Outline

1. Optimal portfolio allocation for pension funds in the presence of background riskJune 2007David McCarthy (Imperial)David Miles (Morgan Stanley)

2. Outline • Overview • Motivation • Framework • Calibration • Results • Conclusions

3. Overview → framework • We examine optimal pension scheme investment in a dynamic context • Changing investment conditions • Uncertain longevity • Rational trustees, but: • We take account of asymmetries in the payoffs to trustees: • a sponsor might add cash to meet the liabilities if fund assets are not sufficient; • a pension insurance fund might pay part of the liabilities if the sponsor defaults; and • surplus might be shared between the sponsor and the fund

4. Overview → results • We find that uncertain longevity has little influence on optimal fund investment • The funding position has a very big effect • The asymmetries in payoffs cause trustees to take more investment risk than they otherwise would, in the form of high equity allocations and short positions in long bonds • this effect is persistent over time, economically significant, and applicable to most schemes • this may provide an explanation for the high equity investment of UK and US DB pension schemes

5. Motivation → the problem • Pension fund trustees face a tough set of decisions which, because of the size of existing portfolios, have potentially major consequences for asset pricing. • The structure of these decisions is not well modeled by standard portfolio theory • Simplifications, while they are informative, are not necessarily useful

6. Motivation → past literature • If the pension scheme is part of a sponsor’s balance sheet, and the sponsor is guaranteed to meet all shortfalls and take all surpluses, then the problem is uninteresting (and unrealistic) • Treynor (1977) • Black (1980) • Exley, Mehta & Smith (1997)

7. Motivation → past literature • If the pension fund stands entirely alone, then standard portfolio tools are useful (i.e. they give an answer) & there are many of them • but their premise is still unrealistic in the presence of the PPF, and the ability of sponsors to reclaim surplus (and some willingness and ability to make good deficits)

8. Motivation → past literature • If payoffs to trustees are asymmetric, then this is likely to have a large impact on pension fund portfolios • Furthermore, it is at least theoretically conceivable that background risk – longevity & political risks are two that pension funds face – should then have an impact on pension fund portfolios (Orszag et al, 2006)

9. Framework → model of pension fund • We suppose a pension fund with a terminal liability in 10 years time • Liability depends on the level of mortality and interest rates at that date • Pension fund has a stock of assets today, which must be invested by the trustees tomeet this liability at the terminal date • We assume no further contributions until the terminal date

10. Framework → sources of uncertainty • We assume the following sources of uncertainty: • Interest rate shocks (which drive returns on bond portfolios, the equity risk premium, and the level of the terminal liability) • Mortality shocks (which affect the level of the liability) • Random equity returns (which have a constant expected value, although the risk premium depends on the level of interest rates) • Sponsor may not meet deficit at terminal date

11. Framework → sources of uncertainty Mortality: , Interest rates:

12. Framework → sources of uncertainty Assets (incl equities and bonds): , .

13. Framework → transfers at terminal date • We assume that at the terminal date, the following occurs: • The sponsor has a certain probability, p, of making up the entire deficit of the fund, if assets are less than liabilities at that time • A pension insurance fund picks up a proportion of the deficit, s, if the sponsor fails to make a contribution • The sponsor reclaims a certain proportion, t, of any surplus

14. Framework → trustee preferences • We assume that trustees are risk averse, and that they measure their utility based on the ratio of assets to liabilities of the fund, once the transfers just described have occurred • For convenience, we assume that trustees have CRRA preferences with a fairly high degree of risk aversion (5 in our numerical analysis)

15. Framework → trustee preferences

16. Framework → trustee decisions • We assume that the trustees review their investment policy each year • Can invest in three assets: equities, long bond (duration matched to liabilities) and short bond • Trustees’ decision based on the new level of the fund’s assets, interest rates and longevity (which affect expected level of future liabilities)

17. Framework → solution technique • Trustees are assumed to maximise the value function – which is the conditional expected discounted present value of their utility at the terminal date. The value function is: • This yields a stochastic dynamic programming model with 3 SV’s, which we solve numerically

18. Calibration → assets • Equity returns have constant expected value of 0.07, s.d. of 0.20 • Bonds priced consistently with no arbitrage, off a yield curve determined by a one-factor discrete-time version of the Vasicek model, roughly calibrated to the UK (long run mean = 0.04, mean reversion = 0.3, sigma = 0.01)

19. Calibration → mortality • Mortality assumed to follow a Lee-Carter model, fitted to mortality of UK over-65’s from 1900 to 2003 • We examine three levels of mortality risk (”low”, “medium” and “high”)

20. Years s.d. of annual longevity shock Calibration → mortality Figure shows 95% confidence interval for life expectancy of 65-year old based on the three scenarios we assume in our model, at a 10-year horizon from 2003

21. Calibration → liability • Liability is assumed to be equal to a pension paid to an individual aged 65 in 10 years time, which increases with the RPI and is paid until the individual dies, based on mortality expectations in 10 years time.

