Downside Risk Management of a Defined Benefit Plan Considering ...

Downside Risk Management of a Defined Benefit Plan ConsideringLongevity Basis Risk

By

Yijia Lin, Ken Seng Tan, Ruilin Tian and Jifeng Yu

Please address correspondence to Yijia LinDepartment of FinanceUniversity of NebraskaP.O. Box 880488Lincoln, NE 68588 USATel: (402) 472-0093Email: [email protected]

Yijia LinUniversity of Nebraska - LincolnEmail: [email protected]

Ken Seng TanUniversity of WaterlooEmail: [email protected]

Ruilin TianNorth Dakota State UniversityEmail: [email protected]

Jifeng YuUniversity of Nebraska - LincolnEmail: [email protected]

DOWNSIDE RISK MANAGEMENT OF A DEFINED BENEFIT PLAN CONSIDERINGLONGEVITY BASIS RISK

ABSTRACT

To control downside risk of a defined benefit (DB) pension plan arising from un-

expected mortality improvements and severe market turbulence, this paper pro-

poses 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. INTRODUCTION

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 bil-

lion (Cowling and Dales, 2008). As such, an unexpected increase in life expectancy among pen-

sioners 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 to-

tal 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) con-

sider supplementary contributions in their generalized optimization problem. Josa-Fombellida and

Rincon-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 man-

age 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 Rincon-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


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 improve-

ment has motivated plan sponsors to implement longevity risk management. The existing studies

advocate plan designs, annuity purchase, and/or longevity securitization to prepare for unantici-

pated 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 dy-

namics 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-age-

specific 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 depar-

ture 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


mortality projections. As a result, we rely on Li and Lee (2005)’s model for our numerical illus-

trations.

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 previ-

ous 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 mar-

ket 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, sub-

ject 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.


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 FRAMEWORK

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 ben-

efit, 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,

PBOt, can be expressed as

PBOt =Ba(x(T ))

(1 + ρ)T−tt = 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 PBOt 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, PBOt 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.


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

aK(x)

=

v1 + v2 + · · ·+ vK(x) if K(x) ≥ 1

0 if K(x) = 0

, (2)

where v = 1/(1 + r) denotes the discount factor with the discount rate r.

Let spx,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 spx,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

spx,T = E [spx,T |p(x, T ), p(x+ 1, T + 1), · · · , p(x+ s− 1, T + s− 1)] , (3)

where p(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 )) =∞∑s=1

vsspx,T . (4)

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 PA0. The accumulated fund of the plan at time t is

PAt, t = 0, 1, . . . , T . It is obtained from time t− 1’s investment in n assets, that is,

PAt =n∑i=1

Ai,t−1(1 + ri,t) i = 1, 2, · · · , n; t = 1, 2, . . . , T, (5)

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∑i=1

Ai,t = PAt + Ct t = 1, 2, . . . , T, (6)


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 ULt at time t:

Ct =

C + SCt if ULt ≥ 0

C −Wt if ULt ≤ 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, ULt, equals:

ULt = PBOt − PAt − 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 =1∑m−1

i=0 (1 + ρ)−i. (9)

Therefore, the supplementary contribution or withdrawal at time t based on k is calculated as:

SCt = max{k · ULt, 0},

Wt = max{−k · ULt, 0}.

With this setup,

ULt = PBOt −

[n∑i=1

Ai,t − (C + k · ULt)

]− C =

1

1− k(PBOt −

n∑i=1

Ai,t).

1Normal contribution or service cost, C, is the cost of additional benefits earned by employees for their serviceeach year, which depends on salary levels, employee turnover and mortality. However, the ultimate cost is usuallyuncertain. To measure this cost, in practice, pension firms often first estimate their future pension obligations usingactuarial assumptions and then attribute these obligations to service years to derive an annual service cost (CompetitionCommission, 2007). In our example, we calculate future pension obligations based on the retirement benefit B andthen determine the optimal annual normal contribution C with our proposed model.


