CHAPTER Pension Fund Asset Allocation under Uncertainty Wilma de Groot and Laurens Swinkels CONTENTS 7.1 I ntroduction 7.2 Asset Allocation Framework with Nontradable Pension Liabilities 7.3 Da ta 7.4 How to Determine a New Asset Allocation? 7.5 C onclusions References 158 158 160 161 166 66 T he o utco mes o f tr a diti ona l m ean–variance a nalysis a re h ighly sensitive t o t he ex pected r eturns t hat a re u sed a s i nput This also holds for mean–variance analysis extensions that take into account nontradable pension liabilities In this chapter, we use uncertainty in expected returns in an asset allocation framework with nontradable pension liabilities This results i n more robust portfolio weights t han t raditional optimization and circumvents the use of arbitrary portfolio restrictions We apply this method to a U.S pension fund that contemplates investing in several new asset classes We illustrate this by showing how the allocation to emerging ma rket equities for a la rge r ange of ex pected returns leads to more robust portfolios when taking the uncertainty in these expected returns into account 157 © 2010 by Taylor and Francis Group, LLC 158 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 7.1 INTRODUCTION Recent developments in regulation require the mark-to-market valuation of pension liabilities This leads pension funds to take into account their pension liabilities when deciding on their asset allocation The framework developed by Sharpe and Tint (1990) allows pension funds to use mean–variance optimization wh ile t aking i nto account nontradable pension l iabilities Much like traditional mean–variance optimization, the portfolio weights that follow from this framework are sensitive to the choice of expected returns that serve as input This lack of robustness with respect to uncertain estimates of expected returns is an important reason why this framework has not often been implemented by pension funds; see Best and Grauer (1991) In this chapter, we describe a robust asset allocation framework that explicitly takes into account uncertainty about the expected returns on several asset classes while simultaneously taking into account pension liabilities We apply this method to an asset allocation problem for a U.S pension fund with nominal pension l iabilities This p ension f und c ontemplates e xtending its current portfolio of domestic equity, international equity, domestic bonds, and real estate with other asset classes, such as emerging market equities, small capitalization stocks, high yield debt, and commodities This chapter is organized as follows Section 7.2 describes the theoretical framework that is used In Section 7.3, we describe the data used in the empirical analysis In Section 7.4, we present how a U.S pension fund could determine its strategic asset allocation Finally, Section 7.5 concludes the chapter 7.2 ASSET ALLOCATION FRAMEWORK WITH NONTRADABLE PENSION LIABILITIES The framework described in this section was first described by De Groot and Swinkels (2008) I nstitutional i nvestors ma ke abundant u se of quantitative techniques to support their asset allocation decisions However, the standard portfolio o ptimization m odels o ften r esult i n e xtremely h igh a llocations to a ssets w ith somewhat bet ter r isk-return cha racteristics M ichaud (1989) describes this problem in more detail One way to circumvent the sensitivity of t he portfolio a llocations to t he expected returns used in t he a nalysis was put forward by Black a nd Litterman (1992) They introduce the notion of uncertainty about the expected future returns on asset classes This new technique s s ince t hen be en u sed f requently t o so lve p ractical po rtfolio optimization problems, and in general leads to better out-of-sample portfolio performance than the standard mean–variance optimization framework In t his cha pter, w e a pply t he Black a nd L itterman (1992) tech nique i n a n © 2010 by Taylor and Francis Group, LLC Pension Fund Asset Allocation under Uncertainty ◾ 159 asset–liability–management framework The mean–variance allocation problem for investors who face nontradable pension liabilities was already solved by Sharpe and Tint (1990) They define the surplus of the pension fund as St (k ) = At − k × Lt where At is the value of the assets at time t L is the value of the pension liabilities at time t ∼t k den otes t he i mportance o f t he l iabilities i n t he po rtfolio ch oice problem ∼ ∼ For k = we are back in an asset-only framework, while with k = we speak about full surplus optimization The return on the surplus is then defined as RtS (k ) = RtA − k × RtL where RAt is the return on the assets RtL is the return on the liabilities k is again a parameter denoting the importance of the pension liabilities In this chapter, we assume a pension fund with full surplus optimization, that is, k = 1.