Refinancing Mortgages

in Switzerland

Karl Frauendorfer and Michael SchurleInstitute for Operations Research, University of St. Gallen

Bodanstrasse 6, 9000 St. Gallenwww.ifu.unisg.ch

Working-Paper 02-09-1

Refinancing Mortgages in Switzerland

Karl Frauendorfer and Michael SchurleInstitute for Operations Research, University of St. Gallen, Switzerland

14.09.2002

Abstract

This paper presents a multistage stochastic programming model for refinancingmortgages with non-contractual maturity under liquidity restrictions in the mar-ket. An extension to the management of other products such as savings accountsis straightforward. The evolution of interest rates is modelled by principal com-ponents for short-term and a two-factor mean reversion model with long rate andspread for long-term planning. Barycentric approximation provides tight lowerand upper bounds for the original problem with relative discretization errors inthe order of one per cent.

1 Introduction

Savings accounts, sight deposits and mortgages make up a significant por-tion of a bank’s assets and liabilities. According to statistics published bythe Swiss National Bank, their average share in the balance sheet totals ofall financial institutions in Switzerland amounts to 20 % for savings and53 % for mortgages in 2000. Typically, a large percentage of these productsconsists of so-called “non-maturing accounts” that can be characterized asfollows: First, there is no contractual maturity, i.e., clients may withdrawtheir investments or repay their mortgages, respectively, at any point in timeat no penalty. Second, the customer rate is not indexed to money or capitalmarket rates but may be fixed by the bank as a matter of policy (in contrastto “adjustable rate mortgages” as they are known in the US). As a conse-quence, the volume of these positions may fluctuate heavily as clients reactto changes in the customer rate, the relative attractiveness of alternativeinvestment or financing opportunities due to rising or falling interest ratesetc.

1.1 Specific problems of non-maturing accounts

In case of variable-rate (non-fixed) mortgages, the change in volume is posi-tively correlated with interest rates. When the latter are low, a sharp drop

2.53.03.54.04.55.05.5

88 90 92 94 96

savings deposits

10152025303540

4.55.05.56.06.57.07.5

88 90 92 94 96

variable mortgages

20253035404550

client rate (%, left) volume (bio. CHF, right)

Figure 1: Negative and positive correlation between client rate and volume

in demand can be observed since clients switch to fix-rate mortgages in or-der to hedge themselves against a future interest increase (prepayment risk).On the contrary, the volume of savings accounts grows since their yields be-come relatively attractive compared to short-term securities, and even someinstitutional investors like pension funds prefer these deposits to direct in-vestments in money market instruments.

During a period of high interest rates, investors shift their assets fromsavings accounts to bonds with long maturities (withdrawal risk). This resultsin a negative correlation between yields and volume change. At the sametime, homeowners’ demand for variable-rate mortgages rises significantly.For example, while non-fixed mortgages represented approximately 90% ofthe entire mortgage volume of the largest Swiss cantonal bank in 1985, theirpercentage decreased with falling interest rates to less than 60% in 1989.Three years later, a new high of 80% was reached after an increase in marketrates.

As the numbers above indicate, a bank is usually unable to refinance itsmortgages directly by savings deposits since in general the latter have a muchsmaller percentage of the balance sheet total. Moreover, the volumes of bothproducts fluctuate “out of phase” as a result of the different correlationswith the level of interest rates (see Figure 1). Therefore, mortgages must berefinanced on the money and capital market at higher rates. The challenge forthe management is to find a mix of fixed-income instruments which minimizesnot only the funding costs but also takes into account the prepayment riskthat a significant portion of the volume under management is withdrawn sincethis would result in a surplus of liabilities over the assets (i.e., the portfolioof instruments with fixed maturities raised in the money and capital marketvs. variable mortgages). As an additional complication, there is a politicalcap on the mortgage rate in Switzerland, and numerous banks were not ableto refinance their mortgages at a positive margin during the early 1990s.

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1.2 Static vs. dynamic approaches

The current practice in the management of non-maturing accounts is thedetermination of a so-called replicating portfolio that mimics the behaviorof the underlying position. To this end, one constructs a portfolio of fixedincome securities whose return replicates the client rate of the relevant assetor liability position plus margin. Transaction costs remain low since liquidmoney market instruments and swaps are used that are held until maturityto avoid rebalancing (“buy and hold”).

While maturing funds are always renewed at the same maturity, instru-ments are bought or sold at constant proportions whenever the volume of theposition under management increases or decreases due to changes in customerdemand. These weights are derived by minimizing the tracking error (i.e.,the difference between the average portfolio rate plus margin and the clientrate) over a historic sample period under the constraint that the volume ofthe replicating portfolio matches that of the target account at all points intime. Therefore, prepayment and withdrawal risks are implicitly taken intoaccount.

In this way, uncertain cash flows are transformed into (apparently) cer-tain ones, allowing the bank to manage them like normal positions with fixedmaturities. However, the considerable risk of an incorrect transformation re-mains. For example, different historic sample periods for the determination ofa replicating portfolio may result in substantially different portfolio weights.While the minimization of the tracking error aims at the stabilization of themargin over time, it does not provide minimal funding costs or even guaran-tee a positive margin at all.

The approach is also static in the sense that it does not incorporatethe possibility of future changes in random data such as interest rates andvolume and their impact on the optimal portfolio composition. Therefore,the question arises if a dynamic policy with active reactions to changes inmarket rates and customer demand might increase the bank’s profit or reduceits refinancing costs, respectively. In particular, it remains to be clarified ifthe correlation between interest rates and volume can be exploited moreappropriately to manage the different risks associated with non-maturingaccount positions.

Multistage stochastic programs take all these aspects into account. Basedon assumptions about the (joint) dynamics of relevant risk factors that areusually described by stochastic processes, representative scenarios for theirfuture outcomes are generated. In the context of asset & liability manage-ment (ALM) problems, the latter quantify the impact of changes in the riskfactors on the return of investment or refinancing strategies. Transactions

3

may take place at discrete points in time over some finite planning horizonfor the correction of an initial policy, given new observations of random datain each scenarios. Furthermore, various constraints can be taken into ac-count. For example, this allows the incorporation of liquidity restrictions inthe market or limits for the risk exposure with respect to certain positions.

In general, stochastic optimization models result in large-scale programssince they have to include a large number of scenarios to reflect the entireuniverse of possible future outcomes of risk factors and cash flows. Sincemultistage programs suffer from an exponential growth in problem size withrespect to the number of periods under consideration, the first models forfinancial planning that appeared in the early 80s (e.g., see Kallberg et al. [25]or Kusy and Ziemba [26]) were restricted to a two-stage structure due tolimitations of the computational resources available at this time.

Today, the dramatic improvement of powerful hardware as well as thedevelopment of efficient algorithms, in particular if they exploit the specialstructure and high sparsity inherent to stochastic programs, provide the basisfor the solution of problems with even some million scenarios like in thepension fund model of Gondzio and Kouwenberg [20]. Moreover, the inclusionof new theoretical models from the financial literature and related empiricalevidence allows a more appropriate modelling of the complex dynamics ofrisk factors.

