A Stochastic Programming Model for Funding

A Stochastic Programming Model for Funding Single Premium Deferred Annuities Soren S. Nielsen Management Science and Information Systems University of Texas at Austin Austin, TX 78712 Stavros A. Zenios Decision Sciences Department The Wharton School University of Pennsylvania Philadelphia, PA 19104 Report 92{08{03 Decision Sciences Department The Wharton School University of Pennsylvania Philadelphia, PA 19104. August 1992 Revised December 1994

 Research partially supported by NSF grants CCR{9104042 and SES{91{00216, and AFOSR grant 91{

0168. Computing resources were made available by AHPCRC at the University of Minnesota, by NPAC at Syracuse University, New York, and by the GRASP Laboratory at Computer Science Department at University of Pennsylvania.

1

Abstract Single Premium Deferred Annuities (SPDAs) are investment vehicles, oered to investors by insurance companies as a means of providing income past their retirement age. They are mirror images of insurance policies. However, the propensity of individuals to shift part, or all, of their investment into dierent annuities creates substantial uncertainties for the insurance company. In this paper we develop a multiperiod, dynamic stochastic program that deals with the problem of funding SPDA liabilities. The model recognizes explicitly the uncertainties inherent in this problem due to both interest rate volatility and the behavior of individual investors. Empirical results are presented with the use of the model for the funding of an SPDA liability stream using government bonds, mortgage-backed securities and derivative products.

2

1 Introduction Interest rate risk in xed-income markets has, traditionally, been managed using a very simple idea: matching the interest rate sensitivity of both sides of the balance sheet. The derivatives | duration and convexity | of security prices with changes in the interest rates are matched and this ensures that, at least for small changes of interest rates, assets and liabilities will move together. This approach is widely known as portfolio immunization, see, e.g. Dahl et al. [1993]. In spite of its numerous shortcomings portfolio immunization remained the method of choice for nancial intermediaries since the interest rate deregulation of the 1970's, Fabozzi and Pollack [1987]. In particular it seems to work reasonably well { especially in various enhanced forms like factor immunization, key-rate duration matching etc. | for a variety of xed income instruments and for moderate interest rate volatilities. However, the immunization approach breaks down when employed in the context of novel and complex nancial instruments, whose price volatility is due to other factors than interest rate shifts. We have here in mind instruments such as corporate bonds with call options, mortgage-backed securities, insurance products, etc. In these cases an integrated asset/liability management strategy needs to recognize explicitly the correlation between both sides of the balance sheet. This approach, which has received some attention in practice, was formalized in the framework of integrated nancial product management by Holmer and Zenios [1994]. Holmer [1994] discusses models for interest rate risk management developed by the Federal National Mortgage Association, FNMA, Worzel, Vassiadou-Zeniou and Zenios [1994] discuss a model developed by Metropolitan Life Insurance, METLIFE, and Mulvey and Zenios [1994] discuss applications to the high-yield bond markets. At the same time we have seen the emergence of a class of dynamic, stochastic programs that accurately capture the dynamics of a wide range of portfolio management problems. The contribution that motivated these developments was the bond portfolio management model of Bradley and Crane [1972]. They formulated the problem as a dynamic programming problem on a decision tree. Stochastic programming formulations for cash- ow management were then suggested by Kallberg, White and Ziemba [1982]. Kusy and Ziemba [1986] applied multiperiod stochastic programming to asset/liability management at a Canadian bank. Mulvey and Vladimirou [1992] proposed | and validated with empirical results | stochastic programs for international asset allocation. Zenios [1993] developed a model for managing portfolios of mortgage backed securities. This model was implemented and tested by a money management rm, as described in Golub et al. [1994]. Hiller and Eckstein [1994] | developed similar models for managing xed income portfolios. A classi cation of asset/liability management techniques | and a comparison of the classical immunization approach with dynamic, stochastic programs | is given in Hiller and Schaack [1990]. In this paper we propose a dynamic, multiperiod model for the management of a portfolio designed to fund Single Premium Deferred Annuity (SPDA) liabilities. SPDAs are policies sold by insurance companies. The insurer assumes the long-term responsibility for providing the insurance taker (annuitant) with income past his or her retirement age, in return for a 3

single premium paid by the annuitant in advance. An SPDA comes with a valuable option: The annuitant may at any time lapse, i.e., withdraw part or all of the balance of the policy, subject to a possible penalty. In the event of rising interest rates, the insurer is forced to increase the rate at which interest is credited to policy accounts, or face policy lapses. Hence, these instruments are particularly sensitive to interest rate changes. It is imperative for the insurer to fund the SPDA liability in such a way that the underlying assets exhibit cash- ow patterns closely matching those of the SPDA liability, under a variety of interestrate scenarios. The model takes the form of a stochastic program and interest-rate scenarios are explicitly taken into account. An empirical model of lapse behavior is used to project the SPDA liability streams for these scenarios. The asset portfolio is then constructed from a speci ed universe using state-of-the-art methods for calculating interest-rate dependent prices and cash- ows for the assets. In particular, mortgage securities and derivative products are used to synthesize a portfolio that is highly correlated with the liability stream. The model assumes an actively managed portfolio, i.e., the asset portfolio is constructed taking into account the opportunities for subsequent rebalancing in response to changes in interest rates, prices etc. A measure of the insurer's nal wealth is maximized while solvency is ensured. In Section 2 we discuss SPDAs. The stochastic model is de ned in Section 3. Section 4 discusses the data requirements of the model, particularly the generation of interest scenarios and the pricing of assets. In Section 5 we discuss criteria for evaluating the model, and solve numerically a set of test problems. Section 6 concludes the paper.

