daniel bauer [email protected] ulm university dfg research training group 1100 frederik weber...

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



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?



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




• 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




• 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



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



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



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





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







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



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



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



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



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.



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.



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?



Assessing Investment and Longevity Risks within Immediate Annuities

Thank you for your attention!

Any questions, remarks etc. are greatly appreciated.

daniel bauer [email protected] ulm university dfg research training group 1100 frederik weber...

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