22. Results → Case 1 • Case 1: p = 1; t = 1; s = 1 • All deficits made good by employer with certainty • All surpluses revert to employer • Asset allocation decision irrelevant to trustees (Black, Treynor, Exley-Mehta-Smith etc) • Decision reverts to shareholders

23. Results → Case 2 • Case 2: p = 0; t = 0; s = 0 • Fund stands alone without an employer to back its liabilities • This is a Merton model • No intertemporal hedging is possible because all sources of uncertainty are assumed to be independent

24. Results → Case 2 • Investment policy independent of time to maturity, funding level, level of mortality and level of mortality risk • Investment policy not independent of level of interest rates (since this determines the equity risk premium)

25. Results → Case 2 → Optimal portfolio Case 2: Optimal investment strategy, all periods Proportion of fund • Equity proportion given by Merton formula • Large allocation to long bonds (~ 2/3, exact level depends on ERP) • Balance in short bonds • Optimal portfolio NOT time varying. FRS 17 funding ratio, time 0

26. Results → Case 3 • Case 3: p = 0.0; t = 0.5; s = 0.5 • No sponsor bailout • Insurance fund makes up 50% of deficit • 50% of surplus reverts to sponsor • No longer a Merton model (economically equivalent to a Merton model with a lump sum transfer)

27. Results → Case 3 → Optimal portfolio Case 3: Optimal investment strategy, one period from end Proportion of fund • Equity proportion falls as funding rises Allocation to long bonds rises • Balance in short bonds • Lump sum transfer causes increasing risk aversion as funding level increases FRS 17 funding ratio, time 0

28. Results → Case 3 → Optimal portfolio Equity proportion falls as funding rises allocation to long bonds rises Balance in short bonds Lump sum transfer causes increasing risk aversion as funding level increases Case 3: Optimal investment strategy, ten periods from end Proportion of fund FRS 17 funding ratio, time 0

29. Results → Case 4 • Case 4: p = 0.0; t = 0.5; s = 0.7 • No sponsor bailout • Insurance fund makes up 70% of deficit • 50% of surplus reverts to sponsor • Kink in payoff function at full funding induced by insurance KINK

30. Results → Case 4 → Final period • Equity proportion same as case 3 (pink line) if likely to be fully funded • Equity proportion higher if likely to be under-funded • Kink in payoffs induces large equity allocation Case 4: Optimal investment strategy, one period from end Proportion of fund FRS 17 funding ratio, time nine

31. Results → Case 4 → First Period Case 4: Optimal investment strategy, ten periods from end Proportion of fund FRS 17 funding ratio, time 0

32. Results → Case 5 • Case 5: p = 0.5; t = 0.5; s = 0.7 • Sponsor bails out entire deficit with 50% prob • Insurance fund makes up 70% of deficit • 50% of surplus always reverts to sponsor • Kink in payoff function more extreme than case 4 Expected funding ratio at maturity after transfers KINK

33. Results → Case 5 → Final Period • Equity proportion • same as cases 3, 4 if likely to be fully funded • Equity proportion • same as case 4 if likely to be underfunded (marginal utility if bail-out occurs is zero) • More investment risk taken on if near the kink Case 5: Optimal investment strategy, one period from end Proportion of fund FRS 17 funding ratio, time nine

34. Results → Case 5 → First period Case 5: Optimal investment strategy, ten periods from end Proportion of fund FRS 17 funding ratio, time zero

35. Results → Case 5 → Effect of longevity risk Proportion of fund in equities -10 periods from end FRS 17 funding ratio, time 0

36. Conclusions • Degree of asymmetry of payoffs is a very important driver of portfolio allocation, not just if funds are near the point of full funding, and not just in the final few periods • Relatively plausible levels of asymmetry can induce apparently very risky investment strategies • Longevity risk has a relatively small effect on optimal portfolios in all situations that we examine, even in the presence of very asymmetric payoffs

37. Conclusions • The stronger the sponsor, and the more generous the insurance system, the riskier optimal investment strategies will be • This result might explain the high equity allocation of UK & US pension schemes • Implications for welfare & asset prices unclear, but important to examine this • The short rate is also an important driver of investment allocation