Asset/liability management under uncertainty for fixed-income securities

Annals of Operations Research 59(1995)77-97 77

Asset/liability management under uncertainty for fixed-income securities

Stavros A. Zenios

HERMES Laboratory for Financial Modeling and Simulation, Decision Sciences Department, The Wharton School,

University of Pennsylvania, Philadelphia, PA 19104, USA

Short-sighted asset/liability strategies of the seventies left financial intermediaries -banks, insurance and pension fund companies, and government agencies - facing a severe mismatch between the two sides of their balance sheet. A more holistic view was introduced with a generation of porO~olio immunization techniques. These techniques have served the financial services community well over the last decade. However, increased interest rate volatilities, and the introduction of complex interest rate contingencies and asset-backed securities during the same period, brought to light the shortcomings of the immunization approach. This paper describes a series of (optimization) models that take a global view of the asset/liability management problem using interest rate contingencies. Portfolios containing mortgage-backed securities provide the typical example of the complexities faced by asset/liability managers in a volatile financial world. We use this class of instruments as examples for introducing the models. Empirical results are used to illustrate the effectiveness of the models, which become increasingly more complex but also afford the manager increasing flexibility.

1. Introduction: The problem and its applications

Government agencies, such as the Federal National Mortgage Association (Fannie Mac), suffered severe losses during the early 1980's: the agencies had issued short-term non-callable bonds in the early 1970's to finance the purchase of long- term assets. However, the maturity mismatch of assets and liabilities exposed the agencies to substantial interest rate risk. As rates rose substantially in the latter part of the decade, Fannie Mae found itself with maturing liabilities, while their assets had a long remaining time to maturity and were priced at a fraction of their original value. Similar problems were faced by insurance carders: these companies used short-term liabilities (like their Guaranteed Investment Contracts) to fund long-term assets. In the upward sloping yield curve of the time, this strategy was deemed reasonable and substantial gains were expected. However, as rates rose they found themselves in the same predicament as the government agencies.

© J.C. Baltzer AG, Science Publishers

78 S.A. Zenios, Asset~liability management

While it is easy to criticize these mistakes after the fact, we should not forget that, at the time, interest rates were regulated and had remained quite stable for a long period of time. In the aftermath of these shocks, however, financial intermediaries took a more holistic view of the asset/liability management problem. A policy of duration matching the assets and liabilities was instituted among insurance and pension fund companies, banks and government agencies. See, for example, Christensen and Fabozzi [5], Holmer [13] or Platt [21].

Portfolio immunization - i.e. selecting portfolios with duration matched assets and liabilities - was instrumental in reducing the gap between the two sides of the balance sheet created by the short-sighted policies of the 1970's. However, as these techniques rose to widespread use throughout the 1980's, their shortcomings started to become apparent. A whole new generation of models started to emerge that could more fully cope with the volatile interest rate environment of the 1990's while dealing with the spectrum of complex financial instruments that were introduced throughout this period.

The objective of this paper is to describe a series of three such models. We use as a base case the problem of managing portfolios containing mortgage-backed securities (abbreviated: MBS). These securities - being some of the most complex instruments available in the financial markets - provide the framework for discussing a complete spectrum of issues that relate to the management of fixed-income portfolios under uncertainty. In particular, they are not only sensitive to changes in interest rates, but they are also volatile due to the embedded call option: A homeowner has the option to prepay the outstanding balance of her mortgage, with no penalty, and hence call the mortgage security. Other fixed-income securities exhibit similar characteristics. We mention, for example, callable bonds issued by corporations, single premium deferred annuities offered by insurance companies, options on bonds, etc.

The remainder of this section characterizes the complexities of the problem and discusses specific applications. Section 2 classifies and specifies three asset/ liability portfolio optimization models. Calculations of the input data required in order to operationalize the models can be obtained using extensions of standard pricing models that are grounded on prevailing financial theories. This is the topic of section 3. Empirical evidence on the performance of the models, and some discussion on further validation, are considered in section 4. The final section, section 5, provides some critical analysis of the models. The models developed here have been applied in several corporate settings for managing portfolios of mortgage-backed securities and callable bonds. These experiences are published elsewhere [10,23,7].

1.1. THE PROBLEM

We loosely define the problem addressed in this paper as follows:

Construct a portfolio of fixed-income securities whose performance measures will remain invariant under a wide range of uncertain scenarios.

S.A. Zenios, Asset~liability management 79

For now, we leave unspecified what we mean by performance measures and the precise nature of uncertainty. The key idea is to decide what goals we want our portfolio to achieve, specify measures that indicate that the goals are achieved, and make sure that these goals are still met when the economic environment changes. The three models we introduce in the next section allow the portfolio managers to specify increasingly more complex goals, and ensure that these goals are met for increasingly more complex scenarios.

