This document discusses optimal portfolio choice in retirement using natural tontines and systematic longevity risk. It introduces tontines as a retirement product that can match the "retirement smile" funding pattern. The model examines optimal investment in risk-free bonds, risky stocks, and risky tontines over a lifecycle, accounting for mortality risk, risk preferences, and bequest motives. Results show increasing tontine investment with age, crowding out bonds, with disinvestment toward end of life if there is a bequest motive.
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The aim of this paper is to answer the following questions:
are risk management practices (RMP) inefficient or beneficial to wheat farming productivity? And if they are beneficial (as advocated by many authors), how much they contribute to
wheat farming productivity when a natural disaster or a market risk occur?
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The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
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These findings sharply contrast with the custom, popular in macroeconomics, that consists in calibrating the output elasticity of energy according to the cost share of energy. In most countries, this practice leads to the postulate that energy elasticity should be close to 0.08% on average.
Poster presentation given by Mauro Vigani at the recent ICAE conference in Milan.
The aim of this paper is to answer the following questions:
are risk management practices (RMP) inefficient or beneficial to wheat farming productivity? And if they are beneficial (as advocated by many authors), how much they contribute to
wheat farming productivity when a natural disaster or a market risk occur?
Conditional probabilities for euro area sovereign default risk - Andre Lucas,...SYRTO Project
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The dangers of policy experiments Initial beliefs under adaptive learningGRAPE
The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
Should Commodity Investors Follow Commodities' Prices?guasoni
Most institutional investors gain access to commodities through diversified index funds, even though mean-reverting prices and low correlation among commodities returns indicate that two-fund separation does not hold for commodities. In contrast to demand for stocks and bonds, we find that, on average, demand for commodities is largely insensitive to risk aversion, with intertemporal hedging demand playing a major role for more risk averse investors. Comparing the optimal strategies of investors who observe only the index to those of investors who observe all commodities, we find that information on commodity prices leads to significant welfare gains, even if trading is confined to the index only.
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The work of the Shift Project focuses on the link between energy consumption and the GDP growth. The econometrical research confirms the standpoint defended by ecological economists and introducing primary energy as a key factor that drives GDP growth. The results show that an increase of 10% of energy use per capita induces, on average, an increase (resp. decrease) of about 6 to 7% of GDP per capita. The research also concludes that the causality relation goes from the consumption of energy to growth in both the short and long-run.
These findings sharply contrast with the custom, popular in macroeconomics, that consists in calibrating the output elasticity of energy according to the cost share of energy. In most countries, this practice leads to the postulate that energy elasticity should be close to 0.08% on average.
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Dierent methods have been used in the literature to mesure and analyze price markup cyclical behavior.
We use a medium-scale DSGE Model with positive trend in
ation, in which aggregate
uctuations are driven
by neutral technology, MEI and monetary policy shocks and, where both price and wage markups vary. We
nd that when raising trend in
ation from 0 to 4 percent, wage markup is more important than price markup
in explaining the dynamics eects of shocks.
Estimating Financial Frictions under LearningGRAPE
The paper studies the implication of initial beliefs and associated confidence under adaptive learning. We first illustrate how prior beliefs determine learning dynamics and the evolution of endogenous variables in a small DSGE model with credit-constrained agents, in which rational expectations are replaced by constant-gain adaptive learning. We then examine how discretionary experimenting with new macroeconomic policies is affected by expectations that agents have in relation to these policies. More specifically, we show that a newly introduced macro-prudential policy that aims at making leverage counter-cyclical can lead to substantial increase in fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect information about the policy experiment.
Shock is a common complication of severe febrile illness, and worldwide aggressive correction with intravenous bolus therapy is recommended as the initial treatment. Nevertheless, the evidence supporting this approach remains weak. The only controlled trial of fluid resuscitation, Fluid Expansion as Supportive Therapy (FEAST), involving 3141 African children with severe febrile illness, including large groups with sepsis and malaria, called into question aggressive fluid resuscitation, demonstrating excess mortality in both bolus arms (albumin and saline) compared to no-bolus control, relative risk of morality in bolus versus control was 1.45(1.13-1.86, p=0.003). Excess mortality was consistent across all subgroups, being greatest in those with the most severe forms of shock and acidosis. Remarkably, despite earlier shock reversal in those receiving fluid boluses the excess mortality in the FEAST trial was caused by subsequent cardiovascular collapse and was not secondary to fluid overload.
These observations are intriguing warranting an in-depth understanding of host responses including those of the myocardium to fluid resuscitation and at the microvasacular level since the two maybe synergistic. Current studies are underway in ovine models of sepsis (‘FEAST-in-Sheep’) in Professor John Fraser’s laboratory, Brisbane to understand the mechanism of harm, gain further insights in host responses to fluid management, and re-define the optimal fluid and supportive inotrope/vasopressor management of septic shock.
Four years have elapsed since the publication of FEAST, yet World Health Organization continues to recommend fluid boluses for children managed in resource-poor hospitals, where there is no access to intensive care. These are the precise settings where the FEAST trial was conducted in order to inform management guidelines. In Africa alone, where one in 10 febrile child admissions present with shock, we have estimated that the current guidelines, if fully implemented, will result in ~5,600 and 33,000 excess deaths each year per million hospital admissions treated for shock.
