Asset Lia Version of CAPm

8/8/2019 Asset Lia Version of CAPm

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A n A s s e t -L ia b il it y V e rsio n ofthe C apita l A sse t Pr ic ing M od e lw i t h a Mul t i -P e r i o d T w o -F undT he o r e m x

^ - , ! / - : . - . . , .

M . B A R T O N W A R I N G AND DuANE W H I T N E Y

econo-He has written many

addressing strategicand invest-

policy.wa nng@a ya.yale.edu

W H I T N E Yat Barclays

Investors inCA .

"Everything should be made as simple aspossible,but not one bit simpler."

Attributed to Albert Einstein'O ne of the most powerful, elegant,and delightfully simple con-structs in modern portfoliotheory is the Two-Fund The-oremsometimes called the Two-Fund Sep-aration Theorem or the Mutual FundTheoremfirst articulated by Tobin [ 1958]. Afew years later, the capital asset pricing model(CAPM) of Sharpe [1964] created a newtheory of market behavior. One of the keyresults in the process was a similar, although dif-ferently motivated, Two-Fund Theorem. There-fore, we will refer to the Two-Fund Theoremfrom time to time as the Tobin-Sharpe con-struct. As one of the key foundation stones ofmodern portfolio theory, the theorem can besummed up by saying that all desirable, or "effi-cient," portfolios can be constructed out of longand short positions in just two funds: one deliv-ering the riskless rate of return, the other con-sisting of the market portfolio of all risky assets.This construct has become all but synonymouswith the conventional CAPM and is as wellaccepted as any fmancial modelat an intel-lectual level. The caveat is that few if any prac-ticing investment policy strategists use the stricttwo-fund form of the CAPM as a basis fordeveloping actual investment policies. It seems

that no strategist wants his clients' portfolios tohave a substantial exposure to cash. Is the Two-Fund Theorem too simple, too elegant, to beuseful in practice?'

Our intent in this article is to go beyondthe asset-only Tobin-Sharpe construct in orderto develop a capital asset pricing model par-allel to Sharpe's CAPM, but incorporating theaggregate plans that investors have for usingtheir assets, that is, the present value of futureconsumption. This present value is a key por-tion of the "economic liability" on the right-hand side of investors' individual economicbalance sheets.

We use the terms "liability" and "eco-nomic liability" interchangeably, with the gen-erality of meaning allowed to economists, butdenied perhaps to lawyers. Our liability isn'tnecessarily debt enforceable in court, but thepresent value of expected fiiture consumption,or spending. In the context of an individualinvestor, the liability might be the investor'sretirement spending needs through his max-imum possible age, including even his plansfor bequests (neither of which is likely to bea legal obligation). For a defmed benefit pen-sion, the liability is the present value of prob-abilistically weighted expected future benefitpayments, likely to be somewhat higher thanthe mere legal obligation (where the latter isusually measured with the limiting assumption

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that the plan is being immediately terminated with nofurther new accruals).Our approach is that of proof hy contradiction. First,we'll incorporate the liabihty using standard single-periodforms of the CAPM and such standard forms of mean-

variance surplus utility functions as have been in commonuse since Leibowitz [1987] and Sharpe [1990]. From these,we will derive our first effort, a three-fund theorem that isappropriate for investors with a liability and is intendedto be consistent with the conventional CAPM. In thistheory, an investor always holds three funds; a liability-match ing asset portfolio (LMAP), the risk-free asset, andthe risky asset portfolio of the conventional CAPM.In the second section of the article, taking a closerlook at the three-fund theorem, we will show that it isinsufficient to describe a world in w^hich investors hold

assets for multi-horizon future use. If all investors held anLMAP, then all investors could not also hold a complete ver-sion of either the risk-free asset portfolio or the risky assetportfolio. This lack of macro-consistency is unacceptable,revealing an inadequate specification of both the risk-freeasset and the risky asset portfolio in conventional approaches.Accepting the premise that holding investment assetsrepresents a choice by the investor to consume tomorrowrather than today, we will resolve this conflict by placingspecial weight on our earlier observation that all assets inthe market are held to fund consumption deferred over

multiple friture periods. In a macro-consistent w orld, theaggregate risk-free asset must be equal in value and char-acter to the aggregate of all of these deferred consump-tion laddersthe aggregate econotnic liabilityand mustmatch the liability in its key financial characteristics,including its term structure. This aggregate risk-free assetis not the single-period risk-free asset of the conventionalCAPM; it is necessarily multi-period in nature. It mightremove some of the mystery from the discussion to saythat the new risk-free asset is a schedule of paym entsabond having the same aggregatefinancialcharacteristicsas the aggregate liability (sometimes called a superbondto reflect the fact that the payments prio r to m aturity arenot all the sam e).This logic leads us to recast the three-frind theoremas a new two-fund theorem in which the multi-periodLMAP is seen as a more fully general kind of risk-freeasset. Investors thus hold the following two funds: 1) aliability-matching asset sized to the full value of theirliability and thus risk free, and 2) a fund held in add itionto (not instead of) the LAAP, consisting of the world

mark et po rtfolio of risky assets, wh ere the size ofinvestors exposure to the risky asset portfolio is dmined by her risk tolerance. Note that the LMAPsists of a full term structure weighted by each invesuniq ue future spending plan and is inclusive of traditi"cash" versions of the risk-free asset.An intriguing corollary affects the second funthe risk-fi^e rate has a term structure, then the risk miu m must reflect it; the risky asset portfoho can noadequately specified in term s of a single-p eriod riskrate of return. This second fund, the risky asset portf{RAP), is a revised version of the world market porin which the risk premia earned by its holdings are msured in excess of the return on a multi-horizo n riskasset, in this case that of the aggregate market. This premium is quite different from that of the conventiasset-only two-fund theorem and of the capital apricing m odel, and it repairs the inadequacy of the sinperiod model revealed by our parsing of the three-ftheorem.

O ur am bition isn't so much to develop a new ehbriun i theory of the C AP M, however, but rather to practitioners a familiar and tractable intertem poral mfor developing investment policy. Although our viewpresented within the framework of the conventiCA PM , using m ean-variance utility, we believe thembe consistent with those found in m ore sophisticated pricing theories (including the arbitrage pricing theorRoss [1976J and the large family of models inspiredMerton s intertemporal model [ 1973]). This article isintended to present a completely general approacdeveloping strategic asset allocation policy, one thapplicable to all investors individuals, defined be(DB) retirement plans, foundations, and endow ment

The Original Two-Fund TheoremAs noted earher, the original Two-Fund Theo

states all optimal portfolios consist simply of some bination of 1) an asset that is risk free over a single pe(unspecified in theory, but in practice often one yearoccasionally one month or one quarter), and 2) the wmarket portfolio of risky assets.^ In effect, on a graph drin risk-return space, such as in Exhibit 1, the theostates that the ideal and "most efficient" efficient frois on the straight line starting at the sing le-period riskrate (on the vertical, or expected return, axis), extenthrough nd beyond the data point indicating the ejq)e

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1

Return

VTypical ConstrainedAsset-only Efficient Frontier

Expected Risk

or C M L. Althou gh it required explanation originally,

The mechanics of the Two-Fund Theorem are usuallyin the form of a budge t constraint that always has

market portfolio (at point A in Exhibit 1), the investor de-

Alternatively, if the investor desires to increase leverage tove a higher-risk position to th e right of the world por t-folio (at point B in Exhibit l},the investor borrows cash anduses it to purchase more than 100% in risky assets.A more genera l a r t i cu la t ion w ^ou ld be in r i s k -portfolio in cash plus any desired exposure to the r iskpremia of the r isky asset portfolio (no cash needed forr isk premia) .Th e elegance o f this theore m is that th e investor needsto make only one desion: the level of aversion to invest-ment r isk. Once decided, r isk aversion direcdy maps to aparticular point on the ideal efficient frontier, and thus to.in optimal mix of cash and risky assetsa policy port-folio. Strategic asset allocation policy? One decision, anddone. What could be s impler?

