metadata: authors: 0: author: Vittas,Dimitri
docna: 0: docna: The simple(r) algebra of pension plans
docty: Policy Research Working Paper
subtopic: Pensions & Retirement Systems,Labor Markets,Social Funds and Pensions,Capital Markets and Capital Flows,Non Bank Financial Institutions,Capital Flows,Inflation
teratopic: Finance and Financial Sector Development
abstracts: cdata!: Chile's success with pension reform
            in the early 1980s and the continuing financial pressures
            facing the social security systems of many developing (and
            some industrial) countries have elicited considerable
            interest in the mechanics of pension systems that are based
            on fully funded individual capitalization accounts. The
            author sets out the simple(r) algebra of both defined
            contribution and defined benefit plans. The author notes the
            following results: (a) unless the real rate of interest
            exceeds the growth rate of real wages by a significant
            margin, payment of a reasonable pension rate requires a high
            contribution rate or a high active worklife ratio (the ratio
            of working life to retirement life); (b) it is more
            difficult for a high-growth economy to provide a high
            pension rate, although the absolute level of the pension
            could be higher than the pension level in a low-growth
            economy with a high pension rate; (c) when pensions are
            indexed to prices, the level of interest rates and growth
            rates affects positively the level of the pension rate. But
            when pensions are indexed to wages, only the difference
            between interest rates and growth rates has an effect on
            pension rates. In all cases, the active worklife ratio has a
            positive effect on the pension rate; (d) the timing of
            contributions, and therefore the pattern of earnings over a
            person's full career, has a significant but not major
            effect on pension rates; (e) employer-based, final-salary,
            defined-benefit plans penalize early leavers and favor late
            high-fliers; (f) full funding with universal coverage and
            full lifetime careers would lead to a massive accumulation
            of capital, especially if the demographic structure
            resembles a pillar rather than the more traditional pyramid.
            This would have major implications for the productivity of
            capital and the functioning of financial systems; (g) a
            mixed system -- combining a redistributive first pillar and
            a fully funded defined-contribution second pillar --
            represents a more prudent, perhaps more equitable, approach
            to reforming pension systems; (h) variable contribution and
            pension rates, within specified limits, might be preferable
            and more consistent with greater reliance on personal choice.
repnme: repnme: The simple(r) algebra of pension plans
display_title: The simple(r) algebra of pension plans
listing_relative_url: /research/1993/06/699005/simpler-algebra-pension-plans
topicv3: Pension Funds m1326066 611,Inflation m1326709 0870F4727ACED1CF8525815C00471347,Population, Aging and Health m1327736 640,Jobs m1503931 2670,Retirement m1504007 2896
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Finance and Financial Sector Development
origu: Financial Sctr Operations & Pol (OPD)
no_of_pages: 40
author_keys: 0: author_key: 92949
model_extractions: A<!-- Start of picture text -->
-<br>Policy Research<br><!-- End of picture text -->

<!-- Start of picture text --> Fnancial Sctor Development<br><!-- End of picture text --> WPS 1145 This paper-a product of the Financial Sector Development Department-is part of a larger effort in the department to study pension systems and contractual savings. Copies of the paper are available free from the World Bank, 1818 H Strmet NW, Washington, DC 20433. Please contact Priscilla Infante, room N9003, extension 37664 (June 1993, 35 pages).

Chile's success with pension reform in the early 1980s and the continuing financial pressures facing the social security systems of many developing (and some industrial) countries have elicited considerable interest in the mechanics of pension systems that are based on fully funded individual capitalization accounts.

Vittas sets out the simple(r) algebra of both defined contribution and defined benefit plans. He notes the following results:

- Unless the real rate of interest exceeds the growth rate of real wages by a sig!.ificant margin, payment of a reasonable pension rate requires a high contribution rate or a high active worklife ratio (the ratio of working life to retirement life).

- It is more difficult for a high-growth economy to provide a high pension rate, although the absolute level of the pension could be higher than the pension level in a low-growth economy with a high pension rate.

* When pensions are indexed to prices, the level of interest rates and growth rates affects positively the level of the pension rate. But when pensions are indexed to wages, only the difference between interest rates and growth rates has an effect on pension rates. In all cases, the active worklife ratio has a positive effect on the pension rate.

career, has a significant but not major effect on pension rates.

> * Employer-based, final-salary, definedbenefit plans penalize early leavers and favor late high-fliers.

* Full funding with universal coverage and full 'lifetime careers would lead to a massive accumulation of capital, especially if the demographic structure resembles a pillar rather than the more traditional pyramid. This would have major implications for the productivity of capital and the functioning of financial systems.

* Full funding may not provide a solution to the financial pressures caused by demographic changes. An increase in the retirement age may be required under both funded and unfunded pension systems to accommodate population aging. This will result in a higher active worklife ratio and a lower dependency ratio.

> * A mixed system - combining a redistributive first pillar and a fully funded defined- contribution second pillar represents a more prudent, perhaps more equitable, approach to reforming pension systems.

* Variable contribution and pension rates, within specified limits, might be preferable and more consistent with greater reliance on personal choice.

* The timing of contributions, and therefore the pattern of earnings over a person's full

The Policy Research Working Paper Series disseminates the findings of work **under** way in theBank. **An** objective of the series is **to** get these findings out quickly, **even** if presentations are less than fully polished. The fndings, interpretations, and conclusions in these papers do not necessarily **represent** officir **I** Bank policy.

Produced by the Policy Research Dissemination Center
## Financial Sector Department World Bank

# The Simple(r) Algebra of Pension **Plans**

# **Dimitri** Vittas
# I. INTRODUCTION'

The success of the Chilean pension reform of the early 1980s and the continuing financial pressures facing the social security systems of many developing and some developed countries have elicited considerable interest in the mechanics of pension systems that are based on fully-funded individual capitalization accounts. Such systems are known as defined-contribution systems because they pay pensions that depend on the contributions made during a person's working life and the investment income generated by investing the accumulated funds.

Fully-funded defined-contribution systems are one of three main forms of organizing the provision and financing of pensions. Many countries, both developed and developing ones, especially in Latin America, have social security systems that operate on a pay-as-you-go basis, i.e. pensions to retired people and their dependents are financed from the current contributions of active workers (in most cases, transfers from general tax revenue are also used to cover any deficit). Such systems are known as unfunded defined-benefit schemes because they do not accumulate any savings to fund future pensions and pension bsnefits are not directly linked to the contributions made on an individual basis. Pay-as-you-go systems face financial difficulties because of the aging of the popilation of many countries and also because of evasion, which not surprisingly is particularly pronounced in countries with high contribution rates.

The third main system of pension finance involves funded defined-benefit plans. The social security systems of some countries operate on a (partially) funded basis, while companybased pension schemes are also often based on funded definedbenefit systems.

The various types of pension schemes have their distinct operating characteristics and raise different economic and regulatory issues<sup>2</sup> . However, a common feature of funded pension plans, whether they are based on defined benefits or on defined contributions, is the accumulation of a capital fund over the working lives of members to provide a desired pension to its beneficiaries after retirement.

I The (r) in the title of this paper reflects the sad fact that pension issues are highly complex and almost nothing is really "simple" about them.

2 Vittas and Skully (1992) provide an overview of the operating characteristics of different types of contractual savings institutions and pension systems, while James (1992) reviews policy issues in providing income security for old age.

1
A basic question facing policymakers<sup>in countries that</sup> contemplate the introduction<sup>of a pension system based on fully-</sup> funded individual capitalization<sup>accounts concerns the</sup> contribution rate that would<sup>be required to cover an adequate but</sup> affordable level of pension.<sup>Pension plan calculations can be</sup> complicated by a plethora of<sup>necessary actuarial assumptions.</sup> However, these actuarial assumptions<sup>somehow obscure a</sup> fundamentally simpler algebra<sup>that underlines the working of</sup> pension plans.

