1

Measuring Investment Risk

in Pension Funds

Shane F. Whelan

20th February, 2004 Department of Statistics,

University College Dublin Belfield, Dublin 4, Ireland.

Shane.Whelan@ucd.ie

This paper accompanies and supports the report of the working party of the Society of

Actuaries in Ireland, Report on Investment Risk (2004). I thank the working party for many

useful discussions in honing the ideas presented here. In particular, I single out Tom Murphy,

the chairman, who had the difficult task of keeping me to the point, and Pat Ryan who

scrutinised and suggested improvements to earlier drafts of the paper.

Outline of Paper

Overview 2

Defining Investment Risk 6

Discussion on the Different Investment Risks for Irish Pension Schemes 13

Identifying the Liability Reference Portfolio 16

Case Studies: Evaluating the extrapolation technique to

members under 55 years of age 22

Time Diversification of Risk Argument 33

Conclusion 37

References 39

Appendix I: Limitations of Proposed Definition

of Investment Variation (and the associated Investment Risk) 42

Appendix II: Relationship between Price and Wage Inflation in Ireland 44

Appendix III: Returns from Irish Capital Markets in the 20th Century 47

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Overview This is not the first paper to treat investment risk in pension schemes. The topic has

been contentious and divisive in the actuarial profession in the UK, the US and

Canada. The beginnings of the debate can be traced to the paper of Exley, Mehta, &

Smith (1997). In this paper, Exley et al. brought to an actuarial audience the approach

pioneered by Modigliani & Miller (1958) to the financial structure of companies and

adapted to the special case of pension funds by, inter alia, Sharpe (1976), Black

(1980), and Tepper (1981). Originally the debate centred on how to value pension

liabilities and invest pension funds but has since grown to polarise the actuarial

community into advocates of financial economics and advocates of the traditional

actuarial approach. Writing of US actuaries and their similar approach to valuing and

investing pension funds, Bader & Gold (2003) put the importance of the issues in

these terms:

“To protect the pension system and the vitality of our profession, we urge

pension actuaries to reexamine and redesign the model. The new model

must incorporate the market value paradigm and reporting transparency

that is rapidly becoming a worldwide minimum standard in finance.”

To simplify the debate to the point of caricature, the ‘financial economist’ group

argue that pension funds should invest primarily in bonds, while the ‘traditional

actuaries’ defend the common high equity exposure. A recent symposium1 gives an

overview of the issues with Gordon & Jarvis (2003) giving its history and Day (2003)

giving a synopsis of the current state of the debate.

The debate has not been confined to a pure professional or academic setting, becoming

critically important to the whole pensions industry as the performance of bonds and

equities diverged significantly since the start of the millennium. Defined benefit

schemes, the backbone of the pensions industry in English-speaking countries, appear

to be in crisis and the actuarial profession appears unable to achieve a consensus on the

way forward.

1 The Great Controversy: Current Pension Actuarial Practice in Light of Financial Economics Symposium, Vancouver, British Columbia, Canada, June 24 – 25, 2003, Organised by the The Society of Actuaries (Retirement Systems Practice Area), The Actuarial Foundation and American Academy of Actuaries. Papers available at www.soa.org/sections/pension_financial_econ.html.

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This paper takes a new line in the debate. First, it defines investment risk in the context

of actuarial investigations generally. This definition is, we believe, just a formalisation

of our intuitive notion of the concept. From the definition, one can quantify the

investment risk inherent in any given investment strategy and thereby identify the

strategy with lowest investment risk. We further define the notion of consistency in the

valuation of assets and liabilities (which coincides with the ‘no-arbitrage principle’ in

finance or the ‘law of one price’ in economics). We show that when assets are valued

at market value then the consistency constraint puts a value on some (but generally not

all) of the liabilities. In particular, the perfect matching portfolio of assets to meet the

liabilities, if it exists, is shown to have zero investment risk irrespective of the

investment assumptions underlying the valuation.

We briefly review the different investment objectives of the different parties to the

pension scheme. In contrast to the approach generally adopted by financial economists,

we take a scheme-centered approach.2 In particular, our approach finds a role for the

trustees, the sponsoring employer and the regulator who, of course, typically require

the actuarial investigation. We argue that recent changes to the regulatory regime in

Ireland have placed the maintenance of the funding level on discontinuance of at least

100% as a key constraint and investment risk must first be judged against this

constraint. These regulations demand assets be valued at market value which, by the

consistency requirement, entails that liabilities be valued on market-based

methodology. This paper concerns itself primarily with outlining a market-based

methodology for the termination liabilities, extending it to the case of liabilities that

have no matching asset and, applying the earlier concept of investment risk, measure

the investment risk inherent in given investment strategy to meet those liabilities. Our

method extends easily to a funding plan when assets are valued at market value and,

for an example, we sketch how this is done under the Defined Accrued Benefit Method

(described in McLeish & Steward (1987)) with a one-year control period. Note that

investment risk can also be monitored relative to any funding plan and whatever the

2 That is, we do not attempt the rather Marxist reduction of the relationships between the various parties to pension schemes into the ultimate struggle of financial interests between shareholder (capitalist) and employee (worker). That reduction, in the manner generally done, leads to the conclusion that bonds are, to a second order, preferable to equities in pension fund investment. This debate, though, is somewhat removed from the topic of this paper, quantifying investment risk.

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approach to valuing assets but, of course, it will typically give a different answer as the

result is dependent on the funding method and valuation assumptions.

Next comes the main body of the paper. We outline how to identify the asset portfolio

that minimizes the investment risk relative to the termination liabilities (the so-called

Liability Reference Portfolio or LRP). It is argued that since 1999, when the broad and

deep euro-markets became our domestic currency markets, it has been possible to find

assets that provide a reasonably close match to pension liabilities, at least for benefits

that will fall due within the next three decades. The mechanics of finding the LRP are

often straightforward, if tedious, for many pension benefits. We extend the approach in

the natural way to value the benefits falling due after three decades. We investigate the

possible investment strategies to meet the benefits and quantify important aspects of

the mismatch distribution (or ‘investment variation’ as we prefer to call it). In all cases

we report that a portfolio of bonds (conventional and index-linked) approximate the

LRP.

We illustrate trial-and-error methods that can be used to find the LRP in a single

important set of case studies, and quantify the extra risks entailed by other investment

strategies. We report the results of these analyses in detail showing that they appear to

be reasonably robust across economies over the last 30 years, and reasonably robust

when termination liabilities are escalating in line with wages. We get an important

insight from this analysis: even long bonds are not long enough to match the liabilities

of young scheme members, and investing in such bonds can be as risky as investing in

equities. So, in particular, a fully funded relatively immature defined benefit scheme

investing in 20-year conventional bonds could quickly develop funding problems on

the statutory basis. Just as much care must be exercised in matching liabilities by

duration as by matching liabilities by asset type.

We point out that finding the LRP for some pension liabilities, especially those defined

as the lesser of two amounts either of which can vary in the future, can be non-trivial

and might involve dynamic matching strategies. As Palin & Speed (2003) candidly

admit:

“Hedging pension liabilities is difficult, more difficult than we thought

when we started to write this paper. We don’t claim to have found a full

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solution to the problem; this paper represents work in progress. However

we believe that our method produces reasonable, sensible results for the

cases described above, even if we can’t explain the answers fully.”

Much work remains to be done in determining the LRP for different pension liabilities.

We end the paper by demonstrating that the argument that the risk of equities dissipates

with time (so that, at some long-term investment horizon, equities are preferable over

other asset classes by any rational investor) is fallacious. This argument, generally

known as the ‘time diversification of risk’, does not hold in that strong a form. True,

the expected return from equities might well be higher than other asset classes but, on

some measures, so is the risk.

So the most closely matching asset for Irish pension fund liabilities is composed

mainly of conventional and index-linked bonds. The LRP is a portfolio that, 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 one investment strategy is preferable to

another. Investors, including pension funds, are routinely tempted to take risks if the

reward (that is, the form of the investment variation distribution) is judged sufficiently

tempting. However, pension funds should appreciate the risks involved in alternative

strategies and, at a minimum, seek to ensure that the investment portfolio is efficient

in the sense that risk cannot be diminished without diminishing reward. To appreciate

the risks and ensure that all risks undertaken are reasonably rewarded requires

knowledge of the investment variation distribution and, in particular, the Liability

Reference Portfolio.

6

Defining Investment Risk

In this section we attempt to define the concept of investment risk in actuarial

investigations. Our intuitive notion that it measures the financial impact when the

actual investment experience differs from that expected, holding all other things

equal, is given a formal expression. This definition of investment risk is sufficiently

general that it can be readily interpreted and applied whether the investigation is into

the on-going funding level of the scheme, its solvency position, or the FRS17 balance

sheet item. Further, the impact of investment risk on the contribution rate or FRS17

pension cost can, with some elementary calculations, also be assessed. Once

investment risk is properly defined, it is straightforward (in theory at least) to measure

and attempt to minimise it.

Actuaries have not been explicitly required to date by GN9(RoI) to measure or report

on the investment risk to which the scheme is exposed, only being required to

comment if the investment policy “is inappropriate to the form and incidence of the

liabilities” and then only on the funding basis (not on the discontinuance basis). So, to

date, Irish actuaries have not needed to refine the notion of investment variation and

investment risk as explicitly as is done here.

Actuarial investigations into the current financial state and future financing of pension

schemes, as envisaged in GN9(RoI), typically require the actuary to use his judgement

to decide on the most appropriate approach under three broad headings:

1 The valuation methodology, suitable for the purpose of the investigation.

2 Demographic assumptions to estimate the type of benefit and when

payable. The quantum of the benefit may depend on the timing of the

payment.

3 Financial assumptions, so as to project, in a consistent way, the amount

and timing of the liabilities and the asset proceeds.

The financial assumptions generally include inflation in any future year and rates of

wage escalation. Further assumptions, compatible with the financial assumptions

underlying the projection of the liabilities, are generally required to value the assets.

7

In particular, the valuation rate of interest – the rate of interest used to discount the

projected future cashflows to the present time – is generally taken to be an estimate

(perhaps erring on the conservative side) of the return over the future that could be

expected to be achieved on a portfolio of assets that are broadly suitable to the nature

and term of the liabilities.3

Once the three components of the valuation (1,2 and 3 above) are specified then the

investigation allows the actuary to report not only on the current state of the finances

but generally also to outline, on the valuation assumptions, the future course of the

pension scheme’s finances. This point is key as it allows future valuations to assess

the divergence of the actual experience from that expected (which might then help

suggest timely corrective actions).4

To readily see this, let us assume that the valuation methodology is the form of a

discounted cashflow approach. When t = 0, we call it the present time, and t > 0 a

future time. Let At be the forecast cashflow from the assets at time t, Lt be the forecast

liability cashflow at time t and i be the valuation rate of interest. Now, the reported

valuation result at time 0, expressing the surplus (excess of the present value of the

assets over the present value of the liabilities), denoted X0, can be written as:

00

0

0

0 0

0

0

( )(1 )

( )(1 ) ( )(1 )

( )(1 ) (1 ) ( )(1 )

( )(1 ) (1 ) ( )(1 )

( )(1 ) (1 )

tt t

t

pt t

t t t tt t p

pt p t p

t t t tt t p

pt p s

t t t tt s

pt p

t t pt

X A L i

A L i A L i

A L i i A L i

A L i i A L i

A L i X i

∞−

=

∞− −

= =

∞− − − +

= =

∞− − −

= =

− −

=

= − +

= − + + − +

= − + + + − +

= − + + + − +

= − + + +

∑

∑ ∑

∑ ∑

∑ ∑

∑

(I)

where

p is the time that the next actuarial valuation falls due

3 A concise overview of the traditional actuarial approach to investigating pension funding, especially the consideration affecting the choice of the valuation rate, is given in Paterson (2003). 4 The so-call ‘actuarial control cycle’.

8

If the experience of the scheme in the inter-valuation period is exactly in line with that

assumed at time 0, and the method and assumptions in the valuation undertaken at

time p are also the same, then the valuation result at time p will be 0 (1 ) pX i+ .5

Consider X0. We shall assume that this is a number.6 So, under this simplifying

assumption, X0 is a constant, representing the surplus (if positive) or deficit (if

negative) of assets relative to liabilities at the present time identified by the specified

(deterministic) valuation methodology.

