Chapter 1 quantifying investment risk in pension funds

1 C H A P T E R Quantifying Investment Risk in Pension Funds Shane Francis Whelan* CONTENTS 1.3 Case Studies Estimating Investment Risk 101.3.1 Pension Saving, Person Aged 55 Years and O

1

C H A P T E R

Quantifying Investment Risk in Pension Funds

Shane Francis Whelan*

CONTENTS

1.3 Case Studies Estimating Investment Risk 101.3.1 Pension Saving, Person Aged 55 Years and Over 121.3.2 Case Study 1: Measurement of Investment Risk

in Pension Funds—Termination Liability 161.3.3 Case Study 2: Measurement of Investment Risk

in Pension Funds—Ongoing Liabilities 23

The co ncept o f i nvestm ent r isk is generalized, wh ich a llows t he

quantifi cation of the investment risk associated with any given ment strategy to provide for a pension Case studies, using historic mar-ket data over the long term, estimate the investment risk associated with diff erent investment strategies It is shown that a few decades ago, when

invest-* Th is chapter is based on my paper, “Defi ning and measuring risk in defi ned benefi t pension

funds,” Annals of Actuarial Science II(1): 54–66 I t hank the copyright holders, the Faculty

and Institute of Actuaries, for permission to reproduce and extend that paper here.

bond markets only extended in depth to 20 year maturities, the ment risk of investing in equities was of the same order of magnitude as the investment risk introduced by the duration mismatch from investing

invest-in bonds for immature schemes It is shown that now, with the extension

of the term of bond markets and the introduction of strippable bonds, the least-risk portfolio for the same pension liability is a bond portfolio of suit-able duration It is argued that the investment risk voluntary undertaken

in defi ned benefi t pension plans has grown markedly in recent decades

at a t ime when t he ability to be ar t he i nvestment r isk ha s d iminished Investment risk in pension funds is quite diff erent to i nvestment risk of other investors, which leads to the possibility that current portfolios are not optimized—that is, there exist portfolios that increase the expected surplus without increasing risk Th e formalizing of our intuitive concept

of investment risk in pension saving is a fi rst step in the identifi cation of more effi cient portfolios

Keywords: Investment risk, defi ned benefi t pension funds,

invest-ment strategies, actuarial investigations

pen-Th is cha pter, ex tending W helan ( 2007), p roposes a defi nition o f investment risk that formalizes our intuitive concept We develop, in

a more technical setting, ideas fi rst presented in Arthur and Randall (1989) and provide, using historic data on the United Kingdom, United States, and Irish capital markets, an empirical assessment of the mag-nitude of risk entailed by diff erent investment strategies and relative to diff erent objectives Th e analysis, through a series of case studies, leads

to a rather simple conclusion: sovereign bond portfolios (of appropriate duration a nd i ndex-linked/nominal m ix) a re t he least-risk portfolios for pens ion s avers, i rrespective of t he a ge of t he pension s aver, i rre-spective of the currency of the pension and, within a reasonable range, irrespective of the precise investment objectives of the pension saver

Th e a nalysis a llows u s to quantify t he r isks i n a ll i nvestment st gies, and we provide fi gures for the risks inherent in investing in equi-ties, conventional long bonds, cash, and the closest matching bonds by duration.

rate-Investment risk is defi ned in Section 1.2 and some of its properties are considered From t he defi nition, one c an quantify t he i nvestment risk inherent in any given investment strategy and thereby identify the strategy with the lowest investment risk Section 1.3 reports the results

of case studies that quantify the investment risk for pension savers from various diff erent investment strategies Th is analysis shows that the rela-tive risk inherent in diff erent strategies appears to be very similar over diff erent t ime per iods a nd d iff erent na tional ma rkets a nd r easonably robust when the objective is to provide pensions in deferment increasing

in line with wages or increasing in line with infl ation subject to a nal cap We get an important insight from this analysis: even conven-tional long bonds are not long enough to match the liabilities of young scheme members, and investing in such bonds can be as risky as invest-ing in equities but without the expected rewards We conclude that just

nomi-as much care must be exercised in matching liabilities by duration nomi-as

in matching liabilities by asset type Section 1.4 demonstrates the laciousness of the argument that the risk of equity investment dissipates with time so that, at some long investment horizon, equities are prefer-able o ver o ther a sset cla sses b y a ny r ational i nvestor Th is argument, generally known as the “time diversifi cation of risk,” does not hold in that strong a form True, the expected return from equities might well

fal-be higher than other asset classes but, on some measures, so too is the risk and this remains true over all time horizons We conclude that the most cl osely ma tching a sset f or pens ion f und l iabilities i s co mposed mainly of conventional and index-linked bonds, which, if history is any guide, has a lower expected long-term return than a predominantly equity portfolio

