robust optimization models for managing callable bond portfolios

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ELSEVIER European Journal of Operational Research 91 (1996) 264-273 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Robust optimization models for managing callable bond portfolios Christiana Vassiadou-Zeniou l, Stavros A. Zenios * Department of Public and Business Administration, University of Cyprus, Nicosia, Cyprus Received June 1994; revised February 1995 Abstract A major sector of the bond markets is currently represented by instruments with embedded call options. The complexity of bonds with call features, coupled with the recent increase in volatility, has raised the risks as well as the potential rewards for bond holders. These complexities, however, make it difficult for the portfolio manager to evaluate individual securities and their associated risks in order to successfully construct bond portfolios. Traditional bond portfolio management methods are inadequate, particularly when interest-rate-dependent cashflows are involved. In this paper we integrate traditional simulation models for bond pricing with recent developments in robust optimization to develop tools for the management of portfolios of callable bonds. Two models are developed: a single-period model that imposes robustness by penalizing downside tracking error, and a multi-stage stochastic program with recourse. Both models are applied to create a portfolio to track a callable bond index. The models are backtested using ex poste market data over the period from January 1992 to March 1993, and they perform constistently well. Keywords: Portfolio management; Callable bonds; Fixed income 1. Introduction A large portion of the U.S. corporate and govern- ment fixed-income markets consists of bonds with em- bedded options. These instruments are contracts in which the issuing institutions promise to pay inter- est and principal on prespecified future dates in ex- change for use of cash today. Most of these contracts are callable by the issuer who may decide to call the bond before its maturity, if interest rates drop substan- tially, and reissue another bond with a lower coupon. * Corresponding author. I Currently with the Bank of Cyprus, Nicosia, Cyprus. Options can significantly affect the value of the in- strument. Since the call option benefits the issuer, the buyer of a callable bond is implicitly receiving a pre- mium for the call option written to the issuer. This premium is paid out in the form of higher coupon pay- ments than a corresponding non-callable bond. A callable bond holder will benefit if the option expires worthless, and lose from any decline in bond price. Ideally, if the bond price would stay at the exer- cise price of the option, the option value would reduce to zero, allowing the bond holder to keep all of the pre- mium and preserve the greatest possible value of the underlying bond consistent with the zero value for the option. In reality the option has value because of the uncertainty of interest rates. If the bond price moves 0377-2217/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 03 77-22 17(95)00283-9

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E L S E V I E R European Journal of Operational Research 91 (1996) 264-273

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

Robust optimization models for managing callable bond portfolios

C h r i s t i a n a V a s s i a d o u - Z e n i o u l , S t a v r o s A . Z e n i o s * Department of Public and Business Administration, University of Cyprus, Nicosia, Cyprus

Received June 1994; revised February 1995

Abstract

A major sector of the bond markets is currently represented by instruments with embedded call options. The complexity of bonds with call features, coupled with the recent increase in volatility, has raised the risks as well as the potential rewards for bond holders. These complexities, however, make it difficult for the portfolio manager to evaluate individual securities and their associated risks in order to successfully construct bond portfolios. Traditional bond portfolio management methods are inadequate, particularly when interest-rate-dependent cashflows are involved. In this paper we integrate traditional simulation models for bond pricing with recent developments in robust optimization to develop tools for the management of portfolios of callable bonds. Two models are developed: a single-period model that imposes robustness by penalizing downside tracking error, and a multi-stage stochastic program with recourse. Both models are applied to create a portfolio to track a callable bond index. The models are backtested using ex poste market data over the period from January 1992 to March 1993, and they perform constistently well.

Keywords: Portfolio management; Callable bonds; Fixed income

1. Introduct ion

A large portion of the U.S. corporate and govern- ment f ixed-income markets consists of bonds with em- bedded options. These instruments are contracts in which the issuing institutions promise to pay inter- est and principal on prespecified future dates in ex- change for use of cash today. Most of these contracts are callable by the issuer who may decide to call the bond before its maturity, if interest rates drop substan- tially, and reissue another bond with a lower coupon.