The Two-Fund portfolios that He on theCM L also share the property of being comp letelydescribed by their (s ingle-factor) CAPM betarelative to the risky asset portfolio. (An asset port-folio wit h a beta of 0.5, for exam ple, has only halfof its assets invested in the world market port-folio of all risky assets, which we designate as Q.)We m ake that point in Exhibit 1 by showing the" p i e " of asset class weights as being the same ,justlarger or smaller than the investable cash value, asbeta changes. T he "slices," or asset classes (m oreformally, the asset classes can be generalized asfactors for a muld-factor mo del representing po rt-folio Q),each have a constant (market-weigh ted)percentage of Q, regardless of the overall beta.M ost of the efficient fix)ntiers used today bypractitioners are still asset-only frontiers ratherthan surplus frontiers and, as such, do not include the eco-nom ic liability or any othe r liability. Moreo ver, as illustratedby the dotted line in Exhibit 1, constraintssuch as "n ocash or small-cap" and "international no greater than 20% "are nearly always placed on the optimization and thus pre-vent the frontier from taking the basic linear shape of theCML. This reflects the discomfort of strategists with theTwo-Fund Theorem's recommendat ion that a por t ion ofthe po rtfolio b e held in the form of cash; the result of theseconstraints is that the portfolio holds the market portfolioof bonds rather than cash, a more desirable result in their eyes.In the constrained world o f curve d efficient frontiers,the relative proportions of asset classes vary with risk levelfrom th ose in portfolio Q, with con servative portfoliosholding a differently weighted mix than aggressive portfo-lios, a result at odds with the constant market-weightedpercentages found in the Two-Fund Theorem. Given thatmany of the constraints used have li t t le theoreticalgrounding, i t is a surprise that such results are widelyaccepted as being "correct" by many or most s trategis ts .This practice has done much to cloud the understandingofin sdtu o nal and retail investors of investment policy, thus

motivating us to f ind a "bett er" two-fund version of policydevelopment for use by practit ioners .C U R R E N T M E T H O D S FO R I N C O R P O R A T I N GT H E L IA B IL IT Y : S U R P L U S O P T I M I Z A T I O N

The mos t commonly accepted , academical ly sup-porte d, and practical fi-amework for inc orpo rating the lia-bili ty is surplus optitnization. This is a var iant of the

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traditional form of asset-only utility used u nder the CAPMin which the Hability is included as an asset held short. Th eoptimal set of investment choices for any investor that issubject to a liability is determined by optimizing a mean-variance surplus utility flinction,^

Max

U f. is the utility ofthe surplus; R^.^ . is the net orsurplus risk-free rate over the period ofthe single-periodoptimization; ^ is the investors surplus beta exposure;and / /Q is the expected equilibrium risk premium of port-folio Q{\.e. ,^^ = R ^ - R ^ , t h e total return of Q less R ).Q \ variance is (j~. Th e optimal risk level is a function oftlie investor's aversion to risk expressed in terms o fth e vari-able lambda, A.Equation (1) is expressed in a succinct and com-pressed "surplus" form; we can unpack it a bit to aid thereader. We can expand surplus beta, s {A ~l%i.)'which is the net or difference in betas between the assetsand the liabilities, with the liabilities being corrected fortheir size relative to the assets. And we can expand thesurplus risk-free rate, likewise a net o fthe risk-free ratesofthe assets and the liabilities, ^ f,s) ~(^/(.4) ~ 1 7 ^ / ^ ( L I ) -The variables A , L , 5, and Q indicate the assets, liability,surplus, and world risky asset portfolio, respectively,whether used as variables, subscripts, or parenthetical notesto those variables. The subscript 0 indicates beginning-of-period values. Also, although we show the source ofeach risk-free rate element in a parenthetical with A , L ,or 5, as appropriate, the rate itself is, of course, the sameregardless of source.

We can make the following substitutions so as tovisualize surplus utility more completely:

M ax

Equation (2) reflects an optimization problem havingas its objective function the maximization ofthe rate ofreturn, or growth, of the surplus (or, if L is greater than A ,

maxim izing the rate of decrease ofth e deficit), while ctrolling the risk, or variance, ofthat rate of return." "

Why Should Investors IncorporateTheir Economic LiabilityWhen Developing Investment Policy?A plan to spend later rather than now is, after all,reason for holding investments in the first place, andfirst objective of those investments must be to cover tfuture spending plans and perhaps provide the meanincrease them. This is simply deferred consum ption, a clEcon 101 trade-off: People are choosing to consum e goand services tomorrow instead of today. This trade-offkey premise of this article. It is convenient to say investors have an economic liability, whether explici

implicit, representing their planned future consum ptioThe value ofthe liability is the present valuedeferred consumption, or, in mo re specific term s, the sent value of expected future consumption; we will these terms interchangeably. For pension plans, it is the sent value of expected future benefit paymen ts (a broameasure than just the accum ulated benefit obligationABO). For foundations and en dowm ents, it is the prevalue of future spendin g. For individuals, it is the p revalue of expected future spending during the periotheir maximum possible life, a fixed-term annuity (oavailable, the value of a risk-free life annuity).In fact, it is fair to make a statement of this relatship as an important identity: Wealth 15 the present vof deferred future consumption, from the tautology the value of an investors econom ic assets necessarily eqthe present value of future consumption. As Rubens[1976] so nicely put it, wealth is defined in terms oclaims to future consumption that it endows.Don't be misdirected, however; liability-relainvesting is still about how to invest the assets. Thesent value of expected fliture consumption is includedthe optimization, but it comes in a fixed value and wfixed factor, or market risk, characteristicshaving scorrelations to the assets, to be sure. The size and chacter of the liability is not subject to change from optim ization. In fact, because the liability's characterisare "givens,"the liability becomes a special sort of benmark that has to be matched or beaten by the assrevealing the risk and return benefits of hedging liability. Th e liability becom es a central part oft he invment policy construct.

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The Investor's Economic Balance Sheet:Can the Investor Be Over- or Underfunded?In the classic economic balance sheet, assets equal lia-bilities plus owner's equity. For an individual investor, one

can imagine a liability (the present value of deferred co n-sumption) that is smaller than the assets, and thus on e canimagine a concept of "own ers equity" and "surplus" torepresent assets not planned for personal consumption(i.e., overfunded). We can move that "owner's equity"into the liability category, if we treat it as the present valueof an investor's final deferred consumption expenditure,or gifts and bequests to heirs. In this general sense, theassets must equal the liabilities at all times.After all, wh at does it really m ean, economically, foran investor to be in deficit? Or in surplus? An investor

cannot owe to herself mo re, or less, than she has. The con-cept of over- or underlunding requires us to accept thatan investor is dealing with only some ofthe elements ofthat balance sheet and not others (as happens routinely forDB retirement plans).^And we can also imagin e a present value of plannedfliture consumption, again for an individual person, that

is greater than that person's current assetsan aspirational.but not currently funded, plan for iliture spending that,in turn, leaves room to imagine a "deficit." But if we w ereto provide room in the asset column for economic assets,however, such as the present value of expected ftiture sav-ings from labor inco me, then it is difficult to believe thatthis view would be very usefiil, as any deficit fim spendingplans that have a present value greater than available wealthis just so much wishful think ing. For a foundation orendou'ment, these additions would include the presentvalue of expected future contributionsfix)mfiind-raisingactivity; for individuals, the present value of expectedfuture savings from labor income; and for a DB pensionplan, the present value of expected future contributions."Further, imagine segregating the liability side ofthebalance sheet into different buckets representing, forexample: 1) a survival level of consumption, 2) an addi-tional amount needed to maintain the standard of livingto which the investor has become accustomed during hisworking life, 3) an additional am ount that would providefor a luxury level of consumption, and 4) another amoun tfor desirable planned bequests. The investor could declarethat he is in surplus or deficit relative to any or all of theseamounts. But, regardless, any portion of this layer cakethat, at the end of the day, exceeds the economic assets

simply vanishes as so much wishful think ing. So it wou ldadd little to the analysis to consider an investor as beingover- or underfunded. YouVe got what you've got!(An analysis of the investor's potential level of deferredconsumption might, however, be relevant to the investor'srisk aversion, according to Merton [2006a].)Of the common types of investors, only DB pen-sion plans appear at first blush to be capable of beingunder- or overfunded; actually this is only in the benefit-security sense of segregating specific physical assets heldin trust to be set against a portion of the present value ofexpected future benefit paym ents, the accrued p ortion ofthat liability.'' But, in truth, there is much more to theeconom ic balance sheet of pension plans, and they also arealways in balance (Waring [2009]).

For purposes of this article, it is necessary to limitthe discussion, keeping this in its simplest form, and notattempting to optimize the full economic balance sheet.We will assume that individuals have no labor income,and thus no present value of fiature savings or additionalpresent value of future consum ption associated with suchfuture savings, nor will we consider t he h ouse or th eexpected bequest from Un cle Da n.'" Ditto for foundationsand endowinents we'll assume no present value for fliturecontributions or o ther revenue sources, nor spending plansconditional on them. We'll assume that the ABO is thebenefit-security measure o fth e DB plan Uability, avoidingdiscussion of the full economic liability or of the inte-grated balance sheet. Th e path to including these mattersis straightforward, however, and may merit a foUow-uppaper. . . , .

