APPROACHES TO LIABILITY-RELATIVE ASSET ALLOCATION: SURPLUS OPTIMIZATION
Learning Outcomes
describe and evaluate characteristics of liabilities that are relevant to asset allocation
discuss approaches to liability-relative asset allocation
recommend and justify a liability-relative asset allocation
Various approaches to liability-relative asset allocation exist. These methods are influenced by tradition, regulations, and the ability of the stakeholders to understand and extend portfolio models that come from the asset-only domain.
There are several guiding principles. The first is to gain an understanding of the make-up of the investor’s liabilities and especially the factors that affect the amount and timing of the cash outflows. Given this understanding, the present value of the liabilities is calculated, along with the surplus and funding ratio. These measures are used to track the results of ongoing investment and funding policies and for other tasks. Next come the decisions regarding the asset allocation taking account of the liabilities. There are a number of ways to proceed. We will discuss three major approaches:
Surplus optimization. This approach involves applying mean–variance optimization (MVO) to an efficient frontier based on the volatility of the surplus (“surplus volatility,” or “surplus risk”) as the measure of risk. Surplus optimization is thus an extension of MVO based on asset volatility.14 Depending on context, surplus risk may be stated in money or percentage terms (“surplus return volatility” is then another, more precise term for this measure).
Hedging/return-seeking portfolios approach. This approach involves separating assets into two groups: a hedging portfolio and a return-seeking portfolio. The reading also refers to this as the two-portfolio approach. The concept of allocating assets to two distinct portfolios can be applied for various funding ratios, but the reading distinguishes as the basic approach the case in which there is a positive surplus available to allocate to the return-seeking portfolio.
Integrated asset–liability approach. For some institutional investors, such as banks and insurance companies and long–short hedge funds, asset and liability decisions can be integrated and jointly optimized.
We cover these three approaches in turn.
Surplus Optimization
Surplus optimization involves adapting asset-only mean–variance optimization by substituting surplus return for asset return over any given time horizon. The quadratic optimization program involves choosing the asset allocation (mix) that maximizes expected surplus return net of a penalty for surplus return volatility at the chosen time horizon. The objective function is����=�(��,�)−0.005��2(��,�)2where ���� is the surplus objective function’s expected value for a particular asset mix m; E(Rs,m) is the expected surplus return for asset mix m, with surplus return defined as (Change in asset value − Change in liability value)/(Initial asset value); and the parameter λ (lambda) indicates the investor’s risk aversion. The more risk averse the investor, the greater the penalty for surplus return volatility. Note that the change in liability value (liability return) measures the time value of money for the liabilities plus any expected changes in the discount rate and future cash flows over the planning horizon.
This surplus efficient frontier approach is a straightforward extension of the asset-only portfolio model. Surplus optimization assumes that the relationship between the value of liabilities and the value of assets can be approximated through a correlation coefficient. Surplus optimization exploits natural hedges that may exist between assets and liabilities as a result of their systematic risk characteristics.
The following steps describe the surplus optimization approach:
Select asset categories and determine the planning horizon. One year is often chosen for the planning exercise, although funding status analysis is based on an analysis of all cash flows.
Estimate expected returns and volatilities for the asset categories and estimate liability returns (expanded matrix).
Determine any constraints on the investment mix.
Estimate the expanded correlation matrix (asset categories and liabilities) and the volatilities.15
Compute the surplus efficient frontier and compare it with the asset-only efficient frontier.
Select a recommended portfolio mix.
Exhibit 22 lists LOWTECH’s asset categories and current allocation for a one-year planning horizon. The current allocation for other asset categories, such as cash, is zero. LOWTECH has been following an asset-only approach but has decided to adopt a liability-relative approach. The company is exploring several liability-relative approaches. With respect to surplus optimization, the trustees want to maintain surplus return volatility at a level that tightly controls the risk that the plan will become underfunded, and they would like to keep volatility of surplus below US$0.25 billion (10%).
Exhibit 22:
Asset Categories and Current Allocation for LOWTECH
Private Equity
Real Estate
Hedge Funds
Real Assets
US Equities
Non-US Equities (Developed Markets)
Non-US Equities (Emerging Markets)
US Corporate Bonds
Allocation
20.0%
12.0%
18.0%
7.0%
15.0%
12.0%
8.0%
8.0%
The second step is to estimate future expected asset and liability returns, the expected present value of liabilities, and the volatility of both assets and PV(liabilities). The capital market projections can be made in several ways—based on historical data, economic analysis, or expert judgment, for example. The plan sponsor and its advisers are responsible for employing one or a blend of these approaches. Exhibit 23 shows the plan sponsor’s capital market assumptions over a three- to five-year horizon. Note the inclusion of the present value of liabilities in Exhibit 23.
