#### 1. Introduction

The differences between others’ contributions and our work are evident. First, different from [911], we include the market regime for the first time into the premium calculations of PBGC’s insurance program. We not only assume that the price dynamics of risky assets depend on the states of the economy but also assume that the proportion of pension fund invested in the risky asset also depends on the states of the economy. It is well known that the popularity of regime switching model has been supported by many empirical evidences for a while. The switching models reflect the changes of macroeconomic environment, such as the adjustment of economic structure, the change of market system, and the business cycle, which are exemplified by a continuous-time Markov chain. The market variable follows one risk model when one state of the economy is specified and transfers to other models when the market scheme transfers between different states. Some applications of regime switching models can be seen in [12, 13]. Second, different from [6, 7], Levis and Pennacchi [8], and others, we consider the risk-based premium that the PBGC should charge. In addition to examining the risk of premature termination for pension fund, we study the financial risk of sponso’s assets as well. Finally, explicit solutions of the premiums are provided and numerical analysis is also carried out to demonstrate our results.

The rest of the paper is arranged as follows. Section 2 presents the problem formulation. Section 3 shows the Esscher transform under regime switching. Section 4 introduces the case when pension is underfunded and discusses the case that sponsor asset suffers from distress termination. Section 5 demonstrates how the premiums are calculated based on the two scenario cases. Section 6 provides some numerical examples to illustrate the effect of regime switching on the premium of PBGC. Section 7 concludes the paper with some further remarks.

#### 2. Formulation

This section presents the formulation of the model of our interest. We begin with notation and assumptions. Given a probability space , we use as a representative of beneficiary’s retirement time. Let be survival probability in which is the random variable representing the time until death for a representative beneficiary retires at age and be long-term interest rate applying to a typical retiree (for example, we can use 30 years’ rate of interest without loss of generality based on the fact that most people retire at age of 60 or older and according to data compiled by the Social Security Administration, the average longevity of people is close to 87) and is the prescribed annual benefit, which depends on employee’s years of -service, age, salary before retirement, and other possible factors. Because the focus of this work is on discussing how the PBGC charges the premiums under different scenario cases, we define the benefit that a typical beneficiary expects to receive similar to that in [10] and express it as below:

Note that we use a constant interest here to define the expected present value of the whole life annuity due starting at age for two reasons. (1) It is both mathematically elegant and practically important to obtain a closed-form solution. (2) It is convenient to choose a conservative interest rate in the first place, which, accordingly, contributes to making proactive provisions regarding all the uncertainties.

We further assume that there are three funding sources for the beneficiary’s annual benefit and the pension fund is the primary funding resource for the pension benefit; sponsor company provides secondary support; then, the PBGC contributes the rest. These assumptions hold throughout the entire paper. We assume further that the portfolio of pension fund consists of two kinds of assets. One is risk free and the other is risky asset. The growth rate of the risk-free asset is risk-free interest rate and that of the risky asset is its expected rate of return. More often than not, at a given time, the expected rate of return of the risky asset is higher than that of the risk-free asset, which gives a positive risk premium for the risky asset. Different from the classical portfolio selection assumption, we want to take the market regime into consideration to reflect that the dynamics of a given product are different under different market regimes. In this paper, we consider such a case that there are two possible market regimes, which are bull and bear markets, respectively. Mathematically speaking, we assume that there is a continuous-time two-state Markov chain taking values in with generator

The Markov chain takes different values when the market is in different regimes. With this practical assumption, the risk-free interest rate and expected rate of return are functions of the market regime . To be specific, the dynamics of the risk-free asset arewhere and is the risk-free interest rate. Moreover, the dynamics of risky asset are given as follows:in which , a standard one-dimensional Brownian motion, is independent of . Using to denote the proportion of pension fund invested in the risky asset and to denote the proportion of pension fund invested in the risk-free asset, we can thus represent the dynamics of the total pension fund as below:where .

The proportion of pension fund invested in the risky asset is dependent on the and it means there will be different proportion of pension fund distributed to risk free as well as risky asset at different times. Note that our interest in this work is about evaluation of premium for the PBGC, so we would assume that there is a given at time for the sake of calculation convenience. As far as the way to find out , once the objective function is formulated, method of stochastic control can be utilized to find out the representation of and this could be one of our further works.

If the pension fund performs well enough, then all the benefit will be paid by the pension fund itself. If the pension fund is underfunded, then company’s asset is a potential resource of paying benefit. The dynamics of employer’s asset are given bywhere and is another standard Brownian motion and is independent of Brownian motion . We assume that the sponsoring corporation’s assets are correlated to the pension fund’s assets with a correlated coefficient and how the correlation coefficient affects the premiums was analyzed in [10].

Note that typically corporation has corporate debt; we therefore assume that at any time , the sponsor company always has to pay its debt in which is its equity-debt ratio and is the predetermined constant to illustrate the growth rate of the corporate debt. It is practically reasonable to assume that company has higher priority to pay back its corporate debt.