Despite a bias that economic assets must equal eco-nomic liabilities, we hLwe continued to include the cus-tomary inequality term in our mathematics, L^/A,^, forthe convenience of readers who desire to treat DB plansor o ther investors as if they are over- or underfunded, andwe'll show how this affects optim al solution s. Th e effectsare quite different and much simpler than they are oftendescribed.FIRST, A THREE-FUND THEOREMFOR INVESTORS WITH A LIABILITY

Let's start by using a CAPM of conventional, asset-only, single-period derivation, optimizing our portfoliousing the surplus utility variant shown in Equation (2). Andbecause the world is messy, let's use the academics' tech-nique of keeping the function artificially cleaner than it

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really is (for the mom ent) by assuming that the factor modelof the liability and thus also of the liabiHty-matching assetportfoho are both on the capital market line and there areno residuals (returns not linear with beta) from any source.This simplification lets us focus on easily unders tood linearsurplus-optimal solutions, which have the convenient prop-erty that they are all completely describable with the CAPMbeta, nicely consistent with the classic Two-Fund Theorem .

In Part 1 of Appendix A, we derive th e first-orderconditions for this single-factor surplus utility function.The optimal beta solution for an investor with a liabilityis described in Equation (A2). The equation is in hnearform (intercept-slope), with risk aversion, A, as the explana-tory variable and the ideal or oprimal asset beta, ' , as thedependent variable.

(3)Intercepi

The terms in parentheses are constants. The riskaversion term, X, is a variable specific to the investor. Theslope term, ^ y /2 a Q , defines the angle of the line relatingX to the optimal asset beta. An asterisk indicates that agiven value is optimal (in this case for beta, but later inthe article for returns and risks).Liabili ty-Matching Asset PortfolioO ne of the first questions to answer is how to findthe asset beta that provides the lowest surplus-risk levelthat can possibly be achieved. To do this assume that theinvestor's aversion to risk, , is so large that when multi-plied by the slope term the product becomes effectivelyzero, in which case a portion of the equation "goes away"in a numerical sense. In this event, the op timal asset betareduces to its minimum value, the beta of the liability.

We'll call this the liability-matching asset portfolio, orW hen holding this portfolio, the asset betas and theliability betas cancel out. and the optimal surplus beta iszero, ^ = .LM.'iP) ~~^L ~ *-* Because both the risk pre -mium and risk are direct functions of beta, this wouldmean that no surplus risk whatsoever would exist in thisportfolio." Therefore, we can also refer to the liability-matching asset portfolio as the minimum surplus varianceportfolio (MSVP).'-

Holding Equities: The Investor's "RiskyAsset Portfolio"Most investors will have some lesser degree of avsion to investment risk than that just described. Th

investors will want to hold some amount of exposureequities and to oth er elem ents of the risky asset portfoin the hope of trying to increase the amount of futuconsumption that can be supported by their assets. Nohowever, that the liability-matching asset portfolio wstill be present (and will still be, at least notionally, same size as the habihty) no ma tter what value is assigto the risk aversion term, A.So, we can firmly conclude that the fund, ^^j^nmust always be held by every investor with a liabilityin addition, the investor will hold whatever amount

desired of the risky asset portfolio.Thus this investor will always hold threefijnds:1) liability-matching asset portfolio of real interest rate ex psures (for most investors; and for DB pension plans, soexposure to the inflation rate), 2) the single-period risk-frate portfolio, and 3) if the sponsor has a tolerance for risk involved, an additional exposure to the risky asset pofoho of risk premia (i.e., portfolio Q, generating surpbeta, ^ . The second and third "funds" are identical to two funds of the Tobin-Sharpe construct.The information provided in Exhibits 2 and 3 illi

trate the geometry of the optimal solution. In Exhibitthe liability (L) is placed at a random but relatively lrisk position situated directly on a negative capital mahne, which represents the other side of the balance sheTh e liability-mate h ing asset portfolio, LMAP, is plaon the positive, or asset, CM L, located a bit further to right than the liability itself in recognition that many pesion plan sponsors anticipate treating the liability as larthan the assets (the L/A ratio is greater than unity); oerwise, the LMAP would be directly above the liabilitTo generate the surplus expected return equati

for the optimal portfolio in three funds, we substitute solution back into the return components of utility froEquation (2),

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E X H I B I T 2Include the Liability

ExpededReturn

" T O I

^ * ] ^ hin

AssetC M L , . - - '

Expscted Ri(

^..______^ Liability

E X H I B I T 3Three Fund TheoremExpectedReturn * AsBst

-""^ SurplusC ML

E)q)ected Risk

LiabBRy

then decompose it into its LMAP and risky asset port-folio {RAP) components, the latter being its beta expo-sure tof t . K, , .,

(5)In Exhibit 3 we complete the picture. The firstpoint to identify for a surplus investment policy is the

single-period net risk-free rate for the surplus,^fis) = i^f{A)~^fiL))-This rate is slightly negative asis expected for a slightly underfunded DB pension plan.Any underfunding means that the sponsor is payingimplied interest on the shortfall (which would be a zeroterm under our general assumption that A = L).Next, the beta of the liability-matching asset port-folio and the liability beta cancel each other out,s(Lu.iP) = A(LVMP) ~ "^ ft ) =. SO that there is no surplbeta exposureneither return nor riskfrom this source.This "surplus LMAP" can then be plotted at the origin.

The span of choices for the remainder, the nonzeroportion of optimal surplus beta representing the risky assetportfolio, .nn^py forms a capital market line as expected.But because the return to beta is only a risk premiumrather than a total return, the total surplus return and riskposition is in the new position anchored to the net risk-free rate and is shown as the "Surplus CML" described byEquation (5). This is the surplus efficient frontier that setsout the true strategic asset allocation policy opportunityset for an investor (one having a liability on the CML),and it is on this line that the total return surplus optimalsolution, *, will sit.

Once the liability has been matchedthe first fundis keysurplus optimization reduces to be just like anasset-only problem in the sense that the decision abouthow much of the risky asset portfolio and how much ofthe risk-free asset to hold is made without further regardto liability considerations. In other words, with the lia-bility matched, the balance of the solution is just as in theTwo-Fund Theorem. . , *

Making This More General: A LiabilityNot on the CMLAt the beginning of this section we made the unre-alistic assumption that the liability portfolio was on theCMLwhich is only possible if all of its factor-beta expo-sures, or asset class factors, are proportional to those ofthe risky asset portfolio Q (i.e., if we can describe it accu-rately with a single factor-beta). Itwould in fact be rare forthis to happen. Liability portfolios are random relative to effi-cient portfolios and are not expected to be on the CML.For ourfirstapproach, it was sufficient to describe therisky asset portfolio with the traditional single factor-beta.But to adequately allow inefficient ttitcrior portfolios, weneed to describe the portfolio and the liability in a moregranular, multiple-asset-class or multi-factor model. F will

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be a unique vector, or list, of asset class weights or factors(more precisely, this approach will work with a set of m ul-tiple factor-betas fi-om any market risk model, of whichasset class weights are just one very basic example).We also need to establish notation for describing the

time dimension of the liability, and later of the assets, toincorporate the spot curve of real yields and flitiire cashflows across the total period of planned consumption. Wecould set up mathematical machinery to represent anoverlay of these longitudinal periodic descriptors of the lia-bility, on top of the cross-sectional asset class weights, butthis notational Rubik's Cube is unnecessary because wecan accomplish much the same purpose more simply.Rather than detailing the cash flows period by period,they can instead be satisfactorily described in terms of cer-tain sunmiary characteristics, their real interest rate dura-tion (and for DB plans, their inflation duration), a form thatcompresses a great deal of information ix)m he un derlyingcash lows nto just those two values. Th e use of durationsis an especially convenient shortcut for our purpose becausedurations can equivalently and more generally be thoughtof as real interest-rate and inflation factor-betas. Durationshave associated returns that can stand in for yields and, inthat sense, are not unlike our asset-class-weight vector andmay, in fact, be included as elements within it.'"*

For individuals, foundations, and endowments, theliability is most simply modeled solely in real interest ratetermsa complex inflation-protected coupon super-bondbecause the key goal of these funds is to protectconsumption power (similar to Rubenstein [1976]). Again,this is a pattern of deferred consumption that is alsodescribable in summ ary form in terms of factor-beta expo -suresin this case a Fj^ vector consisting of only one ele-ment, the real interest rate duration of the deferredconsumption ladder. (An equivalently simple model of aDB pension liability would include both the real interestrate and inflation durations of the liability as the onlynonzero elements of Fj^.)'^

More detailed models of pension liabilities thanthis are possible, of course, and might pick up usefulfactor-betas for equ ity risks, yield curve tw ist, con -vexity, and other exposures, but real interest rates andoccasionally inflation rates appear to have most of theexplanatory power; further, real and nominal risk-freebonds have the valuable property that prices moveinversely to yield, thus protecting consumption powerwhen held.

W ith this setup, we can show how a flexible mulasset-class ormulti-factor optimization of a liabilithat is not on the CML varies in its optimal solutifirom the outcom e of Equation (3). The optimal soluvector (exclusive of the single-period risk-free asset)described in Part 11 of Appendix A, Equation (A5), repeahere,

(Th e ljMiy'.\1iililiiiif;

Asset th e Riitv Atstt

F^ is the optimal vector of asset class exposures, V ^ is tinverse of the variance-covariance matrix of the assclasses, r is the vector describing the exp ected returnseach of these factors; and T is the transpose operator.

