swimming with wealthy sharks (and some...

Swimming with Wealthy Sharks (and some clams):Longevity, Volatility and the Value of Risk Pooling

Moshe A. Milevsky1

1Professor of FinanceSchulich School of Business &

Graduate Faculty of Mathematics and StatisticsYork University, Toronto, CANADA

Montreal: 16 November 2018

Moshe A. Milevsky Swimming with Wealthy Sharks Montreal: 16 November 2018 1 / 29

The Pension Problem: Pooling with Heterogenous Risks

RETIREMENT Simon HeatherCurrent Age: 65 65

Earned Pension Credits: Maximum Maximum

Annual Pension Income: $25,000 $25,000

Life Expectancy (Horizon): 10 years 30 years

Term Annuity PV at 3%: $212,750 $487,250

Funded* System Assets: $700,000

Total Pension Contributions: $350,000 $350,000

Transfer & Subsidy: -$137,250 +$137,250

Question: Why does Simon wan’t to be part of this?


Annuities offer Longevity Insurance → Utility Benefits

Simon (in Blue) might live longer than Heather (in Red)


Empirical (50%) Question: Guided by Theory (50%)

Do the welfare (utility, risk management) gains for Simon outweighthe expected wealth transfers to Heather? Or, are the Simons of theworld better-off not annuitizing?

“Uniform annuitization would favor those with longer expected lives[such as] high earners relative to low earners...but Brown (2003)shows there is much less diversity in the utility value of annuitizationthan previous comparisons.” Peter Diamond (2004), presidentialaddress, American Economics Association.

But, at what point does this no longer hold? How large is themortality heterogeneity gap, before the pooling benefit is destroyed?


Motivation in 2018: The Mortality & Income Gradient

U.S. Death Rates per 1,000 individuals

MALE (age) FEMALE (age)

Income Group 40 50 60 40 50 60

Lowest (1st pct.) 5.8 12.5 22.1 4.3 8.0 12.8

25th percentile 2.0 4.5 10.9 1.2 2.7 5.9

Median (50th pct.) 1.2 2.9 7.3 0.8 2.0 4.5

75th percentile 0.8 1.8 4.9 0.5 1.3 3.5

Highest (100th pct.) 0.6 1.1 2.8 0.3 0.8 2.2

Source: Chetty et al. (2016), period 2001 to 2014.

Next steps...

1 Economics: Evaluating Utility and Welfare Gains of Pooling.

2 Actuarial: A Model for Mortality Consistent with the Data

3 Insurance: Pricing Life Annuities under the Law of Mortality


A Preview of the Main Result: What Happens in the Pool?

Note: CRRA γ = 1..5 and r = 3%


Utility Gains in a Simple Exponential Mortality Framework

Assuming a constant mortality hazard rate λ, if we maximize discountedlifetime utility without annuities, the best we get is:

U∗(w) =

∫ ∞0

e−(r+λ)su(c∗s )ds. (1)

But, if we annuitize (everything), the maximal utility is:

U∗∗(w) =

∫ ∞0

e−(r+λ)su(w(r + λ))ds =u(w(r + λ)

r + λ. (2)

See Yaari (1965). Annuity Equivalent Wealth (AEW) satisfies:

U∗∗(w) = U∗(w(1 + δ)), (3)

where w is initial wealth and δ is the value of longevity risk pooling.


Nice Result and Intuition for δ

It’s relatively easy to show that when remaining lifetimes are exponentiallydistributed, the value of longevity risk pooling satisfies:

1 + δ =

(r + λ/γ

r + λ

)γ/(1−γ), (4)

where γ 6= 1, is the coefficient of RRA, r is the valuation and subjectivediscount rate, and λ is the mortality hazard rate. E.g. when γ = 2,r = 3% and λ = 1/25, the value of pooling is δ = 0.96 and AEW = $1.96.

Interestingly enough, when γ → 1 and r = λ/γ, this converges to:

δ =√e − 1 ≈ 64.9% (5)

See Milevsky and Huang (NAAJ, 2018).