2 Preliminaries: Single Premium Deferred Annuities A Single Premium Deferred Annuity (SPDA) is a contract between an insurer and an annuitant. The annuitant pays a single premium into an account maintained by the insurer. The insurer pays interest on the account, according to a crediting rate. When the annuitant reaches retirement age, the insurer pays out a monthly annuity for the lifetime of the annuitant. An SPDA thus provides a guaranteed income during the retirement years of the annuitant. An additional feature of SPDAs which makes them attractive as investment vehicles is that the interest income received on the account is tax-deferred, i.e., it is not taxable until payments commence upon retirement. The annuitant has the option to lapse, i.e., withdraw the balance of the insurance, or part of it, at any time, possibly subject to penalties. There are many variations on this simple scheme. Some SPDAs, for instance, pay out for at least a xed number of years, even if the annuitant passes away. An SPDA can be called an \upside-down life insurance". Life insurance is typically paid by periodical installments, and pays out a xed amount upon the death of the insured. In contrast, an SPDA is bought with a single premium, and provides an income stream for the insured past retirement age. SPDA purchasers expect to live longer than life insurance purchasers, and actuaries use dierent life expectancy tables to determine the premia. While a life insurance policy usually requires the insured to be in good health, there is, for obvious 4

reasons, no such requirement for the purchase of an SPDA. In order to fund SPDA liabilities, the insurance company needs to understand the uncertainties inherent in the cash- ow stream of each policy. The primary source of uncertainty is the lapse option built into the instrument. At any time, the annuitant may withdraw all or part of the money held in his or her account. Usually, only a fraction of the account (e.g., 10%) can be withdrawn annually without incurring penalties (surrender charges). The applicable surrender charges typically decrease with the age of the account. In addition, there may be a tax penalty if funds are withdrawn before the age of 59 21 and not reinvested into another tax-deferred annuity. The possibility of lapse constitutes a risk to the insurer. Lapse behavior is, to a large extent, determined by economic factors that are common among annuitants, which lead to a high degree of correlation of lapse across policies. This source of uncertainty, together with the interest rate risk, are our primary modeling concerns. Another form of uncertainty with SPDAs, namely the annuitants' life expectation, is well-understood by actuaries who provide accurate projections to insurers. We model lapse behavior following the work of Asay, Bouyoucos and Marciano [1993]. Their model assumes that lapse behavior is dependent upon three factors: 1. Crediting Strategy: The insurer unilaterally determines the crediting strategy, i.e., the interest rate paid on the account. Adjustments are subject to an initial guarantee period, typically one year, a oor, and possibly maximum periodic changes. The crediting rate is typically reset annually. 2. Competitors' Crediting Strategies: Annuitants can withdraw funds without incurring a tax penalty if the funds are reinvested into another annuity. Hence, if competing insurers issue SPDAs with higher yields, lapse behavior will accelerate. 3. Interest Rates: Similarly, if the crediting rates are signi cantly below the risk-free interest rate, lapses will occur, even if reinvestment in risk-free instruments incur tax penalties. Details of the lapse model are presented in Section 4.3. The instruments to be used in hedging the SPDA liabilities are primarily mortgagebacked securities (MBS). Mortgage-backed securities are attractive instrument for consideration in a portfolio since they have had high yields in the past. Another reason for focusing on MBS's is that we have available prepayment models which provide accurate price and cash- ow data under dierent interest rate scenarios, see Kang and Zenios [1992]. The basic MBS is the PT (pass-through), in which the cash- ows from a pool of mortgages (i.e., home owners' monthly payments of principal and interest, and prepayments, for instance due to re nancing) are passed on the owner of the security. However, derivative instruments in which the security buys only the principal part of the cash- ow, principalonly (PO), or only the interest part, interest-only (IO), have been designed, which are much more sensitive to interest rates changes than PTs. In a rising (declining) interest-rate climate, POs will exhibit increased (decreased) duration. Since SPDA liabilities exhibit some of these same characteristics, we might expect a portfolio of mortgage-backed securities designed to match SPDA liabilities to contain some POs. 5

3 The Stochastic Model We present now a stochastic programming model for funding SPDA liabilities. The simplest stochastic programming model is the two-stage model which is introduced next. This model is generalized to more than two stages in Section 3.2. The problem we are modeling is the following: Construct a portfolio to fund a given stochastic SPDA liability stream under a set of interest rate scenarios. Among the possible solutions, nd the portfolio which has the best risk-return pro le over the scenario set. We will make precise the notions of interest-rate scenario sets and risk-return pro le. The portfolio is to be constructed from a universe U of nancial instruments. The model is dynamic, and all events, such as asset trading or coupon payments, occur at discrete time points, t = 0; :::; T . An initial portfolio is constructed at time 0 = 0, and is subsequently rebalanced at time points 1 < 2 < ::: < Y = T . During the periods between rebalancing points (where period p is the time span p?1  t < p ) the portfolio composition remains unchanged, except that cash- ows are reinvested at the short rate. Transaction costs are included, and limited borrowing is allowed. Uncertainty is modeled using a set of interest rate scenarios, generated according to a suitable term structure model (see Section 4.1). Lapse behavior for SPDA annuitants is driven by the short-term rates under each scenario, leading to a stochastic liability stream. On the asset side, the prices of assets in future time periods, as well as the short-term borrowing and lending rates, are also interest rate driven. Their estimation is described in Section 4.2. The objective of the stochastic model is to construct a portfolio whose cash ows match the liabilities in each time period under all scenarios, while having a risk-return pro le consistent with a prescribed level of risk tolerance. In its basic form the model is a two-stage, stochastic program with recourse. The rststage decision is the construction of the initial portfolio. After a realization of interest rates has been observed, the portfolio is rebalanced. The construction of the rebalanced portfolio constitutes the second-stage decision. Of course, rebalancing decisions are contingent upon the realized scenario and the composition of the initial portfolio. For a given interest rate scenario, the model has a network structure, as shown in Figure 1 for two assets (Mulvey and Vladimirou [1992]). Columns of nodes correspond to dierent time points, while rows of nodes correspond to dierent assets. The bottom row of nodes corresponds to cash. Horizontal arcs | between nodes corresponding to the same instrument i 2 U | model the holding of instrument i in the portfolio. Arcs which link nodes in the bottom row model short-term cash reinvestment and borrowing. The vertical arcs which link cash and instrument nodes model changes in the position of each instrument, that is, rebalancing. The initial infusion of cash into the model consists of the SPDA premium and possibly insurer equity, and is used for constructing the initial portfolio. Triangles on arcs designate arc multipliers, i.e., that the value (cash or holding) entering the arc is changed by some proportion (multiplier) before leaving the arc. For arcs modeling 6

Period 1

...

Period 2

t=0

t=τ1

x1is

Period Y

s mpi

t=τp

xpis

t=T

Instrument 1

z1is

y1is

y0i

ypis

zpis

zYis

Instrument 2

1- γ

s v1

Cash

s v2

s u1

s

W

s u2

s

s

L1

L2

s

LY

First Stage Variables (Scenario independent) Second Stage Variables (Scenario dependent)

Figure 1: Network model underlying the two-stage, stochastic SPDA model. This gure includes two instruments, and depicts a 4-period model. Stochastic quantities are denoted by a superscript, s.