The purpose of this paper<sup>is to set out the relationship</sup> between contribution and pension<sup>rates under different</sup> assumptions about the length<sup>of working and retirement life and</sup> about real rates of interest and rates<sup>of growth of real wages3.</sup> In addition, the paper uses this<sup>algebra to illustrate some basic</sup> policy issues.

The paper is divided into four<sup>sections.</sup> Following this introduction, Section II sets out<sup>the simpler algebra of defined</sup> contribution plans. Section III develops the a_gebra<sup>of defined</sup> benefit plans. Finally, Section IV discusses<sup>a number of policy</sup> issues, such as the required<sup>contribution rates for defined</sup> contribution schemes, the treatment<sup>of early leavers and high</sup> fliers under defined benefit plans,<sup>the macrofinancial</sup> implications of funded schemes, and<sup>the contrast between funded</sup> and pay-as-you-go systems in dealing<sup>with demographic pressures,</sup> Although the paper mostly deals<sup>with a mechanical</sup> application of the algebra involved,<sup>and makes no assumptions or</sup> predictions about behavioral changes<sup>or effects, a number of</sup> interesting results are worth noting:

- Unless the real rate of interest exceeds<sup>the growth</sup> rate of real wages by a significant<sup>margin of 2 to 3</sup> percentage points, payment of<sup>a reasonable pension rate</sup> requires a high contribution rate<sup>and/or a high active</sup> worklife ratio4.

- It is more difficult for a high<sup>growth economy to</sup> provide a high pension rate, though the<sup>absolute level</sup> of the pension could be higher there<sup>than in a low</sup> growth economy with a high pension<sup>rate.</sup>

- When pensions are indexed to prices,<sup>the level of</sup> interest and growth rates affects positively<sup>the level</sup> of the pension rate. But when pensions are<sup>indexed to</sup>

- 3 For other papers that examine the<sup>basic algebra of</sup>

- pension plans, see Bonilla (1992) and<sup>Schwarz (1992).</sup>

- 4 I.e. the ratio of working life to<sup>retirement life.</sup>

2
wages, only the difference between interest<sup>rates and</sup> growth rates has an effect on pension rates.<sup>Of</sup> course, irn all cases, the active worklife<sup>ratio</sup><sup>**he-**a</sup> positive effect on the pension rate.

- The timing of contributions, and therefore<sup>the pattern</sup> of earnings over a person's lifetime career,<sup>has a</sup> significant, though not major, effect<sup>on pension rates.</sup>

- Employar-based final salary defined benefit<sup>plans</sup> penalize early leavers (unless their pension<sup>rights are</sup> fully transferable) and favor late high fliers<sup>(who</sup> would in most corporations normally come<sup>from the ranks</sup> of senior management).

- Full funding with universal coverage and<sup>full lifetime</sup> careers would lead to a massive accumulation<sup>of</sup> capital, especially if the demographic structure resembles a pillar rather than the more traditional pyramid.

- A A massive accumulation of pension fund assets<sup>would</sup> have far--reaching implications for the productivity<sup>of</sup> capital and the functioning of financial systems.

- Full funding may not provide a solution to the financial pressures caused by demographic<sup>changes. An</sup> increase in the retirement age may be required under both funded and unfunded pension systems in order to cope effectively with population aging. This will result in a higher active worklife ratio and a lower dependency ratio.

- In view of the considerable uncertainties<sup>involved, a</sup> mixed system combining a redistributive first<sup>pillar</sup> and a fully funded defined contribution second<sup>pillar</sup> would represent a more prudent, and perhaps<sup>also more</sup> equitable, approach in reforming pension<sup>systems.</sup>

- Although most of the analysis is based on<sup>constant</sup> contribution and pension rates, both rates could<sup>in</sup> practice be variable. In fact, allowing workers to set both their own contribution and pension rates,<sup>within a</sup> specified range, might be preferable and<sup>more</sup> consistent with greater reliance on personal<sup>choice.</sup>

3
# **II.** TWE ALGEBRA OF DEFINED CONTRIBUTION<sup>PLANS</sup>

# **2.1** The simplest case

The simplest possible case is<sup>one where both the real rate</sup> of interest and the growth rate<sup>of real wages5</sup> are equal to zero. Pensions are assumed to be indexed<sup>to either prices or wages and</sup> thus inflation has no impact on<sup>the calculations involved</sup> (alternatively, inflation may be assumed<sup>to be zero)6.</sup> The L.bjective of the pension plan<sup>is to accumulate capital</sup> over the working life of workers<sup>which will be used to make</sup> pension payments over their retirement.

where k is the contribution rate W is the gross real wage p is the gross pension (or replacement)<sup>rate, i.e</sup> the ratio of the pension to the gross<sup>real wage</sup> n is the length of the working life<sup>and</sup> E is the length of life in retirement.

Solving equation (1) for the gross<sup>pension rate yields</sup> (2) p = k* n m The gross pension rate is greater, the<sup>greater the</sup> contribution rate and the greater<sup>the ratio of working life over</sup> retirement life. If the working<sup>life is 40 years and the</sup> retirement life is 20 years, and if<sup>the desired gross pension (or</sup> replacement) rate is 50%, the required<sup>contribution rate would be</sup> 25% of the gross wage<sup>7</sup> .

_5_ It is the growth rate of the real wage<sup>of the</sup> individual worker that needs to be<sup>taken into account. This</sup> would reflect both the growth of average<sup>real wages and the</sup> progression of the individual worker<sup>in terms of seniority.</sup> However, to simplify matters the examples<sup>are based on average</sup> representative workers whose wages<sup>increase by the average growth</sup> rate of real wages.

6 This paper assumes throughout that<sup>pensions are indexed</sup> either to prices or to wages. For<sup>a discussion of the effect of</sup> inflation when pensions are not indexed,<sup>see Schwarz (1992).</sup> _7_ It is interesting to note that in this<sup>case the capital</sup> fund accumulated at the time of retirement<sup>after 4C years of</sup> contributions would correspond to<sup>10 times the annual gross wage.</sup> 4
Note, however, that **"Sp"** is the ratio **of** the pension **to** the gross wage before deducting contributions. If **"b"** is the net pension rate, i.e. the ratio of the pension to the net wage received after deducting contributions, then Thus, with a 25% contribution rate, a 50% gross pension rate would correspond to a 66% net pension rate. To achieve a 100l net pension rate, a contribution rate of 33% would be required. The gross pension rate would then amount to 66%.

The following additional simple results can be obtained: if k=10%, n=40 and m=20, then p=20% and b=22% if k=10%, n=45 and m=15, then p=30% and b--33% if k=10%, n-30 and m=10, then p=30% and b=33% if k=20%, n=40 and m=20, then p=40% and b=50% if k=20%, n=45 and m=15, then p=60% and b=75%.

It can be seen that in the case where the real rate of interest and the growth rate of real wages are both equal to zero, a high contribution rate would be required in order to receive a "decent" pension, i.e. a pension that corresponds to a relatively high net replacement ratio. As discussed below, a high contribution rate would be required for a decent pension in all cases, except where the real rate of interest is much greater than the growth rate of the real wage, or the active worklife ratio is very high.

