Quantitative Finance, Vol. 7, No. 2, April 2007, 189–216

Stochastic programming for funding

mortgage pools

GERD INFANGER*yz

yDepartment of Management Science and Engineering,Stanford University, Stanford, CA 94305-4026, USA

zInfanger Investment Technology, LLC, 2680 Bayshore Parkway,Suite 206, Mountain View, CA 94043, USA

(Received 1 January 2006; in final form 12 October 2006)

We present how to use stochastic programming to best fund a pool of similar fixed-ratemortgages through issuing bonds, callable and non-callable, of various maturities. We discussthe estimation of expected net present value and (downside) risk for different funding instru-ments using Monte Carlo simulation techniques, and the optimization of the funding usingsingle- and multi-stage stochastic programming. Using realistic data we computed efficientfrontiers of expected net present value versus downside risk for the single- and the multi-stagemodel, and studied the underlying funding strategies. Constraining the duration and convexityof the mortgage pool and the funding portfolios to match at any decision point, we computedduration and convexity hedged funding strategies and compared them with those from themulti-stage stochastic programming model without duration and convexity constraints. Theout-of-sample results for the different data assumptions demonstrate that multi-stage stochas-tic programming yields significantly larger net present values at the same or at a lower level ofrisk compared with single-stage optimization and with duration and convexity hedging.We found that the funding strategies obtained from the multi-stage model differed signifi-cantly from those from the single-stage model and were again significantly different to fundingstrategies obtained from duration and convexity hedging. Using multi-stage stochasticprogramming for determining the best funding of mortgage pools will lead, in the average,to significantly higher profits compared with using single-stage funding strategies, or usingduration and convexity hedging. An efficient method for the out-of-sample evaluation ofstrategies obtained from multi-stage stochastic programming models is presented.

Keywords: Stochastic programming; Funding; Mortgage pools

1. Introduction

Historically, the business of conduits, like Freddie Mac,Fannie Mae or Ginnie Mae, has been to purchase mort-gages from primary lenders, pool these mortgages intomortgage pools, and securitize some if not all of thepools by selling the resulting Participation Certificates(PCs) to Wall Street. Conduits keep a fixed markup onthe interest for their profit and roll over most of the(interest rate and prepayment) risk to the PC buyers.Recently, a more active approach, with the potential forsignificantly higher profits, has become increasinglyattractive: instead of securitizing, funding the purchaseof mortgage pools by issuing debt. The conduit firm raises

the money for the mortgage purchases through a suitablecombination of long- and short-term debt. Thereby, theconduit assumes a higher level of risk due to interest ratechanges and prepayment risk but gains higher expectedrevenues due to the larger spread between the interest ondebt and mortgage rates compared with the fixed markupby securitizing the pool.

The problem faced by the conduits is an asset-liabilitymanagement problem, where the assets are the mortgagesbought from primary lenders and the liabilities are thebonds issued. Asset liability problems usually are facedby pension funds and insurance companies. Besidesassets, pension plans need to consider retirement obliga-tions, which may depend on uncertain economic andinstitutional variables, and insurance companies need toconsider uncertain pay-out obligations due to unforseenand often catastrophic events. Asset liability models are*Email: infanger@stanford.edu

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online # 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/14697680601065566

most useful when both asset returns and liability pay-outsare driven by common, e.g. economic, factors. Often, theunderlying stochastic processes and decision models aremulti-dimensional and require multiple state variables fortheir representation. Using stochastic dynamic program-ming, based on Bellman’s (1957) dynamic programmingprinciple, for solving such problems is therefore compu-tationally difficult, well known as the ‘curse of dimension-ality’. If the number of state variables of the problem issmall, stochastic dynamic programming can be appliedefficiently. Infanger (2006) discusses a stochastic dynamicprogramming approach for determining optimal dynamicasset allocation strategies over an investment horizonwith many re-balancing periods, where the value-to-gofunction is approximated via Monte Carlo sampling.The paper uses an in-sample/out-of-sample approach toavoid optimization bias.

Stochastic programming can take into account directlythe joint stochastic processes of asset and liability cashflows. Traditional stochastic programming uses scenariotrees to represent possible future events. The trees maybe constructed by a variety of scenario-generationtechniques. The emphasis is on keeping the resultingtree thin but representative of the event distribution andon arriving at a computationally tractable problem, whereobtaining a good first-stage solution rather thanobtaining an entire accurate policy is the goal. Earlypractical applications of stochastic programming forasset liability management are reported in Kusy andZiemba (1986) for a bank and in Carino et al. (1994)for an insurance company. Ziemba (2003) gives a sum-mary of the stochastic programming approach for assetliability and wealth management. Early applications ofstochastic programming for asset allocation are discussedin Mulvey and Vladimirou (1992), formulating financialnetworks, and Golub et al. (1995). Examples of earlyapplications of stochastic programming for dynamicfixed-income strategies are Zenios (1993), discussing themanagement of mortgage-backed securities, Hiller andEckstein (1993), and Nielsen and Zenios (1996). Wallaceand Ziemba (2005) present recent applications of stochas-tic programming, including financial applications.Frauendorfer and Schuerle (2005) discuss the re-financingof mortgages in Switzerland.

Monte Carlo sampling is an efficient approach forrepresenting multi-dimensional distributions. Anapproach, referred to as decomposition and MonteCarlo sampling, uses Monte Carlo (importance) samplingwithin a decomposition for estimating Benders cutcoefficients and right-hand sides. This approach hasbeen developed by Dantzig and Glynn (1990) andInfanger (1992). The success of the sampling within thedecomposition approach depends on the type of serialdependency of the stochastic parameter processes,determining whether or not cuts can be shared or adjustedbetween different scenarios of a stage. Infanger (1994)and Infanger and Morton (1996) show that, for serialcorrelation of stochastic parameters (in the form ofautoregressive processes), unless the correlation is limited

to the right-hand side of the (linear) program, cut sharingis at best difficult for more than three-stage problems.

Monte Carlo pre-sampling uses Monte Carlo samplingto generate a tree, much like the scenario-generationmethods referred to above, and then employs a suitablemethod for solving the sampled (and thus approximate)problem. We use Monte Carlo pre-sampling forrepresenting the mortgage funding problem, and combineoptimization and simulation techniques to obtain anaccurate and tractable model. We also provide an efficientway to independently evaluate the solution strategy fromsolving the multi-stage stochastic program to obtain avalid upper bound on the objective. The pre-samplingapproach provides a general framework of modelingand solving stochastic processes with serial dependencyand many state variables; however, it is limited in thenumber of decision stages. Assuming a reasonable samplesize for representing a decision tree, problems with up tofour decision stages are meaningfully tractable. Dempsterand Thorlacius (1998) discuss the stochastic simulation ofeconomic variables and related asset returns. A recentreview of scenario-generation methods for stochasticprogramming is given by Di Domenica et al. (2006),discussing also simulation for stochastic programmingscenario generation.

In this paper we present how multi-stage stochasticprogramming can be used for determining the bestfunding of a pool of similar fixed-rate mortgages throughissuing bonds, callable and non-callable, of variousmaturities. We show that significant profits can beobtained using multi-stage stochastic programmingcompared with using a single-stage model formulationand compared with using duration and convexityhedging, strategies often used in traditional finance. Forthe comparison we use an implementation of FreddieMac’s interest rate model and prepayment function.We describe in section 2 the basic formulation of fundingmortgage pools and discuss the estimation of expected netpresent value and risk for different funding instrumentsusing Monte Carlo sampling techniques. In section 3 wediscuss the single-stage model. In section 4 we present themulti-stage model. Section 5 discusses duration and con-vexity and delta and gamma hedging. In section 6 wediscuss numerical results using practical data obtainedfrom Freddie Mac. We compare the efficient frontiersfrom the single-stage and multi-stage models, discussthe different funding strategies and compare them withdelta and gamma hedged strategies, and evaluate thedifferent strategies using out-of-sample simulations.In particular, section 6.5 presents the details of the out-of-sample evaluation of the solution strategy obtainedfrom solving a multi-stage stochastic program. Section 7reports on the solution of very large models and givesmodel sizes and solution times. Finally, section 8summarizes the results of the paper.

While not explicitly discussed in this paper, theproblem of what fraction of the mortgage pool shouldbe securitized, and what portion should be retained andfunded through issuing debt can be addressed through a

190 G. Infanger

minor extension of the models presented. Fundingdecisions for a particular pool are not independent ofall other pools already in the portfolio and those to beacquired in the future. The approach can of course beextended to address also the funding of a number ofpools with different characteristics. While the paperfocuses on funding a pool of fixed-rate mortgages,the framework applies analogously to funding pools ofadjustable-rate mortgages.

2. Funding mortgage pools

2.1. Interest rate term structure

Well-known interest rate term structure models in theliterature are Vasicek (1977), Cox et al. (1985), Ho andLee (1986), and Hull and White (1990), based on onefactor, and Longstaff and Schwarz (1992) based on twofactors.

Observations of the distributions of future interest ratesare obtained using an implementation of the interest ratemodel of Luytjes (1993) and its update according to theFreddie Mac Document. The model reflects a stochasticprocess based on equilibrium theory using random shocksfor short rate, spread (between the short rate and theten-year rate) and inflation.

We do not use the inflation part of the model and treatit as a two-factor model, where the short rate and thespread are used to define the yield curve. To generate apossible interest rate path we feed the model at eachperiod with realizations of two standard normal randomvariables and obtain as output for each period a possibleoutcome of a yield curve of interest rates based on theparticular realizations of the random shocks. Given arealization of the short rate and the spread, thenew yield curve is constructed free of arbitrage for allcalculated yield points.

We denote as it(m), t ¼ 1, . . . ,T, the random interestrate of a zero coupon bond of term m in period t.

2.2. The cash flows of a mortgage pool

We consider all payments of a pool of fixed-ratemortgages during its lifetime. Time periods t range fromt ¼ 0, . . . ,T, where T denotes the end of the horizon; e.g.T¼ 360 reflects a horizon of 30 years considering monthlypayments. We let Bt be the balance of the principal ofthe pool at the end of period t. The principal capitalB0 is given to the homeowners at time period t¼ 0and is regained through payments �t and throughprepayments �t at periods t ¼ 1, . . . ,T. The balance ofthe principal is updated periodically by

Bt ¼ Bt�1ð1þ �0Þ � �t � �t, t ¼ 1, . . . ,T:

The rate �0 is the contracted interest rate of thefixed-rate mortgage at time t¼ 0. We define �t to be the

payment factor at period t ¼ 1, . . . ,T. The payment fac-tor when multiplied by the mortgage balance yields theconstant monthly payments necessary to pay off the loanover its remaining life, e.g.

�t ¼ �0=ð1� ð1þ �0Þt�T�1

Þ;

thus,

�t ¼ �tBt�1:

The payment factor �t depends on the interest rate �0.For fixed-rate mortgages the quantity �0, and thus thequantities �t, are known with certainty. However,prepayments �t, at periods t ¼ 1, . . . ,T, depend on futureinterest rates and are therefore random parameters.

