregressive process, independent of the former one. Wederive an asymptotic formula for the finite-time ruin probability and propose a hybrid method,combining simulation with asymptotics, to compute this ruin probability more efficiently. As anapplication, we consider a portfolio optimization problem in which we determine the proportioninvested in the risky stock that maximizes the expected terminal wealth subject to a constraint onthe ruin probability.

1. INTRODUCTION

Classical risk theory may be questioned on two principal grounds. First, it often assumes that claimsizes are independent and identically distributed (i.i.d.) random variables. This assumption is far un-realistic because claims from the same insurance policy during successive reference periods or claimsfrom different policies during the same reference period are generated in very similar situations, and,hence, they should be dependent on each other. Second, it is only recently that researchers have startedto account for investments while modeling insurance business. At this stage of the study, some strongand unrealistic conditions have to be assumed on the investment portfolio. For instance, log-returnrates are often assumed to follow a multivariate normal distribution. This assumption is dubious be-cause data in insurance and finance often show pronounced asymmetry and heavy-tailedness, bothviolating the normality assumption. In particular, the multivariate normal model implies pairwiseasymptotic independence, while data in insurance and finance often show strong asymptotic depend-ence, indicating a much riskier situation than described by the multivariate normal model. See Bing-ham et al. (2002, 2003) for related discussions.

As a consequence, there is a critical need for modifications that can reflect the skewness, heavy-tailedness, and asymptotic dependence of insurance and financial risks. In the absence of such modi-fications, the study could end up with naive solutions or potentially fatal mistakes.

We propose a bivariate autoregressive (AR) risk model in discrete time, which enables us to accom-modate the modifications mentioned above. In this model, one-period claim amounts are assumed tofollow an AR process with i.i.d. heavy-tailed innovations. The insurer is allowed to make investments in

* Qihe Tang is Professor of Actuarial Science at the Department of Statistics and Actuarial Science, University of Iowa, 241 Schaeffer Hall, IowaCity, IA 52242, [email protected].† Zhongyi Yuan is Ph.D. candidate in Statistics with concentration in Actuarial Science/Financial Mathematics at the Department of Statisticsand Actuarial Science, University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, [email protected].

A HYBRID ESTIMATE FOR THE FINITE-TIME RUIN PROBABILITY IN A BIVARIATE AUTOREGRESSIVE RISK MODEL 379

a discrete-time financial market consisting of a risk-free bond and a risky stock. The log-return ratesof the risky stock are assumed to follow another AR process, independent of the former one, with i.i.d.,but not necessarily normal, innovations. To keep the paper short, we assume that both AR processesare of order 1, but we would like to point out that our study can be extended to AR models of higherorder or even to autoregressive moving average (ARMA) models.

AR (or, more generally, ARMA) models have been extensively applied to actuarial science in theliterature. In this regard we refer the reader to the following references, among many others. Gerber(1981, 1982) used AR and ARMA structures for one-period net gains (premiums minus claims). Wilkie(1986, 1987) discussed in great detail the relevance of mean-reverting AR structures in stochasticinvestment models. Mikosch and Samorodnitsky (2000) modeled one-period net losses by a two-sidedlinear process and studied the asymptotic behavior of the ultimate ruin probability for the heavy-tailedcase. Cai (2002) assumed that the return rates of a risk-free investment possess an AR structure. Yangand Zhang (2003) modeled claim amounts by an AR process under a constant risk-free rate. The bookHardy (2003) contains a short review of AR models in the risk management context.

Our idea of assuming a bivariate discrete-time risk model for the stochastic structures of both claimamounts and investment returns stems from the works of Nyrhinen (1999, 2001) and Tang and Tsit-siashvili (2003a, 2004). These authors earlier considered ruin problems in the presence of both insur-ance processes and investment returns in a discrete-time setting but under different assumptions. Thestudy of ruin in a continuous-time setting that integrates insurance processes with investment returnshas a long history. We refer the reader to Norberg (1999), Frolova et al. (2002), Kalashnikov andNorberg (2002), Paulsen (2008), Chen (2011), and Hao and Tang (2012), to name a few. Chapter VIIIof Asmussen and Albrecher (2010) contains a review of the study in both continuous-time and discrete-time models.

The contribution of this paper is threefold. First, we derive an asymptotic formula for the finite-timeruin probability, which quantitatively captures the impact of dependence of the claim amounts and ofthe investment returns. Second, we propose a hybrid method to compute the finite-time ruin proba-bility. This method makes a compromise between the crude Monte Carlo estimate and the asymptoticestimate, and it effectively overcomes their deficiencies. Finally, as an application, we consider a port-folio optimization problem in which we determine the proportion invested in the stock that maximizesthe expected terminal wealth of the insurer subject to a solvency constraint.

Two possible extensions of this study were proposed by anonymous referees. The first is to modelclaim amounts and investment returns by nonlinear AR models, such as threshold autoregressive mod-els; for details of threshold autoregressive models, see the pioneering work by Tong and Lim (1980)and a recent revisit by Tong (2011). The second is to introduce dependence between the two ARprocesses. We shall pursue these interesting and promising extensions in a future project.

The rest of the paper consists of five sections. Section 2 elaborates on the bivariate AR risk model,Section 3 explains our idea of the hybrid method in a general setting, Section 4 first derives anasymptotic estimate for the finite-time ruin probability and then applies the hybrid method to improvethe computational accuracy and efficiency, Section 5 is devoted to an application of our result andmethod to a portfolio optimization problem, and Section 6 is composed of proofs.

2. THE BIVARIATE AUTOREGRESSIVE RISK MODEL

Consider the following discrete-time insurance risk model. Denote the claim amount (plus other ex-penses) paid by an insurer within period i by a nonnegative random variable Bi, i � �, and assume thatthese claim amounts form an AR process of order 1, written as AR(1). Formally, starting with a deter-ministic value B0 � 0, they evolve according to

B � �B � X , i � �, (2.1)i i�1 i

where the autoregressive coefficient � takes value in [0, 1) and the innovations Xi, i � �, are i.i.d.copies of a nonnegative random variable X. Denote by �X the mean of X and by �B the mean of the

380 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 16, NUMBER 3

stationary solution B�. An advantage of the AR(1) model is that it can capture asymptotic dependencebetween claim amounts; see Proposition 6.1.

