619e71e53bd61b3401cd558d_ETH Zurich 2020 Paper on Tontines.pdf

Introduction The Tontine The Model Results Conclusion References Backup
Optimal Portfolio Choice in Retirement
with Natural Tontines and
Systematic Longevity Risk
Irina Gemmo∗
, Ralph Rogalla∗∗
, Jan-Hendrik Weinert∗∗∗
∗
ETH Zurich
∗∗
St. John’s University New York
∗∗∗
Viridium Group
January 17, 2020
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 0 / 28

The Tontine Principle
http://www.simpsonsworld.com/video/320768579741

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Literature Review
Historic development of tontines
McKeever (2009), Milevsky (2015): Use-cases in the past
I Financing the French public sector deficit in the 1650s
I Feudal lords in the middle ages secured their servants with tontines
Li and Rothschild (2019): Adverse selection in the Irish tontines in
the 1770s
Actuarial fair price and optimal payout structure of tontines
Milevsky and Salisbury (2015): optimal tontine payout structure is
flat
Sabin (2010), Milevsky and Salisbury (2016): Mixing different
cohorts in a tontine

Literature Review
Tontine product design
Chen et al. (2019): ”Tonuity” - product that switches from tontine
to annuity at an optimal point
Weinert (2017a): analyzes the implicit costs of a tontine and the
regulatory treatment of tontines
Weinert (2017b): equips the tontine with a surrender option
Tontines in retirement planning
Weinert and Gründl (2016): The suitability of tontines to cover the
”retirement smile” complementary to annuities

The Retirement Smile
Natural tontine vs. tontine annuity
return
age

Average funding requirements in old age are U-shaped (”retirement
smile”)
60 70 80 90 100
0
20,000
40,000
60,000
age
EUR
liquidity need
5/95% Q. forecast
Figure: Liquidity need per age, based on SOEP data for 1984-2013

old age spending categories
60 65 70 75 80 85 90 95
age
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
food
living
health
care
leisure
mobility
other
refurbishment
Consumer spending decreases with age
Need for medical services, care, support in everyday life increase
strongly
The pension gap from the statutory pension insurance will increase
in the coming years

Motivation
Why invest in a (natural) tontine for old-age provision?
Age-increasing payouts can match retirement smile
Can have lower loadings than standard annuities because aggregate
longevity risk stays in the pool (less regulation)
Payout is independent of care level (in contrast to long-term care
products)
Tontines allow an open use of funds such as
I conversion of the apartment (e.g. ground-level bathroom or stair lift)
I financing costly items to maintain the standard of living (e.g.
increased taxi travel, the use of shopping delivery services)
I high-quality care services beyond the level covered by insurance (e.g.,
massages, domestic help)

Research questions
What is the optimal tontine investment pattern in retirement?
What are the effects of risk aversion and bequest motive on the
investment decisions?
What are the welfare effects of the tontine and how do they differ
with risk aversion, tontine size, and bequest motive?

The Natural Tontine
t − 1 t
Nt−1 Nt
Revolving one year tontine setup
Homogeneous tontine members
Risk free invested tontine funds
Return in t per unit invested if one survives: Rf
Nt−1/Nt

The Natural Tontine
0%
20%
105
40%
60%
95
80%
expected
value
100%
40
(a) expected mortality credit per unit invested
age
120%
85
140%
tontine members
20
75
65 0
0%
105
20%
40%
95
60%
standard
deviation
40
80%
(b) standard deviation mortality credit per unit invested
age
85
100%
tontine members
20
75
65 0
Figure: Moments of the tontine
Risky investment with age increasing mean and age and
inverse-of-tontine size increasing standard deviation
Lower bound mortality credit is zero

The Natural Tontine
Figure: Consumption smoothing using tontines
No tontine annuity needed to smooth consumption

The Lifecycle Model
Investor has access to capital markets by investing in risk-free bonds
(B), risky stocks (S) and risky tontines (Υ)
Initial wealth endowment of W0
The individual can spread wealth on hand Wt across the capital
markets and consumption Ct:
Wt = Bt + St + Υt
| {z }
financial wealth
+Ct
Disposable wealth on hand in t + 1 is
Wt+1 = BtRf + StR•
t+1 + ΥtR◦
t+1
| {z }
value of financial wealth in t+1
+ Yt+1
|{z}
retirement
income in t+1
Yt+1 is exposed to an empirically calibrated increasing liquidity need
with P =