Prepayment models or functions represent therelationship between interest rates and prepayments.See, for example, Kang and Zenios (1992) for a detaileddiscussion of prepayment models and factors drivingprepayments.

In order to determine �t we use an implementation ofFreddie Mac’s prepayment function according to Lekkasand Luytjes (1993). Denoting the prepayment ratesobtained from the prepayment function as �t,t ¼ 1, . . . ,T, we compute the prepayments �t in period tas

�t ¼ �tBt�1:

2.3. Funding through issuing debt

We consider funding through issuing bonds, callable andnon-callable, with various maturities. Let ‘ be a bondwith maturity m‘, ‘ 2 L, where L denotes the set ofbonds under consideration. Let f �‘t be the payment factorfor period t, corresponding to a bond ‘ issued at period �,� � t � � þm‘:

f �‘t �

þ1, if t� � ¼ 0,

�ði�ðm‘Þ þ s�‘Þ, if 0 < t� � < m‘,

�ð1þ i�ðm‘Þ þ s�‘Þ, if t� � ¼ m‘,

8>>><>>>:

where i�ðm‘Þ reflects the interest rate of a zero couponbond with maturity m‘, issued at period �, and s�‘ denotesthe spread between the zero coupon rate and the actualrate of bond ‘ issued at �. The spread s�‘ includes thespread of bullet bonds over zero coupon bonds (referredto as agency spread) and the spread of callable bonds overbullet bonds (referred to as agency call spread), and iscomputed according to the model specification given inthe Freddie Mac document (Luytjes 1996).

LetM ‘� denote the balance of a bond ‘ at the time � it is

issued. The finance payments resulting from bond ‘ are

d ‘t ¼ f �‘tM

‘� , t ¼ �, . . . , � þm‘,

from the time of issue (�) until the time it matures(� þm‘) or, if callable, it is called. We consider the

Stochastic programming for funding mortgage pools 191

balance of the bullet from the time of issue until the timeof maturity as

M ‘t ¼ M ‘

� , t ¼ �, . . . , � þm‘:

2.4. Leverage ratio

Regulations require that, at any time t, t ¼ 0, . . . ,T,equity is set aside against debt in an amount such thatthe ratio of the difference of all assets minus all liabilitiesto all assets is greater than or equal to a given value �. LetEt be the balance of an equity (cash) account associatedwith the funding. The equity constraint requires that

Bt þ Et �Mt

Bt þ Et

� �,

where the total asset balance is the sum of the mortgagebalance and the equity balance, Bt þ Et, andMt ¼

P‘ M

‘t is the total liability balance.

At time periods t ¼ 0, . . . ,T, given the mortgagebalance Bt, and the liability balance Mt, we compute theequity balance that fulfills the leverage ratio constraintwith equality as

Et ¼Mt � Btð1� �Þ

1� �, t ¼ 0, . . . ,T:

We assume that the equity account accrues interestaccording to the short rate itðshortÞ, the interest rateof a 3-month zero coupon bond. Thus, we have thefollowing balance equation for the equity account:

Et ¼ Et�1ð1þ it�1ðshortÞÞ þ et�1,

where et are payments into the equity account (positive)or payments out of the equity account (negative). Usingthis equation we compute the payments et to and from theequity account necessary to maintain the equity balanceEt computed for holding the leverage ratio �.

2.5. Simulation

Using the above specification we may perform a (MonteCarlo) simulation in order to obtain an observation of allcash flows resulting from the mortgage pool and fromfinancing the pool through various bonds. In order todetermine in advance how the funding is carried out, weneed to specify certain decision rules defining what to dowhen a bond matures, when to call a callable bond, atwhat level to fund, and how to manage profits and losses.For the experiment we employed the following six rules.

(i) Initial funding is obtained at the level of the initialbalance of the mortgage pool, M0 ¼ B0.

(ii) Since at time t¼ 0, M0 ¼ B0, it follows thatE0 ¼ ½�=ð1� �Þ�B0, an amount that we assume tobe an endowed initial equity balance.

(iii) When a bond matures, refunding is carried out usingshort-term debt (non-callable 3-month bullet bond)

until the end of the planning horizon, each time atthe level of the balance of the mortgage pool.

(iv) Callable bonds are called according to the call rulespecification in Freddie Mac’s document (Luytjes1996). Upon calling, refunding is carried outusing short-term debt until the end of the planninghorizon, each time at the level of the balance of themortgage pool.

(v) The leverage ratio (ratio of the difference of all assetsminus all liabilities to all assets) is � ¼ 0:025.

(vi) At each time period t, after maintaining the leverageratio, we consider a positive sum of all payments asprofits and a negative sum as losses.

According to the decision rules, when funding amortgage pool using a single bond ‘, we assume at timet¼ 0 that M ‘

0 ¼ B0, i.e. that exactly the amount of theinitial mortgage balance is funded using bond ‘. Afterbond ‘ matures refunding takes place using anotherbond (according to the decision rules, short-term debt,say, bond ‘), based on the interest rate and the level ofthe mortgage balance at the time it is issued. If the initialbond ‘ is callable, it may be called, and then fundingcarried out through another bond (say, short-termdebt ‘). Financing based on bond ‘ is continued untilthe end of the planning horizon, i.e. until T� � < m‘,and no more bond is issued. Given the type of bondbeing used for refunding, and given an appropriate callingrule, all finance payments for the initial funding usingbond ‘ and the subsequent refunding using bond ‘ canbe determined. We denote the finance payments accruingfrom the initial funding based on bond ‘ and its conse-quent refunding based on bond ‘ as

d ‘t , t ¼ 1, . . . ,T:

Once the funding and the corresponding liability balanceM ‘

t is determined, the required equity balance Et ¼ E ‘t

and the payments et ¼ e ‘t are computed.

2.6. The net present value of the payment stream

Finally, we define as

P ‘t ¼ �t þ �t þ d ‘

t � e ‘t , t ¼ 1, . . . ,T, P0 ¼ M0 � B0 ¼ 0

the sum of all payments in period t, t ¼ 0, . . . ,T, resultingfrom funding a pool of mortgages (initially) using bond ‘.

Let It be the discount factor for period t, i.e.

It ¼Ytk¼1

ð1þ ikðshortÞÞ, t ¼ 1, . . . ,T, I0 ¼ 1,

where we use the short rate at time t, itðshortÞ, for dis-counting. The net present value (NPV) of the paymentstream is then calculated as

r‘ ¼XTt¼0

P ‘t

It:

192 G. Infanger

So far, we consider all quantities that depend on interestrates as random parameters. In particular, P ‘

t is a randomparameter, since �t, �t, d

‘t , and e ‘t are random parameters

depending on random interest rates. Therefore, the netpresent value r‘ is a random parameter as well. Inorder to simplify the notation we do not label any specificoutcomes of the random parameters. A particular run ofthe interest rate model requires 2T random outcomes ofunit normal random shocks. We now label a particularpath of the interest rates obtained from one run of the inter-est rate model and all corresponding quantities with !.In particular, we label a realization of the net presentvalue based on a particular interest rate path as r!‘ .

2.7. Estimating the expected NPV of the payment stream

We use Monte Carlo sampling to estimate the expectedvalue of the NPV of a payment stream. Under a crudeMonte Carlo approach to the NPV estimation, we sampleN paths ! 2 S, N ¼ jSj, using different observations ofthe distributions of the 2T random parameters as inputto the interest rate model, and we compute r!‘ for each! 2 S. Then, an estimate for the expected net presentvalue (NPV) of the cash flow stream based on initialfunding using bond ‘ is

�r‘ ¼1

N

X!2S

r!‘ :

We do not describe in this document how we useadvanced variance reduction techniques (e.g. importancesampling) for the estimation of the expected net presentvalue of a payment stream. We refer to Prindiville (1996)for how importance sampling could be applied.

2.8. The expected NPV of a funding mix

Using simulation (as described above) we compute thenet present value of the payment stream r!‘ for eachrealization ! 2 S and each possible initial funding ‘ 2 L.The net present value of a funding mix is given by thecorresponding convex combination of the net presentvalues of the components ‘ 2 L, i.e.

r! ¼X‘2L

r!‘ x‘,X‘2L

x‘ ¼ 1, xl � 0,

where x‘ are non-negative weights summing to one.The expected net present value of a funding mix,

�r ¼1

N

X!2S

r!,

is also represented as the convex combination of theexpected net present values of the components ‘ 2 L, i.e.

�r ¼X‘2L

�r‘x‘,X‘2L

x‘ ¼ 1, x‘ � 0:

2.9. Risk of a funding mix

In order to measure risk, we use as an appropriateasymmetric penalty function the negative part of thedeviation of the NPV of a funding portfolio from apre-specified target u, i.e.

v! ¼X‘2L

r!‘ x‘ � u

!�

,

and consider risk as the expected value of v!, estimated as

�v ¼1

N

X!2S

v!:

A detailed discussion of this particular risk measure isgiven in Infanger (1996). The efficient frontier with riskas the first lower partial moment is also referred to as the‘put–call efficient frontier’; see, for example, Dembo andMausser (2000).

3. Single-stage stochastic programming

Having computed the NPVs r!‘ for all initial fundingoptions ‘ 2 L and for all paths ! 2 S, we optimize thefunding mix with respect to expected returns and riskby solving the (stochastic) linear program

min1

N

Xv! ¼ �v,

s.t.X‘

r!‘ x‘ þ v! � u, ! 2 S,

X‘

�r‘x‘ � �,

X‘

x‘ ¼ 1,

x‘ � 0,

v! � 0:

The parameter � is a pre-specified value that the expectednet present value of the portfolio should exceed or beequal to. Clearly, � � �max

¼ max‘f�r‘g. Using the modelwe trace out an efficient frontier starting with � ¼ �max

and successively reducing � until � ¼ 0, each time solvingthe linear program to obtain the portfolio with theminimum risk �v corresponding to a given value of �.

The single-stage stochastic programming modeloptimizes funding strategies based on decision rulesdefined over the entire planning horizon of T¼ 360 peri-ods, where the net present value of each funding strategyusing initially bond ‘ and applying the decision rules isestimated using simulation.

Stochastic programming for funding mortgage pools 193

A variant of the model arises by trading offexpected NPV and risk in the objective, with � denotingthe risk-aversion coefficient:

min �X‘

�r‘x‘ þ �1

N

Xv!,

s.t.X‘

r!‘ x‘ þ v! � u, ! 2 S,

X‘

x‘ ¼ 1,

x‘ � 0,

v! � 0:

For a risk aversion of � ¼ 0, risk is not part of theobjective and expected NPV is maximized. The efficientfrontier can be traced out by increasing the risk aversion �successively from zero to very large values, where the riskterm in the objective entirely dominates.

This approach is very different to Markowitz’s (1952)mean variance analysis in that the distribution of theNPV is represented through scenarios (obtained throughsimulations over a long time horizon, considering theapplication of decision rules) and a downside risk mea-sure is used for representing risk.

4. Multi-stage stochastic programming

In the following we relax the application of decision rulesat certain decision points within the planning horizon,and optimize the funding decisions at these points.This leads to a multi-stage stochastic programmingformulation.