7

sales of assets, this multipler represents a transaction cost on sales. For reinvestment and borrowing, the multiplier represents short-term reinvestment and borrowing rates. Multipliers on the arcs which represent the presence of instruments (holding arcs) model the yield of the instrument for each time period. Of course, these multipliers are dependent on the speci c scenario, S = f1; :::; S g. The arcs in bold type for investments at time 0, i.e., construction of the initial portfolio, are rst-stage decisions, and must be the same regardless of the scenario. The remaining arcs (later period holdings and rebalancings) represent second-stage decisions which are allowed to depend on the scenario realized. This distinction between rst- and second-stage decisions is a key feature of two-stage, stochastic models, where immediate decisions cannot depend on, as yet, unrealized data, but future decisions can.

3.1 Model De nition

We now describe the components of the stochastic model. The model is described in three parts, dealing with security holdings, cash position, and the model's objective. The complete notation and algebraic formulation of the model is given in Appendix A.

Security Holdings: The variables xspi are used to represent the holding (in dollar value)

of instrument i during period p under scenario s. Purchases and sales of securities are represented by the variables ypis and zpis , respectively, where the index p refers to transactions which take place at the end of period p. The constraints which de ne the holdings of securities during each time period under each scenario can be written as:

y0i; for all s 2 S ; i 2 U : = mspi xspi ? zpis + ypis ; for all s 2 S ; i 2 U and p 2 1; :::; Y ? 1:

xs1i =

xs(p+1)i

(1) (2)

Equation (1) states that the initial holdings equal the initial investments. The initial investment, y0i , must not depend on the scenario to be realized, and hence does not have the scenario superscript. These variables are all measured in cash value rather than face value. Equation (2) de nes the changes in the cash value of holdings due to yields, sales and purchases of instruments. The multipliers mspi represent the yield during period p, of instrument i, under scenario s. We assume that the nal portfolio must be liquidated, and this is ensured by the following constraint, which states that sales after the last period must equal holdings after that period:

zYs i = msY i xsY i ; for all s 2 S ; and i 2 U :

(3)

Cash Position Accounting: Next come de nitions of cash positions in each time period. The initial amount of cash available (premium and equity) is denoted by C . Cash is used for purchases of securities and the payment of liabilities, and is generated by sales. During each period, excess cash is invested at the short rate, and borrowing is allowed. We use usp 8

to designate short-term cash investments during each time period, and for each scenario, and vps to denote short-term borrowing. The constraint for the rst time period states that the initial amount of cash available, C , plus any rst-period loan, v1s, must equal the amount invested in instruments, y0i or held in cash during the rst period, u1:

X i2U

y0i + us1 ? v1s = C; for all s 2 S :

(4)

In general (although not for the rst time period), short-term borrowing and cash investment are scenario-dependent, hence the superscript s. The corresponding cash- ow balance constraints for the intermediate time periods are:

X

i2U

((1 ? )zpis ? ypis ) ? usp+1 + (1 + rps )usp ? (1 + rps +  )vps + vps+1 = Lsp ;

(5)

for all s 2 S ; p = 1; :::; Y ? 1;

where is a transaction cost (charged as a fraction of the amount of the transaction) when selling instruments and Lsp is the liability due at the end of period p under scenario s. For simplicity, transaction costs on purchases are not used. Rather, should account for total costs. The cash- ow constraint for the last time period is:

X i2U

(1 ? )zYs i + (1 + rYs )usY ? (1 + rY +  )vYs = LsY + W s for all s 2 S :

(6)

It diers from (5) in not allowing purchases, and in de ning the nal wealth under each scenario, W s , as the surplus cash after the last liability, LsY , has been paid.

Objective Function: The objective of the model is to maximize the expected value of a

measure of nal wealth across scenarios. We maximize the Expected Utility of Return on Equity. Return on Equity, ROE, under scenario s is de ned as rs = W s =E , where E is equity. Utility is measured using the family of iso-elastic utility functions (Ingersoll [1987]):

U(r) =

(

? (r

1

1

log(r)

? ? 1)

1

for  6= 1; for  = 1;

(7)

where   0 is a risk-aversion parameter. Higher values of  implies more risk-aversion, i.e., less tolerance for risk on the investors' part. The value  = 0 results in a linear utility function which corresponds to a risk-neutral attitude, and  = 1 in a logarithmic utility function which corresponds to a moderate level of risk-aversion and which is known as the growth-optimal strategy. The properties of logarithmic utility functions for portfolio selection are discussed in McLean, Ziemba and Blazenko [1992]. Assuming that all scenarios are equally probable, the objective function of the model is de ned by X Maximize Expected Utility = u =: 1 U (W s=E ); (8) 

9

j S j s2S



where u is the expected utility of return on equity. Utility measures in themselves are not meaningful, except for ranking uncertain outcomes. Hence, we employ the more meaningful certainty-equivalent return on equity, CEROE. The certainty-equivalent has the same utility as the expected utility of the investment. The investor with the prescribed risk-attitude is indierent between receiving the (deterministic) certainty-equivalent return and the (stochastic) portfolio return. We use the CEROE value of dierent portfolios when comparing alternative portfolio management strategies. The Certainty-Equivalent is de ned by CEROE = U?1 (u); (9) and is used in comparing investors' preferences among dierent investments. A complete description of the model is given in Appendix A.

3.2 Extension to a Multistage Model

The two-stage model described above allows rebalancing decisions at times t > 0 to depend on future prices and returns, i.e., on data not yet known at time t. This implies that the decision maker has \perfect foresight" in making rebalancing decisions after the rst time period. This is of course unrealistic. To overcome this problem, the two-stage model is extended to a multistage model in which decisions at time t > 0 do not depend on the speci c sequence of events which will be realized during later time periods, but depend only on events observed prior to time t. Multistage models are a much better representation of reality than two-stage models. They have the same data requirements as two-stage models, but are substantially more complex in structure, and can be signi cantly harder to solve. Similarly to the two-stage models, they are based on scenarios of future interest rates. In the two-stage case these scenarios are independent of each other but in the multistage case, they are grouped together such that certain scenarios are indistinguishable from each other up to a certain time point (See Figure 5, where 8 of the 16 interest rate scenarios are indistinguishable up to time 6, groups of 4 are indistinguishable up to time 12, etc.). The lack of foresight mandates that decisions made under such indistinguishable scenarios, up to the time point where they dier, should be the same under each scenario. This leads to the requirement that some variables should be identical across scenarios. For more discussion of the structure and formulation of multistage models, see Gassmann [1990] or Nielsen and Zenios [1992] . Section 5 discusses the solution and evaluation of the multistage models, and Section 5.5 compares the results of the multistage models with the results of simpler asset allocation strategies.