Note that when ULt > 0, the supplementary contribution SCt equals k · ULt and when ULt <

0, the plan has a withdrawal Wt = −k · ULt. That is, ULt 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 Mau-

rer et al. (2009), we define total pension cost TPC as the sum of the present value of normal

contributions C, supplementary contributions SCt and withdrawals Wt:

TPC =T∑t=1

C + SCt(1 + ψ1)−Wt(1− ψ2)

(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 orWt is positive. The penalty

factor ψ1 represents the opportunity cost associated with each unit of unexpected mandatory sup-

plementary 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 PA0 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 · PBOt, (11)

where Ai,0 = wi · PA0 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


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 mor-

tality 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 incor-

porates 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′(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′(x, t) with the following equations:

ln q(x, t) = s(x) +B(x)K(t) + b(x)k(t) + ε(x, t)

ln q′(x, t) = s′(x) +B(x)K(t) + b′(x)k′(t) + ε′(x, t).

(12)

The term s(x) (s′(x)) is an age-specific parameter indicating the US (UK) population’s average

mortality level at age x:

s(x) =

∑Nt=1 ln q(x, t)

Nand s′(x) =

∑Nt=1 ln q′(x, t)

N, (13)

where N is the length of the time series of mortality data.

In (12), both populations have the sameB(x) andK(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′(x, t) to K(t). Li and Lee (2005) model K(t) as a random walk

with drift:

K(t) = g +K(t− 1) + σKe(t), e(t) ∼ N(0, 1) (14)


where g is the drift term and σKe(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 b′(x)k′(t) in (12) account for the short-term difference between the death

rate changes in the two countries. b(x) (b′(x)) denotes the sensitivity of the US (UK) population to

k(t) (k′(t)). Both k(t) and k′(t) are described as the first-order autoregressive (AR(1)) processes:

k(t) = r0 + r1k(t− 1) + σke1(t), e1(t) ∼ N(0, 1),

k′(t) = r′0 + r′1k′(t− 1) + σ′ke2(t), e2(t) ∼ N(0, 1).

(15)

Here, r0, r′0, r1, and r′1 are constants, and σk and σ′k 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 |r′1| < 1 so that the model will yield bounded

short-term trends in k(t) and k′(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′(t) (Li and

Lee, 2005). That is, when t > 0, the predicted values of K(t), k(t) and k′(t) are

K(t) = K(t− 1) + [g + SE(g)µ] + σKe(t),

k(t) = [r0 + SE(r0)ν1] + [r1 + SE(r1)ν1] k(t− 1) + σke1(t),

k′(t) = [r′0 + SE(r′0)ν2] + [r′1 + SE(r′1)ν2] k′(t− 1) + σ′ke2(t).

(16)

2Available at www.mortality.org or www.humanmortality.de (data downloaded on November 22, 2011).


FIGURE 1. Estimates of Parameters in the Li and Lee (2005)’s Two-PopulationMortality Model


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(g), SE(r0), SE(r′0), SE(r1) and SE(r′1) are the standard errors of

the drift term g and the AR(1) coefficients r0, r′0, r1 and r′1:

SE(g) =σK√N,

SE(r0) =σk√N, SE(r′0) =

σ′k√N,

SE(r1) =σk√∑Nt=0 k

2(t), SE(r′1) =

σ′k√∑Nt=0 k

′2(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)],

ln q′(x, t+ 1) = ln q′(x, t) +B(x)[K(t+ 1)−K(t)] + b′(x)[k′(t+ 1)− k′(t)].(18)

We assume the pension plan has the same mortality experience as that of the US female popu-

lation. 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. Follow-

ing Li and Lee (2005)’s model, we first simulate K(t), k(t), and k′(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′(45 + t, t) for the US and UK populations are calculated as:

q(45 + t, t) = q(45 + t− 1, t− 1)eB(x)[K(t)−K(t−1)]+b(x)[k(t)−k(t−1)], t = 1, 2, · · · , T,

q′(45 + t, t) = q′(45 + t− 1, t− 1)eB(x)[K(t)−K(t−1)]+b′(x)[k′(t)−k′(t−1)], t = 1, 2, · · · , T,(19)


where q(45, 0) and q′(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:

p(45 + t, t) = 1− q(45 + t, t),

p′(45 + t, t) = 1− q′(45 + t, t).