1.2. APPLICATIONS

The precise goal of the portfolio manager depends on the underlying application. We describe here three practical applications where one needs to deal with fixed- income securities and their inherent uncertainties:

Indexation: Passive portfolio managers would like to build a portfolio of fixed- income securities that will track a prespecified index. For example, Shearson- Lehman and Salomon Brothers publish a monthly mortgage index that is (presumably) indicative of the overall state of this segment of fixed-income markets. Investors who wish to invest in mortgages may be satisfied if their portfolio closely tracks the index. The performance measure of such a portfolio is the difference in return between the portfolio and the index. This difference has to be very small, for all changes in the index caused by interest rate movements and by variations in prepayment activity.

Liability paybaek: Insurance and pension fund companies are typically heavily exposed to MBS. These instruments are considered as an investment for paying back a variety of the liabilities held by these institutions. The goal of the portfolio manager is to construct a portfolio of MBS that will pay the future stream of liabilities. Uncertainty here appears once more in the form of interest rate changes and changes in the timing of payments from the MBS. Furthermore, the timing of the liability stream may also be subject to uncertain variations: For example, the timing of payments to holders of single premium deferred annuities (SPDA) may change as annuitants exercise the option to lapse.

Debt issuance: Government agencies, such as Fannie Mae and Freddie Mac, fund the purchase of fixed income assets (typically mortgages) by issuing debt. The problem of a portfolio manager is to decide which type of debt - maturity, yield, call-option - to issue in order to fund the purchase of a specific set of assets. Of course, there is no reason to assume that the assets have been prespecified: The model may choose an appropriate asset mix from a large universe of fixed income securities. The timings of both assets and liabilities may be uncertain in this application. The goal of the portfolio manager is to ensure that the payments against the issued debt will be met from the available assets, irrespective of the timing of cash flows and fluctuations in interest rates.

80 S.A. Zenios, Asset/liability management

2. Structured asset/liability management models

We classify the asset/liability management models into: (1) static, (3) single- period, stochastic, and (3) multiperiod, dynamic and stochastic. It is important to understand how the models address increasingly more complex aspects of the asset/liability management problem. Only then can the portfolio manager decide which model may be more appropriate for the application at hand. Of course, this decision has to be weighted against the increasing complexity - both conceptual and computational - of the models.

Static models: Such models hedge against small changes from the current state of the world. For example, a term structure is input to the model which matches assets and liabilities under this structure. Conditions are then imposed to guarantee that if the term structure deviates somewhat from the assumed value, the assets and liabilities will move in the same direction and by equal amounts. This is the fundamental principle behind portfolio immunization. See, for example, Christensen and Fabozzi [5] for a discussion of the finance-theoretic principles behind immunization, and Dahl et aI. [8] for operational models.

Single-period, stochastic models: A static model does not permit the specification of a stochastic process that describes changes of the economic environment from its current status. However, modern finance abounds with theories that describe interest rates, and other volatile factors, using stochastic processes; see, e.g., Ingersoll [15]. Stochastic differential calculus is often used to price interest rate contingencies. For complex instruments, analysts resort to Monte Carlo simulations, an idea pioneered by Boyle [3] for options pricing. See, for example, Hutchinson and Zenios [14] for its application to the pricing of mortgage securities. A stochastic asset/liability model describes the distribution of returns of both assets and liabilities in the volatile environment, and ensures that movements of both sides of the balance sheet are highly correlated. This idea is not new: Markowitz pioneered the notion of risk management for equities via the use of correlations in his seminal papers [18,17]. However, for the fixed-income world this approach has only recently received attention. It has been formalized by Mulvey and Zenios [20], and was applied at Fannie Mac by Holmer [13].

Multiperiod, dynamic and stochastic models: A stochastic model, as outlined above, is myopic. That is, it builds a portfolio that will have a well behaved distribution of error (error = asset return - liability return) under the specified stochastic process. However, it does not account for the fact that the portfolio manager is likely to rebalance the portfolio once some surplus is realized. Furthermore, as the stochastic process evolves across time, different portfolios may be more appropriate for capturing the correlations of assets and liabilities. The single-

S.A. Zenios, Asset/liability management 81

period model may recommend a conservative strategy, while a more aggressive approach would be justified once we explicitly recognize the manager's ability to rebalance the portfolio.