Notice that this solution is completely parallelform toEquation (3) other than being expressed inmulti-factor vector-matrix form inplace of the singfactor-beta form. This means that the interpretationalso parallel to that of the more constrained version; tintercept term is still readily interpretable as the liabilitmatching asset portfolio (and as the minimum surplvariance portfolio) because it is the only term that remawhen the investor's risk aversion is very high.And when risk aversion is not so high that the firterm becomes a zero term, the investor holds (in additito the liability-matching asset portfolio) an amount exposure to a term (in parentheses) that is clearly tvector-matrix analog to /iXo". term of Equation (3the single factor-beta exposure of the inve stors risky asportfolio, /,,(K,4.,FQ.

To generate the optimal surplus return for an inveswith this unconstrained liability, we again substitute osolution back into the return com ponen ts of utihty 6x)Equation (A3) and again decompose it into its LMAPznrisky asset portfolio {RAP) components.

F. + V ' rA.. "- 21 ^ F .

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R.-,.,+ F. A,Tile SuipUa Rnk-FiTT Ru

Equation (8) shows the solution in three funds,parallel to Equation (5): In addition to holding thesingle-period risk-free surplus asset, it is ahmys optimalto hold the liability-matching asset portfolio, regardless ofwhether the liability is on the C M L . And it is always optionalto hold an additional risky asset portfolio exposed tothe rest of the markets. So an investor holding a lia-bility needs, at minimum, a liability-matching assetportfolio, but may also want a portfolio containing riskyassets.

Exhibit 4 illustrates the nature of this more realisticliability-matching asset portfolio. Because the liabilitymode! for an investor, F^ , is an interior portfolio consistingof, say, only TIPS of a 15-year duration (representing thefiiU vector of spending plan cashflows), the optimal lia-bility-matching asset portfolio is also an interior portfolio,(L,,M,,)Fj^, consisting simply of the same exposure, per-haps levered up or down in duration to reflect a belief thatthe assets aren't as large or as small as the liability.

In fact, because no decisionreally needs to be made aboutholding the liability-matchingasset portfolio (the LMAPi% inde-pendent of risk aversion or otherpreferences'*'), the only strategicasset allocation policy decision thatthe investor needs to consider ishow aggressive or conservative tomake surplus beta. In other words,how averse is the investor to riskwhat is his or her lambda?andthus how large or small is the riskyasset portfolio?

Thus, we return to the verysame interesting generality derivedfrom the Tobin-Sharpe construct:Once a liability-matching assetportfolio is held, the rest ofthestrategic asset allocation pohcydecision is made just as it is for anasset-only policy under the classicTwo-Fund Theorem.

E X H I B I T 4Three-Fund TheoremExpectedReturn

THE THREE-FUND THEOREM LEADSTO A NEW VERSION OF THE TWO-FUNDTHEOREM AND THE CAPM

Our three-fund theorem is as good as we can makeit using a conventional CAPM as a base, and atfirstblush,seems eminendy workable and suitable. In fact, it would notdetract from other common, current practices if investorswere to use it as a primary tool of policy development: Itwould substantially improve policy solutions simply by clar-ifying the purpose, nature, size,and role of the LMAP inhedging consumption and spending power; for this reason,we arewilling to encourage its use by practitioners.

But let's take a closer look and decide whether wecan improve it irther. There is a problemif all investorswere to hold an LMAP, then all investors could not alsosimultaneously hold the conventional risky asset portfolio(in the aggregate), because, as the risky asset portfolio isconventionally defined, a portion of the latter portfolio(real bonds at a minimum) would already be held in theinvestors' LMAP^.The sum of all investments can be nogreater than the market or we face a double-countinginconsistency. ..,.

This problem constitutes a critique, but not of anyattempt to incorporate the liability into the CAPM.

with a Liability That Has Residual Beta Exposures

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Ra t he r , it is a cri t ique ofthe underlying specificat ion ofthe risk-free asset and of th e risky asset portfolio. To resolvethis conflict we have to move from a single-period focusto a mul t i -per iod focus and reconsider what we meanw h e n we refer to risk-free assets.The Case for Revisiting theNatureof the Risk-Free AssetThe academy has put much effort over many yearsinto establishing a richly varied body of equilibrium theoryfor risky portfol io and asset prices, but relatively littleeffort into establishing an equi l ibr ium theory for risklessassets and in tegra t ing such theory into more genera lpricing theory. This lapse of attention isn't trivial becauseit is difficultif not impossibleto correcdy develop the

theory for risky asset pricing if the theory of riskless assetsthat underbes it is weak.In his famous CAPM paper, Sharpe [1964] simplyreferred to the "pure ra te of interest ," w ith ou t fu rtherexplanat ion or in-depth discussion, as the rate of returnon the risk-free asset. But it isn't really fair to single outSharpe because his t r e a t me nt of the risk-free asset isc o m m o n to all the writers in that formative era, inc ludingthose who are often credited, along with Sharpe, as havingco-equal ly developed the capital asset pricing model. Forexample,Treynor [1962, l999],Lin tner [1965],and Mossin

[1966] each did something similarly perfunctory; Mossin'squa int turn-of-phrase was that the pure rate of interest isthe "pr ice for wait ing " as opposed to the "pr ice of risk."A decade later, Ro ss '[19 76 ] arbi t rage pricing the ory useda s ingle-per iod r i sk-free ra te , and Me r t on ' s [1973]inte r tempora l model incorpora ted an instantaneous risk-free asset for determining risk premia. With few e xc e p-tions, virtually all papers diat discussed variations of assetpricing models included some variat ion of the same thing,to this date.

Yet as early as 196 6,M odig liani and Sutch were artic-ulating a view that has been widely accepted as far as it goes,but has yet to be fully reconciled with the CAPMnamely,th e preferred habitat theory of interest rates, which is in fact theseed for a more complete theory of risk-free assets. Thistheory suggests that investors choose to own bonds withcash flows that have the same t iming as their deferredspending needs in order to reduce or eliminate risk. To para-phrase Modighani andSutch [1966], a 10-year bond is risk-less to an investor with anobligation 10 years out, but holdingcash is very risky. Since then, this sentiment has been fre-

quently echoed. Campbell and Viceira [2002] emphasizthe same theory in their compact and wonderflilly writtebo ok , asserting that "[ajt a theoretical level, these points habeen understood for many years" (p. 6). Yet, it remains neauniversal to use some arbitrarily sho rt "ca sh" rate as the risfree rate wh en discussing various forms oft he conven tionsingle-period CA PM , and it seems to be completely univewhen specifying risk premia. The situation is some whimproved in the intertemporal CAPM literature, as Merto[1975, 1977] descrihed his hedging portfolio to includemult i -horizon Modigl iani-Sutch-inspired component , bdid not treat it as part of th e risk-free rate. Ditto for Rosarbitrage pricing theoryit can include a mult i-period pofolio of bonds , but doesn' t re-examine the risk-free raTh e views of Mo digliani andSutch [1966] and the LM/Ptime dimen sion suggest a path for developing a better theoof risk-free assets and risk premia, and for resolving thproblem shown in the three-fund theorem.From the perspec t ive of each asset owner, theinvestments exist simply for the primary, and perhaps sopurpose of prov i d i ng for, or "funding," the i r deferrfuture consumpt ionor the inve stors "liabilities." An dthis is t rue for each investor, it is also true for the overmarket , which is simply the aggregate across all investoIn Appendix B,w e use such an agrgate approach to deria simple mod el for the pricing of risk relative to this aggrgate Uability, an equilibrium basis for a variant ofthe CA PM

Pricing Risk Considering Investors'Aggregate Balance SheetEquation (B5) in Appendix B describes optimal assclass or beta-factor holdings for the ith individual investoportfol io (subscript / indexes unique investors,and m reresents the aggregate market).

LVHP RA P(

Simplified, the investor should optimally hold a pofolio o f factor-betas (asset classes) that m atch es the uni qliability ofthe investor, the LV i^P ,plus a r isky asset pofolio of re turns in excess ofthe aggregate l iabi l i ty pofolio. In parallel, for the market as a w h o l e , ' (1

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The single-factor risky asset beta of the market,m' '^ ge'^c'"aUy taken as unity by construction, andso is omitted.

These portfolios are similar to, yet importan tly dif-ferent from, those implied by traditional CAPM models.The parenthetical term redefmes the risky asset portfoliofrom its classic form. Q. which is the portfoho of all assetsless Rr. In contrast, the new risky asset portfolio consistsof all assets excluding the aggregate liability-matchingasset portfolio or, in other w ords, excluding both R .an dthe longer-duration hability-matching assets taken in theaggregatethe latter having traditionally been consid-ered part of the RAP.

