## _wp04230 - 1. Estimated Premium for a One Dollar Annuity

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### Introduction and motivation
- Employer-provided defined-contribution (DC) pension plans are superseding defined-benefit plans in the United States; DC plans are growing internationally via DC individual accounts.
- U.S. statistics cited:
  - nearly 58 million people participated in a DC plan in 1998;
  - contributions totaled $167 billion;
  - assets of these funds totaled $2,085 billion, or about 20 percent of GDP.
- Key concern: variability of retirement income arises from accumulation-phase investment risk and from distribution-phase risk tied to annuity premiums at annuitization.
- Paper objective: gauge potential variability of annuity premiums indirectly by linking premiums to the slope and height of the yield curve (life insurers mainly fund annuities with bond-market investments).

### Annuity features, demand, and insurer risks
- Life annuity considered: single premium immediate life annuity (lump-sum purchase, immediate nominal fixed regular payments until death).
- Gender pricing example:
  - In 2004, a 64-year-old male purchasing a life annuity paying $10,000 a year might have paid about $130,000;
  - a 64-year-old woman would pay an additional $7,000.
- Demand and welfare:
  - Brown (1999) estimates annuities can increase lifetime utility by as much as 30 percent.
  - U.S. Government Accounting Office (2003): only 7.5 percent of DC plan contributors who retired between 1992 and 2000 received distributions as an annuity.
- Adverse selection and money’s worth ratio (MWR):
  - MWR defined as expected discounted value of $1 in payments divided by premium per dollar of income (PPD): MWR = (sum_{i=1}^T P_i /(1+r_j)^j ) / PPD per equation (1).
  - Mitchell, Poterba, Warshawsky (1999) estimated MWRs for 65-year-old male using treasury bond interest:
    - 0.816 for general population;
    - 0.916 for annuity-purchasing population.
- Insurer risks:
  - Mortality risk (idiosyncratic vs. population mortality uncertainty) and investment risk (bond maturities insufficient to match long annuity horizons).
  - For tractability the paper assumes annuities funded 100 percent by fixed interest securities, making premiums functions of level and shape of the yield curve and life expectancies.

### Simulation methodology
- Actuarial and pricing framework:
  - PPD expressed as PPD = sum_{i=1}^T P_i /(1+r_{0j})^j (equation (2)), assuming insurer risk neutrality and MWR = 1.
  - Survival probabilities from U.S. Social Security Administration cohort life table for 64-year-old men and women in 2003; annuity paid for 30 years or until death.
- Term structure and forward rates:
  - Implicit forward one-year rates r_{t-1,t} derived from observed bond yields r_{0,t} via equation (3).
  - One-year forward rates computed if gaps in maturities per equation (4).
- Data and simulation procedure:
  - Daily U.S. Treasury bond yields from January 1990 through January 2002 used.
  - Implied short (one-year forward) rates computed; mean vector μ_t and covariance matrix Σ_{t-1,t-1} estimated.
  - Short rates simulated 100,000 times assuming multivariate normal distribution per equation (5): r_{t,t-1} ~ N(μ_t, Σ_{t-1,t-1}).
  - Mortality scenarios: general population and cohort-adjusted annuitant population to capture adverse selection.

### Simulations — Annuity premium distribution (Table 1)
- Male general population:
  - Mean: $8.99
  - Standard Deviation: $0.68
  - Maximum: $12.37
  - 90th percentile: $9.89
  - Median: $8.96
  - 10th percentile: $8.15
  - Minimum: $6.93
- Female general population:
  - Mean: $9.97
  - Standard Deviation: $0.82
  - Maximum: $14.54
  - 90th percentile: $11.04
  - Median: $9.91
  - 10th percentile: $8.96
  - Minimum: $7.34
- Male annuitant population:
  - Mean: $9.74
  - Standard Deviation: $0.78
  - Maximum: $14.02
  - 90th percentile: $10.76
  - Median: $9.69
  - 10th percentile: $8.78
  - Minimum: $7.15
- Dollar example:
  - For a 65-year-old man, estimated mean premium for an annuity paying $10,000 a year is $89,983 with a standard deviation of $6,824.
- Tail risk:
  - An annuitant has about a 5 percent probability of being charged a premium about $15,000 more than the mean.
- Gender and adverse selection effects:
  - Women pay about 8.3 percent more for the same annuity payment.
  - Standard deviation for women about $8,200.
  - Adverse selection: annuitant population mean premium ($9.74) exceeds male general population mean ($8.99).

