Asset-Liability Modeling in BarraOne May 2007

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1 Asset-Liability Modeling in BarraOne This case study provides an introduction to modeling assets and liabilities for asset owners within BarraOne. We show how to use BarraOne to analyze both assets and liabilities in a shared framework for understanding risk and return. Our example uses zero coupon bond instruments to proxy for liabilities although other proxies may be easily substituted. We illustrate how to analyze the surplus risk of the portfolio focusing on market risk specifically. Other sources of risk nonmarket or biometric risk are not addressed here but are discussed. 1. Introduction Asset-liability management (ALM) concerns how financial institutions such as banks, insurance companies, pension plans, and endowment funds should optimally manage their assets and liabilities. In this case study we approach the asset-liability matching problem from the perspective of an asset owner with long-term liabilities. We consider in particular the challenge faced by defined benefit pension plans though this framework can be extended to other types of institutions. Pension plans are unique amongst the universe of financial institutions because their investment horizon is very long. For defined benefit pension plans, where employees are promised a certain level of cash benefits to be paid after retirement depending on their length of service and ending salary, 1 the asset-liability matching problem is central since the plan sponsor promises a guaranteed rate of return for the retirees. In the past, pension plan management focused on generating cash inflows on the asset side to match the cash outflows on the liability side. While classical immunization (duration matching) was popular in the 1970s, throughout the 1980s and 1990s, pension plans increasingly shifted their asset allocation towards equities, not wanting to miss out on the return opportunities. The increased allocation to equities and other non fixed income assets was also a result of the unavailability of bonds with long enough duration. Moreover, arguments based on time diversification suggested using high risk, high return instruments to match the long term nature of plans liabilities. In , pension plans were hit with a perfect storm of falling interest rates (raising the value of their liabilities) and a decline in equity markets. This has since spawned the new trend of liability-driven investing where instead of constructing the asset portfolio to meet some returnrisk target, the focus is to custom tailor the asset side to perform better than, but not deviate too far from, a benchmark that consists of the plan s liabilities. Flexibility in structure and the ability to exploit strong equity performance while minimizing the difference between the asset and liability sides (the surplus) has become the main objective. This has been helped by the advent of longer term bonds and futures, swaps, and other derivatives which allow for easier and more flexible hedging of interest rate risk/duration mismatches between the asset and liability sides. 1 As opposed to defined contribution plans in which employees contribute a fraction of salary and his or her assets are held in an individual account MSCI Barra. All rights reserved. 1 of 10

2 2. The Barra Integrated Model: An Overview The Barra Integrated Model (BIM) is a multi-asset class risk model that covers global equities, bonds, currencies, commodities, and hedge funds. The model couples broad asset coverage with the detailed analysis of Barra s single country models that focus on particular markets. This makes BIM suitable for a wide range of investment purposes, from conducting an in-depth analysis of a single-country portfolio to understanding the risk profile of a broad set of international investments. First, to provide the needed level of detail, factor models are built for all local equity and bond markets. These models attribute the explainable portion of an asset s return to the local factors at work in each market. These factors include styles and industries for equities, and term structure movements and credit spreads for bonds, and may differ significantly from market to market. Second, the local factors are mapped to global factors, which include equity, fixed income, and currency factors. Therefore, all assets in BIM can be mapped onto several layers of factors, each layer being responsible for an increased degree of granularity; see Puchkov, Stefek and Davis (2005) and Stefek et al. (2005) for a detailed discussion. 2 As we have discussed, tying together the assets and liabilities into one framework is the key challenge for the problem of asset-liability management. On the liability side, in this illustration we have a portfolio that behaves like a mixture of inflation-linked and nominal bonds. The portfolio is subject to both market risk (interest rate risk, inflation risk, market value risk, etc.) and nonmarket risk (salary risk, demographic risk, regulatory risk, etc. 3 ). On the asset side, we have a mixed portfolio of equities, bonds, and possibly other asset classes such as real estate, private equity, and hedge funds. There is no straightforward way to measure the duration of such a portfolio. 4 Thus, the aggregate asset-liability portfolio is exposed to market risk on both the asset and liability sides, including mismatches in duration, and the non-market risk on the liability side. 5 Using the Barra Integrated Model we can address the most important parts of this problem. The strength of BIM is that it ties together both equity and fixed income factors in one framework. Thus, within BIM, all equities have a relationship to fixed income (yield curve) factors (i.e., implicit interest rate shift, twist, butterfly exposures, and credit spread exposure) and all fixed income instruments likewise are related to equity factors (market, industry, style/risk index factors, etc.). The BIM framework thus presents a natural way to begin analyzing the asset-liability problem. 2 Puchkov, Anton V., Dan Stefek, and Mark Davis (2005), Sources of Return in Global Investment, Journal of Portfolio Management. Stefek, D., Hsieh, A., Puchkov, A., and Hemmati, F. (2005), The Barra Integrated Model: Version 204, Barra Research Insight. 3 Salary risk stems from the risk to liabilities if plan payouts are indexed to salaries; demographic risk poses a problem if members live longer than anticipated or employment patterns differ materially from assumptions. Regulatory risk stems from changes to the regulatory environment or accounting changes. 4 One major challenge then is how to accurately measure the interest rate sensitivity of equities and other non fixedincome assets. To date, there is no clear cut way to measure duration for equities and alternative investments. Some models are derived from traditional dividend discount models which typically result in extremely long duration estimates for equities often in excess of 50 years for growth stocks (Lewin and Satchell, 2001). Other models of equity duration try to measure the interest rate sensitivity of stocks empirically (i.e., statistical duration) following the work of Leibowitz (1986). These Leibowitz-type measures imply that equities behave as if they are much shorter duration instruments, typically 5 years or less. These two types of measures are theoretically disconnected and often offer conflicting results. 5 Modeling inflation risk poses unique challenges on its own within this framework. Inflation expectations are used to estimate future salary and wage growth which can impact both expected contributions from employees and expected payouts. In other words, increases in wage growth result in both increased liabilities and increased assets (through larger contribution amounts). The plan s surplus depends on the relative magnitude of these two. Depending on the sensitivity to changes in inflation of the assets versus liabilities, the plan may be subject to inflation risk arising from a mismatch in exposure. An explicit inflation factor or some factor which captures contribution to inflation might be of importance in this case MSCI Barra. All rights reserved. 2 of 10