To avoid confusion with Q or F Q , we will refer tothis new market portfolio of risky assets as Q', or moreformally in terms of its asset class risk-premium expo-sures, F' = F. F, .' In this way, we conside r risk-free real bonds of all horizons (and of appropriate weightsfor each investor, asset, or the market) to be the buildingblocks of a multi-ho rizon risk-free asset for each investorand for the market. Our redefined risky asset portfohois still priced on the basis of its market risk as in con-ventional theory, but only o n that portion of those risksthat isn't hedging the liability side of investors' aggre-gate balance sheet. This statement constitutes a new inter-pretation of the risky asset portfolio and the riskpremium. '

Expected ReturnsWe can again substitute these optimal holdings intothe return components of our surplus utility function.Equation (A3), to get the op timal surplus return of thisnew form of liability-sensitive investor.

A (J lM P ).i Q L

(11)

The Suiphu Rnk-FRc RU F TI K S u p b i U4 I>

We multiply through by A^ ^., changing the expres-sion from surplus return to the change in surplus wealth,and then group and identify the optimal asset-return andliability-return compo nents, as follows:

A R* ^0,r S j 4 - F ' ' F' r 1

(13)

where r is the vector of expected returns in excess of asingle factor R. from the original CAPM across all assetclass or factor-betas (i.e., inclusive of both risk and horizonpremia).The expected optimal return of the asset portfolio,in isolation, in this liability-relative form of the CA PM is

K., = ^y(.-i) +^I(OMP),,'Q + A'^Q^Q- This is the sum ofthe returns for the three funds we have discussedthetraditional risk-free rate, the LA MP retu rn, and the riskyasset portfoho (RAP) return. But the risk premium of therisky asset portfolio is now explicitly redefined in termsof our m ulti-horizon risk premium, which is now definedSo we have redefined the three-fund theorem in amanner that is macro-consistent and workable. Is there away to make this more parallel to the original Two-Fund

Theorem? . _ . , ,- Is It Really Three "Funds" or Just Two?

Combining R. and the L MAPinto a Single Fund - Because of the way we developed this model, startingwith the notation and forms of the traditional single-periodCAPM, the single-period risk-free rate was taken out ofthe LMAP, which is expressed as a risk premium (orhorizon premium). But if we agree that the two together

constitute the complete risk-free asset on the basis that theLMAP at least the real interest rate portionsis requireduniversally by all investors, then the total return for a par-ticular investor is their sum.I? 4- P r (14)

(12)

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To parallel traditional notation, we can definejR,-. a version of the risk-free rate that inco r-porates the m ulti-h orizo n needs of the investor. Similarly,we can define the risk premium of the risky asset port-folio not as / HQ , but as fi'^ = f o'o - W ith this notation, wecan express the optimal a.sset portfolio return extractedfi-oin Equation (13) in scalar algebra, as a CAPM variant,in clear parallel to the form of the classic CAPM,

(15)Not only does this portfolio lie on the CML, hut,by inspection of Equation (15), can be seen to lie also onthe security market tine (SML) discovered by Sharpe [1964](if graphed inbeta, , versus expected return space, we

would see a linear relationship between beta risk andexpected return). We complete this as anasset pricingmodel by noting that if we have proven this latter pointfor the optimal asset portfolio, then we have proven it forall other portfolios and assets. For this conclusion we needonly refer back to tlie same argument originally developedby Sharpe, the key to the CAPM: Like portfolios, assetsmust all lie on the SML and must also have expectedreturns proportional to their betas. Here, because thereare as many risk-free rates as there are investors, there areas many CMLs and SMLs as there are investors and assets,but they all work with the same RAP.While the asset portfolio must carry risk not onlyfix>m the RAP but also from the LMAP, the LMAP riskdisappears when matched to the liability. Thus, the totalsurplus risk if the investor holds this optimal asset port-folio would simply be

' Z 'A(RAF)j F'V' O ' 'A{RAP)j^Q (16)What Will be the Experience of Our Investor?If an investor with a long future of annual consump-tion flows followed this advice (an individual planning forretirement, for example), and actually established an LA.4Pthat corresponded to the consumption plan "liability" (per-haps a ladder of zero-coupon TIPS), with no iM P , then theinvestor's consumption would remain absolutely level in realterms and at the same level as when the LMAP was estab-lished, a result of the unique property of these bonds thatprices move perfectly inversely to returns . The derivative of

both the rate of consumption and of the present value oexpected future consumption, with respect to changesthe real interest rate curve, is of necessity zero; that is. if reinterest rates decreased, increasing the liability, the LMAFvalue would also increase. The new higher value of thLMAP would support the identical periodic annuitpayment for consumption as did the prior smaller LMAwith the higher real rate.But if the investor also held a portfolio of risky assetconsumption would increase or decrease directly with threturns on that portfolio; the derivative of consumptio(the relative proportion of wealth spent ineach futperiod held constant) with respect tochanges inporfolio value from risky asset returns is one.

Discussion and ImplicationsIn this section, we discuss critical elements of thTwo-Fund Theorem and noteworthy imphcations of iapplication.Relationship to Full Intertemporal AssetPricing ModelsA large literature exists that describes other, morsophisticated intertemporal capital asset pricing modelIn this literature it is c o mmo n for the results to

expressed as two-fund, three-fiand, or even -fund theorem s. Th e flagship of this literature is Merton's [1973article, but there are many more articles of importancby Merton and others.'" The models that are presentein these articles have several strong v irtues, whi ch havmade them the exclusive focus of a numbe r of academiworking today. These advantages include, among othertheir ability touse native utility theory inplace of thmean-variance approximation, their ability to explicitlprovide foran investor's flow ot future consumptioneeds, and their capability of dealing with an investorneed to hedge not only real interest rate risk over timbut other types of unique individual intertemporal riskas well. Unfortunately, these models have the disadvantages of being difficult, or in many cases impossible,solve when applied to practical real-world problems anof being constructed in the somewhat-advanced mathematics of stochastic dynamic calculus which is un fotunately impenetrable to most practitioners. Thus, thesmodels have had little practical application despite theflexibility, elegance, and beauty.

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The asset-liability version of the CAPM presented inthis article is for all intents and purposes a simplified versiono such an interteinporal model, one that combines manyof the best features of the highly tractable, single-period,mean-variance, asset-only world of most practitioners today,and the more heady world of the M erton style of intertem-poral model that uses utility theory and continuous-timemathematics.By remaining in a mean-variance framework,it is fully transparent and understandable by many, if notmost, practitioners and is readily usable. If the single mostimportant lesson of the full intertemporai model is that theinvestor can hedge future consumption over dme, then thismodel also accomplishes that task. If additional importantlessons address hedging other more unique risks faced by aninvestor (energy price risk for a utility company or for anairline, for example), then by extending this m odel to includethe remainder of the organization's economic balance sheetbeyond the investment assets already on hand, similar resultscan also be developed. ' .

What this model can't do is to reflect the finer dif-ferences in the optimal portfolio that might come fromuse of the mo re sophisdcated utility and from the smo othevaluation of continuou s-time mathematics. We don't haveany parallel to th e Merton approach's ability to hold secu-rities that hedge "wealth productivity shocks" or certainother risk exposures that may change the investment o ppor-tunity set; see Campbell and Viceira [2006] or consider theterm structure of the risk-return trade-off (long-horizonserial correlation) shown by CampbeU and Viceira [2005].

Are these shortfalls important? Yes, butin eithercase, the prima ry LMAP portfolio will be essentially thesame and the primary RAP portfolio will be the worldmarket portfolio of risky assets (ex-LAAP assets). Otherpotential hedges will be missing, but in most cases theyare relatively smaller in importance. It is hard not to con-clude that the misnamed Pareto s Law, the 80/20 rule, isfully satisfied.

Guidance for a Sensible Spending Rule >The discussion in this article strongly implies thatwe determine our spending rule either with an explicitannuity (see Endnote 11) or by replicating an annuity-like paym ent stream. As a bonu s, a replicated annuitygives the investor the option to hold a personal LMAPand to hold some risky assets. A replicated annuity assuresthat funds will last a lifetime by avoiding the risk of over-consumpDon during periods of low returns. T he replicated

annuity can be adjusted to front- or back-end loadingon c onsum ption, providing allowances for high late-lifemedical costs, a preference to spend more when rela-tively young than when relatively old, and other timepreferences.Th e key idea is that in each period the amount with -drawn from assets and consumed would not exceed theamount that would have been paid out and spent in thesame period if, at the beginning of the period, the annuityholder had purchased a fairly priced level-payment (orany other desired shape) real annuity for her maximumpossible remaining lifespan (for constant consumption,this provides about 3.0% of the portfolio value each yearfor a 40-year future when the real interest rate is 1%, andabout 3.7% when the real rate is 2%). At the beginningof each successive period, the amounts of the payout and

the consumption would be recomputed, given the then-available value of the asset portfoho after returns, whethergood or bad.If the investor holds as part of his portfolio an LMAPportfolio of real rate bonds that h edges the spending plan,the rate o consu mp tion will be prote cted, as was Justdescribed. If the investor also holds a risky asset portfolio,then consum ption can be ratcheted up (or ratcheted dow n)proportionally to the remrn of the risky asset portfolio eachperiod.