Among the first successful commercial multistage applications are theTowers Perrin-Tillinghast ALM system of Mulvey [28], the fixed-income port-folio management models of Zenios et al. [1, 32] or the well-known Russell-Yasuda Kasai financial planning tool of Carino, Ziemba et al. [3, 4, 5], tomention just a few. Numerous related models have been derived from thelatter like the InnoALM system [19] that exploits sophisticated econometricmodels and scenario dependent correlation matrices to specifically considerextreme market events. For an extensive survey on financial applications,see the book edited by Ziemba and Mulvey [33] or the monograph in Part Iof this volume.

Motivated by the difficulties in the management of non-maturing accountpositions and given their significant importance for most banks in Switzer-land, we developed a stochastic optimization model for non-fixed mortgagesthat has been in use by a major Swiss bank since 1995 and was extended tosavings accounts two years later (cf. [16]). Other versions have been imple-mented, e.g., for pension funds or cash management at an insurance companywhere one must deal with the seasonal behavior of premium payments, un-certain investment yields and stochastic liabilities due to claims.

In the next section, we introduce a simplified formulation of the stochasticoptimization model for refinancing variable mortgages. After a discussion of

4

relevant characteristics of the underlying risk factors, i.e., interest rates andvolume, section 3 describes two term structure models for different planninghorizons and an autoregressive process for the volume dynamics. Barycentricapproximation as a special scenario generation method that is particularlywell suited for the given problem structure is outlined in section 4. Practicalexperience and results from a case study are reported in section 5. The finalsection 6 summarizes the most relevant features of the model and gives anoutlook to possible directions for its improvement.

2 Formulation of the optimization model

2.1 Framework for decision making under uncertainty

We assume that the evolution of random data is described by a discrete-time stochastic process ω := (ωt; t = 1, . . . , T ) defined in a probability space(Ω,F , P ). The sample space can be decomposed with respect to time as Ω :=Ω1× . . .×ΩT with Ωt ⊂ R

Mt . The filtration F := Ft; t = 1, . . . , T specifiesthe information structure and satisfies Ft ⊂ Ft+1. Each σ-field Ft := σωtis generated by observations ωt := (ω1, . . . , ωt) of the data process that areknown at time t, and Pt : Ft → [0, 1] denotes the corresponding conditionalprobability measure.

Restricting ourselves to the linear case, the underlying decision problemcan be outlined as follows: At any point in time t = 0, . . . , T , we have todecide upon a portfolio composition xt in order to meet an uncertain targetht+1 in the future (here: the volume of variable mortgages). The value of theportfolio after one period is given by Tt ·xt where the elements of the matrixTt represent, e.g., prices of individual positions that may be affected by theoutcomes of some risk factors ωt+1 ∈ Ωt+1. Since it is likely that the portfoliofails to meet the target, a correction xt+1 may be required after a realizationof ωt+1 has been observed to compensate the discrepancy ht+1 −Tt ·xt. Sucha recourse action xt+1 is penalized by the fixed matrix Wt+1.

The objective is to minimize the cost c′t · xt in the current period t plusthe expected cost Et+1φt+1(x

t, ωt+1) =∫

Ωt+1φt+1(x

t, ωt+1)dPt+1 for future

transactions that depends on the sequence of observations ωt+1 := (ωt, ωt+1)and earlier decisions xt := (x0, . . . , xt). This can be stated recursively fort = T, . . . , 1 in terms of value functions as

φt(xt−1, ωt) := min c′t(ω

t) · xt(ωt) +

∫Ωt+1

φt+1(xt, ωt+1)dPt+1(ωt+1|ωt)

s.t. Wt · xt(ωt) = ht(ω

t) − Tt−1(ωt) · xt−1(ω

t−1), xt(ωt) ≥ 0

(1)

5

with the boundary condition φT+1(·) := 0 whereas the problem for the firststage is

φ0 := min c′0 · x0 +

∫Ω1

φ1(x0, ω1)dP1(ω1)

s.t. W0 · x0 = h0, x0 ≥ 0.

(2)

A decision xt may depend only on information ωt available at time t and noton future outcomes of random data ωt+1, . . . , ωT . This property is knownas nonanticipativity. For the simplicity of notation, we do not stress thedependency of xt on ωt in the sequel but assume implicitly that xt ≡ xt|Ftis adapted to the filtration Ft.

Due to linearity of the objective function and constraints, the multistagestochastic program given by (1) and (2) is convex. Its properties are widelydiscussed in the literature (e.g., see Part I of this volume or the textbook byBirge and Louveaux [2]). Here, we focus on the special case where the vectorsωt ∈ Ωt ⊂ R

Mt of random data can be decomposed into two components: ηt ∈Θt ⊂ R

Kt affects the coefficients in the objective, ξt ∈ Ξt ⊂ RLt influences the

demand on the right-hand-side of constraints with Mt = Kt + Lt. Then, thevalue function given by (1) is a saddle function for all t = 1, . . . , T , and thecorresponding optimization problem can be solved if the following conditionshold:

(i) The support of ωt = (ηt, ξt) is covered by compact and convex setsΩt = Θt × Ξt.

(ii) ct(ηt), ht(ξ

t) are linear affine in their respective random vectors ηt :=(η1, . . . , ηt), ξ

t := (ξ1, . . . , ξt), and the matrices Tt−1 are deterministic.

In other words, the value function (1) takes on the form

φt(xt−1, ωt) := min c′t(η

t) · xt +

∫Ωt+1

φt+1(xt, ωt+1)dPt+1(ωt+1|ωt)

s.t. Wt · xt + Tt−1 · xt−1 = ht(ξt), xt ≥ 0

(1′)

Since φT+1(·) = 0, this implies that the problem for the final stage T isconvex in (xT−1, ξT ) and concave in ηT . When calculating the expectationsEt+1φt+1 in the remaining stages T − 1, . . . , 1, the probability measures Pt+1

depend on ωt. As a consequence, the saddle property of the value function inT may not be “inherited” to the previous stages due to the integration withrespect to Pt+1(·|ωt). However, if the corresponding distribution functionsmay be represented, e.g., in the form Qt+1

(ωt+1 + Ht+1(ω

t)), where Qt+1 is

6

A

B

C

D

ξtξt

ηt

Figure 2: Saddle property of the value function

independent of ωt and Ht+1 is a linear mapping, then it can be shown thatthe relevant expectation functionals Et+1φt+1(x

t, ωt+1) are continuous saddlefunctions (cf. [13, 16]). This leads to the requirement:

(iii) The distribution function of Pt(·|ωt−1) depends linearly on the past.

In particular, this covers the case that the distribution of risk factors isindependent of prior outcomes. If conditions (i)–(iii) are fulfilled, the valuefunctions φt(x

t−1, ηt, ξt) are convex in (xt−1, ξt) and concave in ηt for t =1, . . . , T which is called the entire convex case (see Figure 2 for the case ofKt = Lt = 1). These conditions are sufficient, and the saddle property mayalso be given for more general problem types (cf. [13] for details).

2.2 Specification of the funding problem

Given the description of the mortgage funding problem in section 1.1, the for-mulation of the optimization model is straightforward: Let D = 1, 2, . . . , Ddenote the set of maturity dates for fixed-income instruments held in the cur-rent portfolio from decisions in the past (D is the longest available maturity).Standard maturities traded in the money and capital market which can beused for refinancing are given by the set DS ⊆ D. Due to liquidity restrictionsin the Swiss interbank market which must be observed by large banks, trans-action costs (bid-ask spreads) increase when a certain volume is exceeded.To this end, the amount to be refinanced in each maturity is split into severaltranches. The maximum number of possible tranches for maturity d is givenby Id, Id := 1, . . . , Id is a corresponding index set, and xd

i,t denotes thetransaction volume in tranche i ∈ Id, d ∈ DS.