* This coincides w ith t he u se of t he funding ratio return as introduced by Leibowitz et al (1994) A common way for pension funds to determine their asset allocation, with or without respect to pension liabilities, is to use a se t of expected returns, volatilities, a nd correlations a s a st arting point for t he mean– variance optimization Should the portfolio deviate too much from the current a sset a llocation, it is not u nusual to go back t o t he “parameter drawing board” and change some inputs in such a way that the resulting allocation is more in line with the existing allocation or prior ideas on how t he n ew po rtfolio w eights sh ould be A lthough such a n a pproach may y ield i mportant i nsights i nto t he sens itivity o f a llocations w ith respect to a ssumed ex pected returns, t his i s fa r f rom a t rue optimization A nother solution to obtain less ex treme weights i s to put restrictions on t he portfolio weights; for ex ample, by ma ximizing t he weight ∼ * The parameter k equals k divided by the funding ratio at time t − So with a funding ratio of both parameters are equal to each other © 2010 by Taylor and Francis Group, LLC 160 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling to emerging market equities at 5% This is typically done for alternative assets with relatively short and attractive past returns and good diversification benefits This last method often leads to optimal portfolios fi lled with alternative assets up to the maximum that was specified as a restriction This means that the choice of the maximum is practically an investment advice by itself In this chapter, we make use of a different approach to account for the asset allocation problem that is less sensitive to the expected returns, by explicitly modeling the uncertainty around the expected returns In our analyses, we start with the current asset allocation of a U.S pension fund with nominal pension liabilities We assume that this pension fund currently has a st rategic allocation to U.S government bonds of 50%, to U.S equities of 20%, international equities of 20%, and listed real estate of 10% Other possible asset classes for the pension fund to invest in are emerging market equities, small capitalization stocks, high yield bonds, and commodities We determine the set of expected returns in excess of the liabilities that correspond to the mean–variance problem with the current asset allocation as the outcome This is also called reverse engineering When these implied expected returns not coincide with the views of the pension fund, this may lead to cha nging t he a sset a llocation The f ramework t hat we de scribe i n t his chapter can also be used for the consistent implementation of tactical asset allocation views In essence, we assume that the pension fund has a view, but is not 100% sure about its view We take the level of uncertainty into account for the optimal deviation from the strategic asset allocation 7.3 DATA The d ata we u se for t he a nalyses a re obtained f rom Tho mson Financial Datastream on a quarterly basis over the period January 1985–December 2008 The returns on t he pension liabilities are approximated by a l ongterm U.S nominal government bond, for which we take the Lehman U.S Treasury Long Index For cash, we use the JP Morgan U.S Month Cash Index; for bonds, t he L ehman U.S Ag gregate Bond I ndex; for domestic equities, the Morgan Stanley Capital International (MSCI) U.S Index; for international equities, the MSCI World ex U.S Index; for listed real estate, the National Association of Real Estate Investment Trusts (NAREIT) Index; for emerging market equities, the MSCI Emerging Markets Index; for s mall c apitalization st ocks, t he R ussell 000 I ndex; f or h igh y ield bonds, the Lehman U.S Corporate High Yield Index; and for commodities, the S&P/Goldman Sachs Commodity Index © 2010 by Taylor and Francis Group, LLC Pension Fund Asset Allocation under Uncertainty ◾ 161 Descriptive statistics on these asset classes are presented in Table 7.1 The returns on the fi xed income type asset classes indicate that there has been a subst antial ter m p remium f rom i nvesting i n bo nds, a s r egular bonds outperformed cash with 2.77% and long-dated pension liabilities generated 5.22% extra return over cash Emerging market equity returns were n ot o nly h ighest o ver t his s ample per iod, b ut a lso m ost v olatile, leading t o a m odest Sha rpe r atio o f 24 M oreover, em erging ma rket equities were the only asset class that kept up with the increase is nominal pension liabilities International equities are slightly more risky than domestic equities due to currency risk, but small caps and commodities future returns are even riskier in a stand-alone context The correlations between these asset classes can be found in Table 7.2 Bonds a nd n ominal pens ion l iabilities a re h ighly co rrelated w ith 92 Most nontraditional assets have a relatively high correlation with equities, reducing potential diversification benefits A clear exception is commodities w ith l ow o r so metimes e ven n egative co rrelations The correlation between bonds and equities is close to zero, and we also see that equities not offer protection against pension liability risk in the short run with a correlation of −0.10 In the analyses that follow, we not use the historical averages from Table 7.1 a s i nput, but i nstead rely on ex pected f uture returns f rom t he pension fund or the investment consultant We only use historical returns for t he e stimation o f t he v olatilities a nd co rrelations be tween t he a sset classes under consideration 7.4 HOW TO DETERMINE