Nonobvious Sources of Long-HorizonRisk-Free ReturnsAll asset returns, including risky assets, include anunderlying term structure of mu ltiple-ho rizon risk-freerates, R'.,embedded within their total return. Presumablythese are weighted as a function of the size and timing oftheir expected future dividends (or other payouts toinvestors). This means that equitiesand all other riskyassetshave a multi-period horizon (that can also be sum-marized as duration) and that they are able to supply a por-tion of the LAi>lPportfolio Fj^. to the investor.''' Becauseof the presence of these embedded long -horizo n real risk-less assets within all assets, including all equities and otherrisky assets, investors probably do have holdings that moreclosely approximate their LMAPs. than we might other-wise think. To the exten t that they do, this new two-fundtheorem is a positive theory, not just a normative one.

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An Indifference Proposition: The PresentValue of Euture Consumption Does notChange with Investment PolicyWe have used a m ulti-period risk-free discount rate

on the liability side ofthe balance sheet to determine thepresent value ofthe investors expected future consump-tion. What if the investor is invested on the asset side ina portfolio including not just an LMAP, but also riskyassets-should the discount rate for the liability alsochange?

Well, yes, and in fact it is axiomatic that both sidesofthe balance sheet, when properly displayed, should havethe same discount rate. But changmg the discount rateon the liability side would make no difference to eitherthe present value ofthe hability or to the amount oftheinvestors wealth today: if the investor is invested in riskyassets, for the same initial portfolio value, the expectedlevel of future consumption will be higher than for aninvestor holding only the LMAP. But that higher expectedlevel of future consumption on the liability side wouldbe discounted back to present value using a discount rateincluding the risk premium from the risky assets on theoth er side, a higher discount ratebecause it is, in fact,riskier. Th e b otto m line is that the present value of fritureconsumption would be the same regardless of investmentpolicy.

What this means is that holdings of risky assets,^A(RApjM might make an investor more wealthy and thusable to spend more, and will certainly do so if realizedreturns are close to, or above, their expected value. But thepresence of risk in the risky asset portfolio means thatreturns of the assets may well be less than those of theLMAP alone and, in such a case, the realized level of futurewealth, and thus of future co nsum ption , will be lower, no thigher, than if the investor had invested solely in theLMAP. Assuming that risk is fairly priced, the odds areexactly fair that the more aggressive policy will result inthe investor having a smaller pool of assets from which tofund consumption as the investor having a larger pool.Taking on more risk does not create wealth in and ofitself. Wealth creation (or destruction) depends on uncer-tain future realizations, not current expectations.

It is convenient to start investment planning byassuming that the investor wants his deferred consump-tion to be secure, which calls for use ofth e risk-fi^e rate.But if the investor is willing to accept some level of un cer-tainty, then he can add risky assets to th e p ortfolio, which

increases the return and risk to both sides ofthe balancs h e e t . Regardless, the present mlue of assets mtist equal the prsent value of liabilities and that value uHll not change with invement policy.

So, contrary to common belief among investmeprofessionals, holding a riskier portfolio with a greatexpected return, rather than a more conservative porfolio, does not assure greater friture consum ption. Neithwill a riskier portfoho assure the make-up of a shortfbetween assets and a higher aspirational liability (for indviduals or for e ndow me nts), nor a shortfall betwe en assand a DB retirement plan's liability.

Honoring the Traditional Budget Constraint100% Investment and No MoreThere is an apparent difference worth discussibetween the two-frind theorem and the theorems of Tob[1958] and Sharpe 11964]: the risk-free asset, or LMAis held in th e full value of the assets, and the risky asportfolio, which consists solely of risk premia, is held ovand above the LMAP. This observation has caused somwith whom we have discussed this article to be cocerned that the optimal portfolio must sum to greatthan 100%, a perhaps inappropriate degree of leverageThe reconciliation lies in keeping the amouinvested in the risk-free asset separate from the amou

invested in risky assets. The frrst generates total returnbut the latter only generates risk premia. A portfolio 100% cash is no more or less fully invested than a pofolio of 100% equitiesbut the equities implicidy reprsent 100% cash plus 100% in the equity risk premiuView ed in this way, this solut ion is no different in leverage characteristics than those of Tobin and Sharpand the portfolio is not more than 100% invested (sprevious discussion).CONCLUSION: IMPLICATIONSOF THE NEW TW O-FUND THEOREM

In this article, we have incorporated the Uabilside of the investor's balance sheetthe multi-peri"eco nom ic hab ility" or present value of deferred futuconsumptioninto our derivation o a new form fthe CA PM . It turns out that a single-period form oftrisk-free rate is simply not compatible with the muperiod form of most expected friture consumption. Theliabilities are the reason for which such investmen

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are held, and therefore all investments represent futureconsumption possibihties, deferred in a multi-periodcontext.

In a macro-consistent world, the agrgate risk-free asset, that is, the aggregate liability-matching assetportfolio, must be equal in value and character to theaggregate economic liability or deferred consumptionplan, and must match the hability in its key financial(beta-factor) characteristics. Such a risk-free asset mustof necessity have multiple horizons over timea termstructurerather than a sin^e-period short horizonofthe risk-free asset ofthe conventional CAPM. Andif the risk-free asset must have a term structure, thenthe risk premium must reflect it.

This gives us a useful iteration ofthe CAPM, incor-porating an old theory of risk-free returns, that ofModigliani and Sutch [19661. Our two flinds are no longerthe same as the two flmds of Tobin and Sharpe, in that ourspecification of both the risk-fi-ee asset and of the riskpremium is now multi-period to match our specificationof investors' deferred consumption plans. And our LMAPportfolio of consumption-hedging, but otherwise risky,bonds is clearly priced on a different basis than are other"risky" assets.

This form ofthe two-fund theorem will be muchmore comfortable than the original Two-Fund Theoremto those strategists who never could bring themselves todivide the portfolio between single-period cash and riskyassets, a solution that while elegant never seemed to fitthe real world of investor needs. A risk-free asset con-sisting of an array of liability-cash-flow-matching risk-free real bonds will be much more intuitively acceptable.Strategists can comfortably return to a consistent usage ofthe world market portfolio of risky assets, using it as thebase method of exposing portfolios to the market acrossasset classes, regardless of risk tolerance.

Tomorrow's strategic asset allocation policies needto be much better than today s, for all investors. Today, wespend a great deal of time on distractions, such as justi-fying policy allocations to hedge funds and "exotic assetclasses," when there is an investment strategy elephant inthe room that we are only slowly beginning to acknowl-edge. That elephant, the economic liability, needs to beattended to. If attention is not paid to it, investors of alltypes are focusing on the wrong issues and carrying excessand uncompensated risks. Surplus asset allocation in thecontext of a multi-period risk-free rate is the tool ofchoice for doing so.

Granted, this multi-period CAPM may be more dif-ficult to parameterize than the original. A multi-periodrisk-fi-ee rate forces us to deal with the term structure, orat minimum, the duration, of our client investors and ofthe market as a whole, perhaps a more difficult task thanin a single-period context. Planned future consumptionfor investors may also be difficult to estimate, but rela-tively level future spending plans will likely achieve accep-tance as the default. In both cases, it may turn out that itis sufficient to simply estimate the real rate duration offuture spending, rather than the exact term structure, aswe do in this article.

A more complete solution will include more off-book economic assets, such as labor income, human cap-ital, current consumption, and future savings, forindividuals' retirement planning, and future gifts and con-tributions, as well as the economic value of assets, for pen-sions, foundations, and endowments. Of course, all thecriticisms and weaknesses that have been asserted againstthe CAPM apply equally to this form of that approach.The fact is, however, that all capital asset pricing models,including the sophisticated intertemporal models, can ben-efit from including the present value of a liability expressedover time and tied in value to the available assets, as wellas from writing budget constraints more carefully so asonly to tie the cash position to investment values, not riskpremia. Doing so will give these models an improvedtheory ofthe role ofthe risk-firee asset and ofthe struc-ture of the risk premium. :

A practically implenientable CAPM needs to be assimple as possible/>f(f fiot one hit simpler. It is too simpwhen based on single-period cash without regard to theliabilities of investors. With the bit of complexity we haveadded, we hope that investors will conclude th;it this two-fund theorem is intuitive and practical, and thereforeusable, improving strategic asset allocation practice.

A F P E N D I X ADeriving the Three-Fund Theoremfor Surplus UtilityPart I: Taking the utility function from Equation (2) inthe text, vre evaluate its first derivative with respect to the single-factor asset beta. This tells us how utility changes when beta

changes.