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The coefficient ϕdi,t quantifies the costs for raising $1 in the i-th tranche

of maturity d at time t. It contains the net present value of all interest pay-ments until maturity of the instrument, transaction costs as well as “penaltyspreads” for exceeding the tranche limits. Since the planning horizon mustbe truncated at time T while the bank’s business is (hopefully) continuedbeyond that date, the present value of all cash flows which occur after theterminal stage, i.e., outstanding interest payments and repayment of the facevalue discounted at the yield curve in T , is also included to incorporate endeffects.

For t > 0 these coefficients depend on future interest rates and, hence, areuncertain. It is assumed that the ϕd

i,t for all d ∈ DS at time t are functions ofa random vector ηt ∈ R

Kt . As discussed in the sequel, the elements of ηt arethe state variables of a term structure model which describes the evolution ofthe yield curve by aKt-dimensional stochastic process. A formal specificationof the functional relationship between the interest rate risk factors ηt and ϕd

i,t

is omitted here because its notation becomes rather cumbersome.Analogously, the mortgage volume υt depends on the random vector ξt ∈

RLt . It may be correlated with ηt to reflect a dependency between interest

rates and volume. With the objective to minimize the expected discountedrefinancing costs over the planning horizon T , the corresponding multistagestochastic program is:

min

∫Ω

T∑t=0

∑d∈DS

∑i∈Id

ϕdi,t(ηt) · xd

i,t

dP (ω) (3)

subject to

xdt − xd+1

t−1 = 0 t = 0, . . . , T ;∀d /∈ DS; a.s. (3.1)

xdt − xd+1

t−1 −∑i∈Id

xdi,t = 0 t = 0, . . . , T ;∀d ∈ DS; a.s. (3.2)

∑d∈D

xdt = υt(ξt) t = 0, . . . , T ; a.s. (3.3)

l,di,t ≤ xdi,t ≤ u,d

i,t t = 0, . . . , T ;∀i ∈ Id;∀d ∈ DS; a.s. (3.4)

xdi,t ≥ 0; xd

i,t ≡ xdi,t|Ft t = 0, . . . , T ;∀i ∈ Id;∀d ∈ DS; a.s. (3.5)

xdt ∈ R; xd

t ≡ xdt |Ft t = 0, . . . , T ;∀d ∈ D; a.s. (3.6)

The budget constraint (3.1) ensures that the position xdt maturing after

d periods equals the corresponding value in the previous period for non-traded maturity dates while (3.2) corrects it by the new transactions in t for

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traded ones. Herein, xd−1 denotes the amount in the initial portfolio held in

maturity d. Constraint (3.3) requires that the complete portfolio matchesthe stochastic mortgage volume υt at all points in time.

Lower and upper limits l,di,t and u,di,t for tranches with different penalty

spreads to reflect liquidity restrictions are given by (3.4). Obviously, theabsolute amount that can be refinanced in maturity d can be controlled bya corresponding number of tranches id. All tranches with low indices are“filled” to their limits first in the optimal solution since the spreads arestrictly increasing. Nonanticipativity constraints (3.5) and (3.6) require thatdecisions are adapted to the filtration Ft. Moreover, the restrictions (3.1)–(3.6) at stage t = 1, . . . , T must hold almost surely, i.e., for all observations ofthe uncertain interest rate and volume risk factors ωt := (ηt, ξt) with positiveprobability.

Since we distinguish between stochastic factors ηt that affect only co-efficients in the objective and those ξt that occur solely in the constraintswith deterministic left-hand-sides, it can easily be seen that the mortgagefunding problem above has the same structure as the special type of multi-stage stochastic programs given by (1′) and (2). This is a particularly usefulproperty for the solution method introduced in section 4.

3 Modelling the dynamics of risk factors

Since the optimization problem (3) contains uncertain coefficients, we mustspecify stochastic processes that describe the dynamics of the relevant riskfactors, i.e., interest rates of traded maturities and the volume of the mort-gage position under management. The corresponding process parametersare then estimated by suitable statistical procedures. Alternative modelsshould be compared by means of econometric tests which allow the rejec-tion of inappropriate specifications. In this way, one can identify the modelthat provides the best explanation of the empirically observed data amongdifferent alternatives.

3.1 Term structure models for interest rate scenarios

Interest rates exhibit a number of characteristic features that should be takeninto account to model their dynamics realistically. For example, principalcomponent analysis reveals that three stochastic factors are sufficient to ex-plain more than 95 % of interest rate volatility. These factors can be associ-ated with changes in the level, steepness and curvature of the yield curve (seeFigure 3 for Swiss Franc Euromarket rates, cf. Schurle [31]). Moreover, one

9

-0.4

-0.2

0

0.2

0.4

12 24 36 48 60se

nsit

ivity

[%]

maturity (months)

level

steepness

curvature

Figure 3: Impact of principal components on changes in yields

can observe that interest rates fluctuate around a long-term mean during aneconomic cycle. This mean reversion property is incorporated in most termstructure models that have been introduced in the financial literature overthe last two decades (e.g., see James and Webber [23] for a comprehensiveintroduction).

The relevance of these different aspects depends on the underlying appli-cation and/or the desired planning horizon. For example, when a portfolioof short-term securities is managed for a horizon of up to one year like incash management problems, it should be taken into account that the shortend of the yield curve often shows complex and quickly varying shapes. Thiscan be reflected well using three principal components as risk factors thatare standard normally distributed and orthogonal by construction while themean reversion property may be neglected. The sensitivities of relevant in-terest rates with respect to the principal components can easily be derivedby means of most statistical software tools.

On the other hand, this approach implies a non-stationary distribution ofinterest rates. As a consequence, the model generates values that are outsideof the “usual” range or may even become negative when the planning horizonis extended. A simple way of modelling mean reversion of a risk factor ηt isthe introduction of a drift term

dηt

dt= κ(θ − ηt), κ > 0. (4)

If ηt < θ (ηt > θ) at time t, then the left-hand-side of this equation ispositive (negative) which will cause ηt to increase (decrease) towards its meanreversion level θ at a speed controlled by parameter κ. Rewriting (4) andadding a volatility term where noise is a function of a Wiener process withincrement dzt as usual in modelling financial data yields the (continuous-time) process

dηt = κ(θ − ηt)dt+ σηγt dzt. (5)

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The instantaneous volatility σηγt may depend on the current level of the factor

if γ > 0 to reflect a possible heteroscedasticity of interest rates. Moreover,this prevents the process from becoming negative since in continuous timethe volatility term becomes zero when ηt = 0 and only the positive driftremains (see [23] for details).

Given K stochastic processes of type (5) for one or more risk factors in a“classical” term structure model, Ito’s lemma can be used to derive a processfor the price of a discount bound. By construction of a hedge portfolio con-sisting of K + 1 instruments with different maturities, one obtains a partialdifferential equation (PDE) for the term structure using a no-arbitrage ar-gument. However, the latter can be solved analytically only for some specialcases. Example are models of the affine type where (i) all K factors followa process of the form (5) with identical exponent γ ∈ 0, 0.5, (ii) are or-thogonal and (iii) sum up to the instantaneous rate rt, i.e., the yield for aninfinitely short holding period (see Frauendorfer and Schurle [17]).