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Green accounting is a type of accounting that attempts to factor environmental costs into the financial results of operations. It has been argued that gross domestic product ignores the environment and therefore policymakers need a revised model that incorporates green accounting.
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619e71e53bd61b3401cd558d_ETH Zurich 2020 Paper on Tontines.pdf
1. 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
2. Introduction The Tontine The Model Results Conclusion References Backup
The Tontine Principle
http://www.simpsonsworld.com/video/320768579741
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 1 / 28
3. Introduction The Tontine The Model Results Conclusion References Backup
The Tontine Principle
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Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 2 / 28
4. Introduction The Tontine The Model Results Conclusion References Backup
The Tontine Principle
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Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 3 / 28
5. Introduction The Tontine The Model Results Conclusion References Backup
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 4 / 28
6. Introduction The Tontine The Model Results Conclusion References Backup
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 5 / 28
7. Introduction The Tontine The Model Results Conclusion References Backup
The Retirement Smile
Natural tontine vs. tontine annuity
return
age
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 6 / 28
8. Introduction The Tontine The Model Results Conclusion References Backup
The Retirement Smile
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 7 / 28
9. Introduction The Tontine The Model Results Conclusion References Backup
The Retirement Smile
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 8 / 28
10. Introduction The Tontine The Model Results Conclusion References Backup
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)
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 9 / 28
11. Introduction The Tontine The Model Results Conclusion References Backup
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?
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 10 / 28
12. Introduction The Tontine The Model Results Conclusion References Backup
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 11 / 28
13. Introduction The Tontine The Model Results Conclusion References Backup
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 12 / 28
14. Introduction The Tontine The Model Results Conclusion References Backup
The Natural Tontine
Figure: Consumption smoothing using tontines
No tontine annuity needed to smooth consumption
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 13 / 28
15. Introduction The Tontine The Model Results Conclusion References Backup
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)
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 14 / 28
16. Introduction The Tontine The Model Results Conclusion References Backup
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 15 / 28
17. Introduction The Tontine The Model Results Conclusion References Backup
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
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18. Introduction The Tontine The Model Results Conclusion References Backup
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.
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 17 / 28
19. Introduction The Tontine The Model Results Conclusion References Backup
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
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20. Introduction The Tontine The Model Results Conclusion References Backup
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
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21. Introduction The Tontine The Model Results Conclusion References Backup
No Bequest
(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), 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
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22. Introduction The Tontine The Model Results Conclusion References Backup
Low Risk Aversion
(a) absolute wealth composition
66 75 85 95 105
age
0
2
4
6
8
10
12
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, low CRRA risk aversion
parameter (ρ = γ = 1)
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 21 / 28
23. Introduction The Tontine The Model Results Conclusion References Backup
High Risk Aversion
(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, 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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 22 / 28
24. Introduction The Tontine The Model Results Conclusion References Backup
Higher risk aversion for consumption
than for and bequest
(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/low CRRA risk
aversion parameter (ρ = 4/γ = 1) for consumption/bequest.
Compared to base case: no shift towards safe assets for bequest
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25. Introduction The Tontine The Model Results Conclusion References Backup
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, ...)
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26. Introduction The Tontine The Model Results Conclusion References Backup
Tontine Equivalent Wealth (preliminary)
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
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27. Introduction The Tontine The Model Results Conclusion References Backup
Tontine Equivalent Wealth (preliminary)
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 (↓)
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 26 / 28
28. Introduction The Tontine The Model Results Conclusion References Backup
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
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29. Introduction The Tontine The Model Results Conclusion References Backup
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
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 28 / 28
30. Introduction The Tontine The Model Results Conclusion References Backup
Thank you for your attention
Gemmo, Rogalla, Weinert - Optimal PF Choice with Tontines 28 / 28
31. Introduction The Tontine The Model Results Conclusion References Backup
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).
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32. Introduction The Tontine The Model Results Conclusion References Backup
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) .
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34. Introduction The Tontine The Model Results Conclusion References Backup
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
.
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35. Introduction The Tontine The Model Results Conclusion References Backup
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
.
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36. Introduction The Tontine The Model Results Conclusion References Backup
No bequest, high risk aversion
(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, high CRRA risk aversion
parameter (ρ = 8).
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37. Introduction The Tontine The Model Results Conclusion References Backup
No bequest, small tontine
(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: Small initial tontine size (N0 = 200), without bequest motive (b = 0),
initial wealth endowment W0 = 150, 000 EUR, medium CRRA risk aversion
parameter (ρ = γ = 4)
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38. Introduction The Tontine The Model Results Conclusion References Backup
Higher risk aversion for bequest
than for consumption
(a) absolute wealth composition
66 75 85 95 105
age
0
1
2
3
4
5
6
7
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 = 4) for consumption and high CRRA risk aversion parameter
(γ = 8) for bequest.
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39. Introduction The Tontine The Model Results Conclusion References Backup
High initial wealth endowment
(a) absolute wealth composition
66 75 85 95 105
age
0
1
2
3
4
5
6
7
8
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 = 300, 000 EUR, medium CRRA risk aversion
parameter (ρ = γ = 4).
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40. Introduction The Tontine The Model Results Conclusion References Backup
Low initial wealth endowment
(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 = 50, 000 EUR, medium CRRA risk aversion
parameter (ρ = γ = 4).
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