A NEW ASSET ALLOCATION? In t his sec tion, w e a nalyze t he q uestion h ow t o de termine a n ew a sset allocation for the U.S pension fund with nominal pension liabilities and the st rategic a llocation a s mentioned i n Section 7.2 We first investigate the implied expected returns above liabilities that form the starting point when the pension fund has no other views In the next step, we determine the allocations to nontraditional asset classes when the pension fund has uncertain (tactical) v iews, a nd we compare t his to t he portfolio weights that result from standard optimizations We conclude this section by showing the portfolio weight to emerging market equities when simultaneously cha nging t he expected return a nd t he associated uncertainty of this asset class To de termine t he i mplied ex pected returns of t he c urrent a llocation, we c alibrate t he r isk a version o f t he pens ion f und such t hat t he eq uity © 2010 by Taylor and Francis Group, LLC Annualized Statistics January 1985–December 2008 Return Excess return Volatility Sharpe ratio TABLE 7.2 Liabilities Cash Bonds International Equities U.S Equities Real Estate Emerging Markets Small Caps High Yield Commodities 10.59% 5.22% 10.3% 0.51 5.37% 0.00% 1.1% 0.00 8.14% 2.77% 4.8% 0.58 9.22% 3.85% 19.3% 0.20 9.94% 4.57% 16.7% 0.27 7.41% 2.04% 16.1% 0.13 12.19% 6.82% 28.7% 0.24 8.55% 3.18% 21.3% 0.15 7.67% 2.30% 9.2% 0.25 7.10% 1.73% 24.2% 0.07 Correlation Matrix January 1985–December 2008 (%) Liabilities Cash Bonds International equities U.S equities Real estate Emerging markets Small caps High yield Commodities Liabilities Cash Bonds International Equities U.S Equities Real Estate Emerging Markets Small Caps High Yield Commodities 100 16 92 −6 −10 −2 −34 −15 −30 100 32 18 −12 100 10 −26 −9 20 −19 100 75 41 59 67 52 100 49 63 87 61 −5 100 33 65 62 100 68 47 100 64 −8 100 100 © 2010 by Taylor and Francis Group, LLC 162 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling TABLE 7.1 Pension Fund Asset Allocation under Uncertainty ◾ 163 premium above the pension liabilities equals 3% The first row of Table 7.3 contains t he a sset a llocation of t he pension f und The second row i ndicates t he implied expected returns t hat follow from t his asset a llocation using reverse engineering The interpretation of t hese numbers is t hat if the pension f und is a m ean–variance investor w ith nontradable pension liabilities, t hese ex pected excess returns should be u sed i n t he model to generate the current asset allocation We see in Table 7.3 that of the asset classes that the pension fund currently does not invest in, only emerging markets (4.18%) and small capitalization s tocks ( 3.44%) ve h igher i mplied r eturns t han t he eq uity premium This m eans t hat o nly wh en t he ex pected ex cess r eturns a re higher than these numbers, an allocation to these asset classes should be made These asset classes a re only attractive when t hey have a r elatively high expected return, due to their high volatility and positive correlation with equity markets Commodities are more risky than small capitalization stocks, but because of their low correlation with equity and bond markets they would be i ncluded in the portfolio when their expected excess return is above 2.07% per annum In the Black and Litterman framework, these implied returns are combined with uncertain views on each asset class For each asset class, we take the level of uncertainty equal to t he volatility of t hat asset class, as suggested by He and Litterman (1999) In practice, the board of trustees of the pension funds may want to include their own views on the uncertainty around the expected excess return to obtain the optimal allocation tailored to their wishes Table 7.4 contains for each of the four nontraditional asset classes in the left column the optimal portfolio weight according to t he standard portfolio optimization a s i ntroduced by Sha rpe a nd Tint (1990) for a g iven level of ex pected excess return I n t he r ight column, we show t he optimal portfolio weights when uncertainty is explicitly taken into account In both analyses, the other asset categories are based on the implied expected returns in Table 7.3 From Table 7.4, i t i s cl ear wh y t he st andard m ean–variance o ptimization is not used in practice that much The ex pected ex cess r eturn is extremely important for the optimal allocation For example, if we expect an excess return of 3% instead of 2% for high yield bonds, the allocation would increase from 31.4% to 262.9% of the portfolio, meaning that large negative weights occur in other categories The robust method we describe in t his chapter results i n much more g radual portfolio de viations when assumptions on expected returns are slightly changed © 2010 by Taylor and Francis Group, LLC Implied Expected Returns above Liabilities for Pension Fund (%) Asset allocation Implied return Cash Bonds International Equities U.S Equities Real Estate Emerging Markets Small Caps High Yield Commodities 1.20 50 0.76 20 3.19 20 3.00 10 2.34 4.18 3.44 1.86 2.07 TABLE 7.4 Portfolio Weight Investments according to Standard and Robust Optimization Depending on the Expected Excess Return (above Liabilities) on the Asset Class (%) Expected