Our analysis does not allow us to suggest that one investment strategy

is preferable to another Investors, including pension providers, routinely take risks if the reward is judged suffi ciently tempting However, pen-sion providers should appreciate the risks involved in alternative strate-gies and, at a minimum, seek to ensure that the investment portfolio is effi cient in the sense that risk cannot be diminished without diminishing reward

1.2 DEFINING INVESTMENT RISK

Th ere would be n o concept of risk if a ll the expectations were fulfi lled: risk arises from a clash between reality and expectations Accordingly, one

fi rst needs to formulate and make explicit future expectations before risk can be quantifi ed Note that future expectations at any point in time are dependent to an extent on the experience up to that time, as past experi-ences infl uence future expectations

Our intuitive notion of investment risk is that it measures the fi nancial impact when the actual investment experience diff ers from that expected, holding all other things equal In this section, we formalize this notion Once the investment risk is properly defi ned, it is straightforward (in the-ory at least) to measure and attempt to minimize it

Th e task of formally setting down future expectations when it comes

to investing to generate a ser ies of future cash fl ows is oft en known as

a “valuation” (e.g., t he ac tuarial va luation o f defi ned benefi t schemes)

We adopt this terminology and call the desired series of cash fl ows the

“liabilities.”

Let t = 0 represent the present time and t > 0 be a future time Let A t denote

the forecast cash fl ow from the assets at time t and L t be the forecast liability

cash fl ow at time t We shall assume, for convenience, that the investment

return expected over each unit time period in the future is constant; it is

denoted as i and termed as the “valuation rate of interest.” It will be clear that allowing i to vary with the time period poses no theoretical issues Th e

reported valuation result at time 0, ex pressing the surplus (if positive) or

defi cit (if negative) of assets relative to liabilities, is denoted as X0 Th us

Consider X0 We shall assume that this is a number.* So, under this

simpli-fying assumption, X0 is a constant, representing the surplus at the present time identifi ed by the specifi ed (deterministic) valuation methodology

* If this is allowed to be a nonconstant random variable, then we call the valuation ogy used stochastic otherwise the valuation approach is said to be deterministic Note that a stochastic valuation is representing some part of the assets and/or liabilities as a nontrivial random variable at t ime 0 We shall d iscuss on ly deterministic valuation methods i n t he sequel to si mplify the analysis but, as should be clear, the results carry through (with rela- tively straightforward extensions) when applied to stochastic valuation approaches.

methodol-Let p be t he time that the next valuation falls due methodol-Let 0

p

X represent the

results of the next valuation at time p, using the same underlying assumptions

as used in the valuation at time 0 Th en the relationship between X0 and 0

Th is can be readily seen, as the cash fl ow in the inter-valuation period will

be invested (or disinvested) at the valuation rate of interest, accumulating

amount is to be added t o the discounted value of all the yet unrealized

asset and liability cash fl ows at time p, namely, 0

Equation 1.2 multiplied by (1 + i) p, whence the result

It is generally possible to form a reasonable apportionment of the diff ence of the valuation result at the next valuation date from that expected

er-from the valuation at time 0 (i.e., X0(1 + i) p) into that due to either

1 Th e actual experience over the inter-valuation period diff ering from that assumed, or

2 A changed valuation method or basis applied at time p

In particular, it is possible to form a reasonable assessment of the fi nancial impact of the actual investment experience relative to that expected, other things being held the same