* Corresponding author. I Currently with the Bank of Cyprus, Nicosia, Cyprus.

Options can significantly affect the value of the in- strument. Since the call option benefits the issuer, the buyer of a callable bond is implici t ly receiving a pre- mium for the call option written to the issuer. This premium is paid out in the form of higher coupon pay- ments than a corresponding non-callable bond.

A callable bond holder will benefit if the option expires worthless, and lose from any decline in bond price. Ideally, if the bond price would stay at the exer- cise price of the option, the option value would reduce to zero, allowing the bond holder to keep all of the pre- mium and preserve the greatest possible value of the underlying bond consistent with the zero value for the option. In reality the option has value because of the uncertainty of interest rates. I f the bond price moves

0377-2217/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 03 77-22 17(95)00283-9

c. Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273 265

above the exercise price, the bond holder must deliver the bond to the issuer, and reinvest the proceeds in a lower interest rate environment. In the case of a bond price decline, the option will not be exercised and the bond itself will produce a lower return. Therefore, the value of a callable bond depends on three basic com- ponents: ( 1 ) the term structure of the risk-free interest rates, (2) the credit risk of the issuing institution, (3) the value of the embedded option. Details on callable bonds are given by Fabozzi [2, pp. 791-808].

In this paper we are concerned with the problem of managing portfolios of callable bonds. We fol- low an indexation strategy, a popular approach which seeks to match the performance of the managed port- folio with that of a benchmark. Constructing an in- dexed portfolio that matches the performance of the U.S. government bond sector is not difficult, since most issues do not have a call feature, and previ- ous studies have reported very small tracking error of portfolios to match the Treasury sector. On the other hand, large monthly errors have been reported for a portfolio tracking a corporate bond sector [5]. In this paper we develop models for callable bond in- dexation drawing from recent developments in robust optimization [7]. Section 2 develops both a single- period optimization model that penalizes downside tracking error and a multi-period stochastic program- ming model. The ideas of this section, i.e. integrat- ing robust optimization and asset-pricing models for tracking fixed-income indices, were first introduced by Worzel, Vassiadou-Zeniou and Zenios [9] for port- folios of mortgage securities. This paper extends our earlier work to problems of callable bonds.

The price performance characteristics of callable bonds must first be understood in order to imple- ment effectively a portfolio strategy. This is the scope of Section 3. It describes the procedure for valuing callable bonds, and develops the simulation techniques for generating scenarios of holding period returns.

Finally, a callable bond index is constructed from data provided by BlackRock Financial Management, NY, and the models are backtested in order to evaluate their performance in tracking the index. The backtest- ing uses ex poste data over the 14-month period from January 1992 to March 1993. The results of the bond portfolios obtained by the two models are compared in Section 4. Section 5 concludes the paper.

2. Robust optimization for indexed bond portfolios

Many investors have turned to indexation strategies for managing their assets. Tracking a fixed-income in- dex is a difficult task due, primarily, to the volatile re- turn of the index. Further difficulties arise due to liq- uidity and diversification constraints, and transaction costs, which must be included when choosing a port- folio, but are not present in the index. The problem is also complicated when dealing with fixed-income se- curities which have a finite term: their returns change as they approach maturity and historical observations are of limited value. We defer until Section 3 the is- sue of generating scenarios of returns, and assume that such data are available. We are concerned, in this sec- tion, with the problem of developing indexation mod- els that are robust with changes in the returns. The models presented below address this problem.

2.1. Minimizing downside tracking error

The purpose of an indexed portfolio is to track the performance of the benchmark portfolio. The differ- ence between the performance of the two portfolios is called tracking error. Having estimated scenarios of holding period returns of the securities in the index we build a portfolio whose return differs little from that of the index for any one of the projected scenarios.