Excess Return Asset Class Emerging Markets Standard Robust −90.2 −68.7 −47.1 −25.5 −3.9 17.7 −19.0 −14.5 −9.9 −5.4 −0.8 3.7 © 2010 by Taylor and Francis Group, LLC Portfolio Weight Small Caps High Yield Standard Robust Standard Robust −446.4 −316.6 −186.8 −57.0 72.8 202.7 −31.0 −22.0 −13.0 −4.0 5.0 14.0 −431.5 −200.1 31.4 262.9 494.4 725.8 −55.5 −25.7 4.0 33.8 63.5 93.3 Commodities Standard Robust −32.5 −16.7 −1.0 14.7 30.4 46.1 −10.7 −5.5 −0.3 4.8 10.0 15.2 164 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling TABLE 7.3 Pension Fund Asset Allocation under Uncertainty ◾ 165 Portfolio weight emerging markets as a function of the uncertainty in expected return 50% 40% 20% 10% Portfolio weight 30% –20% retu rn Very uncertain 100.00 ted 1.00 ne xpe c 7% 0.50 nty i 0.20 rtai 0.15 0.01 Un ce 0.10 Very certain –10% 0.05 0.00 0% 3% 6% rn 5% etu s d r arket 4% e t ec m Exp rging e em Portfolio w eight emerg ing ma rkets ac cording to t he rob ust o ptimization depending on different levels of expected excess returns on emerging markets and the uncertainty around these returns FIGURE 7.1 In Table 7.4, we assumed t hat t he u ncertainty of our ex pectations on the a ssets i s equal t o t he volatility of t he a sset cla sses We now i nvestigate how the portfolio allocation changes when the uncertainty increases or decreases In Figure 7.1, we show how this works for emerging market equities Here, we let the expected excess return range between 3%, which is equal to the equity premium above liabilities, and 7% When w e a re v ery u ncertain abo ut t he ex pected r eturn o f em erging market eq uities, t he final portfolio weight w ill be eq ual t o t he st rategic allocation of 0%, i rrespective of t he ex pected r eturn l evel At t he o ther extreme, when we a re very c ertain about t he ex pected return, t he portfolio weight va ries be tween −18% and 41%, depending on t he ex pected return of emerging market equities We find t he robust figures reported in Table 7.4 for the 3%–5% expected excess return levels when we look at © 2010 by Taylor and Francis Group, LLC 166 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling the uncertainty axis at 0.11, which is the historical variance of emerging market equity excess returns This means that the expected return and the uncertainty around it determine the deviations in the strategic allocation by the robust model In Figure 7.1, we only varied the level of (un)certainty on the expected return of emerging market equities If we are simultaneously very certain on the expected return of each asset class, we obtain the standard portfolio optimization weights that are shown in Table 7.4 7.5 CONCLUSIONS In this chapter, we show that using uncertainty in expected returns can support a pension fund to determine the strategic or tactical asset allocation relative to the nontradable pension liabilities The use of uncertainty is a wa y to ma ke t he portfolio weights of t he model of Sha rpe a nd Tint (1990) less sensitive to the expected return input Hence, this model can be used to guide boards of trustees of pension funds in a more systematic way to form their asset allocations We illustrate this by showing how the a llocation to emerging ma rket equities for a la rge range of ex pected returns l eads t o m ore r obust po rtfolios wh en t aking t he u ncertainty i n these ex pected r eturns i nto acco unt This i s a wa y t o l imit t he ex treme allocations that result from a st andard mean–variance optimization We therefore believe that this method serves as a useful quantitative support tool when determining strategic and tactical asset allocation decisions REFERENCES Best, M.J and Grauer, R.R (1991) On t he sensitivity of mean–variance efficient portfolios t o c hanges in ass et me ans: S ome a nalytical a nd co mputational results, Review of Financial Studies 4, 16–22 Black, F and Litterman, R (1992) Global portfolio optimization, Financial Analysts Journal 48(5), 28–43 De Groot, W and Swinkels, L (2008) Incorporating uncertainty about alternative assets in strategic pension fund asset allocation, Pensions 13(1–2), 71–77 He, G and Litterman, R (December 1999) The intuition behind Black–Litterman model p ortfolios, Investment M anagement Re search, G oldman, Sac hs & Company, New York Leibowitz, M.L., Kogelman, S., and Bader, L.N (1994) Funding ratio return, Journal of Portfolio Management 21(Fall), 39–47 Michaud, R (1989) The Markowitz optimization enigma: Is ‘optimized’ optimal? Financial Analyst Journal 45(1), 31–42 Sharpe, W.F and Tint, L.G (1990) Liabilities—A new approach, Journal of Portfolio Management 16(2), 5–10 © 2010 by Taylor and Francis Group, LLC ... 164 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling TABLE 7.3 Pension Fund Asset Allocation under Uncertainty ◾ 165 Portfolio weight emerging markets as a function of the uncertainty... Francis Group, LLC 162 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling TABLE 7.1 Pension Fund Asset Allocation under Uncertainty ◾ 163 premium above the pension liabilities equals... Francis Group, LLC Pension Fund Asset Allocation under Uncertainty ◾ 159 asset liability–management framework The mean–variance allocation problem for investors who face nontradable pension liabilities