Let X denote the result of the valuation at time 0, under the same 0i−p

methodology and assumptions as underlying the valuation result, X0,

at t ime 0 b ut now refl ecting the actual investment experience in the inter-valuation period Th en 0i− − 0

p

time 0 of how the actual investment experience up to time p diff ered

from that assumed in the original valuation at time 0 Obviously, if it turns out that the actual investment experience bears out the assumed experience i n t he i nter-valuation pe riod t hen 0i− = 0

mea-experience up to time p diff ers from the investment assumptions

underly-ing the valuation at time 0 Th is key concept deserves a defi nition

Defi nition of investment variation (up to time p): Th e fi nancial impact at

time 0 created when the actual investment experience up to time p diff ers

from the investment assumptions underlying the valuation at time 0, a ll other t hings being equal In t he notation introduced earlier, t he invest-ment variation is denoted 0i− − 0

p

Th e investment variation up to time p can generally only be measured

at time p, before that it may be modeled as a random variable with an

asso-ciated distribution Viewed in this way, the investment variation at time 0,

up to time p, is a random variable Th e investment variation at time 0 can

be viewed as a stochastic process, 0i− − 0

p

− −

0i p 0

associated distribution Th e mean of this distribution captures the bias

in the original investment assumptions—a positive mean implies that the original investment assumptions were conservative (as, on aver-age, t he ex perienced co nditions t urn o ut be tter t han t hat o riginally forecast)

Note that if t he valuation is t esting the adequacy of the existing folio, and future prescribed contributions, to generate future cash fl ows

port-to meet t he targeted pension payments then other expectations (e.g., on future mortality) must als o b e em bedded in t he lia bility cash fl ows In the defi nition of the investment variation, these noninvestment expecta-tions are held constant, so only the impact of the variation in the invest-ment experience is me asured Th e actual scale of the resultant fi gure for the obs erved investment variation is, t hough, a f unction of t hese other expectations

Some prefer to give a single number to capture the notion of riskiness

in a distribution, oft en using some parameter that measures the spread

of t he d istribution, such a s i ts st andard de viation, i ts sem i-variance,

or its inter-quartile spread Oft en this summary measure is called the

“ investment r isk.” A lternatively, o ne c an a pply so me o ther m easures such as the value below in which there is a specifi ed low probability of the outcome fa lling (the so-called va lue-at-risk).* Th e key point to be made is that the distribution of 0i− − 0

p

con-cept and maintains more information than any summary spread tic We do not enter into the wider discussion of the most appropriate measure t o a pply t o t he i nvestment va riation d istribution t o c apture our intuitive notion of risk but adopt the generally accepted measure of standard deviation So we identify, to a fi rst-order approximation, the investment risk as the standard deviation of the investment variation distribution

of t he (ex ante) i nvestment va riation d istribution For concreteness, we

shall use the standard deviation as our measure of investment risk in the sequel

If the valuer has p erfect foresight, then the investment assumptions would be perfectly in line wi th the future investment experience, and

so t he in vestment va riation distr ibution w ould b e a deg enerate co stant, wi th a st andard de viation o f zer o M ore uncer tainty a bout t he

n-investment variation implies a greater spread of the (ex ante)

distribu-tion, which corresponds to a gr eater investment risk under t he above defi nition

If we have a perfect matching of assets to liabilities,† t hen a ny va ation m ethod w ill a lways r eport t he i nvestment va riation t o be a degenerate d istribution ( i.e., a co nstant) a nd, acco rdingly, t he i nvest-ment risk to be zero Th is c an be se en a s, b y per fect ma tching,

of the assets at time 0 (i.e., 0 (1 ) t)

or higher even moments if they exist.

† In the technical sense that A t = L t , for all t, independent of any investment assumptions.

investment assumptions, it must vary in a direction proportion to

−

t

to that expected is exac tly off set by an increase (decrease, respectively)

in the value of the liabilities relative to that expected In short, the fect matching of t he ass et and liability cash fl ows has zer o investment variation, ir respective o f t he exp erienced o r t he assumed in vestment conditions

per-Let us assume t hat (1) ass ets a re t o b e val ued a t ma rket val ue a nd (2) there exists a p ortfolio of assets that perfectly matches the liabilities Note, from earlier considerations, we know that if the matching asset port-folio was held at time 0 then the investment variation would be 0 (irrespec-

tive of what happened in t he inter-valuation period) Also, at time p, the

present value of the future liabilities must be equal to the market value of the matching asset at that time (by the defi nition of matching asset) Hence the experienced valuation rate in t he inter-valuation period can now be seen as t he market return on the matching asset over the inter-valuation period We see immediately from this that the investment variation is posi-tive only if the increase in the market value of the actual assets held exceeds the increase in the market value of the matching asset.* Th e upshot is that the investment variation is t he present value of the extent to which the increase in the value of the assets exceeds the increase in the liabilities over the inter-valuation period, discounted at the rate of return on the matching asset over the period.†