(20)

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)) =∞∑s=1

vs E [sp65,20 |p(65, 20), p(66, 21), · · · , p(65 + s− 1, 20 + s− 1)] , (21)

where

sp65,20 = p(65, 20) · p(66, 21) · · · p(65 + s− 1, 20 + s− 1).

Furthermore, a′(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 corpo-

rate 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 mo-

tion 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 con-

stant 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.


TABLE 1. Maximum Likelihood Parameter Estimates of Three Pension AssetsBased on Monthly Data from March 1988 to December 2010 (Source: Cox et al.(2012))

Parameter Estimate Parameter Estimate Parameter Estimate

α1 0.1081 α2 0.0794 α3 0.0523σ1 0.1069 σ2 0.0481 σ3 0.0094λ1 0.2946 λ2 0.0080m1 -0.0272 m2 -0.0744s1 0.0536 s2 0.0000 ρ12 0.3380

3. OPTIMIZATION MODEL WITHOUT HEDGING

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 under-

funding liability TUL as the present value of all future underfundings ULt, t = 1, 2, . . . , T

throughout the accumulation phase before retirement T . That is,

TUL =T∑t=1

ULt(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:

Minimizew,C

E

[T∑t=1

(ULt

(1 + ρ)t

)2]

subject to E(TUL) = 0

CVaRαTUL(TUL) ≤ ζ

CVaRαTPC(TPC) ≤ τ

0 ≤ wi ≤ 1, i = 1, 2, ..., n

n∑i=1

wi = 1

C ≥ 0,

(22)


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),

J = E

[T∑t=1

(ULt

(1 + ρ)t

)2],

is similar to Colombo and Haberman (2005) who minimize the variance of the two-tail funding

status. We include the E(TUL) = 0 constraint to avoid both over- and under-fundings following

Delong et al. (2008), Josa-Fombellida and Rincon-Zapatero (2004), and Cox et al. (2012).

Our optimization problem also takes into account costs. The instrument we use to control ex-

cessive 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(TPC), given the left-tail probability αTPC,

to satisfy the plan’s budget constraint. In addition, to control downside risk from unfunded liabil-

ity, following Bogentoft et al. (2001), we further include a CVaR constraint on the total unfunded

liability TUL at some percentile of interest αTUL, denoted as CVaRαTUL(TUL). 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 = nxc = 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


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 PA0 = 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

PA0 =3∑i=1

Ai,0 = 5,

where Ai,0 =5

3, i = 1, 2, 3. Suppose the pension valuation rate is set at ρ = 0.08 and the life

annuity factor discount rate is r = 0.05. The plan amortizes its unfunded liability over m = 7

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%(TUL) CVaR95%(TPC) E(TUL)Initial 1/3 1/3 1/3 2.50 1119.27 45.90 34.54 -2.35Optimal 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%(TUL) and CVaR95%(TPC) equal 45.90 and 34.54, respectively. Notice that in the initial

case, the plan is overfunded with E(TUL) < 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%(TPC) constraint. Based on this predetermined upper bound

3Withdrawals from DB pension plans are often not permitted, or if permitted are subject to excise taxes. As a robust-ness check, we resolve our optimization problems with and without hedging at a higher withdrawal penalty factor ofψ2 = 0.5 that equals the prevailing excise tax rate in the US. Overall, the results confirm the findings based on thewithdrawal penalty factor of ψ2 = 0.2 shown in this paper. To conserve space, we do not report the results. The resultsare available upon request.


of total pension cost, we identify the lowest feasible upper limit ζ for the second downside risk

constraint—the CVaR95%(TUL) constraint. Specifically, given τ = 34.54 that is the same as

CVaR95%(TPC) of the initial case, we find that the lowest feasible upper limit of CVaR95%(TUL)

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(TUL) = 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%(TPC) 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.


