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The Challenge of 401_k_ Plans

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					           Life is Cheap:
      Using Mortality Bonds To
    Hedge Aggregate Mortality Risk
  Leora Friedberg                    Anthony Webb
University of Virginia      Center for Retirement Research
     and NBER                       at Boston College

   Presentation for Shanghai University of Finance and
                     Economics, PRC
                     22 November 2007
 Aggregate Mortality Risk

• Risk that annuitants on average live longer
  than expected
  – CANNOT be eliminated through diversification
    within annuity business
  – Difficult to hedge with life insurance business
 Aggregate Mortality Risk

• Affects annuity providers
  – Insurance companies
     • offering voluntary annuities
  – Employers
     • offering annuitized pensions
  – Taxpayers
     • through Social Security, PBGC
 Aggregate Mortality Risk

• Affects annuity providers
  – Insurance companies
     • offering voluntary annuities
  – Employers
     • offering annuitized pensions
  – Taxpayers
     • through Social Security, PBGC
 Aggregate Mortality Risk

• Affects potential annuitants
  –  price,  quantity in equilibrium
  – only 7.4% in AHEAD voluntarily annuitized
     • between 1993-2000
 Aggregate Mortality Risk

• Of potentially greater importance
  – Decline in generosity of Social Security and
    displacement of annuitized DB by unannuitized
    DC pensions may increase annuity demand
  – ? Increasing uncertainty about potential for
    dramatic medical breakthroughs
 Outline

1. What is the magnitude of this risk?
2. How might this risk affect pricing of annuities?
3. What price should this risk command in
   financial markets?
Aggregate Mortality Risk

 1. What is the magnitude of this risk?
   –   Lack of agreement
       • Actuarial tables
          – yield point estimates only
          – e.g., Society of Actuaries
       • Social Security Administration
          – high, intermediate, low forecasts
          – but no confidence intervals
       • Lee and Carter (1992)
Aggregate Mortality Risk

 1. What is the magnitude of this risk?
   –   Lack of agreement
       • Actuarial tables
          – yield point estimates only
          – e.g., Society of Actuaries
       • Social Security Administration
          – high, intermediate, low forecasts
          – but no confidence intervals
       • Lee and Carter (1992)
Aggregate Mortality Risk

 1. What is the magnitude of this risk?
   –   Lee-Carter model
       • “leading statistical model of mortality in the demographic literature”
         (Deaton-Paxson 2004)
       • adopted by U.S. Census Bureau
       • performs well in sample
       • provides confidence intervals
                perhaps they are too narrow?
Aggregate Mortality Risk

 2. How might this risk affect pricing of annuities?
   –   Required reserves if Lee-Carter is correct
        to reduce probability of insolvency to 5%
              for 3%-real 50%-survivor annuity sold to couple aged 65-85,
              need reserves of 2.7-4.8%

   –   Impact of using SOA projections if Lee-Carter is
       correct
        same annuity will be underpriced under SOA projections
              by 2.3-3.2%
Aggregate Mortality Risk

 3. What price should this risk command in
    financial markets?
   –   Historical covariances, 1959-99
       •   Impact of mortality shocks on longevity bond prices at ages 65+
       •   Covariance with S&P 500 (CAPM)
       •   Covariance with consumption growth (CCAPM)

   –   These covariances are very small
Aggregate Mortality Risk

 3. What price should this risk command in
    financial markets?
   –   Should be able to hedge risk at virtually no cost
   –   How? Mortality-contingent bonds
       •   two short-term bonds issued recently by Swiss Re
       •   one long-term bond proposed by EIB, not issued
           –   according to our calculations, this bond was overpriced,
                unless investors expected lower mortality than U.K. Actuary did
Aggregate Mortality Risk

 3. What price should this risk command in
    financial markets?
   –   Should be able to hedge risk at virtually no cost
   –   How? Mortality-contingent bonds
       •   two short-term bonds issued recently by Swiss Re
       •   one long-term bond proposed by EIB, not issued
           –   according to our calculations, this bond was overpriced,
                unless investors expected lower mortality than U.K. Actuary did
 Outline