It is generally possible to form a reasonable apportionment of the difference of the

valuation result at the next valuation date from that expected from the valuation at

time 0 (i.e., 0 (1 ) pX i+ ) into that due to either

(i) the actual demographic or financial assumptions in the inter-valuation

period differing from that assumed, or,

(ii) that due to a changed valuation method or basis applied at time p.

In particular, it is possible to form a reasonable assessment of the financial impact of

the actual investment experience relative to that expected, other things being held the

same.

Let 0i

pX − denote the result of the valuation at time 0, under the same methodology

and assumptions as underlying the valuation result, X0, at time 0 but reflecting the

actual investment experience of the scheme in the inter-valuation period. Then

5 This can readily be seen, as the cashflow surplus in the inter-valuation period will be invested (or disinvested) at the valuation rate of interest, accumulating at time p to

0 0

(1 ) ( )(1 ) ( )(1 )p p

p t p tt t t t

t t

i A L i A L i− −

= =

+ − + = − +∑ ∑ and this amount is to be added to discounted value of all

the yet unrealised asset and liability cashflows at time p, namely 0pX . The total value at time p is then

0

0( )(1 )

pp t

t t pt

A L i X−

=

− + +∑ , which is just the right-hand side of equation (I) above multiplied by (1 ) pi+ ,

whence the result. 6 If this is allowed be a non-constant random variable then we call the valuation methodology 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 non-trivial random variable at time 0. We shall discuss only deterministic valuation methods in the sequel to simplify the analysis but, as should be clear by the end, the results carry through (with relatively straightforward extensions) when applied to stochastic valuation approaches.

9

0 0i

pX X− − measures the financial impact at time 0 of how the actual investment

experience up to time p differed 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 in the inter-valuation period then 0 0i

pX X− = , so 0 0i

pX X− − takes the value

zero. We shall call 0 0i

pX X− − the ‘investment variation’ up to time p.

Investment variation, so defined, is a non-trivial concept. It measures the financial

impact at time 0 created when the actual investment experience up to time p differs

from the investment assumptions underlying the valuation at time 0.

We also need to formalise what we mean by saying that assets and liabilities are

valued on a consistent basis. By a ‘consistent valuation method’ we mean that the

present value of a cashflow of a given amount at time t is the same up to a change in

sign, whether the cashflow is positive (an asset) or negative (a liability).

We list some of the consequences of investment variation so defined:

(1) Investment variation affects both sides of the balance sheet in general – the

present value of both assets and liabilities - as some assumptions used to value

the assets can also affect the size of the liabilities.

(2) If we have perfect matching of assets to liabilities,7 then any consistent

valuation method will always report the investment variation to be zero.

The present value of the assets at time 0 (i.e.,0

(1 ) tt

tA i −

≥

+∑ ) might vary

with the investment assumptions but

0 0( )(1 ) 0.(1 ) 0t t

t tt t

A L i i− −

≥ ≥

− + = + =∑ ∑ and hence 0

(1 ) tt

tL i −

≥

+∑ must vary

in direction proportion to 0

(1 ) tt

tA i −

≥

+∑ . Hence, in aggregate, a gain

(loss) on the assets relative to that expected is exactly offset by an

increase (decrease, respectively) in the value of the liabilities relative to

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

10

that expected. In short, perfect matching of asset and liability cashflows

has zero investment variation, irrespective of the experienced or the

assumed investment conditions.

(3) Investment variation can be altered by changing the composition of the asset

portfolio, as this changes the nature and timing of the future asset cashflows.

In particular, the closer the asset proceeds are to the liability outgo in timing,

amount, and nature8 then the closer to zero the investment variation.

(4) Other than the uniformly zero variation in (2) above, the magnitude of the

investment variation depends on the valuation approach, the demographic

aasumptions, and the other non-investment financial assumptions as all these

enter into its measurement of both X0 and 0i

pX − .

(5) Of course, the actual valuation at time p of the scheme will generally be

different from 0i

pX − suitably inflated as

(i) the experience of the scheme in the inter-valuation period is unlikely to be

in line with the non-investment assumptions, leading to gains or losses at

the next valuation on the non-investment financial assumptions and the

demographic assumptions.

(ii) The valuation methodology or the underlying assumptions at time p could

change in the light of experience or other factors consequent on the

passage of time so that, at time p, the valuation approach or basis is

changed.

Suppose, we make the following three assumptions:

(1) Assets are to be valued at market value

(2) There exist assets that perfectly match the liabilities (in the technical

sense earlier).

(3) We demand our valuation methodology be consistent (in the technical

sense earlier).

Then, it follows that the investment variation is the present value of the increase in the

surplus, discounted at the rate of interest equal to the return on the market value of the

8 By matching by nature, we mean that unforeseen economic influences (“shocks”) have a similar effect on the magnitude of both assets and liabilities.

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matching asset. That is, the experienced inter-valuation rate of interest is the return on

the market value of the matching asset. To see this, we can rewrite X0 as:

This latter expression can now be used to calculate 0i

pX − with knowledge of what the

actual investment experience will be over the inter-valuation period. The first term of

0i

pX − in the above decomposition is simply the market value of the assets at time p

[by assumption (1)], discounted at the valuation rate of interest experienced over the

period (which has to be determined). At time p, the present value of the future

liabilities (the last term in the above equation) is equal to the market value of the

matching asset at that time [by consistency, assumption (3)]. Now, we need to find the

experienced valuation rate in the inter-valuation period. Note that if there was a

matching asset at time 0 and the scheme held the matching asset then the investment

variation would be 0 (irrespective of what happened in the inter-valuation period).

Hence the experienced valuation rate in the inter-valuation period can now be seen as

the market return on the matching asset over the inter-valuation period. The upshot is

that the investment variation is the present value of the extent to which the increase in

the value of the assets exceeded the increase in the liabilities over the inter-valuation

period, discounted at the rate of return on the matching asset over the period. We see

immediately from this that investment variation is positive only if the increase in the

market value of the actual assets exceeds the increase in the market value of the

matching asset.

Investment variation at time 0 can be viewed as a stochastic process, 0 0i

pX X− − ,

indexed in p, as the term 0i

pX − cannot in general be evaluated at time 0 from the then

available information. Viewed in this way, the investment variation at time 0, up to

time p, is random variable. This is the more practically useful concept - the ex ante

distribution at the present time. However, in order to define it, it was necessary to first

00

0 0

0 0

( )(1 )

(1 ) (1 )

(1 ) (1 ) (1 )

tt t

t

t tt t

t t

pt t t

t t tt t t p

X A L i

A i L i

A i L i L i

∞−

=

∞ ∞− −

= =

∞ ∞− − −

= = =

= − +

= + − +

= + − + − +

∑

∑ ∑

∑ ∑ ∑

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define how to calculate the ex post investment variation. In short, investment variation

up to time p can only be measured at time p, before that it may be modelled as a

random variable with an associated distribution.

So 0 0i

pX X− − , when viewed at time 0 is a random variable, so it has an associated

distribution. Insofar as pension schemes have been unwilling or unable to perfectly

match the asset and liability cashflows, in the technical sense earlier, then investment

variation will have a non-constant distribution. The 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 average, the experienced

conditions turn out better than originally forecast). Some prefer to give a single

number to capture the notion of riskiness in a distribution, often using some parameter

that measures the spread of the distribution, such as its standard deviation, its semi-

variance, or its inter-quartile spread. Often this summary measure is called

‘investment risk’.9 Alternatively, one can apply some other measures such as report

the value below which there is a specified low probability of the outcome falling (the

so-called ‘Value-at-Risk’), which combine the mean and the standard deviation in a

particular way.10 The key point to be made is that the distribution of 0 0i

pX X− − is a

more foundational concept and maintains more information than any summary spread

statistic. We do not enter on the wider discussion of the most appropriate measure to

apply to the investment variation distribution to capture our intuitive notion of risk but

adopt the generally accepted measure of standard deviation. So we identify

investment risk as the standard deviation of the investment variation distribution (to a

first order approximation).

Appendix I draws attention to a major limitation of our definition of investment

variation (and the associated investment risk) for pension funds. Practitioners must

9 The investment variation distribution in those cases where the proceeds from the assets perfectly match the liability cashflows is uniquely characterised as being a constant (i.e., a degenerate distribution), so it will give a value of zero for any metric attempting to measure the spread of a distribution. 10 Of particular importance in the probability distribution is its extreme left tail behaviour, which gives the probability of a reduction to the surplus of any given large amount. Such an event might cause sudden and severe financial strain to the cost-bearer, perhaps threatening the continuance of the scheme. Appropriate measures for such extreme risks include, for symmetric distributions, the kurtosis or higher even moments if they exist.

13

overcome the limitation in each case, the manner being dependent on unique

characteristics of the scheme, sponsoring employer, and the business of the

sponsoring employer.

Discussion on the Different Investment Risks for Irish Pension Schemes

The primary objective of the pension fund is to pay all the promised benefits as they

fall due. There are many ways to achieve this end, so invariably the primary objective

is supplemented with some secondary aim. Members and other beneficiaries of the

scheme with a fixed contribution rate might, for instance, phrase the unique objective

of the pension fund as to pay all the promised benefits as they fall due with this

objective to be achieved with the utmost certainty – that is, they might stress the

financial security of the scheme. The sponsoring employer (if financing the balance of

the cost) might adopt the objective of meeting the promised benefits at the lowest

regular financing cost. The trustees of the scheme could perhaps adopt the twin aims

of wanting the utmost security of payment of the promised benefits and wanting the

scheme to continue indefinitely (hence maximising the payout to current and future

beneficiaries). This trustees’ objective entails that they must pay attention to the desire

of the employer in providing the benefits at the lowest cost, as they do not want to

frustrate the employer’s objective to such a degree that it provokes the employer to

reduce future benefits or terminate the scheme altogether. A formal manner of

modelling this trustees’ objective is to model the employer’s future contribution rate

as an asset of the scheme and the trustees’ wish to maximise the value of this asset,

subject to adequate security of benefits accrued to date.

The main point is that the beneficiaries, the sponsoring employer, and the trustees can

all be envisaged as in agreement on the primary objective but at odds on secondary

aims. This somewhat stylised manner of viewing pension schemes shows that the

trustees could be pictured as arbiters of the inherent conflict of secondary aims

between the cost-bearer and the beneficiary (as to reduce the pace of financing

inevitably reduces the security of the promised payments). The fundamental cause of

the tension arises from the design of such schemes, where it could be argued that the

14

employer enjoys the investment variation if positive but, if negative, retains the option

of winding-up the scheme.11

Aside from the beneficiaries, the trustees, and the employer, there is a fourth party to

pension provision in Ireland: the regulator.12 The environment in which Irish pension

funds operate has been transformed by regulatory changes since 1990.

The Pensions Act 1990 was a watershed in the regulation of Irish defined benefit

pension funds. First, the Pensions Act 1990 required that the early leaver be given a

‘preserved benefit’ (defined in Part A of the Second Schedule to the Act) and that that

part of the preserved benefit that could be deemed to have accrued after the year 1990

was to be revalued each year by the lesser of inflation or 4% (Section 33(5)) until the

sooner of death or normal retirement age. The Pensions (Amendment) Act 2002

further improved the benefits payable to early leavers when it removed the provision

that the revaluation should only apply to that part of the benefit deemed to have

accrued after the year 1990 – now all the benefit is subject to revaluation.

Second, the Pensions Act 1990 introduced the requirement for Irish pension schemes

to have an actuary undertake a periodic review to ensure that the assets of the scheme

taken at market value exceed the liabilities on termination as defined in the Act. If a

shortfall is revealed then it must be disclosed to interested parties and a short-term

funding plan agreed with the newly established regulator, the Pensions Board, to

make good the deficit. The liabilities on termination are defined (Third Schedule) to

be the benefits payable under the Rules of the Scheme (as amended in the case of the

early leaver above) with in-service members assumed to be early leavers at the

valuation date.

11 The opposite view of the tension in Exley et al. (1997), which sees the member as gaining on positive investment variation with the chance that the benefits will be improved but never suffering on a negative investment variation. No doubt, the reality is more complex than either of our summaries with, perhaps, the value of an investment variation to each party being dependent on the relative negotiating strength of each, which varies in time. 12 See, for instance, Whelan (2003), for a fuller discussion of the role of the regulator and the likely long-term effect of the current regulatory regime on Irish defined benefit pension funds. Some of the ideas presented below are a development of arguments presented in that paper.