4. HEDGING PENSION LIABILITY WITH LONGEVITY BASIS RISK

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 spx,T and sp′x,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

as a(x(T )) =∑∞

s=1 vsspx,T and a′(x(T )) =

∑∞s=1 v

ssp′x,T . Again, assume the pension plan has the

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 payBGsp′x,T in year

T + s, s = 1, 2, ..., where BG is the annual survival payment and sp′x,T = E[sp

′x,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

HPG =(1 + δG)BGa′(x(T ))

(1 + ρ)T,

where a′(x(T )) = E[a′(x(T ))] and δG is the unit transaction cost of the longevity security that

covers risk premium, issuance cost and administrative expenses of the longevity risk taker.


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 HPG for the longevity security, the liability of the pension plan at the

end of period t becomes

PBOGt =

Ba(x(T ))−BGa′(x(T ))

(1 + ρ)T−tt = 1, 2, . . . , T.

The ratio hG = BG/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 HPG, the total amount of pension assets available for

investment at t = 0 is

PAG0 =n∑i=1

AGi,0 = PA0 −HPG,

which is lower than PA0 in the no hedge case when hG > 0. Here, AGi,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 TPCG at time 0:

TPCG =δGBGa′(x(T )) + γGBG|a(x(T ))− a′(x(T ))|

(1 + ρ)T+

T∑t=1

CG + SCGt (1 + ψ1)−WG

t (1− ψ2)

(1 + ρ)t,

(23)

where a(x(T )) = E[a(x(T ))]. The term δGBGa′(x(T ))/(1 + ρ)T measures the transaction cost

of the longevity security. The term γGBG|a(x(T ))− a′(x(T ))|/(1 + ρ)T is the penalty to capture

the adverse effect of longevity basis risk where the positive constant γG accounts for the unit basis

risk cost. The higher the basis risk measured by the absolute value of a(x(T ))− a′(x(T )), given a

positive γG, the higher the basis risk cost. This will translate to a higher total pension cost TPCG.

In this case, the unfunded liability at time t equals

ULGt =1

1− k

(PBOG

t −n∑i=1

AGi,t

), (24)

where AGi,t is the amount invested in asset i at time t with the ground-up hedging.


Accordingly, given the objective function,

JG = E

[T∑t=1

(ULGt

(1 + ρ)t

)2],

the plan’s optimization problem with respect to the asset weights wG = [wG1 , wG2 , . . . , w

Gn ], normal

contribution CG and the hedge ratio hG = BG/B can be expressed as:

MinimizewG,CG,hG

E

[T∑t=1

(ULGt

(1 + ρ)t

)2]

subject to E(TULG) = 0

CVaRαTUL(TULG) ≤ ζ

CVaRαTPC(TPCG) ≤ τ

(1 + δG)BhGa′(x(T ))

(1 + ρ)T≤ PA0

Ba(x(T ))−BhGa′(x(T )) ≥ 0

0 ≤ wGi ≤ 1, i = 1, 2, . . . , n

n∑i=1

wGi = 1

CG ≥ 0,

(25)

where TULG =∑T

t=1

ULGt(1 + ρ)t

is the present value of all underfundings across all periods before

retirement T with the ground-up hedging strategy. The budget constraint

(1 + δG)BhGa′(x(T ))

(1 + ρ)T≤ PA0

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

Ba(x(T ))−BhGa′(x(T )) ≥ 0


guarantees that the protection provided by the longevity security does not exceed the pension lia-

bility.