What is needed is a model that explicitly captures both the stochastic nature of the problem, but also the fact that the portfolio is managed in a dynamic, multiperiod context. Mathematical models under the general term of stochastic programming with recourse provide the framework for dealing with this broad problem. Stochastic programming has a history almost as long as l inear programming - Dantzig [9], Wets [22]. However, it was not until the early seventies - Bradley and Crane [4] - that its significance for portfolio management was realized. With the recent advances in high-performance computing, this approach has been receiving renewed interest from the academic literature - Mulvey and Vladimirou [19], Hiller and Eckstein [12], Zenios [24], and Golub et al. [10]. We are also aware of research in several industrial settings for the deployment of such models in practice.

We now continue with a mathematical description of a model from each class. The formulations are general. Our goal is to describe the key components of the models, and then discuss - in section 3 - the computation of the data requirements for each. In order to operationalize each model for the application mentioned earlier, additional specifications are needed. We do not completely specify the details, since those will only distract from the general principles we want to convey.

2.1. PROBLEM FORMULATION

We are given a universe of fixed-income securities, indexed by a set .7, with market prices {P0j}, and a stream of liabilities {Lt}, where t denotes a time index drawn from a discrete set T. Given is also a term structure, specified by a vector of forward rates { rt}, t ~ T. The problem of the portfolio manager is to decide the holdings of each security xj in a portfolio that will match the assets with the liability stream.

For the stochastic models, we also need to specify a set of scenarios S. We assume discrete and equiprobable scenarios. The scenarios can be very general: they can represent a series of term structures drawn from some stochastic process of interest rates, or they can represent levels of prepayment activity for the mortgage securities, or they can represent levels of the liability stream, and so on. Whenever a model parameter is super-scripted by an index s ~ S, it is understood that the value of the parameter is scenario dependent. In this respect, we will use Cjt to denote the cash flow generated b y security j E J (per unit face value), and ~ to denote the discount rate at period t ~ T, under scenario s ~ S.

The interest rate scenarios can be calculated using a variety of term structure models, such as the diffusion process of Cox et al. [6], or binomial lattice models

82 S.A. Zenios, Asset/l iability management

such as the one proposed by Black et al. [2]. These models are designed to generate term structure scenarios that are consistent (i.e. arbitrage free) with the treasuries' yield curve and its volatility.

Most fixed-income securities, however, cannot be priced using the same discount rates implied by the treasuries' curve. In particular, the price of the security has to reflect the credit, liquidity, default and prepayment risks associated with this instrument. In order to value the risks associated with MBS, we compute an option adjusted premium (OAP). The OAP methodology estimates the multiplicative adjustment factor for the treasuries' rates that will equate today's (observed) market price with the "fair" price obtained by applying the expectations hypothesis, see, e.g., Babbel and Zenios [ 1 ]. The discrepancy between the market price and the theoretical price is due to the various risks that are present in most fixed-income securities, but are not present in the treasuries' market. Hence, this analysis will price the risks.

The OAP for a given security is estimated based on the current market price Poj. In particular, it is the solution of the following nonlinear equation in pj:

1 ISt T C~ t p0j= Z Z ,

• =1 t=o I I i= l (1 + pj • rZ) (1)

The computed risk premium appears in several of the models in the following sections.

2.2. A STATIC APPROACH: DURATION MATCHING

Given the term structure, a stream of projected cash flows for the fixed-income security and a stream of liabilities, we can build a dedicated portfolio. That is, a portfolio of least cost - or maximum yield - of fixed-income assets that will match the stream of liabilities. Let Cjt denote the cash flow generated by security j at period t. This stream is projected, conditional on the current term structure. We can write the following optimization model:

Minimize ~ Po j x j (2) x j~ j

subject to ~ t ' Xj >-- Z t , (3) jE j rii__ (1 + ,Er r l i = l O + r/)

xj > O. (4)

This model will choose the least-cost portfolio, with the property that the present value of the portfolio cash flows will be at least equal to the present value of the liabilities. If the timing and magnitude of assets and liabilities do not change, nor the discount factors, then it is easy to see that the portfolio will ensure timely payments


against the liabilities. (It is assumed in this model that unlimited borrowing is allowed, at all time periods, at the prevailing discount rate r i. The model can easily be modified to eliminate borrowing, or permit limited borrowing at a discount rate greater than ri.)

To account for the stochasticity of the security cash flow, and the volatility of the term structure, the model can be extended to match the sensitivities of both assets and liabilities to such stochasticity. For example, the duration of a security measures the sensitivity of its price to small parallel shifts to the term structure. Hence, we extend the model to match the duration of assets and liabilities. In order to capture the complex dependency of the cash flows of a fixed-income security to changes in the term structure, we use a Monte Carlo simulation procedure:

MONTE CARLO SIMULATION FOR OPTION ADJUSTED DURATION CALCULATIONS

Step 0: Initialize the stochastic process of interest rates, based on the current term structure, and use it to compute the option adjusted premia pj for all securities, implied by the current market prices P0j (cf. equation (1)).

