The Future of Pension Plan Funding
London School of Economics
UBS Pensions Research Programme
Paper:
Optimal Portfolio Allocation for Pension Funds in the Presence of
Background Risk
by
David McCarthy (Imperial College) and David Miles (Morgan Stanley)
Discussant:
Francisco Gomes
London Business School
Goal of the Paper
• This paper studies the optimal asset allocation for a
defined benefit pension fund.
• The pension fund has a deterministic finite horizon:
– Terminal date at which its assets are converted into
annuities that are used to pay off its liabilities.
• The fund chooses an investment strategy in every
period before the terminal date:
– It can invest in equities and an “optimally-chosen”
portfolio of short and long-term bonds.
Model Set-up
• Important: The pension fund’s liabilities are known in
advance.
– The fund is closed to new employees.
– There is no accrual of new pension liabilities.
• Important: The fund only manages its initial funds,
since it receives no additional income.
– There are no additional contributions from
existing employees.
Model Set-up (cont.)
• This set-up significantly simplifies the problem, since
there are only two sources of uncertainty:
– Asset returns.
– Shocks to survival probabilities.
• Concern: Ignoring other sources of uncertainty
significantly decreases the degree of “background
risk” faced by the pension fund:
– Potential large positive bias in the optimal equity
allocation.
Model Set-up (cont.)
• Future worker contributions are like riskless asset
holdings. In that case you get the Merton problem
with a riskless income stream:
– The contributions act as an implicit bond holding,
and thus ignoring them will biased the optimal
equity investment downwards.
• So we have two important missing features that might
affect the results (they are opposing biases but
unlikely to cancel out each other).
Model Set-up (cont.)
• Firms usually extract fund surpluses gradually, by
taking “contribution holidays”: this might also affect
the optimal investment incentives of trustees.
Objective Function
• The fund’s trustees are assumed to maximize:
– CRRA power utility preferences.
– Defined over the fund’s final asset position (after
potential adjustments for deficits or surplus).
• Is this the right objective function?
– Study observed trustee behaviour.
– Ask trustees.
Treatment of Deficits and Surpluses
• If the fund’s assets (at the terminal date) exceed its
liabilities then:
– The sponsor will (potentially) collect a fraction (t)
of the surplus.
• If the fund’s assets (at the terminal date) fail to meet
its liabilities then:
– The sponsor will cover any deficits with
probability p.
– If the sponsor defaults (probability 1-p), the
insurer will cover a fraction (s) of the deficit.
• Multiple cases are consider for t, p and s. Good!
Non-convexity in the asset allocation decision
• The paper discusses in detail the non-convexities in
the optimal allocation just before the terminal date.
– These are theoretically interesting but probability
not very important in practice.
• They don’t exist for t=T-10 and probably in other
periods as well: the objective function is made
concave by the probabilities.
• So this would likely disappear as well if we were to
add uncertainty in the terminal period, the size of the
contributions, the size of the liabilities, etc.
Numerical Solution and Related OR Literature
• The non-convexities in the objective function make
this a very hard problem to solve:
– Computationally intensive.
• Suggestion: look at the OR literature on “assetliability management”:
– Scenario optimization tools will probably be
faster.
• This OR literature might have studied similar types of
problems before.
Summary
• Conclusion: This is a very nice paper!
• The authors are tackling a very interesting and
important problem in a realistic model.
• Main suggestion: Think of ways to try to incorporate
even more realism into the analysis (“clever” shortcuts or alternative solution techniques).