daniel bauer [email protected] ulm university dfg research training group 1100 frederik weber...
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Daniel [email protected] UniversityDFG Research Training Group 1100
Frederik [email protected]ät MünchenInstitute for Risk and Insurance Management
3rd Intl Longevity Risk & Capital Market Solutions SymposiumTaipei 2007
Assessing Investment and Longevity Risks within Immediate Annuities
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 2
Agenda
1 Introduction and Motivation
2 Model Setup and Assumptions
3 Results
4 Summary and Conclusion
5 Outlook: Further research
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 3
Introduction and Motivation
• Longevity trends in numerous countries…
• …may pose serious problems for annuity providers.
• Frequent perspective:
- evaluation of annuity products from insured persons‘ point of view (Yaari 1965, Mitchell et al. 1999, Davidoff et al. 2005, Milevsky et al. 2006)
- less often: profitability/riskiness of annuity books (Dowd et al. 2006)
• Common claims:
- longevity risk independent of investment risks and far smaller (Persson et al. 1998, Richards and Jones 2004)
• Missing research:
- joint investigation of capital market and longevity risks:How large or influential are these risks?
- assessment of annuity provider‘s financial position:How risky is the annuity business?
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 4
Model Setup and Assumptions
• Simulate a UK annuity provider‘s position – as realistic as possible:
- Consider a cohort of N males aged x0=60, all buying immediate annuitiesfor typical single premium charged in the UK market in 2005.
- Each annuity paid annually in arrears, until insured actually dies; upon survival beyond age 100 lump sum instead of further payments.
- Provider annually charges realistic fees and expenses.
• Premiums/assets invested pursuing two strategies:
1. annual (re)investment of assets into 1/3 stocks + 2/3 savings account
2. upfront hedging with bundle of bonds (maturing in 1 – 40 years), proportioned to fit expected survival; loss/excess amount financed/invested via portfolio as above
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 5
Model Setup and Assumptions
• Assessing reserve values/surplus distributions…
- individual account at end of year 1
- individual account at end of year t>1
- in large portfolios: reserve at t
• … by looking at reserves R10, R20, and surplus R40
}1{101 1)1)(( PA
}{11 1)1)(( ttttt AA
0)1)(( 11 xttttt pRR
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 6
Model Setup and Assumptions
• Simple, common stochastic models calibrated to 20 years of real data
- mortality: Lee-Carter/Poisson log-bilinear approach (Brouhns et al. 2002);calibrated to male mortality data for England/Wales pop. & UK Pensioners
- interest rates: Cox-Ingersoll-Ross; calibrated to 3-Months-LIBOR
- stocks: geometric Brownian motion; calibrated to FTSE100 index
• …produced 20,000 random paths from which we sampled
- arbitrary combinations
- 10% best/worst*) capital market paths + arbitrary mortality paths and v.v.
*) “good“: low life expectancy or high average rate of return“bad“: high life expectancy or low average rate of return
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 7
ResultsDifferences by selection effects
• bond-hedging, t=10,20,40 years; population vs. pensioners‘ mortality per capita
0
0,005
0,01
0,015
0,02
0,025
0 100 200 300 400 500R40
rel.
fre
q.
Population
Pensioners
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 8
ResultsDifferences by selection effects
• bond-hedging, t=10,20,40 years; population vs. pensioners‘ mortality per capita
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
0 5 10 15 20 25 30 35 40 45 50
R10
rel.
freq
.
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
0 5 10 15 20 25 30 35 40 45 50
R20
rel.
freq
.
0
0,005
0,01
0,015
0,02
0,025
0 100 200 300 400 500R40
rel.
fre
q.