3 3. Modeling Liabilities in BarraOne BarraOne allows you to analyze the risk of a pension plan s liabilities as well as the risk of the surplus, the difference between assets and liabilities. The general framework we present here can be applied to a pension plan, insurance company, or any similar institution with an expected future stream of cash outflows. For this illustration, assume a hypothetical stream of future cash flows. (Typically, actuaries will provide estimated expected future cash flows.) In this illustration, we make two assumptions about the expected future cash flows: a) Expected future cash flows are known today and are adjusted for residual risk (risk other than market risk) b) Expected future cash flows are exposed only to market risk. The first assumption is necessary because the Barra Integrated Model (BIM) is constructed to measure market risk. (Expected cash flows need to be adjusted for other sources of risk than market risk, e.g. biometric or non-market risks. 6 ) The second assumption captures the core tenet of the importance of simultaneous asset-liability optimization - both assets and liabilities are exposed to market risk and thus should be analyzed in a consistent framework. For this case study, let us assume the following schedule of expected (nominal) future cash flows CF(t): Table 1: Hypothetical Future Cash Flow Year of the Future Cash Flow T CF(t) There are a range of fixed income instruments that can be used to proxy liabilities as suggested by the academic literature and standard practice. They are easy to implement and have the advantage of being very transparent: 1) Zero coupon bonds: Model each cash flow, CF(t), as a zero bond with maturity t and face value CF(t) 2) Duration proxy: Calculate the duration of the cash flows and use a duration proxy instrument that is available in BarraOne or a user-defined instrument (a range of proxies including high-quality corporate bonds and interest rate swaps or futures can be used) 3) Risk factors: Proxy liabilities by a shift factor in the respective currency. For this illustration, we use the first alternative, zero coupon bonds, although any of the three options can be implemented. Within BIM, fixed income risk is modeled using a multi-factor risk model that captures term structure risk (Shift/Twist/Butterfly) and spread risk (one swap spread factor per currency to account for the average spread risk, and multiple sector-by-rating factors for corporate bonds) for each market. 7 Within BarraOne, users can create zero coupon bonds 6 One solution might be to add a certain portion of idiosyncratic ( specific ) risk to the liability portfolio to capture the nonmarket risk of liabilities. BarraOne also allows users to import user-defined assets which can include Interest Rate swaps and other non-traded instruments which are widely used in LDI. 7 Term structure curves and spreads are estimated daily in BarraOne MSCI Barra. All rights reserved. 3 of 10

