# Improving Money’s Worth Ratio Calculations: The Case of Singapore’s Pension Annuities

• Joelle H. Fong , Jean Lemaire and Yiu K. Tse

## Abstract

This paper contributes to a better understanding of the risks involved in a life annuity investment. We study the full distribution of weighted annuity benefits and quantify risk measures such as dispersion and skewness, thereby extending the usefulness of the popular money’s worth valuation framework for life annuities. Using data from pension annuities in Singapore, we also introduce several risk measures that might appeal to less financially sophisticated retirees. A more detailed and accurate picture of the risk of investing in life annuities emerges, enabling prospective annuitants to differentiate among products that may appear seemingly uniform in terms of money’s worth, but vary widely in terms of their risk attributes.

## Acknowledgments

All opinions are solely those of the authors who acknowledge research support from the Wharton-SMU Research Center at Singapore Management University and the Pension Research Council/Boettner Center at The Wharton School of the University of Pennsylvania.

## Appendix

For immediate single-life annuities, eq. [1a] can be written as (see Bowers et al. 1997)17:

EPDV= Aat=0pta (m=0tvm)= Aat=0ptaqa+ta¨t+1¯|= t=0qt|aYt

where Yt= Aaa¨t+1¯|.

For immediate single-life annuities with money-back guarantees, such as the Singaporean pension annuities, eq. [1b] can be written as:

EPDV= t=0ptaqa+t (m=1t+1vm) Gt+1 + Aat=Dpta (m=1tvm)= t=0ptaqa+t (m=1t+1vm) Gt+1 + Aat=Dptaqa+t (m=1Dvm) a¨tD+1¯| = t=0qt|aYt

where

Yt= {(m=1t+1vm) Gt+1,                                             for t<D(m=1t+1vm) Gt+1+ (m=1Dvm) Aa a¨tD+1|¯,            fortD.

Definitions

1. a is the age at which the annuity is purchased;

2. t is the number of months beyond the annuity start date;

3. D is the deferred period in months;

4. pta is the probability that an individual of age a survives after t months;

5. qa+t is the monthly probability to die between month a+t and a+t+1;

6. vm is the discount factor for month m based on a riskless term structure;

7. Gt+1 is the death benefit paid: Gt+1=max{0,premium+accrued interesttotal annuity payouts received prior to timet+1},

8. Aa is the monthly level annuity payout for the individual purchasing the annuity at age a,

9. a¨n|¯ is the present value of an n-payment annuity-due.

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J. 1988. “Private Annuity Markets in the United States.” The Journal of Risk and Insurance55:51828.10.2307/253260Search in Google Scholar 1. 1 Details on the term and annuity component of the LIFE plans are discussed extensively in the Report by the National Longevity Insurance Committee (CPF 2008). 2. 2 The lump-sum annuity premium varies from year to year and is based on the so-called prevailing “Minimum Sum” which is the minimum amount that must be set aside in a CPF member’s retirement account before any CPF balances can be withdrawn at age 55. For instance, the Minimum Sum is S$70,000 in 2000, S$99,600 in 2008, S$117,000 in 2009, and S$148,000 in 2013. 3. 3 “Annual Value” is the property value used to calculate the property tax of the CPF member’s home. “Assessable Income” is the full income in a calendar year (sum of employment, dividends, interest, and other income) less allowable expenses, applicable capital allowances, charitable donations, and any loss incurred in trade and business. 4. 4 We use a projected 2% bonus with a projected rate of return of 5.25% per annum. This is in line with the benefit illustrations in NTUC Income prospectus (NTUC 2009). NTUC Income is a home-grown co-operative formed in 1970 and is one of the Singapore’s leading composite insurers offering life, health, general insurance, and investment-linked products. 5. 5 Prior to 2000, some of the commercial pension annuities offer period-certain guarantees instead of dollar guarantees. 6. 6 In the United Kingdom, capital-protected or value-protected annuities feature a partial money-back option where the taxable lump-sum death benefit is permitted only up to age 75 (Boardman 2006). 7. 7 The death benefit Gt+1 is equal to max{0,premium+accrued interest from guaranteetotal annuity payouts received prior to time t+1}. 8. 8 The first year rate is derived from the 1-year Singapore Government Treasury bill; thereafter, the 2-, 5-, 7-, 10-, 15- and 20-year Treasury bond rates as of 2007 are used to estimate the riskless spot rates (MAS 2008). The 20-year rate is applied for periods after year 20; annual spot rates range from 1.4% to 3.44%. 9. 9 The a(1990) tables are constructed based on UK annuitants’ mortality experience from 1967–70 with mortality improvements projected to 1990. Note that annuitant life tables are not available in Singapore due to limited annuitant mortality experience. 10. 10 Some MWR values may not be exactly identical because of different assumptions regarding the potential bonus payouts from the participating annuities and minor corrections. 11. 11 The jagged portions between ages 55 and 62 in Figure 2 are due to stepped increases in the guarantee amount accruing from accrued interest of 0.75% per annum (up to age 62), as well as the non-monotonicity of the discount rates. Accrued interest on the guarantee does not exceed 2.5% for the rest of the annuities; three insurers do not pay such interest. Over time, the guarantee starts to wear off due to discounting and the commencement of payouts. 12. 12 The death benefit wears down as monthly payouts accumulate. 13. 13 This generalization is made possible because the nine annuities (provided by eight different insurers) offered largely similar specifications: a S$99,600 premium and low variation in sex-specific payouts/benefits across insurers.

14. 14

A recent cross-check shows that illustrative payouts (based on the CPF web calculator) have not altered much since September 2009; only small decreases in the range of S\$5–10 were noted.

15. 15

The Income plan channels 100% of the premium into the annuity component. Also, any unused balance (premium less payouts made) is non-refundable. Hence, this product offers no death benefit offering the most attractive monthly payouts as a trade-off. LIFE Plus is similar, except that any unused balance is refundable to one’s estate. This explains why its NPV profile does not dip into the negative region as much as the Income plan in early years.

16. 16

This study extends to 17 countries comprising over 75% of the world’s global premium; it defines risk as the CV of the loss ratio of each book of business.

17. 17

Bowers et al. (1997, 143) provide proof for the following mathematical equivalence: Actuarial present value of an annuity = t=0pta vt =t=0(a¨t+1|¯ptaqa+t).

Published Online: 2014-3-4

©2014 by Walter de Gruyter Berlin / Boston