In order to overcome these restrictions, we implemented several expo-nential functions for an interpolation of the term structure instead of anarbitrage-free PDE. For the ease of exposition, only the rather simple form

R(st, lt, d) = (st + β1 · d) · e−β2·d + lt (6)

is depicted here where R(st, lt, d) denotes the spot rate for maturity d. Whilethe fix parameters β1, β2 control the shape of the yield curve, we model thelong rate lt and the spread st := rt − lt by the stochastic processes

dst = κs(θs − st)dt+ σsdz1,t

dlt = κl(θl − lt)dt+ σllγt dz2,t.

(7)

The volatility of the long rate may depend on the current level of lt whenγ > 0 to incorporate a possible heteroscedasticity. Obviously, the process forthe spread must allow for negative values to reflect normal (st < 0) as well asinverse term structures (st > 0). Although this does not preclude negativeinterest rates when the spread becomes sufficiently large, they are extremelyunlikely for realistic parameter estimates.

The process specification (7) resembles in some sense the model of Schae-fer and Schwartz [30] that uses the same state variables but assumes γ = 0.5and no correlation between both Wiener processes to derive an analyticalapproximation for the term structure PDE. The advantage of our approachis that we have more flexibility in the specification of yield curve functionsthan in conventional term structure models. Furthermore, we may chooseany value for the correlation ρ = dz1,t · dz2,t between the Wiener processesand the volatility exponent γ. Using discrete time approximations of (7)

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st+1 = st + κs(θs − st)∆t+ σs

√∆tε1,t

lt+1 = lt + κl(θl − lt)∆t+ σllγt

√∆tε2,t,

(8)

parameters can easily be estimated for both processes separately with max-imum likelihood (under the assumption that the residuals εi,t ∼ N (0, 1),i = 1, 2, are serially independent and uncorrelated with st and lt). Herein,the long rate lt is approximated by observations of the 5 year CHF Euromar-ket rate as the longest available (liquid) maturity and the spread st by thedifference between the one-month rate and the latter.

In view of the approximation method introduced in the sequel for the so-lution of multistage stochastic programs, we restrict ourselves to fixed valuesγ ∈ 0, 1 when we estimate the parameters of the discrete processes (8).Under these specifications, the saddle property of value functions discussedin section 2.1 will be preserved. An analysis of different historic sample peri-ods reveals that γ = 1 provides higher likelihood values in most cases whichsupports the assumption of heteroscedastic interest rates. After calibrationof the processes, an estimate for ρ is obtained from the cross-correlationbetween the residuals. Finally, the parameters of the yield curve interpola-tion function (here: β1, β2) are determined that allow for the best fit of (6)to the observed rates of all maturities in the historic sample by quadraticminimization.

3.2 Specification of the volume process

At first sight, the specification and calibration of stochastic processes for thevariable mortgage volume seems to be easier than for interest rates since it isdirectly observable and, in contrast to the term structure, does not dependon several “latent” factors. However, data for the estimation and assessmentof alternative model are often difficult to obtain in practice. The SwissNational Bank publishes only an aggregate of variable and fixed mortgagesin its monthly reports, and the percentage of non-fixed mortgages is onlyavailable on a yearly basis since 1996. Furthermore, fluctuations in mortgagedemand often depend on the type of bank. For example, the clientele of largecommercial banks in Switzerland consists mainly of urban population thattends to react more actively to changes in the economy than customers ofsmaller cooperative banks in the countryside. As a consequence, any volumemodel has to calibrated individually to the specific bank situation.

According to the problem description in section 1.1, it seems plausiblethat the demand for non-maturing accounts depends on the level of interestrates. In case of savings accounts, Schurle [31] found that a trend-stationaryprocess with two factors of a term structure model as explanatory variables

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provides a good description for real deposit positions. However, when theimplementation of the mortgage funding model started in 1993, we did nothave a sufficiently large data set at our disposal to support such a hypothesis.Therefore, we simply model the volume by the autoregressive process

υt = a+ ρυυt−1 + ξt (9)

to which a deterministic trend is added, i.e., υt = υt + bt. The stochasticcomponent (9) has a long-term mean of a/(1−ρυ) if the process is stationary(i.e., |ρυ| < 1). After correcting a sample time series by the trend bt, theparameters of the resulting standard AR(1) process can be easily estimatedby means of most statistical standard software. A dependency on interestrates is taken into account by the correlation between the stochastic factorξt and the residuals of the discrete-time interest rate processes (8).

4 Solution of multistage stochastic programs

The numerical difficulty in solving a problem of type (1) and (2) lies in thenested minimization and multi-dimensional integration of value functions.Since the latter are given only implicitly as the solution of stochastic pro-grams with respect to the remaining stages, this integration cannot performedanalytically, and numerical techniques are required that can broadly be clas-sified into simulation-based methods and bound-based approximations.

In the former case, random samples are drawn from the underlying prob-ability distributions, e.g., to derive stochastic quasi-gradients or for the ap-plication of stochastic decomposition algorithms. While the computationaleffort is independent of the dimension of random data, these approaches pro-vide only probabilistic bounds for the discretization error. Loosely speaking,this is the error that results from replacing the entire universe of possibleoutcomes of random data by a relatively small set of scenarios.

In contrast to this, bound-based approximation methods partition the do-main of random data into cells and determine representative points withinthem. The accuracy may be improved by adding new scenarios that resultfrom partitioning the initial cells into smaller ones (refinement). For multi-stage problems, a careful control of the refinement process is necessary sincethe number of scenarios grows exponentially with the dimension size and thedesired accuracy (curse of dimensionality). However, generalizations of thewell-known inequalities of Jensen [24] and Edmundson/Madansky [27] thatexploit the saddle property of value functions discussed in section 2.1 pro-vide exact lower and upper bounds to the original problem (bounds based on

13

these inequalities and their refinements in the context of two-stage stochas-tic programming were introduced, e.g., by Huang et al. [21]). This allowsa deliberate refinement for those scenarios and stages where the largest dis-cretization error is observed to reduce the growth in problem size.

Since the evolution of interest rates and volume is modelled by low-dimensional processes, the underlying saddle property motivates the appli-cation of a bound-based scenario generation approach for the solution ofthe mortgage funding problem (3). In order to deal with the observed cor-relations between the risk factors, approximation schemes must take intoaccount cross moment information in a numerically efficient way. Such tech-niques have been developed, e.g., by Edirisinghe and Ziemba [6, 8, 9, 10] orFrauendorfer [11, 13, 14] (see also Edirisinghe [7] for a survey on bound-basedapproximations).

4.1 Barycentric approximation

In the sequel, we concentrate on the barycentric approximation techniquethat was originally introduced in Frauendorfer [11] for two-stage stochasticprograms and extend it to the multistage case. The basic idea is to replacethe implicitly given value functions of type (1′) by two bilinear functionsthat can easily be integrated. Then, the best points where the original valuefunction must be supported by its bilinear approximations in order to mini-mize the discretization error are the so-called generalized barycenters. Theseare determined with respect to cross-simplices (or briefly: ×-simplices), i.e.,the Cartesian product of regular simplices, that cover the support of randomdata.

