Downside Risk Management of a Defined Benefit Plan Considering
Longevity Basis Risk
By
Yijia Lin, Ken Seng Tan, Ruilin Tian and Jifeng Yu
Please address correspondence to
Yijia Lin
Department of Finance
University of Nebraska
P.O. Box 880488
Lincoln, NE 68588 USA
Tel: (402) 472-0093
Email: [email protected]
Yijia Lin
University of Nebraska - Lincoln
Email: [email protected]
Ken Seng Tan
University of Waterloo
Email: [email protected]
Ruilin Tian
North Dakota State University
Email: [email protected]
Jifeng Yu
University of Nebraska - Lincoln
Email: [email protected]
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING
LONGEVITY BASIS RISK
ABSTRACT
To control downside risk of a defined benefit (DB) pension plan arising from unexpected mortality improvements and severe market turbulence, this paper proposes an optimization model by imposing two conditional value at risk (CVaR)
constraints to control tail risks of pension funding status and total pension costs.
With this setup, we further examine two longevity risk hedging strategies subject
to basis risk. While the existing literature suggests that the excess-risk hedging
strategy is more attractive than the ground-up hedging strategy as the latter is more
capital intensive and expensive, our numerical examples show that the excess-risk
hedging strategy is much more vulnerable to longevity basis risk, which limits its
applications for pension longevity risk management. Hence, our findings provide
important insight on the effect of basis risk on longevity hedging strategies.
Keywords: defined benefit pension plan, downside risk, basis risk, CVaR, longevity
risk hedging.
We are grateful to the Co-Editor, Professor Richard MacMinn, and one anonymous referee for
very helpful comments.
Date: June 24, 2013.
1
2
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
1. I NTRODUCTION
In a defined benefit (DB) pension plan, a firm is responsible for meeting pension obligations no
matter what happens in financial markets and how long its employees live after retirement. Thus,
DB firms are subject to significant pension downside risk with serious financial consequences
driven by severe market disruptions and/or dramatic mortality improvements. On the one hand,
mortality improvements at older ages have increased at a much higher rate than the expectation
of pension plans and annuity providers (Cox et al., 2012). For example, companies in the United
Kingdom FTSE100 index underestimated their aggregate pension liabilities by more than £40 billion (Cowling and Dales, 2008). As such, an unexpected increase in life expectancy among pensioners can trigger pension downside risk and cause serious financial consequences. On the other
hand, extreme negative market events are another major factor that triggers pension downside risk.
The most recent example is the late-2000s recession that brought the average funding ratio of U.S.
DB pension plans down to 75% at the end of 2008. While this ratio slightly went up to 82% by
the end of 2009, it was still so low that the plans had to make additional contributions (Sheikh and
Sun, 2010).
The literature suggests that an effective way to manage pension downside risk is to control total pension cost (i.e., all costs and penalties associated with normal contributions, supplementary
contributions and withdrawals (Maurer et al., 2009)). Along this line, Delong et al. (2008) consider supplementary contributions in their generalized optimization problem. Josa-Fombellida and
Rincón-Zapatero (2004) minimize a convex combination of contribution rate risk and the square
of difference between pension liabilities and funds. Some research goes one step further to manage the tail risk caused by excessively high total pension cost. Maurer et al. (2009), for example,
minimize the variance of plan contributions subject to a conditional value at risk (CVaR) constraint
on total pension cost. While these papers consider the cost of pension contributions, they do not
directly control pension funding risk from extreme events.
Another stream of research in the literature focuses on the hazard of pension underfunding while
not directly controlling total pension cost. Pension underfunding refers to the shortfall of pension
assets to cover pension liabilities, i.e. pension liabilities minus pension assets. The papers in
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
3
this line of research include Haberman (1997); Haberman et al. (2000); Owadally and Habermana
(2004); Habermana and Sung (2005); Josa-Fombellida and Rincón-Zapatero (2006); Ngwira and
Gerrard (2007) and Delong et al. (2008). Some other studies manage the underfunding with the
ratio of pension assets to pension liabilities, such as Chang et al. (2003) and Kouwenberg (2001).
There are only a few papers that directly control downside risk of underfunding. For example,
Bogentoft et al. (2001) use a CVaR constraint on pension underfunding to manage pension tail risk
but they do not explicitly control total pension cost within a firm’s budget constraint. Nevertheless,
specifying a total pension cost constraint is important since the total pension cost includes all costs
and penalties a plan incurs during a period of interest (Cox et al., 2012). This total pension cost
constraint provides an upper bound for a firm to manage its scarce resources.
While most pension papers emphasize only on either pension underfunding or total pension
cost, both issues are important for managing pension downside risk. If we overstate the security of
pension assets to reduce pension underfunding, we have to impose a higher requirement on total
pension costs because low-risk investments such as US Treasury securities generate low returns so
that the plan needs to make higher contributions to accumulate enough pension funds. Making high
pension contributions entails a significant opportunity cost: pension contributions reduce current
wages and benefits, new capital investments or dividend payments to shareholders (Sheikh and Sun,
2010). The low return environment along with an unexpected significant mortality improvement
requires substantial additional pension contributions, leading to downside risk on total pension
cost. On the other hand, if we attempt to reduce total pension cost and advocate highly investing
in equity markets or bond markets, pension plans would be excessively exposed to fluctuations in
either of the markets, leading to high funding downside risk.
To extend the previous analysis, in this paper, we manage both pension funding status and total
pension cost to control pension downside risk. This will provide double insurance against extreme
events. Specifically, we minimize pension risk across all periods before retirement. Not only do we
consider a traditional pension funding variation problem in the pension asset-liability management
setting, but we also impose two CVaR constraints to control downside risk from extreme pension
underfunding and excessive total pension cost. CVaR is a risk measure that has been widely used
4
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
in the recent risk management literature to manage tail risk (Rockafellar and Uryasev, 2000; Tsai
et al., 2010; Tian et al., 2010; Cox et al., 2012).
Given the above proposed model, as the second objective of this paper, we further analyze a
pension plan’s longevity risk hedging strategy. Recently, potential unexpected mortality improvement has motivated plan sponsors to implement longevity risk management. The existing studies
advocate plan designs, annuity purchase, and/or longevity securitization to prepare for unanticipated advancement in life expectancy (Lin and Cox, 2005; Blake et al., 2006; Cairns et al., 2006;
Cox and Lin, 2007; Sherris and Wills, 2008; Lin and Cox, 2008; Brcic and Brisebois, 2010; Wills
and Sherris, 2010). In particular, capital market solutions for longevity risk are attracting more and
more attention from both industries and academia. As a result, a market for longevity instruments
is emerging (Loeys et al., 2007). The payoffs of these instruments, in many cases, are determined
by population mortality indices so they do not provide a perfect hedge for pension plans. While
these “standard” securities provide liquidity and transparency on the one hand, they incur the latent
basis risk on the other. Basis risk is caused by the mismatch between a plan’s actual longevity risk
and the risk of a reference population underlying a hedging instrument.
To evaluate the effect of longevity basis risk, we need a good mortality model to describe dynamics of two-correlated populations. Carter and Lee (1992) use a single time-varying index to
describe changes in mortality rates of two populations. The sensitivity of the log central death rate
of each age in each population to this single time-varying index is captured by a population-agespecific parameter. Li and Lee (2005) introduce a mortality model that captures not only the central
tendencies within two populations but also their population-specific variations. As a major departure from Carter and Lee (1992)’s model, Li and Lee (2005)’s model includes population-specific
common risk factors. To account for the connection between mortality rates of two populations,
Li and Hardy (2011) model population-specific common risk factors as a bivariate random walk
with drift. The variance-covariance matrix of this process picks up the dependence between the
two populations. After evaluating the performance of these three models, Li and Hardy (2011)
conclude that Li and Lee (2005)’s model is preferred in terms of both goodness-of-fit and ex post
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
5
mortality projections. As a result, we rely on Li and Lee (2005)’s model for our numerical illustrations.
There are a few papers that examine mortality/longevity basis risk arising from a hedge with
an instrument of which the payoff depends on countrywide population mortality. Coughlan et al.
(2011), for example, show that basis risk is an important consideration when hedging longevity
risk with mortality indices. Li and Hardy (2011) consider four extensions to the Lee-Carter model
to measure basis risk. Plat (2009) proposes a stochastic model to quantify basis risk. These studies
mainly focus on quantifying basis risk given a certain type of hedging instruments. For instance,
the studies by Plat (2009) and Li and Hardy (2011) are based on a q-forward contract, a hedging
derivative introduced by JP Morgan (Coughlan et al., 2007).
