Part I: Review of the stochastic programming ALM literature
2. Stochastic programming ALM models
This method provides a general-purpose modeling framework that conveniently addresses real world concerns such as transaction costs, taxes, legal and policy constraints. The num- ber of decision variables becomes very large resulting in large scale optimization problems.
The computational costs make it impractical to test the recommendations out of the sample.
We describe various modeling approaches developed within this framework:
2.1. Chance-constrained model
Charnes and Kirby (1966) develop a chance-constrained model that expresses future de- posits and loan payments as jointly distributed random variables, and capital adequacy formula by chance-constraints on meeting withdrawal claims. A drawback of the model is that constraint violations are not penalized according to their magnitude.
The methodology has found applications in various areas: Charnes, Gallegos and Yao (1982) applies this methodology to balance sheet management, Li (1995a, b) uses chance- constrained programming in portfolio analysis of insurance companies, and Dert (1998) develops a multistage chance-constrained ALM model for a defined benefit pension fund.
As opposed to the original approach of Charnes and Kirby, Dert models the uncertainty using scenarios rather than making distributional assumptions.
Dert’s model minimizes the cost of funding while ensuring the stability of contributions and ability to make benefit payments timely with an acceptable level of insolvency risk.
The solvency requirement is the asset level being at least equal to the product of required funding level with the value of the remaining liabilities (constraint (7)). The asset value falling below the required level is modeled as a probabilistic constraint. Since uncertainty is modeled through scenarios, binary variables are needed to formulate the chance constraint explicitly (constraints (8)–(10)). It is assumed that remedial contributions are made in case of under-funding (constraint (6)).
The ALM model is formulated as follows:
minA01+
T−1 t=1
St
s=1
P (t, s)γt sYt s+λ T t=1
St
s=1
P (t, s)γt sZt s subject to
Yt sl Yt sYt su, (1)
yt sl Yt s
Wt s yt su, (2)
Yt s
Wt s − Yt−1,sˆ
Wt−1,sˆ βt, (3)
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At s+Yt s−lt s= N i=1
Xit s, (4)
xit sl (At s+Yt s−lt s)Xit sxit su (At s+Yt s−lt s),
t=0, . . . , T−1, s=1, . . . , St, (5)
At s=Zt s+ N
i=1
eritsXi,t−1,sˆ, (6)
At sαLt s, (7)
Zt sft sMt s, (8)
St
s=1
P
(t, s)|(t−1,s)ˆ
ft sΨt−1,ˆs, (9)
ft s∈ {0,1}, t=0, . . . , T −1, s=1, . . . , St, (10) where,
t=0,1, . . . , T is the time period, s=1,2, . . . , St is the status of the world, i=1,2, . . . , Nis the asset class,
αis the demanded funding level,
βt is the maximal raise in contribution per period as a fraction of the cost of wages at timet,
γt sis the discount factor for a cash flow at timet in states, lt sis the benefit payments and costs to the fund at timet in states, Lt sis the actuarial reserve at timet in states,
λis the penalty parameter to penalize remedial contributions,
rit s is the continuous return on investment of each asset classi during periodt in states,
Mt sis the large constant at timetin states, Wt sis the cost of wages during periodtin states,
At sis the total asset value before receiving regular contributions and making benefit payments at timetin states,
ft sis the binary variable for remedial contributions at timet in states,
Ψt sis the probability of under-funding at timet+1 given the world was in statesat timet,
Xit sis the amount of money invested in asset classiat timetin states, xit sis the fraction of asset value invested in asset classiat timetin states, Yt sis the regular contribution during periodtin states,
yt s is the regular contribution as a fraction of the cost of wages during periodt in states,
Zt sis the remedial contribution at timetin states.
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The first three constraints, namely (1)–(3), limit the regular contribution amount, regular contribution as a fraction of wages and maximal raise in contribution as a fraction of cost of wages, respectively. After receiving regular contributions and making benefit payments, the assets are reallocated (4) considering the upper and lower bounds on the asset mix (5).