We partition the planning horizon h0,T i into nsub-horizons hT1,T2i, hT2,T3i, . . . , hTn,Tnþ1i, whereT1 ¼ 0, and Tnþ1 ¼ T. For the experiment, we considern¼ 4, and partition at T1¼ 0, T2¼ 12, T3¼ 60, andT4¼ 360. We label the decision points at time t ¼ T1 asstage 1, at time t ¼ T2 as stage 2, and at time t ¼ T3 asstage 3 decisions. Funding obtained at the decision stagesis labeled as ‘1 2 L1, ‘2 2 L2, and ‘3 2 L3 according tothe decision stages. At time t ¼ T4, at the end of theplanning horizon, the (stage 4) decision involvesmerely evaluating the net present value of each endpoint for calculating the expected NPV and risk. Inbetween the explicit decision points, at which funding issubject to optimization, we apply the decision rulesdefined above.

Instead of interest rate paths as used in the single-stagemodel, we now use an interest rate tree with nodes at eachstage. We consider jS2j paths !2 2 S2 between t ¼ T1 andt ¼ T2; for each node !2 2 S2 we consider jS3j paths!3 2 S3 between t ¼ T2 and t ¼ T3; for each nodeð!2,!3Þ 2 fS2 � S3g we consider jS4j paths !4 2 S4

between t ¼ T3 and t ¼ T4. Thus, the tree hasjS2 � S3 � S4j end points. We may denoteS ¼ fS2 � S3 � S4g and ! ¼ ð!2,!3,!4Þ. Thus aparticular path through the tree is now labeled as! ¼ ð!2,!3,!4Þ using an index for each partition.Figure 1 presents the decision tree of the multi-stagemodel for only two paths for each period.

The simulation runs for each partition of the planning

horizon are carried out in such a way that the dynamics

of the interest rate process and the prepayment function

are fully carried forward from one partition to the next.

Since the interest rate model and the prepayment

function include many lagged terms and require the stor-

ing of 64 state variables, the application of dynamic

Figure 1. Multi-stage model setup, decision tree.

194 G. Infanger

programming for solving the multi-stage program is not

tractable.Let I�t be the discount factor of period t, discounted to

period �, i.e.

I�t ¼Yt

k¼�þ1

ð1þ ikðshortÞÞ, t > �, I�� ¼ 1,

where we use the short rate at time t, itðshortÞ, fordiscounting.

Let L1 be the set of funding instruments available attime T1. Funding obtained at time T1 may mature or becalled during the first partition (i.e. before or at time T2),during the second partition (i.e. after time T2 and beforeor at time T3), or during the third partition (i.e. after timeT3 and before or at time T4). We denote the set of fundinginstruments issued at time T1 and matured or called dur-ing the first partition of the planning horizon as L!2

11 , theset of funding instruments issued at time T1 and maturedor called during the second partition of the planninghorizon as L

!2!3

12 , and the set of funding instrumentsissued at time T1 and matured or called during the thirdpartition of the planning horizon as L

!2!3!4

13 . Clearly,L1 ¼ L!2

11 [ L!2!3

12 [ L!2!3!4

13 , for each ð!2,!3,!4Þ 2

fS2 � S3 � S4g. Similarly, we denote the set of fundinginstruments issued at time T2 and matured or called dur-ing the second partition of the planning horizon as L

!2!3

22 ,and the set of funding instruments issued at time T2 andmatured or called during the third partition of the plan-ning horizon as L

!2!3!4

23 . Clearly, L2 ¼ L!2!3

22 [ L!2!3!4

23 ,for each ð!2,!3,!4Þ 2 fS2 � S3 � S4g. Finally, we denotethe set of funding instruments issued at time T3 andmatured or called during the third partition of theplanning horizon as L33. Clearly, L3 ¼ L33 ¼ L

!2!3!4

33 ,for each ð!2,!3,!4Þ 2 fS2 � S3 � S4g.

For all funding instruments ‘1 2 L!2

1 initiated at timet¼ 0 that mature or are called during the first partition,we obtain the net present values

r!2

‘1ð11Þ¼XT2

t¼0

P ‘1!2t

I!2

0t

;

for all funding instruments ‘1 2 L!2!3

12 initiated attime t¼ 0 that mature or are called during the secondpartition, we obtain the net present values

r!2,!3

‘1ð12Þ¼

1

I!2

0T2

XT3

t¼T2þ1

P‘1!3t

I!3

T2t

;

and all initial funding instruments ‘1 2 L!2!3!4

13 , initiatedat time t¼ 0 that mature or are called during the thirdpartition, we obtain the net present values

r!2,!3,!4

‘1ð13Þ¼

1

I!2

0T2

1

I!3

T2T3

XT4

t¼T3þ1

P ‘1!4t

I!4

T3t

:

For all funding instruments ‘2 2 L!2!3

22 , initiated at timet ¼ T2, that mature or are called during the secondpartition, we obtain the net present values

r!2,!3

‘2ð22Þ¼

1

I!2

0T2

XT3

t¼T2þ1

P‘2!3t

I!3

T2t

,

and for all funding instruments ‘2 2 L!2!3

23 , initiated attime t ¼ T2, that mature or are called during the thirdpartition, we obtain the net present values

r!2,!3,!4

‘2ð23Þ¼

1

I!2

0T2

1

I!3

T2T3

XT4

t¼T3þ1

P‘2!4t

I!4

T3t

:

We obtain for all initial funding ‘ 2 L33 initiated at timet ¼ T3 the net present values

r!2,!3,!4

‘3ð33Þ¼

1

I!2

0T2

1

I!3

T2T3

XT4

t¼T3þ1

P‘3!4t

I!4

T3t

:

Let N ¼ jSj. Let R!‘1 ¼ r!2

‘1ð11Þþ r

!2!3

‘1ð12Þþ r

!2!3!4

‘1ð13Þ,

R!‘2 ¼ r

!2!3

‘2ð22Þþ r

!2!3!4

‘2ð23Þ, and R!

‘3 ¼ r!2!3!4

‘3ð33Þ. Let x‘1 be the

amount of funding in instrument ‘1 2 L1 issued at timet ¼ T1, x‘2 be the amount of funding in instrument‘2 2 L2 issued at time t ¼ T2, and x‘3 be the amount offunding in instrument ‘3 2 L3 issued at time t ¼ T3. Basedon the computation of the net present values, we optimizethe funding mix solving the multi-stage (stochastic) linearprogram:

min E v! ¼ �v,

s.t.X‘12L1

x‘1 ¼ 1,

�X

‘12L!211

x‘1 þX‘22L2

x!2

‘2¼ 0,

�X

‘12L!2!312

x‘1 �X

‘22L!2!322

x!2

‘2þX‘32L3

x!2!3

‘3¼ 0,

X‘12L1

R!‘1x‘1 þ

X‘22L2

R!‘2x

!2

‘2þX‘32L3

R!‘3x

!2!3

‘3� w!

¼ 0,

v! þ w!� u,

E w!� �,

x‘1 , x!2

‘2,x

!2!3

‘3, v! � 0,

where E w!¼ ð1=NÞ

Pw! is the estimate of the expected

net present value and E v! ¼ ð1=NÞP

v! is the estimateof the risk. As in the single-stage model before, theparameter � is a pre-specified value for the expected netpresent value of the portfolio. Starting with � ¼ �max,the maximum value of � that can be assumed withoutthe linear program becoming infeasible, we trace out anefficient frontier by successively reducing � from � ¼ �max

to �¼ 0 and computing for each level of � thecorresponding value of risk �v by solving the multi-stagestochastic linear program. The quantity �max, the

Stochastic programming for funding mortgage pools 195

maximum expected net present value without consideringrisk, can be obtained by solving the linear program

max E w!¼ �max,

s.t.X‘12L1

x‘1 ¼ 1,

�X

‘12L!211

x‘1 þX‘22L2

x!2

‘2¼ 0,

�X

‘12L!2!312

x‘1 �X

‘22L!2!322

x!2

‘2þX‘32L3

x!2!3

‘3¼ 0,

X‘12L1

R!‘1x‘1 þ

X‘22L2

R!‘2x

!2

‘2þX‘32L3

R!‘3x

!2!3

‘3� w!

¼ 0,

x‘1 , x!2

‘2, x

!2!3

‘3� 0:

Note that the model formulation presented above doesnot consider the calling of callable bonds as subject tooptimization at the decision stages; rather the calling ofcallable bonds is handled through the calling rule as partof the simulation. Optimizing also the calling of callablebonds at the decision stages requires only a minorextension to the model formulation, but this is notdiscussed here.

A variant of the multi-stage model arises by trading offexpected net present value and risk in the objective with �as the risk-aversion coefficient:

min �E v! � E w!,

s.t.X‘12L1

x‘1 ¼ 1,

�X

‘12L!211

x‘1 þX‘22L2

x!2

‘2¼ 0,

�X

‘12L!2!312

x‘1 �X

‘22L!2!322

x!2

‘2þX‘32L3

x!2!3

‘3¼ 0,

X‘12L1

R!‘1x‘1 þ

X‘22L2

R!‘2x

!2

‘2þX‘32L3

R!‘3x

!2!3

‘3� w!

¼ 0,

v! þ w!� u,

x‘1 , x!2

‘2, x

!2!3

‘3, v! � 0:

5. Duration and convexity

Since the payments from a mortgage pool are notconstant, indeed the prepayments depend on the interestrate term structure and its history since the inception ofthe pool, an important issue arises as to how the netpresent value (price) of the mortgage pool changes as aresult of a small change in interest rates. The same issuearises for all funding instruments, namely, how bondprices change as a result of small changes in yield. Thisis especially interesting in the case of callable bonds.In order to calculate the changes in expected net presentvalue due to changes in interest rates, one usually resorts

to first- and second-order approximations, where thefirst-order (linear, or delta) approximation is called theduration and the second-order (quadratic, or gamma)approximation is called the convexity. While the durationand convexity of non-callable bonds could be calculatedanalytically, the duration and convexity of a mortgagepool and callable bonds can only be estimated throughsimulation. We use the terms effective duration andeffective convexity to refer to magnitudes estimatedthrough simulation.

5.1. Effective duration and convexity

Let

p ¼XTt¼0

P poolt

It

be the net present value of the payments from themortgage pool, where P pool

t ¼ �t þ �t and It is the dis-count factor using the short rate for discounting.We compute p!, for scenarios ! 2 S, using Monte Carlosimulation, and we calculate the expected net presentvalue (price) of the payments of the mortgage pool as

�p ¼1

N

X!2S

p!:

Note that the payments Pt ¼ Ptðik, k ¼ 0, . . . , tÞ dependon the interest rate term structure and its history up toperiod t, where it denotes the vector of interest ratesfor different maturities at time t. Writing explicitly thedependency,

�p ¼ �pðit, t ¼ 0, . . . ,T Þ:

We now define

�pþ ¼ �pðit þ�, t ¼ 0, . . . ,T Þ

as the net present value of the payments of the mortgagepool for an upward shift of all interest rates by �%, and

�p� ¼ �pðit ��, t ¼ 0, . . . ,T Þ

as the net present value of the payments of the mortgagepool for a downward shift of all interest rates by �%,where � is a shift of, say, one percentage point in theentire term structure at all periods t ¼ 1, . . . ,T.