4 Data Requirements for the Model We now turn to the problem of data generation. The data requirements are the liability

ows, and prices and cash- ow data for the instruments in the portfolio under each interestrate scenario. We also need to generate scenario-dependent prices of the assets at future 10

time points. In order to generate the model data we couple standard interest rate models with empirical models of the lapse behavior of annuitants, and the prepayment behavior of mortgage pools. The scenario structure is generated rst, using suitable interest rate models such as the binomial lattice of Black, Derman and Toy [1990]. Price scenarios are then computed o the generated interest rate scenarios using fairly standard pricing models, and cash- ow projections for the securities under consideration. For the lapse behavior, for example, we use the model of Asay, Bouyoucos and Marciano [1993]. For prepayment projection we use the model of Kang and Zenios [1992]. The procedure then is to specify, or generate, a set of interest rate scenarios, consistently with the binomial lattice model. For each scenario, the prepayment model is then used to generate mortgage cash ows (and prices), and the lapse model is used to generate insurance liabilities. This framework allows the model to capture very complex correlations between the asset and liability cash ows, under very general assumptions of the term structure. The details of the models are given in the following sections.

4.1 Speci cs of the Term Structure Model and Scenario Generation

The term structure model underlying the scenario generator is the binomial lattice model the Black, Derman and Toy [1990]. The term structure is de ned by a series of base rates, bt, and volatility coecients, kt , for t = 0; :::; T ? 1. These parameters are estimated according to the procedure given in Black, Derman and Toy [1990], based on the input term structure (yield and volatility curves). Scenarios in the set S are constructed as follows. Let !ts , for t = 0; :::; T ? 1 and s 2 S , be independent random variables that P take the values 0 or 1, with equal probability. Then  s = (0s; 1s ; :::; Ts ), where ts = tu?=01 !us , for t = 0; :::; T , is a path through the binomial lattice, see Figure 2. There are hence 2T possible paths through the lattice, each with probability 2?T . The path  s is used to construct an interest rate scenario, rs = (r0s; r1s; :::; rTs ?1), where rts is the short-term rate at time t under scenario s, as follows:

rts = bt(kt)ts ; for t = 0; :::; T ? 1 and s 2 S : For each interest rate scenarios sampled from the binomial lattice, its \mirror image" is also included. This variance reduction technique is known as antithetic sampling, see, for example, Figure 5.

4.2 Estimating Price Scenarios

To estimate the price of a security at a future time point  , under a scenario s, we let S` = fs 2 S j s = `g denote the set of scenarios that go through the state ` of the lattice at time  , see Figure 3. Let Cts denote the cash- ow (per unit of face value) of a security at time t,   t  T , under scenario s 2 S` . Then, according to the expectations hypothesis,

11

state S

CTs

 ? ? @@  ? .. . @@ ? ? ? ?? @? ? ?? C +2 ? @? ?  @ ? ? @? ???? C +1@ ? ? ?    @ ?  ? ? @ ? ? ? ? @ P @ ? ????? @?@R? ?@ ? @ @?? ??@?@? ?@??? ?? ?? R???@R? ? ?  @?  ?  @ ? @ ?? ? ? ? ? ? ? ? @ R @ ? ?     ? ?? ?? ?? ?? ?? ?? ?? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?  ? ? ? ? ? ? ? ? ?  ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 

@

s 

s 

s 

`

s

0

0



T

time

Figure 2: A path through the binomial lattice corresponding to interest-rate scenarios s. a fair price for the security at time  , conditioned by the scenario s = `, is given by

P = j S1` j

T XX

s Qt? C(1t + rs ) : u  u 

 s2S` t=

1 =

(10)

This relationship is the foundation for the pricing models. The prices of mortgage-backed securities at any given time are path-dependent, i.e., they depend not only on the interest rate at time p , but also on the history of rates up to time p. Calculating the prices directly using (10) would require the enumeration of the complete set of samples in S` , which is usually prohibitively expensive. Instead, we sample the set S` . For each sample path, the prepayment model of Kang and Zenios [1992] is used to calculate the cash- ows Cts for t   , and the price is calculated using (10) for the sample set.

4.3 Calculating the Liability Stream

In order to calculate the stochastic liability stream, we need to quantify the lapse behavior for SPDA annuitants. To do this, we adopt the behavioral assumptions of Asay, Bouyoucos and Marciano [1993]. It is assumed that the issuers follows crediting strategies, which are functions of interest rates. It is also assumed that annuitants' lapse behavior is a function of interest rates, as well as of crediting rates. 12

state S

? ? ? ?? ? ??? ?? ? ? ? ?  9 ? ? ? ????? >> ? ? ? ?  > sub-lattice S to ? > ? ? ? ? ? = ? ? price P ? ??  estimate ? ? ? ? ? at time  ? ? ? ? @? ? ? ??? ? ?? >> for state ` @?@ ? ??? ? ?  >>  ? ?? ? ? ?? ; ? ? ? ? @ R @ ? ? ? ?? ?? ?? ?? ?? ?? ?? ?          ?????????????????????????? ?          P0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?  ` t

s 

`

0

0



T

time

Figure 3: Calculation of the fair price. Paths corresponding to scenarios in S` , i.e., which go through (; `), are used in the calculation.

13

Annual Lapse Percent Surrender Charge = 0% Surrender Charge = 3% Surrender Charge = 7%

40.00 35.00 30.00 25.00 20.00 15.00 10.00 5.00

Competitor Less Credited Rate, % -5.00

0.00

5.00

Figure 4: Lapse behavior as a function of the Competitor rate (COt) minus Credited rate (CRt), for Surrender Charges of 0%, 3% and 7%. The insurer has the option to change (reset) the oered crediting rates annually, but cannot let the crediting rate re ect every increase in the interest level, since the premium may be invested in long-duration instruments. The model assumes a crediting policy where the crediting rate is adjusted some given fraction, i , towards the current interest rate once a year, and is unchanged during each year. The fractional adjustment of the crediting rate is assumed given. The crediting rate CRst is given by

CRst = CRst?12 + i (rts ? CRst?12); t = 12; 24; :::; T;

(11)

where rts is the short-term interest rate under scenario s, and CRs0 is the (contractually speci ed) initial crediting rate. While the insurer may not be able to match market rates on existing SPDAs when interest rates increase, there is no such restriction for new SPDAs, where the insurer can invest the initial premium at market rates. We incorporate a separate crediting strategy for new SPDAs, the competition's crediting rate:

COts = maxfCOts?12 + c (rts ? COts?12); rtsg; t = 12; 24; :::; T

(12)

