Optimal Dynamic Longevity Hedge with Basis Risk

Jinggong Zhanga, Ken Seng Tanb, Chengguo Wengb∗

aDivision of Banking and Finance, Nanyang Technological University

bDepartment of Statistics and Actuarial Science, University of Waterloo

Abstract

From a pension plan sponsor’s perspective, we study dynamic hedging strategies for

longevity risk using standardized securities in a discrete-time setting. The hedging secu-

rities are linked to a population which may differ from the underlying population of the

pension plan, and thus basis risk arises. Drawing from the technique of dynamic pro-

gramming, we develop a framework which allows us to obtain analytical optimal dynamic

hedging strategies to achieve the minimum variance of hedging error. The most striking

advantage of the method lies in its flexibility. While q-forwards are considered in the spe-

cific implementation in the paper, our method is readily applicable to other securities such

as longevity swaps. Further, our method is implementable for a variety of longevity models

including Lee-Carter, Cairns-Blake-Dowd (CBD) and their variants. Extensive numerical

experiments are conducted to show the outperformance of our optimal strategy compared

with the standard “delta” hedging strategy from the literature.

∗Corresponding author. Tel: +001(519)888-4567 ext. 31132. Email: c2weng@uwaterloo.ca. Postal address:M3-200 University Avenue West, Waterloo, Ontario, Canada. N2L 3G1.

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1 Introduction

Over the last 100 years, life expectancies have increased at the rate of approximately 2.5 years

per decade. What this means is that one is expected to live 2.5 years longer than someone who

was born 10 years earlier. While the improvement of our life expectancy is one of the greatest

achievements in mankind to celebrate, the unanticipated mortality improvement also has an

undesirable effect on the society. More specifically, the longevity, that is, the fact that we are

living longer, creates additional (and significant) financial burden to individuals, corporations

and governments, as attributed to the greater retirement cost and medical cost, among others.

The challenges with managing longevity risk, that is, the uncertainty associated with future

mortality improvements, stem from the fact that it is a systematic risk and that it cannot be

mitigated via typical diversification strategies. As such longevity risk has become a high profile

risk in recent years. For example, the Chapter 4 of the International Monetary Fund Global

Financial Stability Report (2012) demonstrated that if individuals live three years longer than

expected, “the already large costs of aging could increase by another 50 percent, representing

an additional cost of 50 percent of 2010 GDP in advanced economies and 25 percent of 2010

GDP in emerging economies.”

The corporate pension plan sponsors and the annuity providers are similarly facing the ad-

verse financial effect of the longevity risk and raise considerable concern on their sustainability

and viability. Hence these stakeholders (or hedgers) are constantly seeking for more effective

longevity risk management solutions. Traditional approaches, including pension buy-in, pen-

sion buy-out and reinsurance, are often considered as customization strategies in that actual

longevity risk is effectively transferred to a third party. Another solution that has been advo-

cated is via the capital market whereby hedgers hedge their longevity risk using standardized

longevity securities that are linked to some certain longevity indices. For example, q-forward,

an index-based longevity derivative transaction between investment bank J.P. Morgan and UK-

based insurer Lucida, became the first publicly announced longevity derivative in January 2008

(Blake et al., 2013), and the idea was followed by a number of capital markets and insurance-

based transactions such as longevity swaps (Canada Life) and Kortis Capital bond (Swiss Re).

While the customization based solutions are undoubtedly more effective, it tends to be more

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costly. The longevity index-based solution, on the other hand, can be more cost effective, and

the availability of a trading market on longevity-linked securities will further facilitate the de-

velopment of annuity market and protect the long-term viability of retirement income provision

globally (Blake et al., 2013). The main drawback of this strategy is the presence of basis risk.

Basis risk refers to the mismatch between the mortality experience which underlies the standard

longevity securities and the hedger’s own mortality experience. Basis risk, therefore, diminishes

the effectiveness of the longevity index-based solutions, and the importance to consider basis

risk when constructing longevity hedge was emphasized by many researchers (Li and Hardy,

2011; Coughlan et al., 2011; Zhou et al., 2013, Cairns et al., 2014; Lin et al., 2014; Rosa et al.,

2017; Blake, 2018; Yang et al., 2018; Li et al., 2019).

In recent years there have been some advances in addressing the longevity hedge in the

presence of basis risk. Most of the existing literature, such as Li and Hardy (2011), Ngai and

Sherris (2011), Li and Luo (2012), Cairns (2013), and Yang et al. (2018), focus on static hedge,

as opposed to dynamic hedge. While it is relatively straightforward to implement static hedging

strategy, it can be inefficient due to the following reasons. First, because of the static nature

of the hedging strategy, the strategy does not have the mechanism of updating its strategy as

new information unfolds with time. As a result, such a strategy is vulnerable to future market

changes and, in general, is less effective than a strategy that is able to update dynamically. Sec-

ond, in order to achieve certain financial objectives, the hedger usually has to pay a considerable

initial cost (or reserve for future cost) for a static hedging strategy, which might often be im-

practical. Third, constructing an effective static hedging portfolio generally requires long-dated

hedging instruments which are expensive, unappealing to investors, in addition to exposing to

high counterparty risks. For these reasons, this paper focuses on a dynamic longevity hedging

strategy that can be a more effective strategy relative to the static hedging strategy.

It appears that the most commonly used dynamic hedging strategies for longevity risk in

the presence of basis risk is based on “delta” hedging method as advocated by Cairns (2011,

2013) (see also Luciano et al., 2012; Rosa et al., 2017; Zhou and Li, 2016; Zhou and Li, 2019).

The key principle of the delta hedging method is to construct a hedging portfolio that matches

the sensitivity of the hedger’s (usually an annuity provider’s or a pension plan sponsor’s) future

liability with respect to changes in some underlying mortality indices. Because the method is

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heuristic, it should be emphasized that in the presence of basis risk, there is no guarantee on the

effectiveness of the above delta hedging strategy due to the imperfect correlation between the

hedging portfolio and the underlying mortality risk.