4 that are either exposed to the term structure factors only, or are exposed to the term structure factors plus the swap spread factor. Figure 1 below shows a screenshot from BarraOne with zero coupon bonds that reflect the cash flows from Table 1 on the previous page. Figure 1: Zero Coupon Bond Proxies for Expected Future Cash Flows The column Asset ID in Figure 1 contains a generic identifier for zero coupon bonds where we have assumed the following: Table 2: Description of Identifier for Zero Coupon Bond EUR E C YYYYMMDD Identifies Term Deposits, i.e. zero bonds Identifies the currency of the zero bond: The instrument will be exposed to European fixed income risk factors and valued on the European term structure Stands for an explicit exposure to currency risk factors (not relevant for now) This gives the term deposit an exposure to the (European) swap spread factor Identifies the maturity date The column Holdings contains the par value of the cash flows. The column Mkt Value is calculated by BarraOne and is the present value of the Market Value. Once you have created these zero coupon bond proxies, you can look at the risk profile of these liabilities. Figure 2 shows the time-to-maturity, effective duration, percent contribution to total risk, and total risk (annualized standard deviation) of these four instruments. The liability duration is about 18, and the ex-ante volatility of the liability portfolio is 9.60% annually. Figure 2: Risk Profile of Liabilities 2007 MSCI Barra. All rights reserved. 4 of 10

5 Furthermore, you can view the risk of the liability portfolio decomposed along Barra risk factors using the Risk Decomposition feature in BarraOne shown in Figure 3. Figure 3: Risk Decomposition of Liabilities Using Barra Factors The risk of this liability portfolio is composed of (systematic) factor risk, in turn comprised of spread and term structure risk plus an interaction component, as well as a specific risk component. 4. Integrating Assets and Liabilities in BarraOne Once the liability portfolio is determined, the next step is to combine it with the asset portfolio and integrate the two into one framework. The structure we illustrate here is similar to the one used for benchmark-relative investing. On the asset side, equity positions as well as fixed income positions that do not explicitly match the duration of the liabilities, can be viewed as departures from a liabilities-defeasing portfolio. To understand the logic, consider the following situation: A plan s liabilities consist of one zero coupon bond with a duration of n. If the plan s assets consist of a matching zero coupon bond with the same duration, then there is no surplus risk; the liabilities are perfectly immunized. Next, suppose we add a small equities position to the asset side. This will create exposures to market risk that are not captured by the liability portfolio. We can view this exposure as an active exposure similar to an ex-benchmark asset in a traditional benchmark-relative investment portfolio. In this framework, asset liability management involves three separate portfolios - the liability portfolio, the asset portfolio, and an active portfolio which is exposure-wise the difference between the asset and liability portfolio. If we define the active portfolio holdings as the difference between the asset weights and liability weights, then we can express the risk of the surplus as: 2 σ surplus = h T T ( S ) XVX h( S ) where h( S) = h( Assets) h( Liabilities) X = the matrix of exposures of the assets and liabilities to the BIM risk factors V = the covariance matrix 2007 MSCI Barra. All rights reserved. 5 of 10

6 Of course, one can also combine the assets and liabilities into a single portfolio with the liabilities treated as short positions instead of using a benchmark approach. This would for instance allow for control of the notional value of liability payments. The choice of approach is ultimately a function of the objective and assumptions of the user. 8 Consider a hypothetical asset portfolio with a market value of EUR 33 billion that consists of twothirds fixed income (which is duration-matched with the liabilities) and one-third equities (a passive investment in the MSCI Europe Index). Since the present value of the liability portfolio is EUR 31.8 billion, the surplus is EUR 1.2 billion. The risk of the surplus is shown in Figure 4 below in the columns pertaining to Active Risk. The surplus risk for this hypothetical portfolio then is 6.16% in annualized standard deviation. The risk of the asset and liability portfolios is also shown in Figure 4. Figure 4: Risk Profile of the Asset Portfolio, Liability Portfolio, and the Surplus What does the risk estimate of 6.16% for the surplus mean in Euro-terms? We can derive a value-at-risk-like number using a formula such as the one following: ( a, T ) = EuroAmount * CL( a) * Risk T Var * With a horizon of 1 year (T = 1) and a confidence level of 1.65 (CL(a)), then the surplus at risk is EUR 0.12 billion (given a Euro amount of EUR 1.2 billion and a risk of 6.16%). Of course, one can extend this analysis to the asset portfolio and the liabilities as well. For example, the ex-ante risk of 6.87% for the asset portfolio is lower than the 9.6% ex-ante risk for the liability portfolio. As both risk numbers are calculated using a cross-asset-class risk model, in this particular case, the asset portfolio is less volatile than the liability portfolio. 8 In BarraOne, users can select the Base Value setting to achieve their desired leverage effect. Ordinarily the base value is the net portfolio value which assumes the aggregate value of all positions is used as the reference. For long-short portfolios, net value assumes the portfolio is self-financed or costless. For asset-liabilities, it may be appropriate to use the long side of the portfolio as a reference value ( Long value assumes that the portfolio is leveraged). BarraOne also allows users to use the sum of the absolute long and short portfolios as the reference or choose their own assigned value MSCI Barra. All rights reserved. 6 of 10