Population
Pensioners
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 9
ResultsConditioning on best/worst paths
• bond hedging, t=40 years, pensioners‘ mortality per capitaarbitrary sampling vs. 10% best/worst capital market or mortality paths
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0 100 200 300 400 500
R40
rel.
fre
q.
arbitrary sampling
10% best mortality paths
10% worst mortality paths
10% best capital market paths
10% worst capital market paths
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 10
ResultsConditioning on best/worst paths
• bond hedging, t=40 years, pensioners‘ mortality per capitaarbitrary sampling vs. 10% best/worst capital market or mortality paths
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0 100 200 300 400 500
R40
rel.
fre
q.
arbitrary sampling
10% best mortality paths
10% worst mortality paths
10% best capital market paths
10% worst capital market paths
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 11
ResultsConditioning on best/worst paths
• bond hedging, t=40 years, pensioners‘ mortality per capitaarbitrary sampling vs. 10% best/worst capital market or mortality paths
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0 100 200 300 400 500
R40
rel.
fre
q.
arbitrary sampling
10% best mortality paths
10% worst mortality paths
10% best capital market paths
10% worst capital market paths
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 12
ResultsDifferences by investment strategy
• t=40 years, pensioners‘ mortality per capita, arbitrary samplingopportunistic investment vs. bond hedging
ruin prob. 0.00%
0
0,005
0,01
0,015
0,02
0,025
-100 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
R40
rel.
fre
q.
bond hedging
opport. investment
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 13
ResultsDifferences by investment strategy
• t=40 years, pensioners‘ mortality per capita, arbitrary samplingopportunistic investment vs. bond hedging
ruin prob. 0.00%
ruin prob.0.76%
0
0,005
0,01
0,015
0,02
0,025
-100 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
R40
rel.
fre
q.
bond hedging
opport. investment
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 14
ResultsBad developments with no bond-hedging
• t=40 years, pensioners‘ mortality per capita, no bond-hedging availableinfluence of 10% worst mortality/capital market developments
0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
0,04
-100 0 100 200 300 400 500
R40
rel.
freq
.
R_40 - bad mortality
R_40 - bad capital market
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 15
ResultsBad developments with no bond-hedging
• t=40 years, pensioners‘ mortality per capita, no bond-hedging availableinfluence of 10% worst mortality/capital market developments
Caveat:Negative reserves are possible in our model; shortfall would be financed by borrowing against stocks/savings account portfolio – instead of borrowing at prevailing interest rate.
0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
0,04
-100 0 100 200 300 400 500
R40
rel.
freq
.
R_40 - bad mortality
R_40 - bad capital market
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 16
Summary and Conclusion
• Hedging makes it safe. Under bond hedging, negative reserves/surplus did not occur. Under opportunistic investments, defaults are possible but rare.
• Selection effects are strong. If pensioners‘ (vs. population) mortality is considered results are notably worse but also less volatile.
• Results for reserves R10, R20 show same tendency as surplus R40 but are less pronounced.
• Conditioning on good/bad mortality yields results relatively close to “unconditional“ results, but the spread in the surplus (R40) may reach 5 per unit.
• Distribution of reserves/surplus conditioned on good/bad capital market is considerably different: “bad“ results are smaller, yet less volatile.
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 17
Summary and Conclusion
• Longevity risk is obviously smaller than investment risk,but cannot yet be hedged with market instruments.
• Longevity risk may be less serious in terms of short-falls.Instead, fluctuations of reserves/surplus generate uncertainty.
• Is longevity risk neglectible? Clearly not…
- Though smaller than investment risk, longevity risk itself may cause considerable spreads of provider‘s surplus situation.
- Results indicate existence of (high?) “transaction costs“.
- Availability of instruments for hedging longevity risk may further improve annuity providers‘ position and force to offer lower prices.
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 18
Outlook: Further research
• consider other, better (?) models for mortality and capital markets
• incorporate longevity bonds to hedge longevity risk:
- Does the provider‘s financial position improve further?
- Can annuities be offered at lower prices without exposition to ruin risk?What is the cheapest price with sufficiently low ruin probability?
• investigate mortality (and capital markets) in other countries:
- Are the results similar?
- Is the annuity business equally risky?
- Does longevity risk develop in a different way?
Institute for Riskand Insurance Management
Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 19
Assessing Investment and Longevity Risks within Immediate Annuities
Thank you for your attention!
Any questions, remarks etc. are greatly appreciated.