As opposed to the “delta” hedging method, another possible way of implementing a dy-

namic hedging strategy is to formulate it as an optimization problem. Such an approach has the

advantage that the dynamic hedging strategy attains certain optimality depending on the adopted

objective. For example, Wong et al. (2014, 2015) devised an optimal dynamic longevity hedg-

ing strategy using longevity bonds for an insurer under the mean-variance criterion. They de-

rived a closed-form hedging policy using the Hamilton-Jacobi-Bellman (HJB) framework. A

major limitation of their results is that they used diffusion models for underlying mortality rates

and their derivation was conducted in a continuous-time setting. However, in the context of

longevity risk, discrete-time models such as the Lee-Cater model and the Cairns-Blake-Dowd

(CBD) model are more popularly adopted due to the discrete nature of the problem. In this

paper our proposed optimal strategy is based on a discrete-time framework and hence it is more

consistent with practice.

In this paper, we similarly adopt the mean-variance objective and aim to find an optimal

dynamic trading strategy involving mortality-linked securities such as the q-forward for hedging

a pension plan liability in the presence of population basis risk. We will show that under our

assumptions we can tackle the hedging problem using the stochastic optimal control framework.

Therefore a Bellman equation is derived to obtain a semi closed-form solution that allows us to

efficiently calculate the optimal trading strategy for hedging the longevity risk underlying the

pension plan.

This paper contributes to the existing literature in at least the following three aspects: first,

to the best of our knowledge this is the first time to discuss optimal dynamic longevity hedg-

ing problem using q-forward contracts in the presence of basis risk, and the results obtained

provides an effective solution for longevity risk management involving standardized securities;

second, by some numerical examples we show that our proposed hedging strategy outperforms

the benchmark “delta” method in terms of reducing the hedger’s basis risk, while maintaining

a practical computational effort; third, we provide a general framework for dynamic longevity

hedging problems. Hence our proposed framework can be adapted to other longevity models,

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hedging instruments and other hedging trading structures.

The rest of this paper is organized as follows. Section 2 introduces notation, the longevity

models, q-forward, as well as the formulation of the hedging problem as an optimization prob-

lem. Section 3 derives the optimal control process by adopting the dynamic programming

principle and the Bellman’s equation framework. Section 4 provides numerical examples to

demonstrate the effectiveness and feasibility of our proposed strategy. Section 5 concludes

the paper. Some approximation formula used for numerical computation are provided in the

appendix.

2 Problem setup

2.1 Stochastic mortality model

In this section, we describe the setup of our longevity risk management framework. We em-

phasize that the methodology for deriving an optimal hedging strategy using the dynamic pro-

gramming framework is quite general and it is applicable to many popular longevity models,

including the Lee-Carter model, the Cairns-Blake-Dowd (CBD) model and their variants. For

illustrative purpose, we proceed our mathematical derivation based on the assumption that the

future mortality improvement follows the Augmented Common Factor (ACF) model proposed

by Li and Lee (2005). By labelling two different populations as H and R and that i ∈ {H,R},we consider the following two-population model:

ln(m(i)x,t) = a(i)x +BxKt + b(i)x k

(i)t + ε

(i)x,t. (1)

Here m(i)x,t denotes population i’s central death rate at age x in year t; a(i)x denotes population i’s

average mortality level at age x; Kt represents the mortality improvement to both populations,

Bx is the corresponding coefficient for age x; k(i)t represents the mortality improvement specific

to population i, and b(i)x is the corresponding coefficients for age x specific to population i;

ε(i)x,t are the residual terms modeled by independent and identically distributed (i.i.d.) normal

random variables. The parameters Bx, b(i)x , Kt and k(i)t are subject to∑

xBx =∑

x b(i)x = 1

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and∑

tKt =∑

t k(i)t = 0 to ensure identifiability of the model. Note that the correlation of the

central death rates between both populations H and R is induced via the “common” parameters

Bx and Kt.

The time-varying indices {Kt}t≥0 and {k(i)t }t≥0, i ∈ {H,R}, are further modeled by time

series models, e.g., autoregressive integrated moving average (ARIMA) models. In this paper,

we follow Li and Lee (2005) and assume that {Kt}t≥0 follows a random walk with drift, while

each of {k(i)t }t≥0, i ∈ {H,R}, follows an AR(1) model. In other words, ∀ t = 0, 1, 2, 3, ..., we

have

Kt = C +Kt−1 + ξt, (2)

k(i)t = φ

(i)0 + φ

(i)1 k

(i)t−1 + ζ

(i)t , (3)

where C, φ(i)0 and φ

(i)1 are constants, and {ξt} and {ζ(i)t } are two mutually independent se-

quences of i.i.d. normal random variables with zero mean and constant variance:

var(ξt) = σ2K ,

var(ζ(i)t ) = σ2

k,i i ∈ {H,R}.

Additionally, we assume |φ(i)1 | < 1 to ensure that the two time series {k(i)t }t≥0 are stationary.

Finally, we emphasize that the assumptions on the time-varying indices {Kt}t≥0 and {k(i)t }t≥0,

i ∈ {H,R}, are for illustrative purpose only; our derivation also applies to general ARIMA

(p, d, q) models.

2.2 Pension liability

Before describing the pension plan liability and formulating the longevity risk management

framework, it is useful to introduce the following notation:

• q(i)x,t denotes the probability that an individual aged x at time t− 1 (alive) from population

i dies between time t− 1 and t;

• S(i)x,t(T ) :=

∏Ts=1(1− q

(i)x+s−1,t+s) denotes the probability that an individual from popula-

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tion i aged x at time t (alive) will survive to time t+ T ;

• p(i)x,u(T,Kt, k(i)t ) := E(S

(i)x,u(T )|Ft) = E(S

(i)x,u(T )|Kt, k

(i)t ), u ≥ t, is the expected survival

probability given information up to time t, where {Ft}t≥0 denotes the filtration generated

by {Ku, k(H)u , k

(R)u }{0≤u≤t}. It is denoted as a spot survival probability if u = t or a

forward survival probability if u > t.

In subsequent discussion, we will assume that the time unit is expressed in year for convenience.