We use the modeling framework for the robust op- timization of large scale systems, introduced by Mul- vey, Vanderbei and Zenios [7]. The optimal solution of a mathematical program is called robust if it re- mains "close" to optimal and "almost" feasible for small changes in the input data. To formalize these ideas, consider the general linear programming model:

Maximize cT x ( 1 )

s.t. Ax = b, (2)

x _> 0, (3)

where x denotes the vector of decision variables. In the context of portfolio management, these variables denote portfolio decisions, c, A and b are the coeffi- cients of the model which may be uncertain.

In order to write the model using a robust opti- mization formulation, we introduce a set of scenarios /'2 = {1,2, 3 . . . . . S}. Equal probability of realization of each scenario, p~, is assumed and ~.+~s P.~ = 1. With

266 C. Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273

each scenario s define a set o f coefficients {c s, A' , b'}. The optimal solution of the linear program above is solution robust if it remains close to optimal under all scenarios s E /2, and is model robust if it remains al- most feasible for all realizations of /2 . Since it is un- likely that any solution to the mathematical program will remain both feasible and optimal under all scenar- ios, the model should be built in a manner that controls the tradeoff between solution and model robustness. For this purpose we introduce a set {et, e2 . . . . . es} of error vectors that measure the infeasibility allowed in the constraints under each scenario. The robust opti- mization formulation of the linear program now takes the lorm:

Maximize g~.{lg(c*Tx)} -- w ~ ( e ' ) (4)

s.t. A"x + e '~ = b" for all s E /2. (5)

where/.4 (.) is a utility function, gs ( ') denotes expec- tations with respect to the random variable indexed by s E /2, @(.) is a suitable penalty norm of the error vectors e ' , and ~o is a goal programming weight used to derive a spectrum of answers which tradeoff solu- tion with model robustness.

We now cast the indexation problem in the robust optimization framework. Let {Ri} be a vector random variable denoting rates of return for each security j in the index set J, and let /~j = £ ( R j ) denote the ex- pected value of the random variable. Let also x = {x j} denote the composition of the portfolio. We restrict x to be nonnegative, so that short sales are not allowed. The composition of the index is known and is denoted by a set o f weights {/3i}. The returns of the portfo- lio and the index are given by the random variables R = Y~<iEJ Rjxj and I = ~.i~J Ri~J' respectively. The tracking error is defined by:

e' = R' - 1 ~, (6)

where R s = ~.i~J RiSxj denotes the portfolio return, and 1" = ~.iE~ Ris~i represents the index return under scenario s.

The indexation problem is first written as a linear programming model, in which we maximize the ex- pected return:

Maximize g.,.{R'} (7)

s.t. R ' = 1' for all s E /2. (8)

This model is unlikely to have a solution that sat- isfies R s = I" for all scenarios. Therefore we rewrite the model as a robust optimization program:

Maximize gs{lg( R'~) } - toz (9) x,z E~+ °

s.t. e s = R s - I s for a l l s E /2, (10)

e "~ _> -Z , (11)

where z is a margin of tracking error. The variables of this program are {xj}, via the definition of R', and the margin of tracking error z. The model esti- mates, simultaneously, the portfolio composition and the margin of tracking error. In all experiments we use a large value of w, thus building portfolios with the smallest possible tracking error. This approach is consistent with the principle of indexation, where em- phasis is given on good tracking performance and not on high returns. This is an example of integratedfi- nancial product management, whereby the portfolio management strategy is consistent with the character- istics of the financial product, in this case the index.

Transaction costs, cash infusion or withdrawal and diversification or liquidity constraints can be incorpo- rated easily into this model; see the Appendix in [9] . We also point out that this robust optimization model is equivalent to the asymmetric penalty model intro- duced by Zenios and Kang [3] , which in turn is the limiting case of the mean-absolute deviation (MAD) model of Konno and Yamazaki [4] . We mention these relations here for general background, but they are of no direct consequence to our indexation model.

2.2. The multi-stage stochastic programraing model

Consider now an alternative formulation of the in- dexation problem, cast as a multi-period dynamic de- cision problem with transactions taking place at dis- crete points in time. At each point in time the manager has to assess the prevailing market conditions - such as prices, interest rates, the value of the index - and the composition of the existing portfolio. The man- ager has to decide on which securities to buy or sell taking into account possible future movements of the index. At the next point in time the portfolio manager

C Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273 267

holds a seasoned portfolio and is faced with a new set of possible future movements. Transactions must now be executed with the new information at hand.