15

The regulations have fundamentally altered pension fund financing. Irish pension

schemes now have at least two investment targets. One target is to provide, as before,

from the existing assets and the members’ and sponsoring employer’s future

contribution rates, the benefits promised under the rules when they fall due (with, as

noted earlier, the different interested parties augmenting this primary objective with

different sub-objectives). The other target is to ensure that, at any valuation date, the

value of the assets exceed the value of the liabilities on the termination of the scheme.

Now, prior to the Pensions Act 1990 and subsequent amendments (especially the

Pensions (Amendment) Act 2002), the value of the termination liabilities was in

general a small percentage of the value of the assets of the typical scheme, so that the

second target could be ignored in practice when financing the scheme and setting the

investment strategy. However, as outlined in the accompanying working party report

of the Society of Actuaries in Ireland, Report on Investment Risk (2004), now the

termination liabilities of Irish pension funds form a high percentage of the value of the

assets so that attention must now be given to this latter target. This creates an

investment dilemma because the investment strategy that produces an acceptably low

probability of showing a deficit on future termination valuations may not coincide

with the investment strategy that produces the employer’s desired low regular

financing cost13 and hence might be detrimental to the trustees’ secondary aim in

prolonging the scheme.

The question naturally arises as to how one prioritises the twin investment objectives

identified above. The more stringent requirement is that imposed by the regulator, as

if the scheme can always demonstrate it is at least 100% funded on a discontinuance

basis then, trivially, it can always pay the promised benefits. So the primary objective

of the interested parties is satisfied on the regulatory basis. Also, the secondary

objective of the beneficiaries is also satisfied if the scheme is always 100% funded on

the regulatory basis. Further, if at any time the scheme falls below the 100% funding

level on discontinuance, then the future funding plan is no longer at the sole discretion

of the employer but the plan must be approved by the Pensions Board. So for the

13 From the formal definition of investment variation defined in the last section, it is clear that investment variation depends on the liabilities. As the liabilities change from a valuation assuming the scheme to be on-going and a valuation on assumed termination, clearly the investment variation distribution is altered. In particular, the zero risk investment strategy if it exists, will not in general be the same for a valuation assuming determination and a valuation with the on-going assumption.

16

employer to maintain autonomy in the financing arrangements, it is necessary to

maintain at least a funding rate of 100% on discontinuance. Accordingly, for Irish

pension funds, the two investment objectives can be prioritised with the funding level

of discontinuance being the constraint (i.e., must be met) and minimisation of the

financing cost a target subject to this constraint. Consequently, the focus of

investment variation (and therefore the associated investment risk) will in the first

instance relative to the termination liabilities and the funding level on discontinuance.

This marks a change from the current focus of GN9(RoI), which is primarily

concerned with the on-going valuation and evolution of the on-going funding level.

Identifying the Liability Reference Portfolio

In this section we show how to apply the concept of investment variation in some case

studies. When necessary we use historic data from the Irish, UK and US capital

markets to assess the investment variation associated with different investment

strategies. When no matching asset exists we give several descriptors of the

investment variation distribution from the historic data – including the key measures

of its geometric mean and its standard deviation. These latter two summary measures

give an illustration of the relative rewards of the different strategies and, to a first

approximation, the risks associated with the strategies. We call the asset portfolio that

minimises the variance of the investment variation distribution the Liability Reference

Portfolio (LRP).

Theoretical Considerations

For concreteness, let us consider a person aged 55 who is a member of a scheme that

promises, based on his past service of 20 years, a pension from age 65 of one-third his

salary at the time of retirement, the pension subject to a fixed rate of increase while in

payment. Let us further assume that the man will die on his 85th birthday.

Now, the value of his preserved benefit under the Pensions Acts is to be revalued by

the lesser of inflation or 4% in any year, up to vesting at age 65. Given that the

average revaluation rate can be reliably estimated (a point treated later), then the

17

liability is a series of (estimated) nominal amounts falling in a regular pattern,

beginning in 10 years’ time and ending in 30 years’ time.

From our definition of investment variation and investment risk earlier, it is clear that

to minimise the investment risk then we want to find an asset portfolio that provides

an income that most closely matches this liability stream. Since 1999 we have had a

common currency with many larger economies and can now seek in euro-

denominated fixed interest markets and, in particular, the sovereign guarantee issues,

for such matching securities.

We analyse the maturity profile of the euro-denominated sovereign debt market in

Graph 1 below.

Graph 1: Outstanding Nominal Amount of Euro-denominated Government bonds over 1 year, by Calendar Year of Maturity, € Billions (as at September 2003)

0

50

100

150

200

250

300

350

400

2004

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2032

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2034

Source: Calculated from data kindly provided by the National Treasury Management Agency.

The total size of the euro-denominated sovereign bond market is €2,700 billion while

the total size of Irish pension funds at the same time (September 2003) is of the order

of €50 billion. The euro sovereign bond market is, accordingly, over 50 times the size

of Irish pension funds. The outstanding nominal amount tends to be concentrated in

18

the next 10 years, with a relatively light and patchy volumes outstanding from 2014

onwards (totalling €434 billion or under 10 times the size of Irish pension funds) with

nothing beyond 2034. The graph thus indicates that a pattern of fixed amounts falling

due anywhere within the next three decades can adequately be matched by euro-

denominated bonds, especially now that that many such bond issues are strippable.14

It follows that we can identify a bond portfolio matching the revalued preserved

payments due to the 55 person, subject only to the extent that we can reliably estimate

the rate of revaluation prior to vesting. This points to a predominantly bond-based

portfolio to match the termination liabilities.

However, the benefits due to the same 55 year-old person, assuming the scheme

continues and he stays in service to normal retirement age is a pension of half his

salary at the time of retirement, the pension subject to a fixed rate of increase while in

payment. In order to estimate the payments falling due after 10 years’ time now

requires us to estimate the person’s wage increases over the next decade. This

problem can be decomposed into estimating the general rate of inflation over the next

decade and the real rate of wage increase. The latter might be estimated to a

reasonable accuracy (see Appendix II for a discussion) leaving us to allow for the rate

of inflation over the next decade. Ireland does not have a developed market in index-

linked bonds15 so, it could be argued that a freely traded security directly matching

this rate of escalation is not available. The French government has, since 1998, issued

some bonds with payments linked to French inflation16 and some bonds with

payments linked to eurozone inflation (excluding tobacco).17 The index-linked market

in the eurozone is nowhere near as developed as the conventional bond market, with

few issues of comparatively small nominal value issued relatively recently. For Irish

pension funds, there is also the associated basis risk of how French or eurozone

inflation might differ from Irish inflation. We believe that, at least over the longer

14 Stripping means trading each coupon or principal payment of the bond as a separate asset – each a bullet bond. The 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. 15 There are just two stocks with payments linked to Irish inflation – the (government guaranteed) Housing Finance Agency bonds of 2008 and 2015. Both are small issues, with a combined market value of less than €½ billion. 16 The OATi 2009, 2013, and 2029 with combined nominal outstanding at the start of 2003 of about €14.5 billion. 17The OAT€i 2012 and 2032 (of nominal amount outstanding of approximately €6.5 billion and €4 billion respectively. Source: Agency France Trésor (www.francetresor.gouv.fr/oat/us/indexus.cgi)

19

term, the average inflation rates within the two economies should be reasonably close

so that this basis risk is not significant. Thus the French index-linked market does

provide the most closely matching asset and thus can be used to construct a portfolio

most closely matching the benefit outgo.18

The above considerations have allowed us to identify in general terms the most

closely matching portfolio to the stylised pension liabilities in our example. The

portfolio mix depends on whether the termination liability or on-going benefits are

targeted, but it still comprises only conventional bonds and eurozone index-linked

securities. In particular, a role for equities has not been identified in the most closely

matching portfolio as the proceeds from equities are not known in advance. If we

generalise our example to consider pension payments linked to inflation, or persons

aged over 55, then clearly similar arguments above apply and we again identify

portfolios consisting of just bonds (conventional and index-linked) to be the closest

matching portfolio to the liabilities.

It will be helpful if we briefly go through a case study. Suppose the liability is a level

pension beginning in, say, 5 years’ time and ending in, say. 25 years’ time. Then each

pension payment can be matched from the income from a euro-bond. The sovereign

euro debt yield curve at the time of writing is shown in Graph 2.

18 Exley et al. (1997) indicate how to replicate index-linked stock from equities, cash, and bonds if index-linked stock do not exist but do not give a feel for how robust the replication is in practice. More work needs to be done on this promising idea.

20

Graph 2: Yield Curve: Sovereign Euro Debt Gross Redemption Yield as at 10th

February, 2004

0.0%

1.0%

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6.0%

Feb-

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34

From the underlining bonds that comprise the yield curve, we can calculate a spot or

strip yield curve that gives to the interest rate available over each term for a unit

investment at the start of the start. The strip curve will lie slightly above the yield

curve above. From the strip curve, one reads the rate of interest for each year between

year 5 and year 25 and applies that rate of interest to discount the pension payable in

that year. This gives us the sum of 20 discounted values, each being a year’s pension

being discounted at the appropriate interest rate for that term. Conversely investing an

amount of money equal to each of the 20 discounted values into its corresponding

strip, the proceeds of the investment will exactly match the pension outgo. This

process can be carried out for all payments falling due within the next 30 years as the

yield curve extends out that far (or equivalently, under the demographic assumptions

under our model, for any nominal amount payable from age 55).

The process is exactly similar for index-linked payments, but now one uses the real

spot interest rate for each term derived from the real yield curve. Once again, as the

real yield curve extends to almost 30 years, we can price and match all inflation-

linked liability outgoes up to that term (which, in our example, corresponds to any

index-linked payments payable to someone over 55 years of age).

21

The theoretical method outlined above can be simplified to give a reasonable

approximation to the true answer by, say, discounting the liability outgo falling within

each (of the six) five-year periods over the next 30 years (rather than each year of the

next 30 years).19

So the liability reference portfolio for members over the age of 55 with benefits

subject to fixed or inflation-linked escalation can be estimated by the methods

above.20

For persons younger than 55, there is no sovereign guaranteed security matching

payments falling due after 2032 for inflation-linked benefits or 2034 for nominal

payments. However the market allows us to provide a nominal amount or inflation-

linked amount in three decades’ time and this can be used as a stepping-stone to

provide payments falling due after the three decades. Applying this logic entails that

solving for the most closely matching asset for nominal or index-linked liabilities

after 30 years is perhaps best done by extrapolating the yield curve beyond the present

cut-off and price on the basis that longer dated securities at the extrapolated yield

exist.

The investment strategy to allow for these very distant payments would be to invest

the estimated amount in the longest dated bonds. Of course, extrapolation of the yield

curve introduces another risk, the risk related to the extent of the extrapolation.

However, if the weight of the liabilities of a scheme falls due within the next three

decades21 then this extrapolation technique will produce an acceptable error as a

proportion of the total liability. A key question is how much investment variation is

increased with the extrapolation technique and the associated investment strategy

19 It would not in general be sufficient to approximate the answer by using the average yield over all durations, and then state that the liability reference portfolio is a bond portfolio. We must seek to match the duration of the liabilities more closely, as duration mismatch can be as significant as type mismatch (as illustrated later). 20 We would not like to give the impression that is always straightforward. It can be non-trivial to estimate the liability reference portfolio of some pension payments, particularly those expressed as the lower of two amounts. Much work remains to be done in this area. 21 This is often the case with defined benefit schemes as the liability increases, other things being equal, with the greater the age of the member, the longer the past service and the higher the salary. However, the extent to which it holds true is dependent on the maturity of the scheme.

22

proposed above. When the liabilities are linked to inflation then we cannot,

unfortunately, test historically how well the extrapolation method proposed above

would have worked as index-linked stocks have only been in issue since 1981 in UK,

since 1997 in US, since 1998 in France and, as noted earlier, the market has not yet

developed into a liquid one in Ireland. We can, though, assess the extrapolation

technique when the liabilities are fixed in monetary amount. To make this assessment,

we derived the empirical investment variation associated with different investment

strategies over the last century in the following case studies.