Let us now consider a pension plan that consists of a single cohort of n pensioners all

aged x0 at time 0 from population H . Because of the heightened concern with the longevity

risk, the pension plan sponsor is interested in hedging against the unexpected future mortality

improvement associated with the plan liability. Without any loss of generality, we assume that

the pension benefit to each pensioner is $ 1n

and is payable at the end of each year until death.

This implies that at time 0, the pension plan begins with a total pension benefit of $1. We also

assume that the size of the pension plan, n, is large enough such that there is no sample risk and

thus mortality experience of the underlying cohort is exactly the same as the mortality rate of

population H . As a result, the time-t present value of the future pension liability, denoted by

FLt, is given by

FLt =∞∑

s=t+1

(1 + r)−(s−t)S(H)x0+t,t(s− t),

where r is the risk-free rate and is assumed to be a constant. By setting t = 0 into the above

equation, we have

FL0 =∞∑s=1

(1 + r)−sS(H)x0,0

(s) (4)

which captures the plan sponsor’s time-0 future pension liability. This expression also demon-

strates the adverse effect the longevity risk could have on the plan sponsor’s pension liability.

Hence this calls for the sponsor’s interest in hedging against the unexpected increase in the

future pension liability.

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2.3 q-forward

This subsection describes the mortality index-linked q-forward contract. The discussion on

using this instrument to hedge longevity risk will be relegated to the next subsection. In the

nutshell a q-forward contract is a combination of a zero-coupon bond and a swap. It is a zero-

coupon bond in the sense that it generates only one cash flow at the maturity of the contract.

It mimics a swap in the sense that it has both floating-rate party and fixed-rate party. The

payment of the floating-rate party is proportional to the maturity year’s realized death rate of a

pre-determined mortality-index and for a pre-determined reference age. On the other hand, the

payment of the fixed-rate party is fixed and corresponds to the forward mortality rate determined

at the inception of the contract in such a way that the q-forward has zero initial value. The q-

forward’s net payment, which depends on the relative values of the realized death rate and the

fixed death rate, is settled at its maturity date.

Consider now a q-forward written on population R at time t with time to maturity T ∗ (so

that t+T ∗ is the maturity time) and reference age xf . Let qft0 := qf (t0, Kt0 , k(R)t0 , xf , T

∗) be the

q-forward’s forward mortality rate. Under the assumption that the population mortality model

is specified by (1) with Kt0 and k(R)t0 correspond, respectively, to (2) and (3), we have

qft0 = E(q(R)xf ,t0+T ∗|Ft0) = E(q

(R)xf ,t0+T ∗ |Kt0 , k

(R)t0 ). (5)

If the fixed-rate of the above q-forward is prescribed by (5), then at the inception of the q-

forward, there is no initial cost outlay to both fixed-rate party and floating-rate party for entering

into the contract. After the inception of the q-forward, the value of the contract can be positive

or negative, depending on how the mortality rates unfold over time.

To provide additional insight on how the value of a q-forward evolves over time, let us now

consider an investor, who enters into a q-forward at time t as a floating-rate payer; i.e. pays

floating rate and receives fixed rate. The investor is also known as the fixed-rate receiver. The

counterparty of the q-forward, who acts as the fixed-rate payer or the floating rate receiver,

therefore pays fixed rate and receives floating rate. By denoting Qs(t), t ≤ s ≤ T ∗ + t, as the

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time-s value of the fixed-rate receiver, then we have

Qs(t) = (1 + r)−(T∗+t−s)

[qft − E(q

(R)xf ,t+T ∗|Fs)

]. (6)

Note that for s < T ∗ + t, the term E(q(R)xf ,t1+T ∗|Fs), which corresponds to the forward mortality

rate at time s, can also be interpreted as the fixed rate of a q-forward issued at time s with

maturity at time T ∗ + t. By definition, Qs(t) = 0 for s = t and Qs(t) can be non-zero for

t < s ≤ T ∗ + t. This also implies that in latter case the fixed-rate receiver incurs a loss with

higher future forward mortality rate and generates a profit with lower future forward mortality

rate. In particular, when the q-forward matures at time s = T ∗ + t, the term E(q(R)xf ,t1+T ∗|Fs)

is known and is given by the mortality rate realized at that time. If the realized mortality rate

is smaller than qft , then the fixed rate receiver generates a profit and receives the net amount

from the fixed-rate payer. On the other hand, if the realized mortality rate is larger than qft , then

the fixed-rate receiver is obligated to pay the fixed-rate payer the realized mortality rate, which

is higher than the fixed rate the fixed-rate receiver receives from the fixed-rate payer, hence

resulting in a loss. In either case, both parties close out their q-forward position.

The discussion so far assumes that the investor, i.e., the fixed-rate receiver, has entered into a

single q-forward with notional amount of $1. In practice, the notional amount of the q-forward

can be arbitrary, says ht for a q-forward issued at time t. The value of the q-forward is then

adjusted proportional by the notional amount ht although the initial cost outlay is still zero since

ht ·Qt(t) = 0. If the investor were to close out his position a year later at time t+ 1, then using

(6) leads to

ht ·Qt+1(t)

= ht · (1 + r)−(T∗−1)

[qft − E(q

(R)xf ,t+T ∗|Ft+1)

]= ht · (1 + r)−(T

∗−1)[E(q

(R)xf ,t+T ∗|Ft)− E(q

(R)xf ,t+T ∗|Ft+1)

]= ht · (1 + r)−(T

∗−1)[p(R)xf ,t+T ∗−1(1, Kt+1, k

(R)t+1)− p

(R)xf ,t+T ∗−1(1, Kt, k

(R)t )]. (7)

We emphasize that ht · ·Qt+1(t) can be positive, negative, or even zero.

Another important result for investing in a q-forward is formally stated in the following

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proposition. This proposition asserts that the expected return from investing in a q-forward is

zero. This is not surprising since it costs nothing to enter into a q-forward.

Proposition 1. ∀ t = 0, 1, 2, ..., we have

Et[Qt+1(t)] = Qt(t) = 0, (8)

where Et[ · ] denotes E[ · |Ft].