Using the three points �p�, �p, �pþ, and the correspondinginterest rate shifts ��, 0, þ�, we compute the effectiveduration of the mortgage pool, as

dur ¼�p� � �pþ2��p

,

and the effective convexity of the mortgage pool,

con ¼�p� þ �pþ � 2 �p

100�2 �p:

196 G. Infanger

The quantities of the effective duration and effectiveconvexity represent a local first-order (duration) andsecond-order (duration and convexity) Taylor approxi-mation of the net present value of the payments of themortgage pool as a function of interest. The approxima-tion considers the effects on a constant shift of the entireyield curve across all points t, t ¼ 1, . . . ,T. The number100 in the denominator of the convexity simply scales theresulting numbers. The way it is computed, we expect apositive value for the duration, meaning that decreasinginterest rates result in a larger expected net present valueand increasing interest rates result in a smaller expectednet present value. We also expect a negative value for theconvexity, meaning that the function of price versus yieldis locally concave.

In an analogous fashion we compute the duration dur‘and the convexity con‘ for all funding instruments ‘. Let

p‘ ¼Xm‘

t¼0

d ‘t

It

be the net present value of the payments of the bond ‘,where d ‘

t represents the payments of bond ‘ until maturityor until it is called. We compute p!‘ using Monte Carlosimulation over ! 2 S, and we calculate the expected netpresent value (price) of the payments for the bond ‘ as

�p‘ ¼1

N

X!2S

p!‘ :

We calculate

�p‘þ ¼ �p‘ðit þ�, t ¼ 0, . . . ,T Þ,

and

�p‘� ¼ �p‘ðit ��, t ¼ 0, . . . ,T Þ,

the expected net present values for an upwards anddownwards shift of interest rates, respectively.Analogously to the mortgage pool, we obtain the dura-tion of bond ‘ as

dur‘ ¼�p‘� � �p‘þ2��p‘

,

and its convexity as

con‘ ¼�p‘� þ �p‘þ � 2 �p‘

100�2 �p‘:

Since in the case of non-callable bonds the payments d ‘t

are fixed, changes in expected net present value due tochanges in interest rates are influenced by the discountfactor only. For non-callable bonds we expect a positivevalue for duration, meaning that increasing interest ratesimply a smaller bond value and decreasing interest ratesimply a larger bond value. We expect a positive valuefor convexity, meaning that the function of expected netpresent value versus interest rates is locally convex. In thecase of callable bonds the behavior of the functionof expected net present value versus interest rates isinfluenced not only by the discount rate but also by the

calling rule. If interest rates decrease the bond may becalled and the principal returned. The behavior of callablebonds is similar to that of mortgage pools in that weexpect a positive value for duration and a negativevalue for convexity.

5.2. Traditional finance: matching duration and convexity

Applying methods of traditional finance, one wouldhedge interest rate risk by constructing a portfolio witha duration and a convexity of close to zero, respectively,thus achieving that the portfolio would exhibit no changein expected net present value (price) due to a small shift inthe entire yield curve. Duration and convexity matchingis also referred to as immunization (see, for example,Luenberger (1998) or as delta and gamma hedging.

In the situation of funding mortgage pools, hedging iscarried out in such a way that a change in the price of themortgage pool is closely matched by the negative changein the price of the funding portfolio, such that the changeof the total portfolio (mortgage pool and funding) is closeto zero. Thus the duration of the total portfolio is close tozero. In addition, the convexity of the mortgage poolis matched by the (negative) convexity of the fundingportfolio, such that the convexity of the total portfolio(mortgage pool and funding) is close to zero. We write thecorresponding duration and convexity hedging model as

maxX‘

r‘x‘X‘

dur‘x‘ � dg ¼ dur,

X‘

con‘x‘ � cg ¼ con,

X‘

x‘ ¼ 1,

x‘ � 0,

and

�dgmax� dg � dgmax, �cgmax

� cg � cgmax,

and expected net present value is maximized in theobjective. The variable dg accounts for the durationgap, cg accounts for the convexity gap, dgmax representsa predefined upper bound on the absolute value of theduration gap, and cgmax a predefined upper bound on theabsolute value of the convexity gap. Usually, when aduration and convexity hedged strategy is implemented,the model needs to be revised over time as the mortgagepool and the yield curve changes. In practise, updatingthe funding portfolio may be done on a daily ormonthly basis to reflect changes in the mortgage pooldue to prepayments and interest rate variations.

The duration and convexity hedging model is adeterministic model, uncertainty is considered as a shiftof the entire yield curve, and hedged to the extent of theeffect of the remaining duration and convexity gap.

Stochastic programming for funding mortgage pools 197

5.3. Duration and convexity in the single-stage model

In the single-stage case we add the duration and convexityconstraints

X‘

dur‘x‘ � dg ¼ dur,

X‘

con‘x‘ � cg ¼ con,

and

�dgmax� dg � dgmax, �cgmax

� cg � cgmax

to the single-stage linear program using the formulationin which expected net present value and risk are traded offin the objective. Setting the risk aversion � to zero,only the expected net present value is considered in theobjective, and the resulting single-stage stochasticprogram is identical to the duration and convexityhedging formulation from traditional finance as discussedin the previous section. This formulation allows one toconstrain the absolute value of the duration andconvexity gap to any specified level, to the extent thatthe single-stage stochastic program remains feasible.By varying the duration and convexity gap, we maystudy the effect of the resulting funding strategy onexpected net present value and risk.

5.4. Duration and convexity in the multi-stage model

In the multi-stage model we wish to constrain theduration and convexity gap not only in the first stage,but at any decision stage and in any scenario. Thus, inthe four-stage model discussed above we have one pair ofconstraint for the first stage, jS2j pairs of constraintsin the second stage, and jS2 � S3j pairs of constraints inthe third stage. Accordingly, we need to compute theduration and the convexity for the mortgage pool andall funding instruments at any decision point in all stagesfrom one to three.

In order to simplify the presentation, we omit thescenario indices !2 2 S2 and ð!2,!3Þ 2 fS2 � S3g. Letdurt and cont be the duration and convexity of themortgage pool in each stage t, and dur‘t and con‘t bethe duration and convexity of the funding instrumentsissued at each period t.

Let durFt and conFt be symbols used to convenientlypresent the duration and convexity of thefunding portfolio at each period t. In the firststage, durF1 ¼

P‘12L1

dur‘1x‘1 . In the secondstage, durF2 ¼

P‘22L2

dur‘2x‘2þP

‘12L12dur‘1ð12Þx‘1 , where

dur‘1ð12Þ represents the duration of the fundinginstruments from the first stage that are stillavailable in the second stage. In the third stage, durF3 ¼P

‘32L3dur‘3x‘3 þ

P‘22L23

dur‘2ð23Þx‘2 þP

‘12L13dur‘1ð13Þ x‘1 ,

where dur‘2ð23Þ represents the duration of the fundinginstruments issued in the second stage still available inthe third stage, and dur‘1ð13Þ represents the duration of

the funding instruments from the first stage still availablein the third stage.

Analogously, in the first stage, conF1 ¼P

‘12L1con‘1x‘1 .

In the second stage, conF2 ¼P

‘22L2con‘2x‘2þP

‘12L12con‘1ð12Þx‘1 , where con‘1ð12Þ is the convexity of

the funding instruments from the first stage still availablein the second stage. In the third stage, conF3 ¼

P‘32L3

con‘3x‘3 þP

‘22L23con‘2ð23Þx‘2 þ

P‘12L13

con‘1ð13Þx‘1 ,where con‘2ð23Þ is the convexity of the funding instrumentsissued in the second stage still available in the third stage,and con‘1ð13Þ is the convexity of the funding instrumentsfrom the first stage still available in the third stage.

Thus, in the multi-stage case, we add in each staget ¼ 1, . . . , 3 and in each scenario !2 2 S2 andð!2,!3Þ 2 fS2 � S3g the constraints

durFt � dgt ¼ durt,

conFt � cgt ¼ cont,

and

�dgmaxt � dgt � dgmax

t , �cgmaxt � cgt � cgmax

t :

The variables dgt, cgt account for the duration andconvexity gap, respectively, in each decisions stageand in each of the scenarios !2 2 S2 andð!2,!3Þ 2 fS2 � S3g. At each decision node the absolutevalues of the duration and convexity gap are con-strained by dgmax

t and cgmaxt , respectively. This formu-

lation allows one to constrain the duration andconvexity gap to any specified level, even at differentlevels at each stage, to the extent that the multi-stagestochastic linear program remains feasible.

Using the formulation of the multi-stage model, inwhich expected net present value and risk are tradedoff in the objective, and setting the risk aversion coeffi-cient � to zero, we obtain a model in which the durationand convexity are constrained at each node of the sce-nario tree and expected net present value is maximized.Since the scenario tree represents simulations of possibleevents of the future, the model results in a duration andconvexity hedged funding strategy, where duration andconvexity are constrained at the decision points of thescenario tree, but not constrained at points in between,where the funding portfolios are carried forward usingthe decision rules. We use this as an approximation forother duration and convexity hedged strategies, in whichduration and convexity are matched, for example, atevery month during the planning horizon. One could,in addition, apply funding rules in the simulation thatresult in duration and convexity hedged portfolios atevery month, but such rules have not been applied inthe computations used for this paper.

We are now in the position to compare the resultsfrom the multi-stage stochastic model trading offexpected net present value and downside risk withthe deterministic duration and convexity hedgedmodel on the same scenario tree. The comparisonlooks at expected net present value and risk (repre-sented by various measures), as well as underlyingfunding strategies.

198 G. Infanger

6. Computational results

6.1. Data assumptions

For the experiment we used three data sets, based ondifferent initial yield curves, labeled ‘Normal’, ‘Flat’,and ‘Steep’. The data represent assumptions about theinitial yield curve, the parameters of the interest ratemodel, and the prepayment function, assumptions aboutthe funding instruments, assumptions about refinancingand the calling rule, and the planning horizon and itspartitioning.

Table 1 presents the initial yield curve (correspondingto a zero coupon bond) for each data set. For eachdata set the mortgage contract rate is assumed to beone percentage point above the 10-year rate (labeled‘y10’).

For the experiment we consider 16 different fundinginstruments. Table 2 presents the maturity, the timeafter which the instrument may be called, and the initialspread over the corresponding zero coupon bond (of thesame maturity) for each instrument and for each of thedata sets. For example, ‘y03nc1’ refers to a callable bondwith a maturity of 3 years (36 months) and callable after1 year (12 months). Corresponding to the data set‘Normal’, it could be issued initially (at time t¼ 0) at arate of 6:12%þ 0:36% ¼ 6:48%, where 6.12% is theinterest rate from table 1 and 0.36% is the spread fromtable 2.