0.97 0.03
0 1

, which lowers the pension income (and can
even lead to negative pension income)

The Lifecycle Model
Short selling is not allowed
Bt, St, Υt ≥ 0.
Recursive definition of the value function in t
Vt = u (Ct)
| {z }
utility of con-
sumption in t
+β





pt Et (Vt+1)
| {z }
expected total
utility in t+1
+ (1 − pt) Et (u (Dt+1))
| {z }
expected utility
of bequest





pt: survival probability from t to t + 1
β: subjective discount factor
Bequest in t + 1 is Dt+1 = BtRf + StR•
t+1

Mortality Dynamics
Cairns, Blake, Dowd (2006):
logit qt
x

= log

qt
x
1 − qt
x

= κt
1 + κt
2 (x − x̄)
x: age, t: year
x̄: mean age
κt
1: intercept term
κt
2: slope term
Future stochastic simulations: κt
i , i = 1, 2 follow correlated random
walks
Calibrated based on Human Mortality Database for U.S. males ages
60-105 over the period 1933-2014

Mortality Dynamics
66 75 85 95 105
age
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
conditional
survival
probability
at
age
66
Simulated survival Probability, CBD model
Figure: Simulated distribution of age 66 male t-period survival probabilities
(99%:1%) based on CBD mortality model (N = 10,000 simulations). Darker
areas represent higher probability mass.

Calibration
Variable Description Value Source
N0 Initial tontine size 200;1K;10K
u (·) Utility function CRRA Horneff et al. (2010)
β Subjective discount factor 0.98 Weinert and Gründl (2016)
t Time 0...40 Horneff et al. (2010)
ω̄ Entry age 65 Horneff et al. (2010)
ρ Degree of risk aversion
(consumption)
1, 4, 8 Horneff et al. (2010)
γ Degree of risk aversion
(bequest)
1, 4, 8
Rf Risk free bond return 1.02 Hubener et al. (2014)
µ•
Expected stock return 1.06 Horneff et al. (2010)
σ•
Volatility of stock return 0.18 Horneff et al. (2010)
pt survival probability CBD Cairns et al. (2006)
W0 initial wealth endowment 150,000 e
b bequest motive 0,1
Table: Model calibration

Base Case
(a) absolute wealth composition
66 75 85 95 105
age
0
1
2
3
4
5
6
in
100k
EUR
tontine bond stock consumption savings wealth quartiles
(b) consumption vs savings
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
(c) portfolio composition
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
Figure: Large initial tontine size (N0 = 10, 000), with bequest motive (b = 1),
initial wealth endowment W0 = 150, 000 EUR, medium CRRA risk aversion
parameter (ρ = γ = 4)
Tontine investment increases with age, crowding out bonds
Disinvest tontine towards terminal age because of bequest, shift
towards safe assets

No Bequest
66 75 85 95 105
age
0
1
2
3
4
5
6
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
Figure: Large initial tontine size (N0 = 10, 000), without bequest motive
(b = 0), initial wealth endowment W0 = 150, 000 EUR, medium CRRA risk
aversion parameter (ρ = 4)
Increase tontine investment with age and consume everything in the
final period

Low Risk Aversion
66 75 85 95 105
age
0
2
4
6
8
10
12
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
initial wealth endowment W0 = 150, 000 EUR, low CRRA risk aversion
Invest large fractions in risky stocks
Increase tontine investment in late years
Go out of the tontine because of bequest and replace it with risky
stock again

High Risk Aversion
66 75 85 95 105
age
0
1
2
3
4
5
6
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
initial wealth endowment W0 = 150, 000 EUR, high CRRA risk aversion
parameter (ρ = γ = 8).
High tontine investment from retirement age on
Disinvest tontine towards terminal age because of bequest, shift
towards safe assets

Higher risk aversion for consumption
than for and bequest
66 75 85 95 105
age
0
1
2
3
4
5
6
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
initial wealth endowment W0 = 150, 000 EUR, medium/low CRRA risk
aversion parameter (ρ = 4/γ = 1) for consumption/bequest.
Compared to base case: no shift towards safe assets for bequest

Tontine Equivalent Wealth (preliminary)
What are the utility implications of having access to tontines?
Tontine Equivalent Wealth (TEW)
I Additional initial capital required by investor in no-tontine world to
have the same lifetime utility as investor with access to tontines and
intial wealth W0
Formally:
TEW := W NoTontine
0 | V NoTontine
0 (W NoTontine
0 , ...) = V0(W0, ...)

