Outline

Optimal portfolio allocation for pension funds in the presence of

background risk

June 2007

David McCarthy (Imperial)David Miles (Morgan Stanley)

Outline

• Overview• Motivation• Framework• Calibration• Results• Conclusions

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

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

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

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)


• 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)


• 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)

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 to meet this liability at the terminal date

• We assume no further contributions until the terminal date

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


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Mortality:

Interest rates:

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Assets (incl equities and bonds):

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

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)

Framework → trustee preferences

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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)

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

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max [ ( , , ) | , , ]t t

t t t t t t t tE V A r k A r k

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)

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”)

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

s.d. of annual longevity shock

Years

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.

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

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

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)

0.00

0.20

0.40

0.60

0.80

1.00

0 0.5 1 1.5 2

Equities

Long bonds

Short bonds

Results → Case 2 → Optimal portfolio

• 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

Case 2: Optimal investment strategy, all periods

Proportion of fund

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)

0.00

0.20

0.40

0.60

0.80

1.00

0 0.5 1 1.5 2

Equities

Long bonds

Short bonds


FRS 17 funding ratio, time 0

• Equity proportion falls as funding rises Allocation to long bonds rises


• Lump sum transfer causes increasing risk aversion as funding level increases

Case 3: Optimal investment strategy, one period from end

Proportion of fund



Case 3: Optimal investment strategy, ten periods from end

• Equity proportion falls as funding rises allocation to long bonds rises


• Lump sum transfer causes increasing risk aversion as funding level increases

0.00

0.20

0.40

0.60

0.80

1.00

0 0.5 1 1.5 2

Equities

Long bonds

Short bonds

Proportion of fund

0.5

0.75

1

1.25

1.5

0.5 0.75 1 1.25 1.5

Unadjusted funding ratio

Ad

j fu

nd

ing

ra

tio

c

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

Results → Case 4 → Final period

FRS 17 funding ratio, time nine

• 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


0.00

0.20

0.40

0.60

0.80

1.00

0 0.5 1 1.5 2

Equities

Long bonds

Short bonds

Proportion of fund

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

Equities

Long bonds

Short bonds

Results → Case 4 → First Period



Proportion of fund

0.5

0.75

1

1.25

1.5

0.5 0.75 1 1.25 1.5

Unadjusted funding ratio

Exp

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ed

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just

ed

fun

din

g

ratio

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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

0.00

0.20

0.40

0.60

0.80

1.00

0 0.5 1 1.5 2

Equities

Long bonds

Short bonds

Results → Case 5 → Final Period

FRS 17 funding ratio, time nine

• 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


Proportion of fund

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

Equities

Long bonds

Short bonds

Results → Case 5 → First period


FRS 17 funding ratio, time zero

Proportion of fund

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

low longevity shocks

high longevity shocks

Results → Case 5 → Effect of longevity risk


Proportion of fund in equities -10 periods from end

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

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

Outline

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