(Asay, Bouyoucos and Marciano [1993] use the 5-year treasury rate instead of the short rate rts ). Hence, if short rates increase, competitors are assumed to increase their initial crediting rates correspondingly. To decrease the motivation of annuitants to switch to competitors when rates rise, the annuitant's lapse usually allows the insurer to incur penalties which the annuitant must 14

pay, typically 7% the rst year, 6% the next and so on until year 8, where-after lapse carries no penalty (annuitants can often withdraw up to 10% of the premium annually without incurring this penalty). The annuitants' lapse behavior is modeled by the following formula, which speci es the annual fraction of policy lapses and the eect of the surrender charges, SC : qts = a + b Arctan((COts ? CRst ? SC ) ?  ); (13) where a, b,  and  are parameters used to t the lapse function to empirical data. Figure 4 shows the lapse behavior for dierent levels of surrender charges. Presumably, the propensity to lapse should change with the age of the annuitant. This aspect has not been modeled in the literature. By grouping together SPDAs of similar characteristics, but covering annuitants of dierent ages, we get instruments with an \average" lapse behavior, as modeled here. Let now Hts be the holding of SPDAs at time t = 0; 12; 24; :::; T under scenario s. We express Hts in dollar value. Starting with an initial holding of H0s = P , the recurrence relation for the SPDA holding is

Hts = (1 ? qts)(1 + CRst)Hts?12; t = 12; 24; :::; T: Finally, the liability to be paid at time t under scenario s is then de ned by

Lst = qts (1 + CRst)Hts?1 ; t = 12; 24; :::; T::

(14)

Liabilities are paid once per year, i.e., Lst = 0 when t is not a multiple of 12. At the end of the planning horizon the remaining SPDA pool is assumed to be liquidated. This is modeled by letting qTs = 1. We assume that the liability remaining at time T can be valued at face value, although this may not be the case in reality.

4.4 Calculating the Coecients of the Stochastic Program

We are now ready to describe precisely the calculation of the parameters mspi of the stochastic model, (6){(7). These are the model parameters which capture uncertainty. The calculations are based on the prices, Ppis , and cash- ows, Cpis , of instrument i at time p under scenario s. The multiplier mspi represents the relative change in value of a holding of one unit of instrument i during the period from time p?1 to p , under scenario s. This change in value has two components: 1. Price change: The price of the instrument changes by a factor Fpis Ppis =P(sp?1)i, where Fpis is the face value during period p. 2. Cash ows: Assets will generate cash- ows during period p. For bonds, this cash- ow is coupon payments, and, at maturity, the principal. For mortgage-backed securities and their derivatives, the cash- ow consists of a combination of interest and principal payments. 15

Assuming that intermediate cash- ows are reinvested at the short rate (a very conservative assumption), the value of the cash- ows at time p, adjusted by a factor 1 ? to compensate for the buy-sell spread, is given by p ?1 Ctis
Y s P(sp?1)i t=p?1 +1 1 ? u=t (1 + ru ):

1

 X p

Combining these two sources of changes to the instrument value, we have the following formula for the yield multipliers mspi :

mspi = P s

1

p?1)i

(

0  @Fpis
Ppis + X

1

p ?1 Ctis
Y (1 + rus )A : 1 ?

u=t t=p?1 +1 p

(15)

5 Evaluation of the Model In this section we solve the SPDA model based on various investment strategies and model parameters. In doing this, we seek to answer the following questions: (1) Model robustness: Is the model's solution, i.e., optimal rst-stage portfolios, reasonably insensitive to changes in model parameters, and (2) Model eectiveness: is the model eective compared to other, simpler, strategies. We rst establish a base case model by specifying the investment universe and model parameters used. The base model is then solved using increasingly complex rebalancing strategies, from a buy-and-hold strategy to a 5-stage, stochastic strategy with rebalancing, and using dierent levels of risk-aversion. Then, we investigate the robustness of the model with respect to changes in the transaction costs and the portfolio universe. Finally, we compare the results with those of several simpler portfolio management strategies.

5.1 The Base Model

The basic model used for our experiments has the following features: 1. Model Horizon: The model has a horizon of 36 months. Rebalancing decisions are made every 6 months. The initial portfolio is constructed at time 0, and the performance of the nal portfolio is evaluated at month 36. 2. Objective Function: We solve instances of the model using the linear, risk-neutral utility function U0 (r) = r, and the growth-optimal, logarithmic utility function U1 (r) = log(r). This logarithmic utility function implies some risk-aversion, and generally leads to more diversi ed portfolios with a modest decrease in expected returns. 3. Investment Universe: A set of 27 mortgage-backed securities (MBS) is used, as described in Appendix B. The primary form of MBS is the Pass-Through (PT), where the cash- ows generated by the mortgage-holders are passed through to the MBS 16

Spot Rates, Percent 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 Months 0

5

10

15

20

25

30

35

Figure 5: Antithetic \extreme" interest rate scenarios used for the stochastic models. Rates shown are short rates. owner, but we also use the more volatile derivative instruments, IOs (where the Interest Only is payed to the MBS owner), and POs (which pay the Principal Only). In addition to MBSs, a set of government bonds are available for investing. 4. Term Structure and Interest-Rate Scenarios: The term structure used to generate interest-rate scenarios is estimated based on data from April 26, 1991. The scenario set consist of all combinations of \all up" and \all down" interest paths during the time periods 0-5, 6-11, 12-17 and 18-35 months. These scenarios are illustrated in Figure 5. 5. Borrowing and Reinvestment: Borrowing was disallowed in the base case. Reinvestment was allowed at the short rates under each scenario. 6. Lapse Behavior: The lapse function given in (13) is used, with a = (0:45 + 0:04)=2; b = (0:45 ? 0:04)=;  = 90 and  = 0:18, i = c = 0:2 and SC = 1%, corresponding to a surrender charge of 3%, amortized over 3 years. Lapse ratios vary between a minimum of 4% and a maximum of 45%. This lapse function is shown in Figure 4, together with the corresponding functions for SC = 0% and SC = 7%. 7. Initial Cash Position: The SPDA premium is set to P = $100 (the absolute magnitude of this gure is unimportant), and the equity to E = 2:5, so that the initial cash position is C = 102:5. 8. Transaction Costs: The transaction cost of the base model is = 1%. 17

Strategy Risk-Neutral Growth-Optimal \1 stage" U-U: U-D: D-U: D-D: 2 stages 3 stages 4 stages 5 stages

( = 0)

100.00% 7-PO 17.42% 2-IO 82.58% 7-PO 14.81% 2-IO 85.19% 7-PO 20.59% 2-IO 79.41% 7-PO

( = 1) 100% 2-IO 100% 73-IO 100% 7-PO 100% 70-PO 68.25% 2-IO 31.75% 7-PO 66.34% 2-IO 33.66% 7-PO 61.08% 2-IO 38.92% 7-PO 58.45% 2-IO 41.55% 7-PO

Table 1: The optimal rst-stage portfolios to the base model using 1- to 5-stage stochastic models. Results are shown under both risk-neutrality ( = 0) and risk-aversion ( = 1). The solutions to the \1-stage" program (with perfect foresight) depend on the direction of interest rates during the rst two time periods, up (U) or down (D), and are in this case the same for  = 0 and  = 1. The rst-stage, optimal investment depends on second-period interest rates because of transaction costs. In the following section we solve the base model as stochastic programs with varying numbers of stages. We then investigate the question of the model's robustness, or sensitivity to parametric changes. Finally, the results of the stochastic model are compared to those of simpler asset allocation strategies.