# **2.2 The case where the real rate of interest is positive**

The algebra becomes somewhat more complex if we allow for a positive real race of interest but continue to assume zero growth in real wages. If rontributions are paid at the end of each year, the first year's contributions will accumulate interest for (n-1) years, those of the second year for (n-2) years, and so on. At the time of retirement, the accumulated value of the fund will be equal to the value of pension payments, which will be given by a series of discounted payments, with the last one discounted for "Im" years. The ubjective function of the pension plan will then look as follows: where S is the accumulated value of a stream of payments for n years invested at a real interest rate of r per cent and 5
A is the discounted value of a stream of payments for a years at a real rate of interest of r per cent.

(5) S = (l+r) (-) + (1+r)(> + . . . + (l+r) + 1 and (6) A =J1_ + 1 + ... + I (l+r) (l+r)<sup>2</sup> (l+r)<sup>m</sup> and Equation (2) will then become (7) p = **k*** S A

> Note that the values of **_"8"_** and **"A"** wili be equal to **"Ia"** and

> **"Im"** respectively if the rate of interest is zero. At a positiva rate of interest, "8'" is greater than "n", while "A" is smaller than "Im". Thus, for a given contribution rate, a higher pension rate is obtained.

With a real rate of interest of 3%, "'n" equal to 40 and **"nm"** equal to 20, "S" amounts to 75.40 and "A" to 14.88. A 10% contribution rate would then fund a gross pension rate of 51% and a net pension rate of 56%.

However, if retirement is delayed by 5 years, the working life would increase to 45 years and, assuming no change in life expectancy, the retirement life will decline to 15 years, giving an active worklife ratio of 3. Under these conditions and continuing to assume a 3% real rate of return, "8" will amount to 92.72 and "A" to 11.94. The gross pension rate from a 10% contribution rate would then equal 78%, corresponding to a net pension rate of 87%.

- 2.3 The case where the real interest rate and the growth rate of real wages are positive and equal, and pensions **are indexed** to price inflation

When we allow for a positive rate of growth of the real wage as well as a positive real rate of interest, the formula for the accumulation of the capital fund is more complicated. If the plan aims to provide a pension that is a fraction of the final salary, the objective function of the pension plan has to take 6
account of both the growth in wages and the level of the final wage: (8) where k * W * F - p * W * (1+g)*° * A F is a stream of payments that grow by **"g"** per cent and earn interest at **"r"** per cent. i.e. (9) F = (l+r)(-') + (<sup>1</sup> A+g) (+r)02 + + (1+g)<sup>2</sup> (1+r) (6 + . . . + + (1+g) (-(1+r) + (2+g)(>1 With g=r, (10) F n * (1+g) and equation (8) becomes (11) k * W * n * (1+g)("' p * W * (l+g)(>') * A and (12) p k* n A With g=r>o%, a given contribution<sup>rate produces a pension</sup> rate that is higher than when<sup>g=r=O%, but lower than when r>O%</sup> and g=-0. Thus, with a contribution<sup>rate of 10%, "u" equal to 40</sup> and "nm" equal to 20, the gross<sup>pension rate when r=g=3% will be</sup> 27% and the net pension rate<sup>30%. With "n" equal to 45 and</sup><sup>**"Im"**</sup> equal to 15, the gross pension<sup>rate will be 38% and the net</sup> pension rate 42%.

# **2.4** The case where **"g"** and **"r" are**<sup>positive but not equal and</sup> pensions are indexed to prices

If "g" and "r" are positive<sup>but not equal, the value of the</sup> fund at retirement is given by<sup>the sum of the geometric series of</sup> equation (9).

s If pension plans aim to provide<sup>a pension related to</sup> average actualized lifetime earnings,<sup>the right-hand side of</sup> equation (8) must be multiplied<sup>by the ratio of average wages to</sup> the final wage.

7
(13) r -. (1= r)a tl* (r -g) Equation (11) then becomes (14) k * **_W_** _*_ F p * W * _G_ * A where G = (l+g) *<sup>1</sup> ) and (15) p =k * _G_ *A For a given contribution rate, the pension rate d&pends on the relative values oZ "g" and _"r"_ as well as the values of l'r" and "ma". The greater the growth rate of real wages, the higher the real rate of irterest that is requires to achieve a desired pension rate as a percentage of the final wage. However: a higher "g" also implies a higher final wage so that a lower pension rate may in fact correspond tz a higiher pension in absolute ternms.

Tables la and lb (and Figures la and lb) show the relationship between Lhe gross pension rate and the real interest rate for different growth rates of real wages, when pensions are indexed to prices. It can be seen from Table la that with k=10%, n=40 and m=20, when ge2% and r=3% the gross pension rate amounts to 33%, but when g=3% and r=2% the gross pension rate falls to 20%. A 3 percentage points difference between the real rate of intsrest and the growth rate of real wages is required for a pension rate in excess of 50%.

Table lb provides the same information for **"In"** equal to 45 and 'm" equal to 15. It can be seen that when g=2% and r=3% the gross pension rate is 47%, while when g=3% and r=2% it is 29%. These pension rates are higher than those shown in Table la. They reflect the lonqer working and shorter retirement lives and the effects of compounding. A 2 percentage points difference between "r" and 'g" is adequate to produce gross pension rates in excess of 50%. When r=5% and g=3%, the pension rate amounts to 68%.

When pensions are indexed to prices, the levels of interest rates and wage growth rates also affect the level of pension rates. Thus, as shown in Table la, when r=g=1% the pension rate amounts to 22%, but rises to 27% with r-z=3%. The corresponding pension rates in Table lb are 33% and 38%.

In considering the entries of Tables la and lb (as well as those of following tables) it should be noted that only those combinations that result in reasonable pension rates of between 50% and 70% would be compatible with the "ssumed :ontribution 8
# Table la

# Targeted Pension Rate with Pensions Indexed to Prices k=10%, n=40, m=20

||G<br>|rowth Ra<br>|teofRe|al Wages|||
|---|---|---|---|---|---|---|
||0%|1%|2%|3%|4%|b%|
|Real Interest Rate<br>|||||||
|0%|20.0|16.6|14.0|11.9|10.3|9.0|
|1%|27.1|22.2|18.4|l5.5|13.3|11.5|
|2t|36.9|29.8|24.5|20.4|1.7.2|<br>14.7|
|3%|50.7|40.4|32.7|26.9|22.4|18.9|
|4%|69.9|55.1|44.1|35.8|29.4|24.6|
|5%|96.9|75.5|59.7|47.9|38.9|32.1|
|10%|519.9|387.5|292.0|222.5|171.6|133.9|

# Table lb

Targeted Pension Rate with Pensions Indexed to Prices ks=l0%, n=45, m=15

|||Growth<br>|Rata of<br>|Real Wag<br>|es||
|---|---|---|---|---|---|---|
||0%|1%|2%|3%|4%|5%|
|Real Interest Rate<br>|||||||
|0%|30.0|24.3|20.0|16.8|14.4|12.4|
|1%|40.7|32.5|26.4|21.8|18.3|15.6|
|2*|56.0|43.9|35.0|28.5|23.6|.19.9|
|3%|77.7|59.9|47.1|37.7|30.7|25.5|
|4%|108.9|82.8|64.0|50.5|40.5|33.0|
|5%|153.9|115.4|88.0|68.3|53.9|43.4|
|10%|945.2|672.5<br>9|484.4|353.5|261.6|196.4|
<!-- Start of picture text --> Figure la<br>Targeted Pension Rate with Pensions Indexed to Prices<br>(k=10%, n=43, m=20)<br>160 T<br>120 ~~~~~~~~~~~~~~~~g=-O%<br>120<br>----<br>g=-2%<br>80 O<br>160<br>40<br>g=-5%<br>0 1 2 3 4 5<br>Real Interest Rate (%)<br>Figure lb<br>20.S.-. Targeted Pension Rate with Pensions Indexed to Prices -~. . - ._____ X<br>(k=10%, n-45, m=15)<br>160<br>140<br>120 9--~~~~~~~~~~~~~~~gO%<br>100<br>.2 80<br>40 __<br>g--5%<br>0 1 2 3 4 5<br>Real Interest Rate (%)<br><!-- End of picture text --> 10
rate and working and retirement lives. For instance in Table la, only the combinat, ns r=3% & g=%0, **r=4%** A g=1%, ri5% G **g=2%,** and r=S% g=3%, woulU produce reasonable pension rates. Entries to the top right of these combinations would imply a higher required contribution rate or worklife ratio for the achievement of reasonable pension targets, while entries to the bottom left would imply a lower required contribution rate or worklife ratio. Moreover, some combinations, such as those involving a 10% real rate of interest over a 60 year period, would be unlikely to occur.