While quantifying basis risk is important, a more practical problem is how longevity basis risk
affects a pension plan’s hedging decision with longevity securities. In this study, we extend previous studies by analyzing the impact of basis risk on a plan’s longevity hedging strategy based on
the proposed optimization model. Specifically, we first forecast mortality rates of two correlated
populations with Li and Lee (2005)’s model and project asset returns with stochastic financial market models. Then we set up a pension model for a plan that implements a ground-up or excess-risk
hedging strategy with longevity basis risk. The ground-up hedging strategy transfers a proportion
of total pension liability to the hedge provider, while the excess-risk hedging strategy cedes only
the longevity risk above some predetermined level. Finally, we apply our optimization model to
minimize the expectation of the sum of the squares of the present value of underfunding, subject to conditional value-at-risk (CVaR) constraints on funded status and total pension cost. The
optimization involves three decision dimensions: pension asset allocation, contribution strategy,
and longevity hedging. Advocates for the excess-risk hedging strategy argue that this strategy has
a more attractive structure and a lower cost (Blake et al., 2006; Lin and Cox, 2008). We show,
however, when basis risk unit cost is high, the ground-up hedging strategy may dominate as the
excess-risk hedging strategy is much more sensitive to longevity basis risk than its counterpart.
The paper is organized as follows. Section 2 presents the basic framework for a DB pension
plan. We also introduce a financial market model and a stochastic two-population mortality model.
6
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
In Section 3, we describe the pension fund optimization model. We provide a numerical example
to illustrate how to implement our model for a DB pension plan with a single cohort of employees
during the accumulation phase. To hedge longevity risk with basis risk, we examine the ground-up
hedging strategy and the excess-risk hedging strategy and solve for their optimal hedge ratios with
different basis risk penalty factors in Section 4. Section 5 compares the basis risk and no basis risk
cases for both hedging strategies. The last section concludes the paper.
2. BASIC F RAMEWORK
2.1. Pension Plan Model. Consider a cohort that joins a DB pension plan at the age of x0 at time
0 and retires at the age of x at time T . Following Maurer et al. (2009), we assume the cohort to be
stable across the entire accumulation phase. That is, every member who withdraws is immediately
replaced by a new entrant of the same age.
Under the plan structure, the plan participants are entitled to a guaranteed annual retirement benefit, B, after reaching retirement age x at time T . The benefit, B, is a function of the beneficiaries’
accumulated years of service and projected salaries before retirement. We assume the retirement
benefit, B, is a fixed amount and is not adjusted to cost of living or inflation. This benefit imposes
an obligation or liability on the part of the sponsor. Formally, the plan’s benefit liability at time t,
P BOt , can be expressed as
P BOt =
Ba(x(T ))
(1 + ρ)T −t
t = 1, 2, · · · , T,
(1)
where a(x(T )) is the life annuity factor for age x at retirement T and ρ is the plan’s periodic
discount rate. Importantly, the pension liability P BOt in this paper is an economic equivalent
of projected benefit obligation. Different from the projected benefit obligation, an accounting
measure that is discounted at AA-rated corporate yield, P BOt in our analysis is discounted at the
same interest rates as those underlying a longevity hedging instrument purchased by a pension
plan. Given this assumption, the plan is not subject to interest rate basis risk. This allows us to
focus on the effect of longevity basis risk.
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
7
Given the future curtate lifetime at age x, K(x), the present value of benefits of 1 received by
the participants per annum can be written as



v 1 + v 2 + · · · + v K(x)
a K(x) =


0
if K(x) ≥ 1
,
(2)
if K(x) = 0
where v = 1/(1 + r) denotes the discount factor with the discount rate r.
Let s p̃x,T stand for the probability that a plan member of age x at time T survives to age x + s at
the beginning of year T + s (and gets a benefit payment) given the mortality table at time T . We
will simulate the dynamics of s p̃x,T using the model discussed in Section 2.2.1. So the conditional
expected s-year survival rate for age x at retirement T equals
s p̂x,T
= E [s p̃x,T |p̃(x, T ), p̃(x + 1, T + 1), · · · , p̃(x + s − 1, T + s − 1)] ,
(3)
where p̃(x, T ) is the one-year survival probability of age x at time T . Then the conditional expected
value of life annuity in (1) is derived as
a(x(T )) =
∞
X
v s s p̂x,T .
(4)
s=1
That is, the annuity factor, a(x(T )), is the discounted expected value of payments of 1 per year,
conditional on the survival of the retiree at time T .
We assume that the initial fund at time 0 is P A0 . The accumulated fund of the plan at time t is
P At , t = 0, 1, . . . , T . It is obtained from time t − 1’s investment in n assets, that is,
P At =
n
X
Ai,t−1 (1 + ri,t ) i = 1, 2, · · · , n; t = 1, 2, . . . , T,
(5)
i=1
where Ai,t−1 is the amount invested in asset i at time t − 1 and ri,t is the return of asset i in period
t. For each period, assuming that there are no death benefits payable to members who die before
retirement, the following balance equation holds before retirement:
n
X
i=1
Ai,t = P At + Ct t = 1, 2, . . . , T,
(6)
8
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
where Ct is the plan’s contribution at time t. Specifically, Ct can be decomposed into two parts
depending on the sign of the plan’s underfunding U Lt at time t:



C + SCt if U Lt ≥ 0
Ct =
,


C − Wt
if U Lt ≤ 0
(7)
where C is a constant normal contribution (to be determined by optimization in the later sections),1
SCt ≥ 0 is a supplementary contribution if the plan has unfunded liability at time t, and Wt ≥ 0
is a withdrawal if the plan is over-funded at time t. When the plan is over-funded, it can lower
its annual contribution. So the withdrawal Wt can be viewed as the reduced contribution from the
constant normal contribution C. With this setup, the plan’s underfunding at time t, U Lt , equals:
U Lt = P BOt − P At − C.
(8)
Suppose the plan amortizes its unfunded liability over m > 1 periods at the plan’s periodic
discount rate ρ, which means the pension amortization factor k equals:
k = Pm−1
i=0
1
.
(1 + ρ)−i
(9)
Therefore, the supplementary contribution or withdrawal at time t based on k is calculated as:
SCt = max{k · U Lt , 0},
Wt = max{−k · U Lt , 0}.
With this setup,
U Lt = P BOt −
" n
X
i=1
1
#
n
X
1
Ai,t − (C + k · U Lt ) − C =
(P BOt −
Ai,t ).
1−k
i=1
Normal contribution or service cost, C, is the cost of additional benefits earned by employees for their service
each year, which depends on salary levels, employee turnover and mortality. However, the ultimate cost is usually
uncertain. To measure this cost, in practice, pension firms often first estimate their future pension obligations using
actuarial assumptions and then attribute these obligations to service years to derive an annual service cost (Competition
Commission, 2007). In our example, we calculate future pension obligations based on the retirement benefit B and
then determine the optimal annual normal contribution C with our proposed model.
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
9
Note that when U Lt > 0, the supplementary contribution SCt equals k · U Lt and when U Lt <
0, the plan has a withdrawal Wt = −k · U Lt . That is, U Lt determines the deviation of the
actual periodic contribution Ct from the normal contribution C at t, which can be a supplementary
contribution or a withdrawal.
These contributions and withdrawals determine the plan’s total pension cost. Following Maurer et al. (2009), we define total pension cost T P C as the sum of the present value of normal
contributions C, supplementary contributions SCt and withdrawals Wt :
TPC =
T
X
C + SCt (1 + ψ1 ) − Wt (1 − ψ2 )
t=1
(1 + ρ)t
.
(10)
The constants ψ1 and ψ2 are penalty factors on supplementary contributions SCt and withdrawals
Wt respectively. We emphasize that at any time t, at most one of SCt or Wt is positive. The penalty
factor ψ1 represents the opportunity cost associated with each unit of unexpected mandatory supplementary contributions SCt that could have been invested in positive net present value projects
while ψ2 accounts for the loss of tax benefits when the plan reduces its normal contribution.
Suppose that the plan sponsor invests a proportion wi of the initial accumulated fund P A0 and
the future contributions in asset i, i = 1, 2, . . . , n. Here, wi stays the same throughout the entire
accumulation phase and will be determined by optimization. Therefore, from equations (5), (6),
and (8), Ai,t is calculated as
Ai,t = (1 − k)Ai,t−1 (1 + ri,t ) + (1 + k)wi · C + kwi · P BOt ,
(11)
where Ai,0 = wi · P A0 for i = 1, 2, · · · , n.
2.2. Model Longevity Basis Risk. To control downside risk, a DB pension plan can cede part
of its risk to a third party. In the later sections, we analyze how the plan adjusts its longevity
hedging strategy when basis risk is a concern. Longevity basis risk exists, for example, when a
plan purchases a hedging instrument with an underlying population in a different country. This is
possible when the other country has greater liquidity for longevity hedges, or the home country
does not have reliable data to construct a mortality index. Therefore, we need a model to quantify
10 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
basis risk embedded in this process. To achieve this goal, we use the model developed by Li and
Lee (2005) to describe and forecast mortality rates because their model can measure mortality
interdependence between two (or more) populations. This feature of the model is important as we
observe a global convergence in mortality levels among different countries in the second half of
the past century (United Nations, 1998; White, 2002; Cox et al., 2006; Li and Hardy, 2011; Lin
et al., 2012).
2.2.1. Two-Population Mortality Model. We apply Li and Lee (2005)’s model to describe the mortality interdependence between two populations, say, between the US and UK populations in the
long run and capture the diversity between them in the short term. In particular, this model incorporates the common mortality variation and country- and age-specific mortality variations for all
ages of these two populations. It also incorporates the long- and short-term trends of their mortality
dynamics.