Exhibit 23:
LOWTECH’s Capital Market Assumptions: Expected Annual Compound Returns and Volatilities
Private Equity
Real Estate
Hedge Funds
Real Assets
US Equities
Non-US Equities (Developed Markets)
Emerging Markets
US Corporate Bonds
Cash
PV (Liabilities)
Expected returns
8.50%
7.50%
7.00%
6.00%
7.50%
7.20%
7.80%
4.90%
1.00%
4.90%
Volatilities
14.20%
9.80%
7.70%
6.10%
18.00%
19.50%
26.30%
5.60%
1.00%
5.60%
Typically, in the third step, the investor imposes constraints on the composition of the asset mix, including policy and legal limits on the amount of capital invested in individual assets or asset categories (e.g., a constraint that an allocation to equities must not exceed 50%). In our example, we simply constrain portfolio weights to be non-negative and to sum to 1.
The fourth step is to estimate the correlation matrix and volatilities. We assume that the liabilities have the same expected returns and volatilities as US corporate bonds; thus, the expanded matrix has a column and a row for liabilities with values equal to the corporate bond values. For simplicity, the investor may employ historical performance. Exhibit 24 shows the correlation matrix of asset categories based on historical quarterly returns. Recall that we assume that liability returns (changes in liabilities) are driven by changes in the returns of US corporate bonds. An alternative approach is to deploy a set of underlying factors that drive the returns of the assets. Factors include changes in nominal and real interest rates, changes in economic activity (such as employment levels), and risk premiums. This type of factor investment model can be applied in an asset-only or a liability-relative asset allocation context.
Exhibit 24:
Correlation Matrix of Returns
Private Equity
Real Estate
Hedge Funds
Real Assets
US Equities
Non-US Equities (Developed Markets)
Non-US Equities (Emerging Markets)
US Corporate Bonds
Cash
PV (Liabilities)
Private equity
1
0.41
0.57
0.32
0.67
0.59
0.49
−0.27
0
−0.27
Real estate
0.41
1
0.45
0.41
0.31
0.33
0.17
−0.08
0
−0.08
Hedge funds
0.57
0.45
1
0.11
0.68
0.61
0.54
−0.23
0
−0.23
Real assets
0.32
0.41
0.11
1
0.04
0.06
−0.06
0.34
0
0.34
US equities
0.67
0.31
0.68
0.04
1
0.88
0.73
−0.38
0
−0.38
Non-US equities (developed)
0.59
0.33
0.61
0.06
0.88
1
0.81
−0.39
0
−0.39
Non-US equities (emerging)
0.49
0.17
0.54
−0.06
0.73
0.81
1
−0.44
0
−0.44
US corporate bonds
−0.27
−0.08
−0.23
0.34
−0.38
−0.39
−0.44
1
0
1
Cash
0
0
0
0
0
0
0
0
1
0
PV(liabilities)
−0.27
−0.08
−0.23
0.34
−0.38
−0.39
−0.44
1
0
1
Exhibit 25 shows a surplus efficient frontier that results from the optimization program based on the inputs from Exhibit 23 and Exhibit 24. Surplus risk (i.e., volatility of surplus) in money terms (US$ billions) is on the x-axis, and expected surplus in money terms (US$ billions) is on the y-axis. By presenting the efficient frontier in money terms, we can associate the level of risk with the level of plan surplus, US$0.329 billion. Like the asset-only efficient frontier, the surplus efficient frontier has a concave shape.
Exhibit 25:
Surplus Efficient Frontier
The first observation is that the current mix in Exhibit 25 lies below the surplus efficient frontier and is thus suboptimal.16 We can attain the same expected total surplus as that of the current mix at a lower level of surplus volatility by choosing the portfolio on the efficient frontier at the current mix’s level of expected total surplus. Another observation is that by uncovering the implications of asset mixes for surplus and surplus volatility, this approach allows the deliberate choice of an asset allocation in terms of the tolerable level of risk in relation to liabilities. It may be the case, for example, that neither the surplus volatility of the current mix nor that of the efficient mix with equal expected surplus is the appropriate level of surplus risk for the pension.