For the ease of exposition, we assume that the sets Θt(ωt−1) ⊂ R

Kt andΞt(ω

t−1) ⊂ RLt that cover the support of ηt and ξt are regular simplices. Note

that both may depend on prior observations ωt−1 although this is not stressedin the notation for simplicity. Let the vertices of Θt and Ξt be denoted byuνt , νt = 0, . . . , Kt, and vµt , µt = 0, . . . , Lt. The barycentric weights λt(ηt) =(λt,0(ηt), . . . , λt,Kt(ηt)

)′of ηt with respect to Θt are those nonnegative weights

that allow the representation of ηt as a linear combination of the vertices uνt

and sum up to one:

λt,0 + λt,1 + . . .+ λt,Kt = 1ut,0λt,0 +ut,1λt,1 + . . .+ut,Ktλt,Kt = ηt.

The barycentric weights τt(ξt) =(τt,0(ξt), . . . , τt,Lt(ξt)

)′of ξt with respect to

Ξt are defined analogously. λt(·), τt(·) can be obtained as the solution of

14

Ut · λt =

(1ηt

)with Ut =

(1 1 · · · 1u0 u1 · · · uKt

)(10)

Vt · τt =

(1ξt

)with Vt =

(1 1 · · · 1v0 v1 · · · vLt

). (11)

These weights may be used to derive the generalized barycenters

ξνt =[Mνt(uνt × Ξt)

]−1 ·Lt∑

µt=0

vµt

∫λνt(ηt) · τµt(ξt)dPt(ηt, ξt|ωt−1) (12)

ηµt =[Mµt(Θt × vµt)

]−1 ·Kt∑

νt=0

uνt

∫λνt(ηt) · τµt(ξt)dPt(ηt, ξt|ωt−1) (13)

with respect to the ×-simplex Θt × Ξt where

Mνt(uνt × Ξt)) =

∫τµt(ξt)dPt(ηt, ξt|ωt−1), νt = 0, . . . , Kt (14)

Mµt(Θt × vµt) =

∫λνt(ηt)dPt(ηt, ξt|ωt−1), µt = 0, . . . , Lt (15)

are the mass distributions induced by the probability measure Pt on the Kt-dimensional simplices Θt×vµt and the Lt-dimensional simplices uνt×Ξt,respectively. These mass distributions add up to one, i.e.,

Kt∑νt=0

Mνt(uνt × Ξt) = 1 andLt∑

µt=0

Mµt(Θt × vµt) = 1.

Therefore, we may interpret (14) and (15) as probabilities assigned to thepoints (uνt , ξνt) with probability Mνt(uνt × Ξt) for νt = 0, . . . , Kt and(ηµt , vµt) with probability Mµt(Θt × vµt) for µt = 0, . . . , Lt.

An illustration is given in Figure 4 where the samples represent the jointdistribution of η and ξ for K = L = 1 (the time index is omitted for sim-plicity), indicating a negative correlation between the random data. In theone-dimensional case, the simplices covering the support of η and ξ are in-tervals and, thus, the resulting ×-simplex is a rectangle. For instance, inFigure 4 (a) the edges AB and CD cover the support of η (here: the factorof a term structure model), i.e., A and D correspond to vertex u0 while Band C are equivalent to u1. Analogously, AD and BC represent the domainof (the volume risk factor) ξ and correspond to an interval with vertices v0

and v1.Projecting the distribution mass onto AB and CD as in Figure 4 (b),

taking into account the distance from each sample point to the edges, pro-vides the barycenters η0 and η1. For each simplex, they are determined as the

15

ξ

ηA B

CD

(a) ×-simplex

η1

η0

(b) Barycenters for η

ξ0ξ1

(c) Barycenters for ξ

Figure 4: Approximation of a correlated distribution (K = L = 1)

“center of gravity” of the projected mass, and their probabilities are equiva-lent to the proportion of the latter to a total mass of one. Analogously, thebarycenters ξ0 and ξ1 in Figure 4 (c) result from a projection of the massonto AD and BC, respectively.

An advantageous feature from a computational point of view is that thegeneralized barycenters in (12), (13) and their probabilities (14), (15) arecompletely derived from the first moments of ηt and ξt as well as the bilinearcross moments Et(ηνt · ξµt), νt = 0, . . . , Kt, µt = 0, . . . , Lt. Since the co-variance of two random variables is determined by the first moments and thecorresponding cross moments, the measures Qu

t and Qlt incorporate implicitly

the correlation between ηt and ξt as indicated by the different coordinatesof the corresponding barycenters in Figure 4 (b) and (c). However, crossmoments (or covariances, respectively) between different elements of ηt arenot taken into account (the same holds for the components of ξt). Therefore,no assumptions of independence between the random variables are required.

4.2 Lower and upper bounds for value functions

By application of (12)–(15) at each stage t, the original probability measurePt is approximated by two discrete measures Ql

t and Qut with supports

suppQlt =

(uνt , ξνt)

∣∣νt = 0, . . . , Kt

(16)

suppQut =

(ηµt , vµt)

∣∣µt = 0, . . . , Lt

. (17)

The corresponding probabilities are given by qlt(uνt , ξνt) := Mνt(uνt×Ξt))

and qut (ηµt , vµt) := Mµt(Θt ×vµt). Substituting Pt in (1′) by these approx-

imate measures provides the new value functions

ψt(xt−1, ωt) := min c′t(η

t) · xt +

∫Ωt+1

ψt+1(xt, ωt+1)dQl

t+1(ωt+1|ωt)

s.t. Wt · xt + Tt−1 · xt−1 = ht(ξt), xt ≥ 0

(18)

16

and

Ψt(xt−1, ωt) := min c′t(η

t) · xt +

∫Ωt+1

Ψt+1(xt, ωt+1)dQu

t+1(ωt+1|ωt)

s.t. Wt · xt + Tt−1 · xt−1 = ht(ξt), xt ≥ 0

(19)

for t = 1, . . . , T with ψT+1(·) = ΨT+1(·) := 0. Both ψt and Ψt are bilinearfunctions since the integrand λνt(ηt) · τµt(ξt) in (12) and (13) is bilinear in(ηt, ξt). It is shown in [13] that the following relation holds:

ψt(xt−1, ωt) ≤ φt(x

t−1, ωt) ≤ Ψt(xt−1, ωt). (20)

The meaning of this inequality is that the value function φt(·) with respect tothe original measure Pt (which is a saddle function) is supported from belowand above by the two bilinear functions ψt(·) and Ψt(·) with respect to theapproximate measures Ql

t and Qut . The barycenters ξνt , νt = 0, . . . , Kt, are

the supporting points for the minorant while the majorant is supported in ηµt ,µt = 0, . . . , Lt. This situation is illustrated in Figure 5 (again, the case K =L = 1 is considered and the time index omitted for simplicity). Obviously,the two bilinear functions can easily be integrated since the calculation ofthe expectations reduces to the weighted sums

Etψt(xt−1, ωt) =

∑ωt∈supp Ql

t

ψt(xt−1, ωt) · ql

t

EtΨt(xt−1, ωt) =

∑ωt∈supp Qu

t

Ψt(xt−1, ωt) · qu

t

which was the intention of the approximation. Therefore, the problems thatresult from the substitution of the original conditional measures Pt in (1′)and (2) by Ql

t and Qut can be treated as deterministic multistage programs.