Step 1: Shift the term structure by -50 basis points, and recalibrate the stochastic process of interest rates.

Step 2: Sample interest rate paths { r/-s} from the stochastic process calibrated in step 1, and use the security cash flow projection model to compute option adjusted prices:

1 Isl r C~ (5)

s=l ,=0 n i = l ( l + p j - ~ )

Step 3: Shift the term structure by + 50 basis points, and recalibrate the stochastic process of interest rates.

Step 4: Sample interest rate paths { r/~s } from the stochastic process calibrated in step 3, and use the security cash flow projection model to compute option adjusted prices:

1 Isl r C)St (6) P ; = ~ ' ] Z Z t (1+pj .r/+S)"

s=l t=O ni=l Step 5: The option adjusted duration of the security is given by

Pf - Pf- Aj = 100

and the option adjusted convexity by

(7)

P7 -2e0j +e 7 rj = 502 (8)


An immunized portfolio will match the present values and durations of both assets and liabilities. If A t is the liability duration, we have the following linear

Minimize ~. Po j x j x j~j

subject to jeff 2 I t~T

program:

Cjt "I Lt nt,=, (1 + rt ) x j > ~.~ t t~T I~r=l (1 + rt)

(9)

, ( l O )

2 A j x j = A/ , (11) js_7

xj > O. (12)

The model could be extended further to match asset/liability convexities. Matching higher derivatives is also possible. At the limit, the derivative matched portfolio will be identical to a cash flow matched portfolio.

2.3. A STOCHASTIC APPROACH: CAPTURING CORRELATIONS

Portfolios of fixed-income securities have traditionally been managed using the concepts of duration and convexity matching of the previous section. With the increased volatility of the term structure, following monetary deregulation in the late 70's, this approach appears overly simplistic. The difficulty is further exacerbated with the constant stream of innovations in this market. In a recent report, Mulvey and Zenios [20] observed that returns from corporate bonds have outperformed those of equities, but at the same time exhibited similar or higher volatility. That paper went further to argue that such volatile instruments should be managed in the framework of risk-return tradeoff. In this respect, the tradition of models started with Markowitz's seminal work [18] has much to offer to the managers of fixed-income portfolios. This approach has already received attention from practitioners (Holmer [13]).

In this section, we introduce a stochastic model for managing portfolios of MBS. The model explicitly recognizes the volatility of MBS prices, and the correlation of prices in a portfolio, and develops the tradeoffs between increased return and increased volatility. The optimization model we adopt is based on the mean-absolute deviation (MAD) framework of Konno and Yamazaki [16]. An MAD model is suitable for the fixed-income securities with embedded options that exhibit highly asymmetric distributions of return.

One of the challenges in applying the MAD model - or any other risk-return model for that matter - to fixed-income securities is that these instruments are vanishing, with a fixed term to maturity. Furthermore, the payout function of several kinds of fixed-income securities is path dependent. Hence, at any point in time we


have only one observation of price variations. Therefore, we can not resort to the statistical analysis of historical data in order to capture the volatility and correlation of returns. Furthermore, the "dividends" obtained from the security - in the form of principal and interest payments, as well as prepayment, lapse or call of outstanding balance - can not be reinvested in the same security. Hence, we need to resort to Monte Carlo simulation of the short-term risk-free rates in order to obtain holding period returns of the fixed-income security during the target holding period. The Monte Carlo simulation procedure is explained in section 3. For now, we assume that a random vector of holding period returns has been generated for each security. Let {Rj} denote this vector random variable and let Rj = E(Rj)denote its expected value. Also, let x = {xj} denote the composition of the portfolio, which contains a deterministic liability with return p. The return of the portfolio is R = Y~j~ j Rjxj + p. The mean-absolute deviation of return of this portfolio is defined by

w(R) = E{ I R - E(R)I }, (13)

where 2?(-) denotes expectation. Assume now that a sample of the random variables Rj is available. That is, Rj takes the value {R~} for some scenario s ~ S, and we assume for simplicity that all scenarios in S are equiprobable. Then, an unbiased estimate for the mean-absolute deviation of return of the portfolio is

w(R) = E{I R - E(R)I } (14)

1 I I - I s l + l J I ~ ( R ] - -Rj)xj j~3

The mean-absolute deviation (MAD) model is written as

(15)

(16)

Minimize w(R) (17)

subject to ~ Rjxj > p, (18) j~3

' ~ xj = 1, (19) ja3

0 < xj < u j , for all j E J. (20)