### Simulations — Replacement rates (Table 2)
- Model parameters used:
  - g (rate of growth): 3 percent
  - r (rate of return): 3 percent
  - s (rate of saving): 10 percent
  - N (working years): 40
  - M (potential retirement years): 30
  - n (years at end-career used for RR calculation): 5
- Replacement rate definition derives from equations (6), (7), and (8).
- Simulation draws: 100,000.
- Male general population:
  - Mean: 0.54
  - Standard Deviation: 0.041
  - Maximum: 0.74
  - 90th percentile: 0.59
  - Median: 0.54
  - 10th percentile: 0.49
  - Minimum: 0.35
- Female general population:
  - Mean: 0.49
  - Standard Deviation: 0.040
  - Maximum: 0.65
  - 90th percentile: 0.54
  - Median: 0.48
  - 10th percentile: 0.44
  - Minimum: 0.33
- Male annuitant population:
  - Mean: 0.50
  - Standard Deviation: 0.039
  - Maximum: 0.72
  - 90th percentile: 0.55
  - Median: 0.50
  - 10th percentile: 0.45
  - Minimum: 0.35
- Interpretation:
  - Mean replacement rate for general male population about 0.54 with standard deviation 0.041.
  - Small probability of replacement rate as low as 0.35 or as high as 0.74.
  - Female mean replacement rate about 0.49, about 5 percentage points (10 percent) lower than male mean.
  - Male annuitant population mean replacement rate about 0.50, about 4 percentage points lower than general male population.

### Decreasing risk through staggered purchases and diversification (Table 3)
- Strategy: staggered purchases of multiple annuities over several years to hedge interest rate risk and smooth premium variability.
- Portfolio premium modeled as mean of individual premiums; assumptions include uncorrelated bond yields over time and equal annuity payments.
- Expected premiums for general male population:
  - Portfolio of 2 annuities:
    - Mean: $9.00
    - Standard Deviation: $0.48
  - Portfolio of 5 annuities:
    - Mean: $8.99
    - Standard Deviation: $0.30
  - Portfolio of 10 annuities:
    - Mean: $9.00
    - Standard Deviation: $0.22
- Findings:
  - Buying two annuities reduces standard deviation of the premium by nearly 30 percent relative to a single annuity, without changing expected value.
  - Increasing portfolio size further reduces variance, converging to zero as portfolio grows.
- Caveats:
  - Model ignores fixed transaction costs; multiple annuity purchases may be expensive.
  - Deferred annuities require longer-range mortality projections, potentially explaining market thinness.

### Historical comparison and context
- A.M. Best data for five large firms, 1980–1993, for an annuity paying $12,000 a year to a 65-year-old male:
  - average premium $104,933 with standard deviation $10,841.
- For women in 1980–1993:
  - average premium $113,760 with standard deviation $11,277.
- Explanation: differences reflect higher and more volatile bond yields in the 1980s; simulated premiums using 1990–2002 data differ from historical averages but simulated variances are not hugely different.

### Conclusions, policy observations, and suggested future research
- Main conclusions:
  - Annuity premiums vary with treasury bond yield fluctuations, introducing distribution-phase risk in addition to accumulation-phase risk for DC plan participants.
  - Simulated standard deviation of replacement rates for a pensioner facing only annuitization price risk is about 4 percentage points.
  - Adverse selection increases consumer costs: in the U.S. market an annuitant can pay about 8 percent more for an annuity than in absence of adverse selection; replacement rate is on average about 4 percent lower due to adverse selection.
  - Price discrimination by gender results in lower expected replacement rates for women; potential future discrimination by race or socioeconomic status would raise public policy issues if private annuity income replaces Social Security benefits.
- Policy-relevant observations:
  - Phased (staggered) annuitization can reduce premium variance but faces transaction cost and commitment challenges.
  - Consideration of market instruments (e.g., survivor bonds, TIPS) and public policy could address mortality and inflation risks.
- Suggested future research directions:
  - Incorporate asset risk (especially equity exposure), wage risk, taxes, and income heterogeneity into replacement rate simulations.
  - Investigate correlation between annuity premiums and stock market returns to assess whether such correlation reduces overall retirement income unpredictability.
  - Simulate indexed annuity premiums using developing TIPS data to capture inflation risk.
  - Explore market and policy solutions to mortality risk, including government-issued survivor bonds.