Their solutions provide policies xl := (xl0, . . . , x

lT ) and xu := (xu

0 , . . . , xuT ).

While the decisions for t > 0 correspond to outcomes in the barycentricscenarios that may be seen as “representative” rebalancing actions, only thepolicy for t = 0 will be implemented and is of interest for the user. However,a situation may occur where the first-stage decisions are not unique, i.e.,xl

0 = xu0 . Then, more accurate bounds can be achieved when the support of

random data at time t is partitioned into t(ωt−1) sub-×-simplices with⋃t

i=1 Ωit = Ωt ⊃ suppωt, (21)

Ωit ∩ Ωj

t = ∅, i = j; i, j = 1, . . . , t, (22)

Ωit are regular ×-simplices for i = 1, . . . , t. (23)

17

A

B

C

D

(a) Lower bound

A

B

C

D

(b) Upper bound

Figure 5: Bilinear approximations of the value function

A collection Pt(ωt−1) := Ω1t , . . . ,Ω

tt which satisfies (21)–(23) is called a

partition of the support of ωt, and an approximation can be obtained byapplication of the scheme (12)–(15) to each element Ωi

t individually. Notethat the probabilities assigned to the individual outcomes must be adjustedaccording to the percentage of the distribution mass that is covered by thecorresponding ×-simplex. In case that the accuracy of a bound is not suffi-cient, one may split one of the (sub-) ×-simplices in the initial partition Pt

(refinement, see Figure 6). Then, the solution of the corresponding approxi-mate problem based on the new partition Pt+1 must be as least as good asthe former bound. As t → ∞ and the sub-×-simplices become arbitrarilysmall with respect to diameter, the approximate value functions ψt and Ψt

epi-converge to φt (see Frauendorfer [13] for details).Nevertheless, dividing the elements of a partition without strategy may

dramatically increase the number of scenarios and, hence, the computationalcomplexity of the corresponding deterministic optimization problems. Forexample, one may refine the partition with the largest discretization errorεt(ω

t) := Ψt(ut−1, ωt)−ψt(u

t−1, ωt) until the desired accuracy is achieved. Ifεt(·) = 0, then the approximation of φt is exact, and (further) refinementswill not improve the solution. In this sense, the existence and utilization ofexact bounds may be seen as one of the most important features of barycen-tric approximation for the solution of multistage stochastic programs (seeFrauendorfer and Marohn [15] for refinement techniques in the context ofbarycentric approximation).

18

Figure 6: Split of a ×-simplex and new positions of the barycenters ξνt

4.3 Cross-simplicial coverages and convergence

Obviously, the distributions induced by the stochastic processes we consid-ered in sections 3.1 and 3.2 have unbounded support. For the determinationof a ×-simplicial coverage, Pt(·|ηt−1, ξt−1) must therefore be substituted bya normalized truncation so that Pt(Θt × Ξt|ηt−1, ξt−1) ≥ 1 − ε for some suf-ficiently small ε > 0.

In case that the partition consists of a single ×-simplex only and allK riskfactors are standard normally distributed and uncorrelated, such a coveragemay be constructed as follows: A sphere with radius δ around the origincontains a percentage of 2Φ(δ)− 1 of the total mass distribution (Φ denotesthe c.d.f.). It can be covered by a regular simplex in R

K with K + 1 vertices(note that such a simplex is the polyhedron with the smallest possible numberof independent vertices). In the one-dimensional case, this simplex reduces toan interval [−δ, δ] and for K = 2 to a triangle whose vertices may be chosen,e.g., as u0 = (−√

3δ, δ)′, u1 = (√

3δ, δ)′ and u2 = (0,−2δ)′. For example, withδ = 2 the circle contains more than 95% of the probability mass, and sincethe latter is entirely covered by the simplex, any outcome within a range oftwo standard deviations will be considered in the approximation.

For arbitrary K-dimensional normal distributions with expectation µ andcovariance matrix Σ, we make use of the fact that a standard normallydistributed random vector Z ∈ R

K can be transformed into another oneY ∼ N (µ,Σ) using the lower triangular matrix of the Cholesky decomposi-tion Γ of Σ, i.e., Σ = Γ · Γ′. Given a simplex with vertices uZ

i , i = 0, . . . , K,around the truncated support of an uncorrelated standard normal distribu-tion, the vertices of a simplex that covers the actual correlated distributionare obtained from the transformation µ+Γ ·uZ

i . This procedure is performedseparately for the distributions of ηt and ξt at time t to determine the verticesof the (single) ×-simplex Θt × Ξt in the initial partition.

According to results in [16] where we applied successive refinements toinitial partitions consisting of only one ×-simplex, the convergence is ratherslow when the (lower) bound based on the vertices of Θt and barycenters forξt is refined while the other (upper) bound seems to be stable. The formereffect can be explained geometrically by “too extreme” coordinates of the

19

vertices we selected to cover the support of ηt. As a consequence, the saddlefunction exhibits a relatively large degree of concavity with respect to therisk factors ηt and cannot be approximated well by bilinear functions.

Since the split of a simplex generates only one new point but the existingvertices remain in the partition, the influence of extreme outcomes in theinitial discretization decreases slowly. Furthermore, the split of a ×-simplexin the partition for the first stage provides a better improvement than in caseof the partition where the largest discretization error was observed. This isnot surprising since the approximation of the distribution in t = 0 affects allscenarios and, hence, has the highest impact on the solution.

4.4 Disretization of interest rate processes

To achieve tighter bounds, possibly with less refinement steps, we modifiedthe procedure for the determination of initial coverages in combination withthe discrete-time version of the two-factor mean reversion model (8). While atriangle covers the truncated support of ηt = (st, lt)

′ with only three verticeswhich keeps the scenario size moderate, a better approximation of the circlearound the mass of the (uncorrelated) standard normal distribution can obvi-ously be obtained using polygons with K ′

t > 3 vertices1 (see Figure 7). Moreprecisely, we determine pentagons as approximations of the (unit) circle thatare partitioned into three triangles for the first stages of the stochastic pro-gram. Combining them with intervals that cover the support of ξt results ininitial partitions of t = 3 cross-simplices. After the transformation accord-ing to the expectation and covariance matrix of the actual distributions attime t, generalized barycenters and weights are determined for each of themindividually according to (12)–(15).

Because this discretization yields 9 outcomes for the lower bound (cor-responding to the number of vertices in the partition), we use squares par-titioned into two simplices (t = 2) and then single triangles (t = 1) atlater stages where the approximation has less impact on the solution of thecomplete stochastic program in order to reduce the growth in problem size.The specific size t of the initial partitions for all t = 1, . . . , T depends onthe number of periods that must be achieved. The total number of scenariosat the t-th stage is given by

1General polyhedra to obtain tighter bounds are also exploited in Gassmann andZiemba [18]. However, they develop an upper bound for the case of single polyhedronswith arbitrary number of vertices while we divide the initial polyhedron that covers thesupport of ηt in regular sub-simplices to apply the barycentric approximation scheme.

