Chapter 3 performance and risk measurement for pension funds

Th e fi rst new element is by measuring the performance of a pension fund relative to the value of the liabilities in order to focus on the societal objective of a pension fund as the pr

Performance and

Risk Measurement

for Pension Funds

Auke Plantinga

CONTENTS

3.2 Liability-Driven Investing and Pension Liabilities 72 3.3 Liability-Driven Risk and Performance Attribution 76

This ch a pter p r oposes a n a ttribution m odel f or m onitoring t he

performance and risk of a defi ned benefi t pension fund In order to facilitate easy interpretation of the results, the return is expressed as a percentage of the value of the liabilities Th e model is based on a liability benchmark t hat refl ects the risk and return characteristics of liabilities

As a r esult, t he attribution model focuses t he attention of t he portfolio managers on creating a portfolio that replicates liabilities Th e attribution model allocates diff erences in return between the actual and benchmark portfolio to decisions relative to the benchmark portfolio In addition, the model decomposes risks according to the same structure by using a mea-sure of downside risk

3.1 INTRODUCTION

Pension funds around the world suff ered dramatically from falling stock markets in the period 1999–2003 as well as during the current 2008–2009

fi nancial crisis Many pension funds have become underfunded, and face funding ratios well below 100% In both periods, stock markets performed badly and interest rates dropped considerably Th ese are conditions that have a ser ious impact on pension fund balance sheets, by decreasing the market value of the assets and increasing the market or fair value of the lia-bilities Ryan and Fabozzi (2002) provide evidence of the existence of large mismatches between assets and liabilities in the 200 largest U.S defi ned-benefi t plans prior to the fi rst pension crisis For example, in 1995 the assets

of these 200 pension funds yielded an average positive return of 28.70%, which was insuffi cient to match with the liability return of 41.16%

Liability-driven in vesting i s a n in vestment p hilosophy t hat im plies managing a ssets r elative t o l iabilities.* W ith l iability-driven i nvesting, pension funds can reduce their exposure to fi nancial crises by reducing the mismatches between assets and liabilities

In order to encourage pension funds to adopt liability-driven investment strategies, it is necessary to redesign performance a nd risk measurement systems Th e best way to accomplish this is by adopting a so-called liability-driven benchmark for performance and risk management

In this study, we discuss a performance and risk measurement system that focuses on liability-driven i nvesting Without appropriate per for-mance and risk measurement, liability-driven investing will be diffi cult

to manage as portfolio managers will be tempted to focus on traditional asset-only benchmarks We propose a number of new elements in mea-suring performance and risk for pension funds Th e fi rst new element

is by measuring the performance of a pension fund relative to the value

of the liabilities in order to focus on the societal objective of a pension fund as the provider of reliable pensions Th e second new element is that

we propose to use a decomposition of downside risk consistent with the decomposition of performance

3.2 LIABILITY-DRIVEN INVESTING AND PENSION LIABILITIES

Liability-driven investing is a strategy aimed at reducing a pension plan’s risks by choosing assets that serve as a hedge for the risks implicit in the liabilities Th is i s only pos sible for a l imited se t of r isk fac tors t hat a re

* See, for example, Leibowitz (1986), Siegel and Waring (2004), and Waring (2004).

present both i n t he a sset portfolio a nd t he l iability portfolio Th e most important of these risks are the interest rate risk and the infl ation risk.* Some other risks in the liability portfolio are diffi cult to hedge with tra-ditional asset classes, such a s the mortality risk or the operational risk

Th ose types of risks can be reduced by, for example, reinsurance, or simply

by creating an additional solvency buff er for u nexpected losses I n t his study, we implement liability-driven investing using cash fl ow matching Cash fl ow matching is a robust strategy for dealing with any type of term structure movement Practical considerations may result in choosing for a less robust solution, such as duration matching