We computed the results for a pool of $100M. As thetarget for risk, u, we used the maximum expected netpresent value obtained using single-stage optimization,i.e. we considered risk as the expected net present valuebelow this target, defined for each data set. In particular,the target for risk equals u ¼ 10:5M for the ‘Normal’ dataset, u ¼ 11:3M for the ‘Flat’ data set, and u ¼ 18:6M forthe ‘Steep’ data set.

We first used single-stage optimization using N¼ 300interest rate paths. These results are not presented here.Then, in order to more accurately compare single-stageand multi-stage optimization, we used a tree withN¼ 4000 paths, where the sample sizes in each stage arejS2j ¼ 10, jS3j ¼ 20 and jS4j ¼ 20. For this tree the multi-stage linear program has 8213 rows, 11 377 columns and

218 512 non-zero elements. The program can easily besolved on a modern personal computer in a very smallamount of (elapsed) time. Also, the simulation runs toobtain the coefficients for the linear program can easilybe carried out on a modern personal computer.

6.2. Results for the single-stage model

As a base case for the experiment we computedthe efficient frontier for each of the data sets using thesingle-stage model. Figure 2 presents the result forthe ‘Normal’ data set in comparison with the efficientfrontiers obtained from the multi-stage model. (The sin-gle-stage results obtained from optimizing on the treeclosely resemble the efficient frontiers obtained fromoptimizing on 300 interest rate paths.) The efficient fron-tiers for the ‘Flat’ and ‘Steep’ data sets are similar inshape to that of the ‘Normal’ data set. While ‘Normal’and ‘Flat’ have the typical shape one would expect, i.e.steep at low levels of risk and bending more flat withincreasing risk, it is interesting to note that the efficientfrontier for data set ‘Steep’ is very steep at all levels ofrisk.

As a base case we look at the optimal funding strategyat the point of 95% of the maximum expected net presentvalue. In the graphs of the efficient frontiers this is thesecond point down from the point of the maximumexpected net present value point.

The funding strategies for all three data sets arepresented in table 3. The label ‘j1’ in front of the fundingacronyms means that the instrument is issued in stage 1,and the label ‘j2’ refers to the instrument’s issuancein stage 2. For example, a funding instrument called‘j1y07n’ is a non-callable bond with a maturity of7 years issued at stage 1. While the single-stage modelnaturally does not have a second stage, variables with

Table 1. Initial yield curves.

Interest rate (%)

Label Maturity (months) Normal Flat Steep

m03 3 5.18 5.88 2.97m06 6 5.31 6.38 3.16y01 12 5.55 6.76 3.36y02 24 5.97 7.06 4.18y03 36 6.12 7.36 4.58y05 60 6.33 7.56 5.56y07 84 6.43 7.59 5.97y10 120 6.59 7.64 6.36y30 360 6.87 7.76 7.20

Table 2. Spreads for different funding instruments.

Spread (%)

LabelMaturity(months)

Callable after(months) Normal Flat Steep

m03n 3 0.17 0.15 0.19m06n 6 0.13 0.10 0.13y01n 12 0.02 0.08 0.08y02n 24 0.00 0.11 0.1y03n 36 0.04 0.18 0.12y03nc1 36 12 0.36 0.61 0.22y05n 60 0.08 0.21 0.13y05nc1 60 12 0.65 0.84 0.22y05nc3 60 36 0.32 0.41 0.18y07n 84 0.17 0.22 0.15y07nc1 84 12 0.90 0.95 0.45y07nc3 84 36 0.60 0.73 0.35y10n 120 0.22 0.29 0.22y10nc1 120 12 1.10 1.28 0.57y10nc3 120 36 0.87 0.94 0.48y30n 360 0.30 0.32 0.27

Stochastic programming for funding mortgage pools 199

the prefix ‘j2’ for the second stage represent the amount ofeach instrument to be held at stage 2 according tothe decision rules. This is to facilitate easycomparisons between the single-stage and multi-stagefunding strategies.

In the ‘Normal’ case, the optimal initial funding mixconsists of 85.3% six-month non-callable debt and 14.7%seven-year non-callable debt. The risk level of this strat-egy is 3.1M NPV. In the ‘Flat’ case, the optimal initialfunding mix consists of 91.1% three-month non-callabledebt and 8.9% seven-year non-callable debt. The risklevel of this strategy is 3.1M NPV. In the ‘Steep’ case,the initial funding mix consists of 100.0% three-monthnon-callable debt (short-term debt). The risk level ofthis strategy is 2.6M NPV.

6.3. Results for the multi-stage model

Figure 2 presents the efficient frontier for the ‘Normal’data set. In addition to the multi-stage efficient frontier,the graph also contains the corresponding single-stageefficient frontier for better comparison. The results forthe ‘Flat’ and ‘Steep’ data sets are very similar in shapeto that of the ‘Normal’ data set and therefore arenot presented graphically. The results show substantialdifferences in the risk and expected net present valueprofile of multi-stage versus single-stage fundingstrategies. For any of the three data sets, the efficientfrontier obtained from the multi-stage model issignificantly north-west of that from the single-stagemodel, i.e. multi-stage optimization yields a largerexpected net present value at the same or smaller levelof risk.

In all three data cases, we cannot compare the expectednet present values from the single-stage and multi-stagemodel at the same level of risk, because the risk at theminimum risk point of the single-stage model islarger than the risk at the maximum risk point of themulti-stage model. In the ‘Normal’ case, the minimumrisk of the single-stage curve is about 2.7M NPV. Sincethe efficient frontier is very steep at low levels of risk, weuse the point of 85% of the maximum risk as the ‘lowestrisk’ point, even if the risk could be further decreasedby an insignificant amount. The maximum expected netpresent value point of the multi-stage curve has a risk of

14.0

12.0

10.0

8.0

6.0

4.0

2.0

0.00.0 1.0 2.0 3.0 4.0 5.0

Risk ($M NPV)

Single-stage Multi-stage

Exp

. ret

urn

($M

NP

V)

Efficient frontier, Normal

Figure 2. Efficient frontier for the Normal data set, single- versus multi-stage.

Table 3. Funding strategies from the single-stage model, case95% of maximum expected return, for all three data sets.

Model Normal Flat Steep

Stage 1 Allocation

j1m06n j1y07n j1m03n j1y07n j1m03n

0.853 0.147 0.911 0.089 1.000Stage 2 Allocation

Scenario j2m03n j2m03n j2m03n

1 0.853 0.911 1.0002 0.853 0.911 1.0003 0.853 0.911 1.0004 0.853 0.911 1.0005 0.853 0.911 1.0006 0.853 0.911 1.0007 0.853 0.911 1.0008 0.853 0.911 1.0009 0.853 0.911 1.00010 0.853 0.911 1.000

200 G. Infanger

about 2.4M NPV. At this level of risk, the expected netpresent value on the single-stage efficient frontier is about8.9M NPV, versus the expected net present value on themulti-stage curve of about 13.6M NPV, which representsan improvement of 52.8%. In the ‘Flat’ case, we comparethe point with the smallest risk of 3.0M NPV on thesingle-stage efficient frontier with that with the largestrisk of 2.5M NPV on the multi-stage efficient frontier.The expected net present value at the two points is13.6M NPV (multi-stage) versus 9.6M NPV (single-stage), which represents an improvement of 41.7%. Inthe ‘Steep’ case, we compare the point with the smallestrisk of 2.6M NPV on the single-stage efficient frontierwith that with the largest risk of 1.9M NPV on themulti-stage efficient frontier. The expected net presentvalue at the two points is 20.4M NPV (multi-stage)versus 18.6M NPV (single-stage), which represents animprovement of 9.7%.

As for the single-stage model, we look at the fundingstrategies at the point of 95% of the maximum expectedreturn (on the efficient frontier the second point downfrom the maximum expected return point). The fundingstrategies are presented in table 4 for the ‘Normal’ dataset, and in table A1 and table A2 of appendix A for the‘Steep’ and ‘Flat’ data sets, respectively.

In the ‘Normal’ case, the initial funding mix consists

of 78.5% six-month non-callable debt, and 21.5% five-yea debt callable after one year. After one year the

78.5% six-month debt (that according to the decisionrules is refinanced through short-term debt and is for

disposition in the second stage) and, if called in certainscenarios, also the five-year callable debt are refunded

through various mixes of short-term debt: three-year,five-year and ten-year callable and non-callable debt.

In one scenario, labeled ‘1’, in which interest rates fallto a very low level, the multi-stage model resorts to fund-

ing with 30-year non-callable debt in order to secure thevery low rate for the future. The risk associated with this

strategy is 1.85M NPV and the expected net present value

is 12.9M NPV. The corresponding (95% of maximum

net present value) strategy of the single-stage model,

discussed above, has a risk of 3.1M NPV and a net pre-

sent value of 10.0M NPV. Thus, the multi-stage strategy

exhibits 57.4% of the risk of the single-stage strategy and

a 29% larger expected net present value.In the ‘Flat’ case, the initial funding mix consists

of 50.9% three-month non-callable debt and 49.1%six-month non-callable debt. After one year the entireportfolio is refunded through various mixes of short-term, three-year, and ten-year callable and non-callabledebt. The multi-stage strategy exhibits 68% of the risk ofthe single-stage strategy and a 20.6% larger expected netpresent value. In the ‘Steep’ case, the initial funding con-sists of 100% three-month non-callable debt. After oneyear the portfolio is refunded through various mixes ofshort-term, three-year and five-year non-callable, andten-year callable and non-callable debt. The multi-stagestrategy exhibits 62% of the risk of the single-stagestrategy, and a 4.3% larger expected net present value.

Summarizing, the results demonstrate that multi-stagestochastic programming potentially yields significantlylarger net present values at the same or even lower levelsof risk, with significantly different funding strategies,compared with single-stage optimization. Using multi-stage stochastic programming for determining the fund-ing of mortgage pools promises to lead, in the average,to significant profits compared with using single-stagefunding strategies.

6.4. Results for duration and convexity

Funding a mortgage pool by a portfolio of bonds thatmatches the (negative) value of duration and convexity,the expected net present value of the total of mortgagepool and bonds is invariant to small changes in interestrates. However, duration and convexity give only a localapproximation, and the portfolio needs to be updated as

Table 4. Funding strategy from the multi-stage model, case 95% of maximum expected return, Normaldata set.

Stage 1 Allocation

j1m06n j1y05nc1

0.785 0.215Stage 2 Allocation

Scenario j2m03n j2y03n j2y03nc1 j2y05nc1 j2y10n j2y10nc1 j2y10nc3 j2y30n

1 1.0002 0.597 0.1883 0.125 0.420 0.2404 0.7855 0.508 0.423 0.0696 0.7857 0.7858 0.251 0.7499 0.301 0.035 0.261 0.18810 0.785

Stochastic programming for funding mortgage pools 201

time goes on and interest rates change. The duration andconvexity hedge is one-dimensional, since it considersonly changes of the whole yield curve by the same amountand does not consider different shifts for differentmaturities. The multi-stage stochastic programmingmodel takes into account multi-dimensional changes ofinterest rates and considers (via sampling) the entire dis-tribution of possible yield curve developments. In thissection we quantify the difference between duration andconvexity hedging versus hedging using the single- andmulti-stage stochastic programming models. Table 5gives the initial (first stage) values for duration and con-vexity (as obtained from the simulation runs) for themortgage pool and for all funding instruments for eachof the three yield curve cases ‘Normal’, ‘Flat’, and ‘Steep’.In each of the three yield curve cases the mortgage poolexhibits a positive value for duration and a negativevalue for convexity. All funding instruments havepositive values for duration, non-callable bonds exhibita positive value for convexity, and callable bonds showa negative value for convexity. Note as an exception thepositive value for convexity of bond ‘y07nc1’.