7 5. The Asset Allocation Decision Next we consider the allocation of the asset portfolio with respect to the liability portfolio. You can use BarraOne to understand how changes in the asset allocation affect the surplus risk and locate an optimal asset portfolio with respect to the liabilities. The asset allocation decision should take into account both the expected return and risk characteristics of the asset and liability sides but also the correlation between them. For example you may start with a liabilities-defeasing 100%-fixed-income asset portfolio and shift the allocation into equities and other non-fixedincome asset classes, depending on various factors and assumptions. Including the plan s current funding ratio, expected long-term returns for equities with respect to the liabilitiesdefeasing portfolio, expected population, wage and salary growth, expected mortality rates 9, etc. How does a reallocation on the asset side affect the risk of the surplus? What happens if you change the allocation of assets (fixed income versus equities)? Figure 5 shows the change in the risk (standard deviation) of the asset and surplus portfolios from the previous example, keeping liabilities fixed, as the allocation of fixed income shifts to equity. As expected, decreasing the portion of the asset portfolio in fixed income increases the risk of the asset portfolio and the surplus. Figure 5: The Effect on Surplus Risk Arising from a Change in the Asset Allocation Sensitivity of Risk to Changes in Fixed Income Allocation in Asset Portfolio Annualized Forecast Risk 10% 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% 50% 55% 60% 65% Allocation to Fixed Income 70% 75% Asset Risk Surplus Risk Liability Risk The optimal allocation between fixed income and equities on the asset side will depend on the plan s tolerance for risk and its expected return assumptions. As the portfolio is weighted more heavily towards equities, the risk of the surplus increases but so does the expected return, depending on the expected equity premium. The plan s tolerance for risk should be based in part on its long-term objectives and in part on its current funding status. For instance, a plan that has a surplus may be more likely to have a lower risk tolerance than a plan that is running a deficit if their long term objectives are similar. 9 Milevsky, Ho, and Robinson (1997) find that the risk of outliving your money with low risk, low return investments is very often more serious than the risk of losing money on high risk investments until quite late in life, suggesting the optimal allocation to equity is quite high. See Milevsky, A.M.; K. Ho and C. Robinson (1997): Asset Allocation Via the Conditional First Exit Time or How to Avoid Outliving Your Money, Review of Quantitative Finance and Accounting 9, MSCI Barra. All rights reserved. 7 of 10

8 To illustrate, we assume expected returns of 7% and 3% for our equities and fixed income composite assets, respectively. (For simplicity, we leave the expected return for liabilities equal to zero.) We optimize the portfolio across a range of risk targets; optimal allocations to equities and fixed income components are shown in Table 3, corresponding to different points on the surplus efficient frontier. Table 3: Optimal Portfolio Allocations for Different Risk Targets Risk Target (Active %) Equities (% of Asset Portfolio) Fixed Income (% of Asset Portfolio) % 94.83% % 86.22% % 79.07% % 72.31% % 65.72% Thus in the traditional mean-variance framework, the efficient surplus frontier depends on the risk and expected returns of the liabilities and the investable assets as well as their correlation structure. The optimal allocation for the plan the balances the risk-return tradeoff depending on the plan s long-term objectives and risk appetite. 6. Conclusion This illustration serves as an introduction into how BarraOne can be used to analyze both assets and liabilities in a shared framework for understanding risk and return. We have shown how to use BarraOne to model the risk of the asset and liability portfolios as well as the surplus risk. From there, we demonstrated how minor changes in assumptions on the liability side can affect the overall risk of the aggregate portfolio given its total asset composition. Next we identified how changes in the asset mix may impact the asset-liability mismatch and lastly, we addressed the issue of choosing the optimal asset allocation MSCI Barra. All rights reserved. 8 of 10

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