For simplicity, we assume that the claims are paid out at the end of each period and that the premiumamounts during these periods are a deterministic constant a � 0. Then the one-period net losses areBi � a, i � �. In insurance practice, premiums are usually collected at the beginning of each period.We have ignored insignificant value changes due to investments on premiums.

Note that in (2.1), to ensure the nonnegativity of the claim amounts, the autoregressive coefficient� is not allowed to be negative. This is a slight flaw of the AR(1) process (2.1) from the modeling pointof view despite the fact that a negative autoregressive coefficient is not usual for claim amounts. SeeYang and Zhang (2003) for a similar remark. Therefore, it seems more reasonable to assume an au-toregressive structure directly for real-valued net loss variables i � �, that is,B ,i

˜ ˜ ˜B � � � �(B � � ) � X , i � �, (2.2)˜ ˜i B i�1 B i

where the autoregressive coefficient � takes value in (�1, 1), the innovations i � �, are i.i.d. real-X ,ivalued random variables with mean 0, and the constant is the mean of the stationary solution ˜� B .B �

In this paper we follow the style of (2.1) to model claim amounts, but we remark that our study canbe easily extended to the setting of (2.2).

Suppose that there is a discrete-time financial market consisting of a risk-free bond with a deter-ministic continuously compounded rate of interest r � 0 and a risky stock with a stochastic log-returnrate Ri � � during period i, i � �. These log-return rates are also assumed to follow an AR(1) process.Formally, starting with a deterministic value R0, they evolve according to

(R � � ) � �(R � � ) � Y , i � �, (2.3)i R i�1 R i

or, equivalently,

R � d � �R � Y , i � �, (2.4)i i�1 i

where the autoregressive coefficient � takes value in (�1, 1), the innovations Yi, i � �, are i.i.d. copiesof a real-valued random variable Y with mean 0, the constant �R is the mean of the stationary solutionR�, and the constant d is equal to (1 � �)�R. Assume that {X, Xi, i � �} and {Y, Yi, i � �} are mutuallyindependent, and, hence, so are the two AR(1) processes (2.1) and (2.3).

Suppose that, at the beginning of each period i, the insurer invests a fixed proportion � � [0, 1] ofits current wealth in the stock and keeps the rest in the bond. We remark that the proportion � isassumed to be fixed just for ease of presentation and the extension to a time-dependent investment israther straightforward. Denote by Wm the insurer’s wealth at time m � �. Then with a deterministicinitial value W0 � x � 0, the wealth process {Wm, m � �} evolves according to

r RmW � ((1 � �)e � �e )W � (B � a), m � �. (2.5)m m�1 m

Iterating (2.5) yieldsm m m

r R r Rj jW � x ((1 � �)e � �e ) � (B � a) ((1 � �)e � �e ), (2.6)� � �m ij�1 i�1 j�i�1

where, and throughout this paper, multiplication over an empty index set produces value 1 by conven-tion. Note that investments with a time-invariant � � [0, 1] often appear in continuous-time finance;see, for instance, Merton (1990).

Let n � � be the time horizon of interest. As usual, the probability of ruin by time n is defined as

�(x; n) � Pr min W � 0 �W � x .� �m 00�m�n

To comply with certain risk reserve regulations such as EU Solvency II, the insurer has to hold enoughrisk reserve, the so-called Solvency Capital Requirement, so that it is able to meet its future obligations


with a high probability. This makes the notion of ruin theory particularly relevant. Mathematically, therisk reserve x should be large enough so that the ruin probability satisfies

�(x; n) � 1 � q (2.7)

for some 0 � q � 1 close to 1, for instance, q � 0.995. Therefore, an accurate estimate for the ruinprobability is crucially important for determining the Solvency Capital Requirement.

Introducei 1r � (1 � �)e and � , i � �, (2.8)�i Rj � �ej�1

where i can be interpreted as the overall stochastic discount factor over the time interval (0, i]. Thenwe have

m m mR Rj j�(x; n) � Pr min x ( � �e ) � (B � a) ( � �e ) � 0� � �� i

0�m�n j�1 i�1 j�i�1

m

� Pr min x � (B � a) � 0�� i i0�m�n i�1

m

� Pr max (B � a) � x . (2.9)�� i i1�m�n i�1

At the core of this work is a hybrid method that we create to compute the ruin probability �(x; n),as is shown in the next two sections.

3. THE HYBRID METHOD

In this section we expound our idea of the hybrid method in a general setting. Suppose that we are toestimate

u(x) � E[U(x)], x � �,

where U(x) is a stochastic quantity presumably associated with a rare event so that u(x) → 0 as x →�. A prototype of u(x) is the tail probability Pr(L � x) of a loss variable L, where U(x) corresponds to

the indicator function of (L � x). Such a quantity is important in risk management because1 ,(L�x)

most risk measures, such as Value-at-Risk and Conditional Tail Expectation, target the tail area of theunderlying risk variable. Because of the complex stochastic structure of U(x), very often it is notpossible to obtain the exact value of u(x), and an estimate will be needed.

A common way to estimate u(x) is to use the crude Monte Carlo (CMC) simulation, which gives anestimate as

N1u (x) � U (x),�1 kN k�1

where Uk(x), k � 1, . . . , N, are i.i.d. samples from U(x) and N is the sample size. The CMC simulationis indeed a powerful method prevailing in various computational problems in insurance and finance.However, taking the cost of simulation into consideration, we find that the estimate u1(x) in manycases is efficient only when the true value of u(x) is not too small, or, equivalently, when x is not toolarge.