Let q(x, t) and q 0 (x, t) be the one-year death rate at age x, x = 0, 1, 2, . . . in year t, t =
1, 2, . . . , N for the US and UK populations, respectively. Mathematically, we model the logarithm
of q(x, t) and q 0 (x, t) with the following equations:
ln q(x, t) = s(x) + B(x)K(t) + b(x)k(t) + (x, t)
(12)
0
0
0
0
0
ln q (x, t) = s (x) + B(x)K(t) + b (x)k (t) + (x, t).
The term s(x) (s0 (x)) is an age-specific parameter indicating the US (UK) population’s average
mortality level at age x:
PN
s(x) =
t=1
ln q(x, t)
and s0 (x) =
N
PN
t=1
ln q 0 (x, t)
,
N
(13)
where N is the length of the time series of mortality data.
In (12), both populations have the same B(x) and K(t). K(t) is a time-varying index that drives
changes in the mortality rates for both populations. B(x) is an age-specific parameter indicating
the sensitivity of ln q(x, t) and ln q 0 (x, t) to K(t). Li and Lee (2005) model K(t) as a random walk
with drift:
K(t) = g + K(t − 1) + σK e(t), e(t) ∼ N (0, 1)
(14)
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 11
where g is the drift term and σK e(t) is the error term, i.i.d. normal with zero mean and a standard
deviation of σK .
The terms b(x)k(t) and b0 (x)k 0 (t) in (12) account for the short-term difference between the death
rate changes in the two countries. b(x) (b0 (x)) denotes the sensitivity of the US (UK) population to
k(t) (k 0 (t)). Both k(t) and k 0 (t) are described as the first-order autoregressive (AR(1)) processes:
k(t) = r0 + r1 k(t − 1) + σk e1 (t), e1 (t) ∼ N (0, 1),
(15)
k 0 (t) = r00 + r10 k 0 (t − 1) + σk0 e2 (t), e2 (t) ∼ N (0, 1).
Here, r0 , r00 , r1 , and r10 are constants, and σk and σk0 are the standard deviations of the US and UK
AR(1) processes, respectively. As suggested by Li and Lee (2005), to estimate the US and UK
populations as a group, we require |r1 | < 1 and |r10 | < 1 so that the model will yield bounded
short-term trends in k(t) and k 0 (t). This allows us to accommodate some continuation of historical
convergent or divergent trends for each population (Li and Hardy, 2011).
2.2.2. Data and Estimation. We fit the Li and Lee (2005)’s two-population mortality model (12)
to the annual mortality data of the US and UK female populations from 1950 to 2007. The data are
retrieved from the Human Mortality Database published by the University of California, Berkeley
and Max Planck Institute for Demographic Research.2 The calibrated parameter estimates are
depicted in Figure 1.
2.2.3. Forecasting Future Mortality. When predicting mortality rates after time 0 (i.e. after year
2007), we incorporate estimating errors into forecasted trajectories of K(t), k(t) and k 0 (t) (Li and
Lee, 2005). That is, when t > 0, the predicted values of K(t), k(t) and k 0 (t) are
K(t) = K(t − 1) + [ĝ + SE(ĝ)µ] + σ̂K e(t),
k(t) = [r̂0 + SE(r̂0 )ν1 ] + [r̂1 + SE(r̂1 )ν1 ] k(t − 1) + σ̂k e1 (t),
(16)
k 0 (t) = [r̂00 + SE(r̂00 )ν2 ] + [r̂10 + SE(r̂10 )ν2 ] k 0 (t − 1) + σ̂k0 e2 (t).
2
Available at www.mortality.org or www.humanmortality.de (data downloaded on November 22, 2011).
12 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
F IGURE 1. Estimates of Parameters in the Li and Lee (2005)’s Two-Population
Mortality Model
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 13
Here, e(t), e1 (t), e2 (t), µ, ν1 and ν2 are standard normal variables, which are independent of each
other. In (16), the terms SE(ĝ), SE(r̂0 ), SE(r̂00 ), SE(r̂1 ) and SE(r̂10 ) are the standard errors of
the drift term ĝ and the AR(1) coefficients r̂0 , r̂00 , r̂1 and r̂10 :
σ̂K
SE(ĝ) = √ ,
N
σ̂k
σ̂ 0
SE(r̂0 ) = √ , SE(r̂00 ) = √ k ,
N
N
σ̂ 0
σ̂k
, SE(r̂10 ) = qP k
,
SE(r̂1 ) = qP
N
N
2
02
t=0 k (t)
t=0 k (t)
(17)
where N is the number of sample years. In our case, N = 58 since our data span from 1950 to
2007. Then, subtracting (12) at time t from that at time t + 1, we can simultaneously forecast the
mortality rates of the US and UK populations as
ln q(x, t + 1) = ln q(x, t) + B(x)[K(t + 1) − K(t)] + b(x)[k(t + 1) − k(t)],
(18)
0
0
0
0
0
ln q (x, t + 1) = ln q (x, t) + B(x)[K(t + 1) − K(t)] + b (x)[k (t + 1) − k (t)].
We assume the pension plan has the same mortality experience as that of the US female population. To illustrate the effect of basis risk on the plan’s asset allocation and hedging strategy, we
further assume the hedging instruments are designed based on the UK female population. Following Li and Lee (2005)’s model, we first simulate K(t), k(t), and k 0 (t) based on equation (16) and
then use (18) to calculate future mortality rates. Assume the cohort of interest joins the plan at age
x0 = 45 in year 2007. After setting year 2007 as the base year t = 0, the mortality rates q(45 + t, t)
and q 0 (45 + t, t) for the US and UK populations are calculated as:
q̃(45 + t, t) = q̃(45 + t − 1, t − 1)eB(x)[K(t)−K(t−1)]+b(x)[k(t)−k(t−1)] , t = 1, 2, · · · , T,
(19)
0
0
B(x)[K(t)−K(t−1)]+b0 (x)[k0 (t)−k0 (t−1)]
q̃ (45 + t, t) = q̃ (45 + t − 1, t − 1)e
, t = 1, 2, · · · , T,
14 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
where q̃(45, 0) and q̃ 0 (45, 0) are the historical mortality rates of US and UK female of age 45 in
year 2007. Then the forecasted one-year survival probabilities are:
p̃(45 + t, t) = 1 − q̃(45 + t, t),
(20)
0
0
p̃ (45 + t, t) = 1 − q̃ (45 + t, t).
Based on the simulated survival rates, we calculate the conditional expected value of the life
annuity a(x(T )) for the US population when x = 65 and T = 20 as follows:
a(65(20)) =
∞
X
v s E [s p̃65,20 |p̃(65, 20), p̃(66, 21), · · · , p̃(65 + s − 1, 20 + s − 1)] ,
(21)
s=1
where
s p̃65,20
= p̃(65, 20) · p̃(66, 21) · · · p̃(65 + s − 1, 20 + s − 1).
Furthermore, a0 (x(T )) for the UK populated is similarly calculated.
2.3. Financial Market Model. Following Cox et al. (2012), we assume the plan invests in three
assets: S&P 500 index A1,t , Merrill Lynch corporate bond index A2,t and 3-month T-bill A3,t , with
weights w1 , w2 and w3 respectively. The processes of the S&P 500 index and Merrill Lynch corporate bond index log returns are described as a bivariate Brownian motion with compound Poisson
processes. We further assume the 3-month T-bill evolves according to a geometric Brownian motion uncorrelated with the S&P500 index and the Merrill Lynch corporate bond index. Given this
setup, Cox et al. (2012) obtain the parameter estimates shown in Table 1. In Table 1, the constant α1 (α2 , α3 ) is the drift of the S&P 500 index (Merrill Lynch corporate bond index, 3-month
T-bill); σ1 (σ2 , σ3 ) is the instantaneous volatility of the S&P 500 index (Merrill Lynch corporate
bond index, 3-month T-bill), conditional on no jumps. The parameter λ1 (λ2 ) is the mean number
of arrivals per unit time of Poisson process of the S&P 500 index (Merrill Lynch corporate bond
index). The jump size of the S&P 500 index (Merrill Lynch corporate bond index) is a lognormal
random variable with mean parameter m1 (m2 ) and volatility parameter s1 (s2 ). The parameter ρ12
is the correlation between the S&P 500 index and the Merrill Lynch corporate bond index. In the
later numerical illustration, we use the parameters in this table for projection and optimization.
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 15
TABLE 1. Maximum Likelihood Parameter Estimates of Three Pension Assets
Based on Monthly Data from March 1988 to December 2010 (Source: Cox et al.
(2012))
Parameter Estimate Parameter Estimate Parameter Estimate
α1
σ1
λ1
m1
s1
α2
σ2
λ2
m2
s2
0.1081
0.1069
0.2946
-0.0272
0.0536
0.0794
0.0481
0.0080
-0.0744
0.0000
α3
σ3
0.0523
0.0094
ρ12
0.3380
3. O PTIMIZATION M ODEL WITHOUT H EDGING
In our optimization model, we manage both pension funding status and total pension cost to
control downside risk across all periods before retirement.