# 2.5 **The** case of indexing pensions to wages with g=nr

Indexing pensions to prices maintains their real value. However, if one is concerned to maintain the value of pensions relative to wages and thus ensure that pensioners benefit from continuing growth, pensions must be indexed to wages. Of course, if real wages fall, due to macroeconomic crisis, then real pensions would also decline. Indexing pensions to wages is generally more expensive and requires higher contributions rates, if real wage growth is positive.

When pensions are indexed to wages, the value of pension payments at the time of retirement is given by (16) D = (lea) + ( _1l+a)_<sup>_2_</sup> + _..._ + (1+cf) (-<sup>1</sup> ) + (l+a) (l+r) (l+r)<sup>2</sup> (l+r)(nl) (1+r)<sup>m</sup> If g=r, then equation (16) simplifies to (17) D = m and equation (11) becomes (18) k * W * n *G = p * W * G * m and (19) p = **k*** n m This is identical to equation (2) and provides a simple approximation to the required contribution rate whsn the real rate of interest and the growth rate of real wages are assumed to be equal. Since in real life, "g" and "r" are determined endogenously and cannot deviate much from each other for long periods of time, equation (19) (though very simple) offers a good approximation of the required contribution rate for a targeted pension rate that effectively depends on the active worklife ratio.

11
**"g" and**

# **2.6 The case of indexing pensions to wages with unequal**

If "g" and "r" are not equal, then the value of wage-indexed pensions at the time of retirement is given by the sum of the geometric series of equation (16). This is equal to In this case, equation (14) becomes (21) k _* W_ * F _= p * W * G *_ D and equation (15) (22) p = k* F _G * D_ Tables 2a and 2b (and Figures 2a and 2b) show the relationship between the gross pension rate and the real interest rate for different growth rates of real wages, when pensions are indexed to wages. In general, wage indexation of pensions, when real wages are rising, requires either a higher rate of return or a higher rate of contribution to achieve a given pension rate.

If g<r, "D" is smaller than **"Im"** but greater than "A", so that, for a given contribution rate and when the growth rate of the real wage is assumed to be positive, wage indexation of pensions causes a reduction in pension rates. Table 2a shows that with **k=10%,** n=40, m=20, g=2% and r=3%, the gross pension rate will be 27%. This is lower than the pension rate of 33% achieved when pensions are indexed to prices.

If g>r, then **"D"** is greater than **"Im".** A very high contribution rate would be required to. achieve a decent pension rate. Thus, with g=3% and r=2% the pension rate amounts to only 15%. A contribution rate of 30% w^-)uld be needed to achieve a gross pension rate of 45% that would then correspond to a net pension rate of 64%.

Table 2b shows the same relationships with **k=10%, n=45 and m=15.** The longer working and shorter retirement lives result in higher pension rates as in the case of Table lb.

When pensions are indexed to wages, the pension rate is determined by the difference between the real rate of interest and the growth rate of real wages. The levels of these two variables have little impact. Of course, the cuntribution rate and the length of the working and retirement lives continue as 12
basic determinants of pension rates. Thus, with k=10%, **nx40** and m=20, a difference of 2% between "r" and "g" produces a pension rate of 36-37%, while with k-10%, n=45 and **_m=15,_** the pension rate is 55-56%.

13
# Table 2a

# Targeted Pension Rate with Pensions Indexed to Wages k=10%, n=40, m=20

|||Growth|Rate of|Real Wag|es||
|---|---|---|---|---|---|---|
||0%|<br>1%|<br>2%|<br>3%|<br>4%|5%|
|Real Interest Rate<br>0%|20.0|14.9|11.3|8.6|6.7|5.2|
|1%|27.1|20.0|15.0|11.3|8.7|6.7|
|2%|36.9|27.0|20.0|15.0|11.4|8.7|
|3%|50.7|36.7|26.9|20.0|15.0|11.4|
|4%|69.9|50.2|36.5|26.9|20.0|15.1|
|5%|96.9|69.0|49.7|36.3|26.8|20.0|
|10%|519.9|359.1|250.3|176.0|125.0|89.6|

Table 2b

# Targeted Pension Rate with Pensions Indexed to Wages k=10%, n=45, m=15

|||Growth|Rate of|Real Wag|es||
|---|---|---|---|---|---|---|
||0%|<br>1%|<br>2%|<br>3%|<br>4%|5%|
|Real Interest Rate|||||||
|0%|30.0|22.4|17.1|13.2|10.4|8.2|
|1%|40.7|30.0|22.5|17.1|13.3|10.4|
|2%|56.0|40.6|30.0|22.6|17.2|13.4|
|3%|77.7|55.6|40.5|30.0|22.6|17.3|
|4%|108.9|76.9|55.3|40.4|30.0|22.7|
|5%|153.9|107.4|76.2|54.9|40.3|30.0|
|10%|945.2|631.3|426.4|291.5|201.8|141.6|

14
<!-- Start of picture text --> Figure 2a<br>Targeted Pension Rate with Pensions Indexed to Wages<br>(k=10%, n=40, m--20)<br>160<br>140<br>120 gO<br>g=1%<br>F 00 ~~~~~~~~~~~~~~~~~~~....g-2%<br>80<br>~60<br>40 _ _<br>0 II i<br>0 1 2 3 4 5<br>Real Interest Rate (M)<br>Figure 2b<br>Targeted Pension Rate with Pensions Indexed to Wages<br>(k=10%, n=45, m=15)<br>160<br>140<br>120 g=09t<br>~ 100<br>..... g=2%<br>.2<br>~60<br>40_ _<br>0 1 2 3 4 5<br>Real Interest Rat. (%)<br><!-- End of picture text -->
# III. THE ALGEBRA OF DEFINED BENEFIT PLANS

This paper has so far dealt with plans based on constant contribution rates. These are mainly defined contribution plans, especially when they are established by the authorities and are mandatory on all employees. In such cases, a common contribution rate normally applies to all employees irrespective of their age and years of service, although the contribution rate may be adjusted from time to time.

However, there is no reason why defined contribution plans should not operate with variable rates or with rates that vary for different groups of employees<sup>9</sup> . For instance, in the Chilean pension scheme, the compulsory contribution rate is 10%, but affiliates can make additional voluntary contributions up to 20% of earnings (Vittz,s and Iglesias 1992). In Singapore, the standard contribution rate is a very high 40% of wages, but special lower rates apply for older workers.

Variable contribution rates are more extensively used for defined benefit plans, where the pension rate depends on the years of service of members. Although sponsoring firms make annual contributions on the basis of projections about the group structure of covered members in terms of age, salary level and length of service and in the light of actual and anticipated investment returns (Bodie 1990), in principle the collective contributions are based on contributions that can be calculated separately for each employee.