Let us demonstrate this by considering how many samples are needed for u1(x) to achieve a requiredaccuracy. In statistical terms, accuracy is usually characterized as follows. For some h � 0 close to 0and 0 � p � 1 close to 1, the simulated estimate u1(x) is within 100h% of the true value u(x) withprobability not smaller than p; that is,


Pr( �u (x) � u(x) � � h �u(x) �) � p. (3.1)1

Throughout this paper, we always choose h � 0.05 and p � 0.95.Consider again the example of estimating the tail probability u(x) � Pr(L � x) of a loss variable L.

By the central limit theorem, approximately (3.1) means that

hu(x)� z , (3.2)p

1u(x)(1 � u(x))�N

or, equivalently,2z 1 � u(x)pN � , (3.3)� �h u(x)

where zp � ��1((1 � p)/2) is the quantile of the standard normal distribution at (1 � p)/2. Thus, thesample size N needed for the required accuracy depends heavily on the value of u(x). To offset thenegative effect of u(x) being extremely small, which often arises in rare event simulation, the samplesize N has to be extremely large. For instance, for u(x) � 10�6, according to (3.3) we need as manyas 1.5 � 109 samples to obtain the required accuracy. For simulations of such a scale, both compu-tational time and memory allocation will be big issues. Related discussions can be found in Asmussenand Glynn (2007).

Another question to ask is, given an affordable sample size N, for what values of u(x) � Pr(L � x)the simulated estimate u1(x) can attain the required accuracy. This can be answered also by the for-mulation in (3.2). Straightforwardly, the range for u(x) is

2 �1hu(x) � N � 1 � u . (3.4)� � � � 0zp

As an example, for N � 500,000 the simulated estimate u1(x) meets the required accuracy ifu(x) � 0.003, but it does not if u(x) � 0.003.

Whenever the CMC method breaks down for small values of u(x), or, equivalently, for large values ofx, the asymptotic method emerges naturally as an alternative. There are variance reduction methods,such as importance sampling and antithetic variables, which can help improve the accuracy of MCestimation. In this paper, however, our focus is on the asymptotic method. By this method, one canusually hope for a transparent analytical formula u2(x) such that

u (x)2lim � 1.u(x)x→�

This means that u2(x) will be an accurate estimate for u(x) for large values of x. In this paper we aresatisfied with �u2(x)/u(x) � 1 � � 0.05.

Both estimates u1(x) and u2(x) have their advantages and disadvantages. The simulated estimateu1(x) may meet the accuracy requirement only for relatively large values of u(x). On the contrary, theasymptotic estimate u2(x) performs satisfactorily only for small values of u(x). An overall assessmentsuggests that we should make a compromise between u1(x) and u2(x). Once u1(x) breaks down forsmall values of u(x), we seek a threshold u0 � (0, 1) such that u2(x) starts to achieve the requiredaccuracy when u(x) � u0. Based on this threshold u0, we decide the sample size N through (3.3) sothat u1(x) achieves the required accuracy for u(x) � u0. Then it is natural to use u1(x) when u(x) � u0

and use u2(x) otherwise. In our practice, the condition u(x) � u0 is checked through the value of u1(x).We define the hybrid estimate as

u(x) � u (x)1 � u (x)1 .˜ 1 (u (x)�u ) 2 (u (x)�u )1 0 1 0


Of vital importance for this method to work is the availability of an asymptotic estimate u2(x) foru(x) that starts to work efficiently when u(x) is moderately small. This is usually feasible for heavy-tailed cases. In such cases, losses occur with a relatively small probability but are typically accompaniedby disastrous consequences, which we often call rare events, or, more fashionably, black swan events.

In the next section, we use estimation of the ruin probability as an example to illustrate the procedureof the hybrid method. Nonetheless we would like to stress that the idea behind should work for variousestimation problems involving asymptotics.

4. A HYBRID ESTIMATE FOR THE RUIN PROBABILITY

Hereafter all limit relationships are according to x → � unless otherwise stated. For two positivefunctions f(�) and g(�), we write f(x) � g(x) if lim f(x)/g(x) � 1 and write f(x) � g(x) if limsupf(x)/g(x) � 1. For two positive bivariate functions f(�, �) and g(�, �), we say that the asymptotic relationf(x, y) � g(x, y) holds uniformly for y in a nonempty set if

f(x, y)lim sup � 1 � 0. g(x, y)x→� y�

Let us return to the bivariate AR risk model introduced in Section 2, in which the claim amountsfollow the AR(1) process (2.1) and the log-return rates of the stock follow the AR(1) process (2.3).

4.1 The Crude Monte Carlo EstimateFor the ruin probability �(x; n) given by (2.9), the CMC estimate is

N1� (x; n) � 1 , (4.1)�1 (L �x)kN k�1

where Lk, k � 1, . . . , N, are i.i.d. samples fromm

L � max (B � a).� i i1�m�n i�1

As stated in Section 3, the CMC method works only if �(x; n) is not too small. For instance, supposethat N � 500,000 is the maximal sample size one is willing to work with. Then by (3.4), the thresholdu0 is 0.003. This means that the estimate �1(x; n) given by (4.1) is accurate only if �(x; n) � 0.003(practically, �1(x; n) � 0.003). For large values of x such that �(x; n) � 0.003, we shall rely on theasymptotic estimate constructed in the next subsection.

4.2 The Asymptotic EstimateNow assume that the common distribution function F of the innovations in the AR(1) process (2.1) isheavy tailed. More specifically, there is some � � 0 such that the relation

F(xy) ��lim � y (4.2)F(x)x→�

holds for all y � 0. Denote by �� the class of all distribution functions F on � satisfying the regularvariation property described by relation (4.2). The definition immediately indicates that F � �� ifand only if can be expressed asF(�)

��F(x) � x l(x), x � 0, (4.3)

for some slowly varying function l(�). Hence, a smaller value of � means a heavier right tail of F. Thisclass contains many popular heavy-tailed distributions such as Pareto, log gamma, Burr and Student’st distributions, and it has been extensively used to model heavy-tail phenomena in insurance and fi-


nance. For background knowledge of regular variation, the reader is referred to Bingham et al. (1987),Resnick (1987, 2007), and Embrechts et al. (1997).