The third part of funding resources is the Pension Benefit Guaranty Corporate (PBGC). Let denote the possible contribution that PBGC makes. It is easy to see that depends on the performance of pension fund and sponsor’s assets. We will derive the expression of and the premium that PBGC collects from sponsor company at the end of the following sections. To move forward, we will first find the risk neutral probability measure for the dynamic system with the help of Esscher transform in the following section.

#### 3. Esscher Transform under Regime Switching

Since the additional uncertainty described by regime switching makes the market incomplete, there are infinitely many equivalent martingale measures. Here, we will adopt the Esscher transform to determine an equivalent martingale measure for pricing premium of the Pension Benefit Guaranty Corporation. Esscher transform was first proposed by Gerber and Shiu [14], and it is widely used in the field of finance and insurance. For more details, refer to Bühlmann et al. [15, 16].

We denote as the filtration generated by , the filtration generated by , the filtration generated by , and the filtration generated by , respectively. The regime-switching Esscher transform on with respect to parameters and is given by

Thanks to the well-known result established in [17, 18], the absence of arbitrage opportunities is essentially equivalent to the existence of equivalent martingale, under which the discount price process is a martingale. We know and are martingales under the measure . Thus, we have

According to Girsanov’s theorem, we knoware two independent standard Brownian motions under the measure .

Let ; then, we have

From (10) and (11), we find that, given under the measure , has normal distribution with mean and variance . has normal distribution with mean and variance . Hence, we know that has bivariate normal distribution , with a correlation coefficient .

#### 4. Scenario Case Analysis

In this section, we will focus on the premium calculations under different scenario cases. The first case is analyzing the pension fund has premature termination. We assume that there is a third-party external regulator (like pension actuary), who is in charge of monitoring the performance of pension fund. The pension fund can thus be forced to close prematurely if a certain threshold value is reached. The other case we will study is from the perspective of plan sponsor. When the performance of the sponsor asset is not good enough, the plan provider is not able to cover its debt and thus remains in business, and the distress termination would happen. We choose to examine a threshold value higher than its liability value to include the other possible expenses for the sponsor company to remain in business.

##### 4.1. Premature Termination of Pension Fund

In this section, we consider the case of premature termination of pension fund. The threshold value at time is assumed to be , where is a positive constant less than 1. Therefore, we can define the first hitting time as

Therefore, the entire support provided by sponsor company is

Accordingly, the entire support provided by PBGC is the sum of the following two parts:

Thus, the total contribution supported by PBGC is

##### 4.2. Distress Termination of Sponsor Asset

In this section, we consider the case that sponsor asset can be underfunded and it is called “distress termination” in [11]. Therefore, in this case, we use a stopping time to describe the first time that the sponsor asset falls below or across the threshold , given that the sponsor company has corporate debt at time . We assume , similar to the assumption used in [11]. Under this framework, the definition of is given as

Similar to what we have discussed previously, we consider the case that and in the following paragraphs. Similarly, our analysis breaks into two cases as below.Sponsor asset falls below the threshold before time , . When , the possible outcomes for the pension fund areThe pension fund performs good enough, that is, the value of pension fund is worth more than the discounted promised pension benefit payment. Both sponsor company and PBGC need to do nothing in the case.The pension fund is not sufficient to cover all the discounted pension benefit, i.e., .Hence, we can represent as below:Sponsor asset falls below the threshold not earlier than , ; this implies . When , the pension fund is naturally closed at the maturity date . Note that PBGC collects premium up to time when beneficiary is retired. Therefore, we essentially consider the case of . Also, note that if , both sponsor company and the PBGC do not need to provide anything, and therefore, we just focus on the case that to proceed.

Thus, is described as follows:

Combining what we have above together, the entire support provided by the sponsor company is

Accordingly, the entire support provided by PBGC is

Thus, the entire support provided by PBGC is summarized as

Now we can proceed to find the premium for PBGC. Using the no arbitrage idea, the premium paid by plan sponsor to PBGC is the expected discounted insurance payoff under the risk neutral probability measure. Before we discuss the calculation of premium, let us introduce some notations first. Let denote the occupation time of Markov chain at state 0 during . In the calculation below, we will employ the method mentioned in [19] to analyze the case of stopping time being reached before and after retirement time , and separately due to calculation convenience.

For the sake of simplicity, we first give some symbols. We denote

Then, has bivariate normal distribution, with probability density functionwhere

From Yoon et al. [20], we can obtain the following lemma.

Lemma 1. Let be the probability density of at initial state , and we havewhere , , , , and is the modified Bessel function of the first type such that

Let and denote the probability distribution function of and given on . Then, we have the following lemma.

Lemma 2. Let and denote the condition density functions of and ; and are then given by (32) and (34).

