LIABILITY-DRIVEN RISK AND PERFORMANCE ATTRIBUTION

In developing benchmarks and a performance attribution model, we value the assets based on their market values, and the liabilities as the present value of the expected cash fl ows using the nominal- and real-term struc-tures of interest rates. Th is results in the following stylized balance sheet:

Assets Liabilities

Surplus assets A_s Surplus S

Nominal assets A_nl Nominal liabilities L_n Real assets Arl Real liabilities Lr

Managing the performance and risk of pension fund assets relative to its liabilities ha s i mplications f or t he per formance a nd r isk m easures, t he benchmark and the attribution model.

3.3.1 Performance Measure

Th e performance measure should be chosen consistent with the objective of the pension fund in mind. Th e main objective of a pension fund is to provide benefi ts to its members. Th is suggests that we should focus on measuring the performance from the perspective of the liabilities. We pro-pose to measure the performance by scaling the changes in surplus by the liability value as opposed to scaling by assets or scaling by surplus. From a mathematical point of view, scaling is irrelevant for the decomposition.

However, the major advantage of scaling by liability value is that losses are expressed in terms of liability value, so benefi ciaries can directly relate to the magnitude of losses. A loss of 10% for an underfunded pension plan implies that, roughly speaking, a benefi ciary is likely to receive 10% less in pension payments.* Th is measure of pension fund performance is closely related to the funding ratio return proposed by Leibowitz et al. (1994).

3.3.2 Risk Measures

Since the sponsor and the benefi ciaries have diff erent perspectives on risk, it makes sense to use two risk measures. Th e diff erence in asset and liability

*Th is is diff erent from the surplus approach proposed by Ezra (1991) and Sharpe and Tint (1990).

returns is a general measure of risk, as it focuses on changes in the value of surplus. Th is measure is relevant for both parties involved in the pension fund. We use a downside risk measure as suggested by Sortino and Van der Meer (1991) in order to accommodate deviations from normality in the return distribution. Th e downside risk measure is calculated based on a simulated distribution of possible outcomes:

2 mar 1

DR

n x x

x

S r L

= n

⎛∆ − ⎞

⎜ ⎟

⎝ ⎠

=

∑

^ι _{(3 .4)}

where

ι_x is a dummy variable with value 1 for all ∆S_x/L < r_mar

∆Sx/L is the change in surplus in simulation run x scaled by liability value r_mar = 0

n is the number of simulation runs

If the surplus return is negative, asset returns are lower than liability returns.

Benefi ciaries a re ma inly concerned w ith losses as fa r as t hey decrease the value of their benefi ts. Th is motivates a second downside risk measure, based on a more conservative minimal acceptable rate of return:

mar*= −S

r L (3 .5)

which is t he t hreshold return t hat diff erentiates between ending w ith a positive or a negative surplus value.

In calculating downside risk measures, we need a distribution of asset and liability returns. Using historical numbers is not very meaningful in constructing such a d istribution since the nature of assets and liabilities may change over time. Th erefore, we use a simulation model in the spirit of the value-at-risk models used in banking. Th ere is a vast literature on how to create such m odels, which is beyond the scope of this study. At this point, it suffi ces to state that the measures are calculated over the out-comes of the diff erent simulation runs.

3.3.3 Benchmark

Th e benchmark for the pension fund is a weighted average of the bench-mark return for the surplus investments and the two groups of liability-driven investments (Table 3.1).

Th e benchmarks for the liability-driven assets are based on replicating the return and risk of the liabilities. Given that defi ned-benefi t pensions are fi xed in terms of nominal or real benefi t, we use portfolios of bonds that are cash fl ow–matched with the liabilities. Th e return on the portfo-lio matched with the nominal liability is r , and the return on the portfolio _nl^p matched with the real liabilities is r ._rl^p

3.3.4 Performance Attribution

Th e basis for the performance attribution model is the benchmark speci-fi ed abo ve i n co njunction w ith t he dec ision p rocess w ithin t he pens ion fund. Consistent with our portfolio and benchmark structure, we distin-guish between active decisions within one of the asset portfolios and active decision with respect to the allocation to each of the asset portfolios. Active decisions within one of the asset portfolios refer to the decision to have a portfolio with a diff erent composition as compared to the benchmark. For the surplus-driven assets, this could be the decision to invest in an actively managed fund. For the liability-driven assets, this is the decision to invest in, for example, bonds with diff erent maturities or to invest in bonds with a d iff erent c redit qu ality. Th e a llocation m ismatches a re t he de viations between the market value of the asset portfolios and the value of the liabil-ity portfolios that they aim to cover. In Table 3.2, we summarize the main sources of return. We express the active return and the return on allocation mismatches in terms of our proposed performance measure.