This model can be reformulated into a linear programming problem. (This is a standard reformulation for minimizing absolute values. A minimand I xl is replaced


by y, where y is constrained as y > x and y > -x.) In doing so, it is also possible to differentially penalize the upside from the downside deviation of the portfolio return from its mean. Let #a and ~ denote penalty parameters for the downside and upside errors, respectively. Then the MAD model can be written as the following linear program:

1 Minimize [ S[ + [ J [ ~ y S (21)

s ~ S

subjectto yS + !1 a ~. , (R} - -Rj)xj >_ 0 foral l s ~ S, (22) j~J

yS - ltu ~ ( R j - -Rj)x j >0 , jeJ

for all s E S, (23)

~_~ -Rjxj >- p, (24) j z J

xj = 1, (25) jeJ

0 < xj < u j , for all j E J . (26)

2.4. A MULTIPERIOD, DYNAMIC APPROACH: STOCHASTIC OPTIMIZATION

The multiperiod, stochastic model captures the dynamics of the following situation:

A portfolio manager must make investment decisions facing an uncertain future. After these first-stage decisions are made, a realization of the uncertain future is observed, and the manager determines an optimal second-stage (or recourse) decision. The objective is to maximize expected utility of final wealth.

The first-stage decision deals with the purchase of a portfolio of fixed-income securities. The uncertain future is the level of interest rates and the cash flows that will be obtained from the portfolio. The second-stage decision deals with borrowing (resp. lending) decisions when the fixed-income cash flows lag (resp. lead) the target liabilities. Decisions to rebalance the portfolio at some future time period(s), by purchasing or selling securities, are also included.

2.4.1. Establishing notation

We will be using the following notation. First, parameters of the model:


T

S . r t

spr •

~Sjt

Lt

xj

y#

y j; •

Z~ t "

w);.

y~-S .

: discrcfizafion of the planning horizon, T = {I, 2, 3,..., T} and T o = {0, I, 2, 3 ..... T}. T denotes the end of the planning horizon•

• initial holdings (in face value) of instrument j E .7 and b0 initial holdings in a riskless asset (e.g., cash).

one-year forward interest rate at time period t ~ To under scenario s E S.

spread between the lending and borrowing rates.

cash flow generated from instrument j E .7 at time period t ~ T under scenario s E S, expressed as a percentage of face value. It includes principal and interest payments of the fixed-income security, as well as cash flows generated due to defaults, prepayments, lapse, exercise of the embedded call option, etc.

price per unit of face value of security j ~ .7 sold at period t under scenario s. The cost of the transaction is subtracted from the actual price to obtain this coefficient. The price at t = 0 is independent of the scenario and is denoted by

~jo. • price per unit of face value of security j E.7 purchased at period t under

scenario s. The cost of the transaction is added to the actual price to obtain this coefficient. The price at t = 0 is independent of the scenario and is denoted by ~'j0.

• liability due at time t ~ T. It is assumed to be independent of the realized scenario, although this assumption can easily be relaxed.

Define now the model variables:

• first-stage variable, denoting the face value of instrument j e .7 purchased at the beginning of the planning horizon (i.e. at t = 0).

• second-stage variable, denoting the face value of instrument j E `7 purchased at time period t under scenario s.

• first-stage variable, denoting the face value of instrument j e .7 sold at the beginning of the planning horizon (i.e. t = 0).

second-stage variable, denoting the face value of instrument j E .1 sold at time period t under scenario s (i.e. at t = 0).

second-stage variable, denoting the face value of instrument j E `7 in the portfolio at time period t E T under scenario s. Zjo denotes the starting composition of the portfolio, after first-stage decisions have been made, and is independent of the scenarios.

second-stage accounting variable indicating the cash flow generated by security j at time period t under scenario s.

second-stage recourse variable indicating the amount owed at time period t + 1 due to borrowing decisions made at time period t under scenario s.


yt +s

U(WT ~) •

second-stage recourse variable indicating the surplus invested in the riskless asset at time period t.

denotes the utility of terminal wealth realized under scenario s. Appro- priate choices of utility functions are, for example, the isoelastic utility - 1/y(WTS) r - used in the models of Grauer and Hakansson [11].