But lifetime’s aren’t exponential: Back to Real Mortality.

U.S. Death Rates per 1,000 individuals

MALE (age) FEMALE (age)

Income Group 40 50 60 40 50 60

Lowest (1st pct.) 5.8 12.5 22.1 4.3 8.0 12.8

25th percentile 2.0 4.5 10.9 1.2 2.7 5.9

Median (50th pct.) 1.2 2.9 7.3 0.8 2.0 4.5

75th percentile 0.8 1.8 4.9 0.5 1.3 3.5

Highest (100th pct.) 0.6 1.1 2.8 0.3 0.8 2.2

Lowest / Highest 9.6x 11.3x 7.9x 14.3x 10.0x 5.8x

Source: Chetty et al. (2016), period 2001 to 2014.

(1.) Population’s average mortality grows exponentially, a.k.a Gompertzmodel, and (2.) dispersion between high/low income shrinks over time.


Compensation Law of Mortality (CLaM): A Theory

For a heterogeneous group within a species relatively healthy memberswith low death rates age faster, while those with higher death rates andsicker than average, age slowly. The strong form CLaM states thatinstantaneous hazard rates converge to a constant at some age x∗ ≤ ω.

See Gavrilov & Gavrilov (1979, 1991)


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Compensation Law of Mortality: Remember CLaM


Economic and Insurance Implications of CLaM

Within a Gompertz framework – in which log biological hazard ratesincrease linearly with age – CLaM implies the volatility of remaininglifetime is greater in both relative (%) and absolute (years) terms, thehigher your hazard rate, compared with others at same chronological age x .

Formally, under the Gompertz Makeham law of mortality, if:

λx = λ+ hegx , (6)

and Tx is your random remaining lifetime at age x , then:

SD[Tx ] increases in – and is proportional to – the inverse mortalitygrowth rate: b = 1/g , and

the coefficient of variation: SD[Tx ]/E [Tx ], which I call the individualvolatility of longevity (iVoL), increases in h.

So, yes, life is riskier for those with lower income (and as you age.)


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Coefficient of Variation (or iVoL) over the Lifecycle

Simon (blue, top) vs. Heather (red, bottom) as they age...

Once they reach mortality λ∗, and plateau age x∗, the iVoL → 100%.


Linked: Slope (C1) & Intercept (C0) in Gompertz Model

A strong CLaM places tight restrictions on the relationship between h, andg . In fact, h(g) is now a function of g and

L := ln(λ∗ − λ) = ln h(g) + gx∗, (7)

where L is a (new) convenient constant, and

C0︷︸︸︷ln h(g) = L − x∗

C1︷︸︸︷g . (8)

In the context of Chetty et al. (2016), we have 100 values of ln h and g ,so we can estimate the intercept L and slope (−x∗) on the CLaM line.


Dispersion in estimated mortality growth rates g .

Regress (log) mortality rates on age, for the 100 income percentiles inChetty et al. (2016). See Gompertz slope (g) versus intercept (h).


CLaM Regression Results (US): Big Picture


CLaM Regression Results (US): The Numbers

Using the 100 Gompertz slope and intercept coefficients from Chetty et al.(2016), based on U.S. mortality during the period 2001-2014, I estimate:

100 Data Points: All income percentiles

Variable MALE FEMALECoeff. Std.Er t-val. Coeff. Std.Er t-val.

Intercept (L) -1.234 0.119 -10.4 -2.038 0.129 -15.8

Slope: (−x∗) -99.98 1.284 -77.9 -95.19 1.359 -70.1

Adj. R2 98.39% 98.02%

Plateau λ∗ ∈ [0.2585, 0.3278] λ∗ ∈ [0.1145, 0.1482]

Range: g [i ] (5.961%, 10.491%) (5.454%, 10.621%)

Mean: g [i ] 9.181% 9.415%


CLaM Regression Results (US): Limited Numbers

Using a subset of the values, in which (extremely) low income percentilesare not included, these are the results.