1. What is the magnitude of this risk?
2. How might this risk affect pricing of annuities?
3. What price should this risk command in
   financial markets?
1. Magnitude of aggregate mortality
   risk
 •   Lee-Carter model
      ln (mx,t ) = ax + bxkt + ex,t
      kt = kt-1 – 0.365 + 5.24 flu + et , e = 0.655
 •   Details
      m is mortality by age x, year t
      a, b are parameters that vary with age x
      flu is the 1918 flu epidemic
1. Magnitude of aggregate mortality
   risk
 •   Lee-Carter model
      ln (mx,t ) = ax + bxkt + ex,t
      kt = kt-1 – 0.365 + 5.24 flu + et , e = 0.655
 •   Details
      m is mortality by age x, year t
      a, b are parameters that vary with age x
      flu is the 1918 flu epidemic
1. Magnitude of aggregate mortality
   risk
 •   Lee-Carter model
      ln (mx,t ) = ax + bxkt + ex,t
      kt = kt-1 – 0.365 + 5.24 flu + et , e = 0.655
 •   Implications for mortality trends
      LC estimated that a random walk with drift fits path of k
              implies roughly linear decline in k 
                       decreasing rate of increase in life expectancy
              no mean reversion in mortality trends  current shock to m
              yields almost equal % change in subsequent E[m]
1. Magnitude of aggregate mortality
   risk
 •   Lee-Carter model
      ln (mx,t ) = ax + bxkt + ex,t
      kt = kt-1 – 0.365 + 5.24 flu + et , e = 0.655
 •   Implications within sample
      explains > 90% of within-age variances in mortality rates
      one standard-deviation shock to k
                2-month change in age-65 life expectancy
1. Magnitude of aggregate mortality
   risk
 •   Comparisons of Lee-Carter with other forecasts
     –   more optimistic than SSA
     –   close to SOA at ages 45-79, then more optimistic
 •   Figures 1-3
     –   comparison of mortality forecasts, 2006-54
     –   comparison to recent mortality data, 1989-02
1. Magnitude of aggregate mortality
   risk
 •   Comparisons of Lee-Carter with other forecasts
     –   more optimistic than SSA
     –   close to SOA at ages 45-79, then more optimistic
 •   Figures 1-3
     –   comparison of mortality forecasts, 2006-54
     –   comparison to recent mortality data, 1989-02
                             Future life expectancy at age 60, various mortality forecasts

                            34
                                                                                     LC 95%
                                                 Lee-Carter 95%
                            32
                                                                                     LC m ean
Life expectancy, in years




                            30
                                                       SSA high                      LC 5%
                            28

                            26                                                       SSA high

                            24                                                       SSA
                                                                                     interm ediate
                            22                                                       SSA low

                            20                                                       SOA Scale
                                 2006   2014   2022   2030   2038   2046   2054      AA
                                      Recent actual vs. forecasted mortality declines
                                             Males, 1895-1924 birth cohorts
                              1.10
                                                            Ages 90-94
                              1.0 5
                                                                                         SSA
Mortality relative to 1989




                             1.0 0                                                       intermediate
                                                                                         forecast
                             0 .9 5                                                      LC weighted
                                                                                         forecast
                             0 .9 0

                             0 .85
                                             Actual mortality, ages 65-69

                             0 .80
                                  19 89      19 9 2       19 9 5       19 9 8   20 0 1
                                         Recent actual vs. forecasted mortality declines
                                              Females, 1895-1924 birth cohorts
                              1.10

                                                          Ages 90-94
                              1.05
Mortality relative to 1989




                                                                                        SSA
                             1.00                                                       interm ediate
                                                                                        forecast
                             0.9 5                                                      LC weighted
                                                                                        forecast
                             0.9 0                      Actual mortality, ages 65-69


                             0.85


                             0.80
                                 19 89        19 9 2    19 9 5      19 9 8       2001
 Outline