The price inflation, wage inflation, and asset return scenarios are generated using vec- tor autoregressive model. The characteristics of participants are modeled using a Markov chain. More detailed description of a similar model is given in Dert (1998).
2.2. Dynamic programming
The main idea behind dynamic programming is to solve the problem by dealing with one stage at a time. The procedure produces one solution per possible state in each stage. If there are many state variables or the objective function depends in an arbitrary way on the whole history up to the current period, this method is not very appropriate. It can handle small number of financial instruments simultaneously. Therefore it is of limited use in practice.
Eppen and Fama (1971) model a three-asset portfolio problem using this approach.
At any point in time, they assume that state of the system is described by two vari- ables:mbeing the level of cash balance (m∈N), andb being the level of bond account (b∈ {N−N−}). Decisions concerning the state of the system are made at equally spaced discrete points in time. The stochastic changes in the cash balance between the periods are a sequence of independent identically distributed random variables with the discrete probability mass functionp(d). The functionp(d)is positive only on a finite state space, i.e., there is a finiteKsuch thatp(d)=0 if|d|> K.
The notation is as follows:
T (m, b;m, b) is the minimum transfer cost involved in changing the state from (m, b)to(m, b),
ch is the marginal opportunity cost of starting a period with an additional dollar of cash,
cp is loss of being a dollar short on cash which is incurred at the beginning of the period,
L(m)is the penalty cost of carrying cash:
L(m)=
chm, m0,
−cpm, m<0, αis the discount factor,
fn(m, b)is the discounted expected cost for annperiod problem whose state at the beginning of periodnis(m, b).
The recursive relationship forfn(m, b)is given by:
fn(m, b)=min
m,b
T (m, b;m, b)+Gn(m, b) ,
516 Y. Tokat et al.
where
Gn(m, b)=cbhãb+L(m)+α K d=−K
fn−1(m+d, b)ãp(d).
Gn(m, b)is the current expected holding penalty cost (the first two terms) plus the dis- counted expected cost if a decision is made to start periodnin state(m, b)and an optimal policy is followed in periodn−1 and all future periods.
2.3. Sequential decision analysis
This approach uses implicit enumeration to find an optimal solution. It results in extremely large equivalent linear programming problems since it enumerates all possible portfolio strategies for all scenarios in all periods of consideration. The method ensures feasibility of the first period for every possible scenario, this shrinks the feasible set and gives substantial importance to scenarios with low probabilities of occurrence.
Stochastic decision tree model by Bradley and Crane (1972) overcomes the computa- tional difficulties of the approach by using a decomposition algorithm. The objective is the maximization of expected terminal wealth of the firm. Constraint (11) guarantees that the firm cannot purchase assets that cost more than it has funds available. The second set of constraints balance the inventory. The net realized capital losses in a period are con- trolled by some pre-specified upper bound using (13). Constraint (14) limits the holding of a particular asset.
Their linear programming formulation5is
max
eN∈EN
p(eN) K
k=1 N−1 m=0
ymk(em)+ukm,N(eN)
hkm,N(eN)
+
yNk(eN)+ukN,N(eN) bNk(eN)
subject to K
k=1
bnk(en)− K k=1
n−2
m=0
ymk(em)hkm,n−1(en−1)+ynk−1(en−1)bkn−1(en−1)
− K k=1
n−1
m=0
1+gm,nk (en)
sm,nk (en)=fn(en), (11)
−hkm,n−1(en−1)+sm,nk (en)+hkm,n(en)=0, m=0,1, . . . , n−2, (12)