Appendix 1.A.1 draws attention to a major limitation of our defi nition

of investment variation (and the associated investment risk) for pension investors

1.3 CASE STUDIES ESTIMATING INVESTMENT RISK

Estimating the investment risk has been identifi ed in the last section with

estimating t he st andard de viation of t he (ex ante) i nvestment va riation distribution L et u s a ssume t hat t he ex post investment variation is a reasonable proxy for the ex ante investment variation, that is, make the

* Or, as expressed in Arthur and Randall (1989), “the Main Guiding Principle merely

reaf-fi rms an earlier fundamental principle, namely that if you are mismatched and you get your forecasts wrong then you have to pay the penalty” (Section 2.5).

† Th is expresses, in more technical terms, the “Main Guiding Principle” of Arthur and Randall (1989) that states “that if there is a rectifi able mismatch, a relative change in market values of the matched and mismatched assets should be refl ected in the valuation result” (Section 5.1).

commonplace assumption t hat t he historical experience can be u sed to

assess the realistic ex ante expectations.

Th is section presents two case studies designed to explore the relative investment risk of diff erent investment strategies for those attempting to provide a pens ion However, before delving into the case studies proper,

we begin with by considering the case of a person aged 55 years or over attempting t o p rovide a pens ion—in r eal o r n ominal ter ms—from a ge

65 years Th is provides some insights to identifying the least-risk portfolio for pension savers at all ages which, as it turns out, is confi rmed by the case studies

Th e case studies determine the historic investment risk for a pens ion saver attempting to provide a pension by investing in, alternatively, a broad equity index, a 20 year conventional bond, a 30 year bullet bond, and short-term cash instruments in (a) the U.K markets, (b) the U.S markets, and (c) the Irish markets We give several descriptors of the investment varia-tion distribution from the historic data—including the key measures of its geometric mean and its standard deviation (or investment risk) Th e se two latter summary measures give an illustration of the relative rewards of the diff erent strategies and, to a fi rst approximation, the risks associated with the strategies

Th e fi rst case study takes a relatively low value of the targeted pension,

by assuming that the pension before vesting escalates at infl ation subject

to a nominal cap Th is corresponds to the liability that a defi ned benefi t scheme i n I reland ha s on ter mination to contractual pension promises under current regulations In the second case study, we assume that the pension prior to vesting will increase in line with wage increases, refl ect-ing the pension liability for the fi nal salary-defi ned benefi t schemes on an ongoing basis We treat, in both cases, the position of a 40 and a 30 year old with a pension due from their 65th birthday

A picture of the ex post investment variation distribution associated with

investing in the various asset classes are computed in the following manner

At the valuation date, it is assumed that the market value of the assets equals the value of the liabilities on a market-consistent basis Th e investment over the year subsequent to the valuation is assumed t o be alternatively in e ach diff erent asset class Each investment strategy for each of the two case studies

at each age generates n data points where n is n umber of years in t he

his-toric period studied Each data point gives the present value of the surplus or defi cit arising over the year, expressed as a percentage of the market value of assets at time 0 (termed the “standardized investment variation”) From these

0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 1976 1974 1972 1970 1968 1966 1964 1962 1960 1958

1956

1954

1952

1950

Irish United Kingdom United States

FIGURE 1.1 Long bond gross redemption yield, United States, United Kingdom, and Ireland, year ends, 1950–2000 (inclusive) See text for sources

data, the key summary statistics of the empirical investment variation

distri-bution (p = 1) for each investment strategy are tabulated, such as the mean,

the median, the geometric mean, the standard deviation (which equates to the investment risk up to 1 year), and the higher moments