2.4.2. Defining the model

The model can now be formulated as follows:

1 ~ . U ( W T s) Maximize ~-~ sES

xj subject to zjo + yj ~jo = b j ,

+s 1 Yo ÷ ] ~ x j - ] ~ ( 1 - ~ j ) y j - ( l+r~ +~pr)

j ~ J j ~ J

Z~t_ 1 + gist - mist - zjSt - yjS t =0,

w jSt -- - ~e S z S = O, P J j t j t - I

XJ t S . +S - ~jStYjt + E wJ st - - -7 +(l + rt-l ) Y t - 1 - Yt~l

j ~ a ~ j t

1 - s _ y + S + Yt = L t ,

( l+r t s +spr)

j E J ,

yo s =b O, s E S ,

t E T , j E J, s E S ,

t E T , j E J, s E S ,

t E T , s E S .

w v = E -s j¢ - YT-I" j E j

The first constraint of this mathematical program reflects first-stage decision and is deterministic. Subsequent constraints depend on the realized scenario, as well as the first-stage decision. The terminal wealth WT s is computed by accumulating the total surplus net any outstanding debt at the end of the planning horizon, and liquidating any securities that remain in the portfolio. The complete model has the dual block- angular structure of two-stage stochastic programs with recourse. See figure 1.

3. Model data: Holding-period returns for fixed-income securities

We now turn our attention to the data required in order to implement the models. The portfolio immunization models need preset value, duration and convexity calculations. Those are fairly standard. The stochastic models, however, need a set of scenarios of holding period returns. The calculation of these data is illustrated next.


r£rat-Staqo YarAablea

Seaoad-Stago Var£ablea

I " I I '2'it ~ ll,,i lllml

V//A

eoeo

ileel Ileo

Figure 1. The constraint matrix structure of a two- stage, dynamic stochastic portfolio optimization model.

3.1. PRELIMINARIES

The rate of return of a security j during the holding period is determined by the price of the security at the end of the holding period, and the accrued value of any cash flows generated by the security. For MBS, for example, we need to estimate the accrued value of principal, interest and prepayments during the holding period, and price the unpaid balance of the security at the end of the holding period. To this end, we need a procedure for generating scenarios of the term structure, and a model that predicts the prepayment activity for each scenario. For a given interest rate scenario s, the rate of return of security j is given by

Fjs + Vjs (27) Rjs = Poi

Fjs is the accrued value of the cash flows generated by the security, reinvested at the short-term rates. The cash flow calculation procedures of fixed-income securities, although maybe complex as is the case in MBS, are standard.

Vjs is the value of unpaid balance at the end of the holding period, conditioned on scenario s. This is given by Vjs = BjsPjs, where Bjs is the unpaid balance of the security and Pjs is the price, per unit face value, of the security. Both quantities are computed at the end of the holding period, and are conditioned on scenario s. The estimation of security prices at the end of the holding period is the topic of the next section.

P0j denotes the current market price of the security.

90 S.A. Zenios, Asset~liabil i ty m a n a g e m e n t

3.2. PRICING THE UNPAID B A L A N C E

The pricing models are based on Monte Carlo simulation of the term structure. In particular, following Cox et al. [6], we obtain the equilibrium value of the fixed- income security as the expected discounted value of its cash flow, with discounting done at the risk-free rate. We are particularly interested in pricing the security at some future time period 'r. (This would be the planning horizon for the portfolio management problem.) Possible states cr of the economy at t ime period "r are obtained using the term structure model. From each state of the economy at instance "r, we can observe the possible evolution of interest rates further into the future, until the end of the horizon T. The price of the fixed-income security is the expected discounted value of its cash flow, with expectation computed over the interest rate paths that are emanating f rom that particular state.

In our work, we use the binomial lattice model of Black et al. [2]. A binomial lattice of the term structure can be described as a series of base rates { rot, t = O, 1 ..... T} , and volatilities { kt, t = 0, 1,..., 7"}. The short-term rate at any state of ty of the binomial lattice at some point t is given by

rttr = rt 0 (k t ) tr.

The base rate and volatility parameters are estimated according to the procedure developed by Black et al.

To make the idea of the pricing model precise, let Scr denote a set of interest rate scenarios that emanate from state tr of the binomial lattice at some future t ime period "r. Also, let r[ be the short-term discount rate at t ime period t ('t'< t < T) associated with scenario s ~ Scr, and C]t be the cash flow generated by security j at period t. A fair price for the security at period "r, and condit ioned on the state or, is

Pa_ = 1 Isal r

is lX X j t s=l t=*:

This procedure is illustrated in figure 2.

c; I [ i=x (1 + r[ )"

(28)

Most fixed-income securities, however, cannot be priced using the same discount rates implied by the .treasuries' curve. In particular, the price of the security has to reflect the credit, liquidity, default and prepayment risks associated with this instrument. In order to value the risks associated with a f ixed-income instrument, we compute its option adjusted premium (OAP), cf. equation (1).