*Source: _wp04230 - 1. Estimated Premium for a One Dollar Annuity.................................................................... 10*

### 1. Estimated Premium for a One Dollar Annuity.................................................................... 10

### _wp04230 - 1. Estimated Premium for a One Dollar Annuity.................................................................... 10

### 1. Estimated Premium for a One Dollar Annuity
- Section title: "1. Estimated Premium for a One Dollar Annuity"
- Page reference: 10

### 2. Estimated Replacement Rates and their Variation
- Section title: "2. Estimated Replacement Rates and their Variation"
- Page reference: 13

### 3. Expected Premiums of Portfolios of One Dollar Annuities for the General Male Population
- Section title: "3. Expected Premiums of Portfolios of One Dollar Annuities for the General Male Population"
- Page reference: 15

### Figures
- Figure 1: "Premium for $1.00 Annuity" — page 11
- Figure 2: "United States Yield Curves (1989–2003)" — page 14

*Source: _wp04230 - 1. Estimated Premium for a One Dollar Annuity.................................................................... 10*

### References..............................................................................................................

### _wp04230 - References..............................................................................................................

### Introduction
- Employer-provided defined-contribution (DC) pension plans are superseding defined-benefit plans in the United States; DC plans are growing internationally via DC individual accounts.
- U.S. statistics cited: nearly 58 million people participated in a DC plan in 1998; contributions totaled $167 billion; assets of these funds totaled $2,085 billion, or about 20 percent of GDP.
- Key concern: variability of retirement income arises not only from accumulation-phase investment risk but also from distribution-phase risk tied to annuity premiums at annuitization.
- Annuity premiums are in principle a function of bond yields and can vary considerably over time; therefore, retirement income from DC plans remains uncertain even with fixed accumulation returns.
- Paper objective: gauge potential variability of annuity premiums indirectly by linking premiums to the slope and height of the yield curve (life insurers mainly fund annuities with bond-market investments).

### Basic Features of Annuities and the Market for Annuities
- Life annuity considered: single premium immediate life annuity (lump-sum purchase, immediate nominal fixed regular payments until death).
- Gender pricing:
  - Example: In 2004, a 64-year-old male purchasing a life annuity paying $10,000 a year might have paid about $130,000; a 64-year-old woman would pay an additional $7,000.
  - Social Security and most public pension systems do not discriminate by sex; private market does.
- Demand influences and welfare:
  - Annuities provide longevity insurance and can raise lifetime utility; Brown (1999) estimates annuities can increase lifetime utility by as much as 30 percent.
  - U.S. Government Accounting Office (2003): only 7.5 percent of DC plan contributors who retired between 1992 and 2000 received distributions as an annuity.
  - Factors reducing demand: adverse selection, Social Security provision of indexed annuity, need for lumpy expenditures, attractiveness of alternate investments, desire to leave bequests, irrevocability of annuities.
- Adverse selection and money’s worth ratio (MWR):
  - MWR defined as expected discounted value of $1 in payments divided by premium per dollar of income (PPD): MWR = (sum_{i=1}^T P_i /(1+r_j)^j ) / PPD per equation (1).
  - Estimated MWRs (Mitchell, Poterba, Warshawsky (1999)) for 65-year-old male using treasury bond interest: 0.816 for general population; 0.916 for annuity-purchasing population—substantial gap indicating adverse selection.
- Risks confronting annuity suppliers:
  - Mortality risk (idiosyncratic vs. population mortality uncertainty) and investment risk (bond maturities insufficient to match long annuity horizons).
  - Practical financing: insurers invest in equities and other assets, but for tractability the paper assumes annuities funded 100 percent by fixed interest securities, making premiums functions of level and shape of the yield curve and life expectancies.
- Previous empirical studies:
  - U.K. compulsory annuity market time series used by Cannon and Tonks (2003) and Finkelstein and Poterba (2000).
  - Cannon and Tonks construct 1972–2002 time series; find high correlation between annuity and U.K. consul rates; assume annuity premium proxied by consul rate plus a constant.