In this section we formulate a multi-stage stochastic programming model for the indexation problem. It is a nonlinear program maximizing the expected value of a utility function of excess return generated by the in- dexed portfolio over that generated by the index. The utility of excess return is calculated at the end of the time horizon. Expectations of future prices, cashftow spinoffs and returns along the holding period horizon are computed over a set of postulated scenarios. Cash infusions from coupon payments and any bonds called at the end of the previous holding period are incor- porated in the original wealth at the beginning of the time horizon. Rebalancing decisions made in the first time period do not depend on the realized scenarios, but determine the allowable decisions at future time periods. Decisions made in subsequent time periods, dependent on the scenarios observed thus far, also in- fluence decisions further into the future. The value of the utility function obtained from the solution of the multi-stage program depends on the rebalancing de- cisions made throughout the holding period horizon and on the realized scenarios. We formulate now the complete model.

2.2.1. Notation We develop a model with only three stages for sim-

plicity of notation. There are two points in time where decisions are made (to and tl ), and the portfolio per- lbrmance is evaluated at the end of the holding period r > t~. Extensions to more stages are staightforward.

The model is developed using cashflow account- ing equations. Investment decisions are in number of shares. Define first parameters of the model:

S2o, g2t: sets of scenarios anticipated at to and tl. We use so and s~ to index scenarios from the two sets, respectively. Paths are denoted by (s0, sl ). For sim- plicity, we assume that paths are equally probable with probability 7r.

J: set of available bonds, with cardinality m. co: riskless asset available at to. & transaction cost, paid per unit of the riskless asset

invested in the purchase of new securities. b0: vector of dimension m, denoting the composition

of the initial portfolio.

P0: vector of dimension m, denoting prices at to. These prices are known with certainty.

pl (so) for all so E Do: vector of prices, realized at tl. These prices depend on the scenario so.

p2(so, sl) for all so E /'2o and sl E f21: vector of prices, realized at t2. These prices depend on the scenario (so, sl).

c~0(s0), o~l(so, sj) for all so E Oo and sl c s21: vectors of dimension m, denoting indicator func- tions during the intervals [to, tl ] and [tl, t2] re- spectively. These functions take the value 0 if the bond has been called during the interval, and 1 oth- erwise. The value depends on the scenario.

ko(so), kl(so, sl) for all so E Oo and sj E S2): vectors of dimension m, denoting cash accrual fac- tors during the intervals [ to, tl ] and [ tt, t2 ] respec- tively. These factors indicate cash generated per unit face value of the security due to scheduled payments and exercise of the embedded options, accounting for accrued interest. For example, a corporate secu- rity that is called at the beginning of a l-year inter- val, in a 10% interest rate environment, will have a cash accrual factor of 1.10. These factors depend on the scenarios.

ro(so), rl (so, sl): short term riskless reinvestment rates during the intervals [to, tl ] and [tl, t2], re- spectively. These rates depend on the scenarios.

/ (so, sl ): index return at the end of the time horizon. The index return is scenario dependent.

Now define decision variables. We have three de- cisions at each point in time: (i) how much of each security to buy, (ii) how much to sell, and (iii) how much to invest in the riskless asset. All variables are constrained to be nonnegative.

First stage variables, at to: xo: vector of dimension m, denoting the number of

shares purchased of each bond. Y0: vector of dimension m, denoting the number of

shares sold of each bond. z0: vector of dimension m, denoting the number of

shares held in the portfolio. v0: amount invested in the riskless asset.

Second-stage variables, at tt, for each scenario so: xl (so) : vector of dimension m, denoting the number

of shares purchased of each bond.

268

Yl (so): vector of dimension m, denoting the number of shares sold of each bond.

zl (so): vector of dimension m, denoting the number of shares held in the portfolio.

vl (so): amount invested in the riskless asset.