Once we have priced the various risks associated with the security, we can proceed to price the security at some future time period. The option adjusted price of the security Pj~ can be calculated from

ISa[ T s pj : 1 % s=l t=x I I i = z ( + p j ~s)

(29)


$ T

I 1 t l t ,, 1 ; + I t i ,=u

0 r ~r

Figure 2. Estimating state-dependent prices of fixed-income securities from a binomial lattice model of the term structure.

J u l M ~ t i ~ so

a t i l t

f ~ r , ,zl , g w

We point out that the price Pj~ may depend not only on the state tr but also on the history o f interest rates f rom t = 0 to t = ~" that pass through this state. 1) This diff icul ty can be easily resolved by sampling paths f rom t = 0 that pass through state ( ra t t = "r. Le t S° ' ° denote the set o f such paths, and let PT s(°), s(cr) E S° '~ be the price o f the securi ty at state o- obtained by applying equat ion (29), condi t ioned on the fact that interest rate scenarios s in So originate f rom scenarios s(cr) in S °'°. Then the expected price of the security at cr is given by

pj :ls0 o I E (30 s(o) • S °'°

1) In particular, the cash flows generated by some fixed-income securities at periods after • will depend on the economic environment experienced prior to 7. For example, if the security has experienced periods of high repayments, lapse or defaults, then subsequent changes of interest rates will have less impact on the generated cash flows. Although the short-term rates prior to • do not appear explicitly in the pricing equations, the economic activity prior to • is used in the estimation of the cash flows C~t for t> ~.


4. Do the models work? Some empirical evidence

In this section, we provide some empirical evidence that supports the validity of the models and highlights their advantages. First, we illustrate the performance of an MAD model in the context of an indexation problem. Then we use both portfolio immunization and the MAD model in structuring the funding of an insurance liability claim. In the former case, the model is shown to be very effective in tracking the index. In the latter case, we demonstrate the superiority of stochastic models (even single period ones, such as MAD) over traditional portfolio immunization techniques.

4.1. AN EXAMPLE OF INDEXATION

Worzel et al. [23] built a mean-absolute deviation model to track the Salomon index of mortgage-backed securities. The index consists of a representative of all traded fixed-rate, pass-through securities, issued by FNMA, GNMA and FHLMC. The index is a sanitized image of the mortgage market: For example, cash flows generated by the mortgage pools are assumed to be reinvested in the index itself. There are also holdings in very small pools, but actual investments in such polls may be impossible due to liquidity difficulties. Finally, the composition of the index is changing from month to month without incurring any transaction costs. The Salomon index realized an annual return of 13.96% over the period January 1989-December 1991. Hence, creating a tradeable portfolio that closely tracks the index is of great interest to investors.

The model estimates holding period returns of all securities in the index, and builds a portfolio that minimizes the mean-absolute deviation of the retums of the portfolio from the expected return of the index. Upside and downside risks are penalized differentially, with no penalty on upside risk and infinite penalty on downside risk, (Downside risk is realized when the portfolio under-performs the index by a small margin, set to be -5bp in monthly terms.)

The indexation model was tested over the period January 1989-December 1991. During this period, the index realized an annual return of 13.96%. In back-testing the model, we used the following methodology: At the beginning of each month, a binomial lattice was calibrated based on the term structure of that day. The lattice was used to estimate holding period returns, and the MAD model was used to select a portfolio. The performance of the portfolio was recorded at the end of the month, based on observed market prices, and the process was repeated. Transaction costs of 2/32bp were charged, and cash flows from the mortgage pools were reinvested at the 1-month treasury rate. The very first portfolio (January 1989) was selected using the method outlined above, but no transaction costs were paid.

During the testing period, the portfolio realized an annual return of 14.18%, + 22bp over the index return. The portfolio never under-performed the index by more than -3 .6bp in monthly return, while the over-performance was more substantial; see figure 3. The standard deviation of the index return over the test period was

160

150

140

130

120

110

100 h

, l l t ~ , t J , l t t t i l J l t ~ , , .. ~ .~

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

Portfolio Value Million $


Tracking error in lOOM portfolio

500000

400000

300000

200000

100000 I ! ]

o--h,.llJ.l:.l,.~,l~,,~,~,~i .............. , ~ , . . .

.,oooool ii Figure 3. Return of a $100M investment in the Salomon Brothers Mortgage Index over the period January 1989-December 1991, and tracking error of the indexed portfolio.


0.155, while the portfolio return had a standard deviation of 0.158. The portfolio would typically consist of approximately 25 securities, none of which accounted for more than 12% of the total portfolio.