To both the single-stage and multi-stage stochasticprogramming models we added constraints that, at anydecision point, the duration and convexity of themortgage pool and the funding portfolio are as close aspossible. Maximizing expected return by setting the riskaversion � to zero, we successively reduced the gap induration and convexity between the mortgage pool andthe funding portfolio. We started from the unconstrainedcase (the maximum expected return–maximum risk casefrom the efficient frontiers discussed above) and reducedfirst the duration gap and subsequently the convexity gap,where we understand as duration gap the absolutevalue of the difference in duration between the fundingportfolio and the mortgage pool, and as convexity gap theabsolute value of the difference in convexity between thefunding portfolio and the mortgage pool. We will discuss

the results with respect to the downside risk measure(expected value of returns below a certain target),as discussed in section 2.9, and also with respect to thestandard deviation of the returns. We do not discuss theresults for duration and convexity obtained from the sin-gle-stage model and focus on the more interesting multi-stage case.

6.4.1. Duration and convexity, multi-stage

model. Figures 3 and 4 give the risk–return profilefor the ‘Normal’ case with respect to downside riskand standard deviation, respectively. We compare theefficient frontier (already depicted in figure 2) obtainedfrom minimizing downside risk for different levels ofexpected return (labeled ‘Downside’) with the risk–returnprofile obtained from restricting the duration and con-vexity gap (labeled ‘Delta–Gamma’). For different levelsof duration and convexity gap, we maximized expectedreturn. The unconstrained case with respect to durationand convexity is identical to the point on the efficientfrontier with the maximum expected return. Figure 3shows that the downside risk eventually increases whendecreasing the duration and convexity gap and wassignificantly larger than the minimized downside riskon the efficient frontier. We are especially interestedin comparing the point with the smallest duration andconvexity gap with the point of minimum downside riskon the efficient frontier. The point with the smallestdownside risk on the efficient frontier exhibits expectedreturn of 9.5M NPV and a downside risk of 1.5M NPV.The point with the smallest duration and convexity gaphas expected return of 7.9M NPV and a downside risk of3.1M NPV. The latter point is characterized by a max-imum duration gap in the first and second stage of 0.5and of 1.0 in the third stage, and by a maximum con-vexity gap of 2.0 in the first and second stage and of 4.0

Table 5. Initial duration and convexity.

Normal Flat Steep

Label Dur. Conv. Dur. Conv. Dur. Conv.

mortg. 3.442 �1.880 2.665 �1.548 2.121 �2.601m03n 0.248 0.001 0.247 0.001 0.249 0.001m06n 0.492 0.003 0.491 0.003 0.495 0.003y01n 0.971 0.010 0.964 0.010 0.982 0.011y02n 1.882 0.038 1.861 0.038 1.917 0.039y03n 2.737 0.081 2.686 0.079 2.801 0.084y03nc1 1.596 �0.384 1.376 �0.186 1.338 �0.235y05n 4.278 0.205 4.160 0.197 4.378 0.212y05nc1 1.914 �0.810 1.613 �0.700 1.478 �0.632y05nc3 3.274 �0.012 3.068 �0.111 3.114 �0.241y07n 5.622 0.367 5.448 0.351 5.769 0.380y07nc1 2.378 0.128 1.671 0.959 0.898 0.032y07nc3 3.406 0.269 3.117 �0.298 3.122 �0.230y10n 7.335 0.656 7.083 0.624 7.538 0.681y10nc1 1.555 0.246 1.397 �0.052 0.593 �0.119y10nc3 3.409 �0.160 3.265 �0.346 3.086 �0.446y30n 13.711 2.880 13.306 2.743 14.204 3.011

202 G. Infanger

in the third stage. A further decrease of the convexitygap was not possible as it led to an infeasible problem.The actual duration gap in the first stage was �0:5,where the negative value indicates that the duration ofthe funding portfolio was smaller than that of the mort-gage pool, and the actual convexity gap in the first stagewas 1.57, where the positive value indicates that theconvexity of the funding portfolio was larger than thatof the mortgage pool. Comparing the points with regardto their performance, restricting the duration and con-vexity gap led to a decrease in expected return of 15%and an increase of downside risk by a factor of 2. It is

interesting to note that, for the point on the efficientfrontier with the smallest risk, the first-stage durationgap was �0:97 and the first-stage convexity gap was1.75. Looking at the risk in terms of standard deviationof returns, both minimizing downside risk and control-ling the duration and convexity gap led to smaller valuesof standard deviation. In the unconstrained case,the smallest standard deviation was 3.3M NPV obtainedat the minimum downside risk point, and thesmallest standard deviation in the constrainedcase (3.8M NPV) was obtained when the duration andconvexity gap was smallest.

16

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8

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00 2 4 6 8 10

Risk STD ($M NPV)

2

Downside Delta-Gamma

Exp

. ret

urn

($M

NP

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Risk-return profile, Multi, Normal

1 3 5 7 9

Figure 4. Risk–return profile, multi-stage model, Normal data set, risk as standard deviation.

16

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00 1 2 3 4 5

Risk ($M NPV)

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Exp

. ret

urn

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Risk-return profile, Multi, Normal

Figure 3. Risk–return profile, multi-stage model, Normal data set, risk as downside risk.

Stochastic programming for funding mortgage pools 203

Table 6 gives the funding strategy for the minimumdownside risk portfolio and table 7 the funding strategyfor the duration and convexity constrained case atminimum gap. Both strategies use six-month non-callabledebt, five-year non-callable debt and five-year debt call-able after one year for the initial funding. The minimum

downside risk portfolio used in addition five-year debtcallable after three years, while the duration and convex-ity constrained case used seven-year non-callable debt andten-year non-callable debt. In the second stage, fundingdiffered significantly across the different scenarios forboth funding strategies. In the duration and convexity

Table 7. Funding strategy, case Duration and Convexity Constrained, multi-stage model,Normal data set.

Stage 1 Allocation

j1m06n j1y05n j1y05nc1 j1y07n j1y10n

0.122 0.083 0.534 0.239 0.022

Stage 2 Allocation

Scenario j2m03n j2m06n j2y01n j2y03nc1 j2y05n j2y05nc1 j2y07nc1

1 0.466 0.11723 0.06445 0.506 0.043 0.0446 0.06478 0.5679 0.07210 0.316 0.340

Stage 2 Allocation Call

Scenario j2y10n j2y10nc1 j2y10nc3 j1y05nc1

1 0.0732 0.011 0.1123 0.0584 0.152 0.0295 0.058 0.0046 0.0597 0.1228 0.0899 0.05110 0.534

Table 6. Funding strategy, case Minimum Downside Risk, multi-stage model, Normal data set.

Stage 1 Allocation

j1m06n j1y05n j1y05nc1 j1y05nc3

0.266 0.235 0.218 0.281Stage 2 Allocation

Scenario j2m03n j2y01n j2y03nc1 j2y05n j2y10n j2y10nc1 j2y10nc3 j2y30n

1 0.017 0.148 0.240 0.0792 0.102 0.1653 0.117 0.1494 0.2665 0.031 0.032 0.237 0.1856 0.2667 0.014 0.2528 0.118 0.300 0.019 0.0479 0.012 0.201 0.05310 0.266

204 G. Infanger

constrained case, the multi-stage optimization modelresorted to calling five-year debt callable after one yearin scenario ‘4’ at a fraction and in scenario ‘10’ at thewhole amount. In the minimum risk case, calling ofdebt other than by applying the decision rules did nothappen. In the second stage, interest rates were low inscenarios ‘1’ and ‘8’ and were high in scenarios ‘4’, ‘6’,and ‘10’. The minimum downside risk strategy tendedtowards more long-term debt when interest rates werelow and towards more short-term debt when interestrates were high. The amounts for each of the scenariosdepended on the dynamics of the process and the interestrate distributions. The duration and convexity con-strained strategy was, of course, not in the position totake advantage of the level of interest rates, and fundingwas balanced to match the duration and convexity of themortgage pool.

The results for the case of the ‘Flat’ and ‘Steep’ yieldcurves are very similar. For the ‘Flat’ case, the maximumduration gap in the first and second stage was set to 0.5and to 1.0 in the third stage, and the maximum convexitygap was set to 2.0 in the first and second stage and to 4.0in the third stage. The duration and convexity gap couldnot be decreased further since the problem became infea-sible. The actual duration gap in the first stage was �0:5,and the actual convexity gap in the first stage was 1.5.In appendix A, table A3 gives the initial funding and thesecond-stage updates for the minimum downside riskportfolio, and table A4 gives the funding strategy forthe duration and convexity constrained case. For the‘Steep’ case, the maximum duration gap in the first andsecond stage was set to 0.5 and was set to 1.0 in the thirdstage, and the maximum convexity gap was set to 3.0 inthe first and second stage and to 6.0 in the third stage.Further decrease made the problem become infeasible.The actual duration gap in the first stage was �0:5, andthe actual convexity gap in the first stage was 1.6. Inappendix A, table A5 gives the initial funding and thesecond-stage updates for the minimum downside riskportfolio, and table A6 gives the funding strategy forthe duration and convexity constrained case. Again, inboth the ‘Flat’ and ‘Steep’ case, one could see in theminimum downside risk case a tendency to use longer-term debt when interest rates were low and shorter-termdebt when interest rates were high, and in the durationand convexity constrained case, funding was balanced tomatch the duration and convexity of the mortgage pool.

Summarizing, in each case reducing the duration and

convexity gap helped control the standard deviation of

net present value but did nothing to reduce downsiderisk. Multi-stage stochastic programming led to larger

than or equal expected net present value at each level of

risk (both downside risk and standard deviation of net

present value). There may be reasons for controlling the

duration and convexity gap in addition to controlling

downside risk via the multi-stage stochastic programmingmodel. For example, a conduit might like the results of

the stochastic programming model, but not wish to take

on too much exposure regarding duration and convexity

gap. In the following we will therefore compare runs of

the multi-stage model with and without duration and

convexity constraints.Figure 5 sheds more light on the cause of the perfor-

mance gain of the multi-stage model versus the durationand convexity hedged strategy. The figure presents theduration gap (the difference of the duration of the fundingportfolio and the mortgage pool) versus the interestrate (calculated as the average of the yield curve) foreach of the second-stage scenarios of the data set‘Normal’. When interest rates are very low, the 95%maximum expected return strategy takes on a significantpositive duration gap to lock in the low rates for a longtime. It takes on a negative duration gap when interestrates are high, in order to remain flexible, should interestrates fall in the future. The minimum downside riskstrategy exhibits a similar pattern, but less extreme.Thus, the multi-stage model makes a bet on the directioninterest rates are likely to move, based on the informationabout the interest rate process. In contrast, the durationand convexity constrained strategy cannot take on anyduration gap (represented by the absolute value of 0.5at which the gap was constrained) and therefore mustforsake any gain from betting on the likely direction ofinterest rates.