Proof. The proof of this proposition is straightforward, as shown below:

Et[Qt+1(t)] = (1 + r)−(T∗−1)

{Et

[E(q

(R)xf ,t+T ∗|Ft)

]− Et

[(E(q

(R)xf ,t+T ∗|Ft+1)

]}= (1 + r)−(T

∗−1){

Et

[q(R)xf ,t+T ∗

]− Et

[q(R)xf ,t+T ∗

]}= 0.

To conclude this subsection, we point out that hereafter we will adopt the same notation for

the conditional expectation operator as in Proposition 1. Specifically, we may write Et[ · ] :=

E[ · |Ft] and Vart[ · ] := Var[ · |Ft], ∀ t ≥ 0. Furthermore, we may also use E[ · ] and Var[ · ]to denote E0[ · ] and Var0[ · ], respectively.

2.4 Hedging longevity risk with q-forward

Recall that as discussed in Subsection 2.2, the plan sponsor’s time-0 future pension liability

is given by (4). Instead of dealing a summation with an infinite terms, it is useful to truncate

the summation in (4) to time τ . In practice, τ can be chosen to be arbitrarily large so that

the residual benefits to the pensioners beyond age x0 + τ is negligible. The objective of this

subsection is to propose a trading strategy involving q-forward for hedging the plan sponsor’s

(truncated) future pension liability.

By construction, the pension liability consists of a series of yearly benefit payments and

the obligations to the plan sponsor can be unexpectedly high depending on the severity of the

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longevity risk. Because of the nature of the benefit payments, it is therefore not constructive to

hedge the pension plan liability using only one q-forward involving a single cash flow. Instead,

a more prudent strategy is to use a portfolio of q-forwards which generates yearly cash flow in

such a way that matches, as close as possible, to the unexpected yearly pension benefit payment

over τ years. For this reason, τ can be interpreted as the hedging horizon of our hedging

strategy. While there are many ways of constructing a portfolio of q-forwards, in this paper

we adopt a yearly “rolling” strategy in line with the principle of hedging the pension liability

dynamically. More specifically, our proposed dynamic hedging strategy proceeds as follow (for

a pre-determined reference age of the q-forward): at the initial time t = 0, the plan sponsor (i.e.

the hedger) writes, as the fixed-rate receiver, a q-forward with notional amount h0 and maturity

time T ∗. This transaction does not require any initial cost outlay. Then at each subsequent time

t = 1, 2, . . . , τ−1, the hedger closes out the position constructed from the previous time period,

realises the profit or loss, and concurrently writes a new q-forward with notional amount ht and

maturity time T ∗. At time τ , the hedger closes out the position constructed from time τ − 1 and

terminates the hedging strategy.

As a result of the yearly rebalancing strategy, the time-t cash flow of the q-forward portfolio

is given by ht−1 · Qt(t − 1) for t = 1, 2, . . . , τ . This also implies that the proposed dynamic

hedging strategy is succinctly represented by the process {ht}t≥0, which is adapted to {Ft}t≥0.

By rebalancing hedging portfolio yearly, this strategy enables us to exploit the latest informa-

tion revealed by the market, and then update the hedging position that best matches the future

liability of the pension plan.

Let X(τ) denote the (truncated) time-0 value of the future net cash flow of the plan sponsor

in the presence of the above dynamic hedging strategy with hedging horizon τ . Then X(τ) is

given by

X(τ) =τ∑t=1

(1 + r)−tht−1 ·Qt(t− 1)−τ∑t=1

(1 + r)−tS(H)x0,0

(t) (9)

=τ∑t=1

(1 + r)−tF (t) (10)

11

where

F (t) := ht−1 ·Qt(t− 1)− S(H)x0,0

(t). (11)

We now make the following remarks with respect to the above results:

• The first summation in (9) captures the time-0 value of the hedging portfolio while the

second summation represents the time-0 value of the plan sponsor’s future pension liabil-

ity (recall equation (4)). The difference of these two values, therefore, corresponds to the

plan sponsor’s time-0 net cash flow. Note that X(τ) is expected to be negative since it is

the plan sponsor’s time-0 cost of the future pension liability (together with the hedging

strategy).

• The term F (t) can be interpreted as the net position of the hedger at time t due to the

yearly adjusted q-forward portfolio that is applied to the pension plan.

• The payoff of the q-forward hedging instruments depend on population R, as opposed to

population H . Implicitly we are assuming that q-forward that is written on population H

is not readily available from the market. Because of the imperfect correlation between the

hedging instrument and the pension liability, our proposed hedging strategy is vulnerable

to population basis risk.

• The effect of hedging on the plan sponsor’s future pension liability can be gleaned from

X(τ) in (10). In situation for which plan sponsor does not hedge, this is equivalent to

setting ht = 0 for t = 1, 2, . . . , τ −1; i.e. taking zero position in the hedging instruments.

In this case the plan sponsor’s time-0 net cash flow is given by X(τ) = −∑τ

t=1(1 +

r)−tS(H)x0,0

(t). This is just the negative of (4) since it represents the time-0 cost to the plan

sponsor. However, for ht−1 6= 0, t = 1, 2, . . . , τ , the time-t net cash flow of the plan

sponsor will be distorted by the magnitude of ht−1 · Qt(t − 1), which can be positive or

negative.

• Equation (11) also demonstrates that when using q-forward to hedge longevity risk, the

position of the hedger should be the fixed-rate receiver, rather than the floating-rate payer.

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To see this, let us first note that when there is an unexpected mortality improvement for

population H , this triggers additional cost to the pension plan sponsor. If we assume pop-

ulationH is positively correlated with populationR (not necessarily perfectly correlated),

population R is likely to experience the similar unexpected improvement in mortality. It

follows from (6) that the hedging portfolio generates a profit (assuming positive notional

amount) to the fixed-rate receiver and the resulting gain is then used to offset the higher

cost of the pension liability. Similarly when the mortality rate increases unexpectedly,

then there is a reduction in the pension liability but the hedging portfolio suffers a loss.