90 Data Points: Income percentiles 11 to 100

Variable MALE FEMALECoeff. Std.Er t-val. Coeff. Std.Er t-val.

Intercept: (L) -0.313 0.129 -2.4 -0.629 0.132 -4.76

Slope: (−x∗) -109.5 1.367 -80.1 -109.6 1.363 -80.4

Adj. R2 98.63% 98.64%

Plateau λ∗ ∈ [0.6422, 0.8324] λ∗ ∈ [0.4669, 0.6082]

Range: g [i ] (7.703%, 10.491%) (8.337%, 10.621%)

Mean: g [i ] 9.451% 9.683%

Note: The range of λ∗ is (now) consistent with value of 0.645, andplateau age of 105, reported in Barbi et al. (2018), for Italy.


CLaM Regression Results (Canada):

Milligan and Schirle (NBER, 2018) extracted mortality rates from CPPdata and estimated Gompertz (w/o Makeham) parameters for 1923-1955cohort based on income percentiles. Consistent with a strong CLaM.


CLaM Regression Results (Canada): The Numbers

Using Gompertz slope and intercept coefficients provided by K. Milligan,based on Canadian (CPP) mortality for the 1923-1955 cohort, I estimate:

100 Data Points: All income percentiles

Variable MALE FEMALE*Coeff. Std.Er t-val. Coeff. Std.Er t-val.

Intercept (L) -1.605 0.332 -4.835 -5.004 0.1448 -34.56

Slope: (−x∗) -97.446 4.039 -24.127 -63.072 1.6709 -37.75

Adj. R2 85.44% 93.5%

Plateau λ∗ ∈ [0.1441, 0.2799] λ∗ ∈ [0.0058, 0.00775]

Range: g [i ] (6.758%, 9.078%) (6.7791%, 10.111%)

Mean: g [i ] 9.415% 8.648%

*Yes, the female (parameter) is puzzling.


A bit of history: Note the trend down the CLaM Line


Pricing & Valuing Annuities on the CLaM Line (U.S.)

Recall the annuity factor ax :=∫∞0 e−rsp(s)ds, which is available in

closed-form under the Gompertz-Makeham law of mortality.

The CLM line: ln h = −1.234− (102.5)g , for U.S. Males

Mortality log hazard Mortality Life Exp. FactorGrowth: g at zero: ln h Age 65: λ65 E [T65] a65

5.5% (poor) −6.871 3.701% 14.0 yrs. $10.64

6.5% −7.896 2.544% 16.3 yrs. $12.05

7.5% −8.921 1.748% 18.5 yrs. $13.35

8.0% −9.434 1.449% 19.5 yrs. $13.90

8.5% −9.946 1.202% 20.5 yrs. $14.51

9.5% −10.971 0.826% 22.4 yrs. $15.54

10.5% (rich) −11.996 0.568% 24.1 yrs. $16.43

Assumes: r = 3%, termination at x = 102.5 and λ = 0 (i.e. no Makeham).CLaM line: (L, x∗) calibrated to upper bound of Chetty et al. (2016) data.


Analytic Expression for the Value of Longevity Risk Pooling

Previously I noted that if maximal annuitized utility is: U∗∗(w) andmaximal non-annuitized utility is: U∗(w), for a given level of initialretirement wealth w , then the value of δ that satisfies:

U∗∗(w) = U∗(w(1 + δ)), (9)

known as Annuity Equivalent Wealth (AEW), can be expressed as:

1 + δ =ax(h, g)

11−γ ax(h, g)−1

ax(h/γ, g)γ

1−γ

, (10)

where: ax(h, g) denotes the life annuity factor under individual (h, g)parameter values, and ax(h, g) is the group factor, all assuming a CRRAvalue of γ. Note: h and g are any Gompertz parameters and do not haveto obey the strong (or weak) CLaM.