6.5. Out-of-sample simulations

In order to evaluate the performance of the different stra-tegies in an unbiased way, true out-of-sample evaluationruns need to be performed. Any solution at any node inthe tree obtained by optimization must be evaluated usingan independently sampled set of observations.

For the single-stage model this evaluation is ratherstraightforward. Having obtained an optimal solutionfrom the single-stage model (using sample data set one),we run simulations again with a different seed. Usingthe new independent sample (data set two), we startthe optimizer again, however now with the optimal solu-tion fixed at the values as obtained from the optimizationbased on sample data set one. Using the second setof observations of data set two we calculate risk andexpected returns.

To independently evaluate the results obtained from aK-stage model (having n ¼ K� 1 sub-periods), we needK independent sets of K-stage trees of observations. Wedescribe the procedure for K¼ 4. Using data set one wesolve the multi-stage optimization problem and obtain anoptimal first-stage solution. We simulate again (using adifferent seed) over all stages to obtain data set two.Fixing the optimal first-stage solution at the valueobtained from the optimization based on data set one,we optimize based on data set two and obtain a set ofoptimal second-stage solutions. We simulate again (with adifferent seed) to obtain independent realizations forstages three and four, thereby keeping the observationsfor stage two the same as in data set two, and obtain dataset three. Fixing the first-stage decision at the levelobtained from the optimization using data set one andall second-stage decisions at the level obtained from the

Stochastic programming for funding mortgage pools 205

optimization based on data set two, we optimize again toobtain a set of optimal third-stage decisions. We simulateagain (using a different seed) to obtain independent out-comes for stage four, thereby keeping the observationsfor stage two and three the same as in data sets twoand three, respectively, and obtain data set four. Fixingthe first-stage decision, all second-stage decisions, andall third-stage decisions at the levels obtained from theoptimization based on data sets one, two and three,respectively, we finally calculate risk and returns basedon data set four.

The out-of-sample evaluations resemble how the modelcould be used in practice. Solving the multi-stage model(based on data set one), an optimal first-stage solution(initial portfolio) would be obtained and implemented.Then one would follow the strategy (applying the decisionrules) for 12 months until decision stage two arrives.At this point, one would re-optimize, given that theinitial portfolio had been implemented and that particularinterest rates and prepayments had occurred (accordingto data set two). The optimal solution for the second stagewould be implemented, and one would follow the strategyfor four years until decision stage three arrives. At thispoint, one would re-optimize, given that the initial port-folio and a second stage update had been implementedand that particular interest rates and prepayments hadoccurred (according to data set three). The optimal solu-tion for the third stage would be implemented, and onewould follow the strategy (applying the decision rules)until the end of the horizon (according to data set four).Now one possible path of using the model has beenevaluated. Decisions had no information aboutparticular outcomes of future interest rates andprepayments, and were computed based on model runsusing data independent from the observed realization

of the evaluation simulation. Alternatively, one couldsimulate a strategy involving re-optimizationevery month, but this would take significantly morecomputational effort with likely only little to be gained.

Using the out-of-sample evaluation procedure, weobtain N ¼ jSj out-of-sample simulations of using themodel as discussed in the above paragraph, and weare now in the position to derive statistics aboutout-of-sample expected returns and risk.

Figure 6 presents the out-of-sample efficient frontiersfor the ‘Normal’ data set and downside risk. The figuregives the out-of-sample efficient frontier for the multi-stage model without duration and convexity constraints,the out-of-sample efficient frontier when the duration gapwas constrained to be less than or equal to 1.5%, and theout-of-sample efficient frontier for the single-stage model.The out-of-sample evaluations demonstrate clearly thatthe multi-stage model gives significantly better resultsthan the single-stage model. For example, the pointwith the maximum expected returns of the multi-stagemodel gave expected returns of 10.2M NPV and a down-side risk of 3.5M NPV. The minimum risk point on thesingle-stage out-of-sample efficient frontier gave expectedreturns of 7.6M NPV, and a downside risk of also 3.5MNPV. Thus, at the same level of downside riskthe multi-stage model gave 34% higher expected returns.The minimum downside risk point of the multi-stagemodel gave expected returns of 9.0M NPV and adownside risk of 2.1M NPV. Comparing the minimumdownside risk point of the multi-stage model with thatof the single-stage model, the multi-stage model had19.2% larger expected returns at 61% of the downsiderisk of the single-stage model. The efficient frontier ofthe duration-constrained strategy was slightly belowthat without duration and convexity constraints.

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atio

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p (%

)

Duration gap versus interest rate

4.00 6.00

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Min. downside riskDur. and conv constr.95% max expo. ret.

Figure 5. Duration gap versus interest rate, multi-stage, Normal data set.

206 G. Infanger

Figure 7 gives the out-of-sample risk–return profileswhen measuring risk in terms of standard deviation.In this setting the multi-stage strategy performed signifi-cantly better than the single-stage strategy at every levelof risk, where the difference was between 17.6% and23.6%. The duration-constrained strategy did not spanas wide a range in risk as the unconstrained strategy.But for the risk points attained by the constrainedstrategy, the unconstrained strategy achieved a somewhathigher expected return.

The out-of-sample evaluations for the ‘Flat’ and‘Steep’ data sets gave very similar results qualitatively.The results are included in appendix A. For the ‘Flat’data set, figure A1 presents out-of-sample efficient fron-tiers for downside risk and figure A2 the risk–returnprofile for risk in standard deviations. For the ‘Steep’data set, figure A3 presents out-of-sample efficient fron-tiers for downside risk and figure A4 the risk–returnprofile for risk in standard deviations for the data set‘Steep’. For both data sets, the multi-stage modelgave significantly better results; the downside risk wassmaller and expected returns were larger.

6.6. Larger sample size

All results discussed so far were obtained from solving amodel with a relatively small number of scenarios at eachstage, i.e. jS2j ¼ 10, jS3j ¼ 20 and jS4j ¼ 20, with a totalof 4000 scenarios at the end of the forth stage. This servedwell for analysing and understanding the behavior of themulti-stage model in comparison with the single-stagemodel and with Gamma and Delta hedging. Choosing a

larger sample size will improve the obtained strategies(initial portfolio and future revisions) and therefore resultin better (out-of-sample simulation) results. Of course, theaccuracy of prediction of the models will be improvedalso. In order to show the effect of using a larger samplesize, we solved and evaluated the models using a samplesize of 24 000 scenarios, i.e. jS2j ¼ 40, jS3j ¼ 30,and jS4j ¼ 20. The results for the data set ‘Normal’ arepresented in figure 8 for downside risk and in figure 9 forrisk as standard deviation. Indeed, one can see improvedperformance in both smaller risk and larger expectedNPV compared with the smaller sample size (comparewith figures 6 and 7).

The point with the maximum expected returns for themulti-stage model gave expected returns of 12.9M NPVand a downside risk of 2.4M NPV. The minimum riskpoint for the single-stage model had expected returnsof 9.0M NPV and a downside risk of 2.7M NPV. So,at slightly smaller downside risk the multi-stagemodel gave expected returns that were 43% higher. Theminimum downside risk point of the multi-stage modelhad expected returns of 10.9M NPV and a downside riskof 1.5M NPV. Comparing the minimum downside riskpoints of the multi-stage and single-stage models, themulti-stage model has 18.4% larger expected returns at57% of the downside risk of the single-stage model.Again, the efficient frontier of the duration-constrainedstrategy was slightly below that without duration andconvexity constraints. Measuring risk in terms of stan-dard deviation, results are similar to those when usingthe small sample size: the multi-stage strategy performedsignificantly better than the single-stage strategy at everylevel of risk, with a difference in expected returns between

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ecte

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Out-of-sample efficient frontier, normal

2.00

2.00

0.005.00

Single-stage Multi-stage dur. constr.Multi-stage

12.00

Figure 6. Out-of-sample risk–return profile, multi-stage model, Normal data set, risk as downside risk.

Stochastic programming for funding mortgage pools 207

23.3% and 24.0%. Again, the duration-constrained

strategy exhibited smaller risk (in standard deviations)

at the price of slightly smaller expected returns.Figure 10 gives a comparison of the multi-stage efficient

frontiers predicted (in-sample) versus evaluated out-

of-sample. One can see that when using the larger

sample size of 24 000, the predicted and the out-of-sample

evaluated efficient frontier look almost identical, thus

validating the model. It is evident that using larger sample

sizes will result in both better performance and a more

accurate prediction.

7. Large-scale results

For the actual practical application of the proposedmodel, we need to consider a large number of scenariosin order to obtain small estimation errors regarding

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Out-of-sample risk-return profile, Normal, larger sample

2.00

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Single-stage Multi-stage dur. constr.Multi-stage

6.00

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12.00

Figure 7. Out-of-sample risk–return profile, multi-stage model, Normal data set, risk as standard deviation.

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ecte

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Out-of-sample efficient frontier, Normal, larger sample

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6.00

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12.00

Figure 8. Larger sample, out-of-sample risk–return profile for the multi-stage model, Normal data set, risk as downside risk.

208 G. Infanger

expected returns and risk and accordingly stable results.We now explore the feasibility of solving large-scalemodels. Table 8 gives measures of size for models withlarger numbers of scenarios. For example, model L4has 80 000 scenarios at the fourth stage, composed ofjS2j ¼ 40, jS3j ¼ 40 and jS4j ¼ 50, and model L5 has

100 000 scenarios at the fourth stage composed ofjS2j ¼ 50, jS3j ¼ 40 and jS4j ¼ 50; both models have asufficiently large sample size in each stage.

In the case of problem L5 with 100 000 scenarios thecorresponding linear program had 270 154 constraints,249 277 variables, and 6 291 014 non-zero coefficients.

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6.00

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9.008.005.003.001.00

Figure 9. Larger sample, out-of-sample risk–return profile for the multi-stage model, Normal data set, risk as standard deviation.

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Predicted vs. out-of-sample efficient frontier, Normal, larger sample

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Multi-stage out-of-sample Multi-stage predicted

Figure 10. Larger sample, predicted versus out-of-sample efficient frontier for the multi-stage model, Normal data set.

Stochastic programming for funding mortgage pools 209

Table 9 gives the elapsed times for simulation andoptimization, obtained on a Silicon Graphics Origin2000 workstation.

While the Origin 2000 at our disposition is a multi-processor machine with 32 processors, we did not usethe parallel feature, and all results were obtained usingsingle processor computations. For the direct solution ofthe linear programs, we used CPLEX (1989–1997) as thelinear programming optimizer. We contrast the results tousing DECIS (Infanger 1989–1999), a system for solvingstochastic programs, developed by the author. BothCPLEX and DECIS were accessed through GAMS(Brooke et al. 1988). DECIS exploited the special struc-ture of the problems and used dual decomposition fortheir solution. The problems were decomposed into twostages, breaking between the first and the second stage.The simulation runs for model L5 took less than anhour. The elapsed solution time solving the problemdirectly was 13 hours and 28 minutes. Solving the pro-blem using DECIS took significantly less time, 2 hoursand 33 minutes. Encouraged by the quick solution timeusing DECIS, we generated problem L20 with 400 000scenarios (composed of jS2j ¼ 200, jS3j ¼ 40 andjS4j ¼ 50) and solved it in 11 hours and 34 minutesusing DECIS. Problem L20 had 808 201 constraints,931 271 variables, and 14 259 216 non-zero coefficients.