Hence the gain from the pension plan is used to compensate the loss from the hedging

portfolio. In either case, the fixed-rate receiver of the q-forward is serving as a hedge

against the plan sponsor’s pension liability.

One immediate property associated with X(τ) is given in the following proposition:

Proposition 2. E0[X(τ)] is independent of the control process {ht}t=0,1,2,...,τ−1 and given by

E0[X(τ)] = −τ∑t=1

(1 + r)−tS(H)x0,0

(t). (12)

Proof. The right-hand-side of (12) is independent of the control process. Thus, it remains to

verify equation (12). Indeed, by Proposition 1, we have Et[Qt+1(t)] = 0, ∀ t > 0, and therefore,

E0[X(τ)] = E0

[τ∑t=1

(1 + r)−t(ht−1 ·Qt(t− 1)− S(H)

x0,0(t))]

=

[τ∑t=1

(1 + r)−t(ht−1 · E0 [Qt(t− 1)]− E0

[S(H)x0,0

(t)])]

= −τ∑t=1

(1 + r)−tE0

[S(H)x0,0

(t)].

The above proposition asserts that, on average, the time-0 value of the future pension li-

ability is not affected by the proposed hedging strategy. This is to be expected since there is

13

no upfront cost in implementing the proposed dynamic hedging strategy. While the proposed

hedging strategy has no impact on E0[X(τ)], the variability of X(τ), however, is not invariant

to the choice of {ht, t = 0, 1, . . . , τ − 1}. In fact, it is prudent to choose a hedging strategy that

results in low variability of X(τ) in order to reduce the plan sponsor’s exposure to population

basis risk. In other words, an optimal longevity hedging strategy can be formulated as a solution

to the following mean-variance optimization problem:

max{ht}t=0,1,2,...,τ−1

{E0[X(τ)]− γ

2Var0[X(τ)]

}, (13)

where γ > 0 can be interpreted as the risk aversion coefficient of the hedger.

There are several justifications for seeking an optimal longevity hedging strategy under the

mean-variance framework. First and foremost is the popularity of the mean-variance analysis.

Since the pioneering work of Markowitz in 1952, the mean-variance analysis has remained a

prominent benchmark in balancing the tradeoff between return and risk. Second, formulating an

optimal longevity hedging strategy that reduces the variance of the plan sponsor’s risk exposure

is a commonly adopted criterion in longevity risk management; see, for example, Cairns (2011),

and Zhou and Li (2016). Consistent with the existing literature, the optimal hedging strategy

from our proposed mean-variance framework (13) can be interpreted as one that minimizes the

variance of the plan sponsor’s time-0 net cash flow X(τ) (and independent of the risk aversion

γ > 0). This follows immediately from Proposition (2) that the mean term in (13) is independent

of the control process. This, in turn, implies that the mean-variance optimization problem (13)

is equivalent to the following variance minimization quadratic hedging problem:

{h∗t}t=0,1,2,...,τ−1 = argmin{ht}t=0,1,2,...,τ−1

E0[X(τ)2], (14)

where {h∗t}t=0,1,2,...,τ−1 is adapted to {Ft}t=0,1,2,...,τ−1.

14

3 Derivation of the optimal solution

Since the mean-variance problem (13) shares the same optimal solution with problem (14), we

focus on solving the latter. We will resort to the dynamic programming principle for an optimal

solution.

3.1 The Bellman equation

In this subsection, we resort to the dynamic programming principle in obtaining the Bellman

equation for the value process of problem (14).

Let Jt := Et[(X(τ) −X(t))2] for t = 0, 1, . . . , τ − 1. Then, it follows from (11) and (10)

that, for t = 0, 1, ..., τ − 1,

Jt = Et

( τ∑s=t+1

(1 + r)−sF (s)

)2

= Et

( τ∑s=t+1

(1 + r)−s(ht−1 ·Qt(t− 1)− S(H)

x0,0(t)))2

. (15)

We further denote the time-t value function by

Vt := min{hs}s=t,t+1,...,τ−1

Jt, t = 0, 1, ..., τ. (16)

Before deriving the Bellman equation for problem (14), we apply (8) to obtain the following

result: for t = 0, 1, . . . , τ − 2 and s = t+ 2, . . . , τ , we have

Et+1[F (s)] = Et+1

[hs−1Qs(s− 1)− S(H)

x0,0(s)]

= Et+1 [hs−1Es−1 (Qs(s− 1))]− Et+1

[S(H)x0,0

(s)]

= −Et+1

[S(H)x0,0

(s)]. (17)

15

Consequently, for t = 0, 1, . . . , τ − 1, we can rewrite the objective function Jt as

Jt = Et

Et+1

((1 + r)−(t+1)F (t+ 1) +τ∑

s=t+2

(1 + r)−sF (s)

)2

= Et

{(1 + r)−2(t+1)F (t+ 1)2 + 2(1 + r)−(t+1)F (t+ 1)Et+1

[τ∑

s=t+2

(1 + r)−sF (s)

]

+ Jt+1

}

= Et

{Jt+1 + (1 + r)−2(t+1)h2tQ

2t+1(t) + (1 + r)−2(t+1)S

(H)x0,0

(t+ 1)2

−2(1 + r)−(t+1)htQt+1(t)Et+1

[τ∑

s=t+1

(1 + r)−sS(H)x0,0

(s)

]

+2(1 + r)−(t+1)S(H)x0,0

(t+ 1)Et+1

[τ∑

s=t+2

(1 + r)−sS(H)x0,0

(s)

]}, (18)

where, by convention,∑b

s=a = 0 for a > b.

Combining (11), (16) and (17) yields the Bellman equation for t = 0, 1, . . . , τ − 1:

Vt = minht

Et

{Vt+1 + (1 + r)−2(t+1)h2tQ

2t+1(t) + (1 + r)−2(t+1)S

(H)x0,0

(t+ 1)2

−2(1 + r)−(t+1)htQt+1(t)Et+1

[τ∑

s=t+1

(1 + r)−sS(H)x0,0

(s)

]

+2(1 + r)−(t+1)S(H)x0,0

(t+ 1)Et+1

[τ∑

s=t+2

(1 + r)−sS(H)x0,0

(s)

]}. (19)

For the boundary condition, it is obvious that Vτ = 0.