20

Figure 7: Coverages for the uncorrelated and correlated case with barycenters

slt :=

∏tτ=1(Kτ + 1) · τ = 3t · ∏t

τ=1 τ

sut :=

∏tτ=1(Lτ + 1) · τ = 2t · ∏t

τ=1 τ

for the lower and upper bound, respectively. In our experience, we ob-tain a better accuracy for a similar problem size compared to sequentialrefinements of partitions where we start with a single ×-simplex for the dis-cretization. For example, in case of an 8-stage quarterly planning problem(DS = 3M, 6M, 1Y, 2Y, 3Y, 4Y, 5Y) with 1 = 3, 2 = 2, 3 = . . . = 7 = 1and δ = 2, we have 13122 scenarios in the lower (larger) approximation. Theobjective function values are 5818.6 for the lower bound and 5888.4 for theupper bound2, equivalent to a relative difference of only 1.20%.

In case of the principal component model for short-term planning, wemake use of the fact that the Kt components of ηt are orthogonal (uncorre-lated) by construction. This allows to represent the Θt, t = 1, . . . , T , them-selves as a ×-simplices, i.e., Θt = Θ1

t × . . . × ΘKtt . Each Θi

t = [−δ, δ] ⊂ R isan interval that covers the truncated support of the i-th principal componentηi

t, i = 1, . . . , Kt. The required modified formulae for the barycentric approx-imation scheme can be found in [12] for the two-stage case. For example, toobtain the generalized barycenters ηµt = (η1

µt, . . . , ηKt

µt)′ and corresponding

probabilities, we have to replace λνt(ηt) in (12)–(15) by∏Kt

i=1 λνit(ηi

t) with

νt = (ν1t , . . . , ν

Ktt ), νi

t = 0, . . . , Kt. Note that each principal component ηit

may still be correlated with the volume risk factor ξt.The ×-simplex Θt as the Cartesian product of Kt intervals is a multi-

dimensional rectangle with 2Kt vertices, and the number of scenarios at staget is given by sl

t :=∏t

τ=1 2Kτ for the lower bound. The growth in problem

2Process parameters: κs = 0.8687, θs = −0.0061, σs = 0.0223, κl = 0.1940, θl =0.0507, σl = 0.1532, ρ = −0.3610, φ1 = −0.002689, φ2 = 0.4293 for the term structuremodel (∆t = 0.25) and a = 83.82, b = 0, ρυ = 0.9835, σξ = 166.0 for the volume process.

21

size for the upper approximation with sut :=

∏tτ=1 2Lτ is less dramatic since

we assumed Lt = L = 1. Thus, with Kt = K = 3 relevant factors, the sizeof the scenario set in the former case is multiplied by 8 with each additionalstage and no refinements.

Since this increase may be too restrictive for the number of periods thatcan be taken into account, the growth in problem size may be reduced byconsidering all principal components only in the first stages while the thirdand then the second factor are ignored at later points in time. This is mo-tivated by the empirical observation that they contribute only 3% and 19%,respectively, to the dynamics of interest rates (cf. [31]). As a consequence, thescenarios incorporate complex tilt and humped movements at the beginningbut may reflect only shifts of the yield curve towards the end of the planninghorizon where the outcomes have less impact on the first-stage decision. Inthis way, we achieve a lower and upper bound of 12830.9 and 12873.3, i.e.,a relative difference of 0.33% for a 7-stage monthly planning problem3 withmaturities DS = 1M, 2M, 3M, 6M, 1Y, 2Y, 3Y, 4Y, 5Y and dimension sizeK1 = K2 = 3, K3 = . . . = K6 = 2 which results in 16384 scenarios.

5 Application to the funding problem

Scenarios generated by means of any discretization method may be repre-sented by a tree (e.g., see [14] for a formal description). Nodes of this scenariotree with depth t correspond to outcomes of random data at time t. Using thebarycentric approximation scheme described above results in a total numberof nodes

∑Tt=0 s

lt for the lower bounding problem which is the larger one since

we have at least as many interest rate than volume risk factors (Kt ≥ Lt ∀t).The funding model (3) has D +

∑d∈DS Id variables at each stage. For in-

stance, with traded maturities DS = 3M, 6M, 1Y, 2Y, 3Y, 4Y, 5Y and 8tranches for each of them, we have 76 variables which must be duplicated forall nodes. Thus, the corresponding deterministic program for the exampleabove with 13122 scenarios already consists of more than 1.5 million vari-ables in 19666 nodes. The problem generation and solution with standardoptimization tools such as Cplex requires up to a few hours on a medium-size

3We estimated the factor sensitivities for CHF Euromarket rates as shown in Figure 3with principal component analysis, and the correlations between the factor scores andthe volume risk factor ξt are 0.34, −0.18 and −0.03. The different magnitudes of theobjective values obtained with the mean reversion model above results from the fact thatonly a fraction of the mortgage volume could be included in the optimization there dueto quarterly planning, i.e., the actual portfolio had to be split into four components sincethe period length was extended to three months there.

22

workstation (Sun Ultra 10, 1 GB RAM). This may be seen as tolerable sincethe determination of refinancing policies is performed only once a month.

As a consequence of the exponential growth in problem size, we can dealwith a limited number of stages only although the planning horizon of therefinancing problem is actually infinite. Our experience from various casestudies is that an increase in the number of stages (even at the cost of areduced accuracy of the approximation) in general leads to a higher per-formance. This can be explained by the fact that the model has a greaterflexibility for corrections of the initial portfolio. Moreover, an extension ofthe horizon allows a better consideration of the impact of future changes inthe risk factors (e.g., a sharp drop in volume that might lead to a surplus ofliabilities over the mortgage position) for the first-stage decision. However, inpractice we are not able to solve problems with more than 10 stages with themodel formulation and solution techniques described before. If we wanted toinclude the whole spectrum of traded instruments in the interbank marketfrom 1 month to 5 years and since the period length is given by the shortestmaturity, this would correspond to a planning horizon of less than one yearwhich we consider as insufficient for the long-term model.

Therefore, the determination of refinancing decisions is carried out in twosteps: First, we perform an optimization run with maturities between one andfive years where scenarios are based on the mean reversion model. When thesolution indicates that a short-term policy should be implemented, we use theprincipal component model to analyze if the funding costs can be reducedby shorter maturities than one year. Alternatively, the liability portfoliomay be split into a short- and a long-term component that are optimizedseparately. The corresponding volumes depend, e.g., on the current portfoliocomposition, limits given by the bank’s internal risk management system, orforecasts for interest rates and mortgage demand.

Before the stochastic optimization model was applied to real positions, acase study had been conducted to assess its performance for a period of highyields. The study was based on monthly money market rates and the volumeof a real mortgage position provided by a Swiss bank (see Figure 8, the lagbetween interest rates and volume results from the fact that banks adjust therelevant client rate with some delay to a new market situation). The level ofinterest rates at which mortgages had to be refinanced on the market reachedup to 10% while the client rate that the bank receives never climbed above7% due to the political cap in Switzerland. As a consequence, refinancing themortgage position with a replicating portfolio of 25% six-month, 50% one-year and 25% three-year instruments which serves as benchmark providesa (negative) average margin of −0.21% for the sample period. The staticpolicy required also to invest significant amounts at low yields because the

23

2.0%3.0%4.0%5.0%6.0%7.0%8.0%9.0%

10.0%

7/88 1/89 7/89 1/90 7/90 1/91 7/91 1/92 7/92 1/93 7/93 1/9420.0

25.0

30.0

35.0

40.0

22.5

27.5

32.5

37.5

3M5Y

volume

Figure 8: Interest rates (left) and mortgage volume (right)

share of portfolio positions with longer maturities was still too large whenthe mortgage demand dropped, resulting in a surplus of liabilities.