3.1. Optimization Problem. Following Ngwira and Gerrard (2007), we calculate total underfunding liability T U L as the present value of all future underfundings U Lt , t = 1, 2, . . . , T
throughout the accumulation phase before retirement T . That is,
T UL =
T
X
t=1
U Lt
.
(1 + ρ)t
Then we solve the following optimization problem for the optimal asset investment strategy and
the optimal pension normal contribution in the mean square sense:
"
Minimize
2
T X
U Lt
E
(1 + ρ)t
t=1
subject to
E(T U L) = 0
w,C
#
CVaRαTUL (T U L) ≤ ζ
(22)
CVaRαTPC (T P C) ≤ τ
0 ≤ wi ≤ 1,
n
X
wi = 1
i=1
C ≥ 0,
i = 1, 2, ..., n
16 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
where the weight wi is the percentage of pension assets invested in asset i, i = 1, 2, . . . , n. The
setting of the objective function in (22),
"
2 #
T X
U Lt
J =E
,
t
(1
+
ρ)
t=1
is similar to Colombo and Haberman (2005) who minimize the variance of the two-tail funding
status. We include the E(T U L) = 0 constraint to avoid both over- and under-fundings following
Delong et al. (2008), Josa-Fombellida and Rincón-Zapatero (2004), and Cox et al. (2012).
Our optimization problem also takes into account costs. The instrument we use to control excessive costs is the total pension cost. Consistent with Maurer et al. (2009), we impose a CVaR
constraint on the total pension cost in (22), CVaRαTPC (T P C), given the left-tail probability αTPC ,
to satisfy the plan’s budget constraint. In addition, to control downside risk from unfunded liability, following Bogentoft et al. (2001), we further include a CVaR constraint on the total unfunded
liability T U L at some percentile of interest αTUL , denoted as CVaRαTUL (T U L). The constants ζ
and τ are the pre-specified parameters based on the plan’s downside risk tolerance. In general,
the levels of funding status and total pension cost move in opposite directions: improving funding
status usually entails an increase in total pension cost, while reducing total pension cost increases
funding deficit. Imposing these two CVaR constraints provides double insurance against pension
tail risk arising from these two risk sources.
3.2. Numerical Application without Hedging. Here we use an example to illustrate how to apply
our model (22) to obtain the optimal normal contribution and asset allocation for a DB pension
plan. This model guarantees a minimum funding variation subject to the funding and total pension
cost constraints.
Consider a cohort with all members joining the plan at age x0 = 45 at t = 0, and retiring at age
x = 65 at T = 20. This cohort has the same mortality experience as the US female population. We
estimate that the benefit payment rate is c and the number of survivors at age x is nx , so that the
annual total survival benefit is B = nx c = 10 million. Given the plan will pay benefits to survivors
at times T + 1, T + 2, . . . , the aggregate present value at T is the sum of n i.i.d. discounted
benefits nc · a(x(T )) = 10 · a(x(T )) million. Note that in our examples all contributions, total
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 17
underfunding liability and total pension cost are in million dollars. For brevity, we omit “million”
in the following discussion.
Assume the plan initially sets its annual normal contribution at C = 2.5. Moreover, the plan’s
initial pension fund P A0 = 5 at t = 0 is equally invested in S&P 500 index A1,t , Merrill Lynch
corporate bond index A2,t and 3-month T-bill A3,t , respectively. That is, w1 = w2 = w3 = 1/3.
This implies
P A0 =
3
X
Ai,0 = 5,
i=1
5
, i = 1, 2, 3. Suppose the pension valuation rate is set at ρ = 0.08 and the life
3
annuity factor discount rate is r = 0.05. The plan amortizes its unfunded liability over m = 7
where Ai,0 =
years. Moreover, following Maurer et al. (2009) we specify the penalty factors on supplementary
contributions and withdrawals as ψ1 = ψ2 = 0.2 when calculating total pension cost.3
TABLE 2. Initial and Optimal Pension Strategies without Hedging
w1
w2
w3
C
J
CVaR95% (T U L) CVaR95% (T P C) E(T U L)
Initial
1/3 1/3 1/3 2.50 1119.27
45.90
34.54
-2.35
Optimal 0.14 0.57 0.29 2.65 1018.41
36.44
34.54
0.00
With this setup, we calculate the objective function and the downside risk measures for the
plan based on 1,000 simulations. The results are shown in the row labeled “Initial” in Table 2.
Given the same weights invested in the three assets wi = 1/3, i = 1, 2, 3, and the constant normal
contribution C = 2.50 in each year before retirement T , the sum of squared present value of
underfunding over the accumulation phase is J = 1119.27. The two downside risk measures
CVaR95% (T U L) and CVaR95% (T P C) equal 45.90 and 34.54, respectively. Notice that in the initial
case, the plan is overfunded with E(T U L) < 0.
Now we solve the optimization problem (22) for an optimal pension strategy. Remember that
there is a tradeoff between total pension cost and funding status. As such, we first determine
the upper bound τ for the CVaR95% (T P C) constraint. Based on this predetermined upper bound
3
Withdrawals from DB pension plans are often not permitted, or if permitted are subject to excise taxes. As a robustness check, we resolve our optimization problems with and without hedging at a higher withdrawal penalty factor of
ψ2 = 0.5 that equals the prevailing excise tax rate in the US. Overall, the results confirm the findings based on the
withdrawal penalty factor of ψ2 = 0.2 shown in this paper. To conserve space, we do not report the results. The results
are available upon request.
18 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
of total pension cost, we identify the lowest feasible upper limit ζ for the second downside risk
constraint—the CVaR95% (T U L) constraint. Specifically, given τ = 34.54 that is the same as
CVaR95% (T P C) of the initial case, we find that the lowest feasible upper limit of CVaR95% (T U L)
for (22) is ζ = 36.44. With the combination of ζ = 36.44 and τ = 34.54, the optimal solution for
(22) is shown in the row labeled “Optimal” in Table 2. The optimal strategy reduces the funding
variation J from 1119.27 to 1018.41. The optimal solution suggests that the plan should invest
14% of the strategic portfolio funds in the S&P 500 index, 57% in the the Merrill Lynch corporate
bond index, and the remaining funds in the 3-month T-bill (w3 = 29%). The optimal normal
contribution equals C = 2.65 per year, which ensures E(T U L) = 0 so that the plan is neither
under- nor over-funded on average. We conclude that the optimization problem (22) returns a
lower funding variation than the initial strategy and reduces the downside risk of underfunding
when setting the target tolerance of CVaR95% (T P C) at its initial level.
Recently, it has become possible to hedge the plan’s exposure to longevity risk through the use
of two types of contracts: (1) customized longevity securities, and (2) standard (or index-based)
hedging instruments such as q-forwards (see, for example, Blake et al. (2006), Cox et al. (2010),
Li and Hardy (2011), Cairns (2011) and Milidonis et al. (2011)). Li and Hardy (2011) discuss the
advantages of standard contracts for pension longevity risk hedging. As the contracts of this type
are often based on broad population mortality indexes, they are less costly and more liquid than the
customized contracts. This implies the standard contracts are able to stimulate a greater demand
and facilitate the development of the market for longevity securities. However, those contracts can
not, in general, completely eliminate risk exposures, thus imposing basis risk cost on the plan.
As such, we need a model to capture the basis risk involved in this process. Not only will this
encourage pension plans to hedge with standard contracts, but it will also enable the plans to hedge
properly. In the next section, we develop the key quantities required for hedging pension liabilities
subject to longevity basis risk.
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 19
4. H EDGING P ENSION L IABILITY WITH L ONGEVITY BASIS R ISK
Using data of the US and UK female populations from 1950 to 2007, in Section 2.2, we have
estimated the mortality dependence between the US and UK populations. We have also forecasted
future mortality dynamics of these two populations using Li and Lee (2005) model. With these
mortality projections, here we show how to hedge pension liabilities with longevity basis risk. In
particular, we focus on two longevity hedging strategies: the ground-up hedging strategy and the
excess-risk hedging strategy. We investigate which strategy is more sensitive to longevity basis
risk.
4.1. Ground-Up Longevity Hedging Strategy with Basis Risk. Let s p̂x,T and s p̂0x,T denote the
conditional expected s-year survival probabilities of the US and UK populations at the age of x at
time T , respectively. These conditional expected s-year survival rates can be calculated following
equation (3). So the corresponding life annuity factors of the US and UK populations are calculated
P∞ s 0
P
s
0
as a(x(T )) = ∞
s=1 v s p̂x,T . Again, assume the pension plan has the
s=1 v s p̂x,T and a (x(T )) =
same mortality experience as the US population. There is a longevity security of which payoffs are
based on the UK population mortality experience. This longevity security will pay B G s p̄0x,T in year
T + s, s = 1, 2, ..., where B G is the annual survival payment and s p̄0x,T = E[s p̂0x,T ] is the expected
percentage of the reference UK population still alive at each anniversary T + s, determined at time
0. The protection provided by this security can be viewed as a capital market version of deferred
annuities purchased at time t = 0 and linked to the UK population index.