5.2 Solving the Base Model using Stochastic Programming

The base model is now solved as a dynamic, stochastic program. The \classical" stochastic program is the 2-stage program, where a unique rst-stage decision is made with no knowledge about the rst-period (future) events, but subsequent rebalancing (or recourse) can be made with full knowledge of the future. However, two-stage models use information about the future which is not actually available when making the second-stage decision, so it is more realistic to solve a multi-stage stochastic program, which is prohibited from using information before it is logically available. We hence solve the model with 2 up to 5 stages. We also solve the \1-stage" program, in which rst-stage decisions can be made separately for each scenario, with full knowledge of the future under each scenario. This experiment will allow us to establish the bene ts of perfect foresight. More precisely, the dierence in returns between an n-stage and the n + 1-stage models is a measure of the expected value of perfect information of events during the nth period. 18

5.3 Interpretation of Results

We show in Table 1 the optimal portfolios from the stochastic programs with varying stages and for both risk-neutrality and risk-aversion. The only active instruments | except for the "1-stage" model | are FNSTR-2-IO and FNSTR-7-PO. We observe, based on the option adjusted premium (OAP) given in Appendix B, (for a discussion of OAP refer to Babbel and Zenios [1992]) ) that FNSTR-2-IO trades at the highest discount compared to other IOs, and the FNSTR-7-PO trades at the lowest premium compared to other POs. They are the preferable instruments to add in the portfolio. The proportion of each instrument in the portfolio is chosen in such a way that the IO and PO act as natural hedges against each other | taking also into account the SPDA liability stream | as their returns move in opposite directions with changing interest rates. The PO is preferred by the risk-neutral investor (who has higher tolerance for risk), having higher expected return than the IO. With only two stages, the optimal risk-neutral portfolio consists of only the PO. The solutions to the more realistic 3- to 5-stage models are diversi ed by also investing in the IO. As we would expect, solutions to models with fewer stages are more risky than to those with more stages, since the former take information about the future into account earlier, and rebalances away from bad portfolios. We also observe that the growth-optimal portfolios contain a much higher component of the IO, which further reduces the overall risk in accordance with the higher risk-aversion. The optimal risk-neutral portfolio is contrasted with the growth-optimal and other portfolios from the perspective of dynamic portfolio management in Section 5.5.4.

5.4 Model Robustness

We investigate in this section the issue of model robustness, i.e., how sensitive is the model to changes in model parameters. First, we vary the transaction costs, and observe the changes in the optimal, rst-stage portfolio. Then we exclude the most volatile derivative instruments (IOs and POs) from the portfolio universe to see if their presence makes a dierence in the quality of the portfolio.

5.4.1 Varying the Transaction Costs The reaction of the model to changes in the level of transaction costs is important for the practical use of the model. In this section we solve the model with the growth-optimal utility function while varying the transaction costs. The results appear in Figure 6. As one might expect, for high transaction costs, the optimal rst-stage portfolio will include a cash position to cover early liabilities. This happens at a transaction cost of 1.5%, where 8% of the portfolio value is shifted from the IO and PO into cash. Even for higher transaction costs, the relative proportions of the IO and PO change little, but an increasing amount of cash is invested at the short rate in order to meet the rst liability without incuring a high transaction cost. Solving the stochastic program without transaction costs is easier than solving the program with transaction costs, since it can be solved period-by-period independently, but 19

Portfolio Composition (%) 60.00

FNSTR-2-IO FNSTR-7-PO Cash

50.00 40.00 30.00 20.00 10.00 0.00

Transaction cost (%) 0.00

1.00

2.00

3.00

4.00

5.00

Figure 6: The eect of varying the transaction cost parameter, , on the composition of the growth-optimal portfolio.

20

Universe PTs only PTs and Bonds

First-Stage CEROE Exp. Portfolio Wealth

100.00% 12.00-PT 52.02% Zero-1 47.98% Zero-2 58.45% 2-IO Base Case 41.55% 7-PO

0.05

3.42

2.46

6.72

6.72

18.72

Table 2: Restricting the security universe to include only Pass-Throughs. The bonds used are zero coupon bonds maturing after 1 and 2 years, respectively. The logarithmic, growthoptimal utility function is used. The base case model is included for comparison, and has a signi cantly higher certainty-equivalent and expected return than the restricted models. of course the resulting portfolio is then not optimal in a realistic situation with transaction costs. The optimal no-transaction costs portfolio is 53.36% FNSTR-2-IO and 46.64% FNSTR-7-PO. We investigate in Section 5.5 the performance of this portfolio when transaction costs are assumed.

5.4.2 Using Pass-Throughs versus Stripped Mortgage Securities It appears from the preceding sections that the model never selects hedged portfolios including Pass-Throughs (PTs). This raises the question whether the interest-rate response of IOs and POs is the reason the model is using them, or whether the model is simply using IOs and POs to synthesize cheap PTs. Results of restricting the model to using PTs only, or using only PTs and bonds, are shown in Table 2 based on the 5-stage, growth-optimal model. The performance of the portfolios consisting of only PTs is very poor, as can be seen by comparison with the base case. In fact, when investments in bonds are allowed, bonds are preferred to PTs. This indicates that the less volatile returns of bonds are a better hedge than that provided by PTs. We conclude that IOs and POs are used by the model in a qualitatively dierent way than merely to synthesize PTs. Their inclusion in the security universe makes a substantial dierence in the quality of the resulting portfolios.