Case Studies: Evaluating the extrapolation technique to members under 55 years of

age

Case Study 1: A 40 year old is promised a non-escalating pension from age 65 of a

fraction of his salary at the time of retirement. Let us further assume that the person

will die on his 85th birthday. So the termination liability is the pension amount based

on his current salary to be revalued by the lesser of inflation or 4% in any year, up to

vesting at age 65. From the above discussion, we shall take the valuation rate of

interest equal to the gross redemption yield on the 30 year bullet bond, and the rate of

escalation of the benefit pre-retirement is assumed to be 2½%. Finally, we assume at

time 0 that the valuation result is that the value of the assets, assessed at market value,

is identical to the (discounted) value of the liabilities. We wish to calculate the

investment variation when the investment strategy is to invest totally in either (a) the

Irish equity market, (b) a conventional 20 year bond, (c) a bullet (or stripped) bond

with a single payment in 30 years, (d) short-term cash, or, (f) commercial property.

The period between valuations is taken to be a calendar year (i.e., 1p = ).

The liabilities are the termination liabilities under the Pensions Acts, assuming the

scheme was wound up at time 0. The rate of escalation in the year following the

valuation was assumed to be 2½% (although, as noted later, this assumption is not

material to the result). From before, we know that the investment variation is the

present value of the extent to which the increase in the value of the assets exceed the

increase in the liabilities over the year, the rate of discount (or inter-valuation rate of

23

interest) being the rate at which the liabilities increased over the year. In the example,

the inter-valuation rate of return, iv, is given by:

1

0

65 41@

65 41 201

65 40@

65 40 200

(1.025) (1.025)( )(1 ) 1(1.025) ( )

(1 )

i

vi

Pen aii

Pen ai

−

−

−

−

+= −

+

where,

ij is the valuation rate of interest at time j (that is, the gross redemption

yield on the 30 year bullet bond)

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

The inter-valuation rate of interest can be seen as the hurdle rate of return that assets

must exceed to show a positive investment variation over the year.

Using the historic statistics of the Irish capital markets set out in Appendix III, we

investigated over each calendar year in the 20th century the investment variation for

the 40 year old, assuming the assets to be invested in different asset classes. We

assumed that the yield on the 30 year bullet bond was the same as the yield on the

long bond. The result shows the investment variation in each calendar year for each

investment strategy, standardized by dividing the investment variation by the value of

the liability at time 0.

24

Graph 3: Standardised Investment Variation for 40 Year Old for each Investment

Strategy, in each calendar year (Case Study 1)

-100%

-50%

0%

50%

100%

150%

200%

250%

300%

350%

2000

1997

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1901

Equity Long Bond Return 30 Bond Cash Return Property

The graph tends to be dominated by the large positive investment variation posted by

many mismatching investment strategies over the 1970s and early 1980s (the first and

second oil shocks which markedly raised inflation which, in turn, caused bond yields

to rise). Apparent from the graph, though, is that the spread of the empirical

distribution appears non-stationary – that is, the spread evolves in time.22 The

implication for those attempting to forecast the distribution of the investment variation

for each asset class is that it is difficult and past experience is only a loose guide to the

future experience.

Table 1 sets out summary statistics to describe the key features of the empirical

investment variation, with figures calculated for the whole 20th century and those

reflecting the experience since 1970.

22 This is not surprising as there is considerable evidence that returns from capital markets form a non-stationary time series (e.g., Loretan & Phillips (1994)), even in the weak sense.

25

Table 1: 40 Year Old: Summary Statistics of the Empirical Investment Variation

Distribution, over 20th Century and 1970-2000 (inclusive)

Based on Investment Strategy of 100%…

Equity Long Bond30 Year

Bullet Bond Cash Property20th Century Mean 8.5% 3.4% 0.1% 5.9% N/aMedian 2.5% 1.9% 0.4% 2.7% N/aGeometric Mean 4.2% 0.4% 0.1% 0.2% N/aStan. Dev. 32.5% 27.8% 1.9% 41.4% N/aSkew 1.7 2.9 -0.6 3.6 N/aExcess Kurtosis 5.2 17.3 1.0 22.5 N/a Since 1970 Mean 13.6% 6.6% -0.3% 13.0% 20.8%Median -4.7% -11.9% -0.4% -14.9% -3.4%Geometric Mean 2.2% -2.5% -0.3% -3.9% 1.3%Stan. Dev. 53.7% 49.3% 2.9% 72.5% 76.8%Skew 0.9 1.6 -0.1 2.1 1.5Excess Kurtosis 0.4 4.1 -0.8 6.1 2.3

We can make the following remarks:

(1) The 30-year bullet bond is the closest match to the liability (of those tested) as

the investment variation has lowest standard deviation for this asset type.

Hence the 30-year bullet bond is close to the Liability Reference Portfolio.

Identifying investment risk as the standard deviation of the distribution, then

equities and long bonds have roughly equal risk, while cash is higher again.

(2) Note in particular that a 20 year conventional bond (which, of course, has a

weighted average duration much lower than 20 years) is a term mismatch from

the 30 year bullet bond (which has a weighted duration of 30 years), and on

the historic simulation above, this term mismatch has introduced almost as

much risk as equity investment.23

(3) While the figures change whether one looks at the last 30 years or the last 100

years, the relative ordering of the different asset classes in terms of risk (or

reward) are largely unaltered. The estimated investment risk is very high and

dependent on the sample period for equities, bonds, and cash. This points to

23 Note that the returns from the long bond and the 30-year bullet bond are highly correlated (with a correlation of 0.96 from 1970 to 2000) but the variability of the former is much lower than the latter, which leads the mismatch.

26

the need for considerable judgement in estimating the future risk of the

different classes – history is only a partial guide.

(4) One of the assumptions in calculating the figures in the table above was that

inflation subject to a cap of 4% over the year following the valuation was

2½%. The upper limit of possible outcomes is 4% that, if applied, would

deduct about 1½% from the mean, median, and geometric mean figures above

and leave all the other figures unaffected. This shows that the results of our

analysis are not particularly sensitive to estimating this figure, once deflation

of any severity is considered unlikely.

(5) The skew of the investment variation has been non-negative, which ensured

that the mean exceeds the median. The geometric mean of the data, which

corresponds to the annualised rate over the period, is the more relevant

average for actuarial investigations. The above table shows us that,

historically, investing in the most closely matching asset of those studied (the

30 Year Bullet Bond) has involved a sacrifice to return only in the case of

equity investment.

(6) Note that there is no simple relationship between risk and return, as cash and

the long bond exhibit higher risk and lower return than the matching asset.

This immediately entails that, in actuarial applications, there is not necessarily

a compensation for assuming extra risk.24

Modelling 30 Year Old Member

We can apply the very same model above but now to a 30 year old. The results are as

follows, in graphical and tabular form.

24 Accordingly, actuarial advice can add real value by identifying the idiosyncratic risk of the scheme (that is the deviation with respect to the liability reference portfolio) and exploiting the uniqueness of this risk measure to increase return without increasing risk.

27

Graph 4: Investment Variation for 30 Year Old for each Investment Strategy, in each calendar year (Case Study 1)

-200%

-100%

0%

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600%

2000

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Equity Long Bond Return 30 Bond Cash Return Property

Note the 30 year bullet bond – the longest on the market – is not long enough to

match the liability so we witness investment variation arising from the term mismatch.

The fluctuations in investment variation for the 30 year bullet bond tend, as is

apparent from the table below, to be lower than that of the other asset classes.

Table 2: 30 Year Old: Summary Statistics of the Empirical Investment Variation Distribution, over 20th Century and 1970-2000 (inclusive), Case 1

Based on Investment Strategy of 100%…

Equity Long Bond30 Year

Bullet Bond Cash Property20th Century Mean 12.1% 7.2% 1.1% 11.1% N/aMedian 2.9% 2.8% 1.4% 4.1% N/aGeometric Mean 4.5% 0.7% 0.4% 0.4% N/aStan. Dev. 45.9% 46.1% 12.6% 65.1% N/aSkew 2.0 4.2 1.0 5.0 N/aExcess Kurtosis 6.0 27.5 6.2 34.9 N/a Since 1970 Mean 22.2% 16.1% 0.8% 26.6% 35.2%Median -10.7% -16.4% -5.6% -19.2% -4.8%Geometric Mean 1.1% -3.5% -1.4% -4.9% 0.2%Stan. Dev. 77.4% 81.6% 21.6% 114.5% 117.4%Skew 1.1 2.3 0.8 2.7 2.1Excess Kurtosis 0.2 7.4 0.8 10.0 5.6

28

We note in the historic study above that equities appear better than 20 year

conventional bonds as the risk is (marginally) lower but the reward is higher. As one

would have expected from the earlier discussion, the risk of all asset types studied in

meeting the termination liability is increased when compared with that of the 40 year

old.

The case studies to date have treated the termination liabilities on the assumption that

the scheme is terminated at the valuation date. This then allowed us to consider the

distribution of the investment variation over a year if different investment strategies

were used. However, if the scheme remains open, then under the assumptions in our

case study, the liability will increase by

(a) the excess of the increase in salary over the increase in pension in

deferment,

(b) the increase in pensionable service,

(c) other factors capturing how the unfolding experience differs from

the other financial and demographic assumptions used to estimate

the liabilities (e.g., new entrants).

In practice, of course, almost all schemes will continue so, arguably, the investment

strategy that is best adopted is not the one that best matches the termination liabilities

at one instant in the past but the one that best matches the increase in the termination

liabilities assuming the scheme is not wound up.

We investigated each of the investment strategies previously under this new scenario.

In order to do so we needed to make the following assumptions

(ii) the salary increase in any calendar year matched that of a skilled

craftsman in the building trade, as outlined in Appendix III. Thus

the rate of increase of the termination liabilities assuming the

scheme is not terminated is (1 + wage increase) /(1 plus the lower

of 4% or the rate of inflation over the year) times the rate of

increase of the termination liabilities assuming it is terminated, all

other things being equal.

(iii) The increasing pensionable service can be accurately allowed for in

advance as it deterministic. This creates a factor (greater than

unity) that multiplies the liability factor on scheme termination. We

29

ignore this factor as it varies from scheme to scheme and can be

estimated in advance.

(iv) The experience of the scheme is in line with that assumed in

calculating the termination liabilities in all other matters (e.g., no

new entrants).

Note the similarity between the approach above and the on-going funding plan known

as the ‘defined accrued benefit method’ described and discussed in McLeish &

Steward (1987).

We can redo the previous analysis with these new assumptions, which we term Case

Study 2.

Graph 5: Investment Variation for 40 Year Old for each Investment Strategy, in

each calendar year (Case Study 2)

-100%

-50%

0%

50%

100%

150%

200%

250%

300%

2000

1997

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Equity Long Bond Return 30 Bond Cash Return Property

30

Table 3: 40 Year Old: Statistics of the Empirical Investment Variation Distribution,

over 20th Century and since 1970 (inclusive), Case Study 2

Based on Investment Strategy of 100%…

Equity Long Bond30 Year

Bullet Bond Cash Property20th Century Mean 5.2% 0.1% -3.2% 2.6% N/aMedian 2.1% 1.5% -1.3% 2.2% N/aGeometric Mean 0.3% -3.3% -3.4% -3.8% N/aStan. Dev. 32.5% 27.2% 6.9% 40.3% N/aSkew 1.3 2.0 -1.5 3.0 N/aExcess Kurtosis 4.3 11.2 2.0 18.0 N/a Since 1970 Mean 6.0% -0.9% -7.8% 5.5% 13.3%Median -10.7% -13.5% -5.8% -17.3% -10.0%Geometric Mean -7.0% -10.8% -8.1% -12.8% -7.9%Stan. Dev. 54.1% 48.1% 7.3% 70.9% 75.6%Skew 0.9 1.3 -0.9 1.8 1.3Excess Kurtosis 0.2 2.4 0.8 4.7 1.5

The 30 year bullet bond is still found, of the assert strategies assessed above, to be

closest to the Liability Reference Portfolio, and the ranking of the other asset classes

in terms of risk remains the same as the first case study (in fact the figures for risk are

very close to those earlier). The means and other measures of the central location of

the distribution of the standardised investment variation are altered significantly (as

could be expected) but, again, the relative ranking is very similar to the result of Case

Study 1.

Perhaps the surprise in the results presented above is that equities do not fare better in

the risk comparisons, as equities, if a good inflation hedge, could have been expected

to match liabilities increasing in line with wage inflation (which, as noted in

Appendix II is closely related to inflation). The hypothesis that there is a positive

relationship between inflation and nominal return on stocks (so that they both move

up and down together) is generally known as the Fisher Hypothesis, after the

mathematical economist, Irving Fisher. While there is mixed evidence that the Fisher

hypothesis holds for Irish and UK equities, it has been shown not to hold in equities

markets generally. In particular, equities have not demonstrated themselves an

31

inflation hedge in the US and the major euro equity markets.25 In short, no consistent

positive relationship is evident between equity returns and inflation in most

economies.