4.1.1. Optimization Problem with Ground-up Hedging Subject to Longevity Basis Risk. With the
ground-up hedging strategy, the plan transfers a proportion of its total liability (with an imperfect
hedge) by purchasing this longevity security at a price of
HP G =
(1 + δ G )B G ā0 (x(T ))
,
(1 + ρ)T
where ā0 (x(T )) = E[a0 (x(T ))] and δ G is the unit transaction cost of the longevity security that
covers risk premium, issuance cost and administrative expenses of the longevity risk taker.
20 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
Since the cohort underlying the plan has the same mortality experience as the US population,
the plan is subject to basis risk when it hedges with the longevity security written on the UK
population. After paying HP G for the longevity security, the liability of the pension plan at the
end of period t becomes
P BOtG
Ba(x(T )) − B G a0 (x(T ))
=
t = 1, 2, . . . , T.
(1 + ρ)T −t
The ratio hG = B G /B can be viewed as the hedge ratio.
Meanwhile, the plan invests in the capital market to generate investment income. Given that the
plan pays the longevity security at a price of HP G , the total amount of pension assets available for
investment at t = 0 is
P AG
0
=
n
X
G
AG
i,0 = P A0 − HP ,
i=1
which is lower than P A0 in the no hedge case when hG > 0. Here, AG
i,0 is the amount invested in
asset i at time 0 under the ground-up hedging strategy (i = 1, 2, ..., n).
After purchasing the longevity security based on the UK population, the plan pays a present
value of total pension costs T P C G at time 0:
T
δ G B G ā0 (x(T )) + γ G B G |ā(x(T )) − ā0 (x(T ))| X C G + SCtG (1 + ψ1 ) − WtG (1 − ψ2 )
TPC =
+
,
(1 + ρ)T
(1 + ρ)t
t=1
(23)
G
where ā(x(T )) = E[a(x(T ))]. The term δ G B G ā0 (x(T ))/(1 + ρ)T measures the transaction cost
of the longevity security. The term γ G B G |ā(x(T )) − ā0 (x(T ))|/(1 + ρ)T is the penalty to capture
the adverse effect of longevity basis risk where the positive constant γ G accounts for the unit basis
risk cost. The higher the basis risk measured by the absolute value of ā(x(T )) − ā0 (x(T )), given a
positive γ G , the higher the basis risk cost. This will translate to a higher total pension cost T P C G .
In this case, the unfunded liability at time t equals
1
U LG
t =
1−k
P BOtG −
n
X
!
AG
i,t
,
i=1
where AG
i,t is the amount invested in asset i at time t with the ground-up hedging.
(24)
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 21
Accordingly, given the objective function,
"
2 #
T X
U LG
t
J =E
,
(1 + ρ)t
t=1
G
the plan’s optimization problem with respect to the asset weights wG = [w1G , w2G , . . . , wnG ], normal
contribution C G and the hedge ratio hG = B G /B can be expressed as:
"
Minimize
2
T X
U LG
t
E
(1 + ρ)t
t=1
subject to
E(T U LG ) = 0
wG ,C G ,hG
#
CVaRαTUL (T U LG ) ≤ ζ
CVaRαTPC (T P C G ) ≤ τ
(1 + δ G )BhG ā0 (x(T ))
≤ P A0
(1 + ρ)T
(25)
Bā(x(T )) − BhG ā0 (x(T )) ≥ 0
0 ≤ wiG ≤ 1, i = 1, 2, . . . , n
n
X
wiG = 1
i=1
C G ≥ 0,
U LG
t
is the present value of all underfundings across all periods before
t=1
(1 + ρ)t
retirement T with the ground-up hedging strategy. The budget constraint
where T U LG =
PT
(1 + δ G )BhG ā0 (x(T ))
≤ P A0
(1 + ρ)T
ensures the longevity security price does not exceed the pension fund at t = 0. That is, the plan
does not borrow money to pay for the price of risk transferred. In addition, the constraint
Bā(x(T )) − BhG ā0 (x(T )) ≥ 0
22 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
guarantees that the protection provided by the longevity security does not exceed the pension liability.
4.1.2. Numerical Application with Ground-up Hedging Subject to Longevity Basis Risk. Here we
continue the example in Section 3.2 given ζ = 36.44 and τ = 34.54. We further assume that the
plan implements a ground-up hedging strategy subject to a transaction cost factor δ G and a basis
risk penalty factor γ G . The optimal results are shown in Table 3. As long as the plan chooses
to hedge some of its longevity risk with hedge ratio hG > 0, the plan can achieve a lower upper
bound of CVaR95% (T U L) than ζ = 36.44, the case for which the plan does not hedge. The row
labeled “CVaR95% (T U LG )” shows the lowest feasible upper limit of total underfunding downside
risk under different assumptions of hedge cost and basis risk penalty factor.
When hedging is costless (δ G = 0) with basis risk penalty factor γ G = 0.01, the plan hedges
18.5% of the pension liability and achieves a pension variation J G = 938.24, which is lower than
that in the no-hedge case J = 1018.41 (See Table 2). This can be explained by three effects. First,
the plan retains a lower longevity risk by ceding some of it to a third party. Second, the optimal
asset portfolio is relatively less risky. The risky assets account for 51% (= 10% + 41%) of invested
funds when the plan hedges without cost, compared to 71% (= 0.14% + 57%) in the no-hedge
situation. Third, the plan makes a higher annual normal contribution C G = 2.87 than the no hedge
case C = 2.65. All of these three effects attribute to a lower funding variation.
Consistent with Cox et al. (2012), Table 3 shows the plan chooses to hedge less as the hedging
transaction cost factor δ G goes up. As δ G increases from 0 to 0.07, given γ G = 0.01, the hedge
ratio hG decreases from 18.5% to 6.0%. When δ G goes up to 0.10 and above, no longevity risk will
be ceded. Moreover, as the hedge ratio hG goes down, the plan’s funding variation J G increases
due to, at least, a higher longevity risk.
On the other hand, when the transaction cost factor δ G is low, we observe the hedge ratio hG
is not sensitive to basis risk cost. For example, when δ G = 0 and δ G = 0.05, hG stays almost
the same with different basis risk penalty factors γ G . The basis risk cost has a somewhat notable
effect on hG only when δ G is high. Given δ G = 0.07, hG decreases from 6.0% to 1.4% when γ G
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 23
increases from 0.01 to 0.03. This indicates that a high transaction cost amplifies the adverse effect
of basis risk on the ground-up hedging strategy.
δG
γG
CG
w1G
w2G
w3G
hG
CVaR95% (T P C G )
CVaR95% (T U LG )
JG
0
0.01
0.02
0.03
2.87
2.87
2.87
0.10
0.10
0.10
0.41
0.41
0.41
0.49
0.49
0.49
18.5% 18.5% 18.5%
34.54 34.54 34.54
22.47 22.47 22.47
938.24 938.24 938.25
0.05
0.01
0.02
0.03
2.77
2.77
2.77
0.13
0.13
0.13
0.49
0.49
0.49
0.38
0.38
0.38
17.6% 17.6% 17.6%
34.54 34.54 34.54
26.86 26.86 26.86
996.75 996.76 996.76
0.07
0.01
0.02
0.03
2.68
2.68
2.66
0.13
0.13
0.14
0.56
0.56
0.57
0.31
0.31
0.29
6.0%
5.6%
1.4%
34.54
34.54
34.54
33.71
33.82
35.77
1018.25 1018.25 1018.26
0.1
0.01
2.65
0.14
0.57
0.29
0.0%
34.54
36.44
1018.41
TABLE 3. Optimal Ground-up Hedging Strategies with Longevity Basis Risk Given ζ = 36.44 and τ = 34.54
24 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 25
4.2. Excess-risk Longevity Hedging Strategy with Basis Risk. With the excess-risk hedging
strategy, the plan transfers the high-end longevity risk, i.e. the survival rate above a certain level.
Basically, this strategy is similar to a longevity insurance with a series of options making positive
payments from T + 1 when the realized survival rate at time t is higher than a certain threshold at
that time where t = T + 1, T + 2, · · · . In the absence of basis risk, the net benefit payment after
the excess-risk hedge of the plan to its retirees at time t is capped at this threshold level.
However, suppose there is no longevity security in the market that has the underlying population
the same as the pension cohort (US population). The plan has to purchase a longevity security with
the UK population as the underlying. This security has a series of exercise prices s p̄0x,T + λσs p̂0x,T at
time T + s, s = 1, 2, ... where λ is a constant (e.g., λ = 0, 1) and σs p̂0x,T is the standard deviation of
0
s p̂x,T .
Recall s p̄0x,T = E[s p̂0x,T ] is the expected percentage of the reference UK population still alive
at each anniversary. The plan will exercise the longevity security only when the survival rate s p̂0x,T
exceeds the strike level s p̄0x,T + λσs p̂0x,T at time T + s, s = 1, 2, .... When s p̂0x,T > s p̄0x,T + λσs p̂0x,T ,
the plan will receive a payment
0
s p̂x,T
− (s p̄0x,T + λσs p̂0x,T ), s = 1, 2, ...
from the longevity security seller.
