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The exact number of members that remove idiosyncratic mortality risk in pooled annuity funds

by Thomas Bernhardt, University of Michigan, Nanyang Technological University, Singapore, October, 2020

Thomas Bernhardt The exact number of members that remove mortality risk

Pooled annuity funds

everyone has a fund account

• decreases with income
• increases with mortality

pooled income = annuity income that is purchased with the current fund value

• 70

80 90 100 110 120 100 200 300 400 500 600 700 age monthly value income

member group 100

When does the income become unstable?

Thomas Bernhardt The exact number of members that remove mortality risk 1 / 6

Pooled annuity funds

everyone has a fund account

• decreases with income
• increases with mortality

pooled income = annuity income that is purchased with the current fund value

• 70

80 90 100 110 120 100 200 300 400 500 600 700 age monthly value income

member group 1000

When does the income become unstable?

Thomas Bernhardt The exact number of members that remove mortality risk 1 / 6

Pooled annuity funds

everyone has a fund account

• decreases with income
• increases with mortality

pooled income = annuity income that is purchased with the current fund value

• 70

80 90 100 110 120 100 200 300 400 500 600 700 age monthly value income

member group 10000

When does the income become unstable?

Thomas Bernhardt The exact number of members that remove mortality risk 1 / 6

The income process

homogeneous group of n members aged x, fixed market returns (a financial market gives no insight in how well mortality is pooled) C(t) = C(0) × tpx ˜

tpx

= C(0) × survival probability realized survival we want the income to be: (i) around the initial value with threshold ε, (ii) for given certainty β, (iii) for as long as possible max time t that depends on the group size n such that P

• spx

˜

spx

≥ (1 − ε) ∀s ≤ t

• ≥ β

Thomas Bernhardt The exact number of members that remove mortality risk 2 / 6

Kolmogorov-Smirnov approach

let ˜ Fn be the empirical distribution of F, then ˜ Fn(F −1(t)) is independent

• n F, it is the empirical distribution of uniform random variables, jumps
• f ˜

Fn(F −1(t)) are ordered uniform random variables (U(k))n

k=1

Kolmogorov-Smirnov test (are distributions close to each other?) supt(F(t) − ˜ Fn(t)) < ε criterion ε + k−1

n

≥ U(k) criterion holds for all k Pooled annuity fund infs≤t spx/ ˜

spx > 1 − ε

ε + (1 − ε) k−1

n

≥ U(k) last k for which criterion holds

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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 k/n, with sample size n k−th order statistics KS−line pooled uniform

• Thomas Bernhardt

The exact number of members that remove mortality risk 3 / 6

Output

the number of alive members before the income becomes unstable is independent of the underlying mortality distribution depends on

• initial group size n
• threshold ε
• certainty β

is independent of

• the mortality distribution

can be computed easily, e.g.

70 80 90 100 110 120 100 200 300 400 500 600 700

1000 member group

age monthly value income −10% +10% 275 remaining ε = 10% threshold β = 90% certainty

75% passed away ε = 10% ε = 10% ε = 5% ε = 5% before unstable β = 90% β = 99% β = 90% β = 99% initial group ≈ 1000 2500 5000 10000

Thomas Bernhardt The exact number of members that remove mortality risk 4 / 6

Output

the number of alive members before the income becomes unstable is independent of the underlying mortality distribution depends on

• initial group size n
• threshold ε
• certainty β

is independent of

• the mortality distribution

can be computed easily, e.g.

2000 4000 6000 8000 10000 5 10 15 20 25 30

Times (UK−based life table S1PFL)

initial group size time before unstable (in years) lower bound lower & upper ε=10%, β=90% ε=10%, β=99% ε=05%, β=90% ε=05%, β=99%

75% passed away ε = 10% ε = 10% ε = 5% ε = 5% before unstable β = 90% β = 99% β = 90% β = 99% initial group ≈ 1000 2500 5000 10000

Thomas Bernhardt The exact number of members that remove mortality risk 4 / 6

Approximate formula

Kolmogorov-Smirnov: √n

• t − ˜

Fn(F −1(t))

• d

− − − →

n↑∞ Brownian bridge n−k n

≈

• 1

1−ε

• 1 − 1/
• 1 + 1

n( 1−ε ε )2(Φ−1( 1−β 2 ))2 n

n − k # of alive members before income unstable (only lower bound) Φ−1 the inverse of the standard normal distribution function ⌊u⌋n = max{i/n | i/n ≤ u and i = 0, 1, . . . , n}

70 80 90 100 110 120 100 200 300 400 500 600 700 age monthly value income −10% 201 remaining ε = 10% threshold β = 90% certainty 2000 4000 6000 8000 10000 initial group size remaining % before unstable 0% 20% 40% 60% 80% 100% approximation exact ε=10%, β=90% ε=10%, β=99% ε=05%, β=90% ε=05%, β=99% Thomas Bernhardt The exact number of members that remove mortality risk 5 / 6

Future work: inhomogeneous group

group configuration same age different initial savings total pool size 1000 two groups (poor and rich) account ratio = savings of poor / savings of rich

200 400 600 800 1000 200 400 600 800 initial number of poor members number of deceased before unstable

• nly poor
• nly rich

ratio 0.7 ratio 0.5 ratio 0.3 ratio 0.2 ratio 0.1

How can we predict the curves?

Thomas Bernhardt The exact number of members that remove mortality risk 6 / 6

Future work: inhomogeneous group

group configuration same age different initial savings total pool size 1000 two groups (poor and rich) account ratio = savings of poor / savings of rich

200 400 600 800 1000 200 400 600 800 initial number of poor members number of deceased before unstable

• nly poor
• nly rich

ratio 0.7 ratio 0.5 ratio 0.3 ratio 0.2 ratio 0.1

How can we predict the curves?

Thank you very much! Any questions or feedback? bernt@umich.edu

Thomas Bernhardt The exact number of members that remove mortality risk 6 / 6