We also did the same analysis for the 30 year old and present the results in the

following table.

Table 4: 30 Year Old: Summary Statistics of the Empirical Investment Variation

Distribution, over 20th Century and from 1970-2000 (inclusive), Case Study 2

Based on Investment Strategy of 100%…

Equity Long Bond30 Year

Bullet Bond Cash Property20th Century Mean 8.8% 3.9% -2.2% 7.8% N/aMedian 3.2% 2.4% 0.8% 3.1% N/aGeometric Mean 0.2% -3.4% -3.2% -3.9% N/aStan. Dev. 45.2% 44.9% 13.3% 63.5% N/aSkew 1.7 3.7 -0.5 4.6 N/aExcess Kurtosis 5.0 23.1 1.6 31.8 N/a Since 1970 Mean 14.7% 8.6% -6.7% 19.1% 27.6%Median -10.2% -18.8% -8.0% -21.7% -12.4%Geometric Mean -9.1% -12.7% -9.1% -15.1% -10.3%Stan. Dev. 77.1% 79.9% 21.5% 112.5% 115.8%Skew 1.0 2.1 0.2 2.6 2.0Excess Kurtosis -0.1 6.1 -0.8 9.0 4.8

The above case studies used only historic Irish figures. We did the analysis again

using historic figures from the US and UK capital markets to ensure our findings are

robust across markets. The sources of data for UK and US markets was Barclays

Capital Equity Gilt Study 2003. First, we graph the evolution of the gross redemption

yield on 20 year sovereign bonds in each economy from 1970 to 2000.

25 For Ireland, see the working paper by Ryan (2002). For international evidence across 26 equity markets capturing more than 60% of the capitalisation of all equities in the world over the period 1947-1979, see Gultekin (1983).

32

Graph 6: Long Bond Gross Redemption Yield, US, UK and Ireland, Year Ends,

1970-2000.

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Irish UK US

The performance of the equity markets over the same period is graphed below.

Graph 7: Equity Market Total Return Indices, US, UK, and Ireland, Year Ends,

1969-2000 (Log Scale)

100

1000

10000

100000

1000000

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1996

1995

1994

1993

1992

1991

1990

1989

1988

1987

1986

1985

1984

1983

1982

1981

1980

1979

1978

1977

1976

1975

1974

1973

1972

1971

1970

1969

Irish UK US

Finally, under Case Study 1 above we redid the analysis for the 40 year old but this

time assuming the experience of the US and UK markets from 1970 to 2000. For aid

comparison, we show the figures from the Irish experience alongside.

33

Table 5: 40 Year Old: Summary Statistics of the Empirical Investment Variation

Distribution, 1970-2000 (inclusive), US, UK and Irish Experiences, Case Study 1

Based on Investment Strategy of 100%…

Equity Long Bond30 Year

Bullet Bond CashUS Market Mean 9.6% 3.9% -0.1% 5.6%Median -0.9% -1.1% 0.0% -2.3%Geometric Mean 3.0% -0.4% -0.1% -2.2%Stan. Dev. 38.9% 31.5% 3.1% 42.3%Skew 0.5 1.0 0.4 0.8Excess Kurtosis -0.6 1.9 0.1 0.7 UK Market Mean 8.3% 5.5% -0.3% 10.2%Median 0.6% -2.5% -0.2% -1.6%Geometric Mean 0.8% -2.5% -0.3% -4.1%Stan. Dev. 38.9% 46.5% 2.9% 66.4%Skew 0.2 2.0 0.0 2.4Excess Kurtosis -0.6 6.9 -0.3 8.8 Irish Market (from Table earlier) Mean 13.6% 6.6% -0.3% 13.0%Median -4.7% -11.9% -0.4% -14.9%Geometric Mean 2.2% -2.5% -0.3% -3.9%Stan. Dev. 53.7% 49.3% 2.9% 72.5%Skew 0.9 1.6 -0.1 2.1Excess Kurtosis 0.4 4.1 -0.8 6.1

The tables above bear out the lessons from Irish capital markets.

Time Diversification of Risk Argument

The previous analysis has compared the actual investment experience with that

expected over periods of one year and, for that analysis, reported descriptive statistics

for the empirical distribution of the investment variation. A natural question is

whether the implications of our empirical investigation would significantly alter if the

time period over which the distribution of the empirical variation was assessed

34

increased from one to three or more years. In particular, some have advanced the

argument that equity investment is preferable in the long-term but not necessarily the

short-term, so if our review period was p years, where p is a ‘large’ number, then the

equity investment strategy would have better risk and reward characteristics.

The problem in testing this hypothesis empirically is that we have a limited history of

capitalism so that as p increases the number of non-overlapping intervals quickly

decreases. We have only 10 distinct non-overlapping decades in the 20th century,

which would give just 10 data-points in the empirical distribution.26 However, we can

resolve the problem with a simple model of the investment variation distribution. We

treat one model below but note that the insight it gives applies to a very wide category

of models.

The empirical distribution given in the tables earlier was the standardised investment

variation over one year, or equivalently, the distribution of the percentage change in

the funding level (on termination). Let Y be a random variable with this distribution.

Then the funding level at time 1 1( )F , given it was 100% funded at time 0 is

1 100(1 )F Y= +

A simple model for the funding level at time p ( )pF is

1 2100(1 )(1 )...(1 )p pF Y Y Y= + + +

where each iY is independent of the others and has same distribution as Y. Now,

1ln ln100 ln(1 ) ... ln(1 )p pF Y Y= + + + + +

Let us further assume ln(1 )Y+ is normally distributed with mean µ and variance 2σ .

Then ln pF is normally distributed and pF is lognormally distributed. Then, from the

well-known parameterisation of the lognormal, we can write 21

2[ ] 1E Y eµ σ+= −

(I)

and

26 Worse, we know that the underlying distribution has changed over this time period (that is, is non-stationary over such a long period) thus making such a sample of questionable value to forecast the future.

35

2 22 2 2[ ]Var Y e eµ σ µ σ+ += −

(II)

We have already estimated [ ]E Y and [ ]Var Y and so can solve the above equations

for µ and 2σ . In particular, we can solve for the distribution of pF .

The distribution of the standardised investment variation over one year has been

approximated by our empirical study earlier and we might assume, for concreteness,

that it has a mean of 8% and a standard deviation of 30%. Solving (I) and (II) above

gives µ=0.04 and σ=0.27. The density function of the funding level at time p, where

p=1, p=3 and p=10, is graphed below:

Graph 8: Probability Density Function of Funding Level, when Viewed at end of

1,3 and 10 years, assuming Log-Normal Distribution (see above)

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

1.6%

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

199

208

217

226

235

244

253

262

271

280

289

298

307

316

325

334

343

352

361

370

379

388

397

1 Year 3 Years 10 Years

We note that the distribution of possible outcomes is wider when the review term

increases (‘the expanding funnel of doubt’) and, in particular, that the probability of a

very low funding level is higher the greater the period between reviews. From the

above graph of the funding levels, a rational investor need not necessarily favour the

outcome when p=10 (or, more generally, when p is large) over the outcome when

36

p=1. When p=10, the expected value is increased but so too is the probability of an

extremely poor outcome. A particularly risk averse investor could conceivably prefer

the outcome when p=1 over when p=10.

We can investigate the above remarks in a more formal setting. Given two

distributions, the condition that

xxFxF ∀≤ ),()( 21

is described as the first order stochastic dominance (FSD) of )(1 xF over 2 ( ),F x where

the ( )iF x are distribution functions. A return distribution that first order dominates

another is preferred by any wealth maximiser regardless of their utility function. The

distribution functions of the funding levels for each forecast period are graphed

below.

Graph 9: Cumulative Distribution Function of Funding Level, when Viewed at end

of 1,3 and 10 years, assuming Log-Normal Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

199

208

217

226

235

244

253

262

271

280

289

298

307

316

325

334

343

352

361

370

379

388

397

1 Year 3 Years 10 Years

So, clearly, no distribution for any p stochastically dominates any of the others.

A less stringent condition in comparing two distributions is second order stochastic

dominance (SSD), with )(1 xF said to dominate )(3 xF by SSD if and only if

37

∫ ∫∞− ∞−

∀≤x x

xdyyFdyyF ,)()( 31

It can be shown that investors who are both nonsatiated and risk averse can be shown

to prefer the payoff of )(1 xF over ).(3 xF 27 Again, under our model earlier, we can show

that no distribution for any p stochastically dominates to second order any of the

others. Graph 10, capturing the area under the distribution functions up to the 400%

funding level, demonstrates this.

Graph 10: Area under Cumulative Distribution Function of Funding Level

( ( )x

pF y dy−∞∫ ), when p= 1,3 and 10 years, assuming Log-Normal Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

1 Year 3 Years 10 Years

Conclusion

The above argument, though different in detail, leads to a conclusion very similar to

that advocated for UK pension funds in Speed, Bowie et al. (2003) – that the most

closely matching asset for Irish pension fund liabilities is composed mainly of

27 See, for instance, Eichberger & Harper (1997).

38

conventional and index-linked bonds.28 It also makes clear that there is generally no

simple matching asset for pension fund liabilities and some judgement must be used

in identifying the closest matching portfolio. We note, in particular, that the above

argument leads to a portfolio that, if history is any guide, has a lower expected long

term return than a predominantly equity portfolio.

Note, in particular, that our foregoing analysis does not allow us to suggest one

investment strategy is preferable to another. Investors, including pension funds, are

routinely tempted to take risks if the reward (that is, the form of the investment

variation distribution) is judged sufficiently tempting. However, pension funds should

appreciate the risks involved in alternative strategies and, at a minimum seek to

ensure that the investment portfolio is efficient in the sense that risk cannot be

diminished without diminishing reward. To appreciate the risks and ensure that all

risks undertaken are reasonably rewarded requires knowledge of the investment

variation distribution and, in particular, the Liability Reference Portfolio.

28 Our different arguments hopefully overcome the objections in Hill (2003) to the conclusions in Speed, Bowie et al. (2003).

39

References Bader, L.N & Gold, J (2003) Reinventing Pension Actuarial Science , The

Pension Forum [Pension Section of the Society of Actuaries], January 2003, Volume 15, No.1, 1-13.

Barclays Capital (2003) Equity Gilt Study 2003, 48th Edition. 54 Lombard Street, London.

Black F, (1980). The Tax Consequences of Long-run Pension Policy, Financial Analysts Journal, 36, 21-28.

Caslin, J. (2002) Hedge Funds. Paper presented to Staple Inn Actuarial Society, London, January 2002 and to The Society of Actuaries in Ireland on 9th October 2001.

Chapman, R.J., Gordon, T.J. & Speed, C.A. (2001). Pensions, funding and risk. B.A.J. 7, 605-686.

Day, T. (2003) Financial Economics and Actuarial Practice. Presented at a symposium in Vancouver, British Columbia, Canada, June 24 – 25, Paper available at www.soa.org/sections/pension_financial_econ.html.

Eichberger, J. & Harper, I.R. (1997) Financial economics. Oxford University Press.

Exley, J., Mehta, S. & Smith, A.(1997) The Financial Theory of Defined Benefit Pension Schemes, British Actuarial Journal, 3, IV, 835-939.

Gordon, T. & Jarvis, S. (2003) Financial Economics and Pensions Actuaries – The UK Experience. Presented at a symposium in Vancouver, British Columbia, Canada, June 24 – 25, Paper available at www.soa.org/sections/pension_financial_econ.html

Gultekin, N.B. (1983) Stock Market Returns and Inflation: Evidence from Other Countries. Journal of Finance, 38,1,49-65.

Hill, J. (2003) Submission on the Relationship between Pension Assets and Liabilities. Paper presented to the Staple Inn Actuarial Society, London, May 2003.

Loretan, M., & Phillips, P.C.B. (1994) Testing Covariance Stationarity of Heavy-Tailed

Time Series. Journal of Empirical Finance, 1, 211-248.

McLeish, D.J.D. & Steward, C.M. (1987) Objectives and Methods of Funding Defined Benefit Pension Schemes. Journal of the Institute of Actuaries, 114, 155-199.