Third-stage variables, at t2, for each scenario (so, sl ): x2(so, sl ) : vector of dimension m, denoting the num-

ber of shares purchased of each bond. y2 (so, s l ) : vector of dimension m, denoting the num-

ber of shares sold of each bond. z2 (so, sl ) : vector of dimension m, denoting the num-

ber of shares held in the portfolio. v2(so, sl ): amount invested in the riskless asset.

C. Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273

rO(sO)vO + ZPlj(sO)Ylj(SO) + ~ koj(so)zoj iEJ jEJ

= vl(so) + Z [ p l j ( s o ) + 6]xlj(so), jEJ

for all scenarios so E g2o.

2.2.2. Model formulation The constraints of the model express cashflow ac-

counting for the riskless asset, and inventory balance for each bond. Such equations are specified for each time period, and for each scenario.

First-stage constraints. At the first stage (i.e., at t0) all prices are known with certainty. The cashflow ac- counting equation specifies that the initial endowment, co, plus any proceeds from bond sales, equals the amount invested in the purchase of new bonds plus the amount invested in the riskless asset:

co + ZPOjYO/ = ~-~[POj + 8]xoj + vo. jE J .jEJ

(12)

For each security in the portfolio we have an inventory balance constraint:

boi + xo./- YO.l = z0j, for all j E J. (13)

Second-stage constraints. Decisions made at the sec- ond stage (i.e., at tl ) depend on the scenario so real- ized during the interval [to, tj ]. Hence, we have one constraint for each scenario. These decisions also de- pend on the investment decisions made at the first stage. Cashflow accounting limits the amount invested in the purchase of new bonds and the riskless asset to be equal to the income generated from the existing portfolio during this period, plus any cash generated from bonds being called. We have the following con- straint for each scenario:

(14)

Inventory balance equations constrain the amount of shares of each bond remaining in the portfolio to be equal to the outstanding amount of shares at the end of the first stage, plus any amount purchased at the beginning of the second stage. There is one constraint for each security and for each scenario:

ozoj( so) zoj + xlj( so) = ylj( so) + zl.i( so),

for all j E J, so C/20. ( 1 5)

Third-stage constraints. Decisions made at the third stage (i.e., at t2) depend on the scenario (so, sj ) re- alized during the period [ tl, t2 ] and on the decisions made at tl. The constraints are similar to those of the second stage. The cashflow accounting equation is:

rl (so, sl )vl (so) + Z P 2 j ( s o , Sl )y2./(so, sl ) .]EJ

+ Z klj(s0, sl )Zlj(So) jEJ

= V2(SO, Sl) -+- Z[P2j(SO, Sl) + •]X2j(SO, Sl ), jEJ

(16)

for all scenarios (so, sj ) such that so E /20, sj E s2j. The inventory balance equation is:

O~lj (S0, SI ) Zl j ( So ) "a t- X2j( SO, SI )

= yzj(so, sl ) + Zzj(so, sl ), (17)

for all j E J, and all scenarios (so, sl ) such that so E Oo, sl Es21.

Objective function. The objective function maxi- mizes the expected utility of excess market value of the portfolio over the index. The market value of the portfolio depends on the scenarios (so, sl ). The objective function is:

Maximize C~so,s,) {b/[ W(so, sl ) /I(so, sl) ] } ,

C. Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273 269

where W(so, sl) denotes terminal wealth, and b/{.} denotes the utility function. The terminal wealth at t2 is given by

) = u2(s0, sl ) + ~ p2.i(s0, sl)z2j(so, sl ). W(s0, SI

.jEJ

(18)

A logarithmic utility function is used in all of our nu- merical experiments. This choice of a utility function corresponds to an optimal growth strategy.

these rates is also consistent with the term structure of volatility. The binomial lattice can be described as a series of base rates {rt °, t = 0, 1,2 . . . . . T} and volatil- ities {kt, t = 0, 1,2 . . . . . T}, where T denotes the end of the planning horizon (typically 360 months). The short-term rate at any state o- of the binomial lattice at some time instance t is given by r~ = r°(kD ".