The algebra of pension plans with variable contribution rates is more complex than that for constant contribution rates. This is because a whole series of contribution rates need to be determined rather than the single rate used in plans with constant contribution rates.

# 3.1 The case where r=g=O

As already noted, if g=r=O, the objective function of the pension plan is given by

> 9 In systems with individual capitalization accounts, workers may be given the right to vary their annual contribution rates within an approved range, depending on whether their accumulated balances are adequate for meeting a targeted pension rate. It should also be noted that targeted pension rates need not be constant throughout a person's retirement. If consumption needs decline with age, a phased reduction in pension rates could be built into the plans. Such an approach may be preferable to offering unindexed pensions, the real value of which would decline in an unpredictable way.

16
(23) k * _W_ * n = p * W * m In a defined benefit plan, the pension rate is equal to (24) p **=** n * q where "q" is the rate of accrual of pension benefit as a per cent of salary per year of service. The contribution rate is then constant and given by (25) k **=** q * m With q=1%, n=40 and m=20, a contribution rate of 20% would be needed to pay a pension rate of 40%. A qg2% would require a contribution rate of 40% and would pay a pension rate of 80%.

# **3.2** The case where **"r"** is positive but "g" is zero

If g=O but "r" is positive, and pensions are indexed to prices, the contribution rate will vary from year to year, since contributions will be accumulating interest until the time of retirement. The contribution rate for the first year will be (26) k, c * A (l+r) (a-l) where, as above, "A" is the value at the time of r-tirement of a stream of payments for "Im" years at a real rate of interest of "r" per cent.

The contribution rate for the second year will be given by (27) k2 **=** * A (l+r) **(a-D** and so on until the last contribution rate will be given by (28) kI = q* A In general, (29) K **=A** (l+r) **(n-1)** If r=3%, q=1%, n=40 and m=20, then k,=4.70% and k,=14.88%. If r=5%, then kl=1.86% and k=12.46%. (Figure 3a shows the path of annual contribution rates.) 17
**3.3 The case where both "r"** _and_ "g" **are positive**

If both "r'" and "g" are positive, the calculation of annual contribution rates becomes more complex and depends on whether accumulated or projected benefits are taken into account. Accumulated benefits cover benefits already earned with regard to past service, while projected benefits also include benefits to be earned with regard to future service.

In the economic literature on pension funds there is considerable debate about the usefulness of actuarial assumptions for ascertaining the future liabilities of pension funds and about the extent of underfunding or overfunding of such liabilities (Bodie and Shoven, 1983). In practice, however, actuarial assumptions are required for both methods. Estimating accumulated benefits requires assumptions about the planned retirement age and the expected retirement life of beneficiaries, the growth (i.e. price or wage indexation) of pensions, and the discount rate. For projected benefits, further assumptions about future wage growth are also required.

On the basis of projected benefits, the series of contribution rates with pensions indexed to prices will be (30) k,1 = (1+a) () * q * A (l+r) (-) (31) k2 = (1+q) ( * q * A (34+r) (a2) and (32) k = q * A In general, However, if g=r, then the contribution rates are constant and equal to (34) k = q * A With g=r=3% and q=1%, n=40 and m=20, the contribution rate is equal to 14.88%. With g=r=5%, it is 12.46%. With g=5% and r=3%, contribution rates start with a high **k1** of 31.55% and decline to a k of 14.88%. In contrast, with r=5% and g=3%, k, is a low 5.88% rising gradually to a k, of 12.46%. (Figure 3b shows the path of annual contribution rates for four different combinations of interest and wage growth rates.) 18
If reserves are set aside for **accumulated benefits** only, the fund will need to cover the value of the accrued benefit at the time of retiremert, without any allowance for projected benefits to be accrued. This is given by the product of the accrual factor and the value of annuity at the time of retirement. Allowing for the fact that reserves will be invested for (n-1) years, the first year's contribution rate will be equal to (35) k - c a(+)A (l+r)<s For the second year of service, the contribution rate will be expressed as a fraction of the new wage W*(I+g). After two years of service, the value of the accrued benefit will be equal to twice the accrual factor times the value of the annuity at the time of retirement times the wage growth factor. Accumulated reserves will be invested for (n-2) years. However, the first year's reserves will have accumulated interest for one year so that the second's year contribution will be given by (36) k 2 = T2 (1+g) where (37) T2 = 2q * A * (1+gl - _a * A_ ? 1+r) (2 (l+r) (".

After three years of service, the;contribution rate will be equal to (38) k3 = T (l+g) 2 where _(39) T3 = 3a_ * A * (1+q)<sup>2</sup> 2 _2_ * A _* (1+a)_ (l+r) l 3)( where and after "'n" years of service (40) k = T (1+g) where (41) T. n * q * A * (l+g)*<sup>1</sup> ) - (n-l) * q * A * (l+g) 2 In general, 19
K _= a_ * A * ri (42)

# (i-2A/(l+a)l

( 1r)(* With g=3% and r=S%, k, = 1.86% and k, = **26.62%,** but with g=5% and r=3%, **k 2** = 4.70% and kx = 42.51%. (Figure 3c shows tha path of annual contribution rates for the same four combinations of interest and wage growth rates as in Figure 3b.) Using projected benefits for i-termining annual contribution rates smooths out the impact of future wage growth, but may req.ire very large initial contribution rates if wage growth exceeds the rate of interest. On the other hand, using accumulated benefits as the basis for determining annual contribution rates may involve large changes in contribution rates from one year to the next as a result of wage growth and longer service. Contribution rates may need to increase steeply l1.t.e in the working life of workers.

If pensions are indexed to wages, "A" must be replaced with "D'" in equations (26) through (42). Since with a positive "g", "D" is greater than "A", wage indexation of pensions will require higher annual contribution rates under either method.

20
<!-- Start of picture text --> Figure 3a<br>Required Contribution Rate with Pensions Indexed to Prices<br>(g=O%, q=1%, n=40, m=20)<br>16%<br>14%<br>j12%<br>10%<br>358% r=3%<br>6 % -..-. r=5%<br>2%<br>1 4 7 10 13 16 19 22 25 28 31 34 37 40<br>Year<br>30%<br>Figure 3h<br>,20% Required Contribution Rate with Pensions Indexed to Prices....... g=3<br>(Projected Benefit Basis: q=1%, n=40, m=20)<br>35%<br>30%<br>I25% r.3%, g=5%<br>120% - - - -<br>r=3%, g=3%<br>.115% -- --- --- --- -- r=5%. g=5%<br>10% r=5%, g=3%<br>5%<br>0% . 1 . 4 . 7 10 13 16 19 22 25 28 31 34 37 40 . . . . . . . .. . . . .<br>Year<br>Fgure23c<br>Required Contribution Rate with Pensions Indexed to Pnices<br>(Accumulated Benefit Basis: q=-1% n=40Q, m=-20)<br>45%<br>40%<br>35% ~~~~~~~~~~~~~~~~r--3%, g=-5%<br>Ix 30%<br>25% - - - - r--3% g=3%<br>r-5%, g-5%<br>U15%<br>10 r=5%. g=3%<br>5%<br>0%<br>1 4 7 10 13 16 19 22 25 28 31 34 37 40<br>Year<br><!-- End of picture text --> 21
## **IV.** POLICY ISSUES

The algebra developed in this paper is mechanical in nature and does not purport to shed any light on the effects of changes in behavior arising from the creation of funded and unfunded pension schemes. Nevertheless, it can be used to highlight a number of important policy issues, such as the required contribution rates for defined contribution schemes, the treatment of early leavers and high fliers under defined benefit plans, the macrofinancial implications of funded schemes, and the contrast between funded and pay-as-you-go systems in dealing with demographic pressures.