40

Modigliani, F. & Miller, M.H. (1958). The cost of capital, corporation finance and the theory of investments. American Economic Review, 48, 261-297.

Palin, J. & Speed, C. (2003) Hedging Pension Plan Funding Ratio. Presented at a symposium in Vancouver, British Columbia, Canada, June 24 – 25, Paper available at www.soa.org/sections/pension_financial_econ.html.

Patterson, J.G. (2003) Selection of Valuation Interest Rates for Funding Valuations of Pension Plans – Traditional Pension Plan Approach versus Financial Economics Approach. Member’s Paper (Document 203044). Canadian Institute of Actuaries.

Pensions (Amendment) Act (2002) Available from the office of the Attorney General of Ireland (on-line at www.irishstatutebook.ie).

Pensions Act (1990) Available from the office of the Attorney General of Ireland (on-line at www.irishstatutebook.ie)

Report on Investment Risk (2004) Working Party Report of the Society of Actuaries in Ireland, available at www.actuaries.ie.

Ryan, G. (2002) Irish Stock Returns and Inflation: A Long Horizon Perspective. Presented at the Irish Economic Association Annual Conference, April 2002.

Society of Actuaries in Ireland (2003) Guidance Notes. Available at www.actuaries.ie.

Society of Actuaries in Ireland (2003) Position Paper on Defined Contribution Plans & PRSAs. Society of Actuaries in Ireland, Dublin (available at www.actuaries.ie).

Sharpe, W. (1976) Corporate Funding Pension Policy. Journal of Financial Economics, 3, 183-193.

Speed, C., Bowie, D., et al. (2003) Note on the Relationship between Pension Assets and Liabilities. Paper presented to the Staple Inn Actuarial Society, London, May 2003.

Tepper, I. (1981). Taxation and corporate pension policy. Journal of Finance, 36, 1-13.

Whelan, S.F. (2000) Evolution of the Asset Distribution of Irish Pension Funds. Irish Banking Review, Summer, 42-52.

Whelan, S.F. (2001) Investing the National Pensions Reserve Fund. Irish Banking Review, Spring, 31-47.

Whelan, S.F. (2002) Prudent Pension Planning. Hibernian Investment Managers, Dublin.

41

Whelan, S.F. (2003) Promises, Promises: Defined Pension Schemes in a

Cynical Age. Irish Banking Review, Winter 2003,

48-62.

42

Appendix I: Limitations of Proposed Definition of Investment Variation (and the

associated Investment Risk)

The above definition of investment variation in actuarial valuations (and the

associated investment risk) has some limitations. Many such limitations arise from the

fact that we have ignored the important relationship between the finances of the

sponsoring employer and the scheme finances. A full treatment of the problem would

model, not just the distribution of the difference between the value of the assets and

that of the liabilities at any point in time, but also the coincidence of risk between a

shortfall being revealed at any future date and the ability (and, if possible to model,

the willingness) of the sponsoring employer to fund the shortfall at that time. We do

not treat this fuller formulation as it demands knowledge of the dynamics of the

sponsor’s business (and so is specific to the sponsor) and how this is related to the

dynamics behind the valuation of assets and liabilities.

We can, however, make some general points on this limitation. First, as a hypothetical

case, consider a pension fund with a high exposure to the business of the sponsoring

employer. Such an investment strategy increases significantly the twin risk of a

shortfall in the value of the assets over the liabilities just when the sponsoring

employer is unable to make up the shortfall.29 In fact, in this case, members might

lose their jobs and part of their pension entitlements if the employer fails.30 Now, in a

less extreme case, the performance of the equity-based portfolio could be correlated to

some degree with the fortunes of the sponsoring employer. Consider, for instance, the

difficulties faced by a small company in the high-technology sector, sponsoring a

pension fund over the couple of years since March 2000. Here, we have the same or at

least similar underlying factors creating financial stress in the pension fund and also

causing strains to the profitability of the sponsoring employer. This is an instance of a

significant fall in the value of the portfolio occurring at an inopportune time for the

employer – once again (but now with less certainty) adversely affecting the security of

the members’ pension entitlements just when those pension assets could be called 29 For this reason, the Pensions Acts impose limits on the level of ‘self-investment’ (as this practice is called) allowed by approved pension schemes. 30 The Pensions Act 1990 outlines a priority ranking of members’ pension entitlements on the termination of the scheme. The Act sets out that members in receipt of a pension have first call on the assets, followed by members in active service. Accordingly, any shortfall in the value of assets to liabilities will be felt more acutely by active members.

43

upon.31 The extent to which these points are material to any particular scheme and

sponsoring employer depend, inter alia, on the relative surplus of the value of scheme

assets over the value of its liabilities (as, other things being equal, the greater the

relative surplus the less likely a deficit will be revealed) and the volatility of the

employer’s profits (as, other things being equal, the less volatile the employer’s

profits the less likely they will experience poor profits when the scheme is in deficit).

(Arguably, a bond-based portfolio of suitable maturity profile ensures that the twin

risks of a deficit revealed in the pension funds and, at the same time, the employer is

particularly financially constrained are largely independent or perhaps even negatively

correlated with one another.)

A case can perhaps be made that Irish pension funds to date have not fully exploited

asset types or investment strategies that are uncorrelated or negatively correlated with

the financial health of the sponsoring employer. Whelan (2002) treats the case of the

National Pensions Reserve Fund, outlining an argument that the Fund should

underweight its exposure to indigenous Irish industries and those sectors of the world

equity market in which the Irish economy has already a high exposure (such as the

pharmaceutical and technology sectors). In particular, pension funds could widen the

search of asset types from the traditional categories to include others such as actively

managed currency funds or hedge funds (see, for instance, Caslin (2002)) which have

little correlation with either the other mismatched assets of pension funds or the

financial strength of the employer and might reduce the standard deviation or thin the

tail of the investment variation distribution.

The general point made in this subsection is that the very same portfolio could have

quite different risk characteristics depending on the nature of the business of the

sponsoring employer – account should properly be taken of the relationship between

the value of the portfolio and the financial strength of the sponsoring employer.

31 Indeed, with the new disclosures demanded of companies under the accounting standard FRS 17, a deficit revealed in the pension fund could precipitate a financial crisis for the employer (say, by reducing their credit rating) and, if the deficit was caused by a sudden collapse of equity values, this is likely just at a time when equity capital is expensive and difficult to raise.

44

Appendix II: Relationship between Price and Wage Inflation in Ireland over 20th

Century

The choice of occupation to represent average wage rate over the very long term is not

so obvious. Few occupations have remained unchanged over the centuries. Many

disappeared with disappearing industries, replaced by new ones with short histories.

Even if an occupation has remained in existence in name for a long while, what it

involves and the social standing it entails, has often radically changed.32 Arguably,

carpentry is an occupation, little changed, whose social standing (and therefore pay-

relatives) remains largely unaltered with time. We use the increase in the hourly rate

paid to carpenters as an indication of wage inflation over the 20th century.33

In the graphs and tables below, we relate wage inflation to price inflation in the past to

find if there is a relatively stable relationship between them. One anticipates such a

stable relationship as workers strive to ensure their standard of living does not fall,

thereby setting inflation as a floor on wage negotiations. Workers can be expected to

demand a share of real productivity increases, giving a positive bias to wage rate

above inflation. This positive bias, however small, should accumulate over a century

to have an obvious positive impact on living standards, as has been the case.

Graph : Irish Price Inflation and Wage Inflation, Each Year over the 20th Century

32 Teachers, for instance, have a long history but their social standing – at least judged by wage relatives – has disimproved; politicians wages, on the other hand, have shown the greatest increase over the 20th century – they were not paid in 1900 (but a blind-eye turned to their use of the position for personal gain), so they record an infinitely high annualised rate of increase. 33 Sources and data as detailed in Appendix I.

45

A relatively stable relationship emerges between inflation and wage increases in any

one year, as could be expected. There tends to be a short lag between wage increases

and inflation, particularly noticeable in the early years after World War II.

Graph: Histogram of Real Wage Increases, Each Year, 1900-2001.

The histogram shows that 0-4% is the most common real wage increase, although

occasional individual years have posted real increases outside this relatively narrow

range.

Table: Key Historical Statistics on Irish Wage Rates

Years Ending 2000

Nominal Wage Increase

Real Wage Increase

Of Real Wage Increase Average SD Min. Max.

25 7.9% 1.7% 1.8% 5.5% -13.2% 10.9% 50 8.1% 1.7% 1.8% 5.1% -13.2% 15.1% 75 6.4% 1.3% 1.5% 6.2% -13.2% 27.9% 100 5.7% 1.1% 1.3% 6.5% -18.7% 27.9%

46

The table confirms the reasonably stable relationship, with wage increases being on

average close to 1% to 2% p.a. above inflation over the long-term.34. We can conclude

from this brief analysis that wage increases would be on average 1% to 2% p.a. higher

than inflation in the long term.

34 One can, as a sensibility check, compare the rate of wage increases to productivity gains over the same period. Labour productivity growth in the larger developed countries averaged 4.5% p.a., 1962-1973, and 1.5% p.a. in 1973-1995 with growth in total factor productivity over the same period falling from 3.3% to 0.8%. (The Economist (1999), Economics). These rates are not inconsistent with the 1½% real wage increase noted above.

47

Appendix III

Returns from Irish Capital Markets in the 20th Century

(with sources indicated)

48

Calendar

Year PropertyReturn

Cash interest rate (Year End)

Long Bond Yield (Year End)

Equity Return

Long Bond

Return

Cash Return

Price Inflation*

Wage Increase†

Equity Index

Bond Index

Cash Index

Property Index

Inflation Index

Wage Index§

2000 15.7% 9.3% 4.4% 27.9% 5.9% 21.3% 1000 1000 1000 1000 1000 1090.54 3.0 5.0 1999 2.2% -7.0% 3.0% 31.1% 3.4% 11.2% 864.3 914.6 957.85 781.86 944.29 899.04 3.0 5.6 1998 25.5% 17.4% 5.5% 38.2% 1.7% 2.3% 844.87 983.69 929.96 596.39 913.24 808.84 3.3 4.6 1997 52.7% 20.7% 6.1% 25.3% 1.9% 8.5% 673.13 837.88 881.81 431.54 897.97 791.05 6.1 5.6 1996 26.1% 13.8% 5.5% 19.1% 1.9% 2.0% 440.8 694.21 831.5 344.45 881.26 729.23 6 6.8 1995 25.0% 18.0% 6.3% 13.2% 2.4% 2.0% 349.69 609.94 788.34 289.33 865.05 714.93 5.6 7.5 1994 1.1% -9.8% 6.0% 15.6% 2.4% 3.4% 279.66 516.98 741.48 255.69 844.88 700.92 6.4 8.4 1993 59.0% 40.0% 9.6% 7.0% 1.5% 9.6% 276.54 573.2 699.54 221.26 825.46 678.1 6.4 6.6 1992 -7.8% 3.5% 12.6% -2.0% 2.3% 9.7% 173.92 409.55 638.15 206.84 813.51 618.82 17 9.4 1991 18.9% 17.7% 10.5% -0.3% 3.6% 14.7% 188.65 395.87 566.56 210.96 794.83 563.95 10.6 8.8 1990 -29.2% 1.8% 11.3% 11.5% 2.7% 2.8% 158.64 336.42 512.71 211.65 767.19 491.48 11.5 9.7 1989 30.9% 1.6% 9.9% 35.3% 4.7% 2.8% 224.19 330.62 460.58 189.88 747.02 478.29 12 8.8 1988 42.2% 32.3% 8.1% 17.9% 2.7% 2.8% 171.29 325.43 419.18 140.35 713.67 465.37 8.3 8.1 1987 -3.9% 30.8% 10.8% 9.8% 3.1% 6.7% 120.42 245.91 387.79 119.07 695.14 452.69 8.8 10.5 1986 49.9% 6.4% 11.7% 6.1% 3.2% 7.1% 125.31 188.04 349.95 108.47 674.48 424.39 11 12.7 1985 57.2% 32.8% 12.2% -0.2% 4.9% 4.0% 83.593 176.8 313.24 102.2 653.84 396.26 11.8 11.9 1984 -1.7% 11.6% 13.2% -0.9% 6.7% 9.2% 53.176 133.17 279.21 102.39 623.13 381.01 15 14.4

§ The wage index can only be regarded as giving indicative changes when the time interval is as short as a year – it gives a more reliable indication of average wage increases when the time interval is over several years or decades. The inflation series gives an accurate estimate of year-by-year changes to the underlying cost of living after 1914; prior to 1914 the estimate is a more crude.