Annual returns and yields from the United Kingdom, United States, and

Irish bond, equity, and cash markets were sourced from Barclays Capital

(2003), Dimson et al (2004), Mitchell (1988), and Whelan (2004) Figures 1.1 and 1.2 display, respectively, the 20 year sovereign bond yield and a broad-based equity index, from each national market over the second half

of the twentieth century

Note that prior to 1978 the yield on Irish long bonds was almost cal to the United Kingdom long bonds because of the currency link.1.3.1 Pension Saving, Person Aged 55 Years and Over

identi-Consider a person aged 55 years targeting a pension from age 65 years, the pension subject to either infl ation-linked or fi xed rate increases both prior to retirement and while in payment For concreteness, we shall make the demo-graphic assumption that the person will die on his 85th birthday Accordingly, the liability in this case is a ser ies of real or nominal amounts falling in a regular pattern, beginning in 10 years’ time and ending in 30 years’ time

100 1,000 10,000 100,000 1,000,000 10,000,000

2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 1976 1974 1972 1970 1968 1966 1964 1962 1960

Irish United Kingdom United States

FIGURE 1.2 Equity market total return indices, United States, United Kingdom, and Ireland, year ends, 1950–2000 (log scale) See text for sources

From our defi nition of the investment variation and the investment risk earlier, it is clear that minimizing the investment risk requires investing in

an asset portfolio that provides an income that most closely matches this liability stream Whether these liabilities are nominal or real in sterling, euro, and dollar, there is arguably a su ffi ciently deep market in conven-tional and index-linked sovereign bonds so t hat a near-perfect matching portfolio can be constructed

First, consider the case that the liability cash fl ows are all nominal (i.e., not linked to infl ation) Th e maturity profi le of the euro-denominated sov-ereign debt markets is shown below (Figure 1.3)

Th e g raph i ndicates t hat a pa ttern of fi xed a mounts i n eu ros fa lling due anywhere within the next three decades can adequately be matched

by e uro-denominated s overeign b onds, e specially no w t hat m any s uch bond issues are strippable.* Similar remarks hold for sterling, dollar, and yen bond markets It follows that we can identify a bond portfolio closely

* Stripping means trading each coupon or principal payment of the bond as a separate asset— each a bu llet bond Th e sovereign euro bonds are generally strippable, with France issuing such bonds since 1991, Germany since 1997, followed by many others (including Ireland) in more recent years.

matching a n ominal pension liability in these currencies for the 55 year old person.

Now, consider the case that the liability cash fl ows are real in nature—subject to, say, wage increases prior to retirement and infl ationary increases thereaft er In order to estimate t he payments fa lling due a ft er 10 years’ time, we now require an estimate of the person’s wage increases over the next decade Th is problem can be decomposed into estimating (a) the gen-eral rate of infl ation over t he next decade a nd (b) t he real rate of wage increase Th e latter might be e stimated to a r easonable accuracy leaving

us to allow for the rate of infl ation over the next decade Th e development

of the index-linked bond markets allows for a portfolio to be constructed that match a pattern of such real payments in the U.K., Eurobloc, and U.S economies up to, again, three decades into the future Figure 1.4 illustrates the maturity profi le of the sterling sovereign debt market in both the nom-inal and index-linked bonds

Th e abo ve-mentioned co nsiderations a llow u s t o i dentify, i n g eneral terms, that the most closely matching portfolio to the stylized pension lia-bilities comprises solely of bonds In particular, a role for equities has not been identifi ed in the most closely matching portfolio as the proceeds from equities are not known in advance Clearly a similar procedure applied to

fi nding the closest matching portfolio to the liabilities of persons over age

FIGURE 1.3 Outstanding nominal a mount of eu ro-denominated government bonds over 1 year, by calendar year of maturity, € billions (as in September 2003)

(From Whelan, S.F., Irish Bank Rev., 48, Winter 2003 With permission.)

55 years will again identify portfolios consisting of just bonds tional and index-linked).*

(conven-For persons younger than 55 years, there is as yet no sovereign anteed security matching payments falling due aft er about three decades

guar-in the major economies, whether nomguar-inal or real However, the market allows us to provide a n ominal amount or infl ation-linked a mount i n three decades’ time and this can be used as a stepping-stone to provide payments falling due aft er the three decades Applying this logic entails that solving for the most closely matching asset for nominal or index-linked liabilities aft er 30 years is perhaps best done by extrapolating the yield curve beyond t he present maturity cutoff a nd price on t he basis that longer-dated securities at the extrapolated yield exist Th is suggests that t he i nvestment st rategy t o a llow for t hese very d istant payments would be to invest the estimated required amount in the longest dated bonds available

* We would not like to give the impression that is always straightforward It can be nontrivial

to estimate the closest matching portfolio for some liabilities, particularly those expressed as the lower of two amounts.