TEW (% of W0) increases in tontine size
I 164.87% (ρ = γ = 4, b = 1, N0 = 200)
I 168.87% (ρ = γ = 4, b = 1, N0 = 10, 000)
→ less volatile tontine is valued by the investors
TEW (% of W0) increases in risk aversion
I 104.13% (ρ = γ = 1, b = 1, N0 = 200)
I 164.87% (ρ = γ = 4, b = 1, N0 = 200)
→ the better risk return profile of the tontine (compared to the
stock) is valued higher in increasing risk aversion

For low risk aversion, TEW (% of W0) decreases in bequest
I 112.52% (ρ = γ = 1, b = 0, N0 = 10, 000)
I 105.69% (ρ = γ = 1, b = 1, N0 = 10, 000)
→ Tontine does not contribute to bequest
For high risk aversion, TEW (% of W0) first increases in bequest
and decreases thereafter
I 187.87% (ρ = γ = 8, b = 0, N0 = 10, 000)
I 193.82% (ρ = γ = 8, b = 1, N0 = 10, 000)
I 184.35% (ρ = γ = 8, b = 3, N0 = 10, 000)
Favorable risk return profile of the tontine allows to accumulate
more wealth which can partially be bequeathed in future periods.
(↑) The stronger bequest motive, the less suitable is the tontine (↓)

Conclusion
Natural tontine provides age increasing payout structure
Co-movement with age increasing liquidity need in the frailty state
Natural tontines can help to mitigate the old age underfunding
problem through pooling of mortality risk
Broader fund usage than common LTC-products, allows to finance
”soft aspects”
Tontine investment increases in risk aversion
Bequest reduces tontine investment
Tontine size plays a role if it is too small

Outlook
Tontine investment in stocks instead of bonds
Include LTC-product
I alleviation of the old age liquidity need
More realistic frailty state modeling (Shao et al., 2018)
Include an annuity
Measure the difference of CBD compared to GoMa

Thank you for your attention

Cairns, A. J., Blake, D., and Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: theory and
calibration. Journal of Risk and Insurance, 73(4):687–718.
Chen, A., Hieber, P., and Klein, J. K. (2019). Tonuity: A novel individual-oriented retirement plan. ASTIN Bulletin: The Journal of the
IAA, 49(1):5–30.
Horneff, W., Maurer, R., and Rogalla, R. (2010). Dynamic portfolio choice with deferred annuities. Journal of Banking Finance,
34(11):2652–2664.
Hubener, A., Maurer, R., and Rogalla, R. (2014). Optimal portfolio choice with annuities and life insurance for retired couples. Review of
Finance, 18(1):147–188.
Li, Y. and Rothschild, C. (2019). Adverse selection and redistribution in the irish tontines of 1773, 1775 and 1777. Journal of Risk and
Insurance.
McKeever, K. (2009). Short history of tontines, a. Fordham J. Corp. Fin. L., 15:491.
Milevsky, M. A. (2015). King William’s tontine: why the retirement annuity of the future should resemble its past. Cambridge University
Press.
Milevsky, M. A. and Salisbury, T. S. (2015). Optimal retirement income tontines. Insurance: Mathematics and Economics, 64:91–105.
Milevsky, M. A. and Salisbury, T. S. (2016). Equitable retirement income tontines: Mixing cohorts without discriminating. ASTIN
Bulletin: The Journal of the IAA, 46(3):571–604.
Sabin, M. J. (2010). Fair tontine annuity. Social Science Research Network Working Paper Series.
Shao, A. W., Chen, H., and Sherris, M. (2018). To borrow or insure? long term care costs and the impact of housing.
Weinert, J.-H. (2017a). Comparing the cost of a tontine with a tontine replicating annuity. ICIR Working Paper Series, (31/2017).
Weinert, J.-H. (2017b). The fair surrender value of a tontine. ICIR Working Paper Series, (26/2017).
Weinert, J.-H. and Gründl, H. (2016). The modern tontine: An innovative instrument for longevity risk management in an aging society.
ICIR Working Paper Series, (22/2016).