Proof. It follows from [21, 22] thatBy taking derivative with respect to variable of , we haveBy adopting the same method, we haveThen,

Theorem 1. If the Markov chain initial state , then the premium received by PBGC with premature closure of pension fund under the risk neutral probability measure is given by (A.14).

Proof. The closed-form solution is given in Appendix A. For details, see Appendix A.

Corollary 1. If the Markov chain initial state , then the risk-based premium of PBGC with early termination of sponsor assets under the risk neutral probability measure is given by (B.6).

Proof. The closed-form solution is given in Appendix B. For details, see Appendix B.

Remark 1. In reality, sometimes both sponsor company and the PBGC only provide a capped retirement income when the employee gets retired early or when the pension fund is highly underfunded. In this case, we can model the support from PBGC by a constant with ; by using the similar method as [11], the premium can be calculated.The premium that sponsor company collects from plan participants can also be calculated by the similar method used before.

#### 6. Numerical Analysis

In this section, we make numerical analysis of the explicit formula derived in the previous section. For numerical demonstration, we take the following parameters:

Figures 1 and 2 demonstrate the effects of on premiums under different situations. To be more specific, Figure 1 illustrates how the premium changes against when premature termination of pension fund occurs and Figure 2 shows the impact of on premium for the case of distress termination of sponsor assets. We make a couple of observations regarding the two graphs. First, the premium is an increasing function of in both graphs. Given that is the retirement benefit that one expects to receive, it makes sense to observe that a higher implies a bigger financial responsibility for the employer to hold. Therefore, it is expected for the employer to pay more premium to the PBGC to transfer the pension risk. Second, we notice that the premium increases much more significantly in Figure 1, compared with its trend in Figure 2. Note that pension fund is the first and foremost pool of fund for postretirement payment regardless of financial soundness of the sponsor company, and thus it is reasonable to witness that premium increases more quickly when there is a risk of premature termination of pension fund.

In Figure 3, we demonstrate how the premium changes with respect to while premature termination of pension is under consideration. Recall that is the trigger ratio of the pension fund and is also referred to as the regulatory parameter. We know that, on the one hand, termination would never happen if given that the pension fund is non-negative. On the other hand, does not make sense since it implies that more than enough reserve should be set aside in the pension fund pool. Thus, we assume that . We can see that the dynamics of premium according to are not monotonic with our choice of parameters. The increase of regulatory parameter first leads to a very mild increase of premium and there is a dip of the premium afterwards. The possible reasons behind this interesting pattern are explored as follows. When the regulatory parameter is small, the pension fund value is small while the threshold value is hit. It is more likely that the PBGC needs to step in when pension fund has premature termination. Our conjecture for the mild increase portion of premium is that it is a reflection of the expected compensation that the PBGC predicts for its higher probability of providing the coverage. With the increase of regulatory parameter, the premium starts to go down to adjust for stronger regulation requirement after an “optimal” value of it being reached first.

Figure 4 is about the relation between premium and threshold value for the case of distress termination of the sponsor assets. We have the requirements about the such that it not only satisfies but also meets the condition that . We need to reflect the assumption that the sponsoring company is not in default yet in the beginning. is necessary to take account of the fact that the pension sponsor has the moral obligation to cover some deficits of the claimed pension benefit. Our result shows a decrease of premium in regard to the increase of in the figure. Note that the three sequential lines of protections are assumed to be the pension fund, the sponsor assets, and the PBGC. Higher threshold value of the sponsor company at distress termination shows that the employer is capable to provide more financial support irrespective of performance of the pension fund, and the PBGC thus charges less premium accordingly with less financial burden.

#### 7. Further Remarks

In this work, we focus on finding the closed-form formula for the risk-based premium of the PBGC with regime switching. The explicit solutions of the premiums are derived.

Further efforts can be directed to the portfolio selection for the pension fund in which stochastic control and Markov chain approximation seem to be reasonable methods to use. Further effort in this direction deserves more thoughts and considerations.

#### A. Proof of Theorem 1

By the risk neutral pricing theory, the premium for PBGC with premature closure of pension fund is the expected discounted insurance payoff under the risk neutral probability measure and is given by

Hence, the premium for the insurance of the PBGC can be decomposed into two parts:(i).(ii).

(1) Regarding (i), we calculate it by iterated expectation formula as follows:

Note that here we used the fact that in the above. On the one hand,in which

On the other hand,in which

According to relationships (A.2), (A.3), and (A.5), we can obtain

(2) As to (ii), we have

Therefore, according to Lemma 1, for and , we getin which

On the other hand,in which

Combining (A.9) with (A.11), we have

From (A.7) and (A.13), the pricing formula of the premium of the PBGC with premature closure is given as follows:

#### B. Proof of Corollary 1

Similarly, we get by the risk neutral pricing theory as follows:

We will deal with the two parts similarly as in the previous section. The details are presented with the detailed calculation omitted.in whichin which