Throughout the paper, we write MY(s) � E[esY], s � �, as the moment-generating function of thegeneric innovation Y of the AR(1) process (2.3), and define 00 to be 1 as usual. The asymptotic formulaobtained by the following result will be used to construct the desired asymptotic estimate for the finite-time ruin probability.

Theorem 4.1Consider the bivariate AR risk model introduced in Section 2. Assume that F � �� for some � � 0.Then for every n � �, the relation

�n n ni i�j¯�(x; n) � E F x � (a � � B ) � (4.4)� � � � � � � �i 0 i

i�1 j�1 i�j

holds under either of the following two conditions:

a. � � [0, 1);b. � � 1 and MY(s1) � � for some s1 � �2�(1 � (1 � �)�1).

PROOF

Let us first verify the conditions of Lemma 6.4. If � � [0, 1), then each i defined by (2.8) is boundedby e�ri � �. If � � 1, then for each i � �, by (2.4) there is some constant ci � 0 such that

i i i�11 � � 1 � � � exp � R � c exp � Y � Y � � � � � Y .� � �i j i 1 2 i1 � � 1 � �j�1

By the moment condition on Y, we have � � for � � �s1/2 (1 � (1 � �)) � �. Thus, Lemmas2�E[ ]i

6.4 and 6.5 are applicable for both cases. Applying these two lemmas we complete the proof of Theorem4.1. �

The asymptotic formula (4.4) can certainly be simplified to

�n ni�j¯�(x; n) � F(x) E � .� �� i

j�1 i�j

However, this completely ignores the sum i(a � �iB0), which is an indispensable term in then�i�1

expression for Wn as shown in our derivation of (6.11). Hence, the simplified version would be lessaccurate than (4.4), as remarked after Lemma 6.3.

The expectation on the right-hand side of (4.4) is still estimated by the CMC method; that is,N1 ¯� (x; n) � F(x � � )� , (4.5)�2 0k kN k�1

where (�0k, �k), k � 1, . . . , N, are i.i.d. samples from

�n n ni i�j(� , �) � (a � � B ), � .� � �� 0 i 0 i

i�1 j�1 i�j

However, we would like to remark that this step of estimation incurs little extra error, as explainedbelow. Consider the relative error of �2(x; n) defined by

2¯var [� (x; n)] 1 E[(F(x � � )�) ]2 0� � 1 .2 2¯� �(E[� (x; n)]) (E[F(x � � )�])�N2 0


Let the conditions of Theorem 4.1 be valid except that the moment condition on Y in (b) is strength-ened to MY(s1) � � for some s1 � �4�(1 � (1 � �)�1). It is easy to verify all the conditions of Lemma6.6, by which we have

2 2¯ �var[�]var [� (x; n)] 1 (F(x)) E[� ] 12lim � lim � 1 � � �.2 2 2¯� �(E[� (x; n)]) (F(x)) (E[�]) E[�]�N �Nx→� x→�2

Thus, the CMC estimate �2(x; n) has a bounded relative error, which is usually the best that one candesire for in rare-event simulation.

4.3 Accuracy of the Asymptotic EstimateTo answer whether or not �2(x; n) given by (4.5) is a good estimate for �(x; n), we carry out numericalstudies to examine its accuracy by checking how close to 1 the ratio �2(x; n)/�1(x; n) is. Throughoutthe rest of this paper, all numerical studies are conducted in the R software environment. For a ref-erence manual on R, see R Development Core Team (2008).

For ease of computation, we assume that the generic innovation X of the AR(1) process (2.1) followsa Pareto distribution with shape parameter � � 1 and mean �X, and that the generic innovation Y ofthe AR(1) process (2.3) follows a normal distribution with mean 0 and variance �2.

In Figures 1(a)–(c), the parameters are set to � � 0.2, 0.5, or 0.8, n � 4, � � 1.1, 1.3, or 1.5,� � 0.3 or 0.5, B0 � 5.0, �B � 5.0, �X � (1 ��)�B, a � 5.5, r � 1.242% (so that er � 1 � 1.25%),� � 0.5 or 0.8, R0 � 1.5%, �R � 1.5%, and � � 0.2. By setting n � 4 we mean that each period is aseason. The parameters r, R0, �R, and � are chosen so that the annual risk-free rate and the annualreturn rates are within a reasonable range. The values of �B and a are chosen to yield a safety loadingof 10%. The sample size is N � 1,000,000 for both �1(x; n) and �2(x; n) despite the fact that it ismore than enough for �2(x; n). We compare the two estimates on the left and show their ratio on theright.

From these graphs we see that, for each set of parameters, the ratio �2(x; n)/�1(x; n) quickly con-verges to 1 as the initial wealth x increases. After staying around 1 for a while, the ratio starts tofluctuate. An explanation of the fluctuation is that, as x becomes large, the true value of �(x; n)becomes too small, leading to too low of an accuracy of �1(x; n). In other words, the fluctuation mayresult only from the low accuracy of �1(x; n). Notice that the larger the parameter � is, the smallerthe ruin probability �(x; n) becomes and the more fluctuation the ratio �2(x; n)/�1(x; n) exhibits.Moreover, the convergence of the ratio is robust with respect to �, �, and �.

In the far right area, the ruin probability �(x; n) decreases quickly as � increases. Hence, not sur-prisingly, a larger value of � leads to a less accurate �1(x; n). For instance, let us look at Figure 2(a),for which the parameters are set identical to those in Figures 1(a)–(c) except � � 0.5, � � 2.0, � � 0.3,and � � 0.8. In this graph the ratio �2(x; n)/�1(x; n) fluctuates dramatically and does not seem toconverge to 1 at all. One might start to doubt the validity of the asymptotic estimate.