# **4.1** Required contribution rates

It is clear from Section II that a high targeted pension rate requires a high contribution rate unless the real rate of interest is substantially higher than the rate of growth of real wages.

Mandatory retirement savings schemes, which include both national provident funds and government mandated but privately run systems, such as the Chilean AFP system, impose contribution rates that vary over a wide range. In Singapore, the contribution rate has been in excess of 35% for most of the period since 1968. The long-term target of the Singaporean authorities is for a 40% contribution rate, divided equally between employers and employees. This nominal rate corresponds to an effective contribution rate of 33% (i.e total contributions of 40 out of a total payroll cost' of 120). In Malaysia, Nepal and Sri Lanka the combined nominal contribution rate amounts to 20% (raised to 22% in Malaysia in 1992), while in a number of African countries it is only 10%.

In Singapore, real wages have grown by more than 4% over the past three decades or so, while the real return credited on CPF balances has on average been around 2% (Vittas 1993). Thus, on the basis of Tables la and lb, Singaporean workers on average wages would save enough over their working lives to buy an annuity that could pay a pension rate ranging between 46% (if they worked for 40 years and lived in retirement for another 20) and 62% (if they worked for 45 years and lived in retirement for another 15). If the pension is related to the net wage, then the net replacement rate would be 58% and 78% respectively. One could argue th_refore that the high contribution rate in Singapore is justified by the high growth rate of real wages. However, the real rate of return has been low, while the right given to CPF members to withdraw their funds for investments in housing, approved securities, and other purposes (e.g education) suggests that the CPF is not used to meet solely pension 22
objectives. The balances left on the individual accounts on retirement would effectively attain an even lower pension rate<sup>10</sup> .

The Chilean AFP pension system imposes a 10% compulsory contribution rate, which is entirely paid by employees. To attain a pension rate of 705 of final wages, which is apparently an objective of the scheme, would require a high real rate of return, a low growth rate of real wages or a high active worklife ratio. Table la shows that, with n=40 and m=20, a real rate of interest of 5% would attain a pension rate of 60% if g=2% and 48% if gw3%. Table lb shows that, with n=45 and m=15, the pension rates would rise to 88% and 68%, respectively.

Proponents of the Chilean scheme also argue that the targeted pension rates would be achieved with lower real rates of return if account is taken of the fact that workers experience a higher rate of wage growth early in their ccreer with a substantial deceleration as they grow older. This is because workers will accumulate a ptoportionally larger balance early on that will be available for interest compounding over a longer period. Assuming that wages increase by 3.06% a year over the first 20 years of a person's career and then by 1% over the next 20 years, so that at the end of 40 years the final wage will be the same as with a constant 2% growth rate, a 10% contribution rate will achieve a 63% pension rate if the real rate of interest is 5% throughout. This compares -ith 59% under a constant 2% real growth rate. The pension ra e will be even higher if one assumes a steeper rise early on in a worker's career with a more dramatic deceleration later. If g, is 4% and g2 is 0.15% (so that overall growth is still 2%), the pension rate will increase to 69%. Thus, it is the relationship of the rate of interest and the wage growth rate over time that is important"<sup>1</sup> . The effect of uneven or nonlinear earnings profiles is significant, though not as profound as is sometimes argued.

**10** For most of its existence, the CPF has allowed withdrawals of accumulated balances in one lump suim at age 55. In recent years, however, a minimum balance under the so-called Minimum Sum Scheme must be kept on the account to buy a pension annuity starting at age 60. The minimum amount is fixed in absolute terms, is indexed to inflation, and appears to be adequate to buy a pension of around 25% of average wages.

11 The objective function of the pension plan is k * F, * (l+r) ('") + k * F2 * (1+gl)('.I) = p * (1+g,)(v-l) * (l+g2 ) (v) * A where F1 and F2 are the sinking funds accumulated in the first and second periods and "v" is the length of the first period.

23
# **4.2** The treatment of early leavers and high fliers

Defined benefit plans have a significant advantage over defined contribution plans in that they protect workers from investment risk and, if benefits are indexed, from inflation risk. However, realization of these advantages depends on continuing employment with the same employer and on the solvency and integrity of plan sponsors.

Defined benefit final salary plans suffer from two important disadvantages. These are the unequal treatment of early leavers and long stayers, on the one hand, and high fliers and slow plodders, on the other.

The problem of early leavers is more pronounced when leaving employees are denied any vested rights, or if they have vested rights but they are not allowed to transfer them to their new employer. If the annual accrual factor is 1.5% and average wage growth is 2%, employees who change jobs in midcareer would be entitled to two pensions on retirement, one from their first employer based on 30% of their salary at the time of leaving and one from the second employer based on 30% of their salary on retirement. With a 2% growth rate, a starting salary of $10,000 would amount after 20 years to $14,600, while after another 20 years it would be equal to $21,600. Workers who change jobs in midcareer would receive a pension equal to 50% of their final salary. In contrast, long stayers with a similar salary profile would get 60% of their final salary. The losses suffered by early leavers would be even greater if pension rights are not protected from inflation. With a 5% inflation, the effective pension rate for early leavers would fall to 38%.

The problem caused by high fliers can be seen by considering equation (40) which gives the required contribution rate in the last year of service. With g=3% and r=5%, the required contribution for the last year would be 27%. However, if high fliers are assumed to get a 10% increase in salary in their last active year against a 3% average up to that time, equation (40) shows that their contribution rate should rise in the last year to 57%. A 30% salary rise in the last year would require a 125% contribution rate for that year.

In a defined contribution system, such additional contributions would have to be made explicitly and would be more transparent. But in defined benefit schemes, the effects of salary rises and promotion near the end of a person's career are obscured by the average contribution rates made on behalf of all employees. The unequal treatment between high fliers and slow plodders (or the average worker) is even greater when, as is usually the case, most workers experience a significant decline in the growth rate of their earnings late in their careers. It 24
is argiued by some that the popularity of defined benefit company pension schemes is due to a significant extent to the favorable treatment of senior managers who get big promotions and large salary rises late in their careers.

A solution to the unequal treatment of early leavers and high fliers would involve regulations that mandate minimum vesting periods, full portability of vested rights, and use of average actualized lifetime earnings as the basis for determining pensions.

# 4.3 Macroeconomic and financial implications **of funded schemes**

Funded schemes accumulate long-term savings that are available for investment in long-term instruments, such as government bonds, mortgage bonds and corporate securities as well as real assets. A question of major macroeconomic and financial significance regards the pace and especially the limit of such accumulation in relation to a country's GNP. To calculate the total size of accumulated funds, it is necessary to make assumptions about the income and age distribution of active and passive workers as well as about the coverage of the pension schemes and the career patterns of workers.

Regarding the income distribution, one assumption is to postulate that all workers start with the same income, which grows by a given percentage each year. The income distribution would then reflect the age distribution and the growth rate of wages. An alternative assumption is to postulate that all workers earn the same income, which increases over time by the given growth rate. The income distribution would then be completely flat.

Regarding the age distribution, one formulation is to assume a pillar population structure which would imply a zero population growth. The alternative is to use a pyramidal demographic structure associated with a positive population growth rate.

In mechanical terms, the limit to the accumulation of capital in relation to GNP is given by the algebra set out above. Taking the simple case where r=g=o, it is easy to see that if k=10%, n=40 and m=20, workers with 40 years of contributions would accumulate funds equal to 4 times their annual income, workers with 39 years would have funds equal to 3.9 times their incomes, and so on. Thus, if it is assumed that the income distribution is flat, the population structure resembles a pillar, coverage is universal, and all workers have full careers, the total funds of pension plans for active workers would amount to 205% of labor income.