Using parallel processing, the simulation times and the

solution times could be reduced significantly. Based on

our experiences with parallel DECIS, using six processors

we would expect the solution time for the model L5 with

100 000 scenarios to be less than 40 minutes, and using

16 processors one could solve model L20 with 400 000

scenarios in about one hour. We actually solved a version

of the L3 model with 60 000 scenarios, composed of

jS2j ¼ 50, jS3j ¼ 40 and jS4j ¼ 30, in less than 5 minutes

using parallel DECIS on 16 processors.

8. Summary

The problem of funding mortgage pools represents animportant problem in finance faced by conduits in thesecondary mortgage market. The problem concerns howto best fund a pool of similar mortgages through issuing aportfolio of bonds of various maturities, callable andnon-callable, and how to refinance the debt over time asinterest rates change, prepayments are made and bondsmature. This paper presents the application of stochasticprogramming in combination with Monte Carlo simula-tion for effectively and efficiently solving the problem.

Monte Carlo simulation was used to estimate multiplerealizations of the net present value of all payments whena pool of mortgages is funded initially with a single bond,where pre-defined decision rules were applied for makingdecisions not subject to optimization. The simulationswere carried out in monthly time steps over a 30-yearhorizon. Based on a scenario tree derived from the simu-lation results, a single-stage stochastic programmingmodel was formulated as a benchmark. A multi-stagestochastic programming model was formulated bysplitting up the planning horizon into multiple sub-periods, representing the funding decisions (the portfolioweights and any calling of previously issued callablebonds) for each sub-period, and applying the pre-defineddecision rules between decision points.

In order to compare the results of the multi-stagestochastic programming model with hedging methods

Table 8. Large-scale models, dimensions.

Scenarios Problem size

Model Stage 2 Stage 3 Stage 4 Total Rows Columns Non-zeros

L1 10 40 50 20 000 54 034 49 918 1 249 215L2 20 40 50 40 000 108 064 99 783 2 496 172L3 30 40 50 60 000 162 094 149 623 3 756 498L4 40 40 50 80 000 216 124 199 497 5 021 486L5 50 40 50 100 000 270 154 249 277 6 291 014L20 200 40 50 400 000 808 201 931 271 14 259 216

Table 9. Large-scale models, solution times.

Simul. Direct sol. Decomp. sol.Model Scenarios time (s) time (s) time (s)

L1 20 000 688 1389.75 1548.91L2 40 000 1390 6144.56L3 60 000 2060 14 860.54 5337.76L4 80 000 2740 28 920.69L5 100 000 3420 48 460.02 9167.47L20 400 000 41 652.28

210 G. Infanger

typically used in finance, the effective duration and con-vexity of the mortgage pool and of each funding instru-ment was estimated at each decision node, and constraintsbounding the duration and convexity gap to close to zerowere added (at each node) to the multi-stage model toapproximate a duration and convexity hedged strategy.

An efficient method for obtaining an out-of-sampleevaluation of an optimal strategy obtained from solvinga K-stage stochastic programming model was presented,using K independent (sub-)trees for the evaluation.

For different data assumptions, the efficient frontier ofexpected net present value versus (downside) riskobtained from the multi-stage model was comparedwith that from the single-stage model. Under all dataassumptions, the multi-stage model resulted insignificantly better funding strategies, dominating thesingle-stage model at every level of risk, both in-sampleand by evaluating the obtained strategies via out-of-sample simulations. Also, for all data assumptions,the out-of-sample simulations demonstrated that themulti-stage stochastic programming model dominatedthe duration and convexity hedged strategies at everylevel of risk. Constraining the duration and convexitygap reduced risk in terms of the standard deviation ofnet present value at the cost of a smaller net presentvalue, but failed in reducing the downside risk.

The results demonstrate clearly that using multi-stagestochastic programming results in significantly largerprofits, both compared with using single-stage optimiza-tion models and with using duration and convexityhedged strategies. The multi-stage model is better thanthe single-stage model because it has the option to revisethe funding portfolio according to changes in interestrates and pre-payments, therefore reflecting a morerealistic representation of the decision problem. It isbetter than the duration and convexity hedged strategiesbecause it considers the entire distribution of the yieldcurve represented by the stochastic process of interestrates, compared with the much simpler hedge against asmall shift of the entire yield curve as used in the durationand convexity hedged strategies.

Small models with 4000 scenarios and larger ones with24 000 scenarios were used for determining the fundingstrategies and the out-of-sample evaluations. The out-of-sample efficient frontiers of the larger models wereshown to be very similar to the (in-sample) predictions,indicating a small optimization bias. The larger modelswere solved in a very short (elapsed) time of a few min-utes. Large-scale models with up to 100 000 scenarioswere shown to solve in a reasonable elapsed time usingdecomposition on a serial computer, and in a few minuteson a parallel computer.

Acknowledgements

Funding for research related to this paper was providedby Freddie Mac. The author is grateful to Jan Luytjes

for many valuable discussions on the subject. Editorialcomments by three anonymous referees and by JohnStone on previous versions of this paper helped enhancethe presentation.

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Table A2. Funding strategy from the multi-stage model, case 95% of maximum expected return,Steep data set.

Stage 1 Allocation

j1m03n

1.000Stage 2 Allocation

Scenario j2m03n j2y03n j2y03nc1 j2y05n j2y10n j2y10nc1 j2y30n

1 0.038 0.296 0.544 0.1222 0.717 0.2833 1.0004 1.0005 0.745 0.085 0.1706 1.0007 1.0008 0.609 0.272 0.1199 0.973 0.02710 1.000

Table A1. Funding strategy from the multi-stage model, case 95% of maximum expected return,Flat data set.

Stage 1 Allocation

j1m03n j1m06n

0.509 0.491Stage 2 Allocation

Scenario j2m03n j2y03n j2y03nc1 j2y10n j2y10nc1 j2y10nc3 j2y30n

1 0.397 0.6032 1.0003 0.251 0.590 0.011 0.1484 1.0005 0.514 0.248 0.2386 1.0007 0.984 0.0168 0.620 0.3809 0.521 0.151 0.32810 1.000

Appendix A: Tables and graphs for data sets ‘‘Flat’’ and ‘‘Steep’’

212 G. Infanger

Table A3. Funding strategy from the multi-stage model, case Minimum Downside Risk, Flat data set.

Stage 1 Allocation

j1m06n j1y05nc3

0.698 0.302Stage 2 Allocation

Scenario j2m03n j2m06n j2y03n j2y03nc1 j2y05n j2y05nc3

0.046 0.418 0.05223 0.201 0.2924 0.6985 0.119 0.1456 0.6987 0.6038 0.3339 0.30210 0.698Stage 2 Allocation

Scenario j2y10n j2y10nc1 j2y10nc3 j2y30n

1 0.129 0.0532 0.6983 0.039 0.039 0.12745 0.221 0.21467 0.0958 0.280 0.0859 0.095 0.30110

Table A4. Funding strategy from the multi-stage model, case Duration and Convexity Constrained,Flat data set.

Stage 1 Allocation

j1m06n j1y05n j1y05nc1 j1y07n

0.448 0.225 0.201 0.125Stage 2 Allocation Call

Scenario j2m03n j2m06n j2y02n j2y03n j2y03nc1 j2y10n j2y10nc1 j1y05nc1

1 0.567 0.0822 0.102 0.3463 0.370 0.0784 0.251 0.033 0.1645 0.018 0.6316 0.397 0.0517 0.036 0.4128 0.634 0.0169 0.44810 0.277 0.372 0.201

Stochastic programming for funding mortgage pools 213

Table A5. Funding strategy from the multi-stage model, case Minimum Downside Risk, Steep data set.

Stage 1 Allocation

j1m03n

1.000Stage 2 Allocation

Scenario j2m03n j2y01n j2y03n j2y03nc1 j2y05n j2y10n j2y10nc1 j2y30n

1 0.299 0.389 0.215 0.0972 0.644 0.3563 1.0004 1.0005 0.734 0.088 0.1786 1.0007 1.0008 0.600 0.005 0.246 0.149 5.126743E-49 0.958 0.042 8.23454E-410 1.000

Table A6. Funding strategy from the multi-stage model, case Duration and Convexity Constrained,Steep data set.

Stage 1 Allocation

j1y01n j1y03n j1y10n

0.682 0.305 0.013Stage 2 Allocation

Scenario j2m03n j2y01n j2y02n j2y03n j2y03nc1 j2y10n j2y30n

1 0.598 0.0842 0.053 0.6293 0.049 0.6334 0.531 0.1515 0.6826 0.158 0.5247 0.500 0.1828 0.6829 0.68210 0.392 0.289

214 G. Infanger

14.00

10.00

8.00

4.00

0.00

Risk ($M NPV)

Exp

ecte

d re

turn

($M

NP

V)

Out-of-sample efficient frontier, Flat

1.00

2.00

0.005.00

Single-stage Multi-stage dur. constr.Multi-stage

6.00

2.00 3.00 4.00

12.00

Figure A1. Out-of-sample risk–return profile for the multi-stage model, Flat data set, risk as downside risk.

14.00

10.00

8.00

4.00

0.00

Risk STD ($M NPV)

Exp

ecte

d re

turn

($M

NP

V)

Out-of-sample risk-return profile, Flat

2.00

2.00

0.0010.00

Single-stage Multi-stage dur. constr.Multi-stage

6.00

4.00 6.00 8.00

12.00

9.007.005.003.001.00

Figure A2. Out-of-sample risk–return profile for the multi-stage model, Flat data set, risk as standard deviation.

Stochastic programming for funding mortgage pools 215

20.00

10.00

8.00

4.00

0.00

Risk STD ($M NPV)

Exp

ecte

d re

turn

($M

NP

V)

Out-of-sample efficient frontier, Steep

1.00

2.00

0.00

Single-stage Multi-stage dur. constr.Multi-stage

6.00

18.00

5.004.003.002.00

12.00

14.00

16.00

Figure A3. Out-of-sample risk–return profile for the multi-stage model, Steep data set, risk as downside risk.

20.00

10.00

8.00

4.00

0.00

Risk STD ($M NPV)

Exp

ecte

d re

turn

($M

NP

V)

Out-of-sample risk-return profile, Steep

1.00

2.00

0.00

Single-stage Multi-stage dur. constr.Multi-stage

6.00

18.00

10.008.006.004.00

12.00

14.00

16.00

2.00 3.00 5.00 7.00 9.00

Figure A4. Out-of-sample risk–return profile for the multi-stage model, Steep data set, risk as standard deviation.

216 G. Infanger