Actually, this is an illusion because of the poor performance of the simulated estimate �1(x; n). Werepeat the simulation with the sample size N increased from 1,000,000 to 10,000,000 and draw Figure2(b). Then we observe a much improved convergence of the ratio �2(x; n)/�1(x; n). This confirms thatthe only source of fluctuation is the awful estimate �1(x; n) in the far right area. There is nothingwrong with the asymptotic estimate �2(x; n). For a large value of �, the sample size N usually has tobe very large to make the simulated estimate �1(x; n) acceptable.

4.4 The Hybrid Estimate for the Ruin ProbabilityNow that a good asymptotic estimate for �(x; n) is available, we explain in detail how the hybrid methodintroduced in Section 3 can be implemented in the current situation.

In our practice, with a reasonably large sample size N*, we explore the value of �1(x; n) for whichthe ratio �2(x; n)/�1(x; n) starts to fall into the strip between 0.95 and 1.05. Such a value serves asthe threshold u0 in our hybrid method. Note that, for a fixed �, the threshold u0 is determined once


Figure 1Accuracy of the Asymptotic Estimate �2: (a) for � � 0.2, (b) for � � 0.5, (c) for � � 0.8


Figure 2Accuracy of the Asymptotic Estimate �2 for � � 2.0: (a) N � 1,000,000, (b) N � 10,000,000

and for all owing to the robustness of the asymptotic estimate �2(x; n) with respect to the parameters�, �, and �. In our numerical study, with N* � 1,000,000, � � 0.2, 0.5 or 0.8, n � 4, � �0.3 or 0.5,B0 � 5.0, �B � 5.0, �X � (1 � �)�B, a � 5.5, r � 1.242%, � � 0.5 or 0.8, R0 � 1.5%, �R � 1.5%, and� � 0.2, we see that the threshold u0 can be chosen to be u0 � 0.01 for � � 1.1, u0 � 0.003 for� � 1.3, u0 � 0.003 for � � 1.5, and u0 � 0.0005 for � � 2.0.

Next, we calibrate the sample size N according to (3.3) with u(x) replaced by u0. For instance, inour numerical study, the sample size needed for the required accuracy of �1(x; n) is about N � 140,000for � � 1.1, N � 500,000 for � � 1.3, N � 500,000 for � � 1.5, and N � 3,000,000 for � � 2.0. Wecompute �1(x; n) and �2(x; n) again based on the calibrated sample size N.

Finally, define

�(x; n) � � (x; n)1 � � (x; n)1 , (4.6)1 (� (x;n)�u ) 2 (� (x;n)�u )1 0 1 0

which is the desired hybrid estimate for �(x; n). It is easily understood that gives a globally�(x; n)accurate estimation for �(x; n).


5. PORTFOLIO OPTIMIZATION

Generally speaking, investors aim at maximizing gains while minimizing risks. How to balance betweengains and risks is a question that permeates many areas of finance. Since the seminal work of Merton(1971), many variants of Merton’s problem have been proposed in the finance literature to addressdifferent interests and concerns.

Let us now restrict ourselves to portfolio optimization problems in the insurance context. Due tothe increasing prudence of insurance regulations, a solvency constraint needs to be imposed in portfoliooptimization problems. Such a constraint is relevant in practice, especially after the 2008 financialcrisis. Related discussions on portfolio optimization with a solvency constraint can be found in Paulsen(2003), Irgens and Paulsen (2005), Dickson and Drekic (2006), Kostadinova (2007), He et al. (2008),and Liang and Huang (2011), among others.

Consider the bivariate AR risk model introduced in Section 2. We now aim to determine the pro-portion invested in the stock that maximizes the expected terminal wealth subject to a solvencyconstraint. We stick to all conditions of Theorem 4.1 but require � � 1 and strengthen the mo-ment condition on Y to MY(s1) � � for some s1 � �2�(1 � (1 � �)�1) and MY(s2) � � fors2 � 1 � (1 � �)�1 so as to ensure the finiteness of the expected terminal wealth. By the convexity ofMY(�), it follows that MY(s) � � for s � [s1, s2].

By relation (2.6), the terminal wealth Wn is

n n nR Rj jW � x ( � �e ) � (B � a) ( � �e ).� � �n i

j�1 i�1 j�i�1

Our goal is to determine a value of � � [0, 1] that maximizes E[Wn] subject to the constraint on�(x; n) given by (2.7), namely, �(x; n) � 1 � q for some 0 � q � 1 close to 1.

We employ a recursive procedure to compute E[Wn]. Write

nsR Rn jI(s, i, n) � E e ( � �e ) , s � �, i � 0, 1, . . . , n,� �

j�i�1

so that

n

E[W ] � xI(0, 0, n) � (E[B ] � a)I(0, i, n).�n ii�1

The expectations E[Bi], i � 1, . . . , n, are computed by using (6.2). The terms I(s, i, n) for i � 0,1, . . . , n can be computed recursively. Denote by �0 the trivial �-field specifying the initial values B0,R0, and W0, and by �m the �-field generated by (B1, R1), . . . , (Bm, Rm). Thus, {�0, �m, m � �} servesas the natural filtration of the bivariate AR model. For i � 0, 1, . . . , n � 1, by conditioning on �n�1

and using (2.4), we have

I(s, i, n)n�1

sR R Rn n j� E E[e ( � �e ) �� ] ( � �e )� �n�1j�i�1

n�1s(d��R �Y ) (s�1)(d��R �Y ) Rn�1 n n�1 n j� E E[e � �e �� ] ( � �e )� �n�1

j�i�1

n�1 n�1sd s�R R (s�1)d (s�1)�R Rn�1 j n�1 j� e M (s)E e ( � �e ) � �e M (s � 1)E e ( � �e )� � � �Y Y

j�i�1 j�i�1

sd (s�1)d� e M (s)I(s�, i, n � 1) � �e M (s � 1)I((s � 1)�, i, n � 1). (5.1)Y Y


Figure 3Optimal � for Different Values of the Initial Wealth

Continue this recursive procedure until the third argument decreases from n to i. We see that thecomputation of I(s, i, n) is reduced to that of I(s, i, i) � for i � 0, 1, . . . , n. Similarly, thesRiE[e ]latter can be computed recursively through

sR sd0I(s, 0, 0) � e , I(s, i, i) � e M (s)I(s�, i � 1, i � 1), i � 1, . . . , n.Y

We remark that, for computing E[Wn], it suffices to set the argument s in the first step of (5.1) to 0.In this case the moment-generating function of Y appearing in these recursive formulas is always finite.

