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Tontines with bequest
University of Michigan Seminars,October, 2019
Thomas Bernhardt and Catherine DonnellyRisk Insight Lab https://risk-insight-lab.com/
Thomas Bernhardt Tontines with bequest
Introduction
Project“Minimizing longevity and investment risk while optimizing futurepension plans”, improve or find pension products with
high expected lifelong retirement incomelow income variationaccess to underlying capitaldeath benefits and bequest
The Project
Thomas Bernhardt Tontines with bequest 1 / 10
Introduction
bequest = wealth that is given to heirs upon deathannuity puzzle = customers do not buy annuities
(fear of losing to insurance company)
Today’s agenda:
Tontine - Basics, Bequest, Numerics
Explicit solution to an optimization
Time preference (work in progress)
Thomas Bernhardt Tontines with bequest 2 / 10
Tontine - Basics
surrender savings to a group to get mortality credits
Tontine = mortality credits + investment return
everyone has a fund account- decreases with income- increases with mortality
no guarantees
- no cost for risk margins- free to invest
Thomas Bernhardt Tontines with bequest 3 / 10
Tontine - Bequest
allow to choose α how much to surrender?
in the background mortality credits boost wealth and bequest
Amount
520Longevitycredits
500Investment
returnInvestment
return400
Tontineaccount
Bequestaccount
0Consumption Consumption
−25
(a) Before re-balancing.
Amount
460
Tontineaccount
= α×wealth
Bequestaccount
=(1−α)×wealth
0
(b) After re-balancing.
Thomas Bernhardt Tontines with bequest 4 / 10
Tontine - Bequest
allow to choose α how much to surrender?in the background mortality credits boost wealth and bequest
Amount
520Longevitycredits
500Investment
returnInvestment
return400
Tontineaccount
Bequestaccount
0Consumption Consumption
−25
(a) Before re-balancing.
Amount
460
Tontineaccount
= α×wealth
Bequestaccount
=(1−α)×wealth
0
(b) After re-balancing.
Thomas Bernhardt Tontines with bequest 4 / 10
Tontine - Numerics
mathematical description
mortality credits = additional α-weighted stream of income
in a Black-Scholes market and force of mortality λ...
dXt
Xt= r(1− πt)dt + µπtdt + σπtdWt − ctdt+αλtdt
optimization problem including lifespan τ , bequest motive b, andconstant relative risk aversion 1− γ
supα,c,π
E[ ∫ τ
0
U(s, csXs
)ds + b B
(τ, (1− α)Xτ
)]U(s, x) = B(s, x) = e−ρsxγ/γ
P[τ > x ] = exp(−∫ x
0
λsds)
Thomas Bernhardt Tontines with bequest 5 / 10
Tontine - Numerics
low (1− γ) (risk-seeking)
down and upchanges from 0% to 100%
high (1− γ) (risk averse)
above 80%stable for changes in µ, σ, rand slight changes in ρ, λ
Optimal Constant α
0 1 2 3 4
constant relative risk aversion 1 − γ
in t
he t
onti
ne
0%20
%40
%60
%80
%10
0%
b=1b=2b=3b=6b=7
70 80 90 100 110 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Force of mortality
Age (years)
For
ce o
f mor
talit
y at
Age
Thomas Bernhardt Tontines with bequest 6 / 10
Tontine - Numerics
low (1− γ) (risk-seeking)
down and upchanges from 0% to 100%
high (1− γ) (risk averse)
above 80%stable for changes in µ, σ, rand slight changes in ρ, λ
Optimal Constant α
0 1 2 3 4
constant relative risk aversion 1 − γ
in t
he t
onti
ne
0%20
%40
%60
%80
%10
0%
b=1b=2b=3b=6b=7
70 80 90 100 110 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Force of mortality
Age (years)
For
ce o
f mor
talit
y at
Age
Thomas Bernhardt Tontines with bequest 6 / 10
Tontine - Numerics
low (1− γ) (risk-seeking)
down and upchanges from 0% to 100%
high (1− γ) (risk averse)
above 80%stable for changes in µ, σ, rand slight changes in ρ, λ
Optimal Constant α
0 1 2 3 4
constant relative risk aversion 1 − γ
in t
he t
onti
ne
0%20
%40
%60
%80
%10
0%
b=1b=2b=3b=6b=7
70 80 90 100 110 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Force of mortality
Age (years)
For
ce o
f mor
talit
y at
Age
65 70 75 80 85 90 95 100
1520
2530
3540
Consumption rate = 0.09 and α = 0.8
Age (years)
Beq
uest
acc
ount
val
ue a
t Age
Thomas Bernhardt Tontines with bequest 6 / 10
Tontine - Numerics
low (1− γ) (risk-seeking)
down and upchanges from 0% to 100%
high (1− γ) (risk averse)
above 80%stable for changes in µ, σ, rand slight changes in ρ, λ
Optimal Constant α
0 1 2 3 4
constant relative risk aversion 1 − γ
in t
he t
onti
ne
0%20
%40
%60
%80
%10
0%
b=1b=2b=3b=6b=7
70 80 90 100 110 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Force of mortality
Age (years)
For
ce o
f mor
talit
y at
Age
65 70 75 80 85 90 95 100
1520
2530
3540
Consumption rate = 0.09 and α = 0.8
Age (years)
Beq
uest
acc
ount
val
ue a
t Age
0 1 2 3 4
constant relative risk aversion 1 − γ
in t
he t
onti
ne
0%20
%40
%60
%80
%10
0%
b=1b=2b=3b=6b=7
Thomas Bernhardt Tontines with bequest 6 / 10
Tontine - Numerics
low (1− γ) (risk-seeking)
down and upchanges from 0% to 100%
high (1− γ) (risk averse)
above 80%stable for changes in µ, σ, rand slight changes in ρ, λ
Optimal Constant α
0 1 2 3 4
constant relative risk aversion 1 − γ
in t
he t
onti
ne
0%20
%40
%60
%80
%10
0%
b=1b=2b=3b=6b=7
70 80 90 100 110 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Force of mortality
Age (years)
For
ce o
f mor
talit
y at
Age
Thomas Bernhardt Tontines with bequest 6 / 10
Tontine - Numerics
low (1− γ) (risk-seeking)
down and upchanges from 0% to 100%
high (1− γ) (risk averse)
above 80%stable for changes in µ, σ, rand slight changes in ρ, λ
above 80% is high for a constantaverage value or not (!?)