5.4.3 Discussion Transaction costs in uences the rst-stage decision in that assets are shifted into cash in the presence of high transaction costs. The model uses derivative mortgage instruments for their sensitivity to changes in interest rates, rather than to merely synthesize Pass-Throughs (PT). While PTs are not suitable instruments for hedging SPDA liabilities, portfolios constructed with suitable choices of mortgage derivatives could provide the required hedging. 21

5.5 Comparing Dierent Asset Allocation Strategies

We now contrast the results of the stochastic model with those of several simpler strategies for allocating assets: The buy-and-hold strategy, the case with no transaction costs, the \myopic" portfolio, the \mean-value" program and the \mean-liability" program. These strategies, that are enjoying various levels of popularity in the nance literature, are described below. The optimal solutions to each model, solved using the logarithmic, growthoptimal utility function, appear in Table 3. The results are discussed in Section 5.5.4

5.5.1 Buy-and-Hold vs. Stochastic Programming The simplest strategy when building a portfolio is a buy-and-hold strategy, where the model constructs an initial ( rst-stage) portfolio, which is not rebalanced at later stages. Parts of the portfolio can be sold, but only in order to satisfy liabilities. The optimal Buy-and-Hold portfolio consists of 61.48% 2-IO and 38.52% 7-PO. The certainty-equivalent return of this portfolio, without rebalancing, is 4.02, and its expected return 11.37, as compared with the base case (allowing rebalancing) of 6.72 and 18.72, respectively. Obviously the Buy-and-Hold strategy is very inecient when compared to a dynamic strategy. At the same time, the optimal Buy-and-Hold portfolio is, in this case, very similar to the optimal dynamic portfolio, so when rebalancing is allowed (as in Table 3), it performs almost as well.

5.5.2 The Myopic Program We investigated previously the no-transaction cost portfolio and now contrast it with the solution to the myopic program. The myopic program consists of maximizing expected utility of returns after the rst time period only, ignoring subsequent time periods and rebalancing opportunities. The myopic program can be viewed as a simpli cation of the no-transaction cost program with the additional assumption that return distributions between dierent time periods are independent. Under these assumptions, and further assuming that no liabilities are to be paid, the rst-stage portfolio which is optimal to the multistage, stochastic program coincides with the myopic portfolio, Mossin [1968]. Grauer and Hakansson [1982] formalized the notion of investing according to a myopic approach, and later employed utility functions in a multiperiod framework, Grauer and Hakansson [1985]. The myopic program is obviously much simpler than the stochastic program, since it only has one time period. The question then is how well the myopic solution performs when used in the context of the stochastic base model. For both models we charge a 1% transaction cost, and liabilities are included. We also point out that the data distributions are not independent among time periods. The optimal solution to the myopic program is composed of 10.21% 67-PO and 89.79% 90-IO. The performance of this portfolio is contrasted in Table 3 with the performance of the portfolios resulting from other strategies. The myopic portfolio diers substantially from the other portfolios. It has, of course, very good returns in the rst period, but is overall very risky, and actually had negative nal returns under some scenarios. Obviously, 22

it is not attractive to the risk-averse investor. We conclude that the myopic program is a poor approximation to the multistage, stochastic program.

5.5.3 Mean-Value Programs

Another commonly used approximation to a stochastic program is the mean-value program, which is obtained from the stochastic program by replacing all uncertain data (gains/losses and liabilities) with their expectations, or mean values. This results in a very simple deterministic, but (in contrast with the myopic model) multiperiod model. Solving this model resulted in an optimal rst-period portfolio consisting of only 7-PO. The return pro le of this portfolio (Table 3) is substantially more risky than that of the base solution, and is in fact dominated by the myopic solution (which has higher returns and is less risky). The mean-value program can be a very poor substitute for the multistage, stochastic program, as has already been demonstrated in the context of other nancial applications by Kallberg, White and Ziemba [1982]. In order to establish the extent to which the model exploits the correlations between the asset returns and the liability stream we also solved the mean-liability model. This model is identical to the multistage, stochastic model except that the stochastic liability stream is replaced by its average at each time point. This model corresponds to a portfolio management strategy whereby assets and liabilities separately rather than in an integrated fashion. The optimal portfolio to the mean-liability model consists of 54.03% FNSTR-2-IO and 55.97% FNSTR-7-PO, which is close to the base solution. From Table 3 we see that this solution is only slightly inferior to the base solution. The dierence in expected return is 7bp, and the portfolios have similar risk pro les. These results may explain why, quite often, assets and liabilities are managed independently from each other. (See Holmer and Zenios [1994] for extensive discussion of this topic.) If assets and liabilities are hedged against interest rate risk independently from each other, then the risk exposure of the whole balanced sheet is controlled. However, when viewed in an integrated fashion (assets minus liabilities) the balance sheet may be over-hedged because a natural hedge arising from correlations between assets and liabilities is not exploited. On the other hand, taking such correlations into account using integrated asset/liability management, allows the manager more exibility in the management of the IO and PO positions in the asset portfolio, and hence achieves the same level of risk exposure at a lower cost. This has been one of the main arguments in favor of the framework of integrated nancial product management of Holmer and Zenios [1994].

5.5.4 Discussion The results of the preceding sections clearly show that a dynamic strategy which allows periodical rebalancing of the portfolio is superior to other, simpler but popular strategies, such as buy-and-hold, myopic models, or mean-value models. 23

Solution Strategy

Optimal portfolio

Base 58.45% 2-IO, 41.55% 7-PO Risk-Neutral 20.59% 2-IO, 79.41% 7-PO Buy-and-hold 61.48% 2-IO, 38.52% 7-PO No transaction costs 53.36% 2-IO, 46.64% 7-PO Mean-Liability 54.03% 2-IO, 55.97% 7-PO Myopic 89.79% 90-IO, 10.21% 67-PO Mean-Value 100.0% 7-PO