In our numerical study, the same as in Subsection 4.3, we assume for ease of computation that thegeneric innovation X of the AR(1) process (2.1) follows a Pareto distribution with shape parameter� � 1 and mean �X, and that the generic innovation Y of the AR(1) process (2.3) follows a normaldistribution with mean 0 and variance �2.

We determine the optimal value of � by a grid search over the interval [0, 1]. First, we find a region,called the admissible region, of � satisfying the constraint described by (2.7), where the ruin probabilityis computed according to the hybrid estimate (4.6). Then we compute E[Wn] for each � in the admis-sible region using the recursive formulas above. The optimal � is the one that corresponds to themaximal value of E[Wn].

The optimization algorithm is also implemented in R, but with C�� integrated to speed up. Thepowerful R package Rcpp makes the integration seamless. As stated by Eddelbuettel and Francois(2011), Rcpp defines a class hierarchy and maps every R type (such as vector, function, and environ-ment) to a dedicated class. For instance, numeric vectors are represented as instances of the Rcpp::NumericVector class. Data passing between R and C��, which used to be cumbersome, is now madesurprisingly straightforward by the highly flexible templates Rcpp::wrap and Rcpp::as. For moredetailed descriptions on Rcpp, the reader is referred to Eddelbuettel and Francois (2011). The packageRcppArmadillo is an integration of R and the templated C�� linear algebra library Armadillo. SeeSanderson (2010) and Francois et al. (2011) for more details of Armadillo and RcppArmadillo,respectively. For the purpose of integrating R and C��, the package inline can be a good assistantas it enables one to define R functions dynamically with inlined C�� code. For instance, in our nu-merical study, an R function that takes certain parameters and returns an optimal value of � is definedvia the function cxxfunction in the package inline. See Sklyar et al. (2010) for more details ofinline.


Figure 3 shows how the optimal � changes with respect to the initial wealth x. The parameters areset to n � 4, � � 1.1, 1.5, or 2.0, � � 0.3, B0 � 5.0, �B �5.0, �X � (1 � �)�B, a � 5.5, r � 1.242%,� � 0.8, R0 � 1.5%, �R � 1.5% or 3%, and � � 0.2, most being the same as in Subsection 4.4. Accordingto the threshold u0 determined in Subsection 4.4, we set N � 200,000 for � � 1.1, N � 500,000 for� � 1.5, and N � 5,000,000 for � � 2.0, with N increasing to appropriately offset the negative effectof a small value of the ruin probability. It shows a clear trend of the optimal � increasing in x and �R,all being consistent with our anticipation.

6. PROOFS

First, let us collect some useful preliminaries on the distribution class ��. The first two below areknown as max-sum equivalence and Potter’s bounds, respectively.

Lemma 6.1Let F, F1, and F2 be three distribution functions on �, all belonging to the class �� for some � � 0. Wehave the following:

a. The convolution F1 * F2 still belongs to the class �� and � �F F (x) F (x) F (x).*1 2 1 2

b. For every ε � 0, there is some x0 � 0 such that, for all x, y � x0,

F( y)��ε ��ε ��ε ��ε(1 � ε)((x/y) � (x/y) ) � � (1 � ε)((x/y) � (x/y) ).

F(x)

c. The relations x��ε � and � o(x��ε) hold for every ε � 0.¯ ¯o(F(x)) F(x)d. For every C � 1, the relation � y�� holds uniformly for y � [1/C, C].¯ ¯F(xy) F(x)

PROOF

a. See Lemma 1.3.1 of Embrechts et al. (1997).b. See Theorem 1.5.6 of Bingham et al. (1987).c. This can be proved by applying Proposition 1.3.6 of Bingham et al. (1987) to the representation for

F(�) given by (4.3).d. By Theorem 1.5.2 of Bingham et al. (1987), the convergence in (4.2) holds uniformly for y �

[1/C, �), namely,

F(xy) ��lim sup � y � 0. F(x)x→� y�[1/C,�)

Then one easily sees the uniformity of the relation � y�� for y � [1/C, C]. �¯ ¯F(xy) F(x)

It is sometimes convenient to consider the following augmented version of the AR(1) process (2.1):

B � �B � X , i � �, (6.1)i i�1 i

where, the same as before, the autoregressive coefficient � takes value in [0, 1) and the innovationsXi, i � �, are i.i.d. copies of a nonnegative random variable X. The stationary solution of (6.1) is givenby

�jB � � X , i � �.�i i�j

j�0

By plugging the starting value B0 in the above we obtaini�1 � i�1

j i j�i j iB � � X � � � X � � X � � B , i � �. (6.2)� � �i i�j i�j i�j 0j�0 j�i j�0


Proposition 6.1Consider the AR(1) process (2.1). If F � �� for some � � 0, then the relation

�i �i ��2 1lim Pr(B � x �B � x) � � (6.3)i i2 1x→�

holds for all i1, i2 � �.

PROOF

Relation (6.3) holds trivially for � � 0 or i1 � i2. Moreover, the proofs for i1 � i2 and i1 � i2 are similar.Hence, we prove relation (6.3) only for 0 � � � 1 and i1 � i2.