1. What is the magnitude of this risk?
2. How might this risk affect pricing of annuities?
3. What price should this risk command in
   financial markets?
2. Implications for pricing of
   annuities

•    Two sets of calculations
    A. Required mark-up/reserves if Lee-Carter is correct
       •   impact of variance of expected mortality

    B. Impact of using SOA projections if Lee-Carter is
       correct
       •   impact of differences in expected mortality
2. Implications for pricing of
   annuities
 A. Required mark-up/reserves if LC is correct
   –   10,000 Monte Carlo simulations
       •   each simulation: draw baseline k, then errors to fill in mx,t
       •   construct resulting life tables
           –   compute premium required to break even, on average
       •   compute annuity payments in each simulation
           –   compare to premium
       •   what % mark-up over premium will reduce probability of loss to x%?
           –   or what % of EPV must be held as capital reserve
           –   x = 0.05 or x = 0.01
2. Implications for pricing of
   annuities
 A. Required mark-up/reserves if LC is correct
   –   required mark-up is 2.7% to 4.8%
       •   competing effects of age
           –   uncertainty about mortality at older ages  with time horizon
           –   but, payments at older ages are heavily discounted

   –   impact if eliminate cancer, all circulatory disease,
       diabetes?
       •   increase PV of an annuity by 50%
 Potential Losses Arising From Aggregate Mortality Risk

                                  Loss probability
     Single men        Single women         Married Couples with survivor benefit
                                                   50%                100%
       5%         1%     5%          1%           5%       1%       5%         1%
                               3% interest rate
65   3.94%   5.66%     3.67%     5.22%       2.69%       3.80%   2.69%       3.89%
70   4.17%   5.95%     3.97%     5.60%       2.82%       4.02%   2.92%       4.12%
75   4.43%   6.32%     4.15%     5.96%       3.00%       4.31%   3.10%       4.47%
80   4.49%   6.53%     4.38%     6.27%       3.13%       4.45%   3.27%       4.63%
85   4.85%   6.96%     4.61%     6.57%       3.31%       4.67%   3.46%       5.01%
 Potential Losses Arising From Aggregate Mortality Risk

                                  Loss probability
     Single men        Single women         Married Couples with survivor benefit
                                                   50%                100%
       5%         1%     5%          1%           5%       1%       5%         1%
                               3% interest rate
65   3.94%   5.66%     3.67%     5.22%       2.69%       3.80%   2.69%       3.89%
70   4.17%   5.95%     3.97%     5.60%       2.82%       4.02%   2.92%       4.12%
75   4.43%   6.32%     4.15%     5.96%       3.00%       4.31%   3.10%       4.47%
80   4.49%   6.53%     4.38%     6.27%       3.13%       4.45%   3.27%       4.63%
85   4.85%   6.96%     4.61%     6.57%       3.31%       4.67%   3.46%       5.01%
2. Implications for pricing of
   annuities
 B. Impact of SOA projections if LC is correct
   –   no actual pricing data
   –   and it would be difficult to use prices to back out
       mortality assumptions
       •   without knowing assumptions about expenses, asset returns,
           annuitant characteristics

   –   instead, we focus on recent SOA projections
2. Implications for pricing of
   annuities
 B. Impact of SOA projections if LC is correct
   –   compute EPV of payments for $1/year annuity
       •   EPV if SOA projection scale is correct
       •   EPV is Lee-Carter is correct
           –   Lee-Carter value is always higher
Percentage Underpricing Resulting From Use of Projection Scale AA


                       Male     Female               Couple
   Survivor Benefit                           50%             100%



          Age
            65        1.64%      2.93%       2.31%            3.01%
            70        2.06%      3.04%       2.57%            3.23%
            75        2.52%      3.16%       2.86%            3.45%
            80        2.84%      3.27%       3.07%            3.62%
            85        3.02%      3.34%       3.18%            3.68%
 Outline

1. What is the magnitude of this risk?
2. How might this risk affect pricing of annuities?
3. What price should this risk command in
   financial markets?
3. Pricing of aggregate mortality risk


