life is cheap: using mortality bonds to hedge aggregate mortality risk

Life is Cheap: Using Mortality Bonds To

Hedge Aggregate Mortality Risk

Leora Friedberg Anthony WebbUniversity 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


• Affects annuity providers– Insurance companies

• offering voluntary annuities

– Employers• offering annuitized pensions

– Taxpayers• through Social Security, PBGC


• Affects potential annuitants price, quantity in equilibrium

– only 7.4% in AHEAD voluntarily annuitized• between 1993-2000


• 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?


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)


– 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?



– Required reserves if Lee-Carter is correctto 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%



– 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



– 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




• Lee-Carter modelln (mx,t ) = ax + bxkt + ex,t

kt = kt-1 – 0.365 + 5.24 flu + et , e = 0.655

• Detailsm 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


kt = kt-1 – 0.365 + 5.24 flu + et , e = 0.655

• Implications for mortality trendsLC 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]



kt = kt-1 – 0.365 + 5.24 flu + et , e = 0.655

• Implications within sampleexplains > 90% of within-age variances in mortality rates

one standard-deviation shock to k 2-month change in age-65 life expectancy


• 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


20

22

24

26

28

30

32

34

2006 2014 2022 2030 2038 2046 2054

LC 95%

LC mean

LC 5%

SSA high

SSAintermediateSSA low

SOA ScaleAA

Lee-Carter 95%

SSA high

Life

expect

ancy

, in

years

Future life expectancy at age 60, various mortality forecasts

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1989 1992 1995 1998 2001

SSAintermediateforecast

LC weightedforecast

Actual mortality, ages 65-69

Ages 90-94

Mort

alit

y r

ela

tive t

o

19

89

Recent actual vs. forecasted mortality declines Males, 1895-1924 birth cohorts

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1989 1992 1995 1998 2001

SSAintermediateforecastLC weightedforecast

Actual mortality, ages 65-69

Ages 90-94

Mort

alit

y r

ela

tive t

o

19

89

Recent actual vs. forecasted mortality declinesFemales, 1895-1924 birth cohorts

Outline




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

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


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 womenMarried 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%

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


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




3. Pricing of aggregate mortality risk

• Mortality-contingent bonds

– can be used to pass mortality risk to those who want it

– very recent examples


– 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



– 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)


• 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:

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

Where Rf is the risk-free return

(1):

(2):

The Capital Asset Pricing Model:

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

Rearranging (2):

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”

• Results

– such bonds would not have been very risky

– standard deviation of return is 0.64%• versus 17% for stocks


• 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


• 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



– 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



– 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


• 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

life is cheap: using mortality bonds to hedge aggregate mortality risk

Documents

mortality bonds

lower mortality

aggregate mortality

aggregate mortality

aggregate mortality

impact of mortality

risk command i

annuity businessdifficult