5 The formulation is taken from Kusy and Ziemba (1986).
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−bn−k 1(en−1)+sn−k 1,n(en)+hkn−1,n(en)=0, hk0,0(e0)=hk0,
− K k=1
n−1
m=0
gm,nk (en)sm,nk (en)Ln(en), (13)
k∈Ki
bkn(en)+
n−1
m=0
hkm,n(en)
Cni(en), i=1,2, . . . , I, (14)
ym,nk (en)0, sm,nk (en)0,
hkm,n(en)0, m=1, . . . , n−1, where
en∈En,n=1,2, . . . , N, k=1,2, . . . , K,
enis an economic scenario from period 1 tonhaving probabilityp(en), Enis the set of possible economic scenarios from period 1 ton, Kis the total number of assets,
Ki is the number of assets of typei, N is the number of time periods,
ymk(em)is the income yield per dollar of purchase price in periodmof assetkcondi- tional on scenarioen,
ukm,N(eN)is the expected terminal value per dollar of purchase price in periodmof assetkheld at periodN, conditional on scenarioen,
bkn(en)is the dollar amount of assetkpurchased in periodnconditional on scenario en,
hkm,n(en)is the dollar amount of assetkpurchased in periodmand held in periodn conditional on scenarioen,
sm,nk (en)is the dollar amount of assetkpurchased in periodmand sold in periodn conditional on scenarioen
gkm,n(en)is the capital gain or loss per dollar of purchase price of assetkpurchased in periodmand sold in periodnconditional on scenarioen,
fn(en)is the incremental increase or decrease of funds available for periodn, Ln(en)is the dollar amount of maximum allowable net realized capital losses in pe- riodn,
Cni(en)is the upper bound in dollars on the amount of funds invested in asset typei in periodn.
They use a decomposition algorithm to breakdown the problem and use an efficient tech- nique to solve the sub-problems of the overall portfolio. However, the solution is still com- putationally intractable for real life problems.
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2.4. Stochastic Linear Programming with Recourse (SLPR) The basic formulation of the generalT-stage SLPR model is
minx1
c1x1+Ew1
minx2
c2(w1)x2+ ã ã ã +EwT−1
minxT
cT(wT−1)xT
subject to
A1x1 =b1,
B2(w1)x1+A2(w1)x2 =b2(w1),
B3(w2)x2+A3(w2)x3 =b3(w2), ...
BT(wT−1)xT−1+AT(wT−1)xT =bT(wT−1), ltxtut, t=1,2, . . . , T ,
where
wt is the random vector that generates the coefficientsbt,ct,At, andBt of the deci- sion problem at thet-th stage,t=2, . . . , T,
lt,ut are the vector of deterministic bounds onxtat staget,t=2, . . . , T, b1,c1, andA1are the deterministic first stage coefficient vectors or matrices, and xt is the vector decision variable.
The objective formalizes a sequence of optimization problems corresponding to dif- ferent stages. At stage 1, the outcome completely depends on future realizations of the uncertainty. After the first period, decisions are allowed to be a function of the observed realization(xt−1, wt)only. One first decides onx1, then observesw1,then decides onx2, then observesw2, and so on. The recourse decisions depend on the current state of the sys- tem as determined by previous decisions and random events. The uncertainty is modeled by using finite scenarios which have pre-assigned probabilities. In this case, the problem reduces to a large linear program of a special structure:
min
c1x1+
K2
k2=2
pk2ck
2xk2+
K3
k3=K2+1
pk3ck
3xk3+ ã ã ã +
KT
kT=KT−1+1
pkTck
TxkT
subject to
A1x1 =b1,
Bk2x1+Ak2x2 =bk2, k2=2, . . . , K2, Bk3xa(k3)+Ak3xk3 =bk3, k3=K2+1, . . . , K3,
...
BkTxa(kT)+AkTxkT=bkT, kT =KT−1+1, . . . , KT, ltxkt ut, kt=Kt−1+1, . . . , Kt, t=1,2, . . . , T .
The scenarios used determine the size, form and optimal solution of the linear pro- gram. There are finitely many sequences of possible realizations of the random coefficients
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(ckt,Akt,Bkt,bkt) with path probabilities pkt of the subsequences of these realizations, pkt >0,∀kt,Kt
kt=Kt−1+1pkt =1,t=2, . . . , T, that identify the discrete joint probability distribution ofw= {w1, . . . , wT−1}. In the program,a(kt)denotes the immediate prede- cessor ofkt, for examplea(k2)=k1.