25
In addition to the balances of active workers, account must also be taken of the balances of pensioners or passive workers. Continuing to assume a pillar demographic structure and **r=g--O,** it can be seen that, if the pension rate is equal to 20% of wages (which it would have to be under these assumptions), the balances of pensioners would range from 3.8 times the annual wage for pensioners who are retired for one year, to 3.6 times for those who are retired for two years, and so on. The total funds of pensioners would then amount to 95% of annual wages.

Thus, under these assumptions, the balances of both groups would amount to 300% of the wage bill. Table 3a shows the size of balances for different combinations of wage growth rates and interest rates under the assumption of pensions i]idexed to wages, a constant contribution rate of 10%, working and retirement lives of 40 and 20 years respectively, and zero population growth (i.e. a pillar demographic structure). Table 3b shows the size of balances when the population growth rate is 1.6% and the resulting population structure is a pyramid<sup>1 2</sup> . (Figures 4a and 4b show the total size of accumulated funds as a proportion of the wage bill for different combinations of interest and wage growth rates.) These calculations correspond to those of table 2a where higher interest rates imply higher pension rates and higher growth rates imply lower pension rates. However, it is easy to compute that, for a given targeted pension rate, a positive "r" would imply lower accumulated balances, while a positive "g" would imply the opposite. It is also easy to see that, again for a given pension rate, a longer working life and shorter retirement life would imply smaller accumulated funds, mainly because the required funds for a shorter retirement life would be smaller.

Table 2a shows that with pensions indexed to wages, a working life of 40 years, a retirement life of 20 years, a real wage growth rate of 2% and a real interest rate of 5%, a 10% contribution rate would provide a pension rate of 50%. Table 3a then shows that with a pillar demographic structure, universal coverage and full working careers, the total accumulated balances 1 3 would be equal to just over 500% of the total wage bill .

12 It can be seen from Tables 3a and 3b that if r=g and pensions are indexed to wages, the same rezults obtain as when r=g=o.

> 13 Table 3b shows that with a population growth rate of 1.6% and the resulting pyramidal demographic structure, total accumulated balances would correspond to just over 300% of the total wage bill.

26
Table 3a

# Accumulated Pension Fund Balances* (per cent of total wage bill)

|||Growth|Rate of|Real Wage|s||
|---|---|---|---|---|---|---|
|Real Interest Rate<br>0%|0%<br>300|1%<br>257|2%<br>223|3%<br>196|4%<br>174|5%<br>156|
|1%|355|300|257|224|196|174|
|2%|423|354|300|258|224|197|
|3%|511|422|353|300|258|225|
|4%|624|508|420|353|300|258|
|5%|769|619|506|419|352|300|
|10%|2499|1903|1468|1148|910|731|

* Pensions indexed to wages, k=10%, n=40, m=20, pillar demographic structure (0% population growth), dependency rate 50%.

# Table 3b

Accumulated Pension Fund Balances* (per cent of total wage bill)

|||Growth|Rate of|Real Wage|s||
|---|---|---|---|---|---|---|
||0%t|1%|2%|3%|4%|5%|
|Real Interest Rate<br>0%|188|163|143|127|114|103|
|1%|219|188|163|143|127|114|
|2%|258|219|188|163|144|128|
|3%|308|258|218|188|164|144|
|4%|371|306|257|218|188|164|
|5%|451|368|305|256|218|188|
|10%|1383|1065|831|658|528|430|

* Pensions indexed to wages, k=10%, n=40, m=20, pyramidal demographic structure (1.6% population growth), dependency rate 30%.

27
<!-- Start of picture text --> Figure 4a<br>Accumulated Pension Fund Balances<br>Pillar Population Structure, Pensions Indexed to Wages<br>(k=10%, n=40, m=20)<br>800<br>700<br>g=-O%<br>600<br>~500<br>~ 400<br>-. - - .-. - -. - - -*- ~g=3%<br>200<br>g=5%<br>100<br>0 1 2 3 4 5<br>Real Interest Rate (t)<br>Figure 4b<br>Accumulated Pension Fund Balances<br>Pyramildal Population Structure (p)opulatlon growth 1.6%), Pensions<br>Indexed to Wages<br>m0 L<br>(k=10%, n=45, m=15)<br>800<br>700 g=-O%<br>6500<br>c~400<br>06 300g-4<br>* . - * *~~~--.--------- ~~g=5%<br>ioo~I<br>0 1 2 3 4 5<br>Real Interest Rate (%l)<br><!-- End of picture text --> 28
Assuming a labor income share of 70%, pension<sup>fund assets would</sup> amount to 350% of GNP. A smaller labor<sup>share, say 40%, would</sup> still imply total pension fund assets of 200%<sup>of GNP.</sup> Such an increase in capital funds would<sup>imply an increase in</sup> the rate of saving, at least until the funds<sup>reach maturity.</sup> Historically, the adoption of funded pension<sup>schemes has not been</sup> associated with an increase in the rate<sup>of household and national</sup> saving, probably because of offsetting changes<sup>in household</sup> behavior. But the absence of a clear effect<sup>on the rate of</sup> saving may also be explained by the lack<sup>of universal coverage,</sup> the interrupted careers of many workers,<sup>the absence of full</sup> funding, the relative youth of most funded<sup>pension schemes, the</sup> low level of pension benefits (relatively<sup>few workers are covered</sup> by pension funds that provide a pension<sup>rate equal to 50% and</sup> indexed to wages), the low level of real<sup>returns (it is unclear</sup> whether a 5% real rate of return would<sup>prevail over a 60-year</sup> period in a country with universal coverage)<sup>and the pyramidal</sup> demographic structure (instead of the pillar<sup>structure assumed</sup> here). A forced saving scheme that does not<sup>suffer from these</sup> shortcomings may more than offset the reduction<sup>in voluntary</sup> saving.

These factors may also explain why the total<sup>assets of</sup> pension funds in different countries are<sup>nowhere near the very</sup> high levels quoted above. The highest level<sup>of assets in</sup> relation to GNP of pension funds and life insurance<sup>companies</sup> combined is found in Switzerland, the Netherlands<sup>and the United</sup> Kingdom where they range between 120% and<sup>150% of GNP (Vittas and</sup> Skully 1991) 14. In Singapore, where the CPF has been<sup>in</sup> existence sir.ve 1955 and where contribution<sup>rates are quite high,</sup> the total assets of the CPF correspond to around<sup>70% of GNP. In</sup> some countries, total financial assets, including<sup>assets of banks</sup> and other deposit institutions, exceed 300%,<sup>and even 400% of</sup> GNP, but these totals include a significant amount<sup>of double</sup> counting.

The accumulation of large pension fund<sup>balances would also</sup> imply a domination of the financial system by<sup>pension funds.</sup> This would have far-reaching implications for the<sup>management,</sup> regulation and supervision of their assets<sup>and for the role of</sup> pension fund managers in corporate governance. Failure to properly regulate and supervise pension funds and<sup>ensure that</sup> they are invested safely and profitably would<sup>undermine their</sup> ability to pay the targeted pensions.

14 Davis (1992) has data showing the rapid growth<sup>of total</sup> pension fund and life insurance assets over<sup>the past two decades</sup> or so.

29
Would pension fund assets ever attain such high levels? This would depend on their coverage, the career patterns<sup>of</sup> workers and the demographic structure as well as the relationship between the growth rate of real wages, the real rate of interest, and the age and wage/pension distribution of workers<sup>and</sup> pensioners. But to absorb the vast expansion in the availability of long-term savings implied by Tables 3a and 3b,<sup>capital</sup> intensity would have to increase substantially and<sup>would require</sup> significant technological advances in order to avoid a reduction in the productivity of capital. Otherwise, the implied reduction in real rates of return would question the feasibility of funding high levels of pension provision.