3.2 Solution of the Bellman equation

From the preceding subsection, we know that equation (19) with the boundary condition Vτ = 0

constitute the Bellman equation for problem (14). Let h∗t be the solution from solving (19) for

16

each t = 0, 1, . . . , τ − 1. Below we will verify that {h∗t}t=0,1,...,τ−1 also solves problem (14).

To proceed, we note that, for t = 0, 1, . . . , τ , Jt does not depend on the whole control

process and instead, but only on the truncated process {ht, ht+1, . . . , hτ−1}. In what follows,

we may write Jt as Jht when it is necessary to emphasize the dependence of Jt on the truncated

control process {ht, ht+1, . . . , hτ−1}. We will show that Jh∗t ≤ Jht for any feasible h, t =

0, 1, . . . , τ − 1. Indeed, for t = τ − 1, equation (15) reduces to

Jτ−1 = Eτ−1

[(1 + r)−2τ

(hτ−1Qτ (τ − 1)− S(H)

x0,0(τ))2]

,

while (19) becomes

Vτ−1 = minhτ−1

Eτ−1

[(1 + r)−2τ

(hτ−1Qτ (τ − 1)− S(H)

x0,0(τ))2]

. (20)

This obviously means Jh∗τ−1 = Vτ−1 ≤ Jhτ−1 for any feasible h.

For t = 0, 1, ..., τ − 2, we assume Jh∗s = Vs ≤ Jhs for s = t+ 1, . . . , τ − 1 and any feasible

h. This assumption, along with equations (11), (18) and (19), yields

Jh∗

t = Et

{Jh

∗

t+1 + (1 + r)−2(t+1)(h∗t )2Q2

t+1(t) + (1 + r)−2(t+1)S(H)x0,0

(t+ 1)2

−2(1 + r)−(t+1)h∗tQt+1(t)Et+1

[τ∑

s=t+1

(1 + r)−sS(H)x0,0

(s)

]

+2(1 + r)−(t+1)S(H)x0,0

(t+ 1)Et+1

[τ∑

s=t+2

(1 + r)−sS(H)x0,0

(s)

]}

= minht

Et

{Vt+1 + (1 + r)−2(t+1)h2tQ

2t+1(t) + (1 + r)−2(t+1)S

(H)x0,0

(t+ 1)2

−2(1 + r)−(t+1)htQt+1(t)Et+1

[τ∑

s=t+1

(1 + r)−sS(H)x0,0

(s)

]

+2(1 + r)−(t+1)S(H)x0,0

(t+ 1)Et+1

[τ∑

s=t+2

(1 + r)−sS(H)x0,0

(s)

]}= Vt.

17

Furthermore, for any feasible h the quantity in the second last equality is less than

Jht = Et

{Jht+1 + (1 + r)−2(t+1)h2tQ

2t+1(t) + (1 + r)−2(t+1)S

(H)x0,0

(t+ 1)2

−2(1 + r)−(t+1)htQt+1(t)Et+1

[τ∑

s=t+1

(1 + r)−sS(H)x0,0

(s)

]

+2(1 + r)−(t+1)S(H)x0,0

(t+ 1)Et+1

[τ∑

s=t+2

(1 + r)−sS(H)x0,0

(s)

]}.

Thus, by induction, we can conclude that h∗ is an optimal control for problem (14).

The following proposition gives the optimal control process h∗ that solves the Bellman

equation (19):

Proposition 3. The optimal control process {h∗t}t=0,1,...,τ−1 is given by

h∗t =

Et

{Qt+1(t)

∑τs=t+1(1 + r)−[s−(t+1)]S

(H)x0,0

(s)

}Et

[Q2t+1(t)

] , (21)

where, as shown in equation (7), we have

Qt+1(t) = (1 + r)−(T∗−1)

(p(R)xf ,t+T ∗−1(1, Kt+1, k

(R)t+1)− p

(R)xf ,t+T ∗−1(1, Kt, k

(R)t )). (22)

Proof. The proof follows trivially by noticing that Vt+1 is independent of ht and the right-hand-

side of equation (19) is a quadratic function of ht, t = 0, 1, . . . , τ − 1.

Remark 1. Proposition 3 shows that, for t = 0, 1, ..., τ − 1, the optimal control h∗t does not

depend on the entire history of the mortality improvements, but only on the current state variable

{t,Kt, k(H)t , k

(R)t }. This feature significantly simplifies the computation of the optimal hedging

strategy.

Remark 2. As stated in the above proposition, the determination of the optimal hedging strat-

egy {h∗t}t=0,1,...,τ−1 boils down to evaluating equations (21) and (22), which entails evaluating

18

nested conditional expectations involving two forward mortality rates. While the nested con-

ditional expectations can be evaluated numerically via the so-called “nested” Monte Carlo

method, the underlying approach is inherently computational intensive and hence hinders its

practicality. Motivated by the need of providing a practical and yet efficient way of evaluating

equations (21) and (22), next section will explore an approximation method that enables us

to numerically determine the optimal hedging strategy {h∗t}t=0,1,...,τ−1 using the crude Monte

Carlo method but without the “nested” feature. The numerical studies to be presented confirm

the effectiveness of the proposed approximation coupled with the crude Monte Carlo method.

4 Hedging Canadian mortality rates with UK mortality rates

By using an example which involves hedging Canadian mortality rates based on UK mortal-

ity rates, this section compares and evaluates the effectiveness of various hedging strategies,

including our strategy proposed in Section 3. Subsection 4.1 describes the data source, summa-

rizes our assumptions and introduces the criterion to evaluate hedging performance. Subsection

4.2 describes an approximation method that enables us to overcome the “nested” computational

issue (see Remark 2). Subsection 4.3 compares the hedging performance between our optimal

strategy and other benchmarks, including the “delta” strategy. The numerical results demon-

strate that our proposed hedging scheme consistently outperforms the “delta” strategy in terms

of variance reduction. Subsections 4.4, 4.5 and 4.6 conduct several sensitivity tests to high-

light that our proposed method is generally robust to the changes in the hedging instrument’s

reference age, the hedging instrument’s time to maturity and the underlying mortality model,

respectively.