Similar to (6.2), it holds thati �i �12 1

j i �i2 1B � � X � � B � I � I ,�i i �j i 1 22 2 1j�0

where is expressed by (6.2) with i replaced by i1. Note that the two terms I1 and I2 are independent.Bi1Applying Lemma 6.1(a) and extending by induction, we see that both I1 and I2 follow distributionsbelonging to the class ��.

Now we deal with the joint probability Pr( � x, � x). On the one hand, we haveB Bi i2 1

i �i2 1Pr(B � x, B � x) � Pr(I � I � x, I � � x)i i 1 2 22 1

� Pr(I � I � x, I � x)1 2 2

� Pr(I � x). (6.4)2

On the other hand, still starting from (6.4) we havei �i2 1Pr(B � x, B � x) � Pr(I � I � x) � Pr(I � x)Pr(I � � x)i i 1 2 1 22 1

� Pr(I � x) � Pr(I � x) � Pr(I � x)1 2 1

� Pr(I � x),2

where in the second step we used Lemma 6.1(a). It follows that Pr( � x, � x) � �i �i2 1B B Pr(� Bi i i2 1 1

x). Therefore,i �i2 1Pr(� B � x)i1 (i �i )�2 1lim Pr(B � x �B � x) � lim � � .i i2 1 Pr(B � x)x→� x→� i1

This proves relation (6.3) for i1 � i2. �

The following elementary result is used in proving Lemma 6.3.

Lemma 6.2Let X be a random variable distributed by F � �� for some � � 0, let � be a positive random variablewith E[��] � � for some � � �, let { t, t � �} be a set of random events satisfying Pr( t) � 0limt→t0for some t0 in the closure of the index set �, and let {�, { t, t � �}} and X be independent. Then

Pr(�X � x, )tlim lim sup � 0.F(x)t→t x→�0

PROOF

Choose some �/� � c � 1. Then split Pr(�X � x, t) into two parts asc cI (x, t) � I (x, t) � Pr(�X � x, , 0 � � � x ) � Pr(�X � x, , � � x ).1 2 t t


By Lemma 6.1(b), it holds for arbitrarily fixed 0 � ε � � � (� � �) and all large x thatcx

I (x, t) � � Pr(sX � x)Pr(� � ds, )1 t0

cx��ε ��ε¯� (1 � ε)F(x) � (s � s )Pr(� � ds, )t

0

��ε ��ε¯� (1 � ε)F(x)E[(� � � )1 ]. t

By Markov’s inequality and Lemma 6.1(c), we havec �c� � ¯I (x, t) � Pr(� � x ) � x E[� ] � o(F(x)).2

Putting the two bounds together yields

Pr(�X � x, )t ��ε ��εlim lim sup � (1 � ε) lim E[(� � � )1 ] � 0, tF(x)t→t x→� t→t0 0

where in the last step we used the dominated convergence theorem. �

The following result plays a key role in establishing our main result, and it is interesting in its ownright.

Lemma 6.3Let {Xi, i � �} be a sequence of i.i.d. random variables with common distribution function F on �, let{�0, �i, i � �} be another sequence of not necessarily independent random variables with �0 real valuedand �i positive for i � �, and let the two sequences be independent. If F � �� for some � � 0,Pr(�0 � �x) � o( (x)), � � for some � � � and � o( i � �, then it holds� �¯ ¯F E[� ] E[� 1 ] F(x)),i i (� ��x)0

for every n � � thatn n n

� �¯ ¯Pr � X � � � x � E F(x � � ) � � F(x) E[� ]. (6.5)� � �� i i 0 0 i ii�1 i�1 i�1

Lemma 6.3 gives two asymptotics, E[ (x � �0) and (x) We remark that then � n �¯ ¯F � � ] F � E[� ].i�1 i i�1 i

former should work better for computation purposes because it captures the impact of �0. For thisreason, Lemma 6.4 and Theorem 4.1 are tied to the former rather than the latter.

PROOF

The proof of the second relation in (6.5) is easy. Actually, for arbitrarily fixed 0 � c � 1, we splitE[ (x � �0) into two parts asn �F � � ]i�1 i

n�¯I (x) � I (x) � E F(x � � ) � 1 .� �1 2 0 i (� ��cx)�(� ��cx)0 0

i�1

By the condition E[ ] � o( (x)) for i � �, it holds that� ¯� 1 Fi (� ��x)0

n� ¯ ¯I (x) � E[� 1 ] � o(F(cx)) � o(F(x)).�1 i (� ��cx)0

i�1

For I2(x), noting that (x � �0) � ((1 � c)x) � (1 � c)�� (x), we have, by the dominated convergence¯ ¯ ¯F F Ftheorem,

n n¯I (x) F(x � � )2 0 � �lim � E lim 1 � � E[� ].� � � � �(� ��cx) i i0¯ ¯F(x) F(x)x→� x→� i�1 i�1

This proves the second relation in (6.5).


Now we turn to the first relation in (6.5). Write � (s1, . . . , sn) and � (�1, . . . , �n). Arbitrarily→→s �

choose 0 � c � 1 and C � 1 such that � [1/C, C]n) � 0. By Proposition 5.1 of Tang and→Pr(�Tsitsiashvili (2003b), the relation

n n

Pr s X � x � s � Pr(s X � x � s ) (6.6)� �� i i 0 i i 0i�1 i�1

holds uniformly for s0 � �cx and � [1/C, C]n. According to these constants c and C, we split the→sspace �n�1 into three disjoint parts as

→ →n n � � � ( �s � � cx) � ( �s � � cx, s � [1/C, C] ) � ( �s � � cx, s � [1/C/, C] ).1 2 3 0 0 0

Then the tail probability Pr �i Xi � �0 � x) is equal ton(�i�1

3 3 n →J (x) � Pr � X � � � x, (� , � ) � . (6.7)� � �� j i i 0 0 jj�1 j�1 i�1

By Markov’s inequality and Lemma 6.2, it holds for arbitrarily fixed ε � 0 and all large x that

n

J (x) � Pr � X � x, � � cx � Pr(� � �cx)�� 1 i i 0 0i�1

n x ¯� Pr � X � , � � cx � o(F(cx))� � �i i 0ni�1

¯� εF(x). (6.8)