### Simulations — Methodology
- Annuity premium per dollar of annual income (PPD) expressed as PPD = sum_{i=1}^T P_i /(1+r_{0j})^j (equation (2)), assuming insurer risk neutrality and MWR = 1.
- Survival probabilities derived from U.S. Social Security Administration cohort life table for 64-year-old men and women in 2003; annuity assumed paid for 30 years or until death.
- Term structure and forward rates:
  - Implicit forward one-year rates r_{t-1,t} derived from observed bond yields r_{0,t} via equation (3).
  - If gaps in maturities exist, one-year forward rates between t and t+n calculated by equation (4).
- Data and simulation procedure:
  - Daily U.S. Treasury bond yields from January 1990 through January 2002 used.
  - Implied short (one-year forward) rates computed; mean vector μ_t and covariance matrix Σ_{t-1,t-1} estimated.
  - Short rates simulated 100,000 times assuming multivariate normal distribution per equation (5): r_{t,t-1} ~ N(μ_t, Σ_{t-1,t-1}).
  - Mortality statistics: general population used; also cohort-adjusted mortality for 65-year-old male annuity buyers considered to capture adverse selection effects.

### Simulations — Results (Annuity Premiums)
- Simulations: distribution of forward one-year rates simulated 100,000 times; summary statistics reported in Table 1.

Table 1. Estimated Premium for a One Dollar Annuity
- Male general population
  - Mean: $8.99
  - Standard Deviation: $0.68
  - Maximum: $12.37
  - 90th percentile: $9.89
  - Median: $8.96
  - 10th percentile: $8.15
  - Minimum: $6.93
- Female general population
  - Mean: $9.97
  - Standard Deviation: $0.82
  - Maximum: $14.54
  - 90th percentile: $11.04
  - Median: $9.91
  - 10th percentile: $8.96
  - Minimum: $7.34
- Male annuitant population
  - Mean: $9.74
  - Standard Deviation: $0.78
  - Maximum: $14.02
  - 90th percentile: $10.76
  - Median: $9.69
  - 10th percentile: $8.78
  - Minimum: $7.15

- Dollar example: For a 65-year-old man, estimated mean premium for an annuity paying $10,000 a year is $89,983 with a standard deviation of $6,824.
- Tail risk: An annuitant has about a 5 percent probability of being charged a premium about $15,000 more than the mean.
- Gender differences:
  - Women pay about 8.3 percent more for the same annuity payment.
  - Standard deviation for women about $8,200, a larger fraction of the annual annuity payment; thus women face both higher mean cost and higher relative volatility.
- Historical comparison:
  - A.M. Best data for five large firms, 1980–1993, for an annuity paying $12,000 a year to a 65-year-old male: average premium $104,933 with standard deviation $10,841.
  - For women in 1980–1993: average premium $113,760 with standard deviation $11,277.
  - Differences reflect higher and more volatile bond yields in the 1980s; simulated premiums using 1990–2002 data differ from historical averages but simulated variances are not hugely different.

### Simulations — Replacement Rates
- Model setup:
  - Contributor works N years, income grows at rate g, saves fraction s, savings earn return r, uses accumulated funds at retirement to purchase annuity.
  - Premium determined by equation (6): Premium = W_0 * sum_{i=1}^{N} ( (1+g)^{i-1} * s / (1+r)^{i} ) — (presentation follows equation (6) in source).
  - Annuity payment A from actuarially fair premium per equation (7): Premium = sum_{i=1}^M P_i /(1+r_i)^i .
  - Replacement rate (RR) defined as ratio of annuity payment to average income net of saving during last n years, given in equation (8).
- Parameter values used in simulations:
  - g (rate of growth): 3 percent
  - r (rate of return): 3 percent
  - s (rate of saving): 10 percent
  - N (working years): 40
  - M (potential retirement years): 30
  - n (years at end-career used for RR calculation): 5
- Simulated replacement rates (100,000 draws); summary statistics in Table 2.