 •   Mortality-contingent bonds
     –   can be used to pass mortality risk to those who
         want it
     –   very recent examples
3. Pricing of aggregate mortality risk

 •   Mortality-contingent bonds
     –   Swiss Re
         •   three-year bond, first issued in 2003
         •   if five-country average mortality > 130% of 2002 level
                    principal will be reduced
         •   if it > 150%  principal will be exhausted
3. Pricing of aggregate mortality risk

 •   Mortality-contingent bonds
     –   EIB
         •   25-year bond, proposed in 2004
         •   mortality-contingent payments  proportionally
                 as annual survival rate for U.K. cohort aged 65 in 2003
         •   but EIB bond was not issued as planned
         •   expected yield implied 20-basis point discount
                 (assuming Government Actuary Department’s mortality
                 forecasts are unbiased)
3. Pricing of aggregate mortality risk

 •   We price the EIB bond
     –   had such bonds been available in U.S.
         •   measure mortality shocks
             –   as identified from Lee-Carter model
             –   Berkeley Human Mortality database, 1959-99
             –   Social Security Administration data, 19xx-yy
         •   correlation with S&P 500
             –   compute beta, risk premium from CAPM
         •   correlation with per capita consumption growth
             –   compute risk premium from CCAPM
The Capital Asset Pricing Model:

                   (1):

  Where Ri is the return on asset i and Rm is the market return




            (2):


                Where Rf is the risk-free return
The Capital Asset Pricing Model:

       Rearranging (2):


The expected return on asset i depends on the risk-free return, and the
       covariance of the asset’s return with the market return


Important implication – idiosyncratic risk – the risk of good or bad returns that are
uncorrelated with the market return do not command a risk premium

Why? – Because an investor can diversify away that risk by investing a small amount
in a lot of assets with uncorrelated returns – think of the “law or large numbers”
3. Pricing of aggregate mortality risk


 •   Results
     –   such bonds would not have been very risky
     –   standard deviation of return is 0.64%
         •   versus 17% for stocks
3. Pricing of aggregate mortality risk

 •   Results for CAPM
     –   correlation with S&P 500
         •   varies with age of bond’s reference population
         •   for age-65 mortality bond, beta = 0.005
             –   95% confidence interval of [-0.005, 0.015]
             –   virtually no correlation with stock market
         •   bond would command risk premium of 2.5 bp
             –   for equity premium of 500 bp
3. Pricing of aggregate mortality risk

 •   Results for CCAPM
     –   hypothesis: mortality bonds pay out most when?
         •   when mortality is unexpectedly low
         •   and then resources that are roughly unchanged in quantity
             have to support more people
             –   expect negative correlation with C growth
3. Pricing of aggregate mortality risk

 •   Results for CCAPM
     –   correlation for age-65 bond is -0.1958
         •   significantly different from 0 for all reference ages
     –   mortality bonds should attract risk discount
         •   in contrast with stocks
             –   correlation is about 0.5
             –   should attract risk premium
3. Pricing of aggregate mortality risk

 •   Results for CCAPM
     –   but mortality bond returns, C growth are very
         smooth series
         •   covariance is extremely small, -0.0013
         •   resulting risk discount is 2 basis points
             –   for risk aversion coefficient of 10
         •   contrast with EIB prospectus
             –   proposed risk discount of 20 basis points
3. Pricing of aggregate mortality risk

 •   What explains EIB bond?
     –   apparently overpriced
         •   EIB expected to pass risk further by obtaining reinsurance
             –   Smetters, Dowd: insurance markets are small, constrained compared to
                 financial markets, which can bear large risks better
         •   maybe investors expected better mortality
             –   compared to U.K. Actuary’s forecasts
             –   and might have perceived risk discount as less than 20 BP
    Conclusions

•    Aggregate mortality risk is considerable
•    But uncorrelated with other financial risks
     –   annuity providers should be able to shed aggregate
         mortality risk at virtually no cost

•    Of growing importance
     –   demand for voluntary annuitization might be
         expected to rise

				
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