TABLE 3.1 Benchmark Composition and Return

Portfolio Benchmark Weight Benchmark Return

Surplus-driven assets ws^p=S A/ rs^p

Nominal liability-driven

assets wnl^p =L An/ r_nl^p

Real liability-driven assets wrl^p=L Ar/ r_rl^p

TABLE 3.2 Decision Process

Actual Return Benchmark Return Active Return Surplus-driven

assets

as

r r^ps [r r S Ls^a− s^p] /

Liability-driven nominal assets

nla

r r_nl^p [r r L Lnl^a− nl^p] /n

Liability-driven real assets

rla

r r_rl^p [r r L Lrl^a− rl^p] /r

Allocation

mismatches [(A S rs− )s^a+(A L rn− n nl) ^a+(A L r Lr− r rl) ]/^a

We construct a performance attribution model by analyzing the incre-mental i mpact of i ndividual d ecisions on b enchmark re turns. Th is is accomplished by creating expressions for the realized surplus returns and the benchmark surplus returns and taking the diff erence between these expressions.

Th e realized surplus return in money terms is

p should be managed in the interest of the pension benefi ciaries, we calculate the surplus return relative to the value of the liabilities, which results in the following expression:

∆^Sa = ^Asr_s^a+^{A Ln}− ⁿr_nl^a +^{A Lr}− ^rr_rl^a+^Ln

(

r_nl^a −r_nl^p

)

−^Lr

(

r r_rl^a− _rl^p

)

L L L L L L (3.7)

Th e benchmark is based on cash fl ow matching. As a result, the allocation to the three asset classes equals A_s = S, A_n = L_n, and A_r = L_r. Th e benchmark return is defi ned as

∆ =_L^Sp _L^Sr_s^p+^L_Lⁿ

(

^r_nl^p−^r_nl

)

+^L_L^r

(

^r_rl^p−^r_rl

)

(3.8) where ∆S^p is the change in surplus based on the benchmark portfolio. With perfect ma tching, rnl^p−rnl = and 0 rrl^p− = . I n p ractice, de viations a re rrl 0 likely to occur for several technical reasons, such as mortality risk and small mismatches between the liabilities and the benchmark portfolios may result in small positive or negative returns. For analyzing the investment perfor-mance of a pension fund, this is not a major concern. By calculating the diff er-ence between Equations 3.7 and 3.8, we calculate the diff erer-ence between the return on the actual portfolio and the benchmark. Since the liability returns are present in both Equations 3.7 and 3.8 it cancels out in the diff erence. As a result, we are able to construct the following performance attribution:

−

where component (i) is the return added by means of active management with the surplus portfolio, component (ii) is the funding allocation mis-matches, and component (iii) is the return on the maturity mismatches of both nominal and real duration mismatches.

In addition to performing a return attribution model, we present a risk attribution model where we decompose the diff erence in downside risk between the actual and the benchmark portfolio into components con-sistent with Equation 3.9. In order to do this, we use the decomposition of downside risk proposed by Reed et al. (2008).

Reed et al. (2008) have shown how each asset i contributes to the total downside risk of a portfolio p:

=

− −

= ¹

Σ

₁ ^{(mar , )(mar , )}

DR DR

n i i x p x

i x

w r r r r

n (3 .10)

where

x is an integer identifying a particular scenario

w_i is the market value of asset i as a fraction of total assets r_i,x is the return of asset i

r_p,x is the portfolio return in scenario x n represents the total number of scenarios

Th e sum of t he individual contributions is equal to total downside risk DR. A similar decomposition can be made for the decisions in the pension fund by recognizing that each decision is a long-short portfolio reallocating money from one part of the portfolio to another.

In document 10610.Pension Fund Risk Management Financial and Actuarial Modeling (Chapman & HallCrc Finance Series) (Page 113-117)