Table 2. Estimated Replacement Rates and their Variation
- Male General Population
  - Mean: 0.54
  - Standard Deviation: 0.041
  - Maximum: 0.74
  - 90th percentile: 0.59
  - Median: 0.54
  - 10th percentile: 0.49
  - Minimum: 0.35
- Female General Population
  - Mean: 0.49
  - Standard Deviation: 0.040
  - Maximum: 0.65
  - 90th percentile: 0.54
  - Median: 0.48
  - 10th percentile: 0.44
  - Minimum: 0.33
- Male Annuitant Population
  - Mean: 0.50
  - Standard Deviation: 0.039
  - Maximum: 0.72
  - 90th percentile: 0.55
  - Median: 0.50
  - 10th percentile: 0.45
  - Minimum: 0.35

- Interpretation:
  - Mean replacement rate for general male population about 0.54 (54 percent) with standard deviation 0.041 (4.1 percentage points).
  - Small probability of replacement rate as low as 0.35 (35 percent) or as high as 0.74 (74 percent).
  - Female mean replacement rate about 0.49 (49 percent), about 5 percentage points (10 percent) lower than male mean; standard deviation nearly equal.
  - Male annuitant population mean replacement rate about 0.50 (50 percent), about 4 percentage points lower than general male population.

### Decreasing Risk — Staggered Purchases and Diversification
- Strategy: staggered purchases of multiple annuities over several years to hedge interest rate risk and smooth premium variability (Alier and Vittas (2001)).
- Simulation of portfolio of annuities (assumptions: each annuity pays same amount, annuities begin payments at same time, bond yields not correlated over time; portfolio premium is mean of individual premiums).
- Results for general male population (Table 3):

Table 3. Expected Premiums of Portfolios of One Dollar Annuities for the General Male Population
- Number of annuities in portfolio: 2
  - Mean: $9.00
  - Standard Deviation: $0.48
- Number of annuities in portfolio: 5
  - Mean: $8.99
  - Standard Deviation: $0.30
- Number of annuities in portfolio: 10
  - Mean: $9.00
  - Standard Deviation: $0.22

- Findings:
  - Buying two annuities reduces standard deviation of the premium by nearly 30 percent relative to a single annuity, without changing expected value.
  - Increasing portfolio size further reduces variance, converging to zero as portfolio grows.
- Caveats:
  - Model ignores fixed transaction costs; multiple annuity purchases may be expensive.
  - Optimal number of annuities requires modeling transactions costs and other frictions.
  - Deferred annuities require longer-range mortality projections, potentially explaining market thinness.

### Conclusions and Directions for Future Research
- Main conclusions:
  - Annuity premiums vary with treasury bond yield fluctuations, introducing distribution-phase risk in addition to accumulation-phase risk for DC plan participants.
  - Simulated standard deviation of replacement rates for a pensioner facing only annuitization price risk is about 4 percentage points.
  - Adverse selection increases consumer costs: in the U.S. market an annuitant can pay about 8 percent more for an annuity than in absence of adverse selection; replacement rate is on average about 4 percent lower due to adverse selection.
  - Price discrimination by gender results in lower expected replacement rates for women; potential future discrimination by race or socioeconomic status would raise public policy issues if private annuity income replaces Social Security benefits.
- Policy-relevant observations:
  - Phased (staggered) annuitization can reduce premium variance but faces transaction cost and commitment challenges.
  - Consideration of market instruments (e.g., survivor bonds, TIPS) and public policy could address mortality and inflation risks.
- Suggested future research directions:
  - Incorporate asset risk (especially equity exposure), wage risk, taxes, and income heterogeneity into replacement rate simulations.
  - Investigate correlation between annuity premiums and stock market returns to assess whether such correlation reduces overall retirement income unpredictability.
  - Simulate indexed annuity premiums using developing TIPS data to capture inflation risk.
  - Explore market and policy solutions to mortality risk, including government-issued survivor bonds.