Conventional gilts Index-linked gilts

FIGURE 1.4 Outstanding n ominal a mount o f s terling-denominated c tional and index-linked (infl ation-adjusted) government bonds, by calendar year

onven-of maturity, Stg£ billions (as at end onven-of March 2004)

Of co urse, t he ex trapolation o f t he y ield c urve i ntroduces a nother risk, the magnitude of the risk related to the extent of the extrapolation However, if the weight of the liabilities falling due occurs within the next three decades,* then this extrapolation technique will produce an accept-able er ror a s a p roportion o f t he t otal l iability A ke y q uestion i s h ow much investment risk is increased with the extrapolation technique and the associated investment strategy proposed earlier When the liabilities are l inked to i nfl ation t hen we c annot, u nfortunately, reliably back-test how well the extrapolation method proposed earlier would have worked

as sovereign index-linked stocks have only been in issue since 1981 in the United Kingdom, since 1997 in the United States, and since 1998 in France However, we can derive the empirical investment variation associated with other investment strategies over the last century, and this is done in the following case studies

1.3.2 Case Study 1: Measurement of Investment Risk

in Pension Funds—Termination Liability

Consider the pension liability to a 40 year old who is due a nonescalating pension from age 65, expressed as a f raction of his salary at the time of retirement Let us further assume that the person will die on his 85th birth-day Th e minimum reserve that must be held for such a pension liability, as required by current legislation in Ireland, is that amount determined if the pension based on his current salary is to be revalued by the lesser of infl a-tion or 4% in any year, up to vesting at age 65 Let us take this approach in valuing the termination liability to this pension

Given that we want our valuation method to be market-based, we take the valuation rate of interest equal to the gross redemption yield on the bond closest in cash fl ow to the liability—in this case, given the restricted maturities on t he bond ma rkets a nd a ssuming no i ndex-linked bonds, the yield of a 30 year bullet bond is taken Th e annual rate of escalation

of the benefi t preretirement is assumed to be 2 5% (this latter assumption

is not a ma terial, as discussed later) Finally, we assume at time 0 t hat t he valuation shows that the value of the assets, assessed at market value, is identical to the (discounted) value of the liabilities We wish to estimate the investment variation when the investment strategy is to invest totally

* Th is is oft en t he case w ith defi ned benefi t schemes as t he l iability i ncreases, ot her t hings being e qual, w ith t he g reater t he a ge of t he me mber, t he lon ger t he p ast s ervice a nd t he higher the salary However, the extent to which it holds true is dependent on the maturity of the scheme.

in either (a) the equity market, (b) a conventional 20 year bond, (c) a let (or stripped) bond with a single payment in 30 years, or (d) short-term cash Th e period between valuations is taken to be a c alendar year (i.e.,

bul-p = 1 in our formal defi nition of investment variation earlier).

From Section 1.2, we know that the investment variation is the ent va lue of t he ex tent to wh ich t he i ncrease i n t he va lue of t he a ssets exceeds the increase in the liabilities over the year, the rate of discount (or the inter- valuation rate of interest) being the rate at which the liabilities increased over the year In the example, the inter-valuation rate of return,

65 41 1

at 20

65 40 0

a i

i

a i

iv is the valuation rate of interest at time j (i.e., the gross redemption

yield on the 30 year bullet bond at that time)

Pen is the pension on termination at time 0, payable from age 65

Th e i nter-valuation r ate of i nterest c an be se en a s t he hurdle r ate of return that assets must exceed to show a positive investment variation over the year

Using h istoric st atistics o f t he U.K c apital ma rkets, w e i nvestigated

over each calendar year in the twentieth century the ex post investment

variation, assuming that the assets are invested in diff erent asset classes.*

Th e result shows the ex post investment variation in each calendar year for

each investment strategy, standardized by dividing the investment tion by the value of the liabilities at time 0

varia-Figure 1.5a and b is dominated by the large positive investment tion posted by many mismatching investment strategies over the 1970s

varia-* Returns and yields for the U.K market were sourced as follows: 20 year gilt yields and returns

and a lso c ash re turns were s ourced f rom B arclays C apital Equity Gilt Study 2003 for t he

period aft er 1962 Prior to 1962, yields at the year end and interest rates during the year were sourced from Mitchell (1988) and the return on a notional 20 year bond and cash calculated

as outlined in Whelan (2004) Th e annual U.K equity market returns were sourced from Dimson et al (2004) We assume that the yield on the 30 year bullet bond is the same as the yield on the long bond.

