1. Daniel Bauer daniel.bauer@uni-ulm.de Ulm University DFG Research Training Group 1100 Frederik Weber fweber@lmu.de Ludwig-Maximilians-Universität München Institute for Risk and Insurance Management 3rd Intl Longevity Risk & Capital Market Solutions Symposium Taipei 2007 Assessing Investment and Longevity Risks within Immediate Annuities

2. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 1 Agenda 1Introduction and Motivation 2Model Setup and Assumptions 3Results 4Summary and Conclusion 5Outlook: Further research

3. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 2 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?

4. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 3 Model Setup and Assumptions Simulate a UK annuity provider‘s position – as realistic as possible: -Consider a cohort of N males aged x 0 =60, all buying immediate annuities for 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

5. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 4 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 R 10, R 20, and surplus R 40

6. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 5 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

7. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 6 Results Differences 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 0100200300400500 R 40 rel. freq. Population Pensioners

8. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 7 Results Differences 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 05101520253035404550 R 10 rel. freq. 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 05101520253035404550 R 20 rel. freq. 0 0,005 0,01 0,015 0,02 0,025 0100200300400500 R 40 rel. freq. Population Pensioners

9. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 8 Results Conditioning on best/worst paths bond hedging, t=40 years, pensioners‘ mortality per capita arbitrary 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 0100200300400500 R 40 rel. freq. arbitrary sampling 10% best mortality paths 10% worst mortality paths 10% best capital market paths 10% worst capital market paths

10. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 9 Results Conditioning on best/worst paths bond hedging, t=40 years, pensioners‘ mortality per capita arbitrary 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 0100200300400500 R 40 rel. freq. arbitrary sampling 10% best mortality paths 10% worst mortality paths 10% best capital market paths 10% worst capital market paths

11. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 10 Results Conditioning on best/worst paths bond hedging, t=40 years, pensioners‘ mortality per capita arbitrary 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 0100200300400500 R 40 rel. freq. arbitrary sampling 10% best mortality paths 10% worst mortality paths 10% best capital market paths 10% worst capital market paths

12. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 11 Results Differences by investment strategy t=40 years, pensioners‘ mortality per capita, arbitrary sampling opportunistic investment vs. bond hedging ruin prob. 0.00% 0 0,005 0,01 0,015 0,02 0,025 -1000100200300400500600700800900100011001200130014001500 R 40 rel. freq. bond hedging opport. investment

13. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 12 Results Differences by investment strategy t=40 years, pensioners‘ mortality per capita, arbitrary sampling opportunistic investment vs. bond hedging ruin prob. 0.00% ruin prob. 0.76% 0 0,005 0,01 0,015 0,02 0,025 -1000100200300400500600700800900100011001200130014001500 R 40 rel. freq. bond hedging opport. investment

14. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 13 Results Bad developments with no bond-hedging t=40 years, pensioners‘ mortality per capita, no bond-hedging available influence of 10% worst mortality/capital market developments 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 -1000100200300400500 R 40 rel. freq. R_40 - bad mortality R_40 - bad capital market

15. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 14 Results Bad developments with no bond-hedging t=40 years, pensioners‘ mortality per capita, no bond-hedging available influence 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 -1000100200300400500 R 40 rel. freq. R_40 - bad mortality R_40 - bad capital market

16. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 15 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 R 10, R 20 show same tendency as surplus R 40 but are less pronounced. Conditioning on good/bad mortality yields results relatively close to “unconditional“ results, but the spread in the surplus (R 40 ) 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.

17. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 16 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.

18. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 17 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?

19. Institute for Risk and Insurance Management Frederik Weber · Munich School of Management · LMU München Taipei · July 20-21, 2007 18 Assessing Investment and Longevity Risks within Immediate Annuities Thank you for your attention! Any questions, remarks etc. are greatly appreciated.