Pension claims in a defi ned-benefi t plan are largely determined by the wages earned by the benefi ciaries and the number of years of employment With a fi nal pay system, the pension claims are based on the wage earned

in the last year However, it is also common to have claims as a function

of past wages earned, such a s the average wage earned during the years

of employment A co rrect va luation of t he l iabilities i s c rucial i n ma n-aging pension f unds Th e va lue of l iabilities ser ves a s a ben chmark for the level of assets needed to service the future cash fl ows of the pension fund Th e valuation starts with an appropriate estimate of the future cash

fl ows resulting from the current promises made to benefi ciaries Next, the present value of these cash fl ows is calculated using a discount rate Th is discount rate is a c rucial element in the liability-driven benchmark We propose to use the risk-free rate on government bonds as the discount rate

In combination with a matching strategy and suffi cient assets, this means that the liabilities will be met with certainty.† From the perspective of the benefi ciary, a va luation based on the risk-free government bond rates is the preferred alternative since it provides the opportunity to create risk-less pension claims It is important to see that the resulting value is not the market value of the liabilities, but the value of a risk-free cash fl ow desired

by the benefi ciaries If the present value of the liabilities based on govern-ment bond rates exceeds the value of the assets, it is clear that the true market value of the liabilities is less, unless the sponsor provides a credible guarantee to make up for any defi cits

Th e return on the cash fl ow matching strategy is the minimum accept-able return (MAR) to be achieved on the assets (see Sortino et al., 1999)

If the assets do not yield this return, the future pension benefi ts cannot be

* Another factor is currency risk, which we ignore in this chapter by assuming that the pension fund has single currency pension liabilities.

† Th e conditions for creating an immunized portfolio are specifi ed by Redington (1952).

realized without further sponsor contributions Th e actual realization of reliable pensions depends on the investment strategy In order to keep the focus of portfolio managers on the objective of the pension benefi ciaries, the cash fl ow matching strategy, which gives the best chance of realizing the objective, has to be embedded in the asset benchmark

Elton and Gruber (1992) provide a t heoretical justifi cation for the use

of matching strategies in the context of a mean–variance framework For

a pension fund with assets A and present value of liabilities L, the return

on surplus is

−

t

S A r L r r

Th e riskless strategy, F, is to invest an amount, L, in a po rtfolio of cash

fl ow–matched assets, a nd to i nvest t he remainder i n t reasury bills Th e return of this strategy is

−

f

Similarly, the return on a risky strategy R is

Any co mbination o f t hese t wo po rtfolios r esults i n a po rtfolio o n a

straight line connecting F and R in the mean standard deviation space,

similar to the capital market line in the CAPM However, the underlying portfolios are diff erent and depend on the leverage and liability returns Consequently, diff erent investors have opportunity sets Th e actual choice for an asset portfolio depends on the risk aversion of the pension fund Pension funds have a multitude of participants that also have diff erent interests Th erefore, t he question how to choose a l evel of r isk aversion for the pension fund that is representative for all participants is not easy and perhaps even impossible to answer In many of the surplus optimiza-tion models, the surplus is owned by the plan sponsor, and the criterion for choosing an asset portfolio is to maximize the utility of the surplus.*

As stated by Erza (1991): “A defi ned benefi t pension plan is eff ectively an operating d ivision of t he sponsor.” With pension f unds bei ng operating divisions of sponsors, confl icts of interest with the benefi ciaries may arise

* See, for example, Elton and Gruber (1992), Erza (1991), or Sharpe and Tint (1992).

Benefi ciaries have a d iff erent perception of risk From a ben efi ciary’s point of view, any asset portfolio is risk-free as long as the pension fund

allocates an amount Al ≥ L to a cash fl ow–matched portfolio In addition, the strategy for the surplus assets As should return above the threshold

level −As/S If returns on the surplus asset are below the threshold value,

the surplus assets become negative and this decreases the value of liability-driven in vestments B enefi ciaries ha ve o nly l imited i nterest i n r eturns above that required to service liabilities, since they usually do not share in the surplus returns

Th e diff erent perspectives on risk by the sponsor and the benefi ciaries motivate a d istinction be tween su rplus a ssets a nd l iability-driven a ssets