The price of a future stream of risk free cashflows - such as those generated, for example, by a U.S. Treasury security - can be obtained as the net present value of these cashflows, with discounting done at the rates r~.

3. Generating scenarios of holding-period returns

A simulation procedure for computing scenarios of holding-period returns of fixed-income assets, such as mortgage-backed securities, was discussed in our pre- vious publications [8,9]. Here we describe a similar procedure, with a few modifications, in order to com- pute scenarios of returns for callable bonds. First it is necessary to develop a pricing model for valuing a callable bond. Once the price of the bond has been determined, the generation of holding period returns is straightforward.

3. I. Pricing a callable bond

A bond with an embedded option may be considered as a portfolio of two securities: a bullet bond without an option feature and an option. In order to price these bonds, we need to: ( 1 ) price the underlying bond using a proper discounted cash-flow method, and (2) price the option by means of an option pricing method. The net value of the callable bond is the difference between the value of the bullet and the value of the option. We first discuss the three factors that determine the value of a callable bond, namely, the term-structure of the riskless rates, credit risk of the issuing institution and the value of the embedded option.

3. I. 1. The term structure model Risk-free interest rates are generated by the bino-

mial lattice model of Black, Derman and Toy [ 1 ]. This model generates distributions of future interest rates that are arbitrage free, and that price correctly the points on the spot yield curve. The volatility of

3.1.2. Credit risk Market participants assume the credit, default, liq-

uidity and other risks inherent in callable bonds issued by corporations, utilities or government agencies. The term structure of credit risk is an important component of callable bond valuation. In order to identify the rel- ative value among callable bonds, the investor must understand the relationship between the default risk of the issuing institution and the yield spreads of its bonds over Treasuries. The magnitude of the spread of a specific corporation depends on the bond's maturity and coupon, its call structure and the expected future volatility of interest rates. The investor has to recog- nize that corporate bond spreads vary with maturity, and that different discount rates should be applied to cashflows at different dates. The cashflows generated by a callable bond can not be priced simply by dis- counting them at the risk-free rate. Instead, the risk free rates must be adjusted by appropriate risk premia in order to derive discount rates which reflect credit risk. A model for the term-structure of credit risk is developed by Litterman and Iben [ 5 ]. The model de- termines a vector of risk premia {pt} that are used to adjust the risk-free rates to reflect the term structure of credit spreads.

3.1.3. Value of the option and of the callable bond We are now ready to value the bond. A callable

security can be viewed as a long position in a non- callable bond, and a short position in a call option on that underlying bond. The price of the callable security is the difference between the non-callable bond value and the option premium.

The non-callable bond can be priced starting at the

270 C. Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273

(a) PRICING A NON-CALLABLE BOND

P N l , l _ l ICO+I00 J c - ~ (l+~Ip~ / ~ f

p ' " ,.l 1.o

• nc - ~ ~ Credit risk ~remia: - - - . . . . .

Pl

100

100 Bond price at maturity

100

(b) PRICING A CALL OPTION, CALLABLE AT R~ IN ONE YEAR, AND AT R2 = 100 IN TWO YEARS

po = max¢0 , - 0

al,I pl.0 p0.O = max{l r + 0,0 • o

1.0 1,0 Pg~-- max{0, P~' - R,} ~ 0

0 Call Option at maturity

(c) SUBTRACTING THE VALUES OF THE CALL OPTION FROM THE PRICE OF THE NON-CALLABLE BOND AT EACH NODE OF THE TREE WE GET THE PRICE OF THE CALLABLE BOND

J Pcl,l _ pl.l _ DI,I -- " N C ~

pO,O pO.O p~ • C = " NC --

~l ,0 _ p l , O _ pl,0 r C' = rNC -- r O

I 0 0 - 0

100-0 Callable bond price at maturity

100-0

Fig. 1. Pricing a non-ca l lable bond (a), a call option (b) and a cal lable bond ( c ) us ing a b inomia l lattice.