CEROE Exp. Final ROE Wealth Min Max 6.72 4.64 6.71 6.69 6.68 N/A N/A

18.72 3.27 14.84 19.21 0.89 48.49 18.67 3.50 14.47 18.83 2.89 15.45 18.65 3.12 14.18 N/A N/A N/A N/A N/A N/A

Table 3: The performance of the base portfolio contrasted with portfolios obtained using simpler strategies. N/A: Results for these portfolios are not available because of negative nal wealth under some scenarios. On the other hand, the actual portfolios which resulted from the simpler strategies are often very similar to the optimal solution to the stochastic program. Does this mean that the solution to a simpler strategy is useful in a dynamic setting, although following the strategy itself is not? To answer this question, we modi ed the multistage, stochastic model to select a speci ed, initial portfolio, and then optimize the remaining decisions. The initial portfolios used are shown in Table 3, together with the certainty-equivalent returns resulting from their implementation, in the dynamic, multistage context, the expected nal wealth, and the range of return outcomes realized. The portfolios resulting from the buy-and-hold, the no-transaction costs, and the MeanLiability models are similar to each other and to the base portfolio, containing from 53% to 61% FNSTR-2-IO and the rest FNSTR-7-PO, and hence perform about equally well. This also indicates that the results of the model are robust, i.e., not overly sensitive to the exact composition of the rst-stage portfolio. The risk-neutral portfolio consists of the same two instruments, but even though it has (of course) the highest expected return, it is quite risky, with a certainty-equivalent return of 4.64. The Myopic and Mean-Value portfolios could not be evaluated ex-post using the logarithmic (risk-averse) utility function. For each portfolio, there was a scenario under which an initial investment in that portfolio would lead to negative nal wealth, where the log utility function is unde ned. For the Myopic portfolio, this happens because, although the portfolio contains a high proportion of IOs, its value remains reasonable high after one period of decreasing rates. However, when interest rates continue to drop during the remaining periods (which the myopic model could not take into account), the position after the rst period is too low to prevent insolvency. A similar explanation holds for the MeanValue portfolio: The mean returns of this portfolio do not lead to insolvency, but, under some scenario, the actual returns do. But even if the adverse scenario is excluded from the optimization, these portfolios lead to poor results: a CEROE of 5.03 for the Myopic, and a CEROE of only 2.14 for the Mean-Value portfolio. Both of these approaches to the 24

multistage problem can clearly lead to very poor solutions.

6 Conclusion A multistage, stochastic model has been developed for the problem of funding a stochastic SPDA liability stream. The model is a realistic approach to the problem, and results in solutions which eectively account for the active management of the asset portfolio and varying levels of risk-aversion among managers, while also being computationally tractable. The solutions of the model are relatively insensitive to changes in a wide range of model parameters. The model returns far less risky portfolios when solved under risk-aversion than when solved using a risk-neutral objective function. Returns of the model are substantially better than those obtained by solving either the mean-value or the myopic programs, which both ignore important aspects of the dynamic situation. The model appears promising in accounting for the important feature of future interest rate scenarios in a dynamic environment while allowing for actual period-by-period matching of asset and liability cash- ows, a hedging approach which is potentially much more powerful than more traditional approaches (such as scenario analysis or duration and convexity matching).

Acknowledgements. The authors bene ted from the comments of M. Holmer, D. Babbel,

W.T. Ziemba and of the referees. We also thank R. McKendall for technical assistance.

7 Appendix A: Summary of the Stochastic Model Below is given the complete, algebraic formulation of the stochastic model describe in Section 3, and the notation and variables of the model are de ned.

Two-Stage Stochastic Model for Funding SPDA Liabilities: X Maximize S1 U (W s =E ) Xs2S s s Subject to y i+u ?v =C for all s 2 S , iX 2U ((1 ? )zpis ? ypis ) ? usp + (1 + rps )usp i2U ?(1 + rps + )vps + vps = Lsp for all s 2 S ; p = 1; :::; Y ? 1; X s s s (1 ? )zY i + (1 + rY )uY i2U ?(1 + rY + )vYs = LsY + W s for all s 2 S ; s x i=y i for all s 2 S ; i 2 U ; xsp i = mspixspi ? zpis + ypis for all s 2 S ; i 2 U ; p = 1; :::; Y ? 1 0

1

1

(16) (17)

+1

+1

1

0

( +1)

25

(18) (19) (20) (21)

for all s 2 S ; i 2 U :

zYs i = msY i xsY i

All variables are non-negative.

(22)

Portfolio Construction and Rebalancing Variables: y0i : Contents of the initial portfolio (in cash value) of instrument i ( rst-stage variables). ypis : Purchase of instrument i at time p under scenario s. zpis : Sale of instrument i at time p under scenario s.

Auxiliary Variables: xspi : usp: vps : W s:

Holdings (in cash value) of instrument i during period p, under scenario s. Amount of cash invested at the short rate in period p. Amount of cash borrowed in period p. Final wealth under scenario s.

Model Data: C: rps : mspi: Lsp:

: :

The initial cash invested, consisting of the SPDA premium, P , and equity, E . The short rate during period p under scenario s. The change in value of a $1 holdings of instrument i in period p under scenario s. The liability due at time p under scenario s. Transaction cost (as a fraction of the transaction). The spread between borrowing and reinvestment rates.

26

8 Appendix B: Securities Data The universe of mortgage-backed securities used in this study consists of the instruments listed below. The securities consisted of Pass-Through (PTs) where all of the mortgage payments are passed on the the investor, and of interest-only (IOs) and principal-only (POs), where only the interest or principal parts of the payments are passed on to the investor. WAC is the weighted average coupon rate of the mortgages in the security and WAM is the weighted average maturity, in months. OAP is the option-adjusted premium, see Babbel and Zenios [1992]. The data shown are from April 26, 1991.

Security

FNMA-8.00-PT FNMA-8.50-PT FNMA-9.00-PT FNMA-9.50-PT FNMA-10.00-PT GNMA-12.00-PT FNSTR-1-PO FNSTR-1-IO FNSTR-2-PO FNSTR-2-IO FNSTR-4-PO FNSTR-4-IO FNSTR-4a-IO FNSTR-7-PO FNSTR-7-IO FNSTR-18-IO FNSTR-24-IO FNSTR-31-IO FNSTR-32-IO FNSTR-39-IO FNSTR-42-IO FNSTR-67-PO FNSTR-67-IO FNSTR-70-PO FNSTR-70-IO FNSTR-73-IO FNSTR-90-IO

WAC WAM 8.75 9.25 9.75 10.25 10.75 12.50 9.69 9.69 10.58 10.58 10.08 10.08 10.08 9.60 9.60 9.85 9.15 9.64 9.16 10.20 10.13 10.58 10.58 10.50 10.50 10.50 9.71

330 330 354 354 354 292 298 298 294 294 297 297 297 120 120 297 303 306 305 324 315 336 336 342 342 344 345

OAP

Price

1.067180 94.8438 1.082560 97.2188 1.087300 99.8438 1.093310 102.1562 1.102980 104.0625 0.960422 111.7812 0.815915 58.0940 1.467120 42.2380 0.776155 64.7190 1.584240 39.7390 0.803718 60.7810 1.526700 41.1250 1.474250 41.8480 0.886932 55.5000 1.457880 42.3770 1.566520 40.7813 1.465360 41.8750 1.500770 41.7500 1.439150 42.2813 1.434310 42.2500 1.441650 42.2500 0.856420 62.5010 1.416800 41.8950 0.957392 59.4060 1.165050 45.8330 0.931394 50.0960 1.286940 45.0400

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29