4.2.1. Optimization Problem with Excess-risk Hedging Subject to Longevity Basis Risk. If the
annual survival benefit of the longevity security is B E , the price of this security is determined as
follows
HP E =
B E (1 + δ E )E
hP
∞
s=1
v s max
h
ii
0
0
0
p̂
−
(
p̄
+
λσ
),
0
s x,T
s x,T
s p̂x,T
(1 + ρ)T
,
where δ E is the unit transaction cost of the longevity security under the excess-risk hedging strategy.
As a result, the liability of the pension plan at the end of period t, P BOtE , becomes
h
i
P
s
0
0
0
Ba(x(T )) − B E ∞
v
max
p̂
−
(
p̄
+
λσ
),
0
s x,T
s x,T
s p̂x,T
s=1
P BOtE =
t = 1, 2, . . . , T.
(1 + ρ)T −t
(26)
26 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
The total amount of pension asset P AE
0 available for investment at t = 0 is:
P AE
0
=
n
X
E
AE
i,0 = P A0 − HP ,
(27)
i=1
where AE
i,0 is the amount invested in asset i at time 0 under the excess-risk hedging strategy.
When the plan implements the excess-risk hedging strategy with longevity basis risk, the present
value of the total pension costs, T P C E , at time 0 is calculated as follows:
T
X
C E + SCtE (1 + ψ1 ) − WtE (1 − ψ2 )
TPC = F +
,
t
(1
+
ρ)
t=1
E
(28)
where
E E
B δ E
F =
hP
∞
s=1
s
v max
h
0
s p̂x,T
−
(s p̄0x,T
+ λσ
0
s p̂x,T
ii
P
s
0
), 0 + γ E B E ∞
s=1 v |s p̄x,T − s p̄x,T |
(1 + ρ)T
,
(29)
where s p̄x,T = E[s p̂x,T ]. The term γ E B
P∞
E
s=1
v s |s p̄x,T − s p̄0x,T |/(1 + ρ)T captures the adverse
effect of longevity basis risk with the penalty factor γ E .
With this setup, the unfunded liability at time t equals
U LE
t
1
=
1−k
P BOtE
where AE
i,t is the amount invested in asset i at time t.
−
n
X
i=1
!
AE
i,t
,
(30)
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 27
With all major items redefined under the excess-risk hedging scenario with longevity basis risk,
the optimization problem with respect to the asset weights wE = [w1E , w2E , . . . , wnE ], normal contribution C E and the hedge ratio hE = B E /B can be expressed as:
"
Minimize
2
T X
U LE
t
E
(1 + ρ)t
t=1
subject to
E(T U LE ) = 0
wE ,C E ,hE
#
CVaR(T U LE )αTUL ≤ ζ
CVaR(T P C E )αTPC ≤ τ
ii
h
hP
∞
0
0
s
E
E
0
Bh (1 + δ )E
s=1 v max s p̂x,T − (s p̄x,T + λσs p̂x,T ), 0
(1 + ρ)T
"
BE
∞
X
≤ P A0
#
s
v max s p̂x,T − (s p̄x,T
(31)
+ λσs p̂x,T ), 0
s=1
"
− BhE E
∞
X
v s max
h
0
s p̂x,T
− (s p̄0x,T
#
i
+ λσs p̂0x,T ), 0 ≥ 0
s=1
0 ≤ wiE ≤ 1, i = 1, 2, . . . , n
n
X
wiE = 1
i=1
C E ≥ 0,
U LE
t
is the present value of all underfundings across all periods before
t=1
(1 + ρ)t
retirement T with the excess-risk hedging strategy. The problem (31) is to minimize the expected
where T U LE =
PT
funding variation over T periods,
"
2 #
T E
X
U
L
t
JE = E
.
(1
+
ρ)t
t=1
The constraint
"∞
#
"∞
#
h
i
X
X
BE
v s max s p̂x,T − (s p̄x,T + λσs p̂x,T ), 0 −BhE E
v s max s p̂0x,T − (s p̄0x,T + λσs p̂0x,T ), 0 ≥ 0
s=1
s=1
28 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
ensures that the protection provided by the longevity security does not exceed the high-end longevity
risk of the pension plan.
4.2.2. Numerical Application with Excess-risk Hedging Subject to Longevity Basis Risk. Here we
continue the example in Section 3.2. Assume the plan adopts the excess-risk hedging strategy with
the strike level s p̄0x,T , s = 1, 2, . . . and T = 20. In this case, λ = 0. We solve problem (31) by
setting ζ = 36.44 and τ = 34.54, the same upper bounds of CVaR95% (T U L) and CVaR95% (T P C)
as in Section 3.2 and Section 4.1.2. Table 4 summarizes the results.
When γ E = 0.01, as long as δ E is not greater than 0.1, the plan will cede at least hE = 45.3% of
longevity risk above the strike level. In addition, the expected funding variations J E are all lower
than those without hedging (see Table 2).
On the other hand, when the basis risk cost is low (e.g. γ E = 0.01) and δ E = δ G , consistent
with Cox et al. (2012), the hedge ratios hE in Table 4 are higher than hG in Table 3. This is
explained by the more attractive structure and the lower capital requirement of the excess-risk
strategy when longevity basis risk is not a big issue. However, the excess-risk hedging strategy
loses its attractiveness when the basis risk cost is high. For example, when the basis risk penalty
factor γ E increases from 0.01 to 0.03 given δ E = 0.05 in Table 4, hE decreases dramatically from
95.9% to 21.9%. In contrast, with the same change in the parameter values, the hedge ratio of the
ground-up hedging strategy hG stays at 17.6% (see Table 3). This suggests that, when the basis risk
cost is not negligible, the ground-up hedging strategy can be superior to the excess-risk strategy.
As indicated in Table 4, the excess-risk hedging does not do anything good when δ E = 0.1 and
γ E = 0.03, so the plan chooses not to hedge (hE = 0.0%).
As a robustness check, we next turn to another problem with a different strike level. Suppose the
plan increases its strike level from s p̄0x,T to s p̄0x,T + σs p̂0x,T , s = 1, 2, . . . and T = 20. Given λ = 1,
Table 5 shows the optimal results with ζ = 36.44 and τ = 34.54. Table 5 presents a picture similar
to that in Table 4: (1) the hedge ratio hE is negatively related to the transaction cost factor δ E and
the basis risk penalty factor γ E ; and (2) the excess-risk hedge ratio is very sensitive to a change in
γE .
0
0.01
0.02
0.03
2.65
2.65
2.65
0.14
0.14
0.14
0.56
0.56
0.57
0.30
0.30
0.29
96.3%
90.8%
65.8%
34.54
34.54
34.54
36.31
36.34
36.39
1017.79 1018.03 1018.29
0.05
0.01
0.02
0.03
2.65
2.65
2.65
0.14
0.14
0.14
0.56
0.56
0.56
0.30
0.30
0.30
95.9%
77.9%
21.9%
34.54
34.54
34.54
36.33
36.36
36.42
1018.05 1018.26 1018.38
0.1
0.01
0.02
0.03
2.65
2.65
2.65
0.14
0.14
0.14
0.56
0.56
0.57
0.30
0.30
0.29
45.3%
15.7%
0.0%
34.54
34.54
34.54
36.39
36.42
36.44
1018.32 1018.38 1018.41
δE
γE
CE
w1E
w2E
w3E
hE
CVaR95% (T P C E )
CVaR95% (T U LE )
JE
0
0.01
0.02
0.03
2.65
2.65
2.65
0.14
0.14
0.14
0.56
0.56
0.57
0.30
0.30
0.29
73.5%
16.5%
0.0%
34.54
34.54
34.54
36.39
36.43
36.44
1018.26 1018.38 1018.41
0.05
0.01
0.02
0.03
2.65
2.65
2.65
0.14
0.14
0.14
0.56
0.56
0.57
0.30
0.30
0.29
62.1%
12.4%
0.0%
34.54
34.54
34.54
36.40
36.43
36.44
1018.31 1018.38 1018.41
0.1
0.01
0.02
0.03
2.65
2.65
2.65
0.14
0.14
0.14
0.56
0.56
0.57
0.30
0.30
0.29
34.0%
9.7%
0.0%
34.54
34.54
34.54
36.41
36.43
36.44
1018.35 1018.38 1018.41
TABLE 5. Optimal Excess-risk Hedging Strategies with Longevity Basis Risk Given ζ = 36.44, τ = 34.54 and λ = 1
δE
γE
CE
w1E
w2E
w3E
hE
CVaR95% (T P C E )
CVaR95% (T U LE )
JE
TABLE 4. Optimal Excess-risk Hedging Strategies with Longevity Basis Risk Given ζ = 36.44, τ = 34.54 and λ = 0
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 29
30 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
4.3. Discussion. The existing literature supports the excess-risk hedging strategy as it is less capital intensive and less costly (Blake et al., 2006; Lin and Cox, 2008; Cox et al., 2012). Without basis
risk, the plan tends to hedge much more under the excess-risk strategy than under the ground-up
strategy (Cox et al., 2012). However, the results shown in this paper provide an opposite conclusion when a notable basis risk exists. In this case, the plan hedges much less or even does not
hedge at all with the excess-risk strategy. In contrast, the ground-up hedging strategy is not very
sensitive to basis risk. Its hedge ratio is much more stable with different values of the basis risk
penalty factor γ G .