4.1 Data and assumptions

In this subsection we describe the data and model assumptions pertaining to our numerical

illustration. In Subsections 4.1 to 4.5, we assume that future mortality improvements follow

the ACF model introduced in Subsection 2.1, and model parameters are calibrated from the

mortality data of Canadian unisex population and UK unisex population aged 60 to 89 over

19

the period of 1966 to 20051. The calibration process follows the procedures escribed in Zhou

and Li (2016), which is based on a first-order singular value decomposition (SVD) procedure.

In this example, Canadian unisex population is referred as population H , and the UK unisex

population is referred as population R. Other main assumptions of our studies are:

• The pension liability is from a single cohort of individuals aged x0 = 60 at time 0 from

population H . The total notional amount is $1.

• The scale of the pension plan is large enough so that we do not consider sample risk.

• The hedging horizon is τ = 30 years.

• The q-forwards used as hedging instruments are linked to death rates of populationR. The

reference age for all the contracts is initially fixed at xf = 75 and the time to maturity at

inception is T = 10 years.

• The risk free rate is r = 4%.

We use Hedge Effectiveness (HE), which measures the proportion of variance reduction of

a hedging strategy relative to the unhedged position, to quantify the performance of a hedging

strategy. Formally, HE is defined as

HE = 1− Var[time-0 value of hedged portfolio]

Var[time-0 value of unhedged portfolio]

= 1− Var[X(30)]

Var[∑30

t=1(1 + r)−tS(H)x0,0

(t)], (23)

where X(30) is defined by (10). By definition, whenever the hedging strategy has a non-

negative impact on the hedger’s position, HE is a real number between 0 and 1. A larger

HE implies that the hedging strategy is more effective at reducing the time-0 variance of the

hedger’s position, and vice versa. HE = 1 will be achieved only if the hedge is perfect. Such

an ideal case, in general, is impossible in view of the mismatch in mortality rates between the

Canadian population and the UK population. Once we obtain the optimal hedging strategy us-

ing equations (21) and (22) based on a numerical procedure, we can use Monte Carlo simulation1Data source: http://www.mortality.org

20

to estimate variance of the time-0 value of both hedged and unhedged cash flows, and thus the

HE, to assess its performance.

4.2 An approximation to forward mortality rates

As alerted in Remark 2, it is computationally intensive to use the “nested” Monte Carlo method

to estimate the optimal strategy {h∗t}t=0,1,2,...,29. As a result, this calls for more efficient nu-

merical procedure of evaluating both equations (21) and (22). One possible approach to avoid

evaluating the nested conditional expectations is to exploit the first-order approximation for

forward mortality rates as proposed by Cairns (2011) and Zhou and Li (2016). This approxima-

tion method is facilitated by the first-order Taylor expansion of estimating the forward survival

rates, instead of calculating the mean of their true distributions. Under this approximation,

equation (22) can be evaluated analytically so that we only need to numerically evaluate equa-

tion (21). This approximation circumvents the “nested” issue of the Monte Carlo method and

therefore significantly reduces the computational burden for calculating the hedging strategies.

The adopted approximation formula for the longevity hedging problem corresponds to the ACF

model is provided in Appendix A.1 (see Zhou and Li (2016) for the proof). A similar approxi-

mation formula but for the CBD model is derived in Appendix A.2. This formula will be used

later in Subsection 4.6.

4.3 Baseline result

The preceding subsection has provided an alternate way of estimating the optimal hedging strat-

egy {h∗t}t=0,1,2,...,29 by combining Monte Carlo method with an approximation formula to for-

ward mortality rates. As argued in the preceding subsection the proposed alternate Monte Carlo

method is computational more efficient (relative to the “nested” Monte Carlo method), though

we have yet to assess its effectiveness at variance reduction. To do this we rely on numerical

examples and benchmark our proposed Monte Carlo method against two other methods. The

first method is based on the “nested” Monte Carlo and the second method corresponds to the

“delta” hedging strategy proposed by Cairns (2011) and Zhou and Li (2016).

Let h̃t, t = 0, 1, ..., 29 represent the time-t q-forward position under the “delta” hedging

21

strategy. The position h̃t is determined in such a way that the hedging portfolio and the pension

plan’s future liability have the same sensitivity to the mortality index Kt. In other words, h̃t is

the solution to

∂[∑τ

s=1(1 + r)−sp(H)x0+t,t(s,Kt, k

(H)t )

]∂Kt

= h̃t ·∂Qt(t− 1)

∂Kt

. (24)

While the above delta hedging strategy has the advantages of simplicity and computationally

less intensive to implement, its downside is that the hedging strategy is based on a heuristic idea

and there is no assurance of any optimality in terms of mitigating longevity risk.

We now describe our numerical experiments. In particular, we consider the following five

implementations to determine the optimal hedging strategies:

• Method 1a: This method is implemented in two steps. In the first step, we analyti-

cally calculate Qt+1(t), which is defined in equation (22), using the approximation equa-

tion (A.26) for the two forward mortality rates. Accordingly, no nested simulation is

necessary for the computation of the two conditional expectations in the expression of h∗t .

In the second step, we simulate M = 1, 000 paths of the mortality rate process and com-

pute the two conditional expectations in equation (21) by averaging over the M = 1, 000

simulated paths of the mortality rate process. We then estimate h∗t by the ratio of the two

computed conditional expectations.

• Method 1b: The same simulation procedure as Method 1a except with M = 10, 000.

• Method 1c: The same simulation procedure as Method 1a except with M = 100, 000.

• Method 2: The “delta” hedging method based on equation (24).

• Method 3: This method directly apply a brutal “nested” Monte Carlo procedure. Recall-

ing equations (21) and (22), we notice that h∗t is the ratio of two conditional expectations

of quantities depending on Qt+1(t) and that Qt+1(t) depends on two forward mortality

rates which by their definitions are conditional expectations. In other words, both the nu-

merator and denominator in the expression of h∗t are given by double conditional expecta-

tions. In this brutal “nested” Monte Carlo simulation method, we simulate M = 10, 000

22

paths of mortality rates for the evolution of the outer conditional expectation and based

on each of the simulated path we further independently simulate M̃ = 10, 000 paths of

mortality rates for the evaluation of the inner conditional expectation.