Similarly, by Lemma 6.2 again, it holds for all large x and C thatn 1 � c → ¯J (x) � Pr � X � x, (� , �) � � εF(x). (6.9)� � �2 i i 0 2ni�1

By conditioning on (�0, and applying the uniform asymptotic relation (6.6),→�)

n n

J (x) � � � � � � Pr(s X � x � s ) Pr � (� � ds )� � �3 i i 0 i ii�1 i�03

n n� ¯� � � � � � s F(x � s ) Pr � (� � ds )� � �i 0 i i

i�1 i�03

n n� �¯ ¯ →� E[F(x � � )� ] � E[F(x � � )� 1 ], (6.10)� �0 i 0 i (� ,�)� � 0 1 2

i�1 i�1

where in the second step we applied the uniform asymptotic result in Lemma 6.1(d). The last sum in(6.10) is negligible. Actually, it holds for all large x and C that

�¯ →E[F(x � � )� 1 ]0 i (� ,�)� � 0 1 2

� � �¯ ¯ →� E[� 1 ] � F(x)E[� 1 ] � F((1 � c)x)E[� 1 ]i (� ��cx) i (� �cx) i (� ,�)� 0 0 0 2

¯� εF(x).

Substituting this into (6.10), then combining (6.7) with (6.8), (6.9), and (6.10), we obtainn n n

� �¯ ¯ ¯ ¯E F(x � � ) � � nεF(x) � Pr � X � � � x � E F(x � � ) � � 2εF(x).� � � � � � �0 i i i 0 0 ii�1 i�1 i�1


The first relation in (6.5) follows by the arbitrariness of ε and the second relation in (6.5). �

Recall the bivariate AR model introduced in Section 2, in which Bi denotes the claim amount, adenotes the premium amount, and, hence, Bi � a denotes the net loss during period i. In particular,as shown in (2.9), the ruin probability �(x; n) is the tail probability of the maximum of n randomlyweighted sums. The following result targets the tail probability of a randomly weighted sum, and itforms the essence of the proof of Theorem 4.1.

Lemma 6.4Let the notation in Section 2 be valid and consider the randomly weighted sum

n

S � (B � a), n � �.�n i ii�1

If the i.i.d. innovations Xi, i � �, of the AR(1) process (2.1) follow a common distribution functionF � �� for some � � 0 and the random weights satisfy � � for some � � � and all i � �,2�E[ ]i

then for every n � �, the sum Sn still follows a distribution in the class �� and the relation�n n n

i i�j¯Pr(S � x) � E F x � (� � � B ) �� n i 0 ii�1 j�1 i�j

holds. In case a � �B0 � 0, the moment condition can be weakened to � � for some � � � and�E[ ]i

all i � �.

PROOF

By (6.2) we haven i�1 n

j iS � � X � � B � a � � �� n i i�j 0 ii�1 j�0 i�1

n n ni�j i� � X � (a � � B )� � �� i j i 0

j�1 i�j i�1

n

� � X � � . (6.11)� j j 0j�1

It is easy to verify all the conditions of Lemma 6.3. For instance, � (x)) holds because� ¯E[� 1 ] o(Fj (� ��x)0

� �� E[� 1 ] � x E[ �� ],j (� ��x) 0 j0

while x�� by Lemma 6.1(c) and, for some C � 0,¯o(F(x))n1

� � 2� 2� 2� 2�E[ �� ] � E[ �� ] � C E[ � ] � ��0 j 0 j i i2 i�1

by cr-inequality. Finally, an application of Lemma 6.3 concludes the proof. �

The following lemma is interesting in its own right.

Lemma 6.5Let Z1, . . . , Zn be n not necessarily independent random variables with partial sums Sm � Zi form�i�1

m � 1, . . . , n. If the last sum Sn follows a distribution in the class �� for some � � 0 and Pr(Zi ��x) � o(Pr(Sn � x)) for each i � 1, . . . , n, then

Pr max S � x � Pr(S � x).� �m n1�m�n


PROOF

It suffices to prove the asymptotic inequality Sm � x) � Pr(Sn � x) because the other onePr(max1�m�n

is trivial. For A � � {1, . . . , n}, define

� � {Z � 0 whenever i � A and Z � 0 whenever i � �\ A}.A i i

Then, for every 0 � ε � 1/n,

Pr max S � x� �m1�m�n

� Pr Z � x, �� i AA� i�A

� Pr Z � x, � , � (Z � �εx) � Pr Z � x, � , � (Z � �εx)� � � �� i A i i A iA� i�A i��\ A A� i�A i��\ A

� Pr(S � (1 � nε)x, � ) � Pr(Z � �εx)� � �n A iA� A� i��\ A

� Pr(S � (1 � nε)x) � o(1)Pr(S � εx)n n

�� (1 � nε) Pr(S � x).n

The arbitrariness of ε concludes the proof. �

Lemma 6.6Let F � �� for some � � 0 and let �0 and � be two random variables with �0 real valued and �positive.

a. If E[�] � � and � then (x � �0)�] � (x)E[�].¯ ¯ ¯E[�1 ] o(F(x)), E[F F(� ��x)0

b. If E[�2] � � and � (x))2), then (x � �0)�)2] � (x))2E[�2].2 ¯ ¯ ¯E[� 1 ] o((F E[(F (F(� ��x)0

PROOF

Both can be proved by following the proof of the second relation in (6.5). �

7. ACKNOWLEDGMENTS

The authors thank the two anonymous referees for their helpful comments on an earlier version of thispaper. The research for this project is sponsored by The Actuarial Foundation and the Society of Ac-tuaries through the 2011 Individual Grants Competition. The source code for the numerical studies inthis paper is available on both authors’ homepages or upon request.

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