Why the excess-risk hedging strategy is so sensitive to basis risk? For a pension plan, basis risk
arises from the difference in mortality experience between the pension cohort and the population
underlying the hedging instrument. A good hedging strategy should select a contract of which
mortality dynamic is as highly correlated as possible with that to be hedged. The excess-risk
hedging strategy aims at transferring longevity risk above s p̄0x,T +λσs p̂0x,T at time T +s, s = 1, 2, ....
To examine its hedge effectiveness, we calculate the correlation between the high-end risk of the
US population
∞
X
v s max s p̂x,T − (s p̄x,T + λσs p̂x,T ), 0
s=1
and that of the UK population
∞
X
v s max
h
i
0
0
0
p̂
−
(
p̄
+
λσ
),
0
,
s x,T
s x,T
s p̂x,T
s=1
where λ = 0, 1, .... When λ = 0, this correlation is only 0.0217, indicating that the US and
UK population mortality dynamics are weakly correlated when the survival rates go above the
expectation. This low correlation translates to a high basis risk cost especially when the basis risk
penalty factor γ E is high, thus the plan optimally hedges less. This low correlation further goes
down in the tail. When we focus on the survival rates higher than one standard deviation above the
mean with λ = 1, the correlation drops to 0.0102. As such, the plan further decreases its hedge
ratio (compare Table 4 with Table 5).
On the other hand, the ground-up strategy is based on the entire annuity payment a(x(T )). It
turns out that the correlation between the US a(x(T )) and the UK a0 (x(T )) is as high as 0.9695,
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 31
TABLE 6. Optimal Ground-up Hedging Strategies without Longevity Basis Risk
Given ζ = 36.44 and τ = 34.54
δG
0
0.05
0.07
0.1
G
C
2.87
2.77
2.68
2.65
w1G
0.10
0.13
0.13
0.14
G
w2
0.41
0.49
0.56
0.57
w3G
0.49
0.38
0.31
0.29
G
h
18.5% 17.6%
6.2%
0.0%
CVaR95% (T P C G ) 34.54 34.54
34.54
34.54
CVaR95% (T U LG ) 22.46 26.85
33.54
36.44
JG
938.21 996.70 1018.24 1018.41
implying an efficient hedge with basis risk. As the basis risk is so low, even with a high basis risk
penalty factor γ G , it does not impose a high basis risk cost. Thus, the plan does not significantly
modify its hedge ratio as the basis risk penalty factor γ G changes.
5. C OMPARISON OF BASIS R ISK AND N O BASIS R ISK C ASES
How the above results would be different if the plan hedges its longevity risk using a customized
longevity security tailored to the plan’s cohort? In particular, we are interested in how the hedge
ratio would change without basis risk. If we replace a0 (x(T )) with a(x(T )) in all expressions in
Section 4.1.1, we can convert our ground-up hedging problem with basis risk to the one without
basis risk. In other words, the no basis risk model is a special case of the ground-up hedging
model. Table 6 shows the results when we solve the optimization problem (25) without basis risk
by substituting a0 (x(T )) with a(x(T )) based on the CVaR constraint parameters ζ = 36.44 and
τ = 34.54. If we compare Table 6 with Table 3 when the plan cannot perform a perfect hedge,
given the same δ G , we do not find a notable difference in the ground-up hedge ratio between basis
risk and no basis risk cases due to a high mortality correlation between the pension cohort and the
UK population underlying the hedging security.
Next we turn to the excess-risk hedging strategy. To investigate the no basis risk case, we replace
all s p̂0x,T with s p̂x,T . Given λ = 0, the optimal results are reported in Table 7 with ζ = 36.44 and
τ = 34.54. Compared to the ground-up hedge ratios, the excess-risk hedge ratios exhibit a much
bigger difference between the basis risk case and the no basis risk case given the same δ E . For
32 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
TABLE 7. Optimal Excess-risk Hedging Strategies without Longevity Basis Risk
Given ζ = 36.44, τ = 34.54, and λ = 0
δE
0
0.05
0.1
0.15
0.2
E
C
2.65
2.65
2.65
2.65
2.65
w1E
0.14
0.14
0.14
0.14
0.14
E
w2
0.56
0.56
0.56
0.56
0.57
w3E
0.30
0.30
0.30
0.30
0.29
E
h
100.0% 100.0% 99.1% 22.6%
0.0%
CVaR95% (T P C E ) 34.54
34.54
34.54
34.54
34.54
CVaR95% (T U LE ) 36.30
36.32
36.34
36.41
36.44
JE
1017.67 1017.93 1018.19 1018.38 1018.41
TABLE 8. Optimal Excess-risk Hedging Strategies without Longevity Basis Risk
Given ζ = 36.44, τ = 34.54, and λ = 1
δE
0
0.05
0.1
0.15
0.2
CE
2.65
2.65
2.65
2.65
2.65
0.14
0.14
0.14
0.14
0.14
w1E
0.56
0.56
0.56
0.56
0.56
w2E
E
0.30
0.30
0.30
0.30
0.30
w3
hE
100%
100%
99.9% 73.3% 33.7%
CVaR95% (T P C E ) 34.54
34.54
34.54
34.54
34.54
CVaR95% (T U LE ) 36.40
36.40
36.41
36.42
36.42
E
J
1018.23 1018.28 1018.34 1018.38 1018.39
example, when δ E = 0.1, Table 7 shows the hedge ratio hE is 99.1% while hE is merely 45.3%
in Table 4 when the basis risk exists with the penalty factor γ E = 0.01. This difference further
widens as γ E increases.
As a robustness check, we raise the strike level and Table 8 shows the results given λ = 1. After
comparing them to the basis risk cases shown in Table 5, we confirm our earlier conclusion that
a high basis risk arising from the weak interdependence of mortality rates in the tail between the
two populations dramatically decreases the excess-risk hedge ratio, leaving the excess-risk strategy
unattractive.
An interesting finding shown in Cox et al. (2012) is that the excess-risk hedge ratio increases
with the strike level since the higher-end risks are more difficult to predict and if they occur, they
will cause serious financial consequences. Therefore, a risk averse plan tends to hedge more to
control extreme longevity risk. Their conclusion is built on the no basis risk assumption. Without
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 33
basis risk, we also reach the same conclusion when comparing Table 7 with Table 8. For example,
in Table 7, the plan chooses not to hedge (hE = 0) when δ E = 0.2 and λ = 0 while the plan still
hedges hE = 33.7% when the strike level is higher with λ = 1 in Table 8.
However, this positive relationship between the hedge ratio and the strike level reverses when
the basis risk is present. If we compare Table 4 with Table 5, we find the plan hedges less with
λ = 1 than with λ = 0. A high basis risk in the tail drives this reverse relationship, diminishing
the advantage of the excess-risk hedging strategy in longevity risk management.
6. C ONCLUSIONS
The current pension literature focuses on the benefits of hedging with standard longevity securities but abstracts from the impact of basis risk cost embedded in this process. While conventional
wisdom often prefers an excess-risk hedging contract to a ground-up hedging contract as the former is less capital intensive and cheaper, we go one step further by presenting an optimization
model to study the impacts of basis risk on the optimal hedging strategies with these two types of
contracts.
This paper first shows, if properly designed, both types of longevity hedging contracts can
achieve lower funding variation than the no hedge case given the same pension downside risk
constraints. Second, our numerical examples indicate that the excess-risk hedging strategy is
much more vulnerable to basis risk than its counterpart within the context of a two-population
framework—Li and Lee (2005)’s model. In Li and Lee (2005)’s model, both the common risk driver (B(x)K(t)) and the country-specific components (b(x)k(t) and b0 (x)k 0 (t)) are used to model
deviations from average mortality rates (s(x) and s0 (x)) of two populations. Although the common risk driver B(x)K(t) imposes larger fluctuations in mortality rates than the country-specific
components b(x)k(t) and b0 (x)k 0 (t) estimated from the US and UK population mortality data, it
neither creates basis risk nor impacts the effectiveness of longevity hedging because it affects both
populations equally. Indeed, it is the country-specific components that cause the longevity basis
risk because the country-specific components b(x)k(t) and b0 (x)k 0 (t) are not correlated. To transfer a plan’s liabilities, the ground-up strategy targets the longevity risk from the whole mortality
34 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
rate distribution, while the excess-risk hedging strategy cedes only the high-end longevity risk.