As long as the sponsor is willing to choose an asset portfolio that is risk-free from the perspective of the benefi ciaries, the potential confl ict of interest

is not relevant By separating the assets in two portfolios, one dedicated to liability-driven investments and one to surplus-driven investments, the par-ticipants can easily monitor the value of the portfolio dedicated to service their claims Participants or their representatives should be able to observe whether large diff erences exist between the value of liabilities and liability-driven asset portfolios If large diff erences do exist, this is a sign that the fund has potentially a la rge exposure to risk In combination with a r isk-monitoring system, this safeguards the continuity of the pension fund

In constructing a liability-driven investment portfolio, it is important to make a distinction between nominal and real liabilities Th e liabilities of pen-sion funds can be nominal, real, or a mixture of nominal and real Nominal liabilities are promises to pay future cash fl ows expressed in nominal terms Real liabilities are promises to pay future cash fl ows with a fi xed purchasing power Each type of liability should be matched with the appropriate type of asset Nominal liabilities can be matched with nominal government bonds and real liabilities can be matched with infl ation-linked bonds such as TIPS Since each type of liability requires a diff erent type of instrument, it is con-venient to create separate asset portfolios for each type of liability

We use cash fl ow matching as the strategy to construct the benchmark

In practice, duration matching and other factor immunization strategies are f requently used as a n a lternative for reasons of fl exibility Duration matching opens the possibility to create a ma tched portfolio even if the sum of the maturities of the liability cash fl ows exceed the largest avail-able asset maturity However, matching based on duration may not shield against c ertain n onparallel s hift s o f t he ter m st ructure Th er efore, we believe t hat i n co nstructing t he ben chmark, c ash fl ow matching is the best alternative as it provides the best track of the value of the liabilities

Duration matching can be used to create the actual asset portfolio As a result, deviations between t he return of t he duration and t he cash fl ow matching strategy are measured in the performance attribution model

3.3 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:

Nominal assets Anl Nominal liabilities Ln

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

x

n x x

S r L n

=

∆

where

ιx is a dummy variable with value 1 for all ∆S x /L < rmar

∆S x /L is the change in surplus in simulation run x scaled by liability value

rmar = 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:

= −

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 p

nl

r , and the return on the portfolio

matched with the real liabilities is p

rl

r

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

s /

s

r

Nominal liability-driven

p n

nl /

nl

r

r

rl /

rl

r

TABLE 3.2 Decision Process

Actual Return Benchmark Return Active Return

Surplus-driven

assets

a s

s

s s

[r r S L] / Liability-driven

nominal assets

a nl

nl

nl nl n

[r r L L] / Liability-driven

real assets

a rl

rl

rl rl r

[r r L L] / Allocation

s s n n nl r r rl

[(A S r) (A L r) (A L r L) ]/

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

S A r A r A r L r L r

where ∆Sa i s t he r ealized cha nge i n su rplus va lue S ince pens ion f unds 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:

S A r A L r A L r L r r L r r

Th e benchmark is based on cash fl ow matching As a result, the allocation

to the three asset classes equals As = S, An = Ln, and Ar = Lr Th e benchmark return is defi ned as

S S r L r r L r r

where ∆Sp is the change in surplus based on the benchmark portfolio With perfect ma tching, p

nl

r −r = and p

rl

r − = I n p ractice, de viations a re r

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:

−

(i) (ii) (iii)

( )

( ) ( )

S

r r

L

S S A S r A L r A L r

L r r L r r

L L

(3.9)

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

1 DR DR

n

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

3.4 AN APPLICATION OF THE MODEL

In this section, we provide an illustration of the performance and risk attri-bution model proposed in Section 3.3 We consider a hypothetical pension fund with a mix of real and nominal liabilities Consistent with our attribu-tion model, the fund has divided its assets in three diff erent portfolios: sur-plus assets, nominal, and real liability-driven assets Th e sursur-plus assets are invested in risky investments such as stocks Nominal and real assets are

fi xed income securities in terms of nominal or real terms Th e duration of the assets is shorter than the duration of the corresponding liabilities Th e market value of assets and the value of the liabilities are presented in Table 3.3 From Table 3.3 we can infer that the funding ratio is 133%, which indi-cates a well-funded pension plan Participants in the pension fund should