Optimal Constant α
0 1 2 3 4
constant relative risk aversion 1 − γ
in t
he t
onti
ne
0%20
%40
%60
%80
%10
0%
b=1b=2b=3b=6b=7
70 80 90 100 110 120
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Force of mortality
Age (years)
For
ce o
f mor
talit
y at
Age
Thomas Bernhardt Tontines with bequest 6 / 10
Explicit solution
John Dagpunar (forthcoming), ResearchGate:“Closed-form Solutions foran Explicit Modern Ideal Tontine with Bequest Motive”
supα,c,π
E[ ∫ τ
0
e−ρs(csXs)γ/γ ds + b e−ρs((1− ατ )Xτ )γ/γ]
HJB yields explicitly solvable ODE’s when α = αt
1− αt =b
11−γ Sβ(t)∫∞
t(1 + λ(s)b
11−γ )Sβ(s)ds
b bequest motive, λ force of mortality, Sβ(t) = exp(−∫ t
0λ(s) + β ds)
and β = r + ρ−r1−γ −
γ2 ( 1
1−γ )2(µ−rσ )2
be aware
restriction 1− b1
1−γ β > 0
nice interpretation b = ( monetary bequestmonetary consumption )risk aversion
Thomas Bernhardt Tontines with bequest 7 / 10
Explicit solution
at fixed times like before
up and down for low (1− γ)stability for high (1− γ)
decreasing function
for all r , µ, σ and λ, ρ and b, γα high when death unlikelybest decision in likely events
amplifies annuity puzzle
punishes early deathdouble reward for living longer(highest bequest %,most mortality credit)
Optimal α over time
70 80 90 100 110
age
in the tontine
0%
20%
40%
60%
80%
100%
b=1
b=2
b=3
b=6
b=7
Thomas Bernhardt Tontines with bequest 8 / 10
Time preference (work in progress)
Reciprocal relation (λ ↓ bequest motive ↑, insurance in unlikely events)and α0 = 0 (nothing for nothing in return)
supα,c,π
E[ ∫ τ
0
e−ρs(csXs)γ/γ ds + e−ρs(b/λ(τ))|γ|((1− ατ )Xτ )γ/γ]
HJB yields explicitly solvable ODE’s
1− αt =(b/λ(t))
|γ|1−γ Sβ(t)∫∞
t(1 + λ(s)(b/λ(s))
|γ|1−γ )Sβ(s)ds
with b chosen so that α0 = 0
no restrictions
b is a result not a parameter
Thomas Bernhardt Tontines with bequest 9 / 10
Time preference (work in progress)
increasing function (special caser = 0, γ ↓ −∞)
link to famous mean excessloss function e
1− αt =λ(0) e(0)
λ(t) e(t)
λe increasing ⇐⇒ αincreasing
e convex ⇒ α increasing
forthcoming research:
is the picture right now?
implications for annuities?
Optimal α over time
70 80 90 100 110
age
in the tontine
0%
20%
40%
60%
80%
100%
initial age=65
initial age=75
initial age=85
Thank you very much!Any questions or feedback?
Thomas Bernhardt Tontines with bequest 10 / 10
Time preference (work in progress)
increasing function (special caser = 0, γ ↓ −∞)
link to famous mean excessloss function e
1− αt =λ(0) e(0)
λ(t) e(t)
λe increasing ⇐⇒ αincreasing
e convex ⇒ α increasing
forthcoming research:
is the picture right now?
implications for annuities?
Optimal α over time
70 80 90 100 110
age
in the tontine
0%
20%
40%
60%
80%
100%
initial age=65
initial age=75
initial age=85
Thank you very much!Any questions or feedback?
Thomas Bernhardt Tontines with bequest 10 / 10
Time preference (work in progress)
increasing function (special caser = 0, γ ↓ −∞)
link to famous mean excessloss function e
1− αt =λ(0) e(0)
λ(t) e(t)
λe increasing ⇐⇒ αincreasing
e convex ⇒ α increasing
forthcoming research:
is the picture right now?
implications for annuities?
Optimal α over time
70 80 90 100 110
age
in the tontine
0%
20%
40%
60%
80%
100%
initial age=65
initial age=75
initial age=85
Thank you very much!Any questions or feedback?
Thomas Bernhardt Tontines with bequest 10 / 10