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 δG and a basis

risk penalty factor γG. 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%(TUL) than ζ = 36.44, the case for which the plan does not hedge. The row

labeled “CVaR95%(TULG)” 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 (δG = 0) with basis risk penalty factor γG = 0.01, the plan hedges

18.5% of the pension liability and achieves a pension variation JG = 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 CG = 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 δG goes up. As δG increases from 0 to 0.07, given γG = 0.01, the hedge

ratio hG decreases from 18.5% to 6.0%. When δG 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 JG increases

due to, at least, a higher longevity risk.

On the other hand, when the transaction cost factor δG is low, we observe the hedge ratio hG

is not sensitive to basis risk cost. For example, when δG = 0 and δG = 0.05, hG stays almost

the same with different basis risk penalty factors γG. The basis risk cost has a somewhat notable

effect on hG only when δG is high. Given δG = 0.07, hG decreases from 6.0% to 1.4% when γG


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.


TAB

LE

3.O

ptim

alG

roun

d-up

Hed

ging

Stra

tegi

esw

ithL

onge

vity

Bas

isR

isk

Giv

enζ

=36.4

4an

dτ

=34.5

4

δG0

0.05

0.07

0.1

γG

0.01

0.02

0.03

0.01

0.02

0.03

0.01

0.02

0.03

0.01

CG

2.87

2.87

2.87

2.77

2.77

2.77

2.68

2.68

2.66

2.65

wG 1

0.10

0.10

0.10

0.13

0.13

0.13

0.13

0.13

0.14

0.14

wG 2

0.41

0.41

0.41

0.49

0.49

0.49

0.56

0.56

0.57

0.57

wG 3

0.49

0.49

0.49

0.38

0.38

0.38

0.31

0.31

0.29

0.29

hG

18.5

%18

.5%

18.5

%17

.6%

17.6

%17

.6%

6.0%

5.6%

1.4%

0.0%

CV

aR95%

(TPCG

)34

.54

34.5

434

.54

34.5

434

.54

34.5

434

.54

34.5

434

.54

34.5

4C

VaR

95%

(TULG

)22

.47

22.4

722

.47

26.8

626

.86

26.8

633

.71

33.8

235

.77

36.4

4JG

938.

2493

8.24

938.

2599

6.75

996.

7699

6.76

1018

.25

1018

.25

1018

.26

1018

.41


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 sp′x,T +λσsp′x,Tat

time T + s, s = 1, 2, ... where λ is a constant (e.g., λ = 0, 1) and σsp′x,Tis the standard deviation of

sp′x,T . Recall sp′x,T = E[sp

′x,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 sp′x,T

exceeds the strike level sp′x,T + λσsp′x,Tat time T + s, s = 1, 2, .... When sp

′x,T > sp

′x,T + λσsp′x,T

,

the plan will receive a payment

sp′x,T − (sp


), 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 BE , the price of this security is determined as

follows

HPE =BE(1 + δE)E

[∑∞s=1 v

s max[sp′x,T − (sp


), 0]]

(1 + ρ)T,

where δE is the unit transaction cost of the longevity security under the excess-risk hedging strat-

egy.

As a result, the liability of the pension plan at the end of period t, PBOEt , becomes

PBOEt =

Ba(x(T ))−BE∑∞

s=1 vs max

[sp′x,T − (sp


), 0]

(1 + ρ)T−tt = 1, 2, . . . , T.

(26)


The total amount of pension asset PAE0 available for investment at t = 0 is:

PAE0 =n∑i=1

AEi,0 = PA0 −HPE, (27)

where AEi,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, TPCE , at time 0 is calculated as follows:

TPCE = F +T∑t=1

CE + SCEt (1 + ψ1)−WE

t (1− ψ2)

(1 + ρ)t, (28)

where

F =BEδEE

[∑∞s=1 v

s max[sp′x,T − (sp


), 0]]

+ γEBE∑∞

s=1 vs|spx,T − sp

′x,T |

(1 + ρ)T,

(29)

where spx,T = E[spx,T ]. The term γEBE∑∞

s=1 vs|spx,T − sp

′x,T |/(1 + ρ)T captures the adverse

effect of longevity basis risk with the penalty factor γE .