An important application of stochastic linear programming with simple recourse model is given by Kusy and Ziemba (1986). The model was developed for the Vancouver City Savings Credit Union for a 5-year planning period. The formulation has the following features:
(1) Changing yield spreads across time, transaction costs associated with selling assets prior to maturity, and synchronization of cash flows across time are incorporated in a multiperiod context.
(2) Assets and liabilities are considered simultaneously to satisfy basic accounting princi- ples and match liquidities.
(3) Transaction costs are included.
(4) Uncertainty of withdrawal claims and deposits is reflected in uncertain cash flows.
(5) Uncertainty of interest rates is explicitly recognized.
(6) Legal and policy constraints are taken into account.
Their two-stage model did not contain end effects. Three possible scenarios that are inde- pendent over time were considered to keep the computations tractable. Their results indi- cate that their model generates policies that are superior than stochastic decision analysis.
Another milestone after the Kusy and Ziemba model is the Russell–Yasuda Kasai model by Carino et al. (1994). The model builds on the previous research to design a large scale SLPR model with possibly dependent scenarios, end effects, and all the relevant institu- tional and policy constraints. We present their model next.
Decision variables are
Vt: total fund market value at timet, Xnt: market value in assetnat timet, wt+1: income shortfall at timet+1, and vt+1: income surplus at timet+1.
Random coefficients are
RPnt+1: price return of assetnfrom end oftto end oft+1, RInt+1: income returns of assetnfrom end oft to end oft+1, Ft+1: deposit inflow from end oftto end oft+1,
Pt+1: principal payout from end oftto end oft+1, It+1: interest payout from end oft to end oft+1,
gt+1: rate at which interest is credited to policies from end oftto end oft+1, Lt: liability valuation att.
The objective is to maximize the expected market value of the firm at the horizon net of penalties for the shortfalls. Expected amount by which goals are not achieved is a more tan- gible risk measure than variance. The penalty costs of shortfalls may be based on expected financial impact or psychological costs. The piecewise linear convex cost function for the shortfall is denoted byct(wt). (15) is the budget constraint. The return on assets and in- flow of deposits net of principal and interest payout gives the total fund market value (16).
520 Y. Tokat et al.
Liability balances and cash flows are computed to model liability accumulation (18). If Yasuda does not achieve adequate income, recourse action must be taken at a cost. The income generation is modeled as a soft constraint (17), which permits surpluses or deficits.
maxE
Vt− T t=1
ct(wt)
subject to
n
Xnt−Vt=0, (15)
Vt+1−
n
(1+RPnt+1+RInt+1)Xnt=Ft+1−Pt+1−It+1, (16)
n
RInt+1Xnt+wt+1−vt+1=gt+1Lt, (17) Lt+1=(1+gt+1)Lt+Ft+1−Pt+1−It+1, (18) Xnt0, wt+10, vt+10.
The abbreviated formulation does not include some elements of the model. There are additional types of shortfalls, indirect investment types, regulatory restrictions, multiple accounts, loan assets, tax effects and end effects that are included in the original model. See Carino and Ziemba (1998) for the details of the formulation. Carino, Myers and Ziemba (1998) discusses the concepts, technical issues and uses of the model.
Korhonen (1987) applies SLPSR to multicriteria decision making. Oguzsoy and Guven (1997) use the SLPSR methodology for a bank ALM model in Turkey. Geyer et al. (2002) describes a pension fund planning model that utilizes this approach.
Some authors argue against linearizing the objective function. Bai, Carpenter and Mul- vey (1997) demonstrates that nonlinear programs are not much more difficult than their linear counterparts. Zenios et al. (1998) applies multistage stochastic nonlinear program- ming with recourse to fixed income portfolio management.
2.5. Dynamic generalized networks
Multistage stochastic nonlinear programs with recourse can be represented by generalized network formulations. This framework can be used to account for the dynamic aspects of ALM problems while considering uncertainty in all relevant parameters and accommodat- ing random parameters by means of a moderate number of scenarios.