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Su I (Al)Setting this equal to zero, and m anipulating, we learn thatthe solution for the u tility-maxim izing, or optimal beta, ^ . in

the presence of a liability is(A2)

rearrange to get a result that is very much parallel to Eqtion (A2),

(A

A P P E N D I X BPart II: When discussing liability models that are no t onthe capital market line, we have to discuss asset class weights(factor-beta exposures) that are weighted differently than marketweights (i.e., they are not propo rtional to portfolio Q).Most liability models, being largely or solely combinations

of long risk-free real and sometimes nominal bonds, have assetclass weights (mixes of beta factors) that are not at ail propor-tiona l to those in Q. In such a case, both the liability mod el andthe LMAP will be off the CML and below it. In preparation,we can re-express Equation (2) in vector-matrix form, whichallows us to move from the single-factor CAPM of Sharpe1964] to a simple multi-asset-class (or multi-factor) model, sothat the asset class weights can vary from those in Q,Max F. ^1* A.

(A3)

Our asset class vectors, F, are row vectors of weights forthe various asset classes or oth er factor-beta exposures; all othervec ton are colum nar. Specifically, F Q is the unique vector of assetclass weights composing the market portfolio Q, and F^^ andFj^ are the weights of the asset portfolio and of the liability. r_is the set of equilibrium expected returns for these asset classesand V Q is their variance-covariance matrix.To get the optimal solution, we take the first derivativeof surplus utility. Equation (A3), with respect to the vector ofasset classes.(A4)

An d to solve for the optimal vector of asset class expo -sures, F^, we set the right side of the equation equal to zero,multiply through by the inverse of the covariance m atrix, and

A New Form of the Two-Fund TheoremWe know the optimal holdings for an individual invesfrom Equation (A5).But a single investor does not describe twhole market. The market, m, is the summation of nil of theNoptimal individual portfolios, including both therisky asset portfolio portions (i.e., across Equation (A5)),

(BBecause the market is simply the sum of all investorsit, we can move the first portion of the term in parenthesoutside of the summ ation simply by replacing X with the riaversion of the market, A^ , and we can drop the asterisk FA. as the m arket portfolio does not need a reminder of its opniality. And the aggregate across all investors' lA M ft , the seco

term, is th e market liability-matching asset portfolio, Fj^ ^^liability/asset ratio for the aggregate is unity, as every asssomeone else's liability. This gives us the market portfoldescribed in two componentsa risky asset component andI lability-matching com pone nt,Q Q ' (B

The reason for aggregating optimal investment stratgies across all investors is so that we can determine the m arkclearing price of risk. We can solve for it by rearrangiEquation (B2) as follows:

(BWe substitute Equation (B3) into Equation (A5) to gemodel of the optimal individual portfolio, but now using marequilibrium-risk pricing,

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p" =Aj X'^ f F - F5 \ A L {B4)

The ratio of che market and individual lambdas is theinvestor's single factor-beta for the risky asset portfolio. There-fore, we can make this look more familiar with a form parallelto the classic CAPM, but with a revised risk-free asset.

F : = F : . +, (B5)

ENDNOTES . , _, "'See http://www.quotationspage.com/quote/2927.html,

accessed on February 27,2008.would like to thank Laurence Siegel for his advice,comments, and suggestions, a.s well as Mary Brennan for her

editing. And a special thanks to Paul Kaplan for not only sug-gesting that we challenge the notion that the liability-matchingportfoho can be a third fund (after he reviewed an earlier ver-sion in which only the three-fiind theorem was discussed),butfor also suggesting later the important transformation in Equa-tions (Ii3) and (B4), which is key to making this a fully gen-eral pricing model. . - .

^In this reference to the "original" Two-Fund Theorem(which we'll capitalize), we're including the generally acceptedbody of knowledge that fleshed it out over the years. Forexample, Sharpe characterized the risky asset portfolio as con-sisting only of equities, but Roll [1977] showed why this port-folio should consist of all risky assets; we include such generallyaccepted incremental refinements.

"it relies on an assumption (among others) that the amountof cash in the portfolio is unconstrained; that is, cash can beborrowed or lent at its risk-free rate in any amount the investorchooses.

^For more detailed discussions of the derivation of sur-plus-optimization utility functions, and to see alternative for-mulations, see Waring |2()()4a, 2004b]. In general, these articlesexpanded on the basic premise of surplus optimization, inte-grating several ideas and derivations originally articulated sep-arately by others (cited there, but particularly Leibowitz [1987]and Sharpe [1990]), and showed how to put the basic theoret-ical idea of surplus optimization into a practical and usable form.It described a total return (beta plus alpha) surplus efficientfrontier using an objectively descrihable measure of the pen-sion liability. For other approaches, see Campbell and Viceira[2002], who provided a summary of the literature of invest-ment strategy technology, quite complete up to its date, withan emphasis on power utility of consumption (in both discreteand continuous time).

^By convention we generally refer to A - Lus the "sur-plus" even if it is a negative number (a deficit).

^To make an analogy to a pension plan,an individuals lia-bility is ahifctys "fiilly funded" (i.e., with an L/A ratio of unity).Unfortunately, this happy state is not all that it might seeni, as theimplied benefit level is all too often too low to be comfortable.

*See Waring [2()04a] for a discussion of U.S. equity realinterest rate and inflation durations and of how one can incor-porate them into the surplus-optimization policy developmenteffort.

^Today the benefit security measure of the liability is gen-erally a conventionally measured ABO, a very unsatisfactorymeasure for that purpose. See Waring [2009].

'"This doesn't merely restrict use to individuals who areretired or otherwise have no labor income, as it is equivalent toan assumption that the present value of future consumption asso-ciated with labor income is perfectly correlated with the presentvalue of future savings from labor income (and definitionallyequal). Thus, there is a hedge with respect to the surplus., but notwith respect to retirement consumption; it carries whatever risksare associated with the investor's labor income and thus his orher savings. To the extent that these are market-like, the strate-gist may want to take those risks into account as adjustments tothe risky asset portfolios we develop later in the article. Whilean optimization of the total economic balance sheet wouldimprove on this "Kentucky windage" approach, our simplifica-tion isn't a bad working assumption even for a young personwith much more human capital than investment capital. For acomplete discussion, see Bodie, Merton, and Samuelson [1992]and Merton [1993].

"Merton,likewise, su^ests that a life annuity is the idealrisk-free asset for individuals [2005]. but here we use the max-imum possible life, and a rcalxf annuity, rather than life expectancyand a real life annuity, because at the current time the issuerdefault risk for available commercial annuities seems highenough to discourage their use. Also, investing in annuitiesrequires funding in cash, making it difficult and impractical tolayer any risky asset exposures on top of the "portfoho."

'' The comment that all risk from the liability goes awayis confined to market-related risk capable of being hedged. Notall liability characteristics can be hedged. As discussed in Waring[2004b], "liability alphas" are those parts of the liability returnthat have risks not hedgeable through any financial instrument.These include mortality risks and other demographic uncer-tainties unrelated to market movements. Further, one person'sinflation rate might well be different from that of another givendifferences in their consumption baskets (see, for example,Jen-nings [2006]). And this, in turn, means that one person's risk-free real bond ladder is different from that of another, not onlyin duration, but in its consumption-hedging dimension (noinstruments yet exist to make this observation actionable).Because there are no investment solutions to these risks, we

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don't address them here. Th e only solutions come in the formof plan design alternatives, that is, building Hability structures thatdon't create as many unhedgeable uncertainties. Fortunately, itappears that the contribution of risk from unhedgeable d em o-graphic uncertainties is much smaller than the risk that is con-tribute d by holding assets not w ell matched to the liabilities, thecurrent norm . So if wefixhe latter problem, we will have fixedthe bigger part of the problem.'Mn Equation (5) and in Equation (8) that follows, weconform to usual practice and consider the optimal RAP to bethe portfolio's exposure to the market risky asset portfolio, asconventionally defined, in our notation Pf^.n^^.u- I" the dis-cussion following Equation (10), we will show that this pre-sumption needs refinement."We are aware that this simplification, using a single dura-tion rather than the full span of consumption-weighted cashflows with their individual zero-coupon durations and yields.gives a higher yield than would the true underlying superbond,but at the p rice of missing important convexity and nonparallel-shift hedging requirements. This may be a fair trade-off forsmall investors with high trading costs, but larger investors willwant to m odel their hability and imp lement their LMAP in th emore granular form. For an excellent approach to building m oredetailed and comp lete m ode b of pension liabilities, see Fabozzi.Martellini, and Priaulet [2006]. Also, if future consumption isin more than o ne currency, that fact should be reflected by thepresence of real-rate instruments matching those cash flows inthose currencies (or effective equivalents using hedges) whenbuilding the liabihty model that will include similar positionsin the LM/^P portfolio and thus in the optimal solution. Thisis similar to Solnik 11974], but without requiring an additionalfund.