As such, this independence between the country-specific components for the high-end longevity
risk adversely affects the effectiveness of the excess-risk hedge to a greater extent, making the
excess-risk strategy more vulnerable to longevity basis risk than the ground-up strategy. As the
basis risk unit cost increases, the excess-risk hedging strategy has a much more significant drop in
the hedge level than the ground-up strategy. This implies that a poorly implemented excess-risk
hedging strategy will add more risk when there is a low mortality correlation between the pension
cohort and the underlying population of the longevity security used for hedging, which could be
detrimental to the pension plan. The incorporation of basis risk in longevity hedging provides important implications to the pension and longevity securitization literatures in that it complements
the longevity hedge model without basis risk (Cox et al., 2012) and offers a more balanced view
of the excess-risk hedging.
This paper provides a new insight on the effect of basis risk on longevity hedging strategies.
As such, it leaves some questions unanswered and in turn opens lines for future research. For
example, notice that our model is developed under the assumption of one cohort of participants.
We would likely obtain richer results from a model that incorporates more cohorts of workers who
continuously join the plan. Another potential extension of our paper is to dynamically solve our
optimization problem in each period by using the method of, for example, Bogentoft et al. (2001).
Third, due to data availability, we illustrate the effect of longevity basis risk based on the US and
UK populations. Longevity hedging, however, is generally more likely to arise in a situation where
two populations come from the same country. In this case, the longevity basis risk might be lower
(while a higher basis risk is also possible). How will it affect the hedge ratios of the two longevity
hedging strategies? Finally, our results rely on Li and Lee (2005)’s two-population mortality model
to capture longevity basis risk. Will these findings hold with other two-population models and be
generalized out of sample? We leave these questions for future research.
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 35
R EFERENCES
Blake, D., A. J. G. Cairns, and K. Dowd (2006). Living with mortality: longevity bonds and other
mortality-linked securities. British Actuarial Journal 12, 153–197.
Blake, D., A. J. G. Cairns, K. Dowd, and R. MacMinn (2006). Longevity bonds: financial engineering, valuation, and hedging. Journal of Risk and Insurance 73(4), 647–672.
Bogentoft, E., H. E. Romeijn, and S. Uryasev (2001). Asset/liability management for pension
funds using CVaR constraints. Journal of Risk Finance 3(2), 57–71.
Brcic, J. and C. Brisebois (2010). An age-old story. http://www.benefitscanada.com. April, 2010.
Cairns, A. J. (2011). Modelling and management of longevity risk: Approximations to survivor
functions and dynamic hedging. Insurance: Mathematics and Economics 49, 438–453.
Cairns, A. J. G., D. Blake, and K. Dowd (2006). Pricing death: frameworks for the valuation and
securitization of mortality risk. ASTIN Bulletin 36(1), 79–120.
Carter, L. R. and R. D. Lee (1992). Modeling and forecasting us sex differentials in mortality.
International Journal of Forecasting 8(3), 35–45.
Chang, S. C., L. Y.Tzeng, and J. C. Miao (2003). Pension funding incorporating downside risks.
Insurance: Mathematics and Economics 32, 217–228.
Colombo, L. and S. Haberman (2005). Optimal contributions in a defined benefit pension scheme
with stochastic new entrants. Insurance: Mathematics and Economics 37, 335–354.
Competition Commission (2007). A report on the economic regulation of the London airports companies (Heathrow Airport Ltd and Gatwick Airport Ltd). www.competition-commission.org.uk.
September 2007.
Coughlan,
G.,
Derivatives
D.
for
Epstein,
transferring
A.
Sinha,
longevity
and
and
P.
Honig
mortality
(2007).
risk.
q-forwards:
Available
at
http://www.jpmorgan.com/pages/jpmorgan/investbk/solutions/lifemetrics/library.
Coughlan, G., M. Khalaf-Allah, Y. Ye, S. Kumar, A. Cairns, D. Blake, and K. Dowd (2011).
Longevity hedging 101: A framework for longevity basis risk analysis and hedge effectiveness.
North American Actuarial Journal 15(2), 150–176.
36 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
Cowling, C. and R. Dales (2008, January). The FTSE 100 and their pension disclosures. Technical
report, Pension Capital Strategies.
Cox, S. H. and Y. Lin (2007). Natural hedging of life and annuity mortality risks. North American
Actuarial Journal 11(3), 1–15.
Cox, S. H., Y. Lin, and H. Petersen (2010). Mortality risk modeling: Applications to insurance
securitization. Insurance: Mathematics and Economics 46(1), 242–253.
Cox, S. H., Y. Lin, R. Tian, and J. Yu (2012). Managing capital market and longevity risks in a
defined benefit pension plan. Journal of Risk and Insurance. forthcoming.
Cox, S. H., Y. Lin, R. Tian, and L. F. Zuluaga (2012). Mortality portfolio risk management. Journal
of Risk and Insurance. forthcoming.
Cox, S. H., Y. Lin, and S. Wang (2006). Multivariate exponential tilting and pricing implications
for mortality securitization. Journal of Risk and Insurance 73(4), 719–736.
Delong, Ł., R. Gerrard, and S. Haberman (2008). Mean-variance optimization problems for an
accumulation phase in a defined benefit plan. Insurance: Mathematics and Economics 42(1),
107–118.
Haberman, S. (1997). Stochastic investment returns and contribution rate risk in a defined benefit
pension scheme. Insurance: Mathematics and Economics 19(2), 127–139.
Haberman, S., Z. Butt, and C. Megaloudi (2000). Contribution and solvency risk in a defined
benefit pension scheme. Insurance: Mathematics and Economics 27, 237–259.
Habermana, S. and J.-H. Sung (2005). Optimal pension funding dynamics over infinite control
horizon when stochastic rates of return are stationary. Insurance: Mathematics and Economics 36, 103–116.
Josa-Fombellida, R. and J. P. Rincón-Zapatero (2004). Optimal risk management in defined benefit
stochastic pension funds. Insurance: Mathematics and Economics 34(3), 489–503.
Josa-Fombellida, R. and J. P. Rincón-Zapatero (2006). Optimal investment decisions with a liability: The case of defined benefit pension plans. Insurance: Mathematics and Economics 39,
81–98.
DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK 37
Kouwenberg, R. (2001). Scenario generation and stochastic programming models for asset liability
management. European Journal of Operational Research 34, 279–292.
Li, J. S.-H. and M. R. Hardy (2011). Measuring basis risk in longevity hedges. North American
Actuarial Journal 15(2), 177–200.
Li, N. and R. Lee (2005). Coherent mortality forecasts for a group of population: An extension of
the Lee-Carter method. Demography 42(3), 575–594.
Lin, Y. and S. H. Cox (2005). Securitization of mortality risks in life annuities. Journal of Risk
and Insurance 72(2), 227–252.
Lin, Y. and S. H. Cox (2008). Securitization of catastrophe mortality risks. Insurance: Mathematics and Economics 42(2), 628–637.
Lin, Y., S. Liu, and J. Yu (2012). Pricing mortality securities with correlated mortality indices.
Journal of Risk and Insurance. forthcoming.
Loeys, J., N. Panigirtzoglou, and R. M. Ribeiro (2007). Longevity: A market in the making.
Available at http://www.jpmorgan.com/pages/jpmorgan/investbk/solutions/lifemetrics/library.
Maurer, R., O. S. Mitchell, and R. Rogalla (2009). Managing contribution and capital market risk
in a funded public defined benefit plan: Impact of cvar cost constraints. Insurance: Mathematics
and Economics 45, 25–34.
Milidonis, A., Y. Lin, and S. H. Cox (2011). Mortality regimes and pricing. North American
Actuarial Journal 15(2), 266–289.
Ngwira, B. and R. Gerrard (2007). Stochastic pension fund control in the presence of poisson
jumps. Insurance: Mathematics and Economics 40, 283–292.
Owadally, M. and S. Habermana (2004). Efficient amortization of actuarial gains/losses and optimal funding in pension plans. North American Actuarial Journal 8(1), 21–36.
Plat, R. (2009). Stochastic portfolio specific mortality and the quantification of mortality basis
risk. Insurance: Mathematics and Economics 45, 123–132.
Rockafellar, R. T. and S. Uryasev (2000). Optimization of conditional value-at-risk. Journal of
Risk 2(3), 21–41.
38 DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERING LONGEVITY BASIS RISK
Sheikh, A. Z. and J. Sun (2010). The blind side: Managing downside risk in corporate defined
benefit plans. Technical report, J. P. Morgan Asset Management.
Sherris, M. and S. Wills (2008). Financial innovation and the hedging of longevity risk. AsiaPacific Journal of Risk and Insurance 3(1), 52–64.
United Nations (1998). World Populations Prospects: The 1996 Revision. United Nations, New
York: Population Division.
Tian, R., S. H. Cox, Y. Lin, and L. Zuluaga (2010). Portfolio risk management with CVA R-like
constraints. North American Actuarial Journal 14(1), 86–106.
Tsai, J. T., J. L. Wang, and L. Y. Tzeng (2010). On the optimal product mix in life insurance
companies using conditional value at risk. Insurance: Mathematics and Economics (46), 235–
241.
White, K. M. (2002). Longevity advances in high-income countries, 1955–96. Population and
Development Review (28), 59–76.
Wills, S. and M. Sherris (2010). Securitization, structuring and pricing of longevity risk. Insurance: Mathematics and Economics 46(1), 173–185.