The network structure is exploited in the solution procedure. The problem is decomposed into its constituent scenario subproblems. Preserving the network structure of each sub- problem is challenged by the existence of non-anticipativity constraints. These constraints dictate that scenarios that share common information history up to a specific period must yield the same decision up to that period, i.e., dependence on hindsight is avoided.
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The desired decomposition is achieved by dualizing the non-anticipativity constraints.
The algorithm by Rockafeller and Wets (1991) operates on the split-variable form of the original problem. The problem is solved by progressively enforcing the non-anticipativity constraints.
Mulvey (1994) utilizes this methodology in designing an asset allocation model for the Pacific Financial Asset Management Company. The single period portfolio model is formu- lated as a network model. The arcs can be constrained to impose legal or policy constraints.
The objective function is the expected utility of surplus at the end of the planning horizon.
The model was implemented in a PC environment with acceptable accuracy and efficiency.
Mulvey and Vladimirou (1989, 1991, 1992) present several aspects of stochastic gener- alized network models. See also the review of Mulvey and Ziemba (1995) which discusses the model in a general context.
2.6. Scenario optimization
According to the scenario optimization approach, one computes a solution to the determin- istic problem under all scenarios then solves a coordinating model to find a single feasible policy. This approach can be compared to the scenario aggregation method suggested by Rockafeller and Wets (1991). It handles multistage stochastic programming problems, and allows for decisions to depend on future outcomes. On the other hand, scenario optimiza- tion is designed for two-period models only. It is assumed that scenario probabilities are functions of time, and estimates of the random parameters in the future stages are poor.
Hence one only selects a policy for the immediate future.
Suppose the scenario subproblem is vs=mincsTx subject to
Asx=bs, (19)
Adx=bd, (20)
x0,
where the objective function is a particular realization of the uncertainty under scenarios, (19) is also a particular realization of the uncertain constraints under scenarios, and (20) is the deterministic constraints.
A possible coordinating model could be
min
s
pscTsx−vs2+
s
psAsx−bs2 subject to Adx=bd,
x0.
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The coordinating model tracks the scenario solution as close as possible while still main- taining feasibility. Alternative coordination models are discussed in Dembo (1991, 1993).
Illustrative applications in portfolio immunization and dedication are also presented therein.
2.7. Robust optimization
Robust optimization approach integrates goal programming formulations with a scenario based description of the uncertainty in the data. The aim is to produce solutions that are relatively less sensitive to the realizations of different scenarios. The objective function, in its most general form, is composed of two terms: the first term trades off between mean value and the variability in the mean; the second term is a feasibility penalty function.
Consider the following formulation.
min σ (x, y1, . . . , ys)+wρ(z1, . . . , zs) subject to Ax=b,
Bsx+Csys+zs=es, ∀s∈Ω, x0, ys0, ∀s∈Ω, where
x is the vector of decision variables whose value cannot be adjusted once a specific realization of the data is observed,
y is the vector of decision variables that are subject to adjustment once uncertain parameters are observed,
zis the vector of decision variables that measure infeasibility allowed, s∈Ω= {1, . . . , S}is the set of possible scenarios,
A,b,Bs,Cs,es are the coefficients related to the variables,
wis the goal programming weight that is used to derive a spectrum of answers that trade-off the two objectives.
The inclusion of higher order moments in the objective function reduces the variability of the solution. Hence, few adjustments become necessary as scenarios unfold. The model recognizes that it may not always be possible to find a feasible solution to the problem un- der all scenarios. The penalty function is used to detect the least amount of infeasibilities to be dealt with outside the model. See Mulvey, Carpenter and Mulvey (1995) for possible ob- jective function choices and their applications. Bai, Vanderbrei and Zenios (1997) argues that linear objective functions fail to identify robust solutions and concave utility func- tions produce much better results for risk averse decision makers even when penalty term is not used. Both papers compare robust optimization with stochastic linear programming approach (SLP). Since SLP optimizes only the first moment of the distribution of the objec- tive value, more adjustment is needed as scenarios are realized. However, there is no mech- anism for choosingw, and the cost of the robust solution may be higher than that of SLP.
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