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)
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)
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)
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
Framework → sources of uncertainty
Mortality:
log(mx ,t )  ax  bx kt
k

t
kt 1  kt  k   tk
N (0,  k2 )
,
Interest rates:
drt  a(b  rt )dt   r dBt
 as
B( s)  (1  e )
1
a
P (i, rt )  C (i )e  B ( i ) rt
 r2
 r2
log(C ( s))  (b  2 )( B( s)  s)  B( s) 2
2a
2a
S
Pt (rt )  P (1, rt )
Pt L (rt )  P(20  t , rt )
Framework → sources of uncertainty
Assets (incl equities and bonds):
Pt L1 (rt 1 )
1
At 1  At ( t e  t L
 (1   t  t ) S )
Pt (rt )
Pt (rt )
 tZ
,
 tZ
N ( e  12  e2 ,  e2 )
.
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
max Et [Vt 1 ( At 1 , rt 1 , kt 1 ) | At , rt , kt ]
 t , t
V (kT , rT , AT )  U (kT , rT , AT )  Eu ( A / LT )
*
T
X 1
u( X ) 
1 


LT  (1  t )( AT  LT ) if AT  LT


AT*  
AT if AT  LT

LT with probability p

if AT  LT


 AT  s ( LT  AT ) with probability 1- p
 i

LT   P(i  1, rT )   (1  m(65  j , j , kT )) 
i 0
 j 0

35
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:
max Et [Vt 1 ( At 1 , rt 1 , kt 1 ) | At , rt , kt ]
 t , t
• This yields a stochastic dynamic programming model
with 3 SV’s, which we solve numerically
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
Years
s.d. of annual longevity shock
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)
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.
Case 2: Optimal investment strategy, all periods
Proportion of fund
1.00
Equities
Long bonds
0.80
Short bonds
0.60
0.40
0.20
0.00
0
0.5
1
1.5
2
FRS 17 funding ratio, time 0
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)
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, one period from end
Proportion of fund
1.00
Equities
0.80
Long bonds
Short bonds
0.60
0.40
0.20
0.00
0
0.5
1
1.5
2
FRS 17 funding ratio, time 0
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
1.00
Equities
Long bonds
0.80
Short bonds
0.60
0.40
0.20
0.00
0
0.5
1
1.5
2
FRS 17 funding ratio, time 0
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
1.5
Adj funding ratio
•
1.25
1
c
0.75
KINK
0.5
0.5
0.75
1
1.25
Unadjusted funding ratio
1.5
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
1.00
Equities
Long bonds
0.80
Short bonds
0.60
0.40
0.20
0.00
0
0.5
1
1.5
FRS 17 funding ratio, time nine
2
Results → Case 4 → First Period
Case 4: Optimal investment strategy, ten periods from end
Proportion of fund
1
Equities
Long bonds
0.8
Short bonds
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
FRS 17 funding ratio, time 0
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
Expected adjusted funding
ratio
•
1.5
1.25
1
c
KINK
0.75
0.5
0.5
0.75
1
1.25
Unadjusted funding ratio
1.5
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
Equities
1.00
Long bonds
Short bonds
0.80
0.60
0.40
0.20
0.00
0
0.5
1
1.5
2
FRS 17 funding ratio, time nine
Results → Case 5 → First period
Case 5: Optimal investment strategy, ten periods from end
Proportion of fund
Equities
1
Long bonds
0.8
Short bonds
0.6
0.4
0.2
0
0
0.5
1
1.5
2
FRS 17 funding ratio, time zero
2.5
Results → Case 5 → Effect of longevity risk
Proportion of fund in equities -10 periods from end
low longevity shocks
1
high longevity shocks
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
FRS 17 funding ratio, time 0
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