For simplicity, we did not follow the “2-step procedure” outlined beforewith separate runs of the long-term and the short-term planning models.Instead, we used the former only with maturities from 3 months to 5 years.With 9 stages, this quarterly planning is equivalent to a time horizon ofonly two years. According to our experience, the performance of the modelimproves significantly for longer periods. In order to achieve this number ofstages, we did not split the initial partitions.

Liquidity limits for transactions without penalties ranged from 400 mio.for the shortest to 100 mio. for the longest maturity. Transactions costsincreased by one basis point (BP) with each additional tranche of 50 mio.except for 3 months where tranches are twice as large. For the initial portfoliocomposition, it was assumed that the static 6M/1Y/3Y-mix had been imple-mented in the past, i.e., all positions had to be renewed within the first threeyears. Parameters for the stochastic processes were updated semi-annuallybased on observations of the previous five years.

ref. costs [%] margin [%]method

avg. std.dev. avg. std.dev.SP model 6.13 1.38 0.16 0.82repl. portf. 6.50 1.80 −0.21 1.26

Table 1: Comparison of dynamic policies with the replicating portfolio (RP)

After a refinancing decision had been found, the portfolio positions wereupdated and a new optimization was started with the next set of interest ratesand mortgage volume out of the sample period (“roll-over planning”). Theresult after all 73 runs is summarized in Table 1. Compared to the replicatingportfolio benchmark, the average refinancing costs for the renewed positions

24

1.0%1.5%2.0%2.5%3.0%3.5%4.0%4.5%

1/96 7/96 1/97 7/97 1/98 7/98 1/99 7/99 1/00 7/00

SP modelbenchmark

1Y rate

Figure 9: Refinancing costs for dynamic and static portfolios since 1995

could be reduced from 6.50% to 6.13%. The margin as the difference betweenthe client rate and the refinancing costs increased correspondingly by 37 BPto 0.16%. While this number implies only a small profit at first sight, onemust take into account that the static approach was not able to provide apositive margin at all for the specific market situation in the case study whichresults in significant losses for the bank. Moreover, the standard deviationwas also reduced noticeably compared to the replicating portfolio althoughthe latter was constructed to minimize the volatility of the margin.

6 Conclusions

An extended version of the multistage stochastic programming model de-scribed in this paper has been in use now by a major Swiss bank since 1995for refinancing variable mortgages. The modifications are, e.g., additionalconstraints for the portfolio duration to achieve a target that is frequentlydefined by the bank’s board of directors. While the two-factor mean rever-sion model provides a good description of both ends of the term structuredue to the selection of state variables, the error of fit is slightly worse formedium-term maturities. Therefore, a third factor has been introduced thatcontrols the curvature of the yield curve. Unfortunately, we cannot reveal alldetails of this commercial model in publications.

As evidence for the performance of the model since it has been appliedin practice, the average funding rate compared to the static mix that wasexploited by the bank before as benchmark is shown in Figure 9. With anaverage margin of approximately 70 BP that could be achieved with thereplicating portfolio approach in the long run, an increase in the order of37 BP represents a significant improvement of the bank’s profits. Before theapplication of another version of the model to savings accounts started in

25

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

7/89 1/90 7/90 1/91 7/91 1/92 7/92 1/93 7/93 1/94 7/94 1/95 7/95

SP modelrepl. pf. 1repl. pf. 2

Figure 10: Evolution of margin for dynamic vs. static investment strategies

1997, another case study with data of a real position was conducted (seeFrauendorfer and Schurle [16] for a detailed description). The results werecompared to the performance of two replicating portfolios that had been usedby the bank for the management of different deposit positions.

Figure 10 shows the evolution of the margin between the return of theinvested funds and the client rate for the dynamic policies determined bythe stochastic optimization model and the static benchmark policies. Ac-cording to Table 2, the average margin over the sample period could be im-proved by 25 BP compared to the better replicating portfolio while volatilityis significantly reduced. The increase in performance at lower risk can beexplained by the possibility of rebalancing transactions in the dynamic ap-proach. Similar observations have also been reported for other multistagestochastic programming models (e.g., see Carino and Ziemba [5]).

weights repl. portf. margin [%]method1 y 2 yrs 5 yrs mean std. dev.

SP model – – – 2.66 0.19repl. portf. 1 0.0 0.5 0.5 2.41 0.36repl. portf. 2 0.35 0.35 0.3 2.40 0.70

Table 2: Performance of stochastic optimization model for savings accounts

There are many possible ways to improve the model in the form presentedhere, e.g., with respect to the volume process specification (9). However,an empirical investigation of alternative models failed because of the lack ofpublicly available data. Moreover, new mortgage products have recently beenintroduced in Switzerland that led to a noticeable change in the demand forvariable mortgages, and it is difficult to correct the structural break inducedby such a “non-stochastic event” in the data. We are also still investigatingother term structure models with respect to their practical use for scenariogeneration.

26

Some authors derive interest rate scenarios from (binomial) lattice modelsthat have been developed for pricing derivative securities. In a continuous-time framework, this is equivalent to a one-factor specification with time-dependent parameters. Since these models are calibrated only to the termstructure observed at present time, we are not confident if such an approachcan provide a suitable description of long-term interest rate behavior butshould rather be seen as an interpolation technique for the initial yield curve.As Hull and White [22] point out: “It is important to distinguish between thegoal of developing a model that adequately describes term-structure move-ments and the goal of developing a model that adequately values most of theinterest-rate-contingent claims that are encountered in practice. It is quitepossible that a two- or three-state variable model is necessary to achieve thefirst goal”.

According to our experience, models for a planning horizon of severalyears should be calibrated to a historical data set that covers at least aneconomic cycle. In can also be helpful to assess term structure model bycomparing scenarios generated in a simulation study with the characteristicsof empirically observed data as suggested by Frauendorfer and Schurle [17].Interest rate scenarios based on the principal component and the two-factormean reversion models in section 3.1 may not be free of arbitrage. Thisaspect should be taken into account in applications with simultaneous in-vesting and refinancing. Otherwise, the model could try to exploit such“spurious” arbitrage opportunities in its decisions although they result onlyfrom a miss-specification (e.g., see Pflug [29] for arbitrage-free scenario gen-eration methods).

Using barycentric approximation, we are able to obtain tight bounds forthe original problem when the support of random data is partitioned intoseveral ×-simplices. Despite the low discretization error, different first-stagedecisions for the lower and upper approximate problems may still occur.This requires an improvement of the scenario selection, and the investigationof efficient refinement techniques based on suitable error measures is stillongoing. However, the example of the funding problem presented here andmany other approaches described in this book show that multistage stochasticprogramming models have already passed the state of research and may beapplied successfully to financial decision making under uncertainty.

Acknowledgement: We are grateful to W.T. Ziemba for his helpful commentson an earlier version of this paper that helped improve the exposition signif-icantly.

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