49

Calendar Year

Equity Return

Long Bond

Return

Cash Return

PropertyReturn

Price Inflation*

Wage Increase†

Equity Index

Bond Index

Cash Index

Property Index

Inflation Index

Wage Index

Short Interest Rate (Year End)

Long Bond Yield (Year End)

1983 81.2% 18.2% 14.2% 4.3% 10.3% 5.0% 54.096 119.28 246.75 103.37 583.96 348.79 12.3 14 1982 -5.1% 41.9% 17.6% 10.1% 12.3% 15.5% 29.854 100.89 216.16 99.13 529.42 332.19 15.5 14.6 1981 1.9% 2.6% 16.4% 21.8% 23.3% 7.0% 31.459 71.1 183.75 90 471.41 287.61 18.8 18.2 1980 15.4% 20.5% 16.5% 30.8% 18.2% 31.1% 30.872 69.32 157.92 73.87 382.28 268.8 13.6 15.7 1979 -4.1% -5.0% 15.8% 33.7% 15.9% 6.2% 26.752 57.55 135.58 56.46 323.3 204.97 18.5 16.4 1978 46.8% -4.2% 9.6% 37.2% 7.9% 12.8% 27.896 60.58 117.12 42.24 278.83 192.99 12.8 13.2 1977 88.3% 47.6% 8.9% 25.0% 10.8% 10.0% 19.008 63.22 106.82 30.78 258.33 171.16 7.1 10.9 1976 -16.4% 13.6% 11.6% 11.5% 20.6% 18.4% 10.094 42.84 98.09 24.62 233.19 155.56 14.7 15 1975 80.8% 31.6% 11.1% -3.7% 16.8% 16.6% 12.078 37.72 87.86 22.08 193.36 131.43 11.1 14.8 1974 -45.1% -17.3% 14.7% 8.0% 20.0% 29.3% 6.68 28.67 79.06 22.92 165.49 112.74 13.9 17 1973 -12.7% -7.6% 12.0% 35.1% 12.6% 9.3% 12.176 34.69 68.91 21.23 137.88 87.23 16.5 11.9 1972 77.0% -2.8% 6.9% 19.4% 8.2% 10.2% 13.952 37.56 61.52 15.71 122.4 79.78 9.3 9.6 1971 13.3% 19.0% 6.6% 19.2% 8.6% 11.9% 7.8815 38.66 57.57 13.16 113.08 72.42 4.6 8.3 1970 -6.2% 1.3% 8.3% 13.0% 10.0% 12.8% 6.957 32.48 53.99 11.04 104.11 64.74 8.1 9.3 1969 -8.2% -0.9% 8.5% 7.6% 23.8% 7.4154 32.07 49.87 9.76 94.62 57.39 8.9 8.5 1968 41.1% -0.1% 7.4% 5.4% 4.7% 8.0796 32.37 45.97 87.94 46.35 7.3 7.6 1967 27.1% 1.1% 6.1% 2.6% 3.4% 5.7276 32.41 42.79 83.4 44.26 7.8 6.9 1966 -1.7% 4.0% 6.4% 3.9% 7.3% 4.5056 32.05 40.34 81.28 42.79 6.9 6.4 1965 -3.6% 5.0% 6.3% 3.2% 6.3% 4.5815 30.81 37.9 78.26 39.88 5.9 6.2 1964 15.3% -1.1% 4.8% 6.9% 0.0% 4.7538 29.34 35.66 75.82 37.53 6.8 6.1 1963 33.0% 4.2% 3.8% 4.5% 12.5% 4.124 29.66 34.02 70.91 37.53 3.9 5.5 1962 24.4% 19.8% 4.4% 3.7% 3.0% 3.1015 28.45 32.77 67.88 33.35 3.9 5.4

50

Calendar Year

Equity Return

Long Bond

Return

Cash Return

PropertyReturn

Price Inflation*

Wage Increase†

Equity Index

Bond Index

Cash Index

Property Index

Inflation Index

Wage Index

Short Interest Rate (Year End)

Long Bond Yield (Year End)

1961 19.2% -4.2% 5.3% 2.5% 11.9% 2.4933 23.74 31.39 65.45 32.38 5.6 6.5 1960 23.0% -1.9% 5.1% 2.8% 3.5% 2.0908 24.79 29.8 63.82 28.94 4.6 5.6 1959 45.1% 2.4% 3.5% -1.7% 6.0% 1.7001 25.27 28.37 62.1 27.96 3.7 5 1958 23.1% 11.7% 4.7% 2.7% 4.4% 1.1718 24.69 27.41 63.17 26.38 3.3 4.8 1957 0.5% -2.5% 5.0% 5.9% 0.0% 0.9518 22.11 26.17 61.51 25.26 6.7 5.3 1956 -13.3% 0.6% 5.0% 2.4% 2.9% 0.9467 22.68 24.93 58.1 25.26 5.1 4.7 1955 5.7% -3.8% 3.0% 4.5% 6.4% 1.092 22.53 23.73 56.75 24.53 4.2 4.4 1954 15.7% 5.2% 1.8% 0.4% 0.0% 1.0334 23.41 23.04 54.31 23.06 1.8 3.8 1953 6.7% 8.3% 2.8% 1.8% 0.0% 0.8935 22.25 22.62 54.09 23.06 2.2 3.9 1952 -23.6% 1.3% 2.7% 8.9% 9.4% 0.8374 20.55 22.01 53.13 23.06 3 4.2 1951 10.8% -3.2% 0.9% 10.8% 11.6% 1.0963 20.28 21.43 48.79 21.08 1.5 4 1950 8.9% 3.5% 0.7% 1.9% 0.0% 0.9897 20.94 21.24 44.05 18.89 0.7 3.5 1949 -2.7% -2.5% 0.6% 0.9% 0.0% 0.9086 20.24 21.09 43.24 18.89 0.7 3.5 1948 -3.2% 1.6% 0.6% 2.3% 8.4% 0.9337 20.74 20.96 42.83 18.89 0.6 3.1 1947 3.2% -4.5% 0.5% 5.5% 0.0% 0.9642 20.42 20.85 41.88 17.42 0.5 3 1946 20.0% 5.6% 0.5% -1.7% 25.7% 0.9341 21.38 20.74 39.71 17.42 0.5 2.5 1945 12.5% 7.4% 0.9% 0.7% 4.6% 0.7787 20.24 20.63 40.39 13.86 0.5 2.7 1944 9.5% 4.5% 1.0% 0.7% 7.6% 0.692 18.85 20.43 40.12 13.24 1 3 1943 13.6% 1.6% 1.0% 7.7% 1.5% 0.6317 18.03 20.22 39.85 12.31 1 3.1 1942 18.4% 3.0% 1.0% 15.2% 1.5% 0.5558 17.75 20.02 37 12.13 1 3 1941 8.5% 6.0% 1.0% 10.7% 1.5% 0.4693 17.23 19.81 32.12 11.95 1 3 1940 9.6% 9.3% 1.0% 11.5% 4.3% 0.4326 16.25 19.61 29 11.77 1 3.2 1939 -6.3% 2.1% 1.2% 9.1% 0.0% 0.3947 14.86 19.41 26.02 11.29 1.2 3.6

51

Calendar Year

Equity Return

Long Bond

Return

Cash Return

PropertyReturn

Price Inflation*

Wage Increase†

Equity Index

Bond Index

Cash Index

Property Index

Inflation Index

Wage Index

Short Interest Rate (Year End)

Long Bond Yield (Year End)

1938 -1.9% 0.5% 0.6% -0.6% 0.0% 0.4211 14.56 19.18 23.85 11.29 0.9 3.5 1937 1.7% -2.6% 0.6% 6.6% 9.6% 0.4291 14.48 19.06 23.99 11.29 0.6 3.3 1936 17.0% 2.9% 0.6% 2.5% 0.0% 0.4219 14.87 18.94 22.5 10.3 0.6 2.9 1935 10.6% 0.0% 0.6% 3.2% 0.0% 0.3606 14.45 18.83 21.96 10.3 0.6 2.9 1934 1.9% 11.8% 0.8% 0.6% -1.2% 0.3262 14.46 18.72 21.28 10.3 0.8 2.7 1933 9.6% 3.3% 0.7% 0.6% -1.2% 0.32 12.94 18.57 21.14 10.42 0.7 3.3 1932 -7.9% 21.2% 1.9% -6.1% -1.2% 0.292 12.52 18.44 21.01 10.55 1.9 3.3 1931 -2.2% 1.8% 3.6% -1.8% -1.1% 0.317 10.33 18.1 22.36 10.67 3.6 4.5 1930 13.1% 9.8% 2.6% -6.1% 2.7% 0.324 10.15 17.47 22.77 10.79 2.6 4.3 1929 2.3% 0.7% 5.3% 1.7% 2.8% 0.2864 9.24 17.04 24.26 10.51 5.3 4.7 1928 16.9% 5.8% 4.2% 0.6% 2.8% 0.2801 9.18 16.18 23.85 10.23 4.2 4.4 1927 5.4% 5.9% 4.3% -7.4% 2.9% 0.2397 8.68 15.54 23.72 9.95 4.3 4.5 1926 5.5% 3.3% 4.5% 0.5% 3.0% 0.2274 8.19 14.9 25.62 9.66 4.5 4.6 1925 1.0% 1.8% 4.1% -2.6% 3.5% 0.2156 7.94 14.27 25.48 9.38 4.1 4.5 1924 6.0% 7.0% 3.5% 3.8% 2.7% 0.2133 7.79 13.7 26.16 9.06 3.5 4.3 1923 13.5% 3.2% 2.7% -1.6% 3.3% 0.2012 7.28 13.24 25.21 8.83 2.7 4.5 1922 15.4% 11.2% 2.6% -16.4% 3.7% 0.1774 7.06 12.89 25.62 8.55 2.6 4.4 1921 12.2% 13.0% 5.2% -9.2% 3.2% 0.1537 6.34 12.56 30.63 8.24 5.2 4.9 1920 -10.6% -3.5% 6.4% 15.8% 3.7% 0.137 5.61 11.94 33.75 7.98 6.4 5.5 1919 -2.1% -3.2% 3.9% 5.9% 3.8% 0.1532 5.81 11.22 29.14 7.7 3.9 4.8 1918 22.8% 9.9% 3.6% 15.3% 8.6% 0.1565 6.01 10.8 27.51 7.42 3.6 4.2 1917 6.7% 0.4% 4.8% 20.5% 19.6% 0.1274 5.46 10.43 23.85 6.83 4.8 4.6 1916 0.7% -2.7% 5.2% 18.7% 24.4% 0.1194 5.44 9.95 19.79 5.71 5.2 4.3

52

Calendar Year

Equity Return

Long Bond

Return

Cash Return

PropertyReturn

Price Inflation*

Wage Increase†

Equity Index

Bond Index

Cash Index

Property Index

Inflation Index

Wage Index

Short Interest Rate (Year End)

Long Bond Yield (Year End)

1915 -6.8% -3.4% 3.7% 23.0% 0.0% 0.1186 5.59 9.46 16.67 4.59 3.7 3.8 1914 0.9% 4.8% 2.9% -2.0% 4.0% 0.1273 5.79 9.13 13.55 4.59 2.9 3.3 1913 0.5% 1.9% 4.4% 2.0% 0.7% 0.1261 5.52 8.87 13.82 4.41 4.4 3.4 1912 -1.6% 1.8% 3.6% 3.1% 0.7% 0.1255 5.42 8.5 13.55 4.38 3.6 3.3 1911 -3.8% 1.7% 2.9% 1.0% 2.5% 0.1276 5.32 8.2 13.15 4.36 2.9 3.2 1910 3.7% 1.6% 3.2% 2.1% 0.0% 0.1326 5.23 7.97 13.01 4.25 3.2 3.1 1909 7.9% 1.5% 2.3% 1.1% 0.0% 0.1278 5.15 7.73 12.74 4.25 2.3 3 1908 9.5% 4.5% 2.3% -2.1% 0.0% 0.1185 5.08 7.55 12.6 4.24 2.3 2.9 1907 -1.2% -0.1% 4.5% 2.2% 0.0% 0.1082 4.86 7.39 12.88 4.24 4.5 3 1906 2.5% 2.8% 4.0% 1.1% 0.0% 0.1095 4.87 7.07 12.6 4.24 4 2.8 1905 0.1% 2.8% 2.6% 0.0% 0.0% 0.1068 4.73 6.8 12.47 4.24 2.6 2.8 1904 -0.4% 2.8% 2.7% 1.1% 0.0% 0.1067 4.6 6.62 12.47 4.24 2.7 2.8 1903 0.3% 0.6% 3.4% 1.1% 0.0% 0.1071 4.48 6.45 12.33 4.24 3.4 2.8 1902 3.9% 2.8% 3.0% 0.0% 0.0% 0.1068 4.45 6.24 12.2 4.24 3 2.7 1901 -3.0% 0.4% 3.2% -1.1% 0.0% 0.1029 4.33 6.06 12.2 4.24 3.2 2.7 1900 1.7% -3.1% 3.7% 5.8% 2.7% 0.106 4.31 5.87 12.33 4.24 3.7 2.5 1899 0.1043 4.45 5.66 11.66 4.12 3.4 2.2