''In a paper that is closely related in subject matter, butthat comes at the topic from a completely different direction,Hoevenaars et al. [2007] built a sophisticated structural e co no -metric model of the key asset classes and concluded, consistentwith our observation, that long-term real bonds should befavored by liabihty-relative investors."This is consistent with the results of Ru do lf and Ziemba[2004] in a full intertemporal model (p. 976).''This discussion of F Q clarifies th e caveat made in En d-note 13.'^See CampbeD and Viceira [2002] for a general overviewof this literature. Mer ton has been prolific; in addition to his otherarticles cited elsewhere herein see 12006b]. Others of noteinclude Breeden [1979], and Ru benste in [1976], as well asauthors cited elsewhere in the article. Of these mo dels, all takefuture consumption into account, but Merton [1993], Rudolfand Ziemb a [2004],and M artellini [2006] do so using an explicitHability. The last paper is the closest to ours in its results.'''The durations of risky assets have historically be en dif-ficult to estimate, but with a stream of future cash flows and a

discounted price today, this suggests there must indeed be psensitivity to changes in the unde rlying risk-free rate. Th e dtion of a risky asset is usually debated as if it were in nomform, but W aring [2004al and Siegel and W aring [2004],lowing Goodman and Marshall [1988], separated nominal intion into real interest rate duration and inflation duratio n. Thauthors suggested that the inflation duration of equities is rougzero and that the real interest rate duration is in the high teToday, we would argue that the real interest rate durationsomewhat lowerin the high single digits, say 8, taking ibetter account the correlations between the real interest ratthe denominator and the dividend stream in the numeratothe pricing equation.REFERENCESBodie, Zvi, Robert C. Merton, and William F. Samuels"Labor Supply Flexibility and Portfolio Choice in a Life-CyModeV Journal of Economic Dynamics and Control, 16 (19pp. 4 2 7 ^ 4 9 .Breeden, Douglas T. "An Intertemporal Asset Pricing Mowith Stochastic Consumption and Investment OpportunitiJournal of Financial Economics, Vol . 7, No. 3 (September 1pp. 2 6 5 - 2 9 6 .Cam pbel l ,John Y. ,and Luis M . Vicei ra . Strategic Asset AllocPortfolio Choice for Long-Term Investors. Oxford, U.K.: OxUniversity Press, 2002.

. "The Term Structure of the Risk-Return TradeoFinancial AnalystsJournal,Vol. 61 , No. 1 (January/Februapp. 34-44 .

. "Strategic Asset Allocation for Pension Plans." In OxHandbook of Pensions and Retirement Income, ed i t ed by GoL. Clark, Alicia H. Munnell, and J. Michael Orszag. OxfoU.K.: Oxford University Press, 2006 .Evans,John L., and Stephen H. Archer. "Diversification andReduction of Dispersion: An Empirical Anaysis."_/t)MrttFinance, Vol. 23 , No . 5 (December 1971), pp. Ihl-ldl,Fabozzi, Frank J., Lionel Martellini, and Philippe Priau"Hedging Interest Rate Risk with Term Structure FacMod els." In Advanced Bond Portfolio Mattagement, Frank J. FSeries. Ho bok en, NJ: Joh n Wiley & Sons, Inc., 2006.Goodm an, Laurie S.,and William J. Marshall. "Inflation, InteRa tes, and Pension Liabilities." In Asset Allocation: A Hanof Portfolio Policies, Strategies, and Tactics, edited by Roberand Frank J. Fabozzi. Chicag o: Probus, 1988.

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Hoevenaars, Roy P.M.M, Roderick D.J. Molenaar. Peter C.Schotman, and Tom B.M. Steenkamp. "Strategic Asset Alloca-tion with Liabilities: Beyond Stocks and Bonds." White paper,2007. Available at SSRN, http://SSRN.com/abstract=675681.Jennings, William. "Disa^egated TIPS: The Case for Disag-gregating Inflation-Linked Bonds into Bonds Linked to Nar-rower CPl Components."y(iHrHii/ of Pension Economics and Finance,Vol. 5, No , 3 (2006 ). pp. 325-343.Leibowitz, M artin L. "Asset Allocation through Surplus M an-agement." Financial AnalystsJournal,Vol. 43 , No. 2 (March/April1987). pp. 29-40.Lintner.John. "The Valuation of PJsky Assets and the Selec-tion of Risky Investments in Stock Portfolios and Capital Bud-gets." The Review of Economics and Statistics, Vol . 47 . No . 1(February 1965), pp. 13-37.Martellini. Lionel. "Managing Pension Assets: From SurplusOptimization to Liability-Driven Investment." White paper,2006. Accessed at ht tp: / /w ww .edhe c-r i sk.co m/A LM /managing_pension_assets/index_html/attachments/Managing_Pension_Assets.pdf on February 13.2008.Merton, Robert C. "An Intertemporal Capital Asset PricingModel." Econometrica, Vol. 41 . No . 5 (September 1973).pp . 867-887.

. "Th eory of Finance from the Perspective of Co ntinuou sTime."7Mmi of Financial and Quantitative Analysis, 10 (November1975). pp . 373-413 . , .

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. "Allocating Shareholder C apital to Pension Plans.''gof Applied Corporate Finance, Vol. 18, N o. 1 (2006a). pp . 15-24.

. "Observations on Innovation in Pension Fund Man-agement in the Impending Future." PREA Quarterly (2006b),pp. 61- 67. Available at http://www .people.hbs .edu/rmerto n/PRA%20Journal%20Article%202%2006.pdf.Mod igliani , Franco, and R ichard Sutch. "Innovations inInterest Rat e Pohcy!' American Economic Review, 56 (May 1966),pp . 178-197.

. "Debt Management and the Term Structure of InterestRates: An Empirical Analysis of Recent Experience."Jdwrnii/o/^Political Economy, Vol. 75. No . 4 (1967), pp . 569-589.Mossin.Jan. "Equilibrium in a Capital Asset Market." Econo-metrica, Vol. 34, No. 4 (O ctober 1966), pp. 768-783.Roll. Richard. "A Critique ofthe Asset Pricing Theory's Tests,Part I:On Past and Potential Tesubility ofth e Theory." Viejournalof Financial Economics, Vol. 4. No . 2 (March 1977), pp. 129-176Ross, Stephen A. "The Arbitrage Pricing Theory of CapitalAsset Pric ing ." Jonriid/ of Economic Theory, Vol. 13, No. 3Pecember 1976). pp. 341-360.Rbenstein,Mark. "The Valuation of Uncertain Income Streamsand the Pricing of Options." Tlie BellJournal of F.cononiics, VolN o. 2 (Autumn 1976), pp. 407 -42 5.Rudolf, Marcus, and William T. Ziemba. "Interteniporal Sur-plus Man agem ent ." Jc 'urnd/ of Economic Dynamics and Control,28 (2004). pp . 975-990.Sharpe. William F "Capital Asset Prices: A Theory of MarketEquilibrium under Conditions ofKskyJournal of Finance,Vol. 19,N o. 3 (September 1964), pp. 42 5-4 42.

. "Asset Allocation." In Managing Investment PortfoliosA Dynamic Process, edited by John L. Maginn and Do nald L.Tuttle. New York: Warren, Gorham & Lam ont, 1990.

. "Equilibrium." In Macro-Investment Analysis available aht tp: / /www.stanford.edu/~wfsharpe/mia/eq/mia_eql .htm.Accessed on February 2 4.2 00 8, and viewed previously over aperiod of several years.Siegel. LaurenceB., and M. Barton Waring. "TIPS, theDual Dura-tion, and the Pension Plan." Financial Analysts Journal, Vol. 60N o. 5 (September/October 2004), pp . 52-64.Solnik, Bruno H. "An Equil ibrium Model of the Interna-tional Cap ita] M arkets ."JuHrna/ of Economic Tlieory, 8 (1974),pp . 500-524 .Tob in, James. "Liqu idity Preference as Beha vior towardsRisk." The Review of Economic Studies, Vol. 25. N o. 2 (1958),pp . 65-86. JTreynor,Jack L. "Toward a Theory of Market Value of RJskyAssets ." In Asset Pricing and P ortfolio Performance, e d i t e d b y R o b e r tA. Korajczyk. Lond on: PJsk Books, 1999. This otherwise un pu b-lished paper has been widely circulated since its original w ritingin 1962. . . .

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Wagner, W H ., and S.C. Lau. "T he Effect of Diversification onRisk." Finanat Analysts Journal, Vol. 27, No. 6 (Nove mber/December 1971), pp. 48-53 .Waring, M. Barton. "Liability-Relative Investing 1: Be D ualDuration Matched and on the Surplus Efficient Frontier."Vie Journa l of Portfolio Management, Vol. 30, No. 4 (20()4a),pp . 8-20 .

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Asset Lia Version of CAPm

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