*Source: _wp04230 - References...............................................................................................................................*

### REFERENCES

### REFERENCES

### Cited works
- Alier, Max, and Dimitri Vittas, 2001, "Personal Pension Plans and Stock Market Volatility," in New Ideas About Old Age Security: Toward Sustainable Pension Systems in the 21st Century, ed. by Robert Holzman and Joseph E. Stiglitz (Washington: World Bank).
- Benitez-Silva, Hugo, 2003, “The Annuity Puzzle Revisited,” MRRC Working Paper WP2003–05 (Ann Arbor, MI: Michigan Retirement Research Center).
- Blake, David, 1999, “Annuity Markets: Problems and Solutions,” The Geneva Papers on Risk and Insurance—Issues and Practice, Vol. 24, No. 3 (July).
- Brown, Jeffrey R., 1999, "Private Pensions, Mortality Risk, and the Decision to Annuitize," NBER Working Paper No. 7191 (Cambridge, MA: National Bureau of Economic Research).
- ———, Olivia S. Mitchell, James M. Poterba, and Mark J. Warshawsky, 2001, The Role of Annuity Markets in Financing Retirement (Cambridge, MA: MIT Press).
- Burtless, Gary, 2002, "How Would Financial Risk Affect Retirement Income Under Individual Accounts?" An Issue in Brief Center for Retirement Research at Boston College No. 5 (Chestnut Hill, MA: Center for Retirement Research Boston College).
- Cannon, Edmund, and Ian Tonks, 2003, "UK Annuity Rates and Pension Replacement Ratios 1957–2002," UBS Pensions Research Program Discussion Paper No.8 (London: London School of Economics).
- Dushi, Irena and Tony Webb, 2004, “Annuitization: Keeping Your Options Open,” CRR Working Paper 2004–04 (Chestnut Hill, MA: Center for Retirement Research Boston College).
- Feldstein, Martin and Elena Ranguelova, 2001, “Individual Risk in an Investment-Based Social Security System,” NBER Working Paper No. 8074 (Cambridge, MA: National Bureau of Economic Research).
- Finkelstein, Amy and James Poterba, 2000, “Adverse Selection in Insurance Markets: Policyholder Evidence from the UK Annuity Market,” NBER Working Paper No. 8045 (Cambridge, MA: National Bureau of Economic Research).
- Friedman, Benjamin M. and Mark Warshawsky, 1990, "The Cost of Annuities: Implications for Saving Behavior and Bequests," Quarterly Journal of Economics, Vol. 105, No. 1, pp. 135−154.
- International Monetary Fund, 2004, Global Financial Stability Report, World Economic and Financial Surveys (Washington, April).
- James, Estelle and Dimitri Vittas, 1999, "Annuity Markets in Comparative Perspective: Do Consumers Get Their Money's Worth?" paper prepared for World Bank Conference, "New Ideas on Old-Age Security" (Washington).
- Milevsky, Moshe, 1998, “Optimal Asset Allocation Towards the End of the Life Cycle: To Annuitize or Not to Annuitize?" The Journal of Risk and Insurance, Vol. 65, No. 3, pp. 401−426.
- Mitchell, Olivia, James Poterba, and Mark Warshawsky, 1999, "New Evidence on the Money's Worth of Individual Annuities," American Economic Review, Vol. 89, No. 5, pp. 1299−1318, reprinted as Chapter 4 in Brown et al., 2001.
- Poterba, James, 1997, "The History of Annuities in the United States," NBER Working Paper No. 6001 (Cambridge, MA: National Bureau of Economic Research).
- Samwick, Andrew, and Jonathan Skinner, 2003, “How Will 401(k) Pension Plans Affect Retirement Income?” American Economic Review, Vol. 94, No. 1, pp. 329−343.
- U.S. Government Accounting Office, 2003, "Private Pensions: Participants Need Information on Risks They Face in Managing Pension Assets at and During Retirement," Report to Congressional Requestors GAO-03-8 (Washington).
- U.S. Department of Labor, Pension, and Welfare Benefits Administration, 2001–2002, "Private Pension Plan Bulletin Abstract of 1998 Form 5500 Annual Reports," Number 11 (Winter).
- Warshawsky, Mark, 1988, "Private Annuity Markets in the United States," Journal of Risk and Insurance, Vol. 55, No. 3, pp. 518−528, reprinted as Chapter 3 in Brown et al., 2001.

*Source: _wp04230 - REFERENCES*

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