53

Sources of Data

The Irish Equity Market

From 1988 to 2000 (inclusive) returns are derived from the Irish Stock Exchange (ISEQ) Return-Overall Index. The official equity index series

of the Irish Stock Exchange is the ISEQ Indices. These comprise the ISEQ Overall Index, the ISEQ Return-Overall Index and two sub-indices of

each, capturing the financial stock (Financial) and all other stock (General). The ISEQ and ISEQ Return indices differ only in that the latter

allows for the reinvestment of gross dividends in the index on the ex-dividend date (that is, dividend grossed up by the largest published tax

credit of the Irish registered company). All companies on either the Official List or the Developing Companies Market with a registered office in

the Republic of Ireland are included in the index series with, from 1st January 1997, the indices extended to include also those companies on the

Official List with a registered office in Northern Ireland. The indices are arithmetic indices, the share price weighted by the issued share capital.

The weights are now updated daily as and when the officially listed issued share capital changes. The set of indices began on 31st December

1987 with, uniformly, 1000.00 as the radix at the close of business that day. A history was calculated by the Irish Stock Exchange for the ISEQ

Overall Index, daily back to 31st December 1986 and weekly back to 5th January 1983 (source: Irish Stock Exchange).

From 1979 to 1987 (inclusive), the return of the Irish equity market is based on the Goodbody Total Market Index with allowance for the gross

income reinvested. This index is identical in construction to the ISEQ Overall Index but coverage includes shares traded on the (former)

Unlisted Securities Market. The yield used to allow for the income is the published twelve month trailing gross dividend yield which we assume

was payable at the end of the calendar year.

54

From 1934 to 1978 (inclusive), the return on the Irish equity market is based on the CSO Price Index of Ordinary Stocks and Shares of

Companies incorporated in Ireland with an approximate allowance made for dividend income. From January 1934 to the mid-1980s the Irish

Central Statistics Office (CSO) and its forerunner compiled a capital return index of Irish companies, the CSO Price Index of Ordinary Stocks

and Shares of Companies incorporated in Ireland (except Railways). Details on the construction of the index are rather scant with, for instance,

official sources such as the CSO itself, the annual Statistical Abstract of Ireland or its forerunner, the Irish Trade Journal, providing minimal

descriptions. However, Geary (1944) describes it as an arithmetic, market-capitalisation weighted index with (at that time) complete coverage of

the 88 non-railway Irish registered stocks listed on the two Irish exchanges of Dublin and Cork.

The CSO Index is calculated from share prices quoted on the Irish Stock Exchange on the first trading day of each month. There have been a

few changes in its method of construction since 1934. Each January beginning in January 1958, the index was adjusted to include only those

shares that had been dealt in the previous twelve months. This entailed a reduction of the number of companies covered from 118 in January

1957 to 101 in January 1958 (Murray (1960)). In 1967 the index was again adjusted to include only companies with a market capitalisation in

excess of IR£0.5 million (Kirwan & McGilvary (1983)). Finally, the index was later superseded in the January 1988 (CSO Statistical Abstract

1988) by the more comprehensive Irish Stock Exchange Equity (ISEQ) series of indices (see above).

Values of the CSO Index were obtained from successive issues of the Statistical Abstract of Ireland or its forerunner, the Irish Trade Journal,

from which the complete series could be reconstructed. The CSO index was used up to the end of 1978.

Ideally, of course, we want an index that includes the reinvestment of dividends. A study of the yield differences and ratios between the Irish and

UK markets from 1965 to the start of 1979 (when exchange controls were introduced and the parity link of the Irish pound with sterling was

55

severed) reveals little difference between the yields (Whelan (1999)). Accordingly, to a reasonable approximation, we assume that the yield on

the Irish market equalled that of the UK market between 1934 and 1978. We use the dividend yield of the UK market estimated by Dimson et al.

(2002) and their calculation of the total return from the Irish equity market based on this yield approximation after correcting a mis-statement in

calendar years 1940 and 1941.

From 1900 to 1933 (inclusive), we use the capitalisation-weighted index of 70 Irish stocks with an estimate of the dividend yield based on the

UK stock market described in Dimson, Marsh & Staunton (2002).

Irish Cash Returns

From 1999 to 2000, we use the Irish Treasury Bill Return as in Dimson, Marsh & Staunton (2002). From 1971 to 1998, we use the average Irish

inter-bank 3 month rate over the year, kindly provided by Goodbody, Stockbrokers. From 1900 to 1970, we use the “Three Month Bank Bill

Rates for Sterling” as reported in Mitchell (1988). The cash return is the average rate for the year calculated from monthly rates and, where

monthly figures were not available (that is, before 1938), the average rate is assumed to be the rate ruling at the end of the year. Honohan &

Conroy (1994) have attempted to construct a comparable series from Irish historical sources. However the observed differences between their

tabulated Irish rates and English rates of roughly ½% in the century to 1922 can perhaps principally be attributed to differences in either default

risk or the lesser liquidity of the Irish instruments. We prefer to use sterling interest rates in this period on the justification that sterling was the

legal tender in Ireland from 1826 to 1927 and from 1928 to 1979 a one-for-one no-margins parity exchange rate was maintained. We make just

one minor adjustment to the value of the cash return for 1955. At the start of 1955 the Irish banks failed to match a 1.5% increase in rates in the

UK. The interest rate differential and freedom of capital movements ensured an enormous outflow of funds from Ireland to the UK,

56

necessitating the inevitable closing of the differential before the end of the year (Honohan (1997)). The average interest rate over 1955 was

accordingly adjusted to be 0.75% below that of the UK.

Irish Inflation

From 1953 to 2000, we use the Consumer Price Index (All Items) (CPI), published by the Central Statistics Office (CSO). This measures the

change in the average price paid by Irish households for consumer goods and services. The Index is available monthly from January 1997 and

quarterly before that date. The annual CPI figures reported relate to the year ending mid-November up to 1998 and from then year ending mid-

December. From 1947 to 1952, we use the Interim Cost of Living Index (Essential Items), published by the CSO. This is the forerunner of the

CPI and differs in that the weights applied to each item reflect the weight that item bore in a basket of essential items purchased by the non-

agricultural employed class. The index is available quarterly and, as above, we report to the year ending mid-November. From 1922 to 1946, we

use the Cost of Living Index (Ireland), which was an Irish government publication, now published by the CSO. The weights of this price index

were based on a small survey of the expenditure pattern of working class families in 1922 (Meghen (1970)). This index was published quarterly,

with the figure quoted relating to the year ending mid-November from 1931 to 1946, and the year ending mid-October from 1922 to 1930. From

1914 to 1921, we use the Working Class Cost of Living Index (All Items) (UK) as representative of Irish inflation over this time. This was the

first official cost of living index published in the UK and is based on the expenditure pattern of the working class derived from a survey in 1904

(sourced from Mitchell (1988)). From 1900 to 1913, we use Bowley’s Cost of Living Index (UK). This is an estimate of UK inflation made by

A.L. Bowley based on the weights used in the first official Working Class Cost of Living Index (UK), backdated with prices from various sources

(again sourced from Mitchell (1988)).

57

The Irish Bond Market

From 1999 to 2000, we use the return on a notional 10 year gilt as in Dimson, Marsh & Staunton (2002). From 1979 to 1998, we base the return

calculations on the coupon and changing yield to redemption on a representative 20 Year Irish Gilt over the year, based on end year yield figures

kindly provided by Riada Stockbrokers (see Whelan (1999)). The coupon (reinvested gross) is assumed to be equal to the average yield to

redemption on 20 year gilts over the previous 5 years. From 1963 to 1978, the return is based on the changing yield to redemption on a 20 Year

British Government Gilt, sourced from BZW (1997), with coupon assumed as above. The justification for using sterling rates is outlined in the

considerations affecting the choice of short-term interest rates. From 1918 to 1962, the return is based on the changing yield to redemption on

irredeemable British Government Gilts from BZW (1997), assumed to apply to a 20 year gilt with coupon equal to the average yield over the

previous 5 years. From 1900 to 1917, we use the yield on British Government Consols as aggregated from various sources in Mitchell (1988),

amended when the price was above par, to calculate the notional return from a 20 year gilt in the same manner as above.

58

Sundry Other Irish Market Returns and Related Economic Series

It was possible to trace back representative returns from Irish institutions’ investment in the Irish commercial property sector. From 1984 to

2001, we use the Society of Chartered Surveyors/Investment Property Databank Index of Total Return. From 1970 to 1983, we use the Jones

Lang Wootton Irish Property Index, Overall Index.

The Irish wage index is based on the increase in the hourly wage of carpenters over the calendar years, sourced from the Building & Allied

Trades Union from 1919 to 1999 (inclusive), prior to 1919, from D’Arcy (1989), and after 1999, from the Construction Industry Federation. The

hourly wage rate used to construct the wage index can be viewed as the minimum rate agreed by the trade union. Accordingly, the wage index

gives a more reliable indication of the average wage increase over the longer term than over short terms.

References D’Arcy, F.A. (1989) Wages of Skilled Workers in the Dublin Building Industry, 1667-1918. Saothar 15 (Journal of the Irish

Labour History Society), pp.21-36.

Dimson, E., Marsh, P, & Staunton, M. (2002) Triumph of the Optimists. Princeton University Press.

Geary, R. (1944) Some Thoughts on the Making of Irish Index Numbers, Journal of the Statistical and Social Inquiry Society of Ireland, Vol. XVII, 345-380.

Hall, F.G. (1949) Bank of Ireland 1783 - 1946. Hodges Figgis, Dublin & Blackwell, Oxford.

59

Honohan, P. & Conroy, C. (1994) Irish Interest Rate Fluctuations in the European Monetary System. The Economic & Social Research Institute, General Research Series, Paper No. 165.

Honohan, P. (1997) Currency Board or Central Bank? Lessons from the Irish Pound’s Link with Sterling, 1928 - 1979. Bank Nazionale del Lavoro Quarterly Review, No. 200.

Kelly, J (1993) The Development of Money and Foreign-Exchange Markets in Ireland. Central Bank of Ireland Annual Report 1992, 119 - 129.

Kirwan, F. & McGilvray, J. (1983) Irish Economic Statistics. Institute of Public Administration, Dublin.

McGowan, P. (1990) Money and Banking in Ireland: Origins, Development and Future. Institute of Public Administration on behalf of the Statistical and Social Inquiry Society of Ireland.

Meghen, P.J. (1970) Statistics in Ireland. Institute of Public Administration.

Mitchell, B.R (1988) British Historical Statistics. Cambridge University Press.

Moynihan, M. (1975) Currency and Central Banking in Ireland, 1922-1960. Central Bank of Ireland & Gill and Macmillan, Dublin.

Murray, C. H. (1960) Some Aspects of the Industrial Capital Market in Ireland. Journal of the Social and Statistical Enquiry Society of Ireland, XX, 97-131.

Thomas, W.A. (1986) The Stock Exchanges of Ireland. Francis Cairns Publications, Liverpool.

Whelan, S.F. (1999) From Canals to Computers: The Friends First Guide to Long Term Investment in Ireland. Friends First, Dublin. ISBN 0-9535887-0-X.

Whelan, S. (2002) Prudent Pension Planning. Hibernian Investment Managers, Dublin.