end of the binomial lattice - when the security is priced at par - discounting its price backwards, and comput- ing the average value. Fig. 1 illustrates the procedure on a s imple binomial lattice. The price of the non- callable bond at t ime period t and state o- (with risk- free discount rate r 7 = r ° ( k t ) ~r and risk premium Pt), is denoted by P~'c '~ and can be computed by applying the recursive equation:

/:~t+ I ,¢r p t + l ,tr+l p ~ = 1 , NC '~-* Nc , (19)

2 1 + rTpt

where P~" = 100 (par) for all states o- at maturity T. To calculate the value of the call option we use the

same discounting method as the one above for the non-callable bond. After the lockout period - when the call may be exercised - its value can not fall below

C. Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273 271

the difference between the non-callable bond and the redemption price Rt (see Fig. 1 (b ) ) . Let P ~ denote the price of the call option at time period t and state o-, and R t denote the redemption price. Then P~'~ can be computed by applying the recursive equation:

+P; R,) P~!~'~ = max 2 1 + rTp,

(20)

We mention that the redemption price is infinitely large before the lockout period expires, and is par at maturity.

Next we compute the value of the callable bond P~."~ at each time period t and state ~r by subtructing the value of the call option from the non-callable bond price:

P U = - P+ (21)

3.2. Holding period returns

Having resolved the price of a callable bond, we proceed with the generation of holding period returns. The return of a callable bond during a holding period ( r ) , under an interest rate scenario s, is given by

p~ + F .~ R . . . . . (22)

P0

where P* is the value of the bond at the end of the holding period (assuming it has not been called). The value is zero if the bond has been called. F s is the ac- crued value of any coupon payments made during the holding period plus the accrued value of the cashflow generated if the bond has been called. P0 is the current market price.

In the evaluation of the holding period return the interest rate scenario (s) realized during the holding period is specified. This scenario determines whether the bond is called or not during the holding period.

To generate scenarios of holding period returns we sample paths from the binomial lattice starting at the origin t = 0 and ending at t = r. Each path corre- sponds to a scenario s. We then traverse the path on the binomial lattice, check whether the call option has been exercised or not, and compute the value of F s. All relevant information needed for these calculations has already been generated as part of the bond pricing

calculations described above. The price of the bond at the end of the holding period under the given scenario ps, is also available from the pricing calculations.

4. Model validation: Tracking an index of callable bonds

Both indexation models were backtested using market data for the period from January 1992 to February 1993. The data were provided to us courtesy of BlackRock Financial Management, NY. The data include historical information for purchase prices, coupon payment dates, call dates, redemption prices of bonds etc. Term structure data was also collected, on a monthly basis, for the 14-month period. Holding period returns were derived from the data using the models of the previous section.

4.1. Index composition

Since there was no single callable bond index read- ily available, one was constructed from the large set of bond data available as follows: - A subset of about 230 securities was selected out

of a total of 600 securities. - Junk bonds (below BBB investment grade) were

excluded due to their speculative nature. - Although the group includes some AAA bonds, it

mostly consists of bonds with ratings ranging from AA to BBB, issued by a variety of institutions such as financial corporations (18%), credit and banking institutions (17%), telecommunications (18%), utilities (17%), department stores (7%), food chains (7%), chemical companies (4%), clothing (5%), automobile production (5%) and other industries (2%).

- The 230 bonds chosen carried equal weights in the composition of the index.

4.2. Minimizing downside tracking error

The backtesting of the model over the sample pe- riod proceeded as follows. First, the term structure available on January 1, 1992 was used to fit a bino- mial lattice and calculate holding period returns until February 1, 1992. The optimization model was then ran and the selected portfolio was kept for a month.

2 7 2 C. Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273

- - - 4 - - INDEX "-- I- - ROBUST- -o- - STOCH

35 t return(%) 3O

25 I

~o t 1 5

10

5

0

-5

-10

^ , , ell

, '., ;/ , ,Y ..... ....,,

6 • ~ y / \ .