The surplus efficient frontier in Exhibit 25 shows efficient reward–risk combinations but does not indicate the asset class composition of the combinations. Exhibit 26 shows the asset class weights for surplus efficient portfolios.
Exhibit 26:
Surplus Efficient Frontier Asset Allocation Area Graph
Exhibit 27, showing weights for portfolios on the usual asset-only efficient frontier based on the same capital market assumptions reflected in Exhibit 26, makes the point that efficient portfolios from the two perspectives are meaningfully different.17
Exhibit 27:
Asset-Only Efficient Frontier Asset Allocation Area Graph
The asset mixes are very different on the conservative side of the two frontiers. The most conservative mix for the surplus efficient frontier (in Exhibit 26) consists mostly of the US corporate bond index (the hedging asset) because it results in the lowest volatility of surplus over the one-year horizon. Bonds are positively correlated with changes in the present value of the frozen liability cash flows (because the liabilities indicate negative cash flows). In contrast, the most conservative mix for the asset-only efficient frontier (in Exhibit 27) consists chiefly of cash. As long as there is a hedging asset and adequate asset value, the investor can achieve a very low volatility of surplus, and for conservative investors, the asset value at the horizon will be uncertain but the surplus will be constant (or as constant as possible).
The two asset mixes (asset-only and surplus) become similar as the degree of risk aversion decreases, and they are identical for the most aggressive portfolio (private equity). Bonds disappear from the frontier about halfway between the most conservative and the most aggressive mixes, as shown in Exhibit 26 and Exhibit 27.
To summarize, the current asset mix is moderately aggressive and below the surplus efficient frontier. Thus, a mean–variance improvement is possible: either higher expected surplus with the same surplus risk or lower surplus risk for the same expected surplus. The current portfolio is also poorly hedged with regard to surplus volatility; the hedging asset (long bonds in this case) has a low commitment.
The LOWTECH plan has been frozen, and the investment committee is interested in lowering the volatility of the surplus. Accordingly, it seems appropriate to choose an asset allocation toward the left-hand side of the surplus efficient frontier. For instance, a surplus efficient portfolio with about 60% bonds and the remainder in other assets (as can be approximately identified from Exhibit 26) will drop surplus volatility by about 50%.
In the end, the investment committee for the plan sponsor and its advisers and stakeholders are responsible for rendering the best decision, taking into account all of the above considerations. And as always, the recommendations of a portfolio-modeling exercise are only as good as the input data and assumptions.
Multi-Period Portfolio Models
The traditional mean–variance model assumes that the investor follows a buy-and-hold strategy over the planning horizon. Thus, the portfolio is not rebalanced at intermediate dates. A portfolio investment model requires multiple time periods if rebalancing decisions are to be directly incorporated into the model. Mulvey, Pauling, and Madey (2003) discuss the pros and cons of building and implementing multi-period portfolio models. Applicable to both asset-only and liability-relative asset allocation, multi-period portfolio models are more comprehensive than single-period models but are more complex to implement. These models are generally implemented by means of the integrated asset–liability methods discussed in Section 11.
EXAMPLE 5
Surplus Optimization
Explain how surplus optimization solutions differ from mean–variance optimizations based on asset class risk alone.
Solution to 1:
The surplus optimization model considers the impact of asset decisions on the (Market value of assets − Present value of liabilities) at the planning horizon.
What is a liability return?
Solution to 2:
Liability returns measure the time value of money for the liabilities plus any expected changes in the discount rate over the planning horizon.
Compare the composition of a surplus optimal portfolio at two points on the surplus efficient frontier. In particular, take one point at the lower left of the surplus frontier (surplus return = US$0.26 billion) and the other point higher on the surplus efficient frontier (surplus return = US$0.32 billion). Refer to Exhibit 26. Explain the observed relationship in terms of the use of corporate bonds as the hedging asset for the liabilities.
Solution to 3:
Whereas the portfolio at the US$0.26 billion surplus return point on the efficient frontier has a substantial position in corporate bonds, the efficient mix with US$0.32 billion surplus return does not include them. The observed relationship that the allocation to corporate bonds declines with increasing surplus return can be explained by the positive correlation of bond price with the present value of liabilities. The hedging asset (corporate bonds) is employed to a greater degree at the low end of the surplus efficient frontier.
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