With this setup, the unfunded liability at time t equals

ULEt =1

1− k

(PBOE

t −n∑i=1

AEi,t

), (30)

where AEi,t is the amount invested in asset i at time t.


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 = [wE1 , wE2 , . . . , w

En ], normal con-

tribution CE and the hedge ratio hE = BE/B can be expressed as:

MinimizewE ,CE ,hE

E

[T∑t=1

(ULEt

(1 + ρ)t

)2]

subject to E(TULE) = 0

CVaR(TULE)αTUL ≤ ζ

CVaR(TPCE)αTPC ≤ τ

BhE(1 + δE)E[∑∞

s=1 vs max

[sp′x,T − (sp


), 0]]

(1 + ρ)T≤ PA0

BE

[∞∑s=1

vs max[spx,T − (spx,T + λσspx,T ), 0

]]

−BhEE

[∞∑s=1

vs max[sp′x,T − (sp


), 0]]≥ 0

0 ≤ wEi ≤ 1, i = 1, 2, . . . , n

n∑i=1

wEi = 1

CE ≥ 0,

(31)

where TULE =∑T

t=1

ULEt(1 + ρ)t

is the present value of all underfundings across all periods before

retirement T with the excess-risk hedging strategy. The problem (31) is to minimize the expected

funding variation over T periods,

JE = E

[T∑t=1

(ULEt

(1 + ρ)t

)2].

The constraint

BE

[∞∑s=1


]]−BhEE

[∞∑s=1



), 0]]≥ 0


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 sp′x,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%(TUL) and CVaR95%(TPC)

as in Section 3.2 and Section 4.1.2. Table 4 summarizes the results.

When γE = 0.01, as long as δE 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 JE are all lower

than those without hedging (see Table 2).

On the other hand, when the basis risk cost is low (e.g. γE = 0.01) and δE = δG, 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 γE increases from 0.01 to 0.03 given δE = 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 δE = 0.1 and

γE = 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 sp′x,T to sp

′x,T + σsp′x,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 δE and

the basis risk penalty factor γE; and (2) the excess-risk hedge ratio is very sensitive to a change in

γE .


TAB

LE

4.O

ptim

alE

xces

s-ri

skH

edgi

ngSt

rate

gies

with

Lon

gevi

tyB

asis

Ris

kG

ivenζ

=36.4

4,τ

=34.5

4an

dλ

=0

δE0

0.05

0.1

γE

0.01

0.02

0.03

0.01

0.02

0.03

0.01

0.02

0.03

CE

2.65

2.65

2.65

2.65

2.65

2.65

2.65

2.65

2.65

wE 1

0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

wE 2

0.56

0.56

0.57

0.56

0.56

0.56

0.56

0.56

0.57

wE 3

0.30

0.30

0.29

0.30

0.30

0.30

0.30

0.30

0.29

hE

96.3

%90

.8%

65.8

%95

.9%

77.9

%21

.9%

45.3

%15

.7%

0.0%

CV

aR95%

(TPCE

)34

.54

34.5

434

.54

34.5

434

.54

34.5

434

.54

34.5

434

.54

CV

aR95%

(TULE

)36

.31

36.3

436

.39

36.3

336

.36

36.4

236

.39

36.4

236

.44

JE

1017

.79

1018

.03

1018

.29

1018

.05

1018

.26

1018

.38

1018

.32

1018

.38

1018

.41

TAB

LE

5.O

ptim

alE

xces

s-ri

skH

edgi

ngSt

rate

gies

with

Lon

gevi

tyB

asis

Ris

kG

ivenζ

=36.4

4,τ

=34.5

4an

dλ

=1

δE0

0.05

0.1

γE

0.01

0.02

0.03

0.01

0.02

0.03

0.01

0.02

0.03

CE

2.65

2.65

2.65

2.65

2.65

2.65

2.65

2.65

2.65

wE 1

0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

wE 2

0.56

0.56

0.57

0.56

0.56

0.57

0.56

0.56

0.57

wE 3

0.30

0.30

0.29

0.30

0.30

0.29

0.30

0.30

0.29

hE

73.5

%16

.5%

0.0%

62.1

%12

.4%

0.0%

34.0

%9.