Hypothetical Numerical Examples for δ in a Mixed Pool

U.S. Males at Age 65, Under γ = 1 (CRRA)Mortality (Swimming) PoolGrowth: g Fair Group Sharks

5.5% δ = 63.5% δ = 24.7% δ = 5.9%

8.0% δ = 40.6% δ = 40.6% δ = 19.4%

10.5% δ = 27.2% δ = 49.8% δ = 27.2%

Annuity Equivalent Wealth (AEW) is: (1 + δ), where δ is the value oflongevity risk pooling. The annuity factors are priced under an: r = 3%(real) interest rate. The Gompertz mortality curve (and h) is based on theCLaM value for the relevant g .


Final Results: Females with a CRRA: γ = 3, r = 3% Real

Income Gomp. (h65, g) iVoL Fair ax δ (Fair) δ (Group)

Low. (1.64%, 5.29%) 56.73% 15.23 62.18% 46.52%

5th (1.22%, 6.68%) 51.05% 15.72 53.59% 43.18%

10th (1.15%, 8.08%) 48.63% 14.97 52.45% 35.39%

20th (1.00%, 8.9%) 46.33% 15.08 49.20% 33.46%

30th (0.86%, 8.61%) 45.24% 15.96 45.87% 38.06%

40th (0.78%, 8.84%) 44.11% 16.23 43.93% 38.57%

50th (0.69%, 8.73%) 43.10% 16.86 41.46% 41.46%60th (0.69%, 10.06%) 41.91% 15.94 41.89% 34.20%

70th (0.55%, 9.08%) 40.95% 17.51 37.82% 43.18%

80th (0.50%, 10.35%) 39.15% 17.00 36.81% 37.97%

90th (0.45%, 10.49%) 38.15% 17.36 35.12% 39.12%

95th (0.38%, 9.74%) 37.46% 18.53 32.44% 45.61%

High. (0.34%, 9.89%) 36.46% 18.89 30.89% 46.66%


Final Results: Males with a CRRA: γ = 3, r = 3% Real

Income Gomp. (h65, g) iVoL Fair ax δ (Fair) δ (Group)

Low. (3.02%, 6.56%) 61.24% 11.15 84.26% 38.25%

5th (2.10%, 6.63%) 56.97% 12.96 70.29% 48.52%

10th (2.00%, 7.46%) 55.13% 12.72 68.20% 43.96%

20th (1.75%, 8.31%) 52.56% 12.89 63.64% 41.90%

30th (1.50%, 8.78%) 50.41% 13.33 59.15% 42.76%

40th (1.18%, 8.43%) 48.41% 14.65 52.95% 50.76%

50th (1.06%, 8.83%) 46.97% 14.86 50.53% 50.53%60th (0.89%, 8.68%) 45.45% 15.77 46.54% 55.47%

70th (0.82%, 9.31%) 44.08% 15.70 45.01% 53.18%

80th (0.70%, 9.49%) 42.58% 16.23 42.14% 55.20%

90th (0.60%, 9.8%) 40.98% 16.67 39.45% 56.39%

95th (0.51%, 9.68%) 39.83% 17.38 36.87% 60.09%

High. (0.42%, 8.74%) 39.01% 18.93 33.24% 69.77%


Conclusion: Ok, what did I learn?

1 Mortality Modeling can be cool (hopefully)...

2 Mandatory pension annuity pooling results in an expected subsidy(transfer) from low (poor) income to high (rich) income. Is the utility(insurance) benefit worth it? Take mortality data to a utility model.

3 Compensation Law of Mortality (CLaM) links low (high) mortalityrates with high (low) mortality growth. (e.g. Rich women age fast.)

4 Utility gains are still there (2018) because (per CLaM) the low-incomegroup have a higher volatility of longevity, and value pooling more.

5 Analytic expression to easily test if the value of pooling is positive.

6 The situation is (much) more complicated when exogenous income(i.e. pre-existing annuity wealth) is included.

7 With pre-existing pension income, Simon may not want anymore(expensive, loaded) annuities.


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