This question about the sufficiency of capital productivity and investment returns is often expressed as a concern about the adequacy of productive assets as collateral in which pension plans can invest their funds. Manufacturing assets are usually<sup>a</sup> small fraction of GNP, no higher than 40% to 50%. However, other potential assets include inventory, infrastructure, houses and land. Financial claims on these assets may be represented by mortgage and government bonds as well as corporate securities, in which accumulated funds may be invested. As already noted, there are several countries (e.g. Switzerland, the Netherlands and the United Kingdom) where pension funds and insurance companies already hold assets that correspond to well over 100% of GNP. .-Nevertheless, accumulation of pension fund balances on the scale suggested above would exert a downward pressure on real returns unless technological progress prevented a reduction in the productivity of capital.

Another, closely related, issue concerns the growing reliance of pension funds on capital gains for achieving their target rates of return. This has adverse implications for the stability and even sustainability of real returns and about the long-term feasibility of funded pension schemes with universal coverage. Evidence from OECD countries with relatively large funded pension sectors shows considerable variation in real rates of return over prolonged periods, ranging from negative levels between 1966 and 1980 to strongly positive ones over the 1980s (Table 4).

Thus, unless economic productivity were to increase substantially as a result of the increased availability of longterm savings, a fully funded pension systsm with universal coverage may not be able to pay targeted pensions.

30
# Table 4

||Real|Pension F|und Returns||
|---|---|---|---|---|
||US|UK|Netherlands|Switzerland|
|1996-70|-5.4|4.2|1.7|0.8|
|1971-75|-0.8|-2.8|-1.4|-0.5|
|1976-80|-1.9|4.9|2.0|4.0|
|1981-85|8.1|12.4|10.5|3.0|
|1986-1990|11.2|10.1|6.3|-0.2|
|1966-90|2.2|5.8|4.0|1.5|

Source: Davis (1992)

**4.4** Demographic pressures and funded and pay-as-you-go **schemes**

The question of the adequacy of capital productivity and investment returns tor fully funded schemes with universal coverage is also related to the issue of whether funded schemes can cope better than pay-as-you-go schemes with aging populations and the resulting demographic pressures. As shown in Tables 3a and 3b, a pillar demographic structure (zero population growth) implies a higher level of pension fund assets than a pyramidal structure.

This issue can be addressed by comparing the simple algebra of defined contribution plans with that of pay-as-you-go schemes. The required contribution rate for a funded scheme, when pensions are indexed to wages, is given by where "G", "D" and **"F"** are defined as above. If "g" is equal to "r", equation (43) simplifies to where the fraction m/n can be seen as the passivity ratio, i.e. 5 the ratio of the retirement life over the working life". The level of this ratio will depend on the age at which workers start _15_ The passivity ratio is the inverse of the active worklife ratio. The passivity ratio corresponds to the dependency ratio in social security parlance, while its inve.se, the active worklife ratio, corresponds to the support ratio.

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to contribute to a pension fund, their retirement age, and their expected retirement life.

For pay-as-you-go schemes, the required contribution rate when pensions are indexed to wages is given by (45) c = p ***** B = p * d C where "B" is the number of beneficiaries, 'IC" the number of contributors, and "d"' the dependency ratio.

The value of the dependency ratio is determined by the same factors as the passivity ratio, except that it is also affected by the shape of the demographic structure. In a society with a demographic structure that resembles a pillar, the two ratios would have the same value. If people work for 40 years and live in retirement for 20 years, the dependency ratio would be equal to 50%, the same as the passivity ratio. If working life is extended to 45 years and retirement life is reduced to 15, both ratios would fall to 33%.

Thus, if the rate of interest and the growth rate of real wages are equal, funded and pay-as-you-go systems would require the same contribution rate if the demographic structure resembles a pillar. Pay-as-you-go systems would require a lower contribution rate if the demographic structure resembles a pyramid and a higher contribution rate if it resembles an inverted pyramid<sup>16</sup> . However, the same pattern could also apply to funded schemes.

This is so because the relationship between real interest rates and real wage growth rates would also be affected by the shape of the demographic structure. An economy with a young population and a low dependency ratio would tend to be characterized by a relative scarcity of capital. In such an economy, the real rate of interest would tend to be high, while the growth rate of real wages would tend to be depressed. A low contribution rate would then be needed for funded schemes as in 16 Historically, demographic structures resembled pyramids with a large base. This translated into low dependency rates so that a given pension rate could be achieved with a low contribution rate. For instance, with a dependency ratio of 10%, a pension rate of 80% required a contribution rate of only 8%. But with the aging of the population, the demographic structure changes shape, first to a pyramid with a smaller base, and then to a pillar, requiring a substantial rise in contribution rates to achieve the same pension rate.

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the case of pay-as-you-go schemes. With the aging of the population and increase in the dependency ratio, there would be a shift towards a relative scarcity of labor, which would tend to produce a higher growth rate of real wages and a lower real interest rate, causing a rise in the required contribution rate.

The implication of this analysis is that funding may not eliminate intergenerational transfers when demographic structures change, although a lot will depend on what happens to economic productivity and the real rate of interest when the capital intensity of production increases. In response to demographic pressures from population aging, the retirement age may need to be raised, regardless of whether pensions are funded or not. This will result in a higher active worklife ratio (a lower passivity ratio) and therefore a lower dependency ratio.

# 4.5 Advantages of a Multi-Pillar Structure

The uncertainties regarding investment returns, capital productivity, demographic trends, not to mention political and economic stability, suggest that an approach based on a multipillar system may be preferable on economic, financial and social grounds<sup>17</sup> . A multi-pillar approach would combine a pay-as-you-go (or partially funded) first pillar offering a pension equal to, say, 30% of net wages with a fully funded second pillar based on individual capitalization accounts offering an additional 40% of net wages. Such a system may combine the main advantages of the pay-as-you-go and funded systems, while avoiding the disadvantages of extreme solutions.

The first pillar could be redistributive, replacing a higher proportion of income for low income workers and a lower proportion for high income workers<sup>1</sup> ". It could consist of two parts, one offering a flat pension and the other an earningsrelated pension. The contribution rate for the first pillar, assuming it was financed from payroll taxes, would depend on the demographic structure (and the level of any partial pre-funding). For countries with mature populations, a contribution rate of 10% for the first pillar would be adequate for paying a 30% net pension rate. But for countries with younger populations and lower dependency ratios, the contribution rate for the first pillar could be significantly lower.

17 A proposed strategy for pension reform based on a multi-pillar structure is set out in Vittas (1993).

18 A redistributive system, funded from payroll or general taxes, would be needed to cover workers with limited and interrupted careers as well as their dependents. This is<sup>a major</sup> issue in all countries, including both Chile and Singapore.

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The second pillar would require a contribution rate of 10% to 15%, depending on what assumptions are made about the future combination of interest rates and wage growth. The net pension rate of 40% that might be targeted in the second pillar would be easy to achieve with the suggested contribution rates as can be verified from Tables 2a and 2b. Thus, the total contribution rate could vary from 15% in the case of young countries to 25% in the case of old ones.

Before concluding this paper it is worth repeating the point made in footnote 9 above. While most of the analysis is based on constant contribution and pension rates, both rates could in practice be variable. In fact, allowing workers to set both their own contribution and pension rates, within a specified range, might be preferable and more consistent with greater reliance on personal choice.

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