,'/ ~k %, ~ mw

o / / • ,.,r"

' r ' I n /

month

- lS •

Fig. 2. Monthly annualized returns for the callable bond (INDEX), the portfolio generated using robust optimization (ROBUST) and the portfolio selected using the stochastic program (STOCH).

STOCH 116 - n •

114 ~ ......... ROBUST

7112 .... DEX = ~Ell 0 I I I I I ~ . ~

=~ 108 . , ~

9_ 106 / / 1 ~

(I. I /'/ 102 t

100 ~ ~ " ' ' ' ' ' ' 0 2 4 6 8 10 12 14

month

Fig. 3. Value of a $100M investment in the callable bond (INDEX), the portfolio generated using robust optimization (ROBUST) and the portfolio selected using the stochastic program (STOCH),

At the end of the month the binomial lattice was recal- ibrated using the term structure of February 1, 1992, holding period returns were calculated until March 1, 1992, and the optimization model was re-run. For any bonds added to the portfolio a transaction fee of 1 / 16th bp was charged. The actual return of the portfolio on February 1, 1992 was determined based on the re- turn of the bonds available ex poste using real market prices and coupon payments. The very first portfolio was constructed using an initial cash investment of $100M but without any payment of transaction costs.

Fig. 2 shows the annualized monthly returns pro- duced by the callable bond index portfolio as well as the robust optimization portfolio, over a 14-month period. The index realized an annualized return of 11.63% over the testing period, while the portfolio achieved a return of 12.36%.

The cumulative value of a $100 million portfolio invested in the whole index and in the robust opti- mization portfolio over the testing period is displayed in Fig. 3. The cumulative tracking error for the same investment is shown in Fig. 4. The index and the port- folio have a Sharpe measure of 0.84 and 0.76 respec- tively. This suggests that the portfolio is slightly less desirable than the index, when comparing the rewards gained by undertaking extra risk over the risk-free as- set. The index, however, is composed without account- ing for transaction costs, and liquidity contraints, and consists of a prohibitively large number of securities.

4.3. Testing the stochastic programming model

The three-stage model determines the composition of the portfolio at to (i.e., xo,Yo, zo) taking into ac-

C. Vassiadou-Zeniou, S.A. Zenios/European Journal of Operational Research 91 (1996) 264-273 273

3000000 ~ I~1

2500000

~2000000

.~ t 5ooooo I ~1000000 { I I

5 0 0 0 0 0

0 - -

1 2 3 4 5 6 7 8 9 10 11 12 13 14 month

Fig. 4. Cumulative surplus of a $100M investment between the portfolios generated using robust optimization (ROBUST) and the stochastic program (STOCH), and the INDEX.

count potential portfolio rebalancing decisions at t] (at the end of one month) and t2 (at the end of two months).

The holding period is two months, with intermedi- ate returns after the first month used for portfolio re- balancing decisions. The objective function is to max- imize the expected utility of e x c e s s r e t u r n expressed as the ratio of portfolio return to index return at the end of the two-month holding period. The model was backtested using data for the same period of January 1992-February 1993 using the same procedure em- ployed above. The results of the backtesting are also shown in Figs. 2, 3 and 4. The stochastic portfolio achieved an annualized return of 13.79%, compared to 11.63% for the index and 12.36% for the robust optimization portfolio. The Sharpe measure for the stochastic model is 0.97. This suggests that the prof- its realized by the stochastic portfolio against the risks undertaken, are better than the ones realized by the index which has a Sharpe measure of 0.84.

5. Conclusions

We have developed and tested robust optimization models for managing portfolios of callable bonds us- ing an indexation strategy. Both models have tracked closely the callable bond index during the 14-month testing period. The stochastic portfolio achieved a greater annual overperformance compared to the ro- bust optimization portfolio. It is also more attractive in terms of the higher "reward to volatility" trade-off suggested by their respective Sharpe measures. It is

clear from the results of this study that the multi- stage, stochastic programming model with recourse is significantly superior to the single-period model. Both models provide a credible and effective methodology for managing portfolios of fixed income securities, such as callable bonds.

References

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