Expected tontine return
Expected tontine return in t based on the information available in
t − 1 if the individual is alive in t is
µ◦
t = Et−1

Nt−1
Nt
Rf

= Rf Nt−1Et−1

1
N̂t + 1

where Nt ∼ Bin (Nt−1 − 1, pt) + 1 and N̂t ∼ Bin

N̂t−1, pt

with
N̂t−1 = Nt−1 − 1.
µ◦
t = Rf Nt−1
N̂t−1
X
k=0
1
1 + k

N̂t−1
k

pt
k
(1 − pt)(N̂t−1−k)
= Rf Nt−1
1

N̂t−1 + 1

pt
N̂t−1
X
k=0

N̂t−1 + 1
k + 1

pt
(k+1)
(1 − pt)(N̂t−1−k) .

Expected tontine return (cont)
By substituting l = k + 1
µ
◦
t = Rf Nt−1
1

N̂t−1 + 1

pt
N̂t−1
X
l=1
N̂t−1 + 1
l

pt
l
(1 − pt )

N̂t−1−l+1

= Rf Nt−1
1

N̂t−1 + 1

pt




N̂t−1+1
X
l=0
N̂t−1 + 1
l

pt
l
(1 − pt )

N̂t−1−l+1

− (1 − pt )

N̂t−1+1





= Rf Nt−1
1

N̂t−1 + 1

pt

(pt + 1 − pt )

N̂t−1+1

− (1 − pt )

N̂t−1+1
#
= Rf Nt−1
1

N̂t−1 + 1

pt

1 − (1 − pt )

N̂t−1+1
#
= Rf Nt−1
1 − (1 − pt )

Nt−1−1+1

Nt−1 − 1 + 1

pt
=
1 − (1 − pt )
Nt−1
pt
Rf .

Variance of tontine return
Variance of tontine return in t based on the information available in
t − 1 if the individual is alive in t is
(σ◦
t )
2
= Var

Nt−1
Nt
Rf

= Rf
2
Nt−1
2
Var

1
N̂t + 1

where Nt ∼ Bin (Nt−1 − 1, pt) + 1 and N̂t ∼ Bin

N̂t−1, pt

with
N̂t−1 = Nt−1 − 1. From the linearity of expected values and the
definition of variance follows
(σ◦
t )
2
= Rf
2
Nt−1
2


E



1

N̂t + 1
2


 − E

1
N̂t + 1
2



= Rf
2
Nt−1
2


E



1

N̂t + 1
2


 −

1 − (1 − pt)
N̂t−1+1

N̂t−1 + 1

pt


2


 .

Variance of tontine return (cont)
E

1
(N̂t +1)
2

can be written as
E



1

N̂t + 1
2


 =
N̂t−1
X
k=0
1
(1 + k)2
N̂t−1
k

pt
k
(1 − pt )

N̂t−1−k

=
N̂t−1
X
k=0
k + 2
k + 1
1
N̂t−1 + 1
1
N̂t−1 + 2
N̂t−1 + 2
k + 2

pt
k
(1 − pt )

N̂t−1−k

= (1 − pt )
N̂t−1
i Fj (x; y; z)
where i Fj (x; y; z) is the the generalized hypergeometric function with
x = {x1, . . . , xi } = {1, 1, − (Nt−1 − 1)}, y = {y1, . . . , yj } = {2, 2} and
z = pt
pt −1 . Finally,

σ
◦
t
2
= Rf
2
Nt−1
2


(1 − pt )

Nt−1−1

3F2 1, 1, −

Nt−1 − 1

; 2, 2;
pt
pt − 1
!
−


1 − (1 − pt )
Nt−1
Nt−1pt


2


 .

No bequest, high risk aversion
66 75 85 95 105
age
0
1
2
3
4
5
6
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
initial wealth endowment W0 = 150, 000 EUR, high CRRA risk aversion
parameter (ρ = 8).

No bequest, small tontine
66 75 85 95 105
age
0
1
2
3
4
5
6
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
Figure: Small initial tontine size (N0 = 200), without bequest motive (b = 0),

Higher risk aversion for bequest
than for consumption
66 75 85 95 105
age
0
1
2
3
4
5
6
7
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
parameter (ρ = 4) for consumption and high CRRA risk aversion parameter
(γ = 8) for bequest.

High initial wealth endowment
66 75 85 95 105
age
0
1
2
3
4
5
6
7
8
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%

Low initial wealth endowment
66 75 85 95 105
age
0
1
2
3
4
5
6
in
100k
EUR
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%
66 75 85 95 105
age
0%
20%
40%
60%
80%
100%

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