4.2. AN EXAMPLE OF LIABILITY FUNDING

We apply the MAD model for funding a liability stream obtained from a major insurance corporation. Using the term structure of April 26, 1991, we calculated the following descriptors of the liability:

T e r l n

Present value

Modified duration

100 months

$166,163,900.00

4.1792 years

4.2.1. An immunized portfolio

Using the portfolio immunization models, we built a portfolio that was duration and convexity matched against the liabilities. The portfolio was built from a universe of both MBS and US Treasury securities. Different levels of exposure to the mortgage market were imposed on an ad hoc basis. We list below the percentage savings realized when the liability is funded using a portfolio of MBS and treasuries, over the cost of funding the liabilities using only the risk-free rate.

Cost of portfolio using Treasuries only (Savings)

Cost of portfolio using up to 25% MBS (Savings)

Cost of portfolio using up to 50% MBS / (Savings)

Cost of poL'tfolio using up to 100% MBS (Savings)

Cost of mixed US Treasury-MBS portfolio (Savings)

$166,163,861.00 0.00%

$152,993,690.99 7.92%

$142,529,529.00 16.58%

$137,489,656.00 21.07%

$136,124,130.99 22.07%

While it is clear from this example that using MBS in an integrated asset/ liability management system produces substantial gains, the savings summarized above will not be necessarily realized in practice. They will be realized only if the term structure shifts in parallel and in small levels from that of April 16, 1991. To


Table 1

Performance of immunized and MAD portfolios.

Model 100% MBS portfolio Mixed portfolio

Exp. return Std. dev. Exp. return Std. dev.

Immunized 10.469 0.406 10.448 0.293

MAD (equal risk) 10.783 0.405 10.692 0.293

MAD (equal return) 10.469 0.234 10.448 0.206

assess the return of the portfolio in different environments, we conducted a Monte Carlo simulation of the returns of the immunized portfolio over the holding period; see table 1. The 100% MBS portfolio produced expected returns of 10.469%, with a standard deviation of these returns of 0.406. The mixed portfolio of US Treasuries and MBS has a slightly reduced expected return of 10.448%, but a substantially reduced standard deviation of 0.293.

4.2.2. An MAD portfolio

We also developed an MAD portfolio for funding the insurance liability. The efficient frontier is shown in figure 4. In the same figure, we show the results of the simulation of the immunized portfolio. First, we observe from this figure that substantial rates of return can be realized with relatively little risk. Second, we observe that the portfolio obtained using standard immunization techniques lies below the efficient

11

R10

e 9 t . .

u 8 r

n 7

6 • : : : : : : : : : : : : : : : : : : :

O. O. O. O. O. O. O. O. 08 08 09 11 13 16 22 33

Variance

Figure 4. Efficient frontier of the mean-absolute deviation portfolio of MBS and the risk-return profile of an immunized portfolio.


frontier. These results illustrate the superiority of the MAD model in managing portfolios of complex instruments, such as MBS, as opposed to traditional fixed- income management tools. Table 1 summarizes the expected return and standard deviation of both immunized and MAD portfolios.

5. Critique and conclusions

We have presented a sequence of asset/liability management models that capture the increasing complexities of the fixed-income markets. Empirical testing has illustrated the validity of the models. Two observations are particularly interesting. First, it is possible to use optimization models, such as MAD, to solve complex problems in a very volatile environment. The performance of the index-tracking portfolio has been particularly encouraging. Second, the use of stochastic models is well justified given their superior performance over traditional immunization techniques. While a bond portfolio manager may find immunization techniques adequate - and we concur with this approach - the complexities of other fixed-income instruments leaves little choice. The volatility and co-movements of these securities must be explicitly modeled. The direct comparison of an immunized with an MAD portfolio has clearly demonstrated this point.

It is still an open question whether the multiperiod, dynamic stochastic programs offer any real advantage to portfolio managers. The state-of-the-art has reached the point where large-scale models of this form can be built and solved efficiently. However, their real advantage over either traditional immunization techniques, or the single-period stochastic models, remains to be established. We are currently completing a large-scale experimentation that pits the MAD model against several variants of the stochastic programming models in managing portfolios of MBS and callable bonds. The outcome of these studies are reported elsewhere [10,7].

Acknowledgements

The research leading to this paper was funded by NSF University/Industry Research Collaboration Grant No. SES-91-00216. The contributions of B. Golub and L. Pohlman from Blackstone Financial Management, and M. Holmer from the Federal National Mortgage Association are gratefully acknowledged. Partial funding was also provided by NSF Grant No. CCR-9104042, AFOSR Grant No. 91-0168, and a research contract from the Metropolitan Life Insurance Company. Computing resources were made available by the Army High-Performance Computing Research Center at the University of Minnesota, and the North East Parallel Architectures Center of Syracuse University,

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Asset/liability management under uncertainty for fixed-income securities

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