7%0.

0%C

VaR

95%

(TPCE

)34

.54

34.5

434

.54

34.5

434

.54

34.5

434

.54

34.5

434

.54

CV

aR95%

(TULE

)36

.39

36.4

336

.44

36.4

036

.43

36.4

436

.41

36.4

336

.44

JE

1018

.26

1018

.38

1018

.41

1018

.31

1018

.38

1018

.41

1018

.35

1018

.38

1018

.41


4.3. Discussion. The existing literature supports the excess-risk hedging strategy as it is less capi-

tal 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 conclu-

sion 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 γG.

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 sp′x,T +λσsp′x,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∞∑s=1


]and that of the UK population

∞∑s=1



), 0],

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 γE 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 a′(x(T )) is as high as 0.9695,


TABLE 6. Optimal Ground-up Hedging Strategies without Longevity Basis RiskGiven ζ = 36.44 and τ = 34.54

δG 0 0.05 0.07 0.1CG 2.87 2.77 2.68 2.65wG1 0.10 0.13 0.13 0.14wG2 0.41 0.49 0.56 0.57wG3 0.49 0.38 0.31 0.29hG 18.5% 17.6% 6.2% 0.0%

CVaR95%(TPCG) 34.54 34.54 34.54 34.54CVaR95%(TULG) 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 γG, 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 γG changes.

5. COMPARISON OF BASIS RISK AND NO BASIS RISK CASES

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 a′(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 a′(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 δG, 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 sp′x,T with spx,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 δE . For


TABLE 7. Optimal Excess-risk Hedging Strategies without Longevity Basis RiskGiven ζ = 36.44, τ = 34.54, and λ = 0

δE 0 0.05 0.1 0.15 0.2CE 2.65 2.65 2.65 2.65 2.65wE1 0.14 0.14 0.14 0.14 0.14wE2 0.56 0.56 0.56 0.56 0.57wE3 0.30 0.30 0.30 0.30 0.29hE 100.0% 100.0% 99.1% 22.6% 0.0%

CVaR95%(TPCE) 34.54 34.54 34.54 34.54 34.54CVaR95%(TULE) 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 RiskGiven ζ = 36.44, τ = 34.54, and λ = 1

δE 0 0.05 0.1 0.15 0.2CE 2.65 2.65 2.65 2.65 2.65wE1 0.14 0.14 0.14 0.14 0.14wE2 0.56 0.56 0.56 0.56 0.56wE3 0.30 0.30 0.30 0.30 0.30hE 100% 100% 99.9% 73.3% 33.7%

CVaR95%(TPCE) 34.54 34.54 34.54 34.54 34.54CVaR95%(TULE) 36.40 36.40 36.41 36.42 36.42

JE 1018.23 1018.28 1018.34 1018.38 1018.39

example, when δE = 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 γE = 0.01. This difference further

widens as γE 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


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 δE = 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. CONCLUSIONS

The current pension literature focuses on the benefits of hedging with standard longevity securi-

ties 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 for-

mer 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 dri-

ver (B(x)K(t)) and the country-specific components (b(x)k(t) and b′(x)k′(t)) are used to model

deviations from average mortality rates (s(x) and s′(x)) of two populations. Although the com-

mon risk driver B(x)K(t) imposes larger fluctuations in mortality rates than the country-specific

components b(x)k(t) and b′(x)k′(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 b′(x)k′(t) are not correlated. To trans-

fer a plan’s liabilities, the ground-up strategy targets the longevity risk from the whole mortality


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 im-

portant 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.


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