–100% –50% 0%

(a)

50% 100% 150% 200% 250% 300%

2000 1996 1992 1988 1984 1980 1976 1972 1968 1964 1960 1956 1952 1948 1944 1940 1936 1932 1928 1924 1920

Equity Long bond return

30 Bond Cash return

–100% –80% –60% –40% –20% 0%

(b)

20% 40% 60% 80% 100%

2000 1996 1992 1988 1984 1980 1976 1972 1968 1964 1960 1956 1952 1948 1944 1940 1936 1932 1928 1924 1920

Equity Long bond return

30 Bond Cash return

FIGURE 1.5 (a) St andardized i nvestment v ariation f or 4 0 y ear ol d f or e ach investment s trategy, i n e ach c alendar y ear, U K ma rket ( case s tudy 1 ) (b) Standardized investment variation for 40 year old for each investment strat-egy, in each calendar year, U.K market (case study 1) [rescaled] (From Whelan,

S.F., Ann Actuar Sci., II, 54, 2007 With permission.)

and early 1980s (coincident w ith t he fi rst and second oil shocks, which raised infl ation markedly leading, in turn, to large rises in bond yields) In particular, it shows that 1974, regarded as a bad y ear for the U.K equity investment because of the market crash, was, from the perspective of the immature defi ned benefi t schemes, one of the better years, as the rise in long bond yields over the year reduced the present value of the liabilities

by a considerably greater amount than equities fell Figure 1.5 gives a very diff erent history of the rewards from investing in the diff erent asset classes

to the traditional version of historic returns based on annual real or nal returns on a unit invested

nomi-Th e sp read o f t he em pirical d istribution a ppears n onstationary i n t he graph—that i s, t he spread appears to cha nge w ith t ime.* Th e implication

of this observation for those attempting to forecast the distribution of the investment variation for each asset class is that it is especially challenging and pa st ex perience i s o nly a l oose g uide t o t he f uture ex perience (see Whelan (2005) for further discussion on this point)

Table 1.1 sets out the summary statistics to describe the key features

of the empirical investment variation based on historic experience, with

fi gures calculated for the whole twentieth century, the second half of the twentieth century, and those refl ecting the experience since 1970

We can draw the following conclusions from Table 1.1:

1 Th e 30 year bullet bond is, of those tested, the closest match to the liability as the investment variation distribution for this asset type exhibits the lowest standard deviation Hence the 30 year bullet bond

is close to the hedging portfolio Equities and long bonds have lar investment risk, while cash is considerably higher

2 While the fi gures change whether one looks at the 30 y ear period, the 50 year period, or the whole century, the relative ordering of the diff erent asset classes in terms of this new defi nition of investment risk is largely unaltered However, the estimated fi gure for the invest-ment risk is very high and dependent on the sample period for equi-ties, conventional long bonds, and cash Th is points to the need for considerable judgment in estimating the future investment risk of the diff erent classes

* Th is is not s urprising as t here is considerable evidence t hat returns f rom capital markets form a nonstationary time series (e.g., Loretan and Phillips, 1994).

3 Note in particular that a 20 year conventional bond (which, of course, has a w eighted a verage d uration less t han 20 y ears) is a d uration mismatch for the 30 y ear bullet bond (which has a w eighted dura-tion of 30 years), and on the historic simulation, this term mismatch has introduced as m uch risk as eq uity investment.* Th e implication

of this fi nding is t hat if p ension funds could invest only in co tional, nonstrippable bonds with a t erm to maturity no lo nger than

nven-* Note that the returns from the long bond and the 30 year bullet bond are highly correlated but the variability of the former is much lower than the latter, which leads to the mismatch.

TABLE 1.1 40 Year Old: Summary Statistics of the Empirical Investment Variation Distribution, U.K Markets in Twentieth Century (Case Study 1)

Based on an Investment Strategy of 100% in… Equity Bond Long 30 Year Bullet Bond Cash