It is worth stressing that the description in the above is only for the computation of h∗t for

a given t and the specific simulation procedure depends on the initial state of mortality process

at time t, i.e., (Kt, k(H)t , k

(R)t ). In other words, we need to simulate a path of mortality rates

before we can numerically apply any of the above methods for the computation of the hedging

strategy {h0∗, h∗1, . . . , h∗τ−1}, and different simulated paths (which should be independent of the

simulation of mortality rate process involved in the above five methods) may lead to different

values of {h∗0, h∗1, . . . , h∗τ−1}.In order to investigate the performance of each of the above five method in reducing the

variance of the pension liability, we simulate N = 2, 000 mortality rate paths according to

the ACF model, and consequently apply each of the above five methods to construct hedge

portfolios and compute the time-0 value of the pension liability with the hedging strategies.

In the meanwhile, we also calculate the time-0 value of pension liability without a hedging

strategy. These leads to N = 2, 000 time-0 values of pension liability for both the hedged

and unhedged portfolios, respectively. We can then use these N = 2, 000 time-0 values to

estimate the variance of the time-0 values of pension liability for both the hedged and unhedged

portfolios, and we can consequently use equation (23) to compute the HE. The resulted values

for the HE from each method are reported in the row ”Mean” of Table 1.

To quantify the stability of the obtained HE, we further apply a bootstrap procedure with

resampling size of Nb = 100, 000 to estimate the variance of the estimator from the above

procedure for the HE. Specifically, based on these N = 2, 000 time-0 values of pension liability

we obtained from a simulation procedure, we independently draw Nb = 100, 000 resamples of

size 2, 000 with replacement, and for each resample, we can compute a value of the measure HE.

So we eventually obtain Nb = 100, 000 values of HE and we report their minimum, variance,

first quartile, second quartile, third quantile and maximum values in the corresponding rows of

Table 1. Further, we also report a 95% confidence interval of the HE for each method in the

table. Such confidence interval is composed by the 2.5-percentile and the 97.5-percentile of

these obtained Nb = 100, 000 values of HE.

23

HE Method 1a Method 1b Method 1c Method 2 Method 3

Mean 0.9149 0.9147 0.9213 0.8938 0.9153Variance 1.5812e-005 1.3876e-005 1.2142e-005 2.3656e-005 1.4084e-005

Minimum 0.8962 0.8979 0.9035 0.8699 0.8968First quartile 0.9123 0.9122 0.9190 0.8906 0.9128

Second quartile 0.9150 0.9148 0.9214 0.8939 0.9154Third quartile 0.9177 0.9172 0.9237 0.8971 0.9179

Maximum 0.9299 0.9301 0.9341 0.9127 0.930095% confidence interval (0.9069,0.9225) (0.9072,0.9218) (0.9143,0.9279) (0.8839,0.9030) (0.9077,0.9224)

Computational time 0.27 1.56 12.18 0.16 117.51(hours)

Table 1: Hedge effectiveness of five hedging strategies

An immediate observation can be made from Table 1 is that while Method 3 is the theoreti-

cally preferred method of determining the optimal hedging strategy, the nested implementation

of the Monte Carlo method leads to huge computational cost of 117.5 hours. This is imprac-

tical, especially its computational cost is unacceptably high when compared to Method 1a and

Method 2 which, respectively, require only 0.27 hours and 0.16 hours of computational time.

Recall that Methods 1a, 1b and 1c are based on the Monte Carlo method combined with

the approximation formula to forward mortality rates. The only difference among them is the

number of sample paths M used to estimate the optimal hedging strategy. More specifically,

Method 1a uses M = 1, 000 sample paths while Methods 1b and 1c use 10 times and 100

times, respectively, of the number of paths. A larger sample size implies a higher precision in

estimating HE, as we can tell from the smaller variance estimates for Methods 1b and 1c. The

benefit of a larger sample size, however, does not come at no price. It comes with a higher

expense in term of computational time.

Comparing Method 3 (i.e., the brutal “nested” Monte Carlo method) with Methods 1a, 1b

and 1c (i.e., the methods resulted from an approximation to forward mortality rates), we can

conclude that the the approximation to forward mortality rate works quite efficiently. Indeed,

the average HE (see row ”Mean” in Table 1) from Methods 1a, 1b and 1c are 91.49%, 91.47%

and 92.13%, respectively, and these are all favourable compared with an average of HE at

91.53% from Method 3. In particular, despite the huge computational time used in Method

3, Method 1c outperforms Method 3 in terms of mean, variance, minimum, maximum and

24

quartiles of the N = 2, 000 simulated HE’s.

Finally, let us benchmark our proposed Methods 1a, 1b and 1c against the “delta” hedging

of Method 2. First note that Method 2 is computationally efficient, it is the fastest one among

the five methods. In term of HE, the “delta” hedging method, however, yields the worst per-

formance. Among the rest four methods, Method 1a applies the smallest number of simulated

paths M = 1, 000 in approximating the outer expectation in the formula of h∗t , but it still yields

an average HE significantly larger than that of 2the “delta” hedging method (i.e., Method 2).

Moreover, the 95% bootstrap confidence interval of the HE for Method 2 is located on the left

of those for the other methods and it does not have any overlapping with those for the other

methods. In order to better demonstrate the huge difference in HE between Methods 1 and 2,

we conduct a two-sample t-test with unequal variances on the HE between Method 1a (which is

the least precise method among Methods 1a, 2a and 3a) and Method 2. The t-test result rejects

the null hypothesis with a p-value less than 10−8, and therefore strongly suggests the significant

efficiency of our proposed strategy relative to the “delta” strategy. Additionally, all the HE from

Method 1 have smaller variance than Method 2, which again means that the results calculated

from our proposed method are more stable. Finally, it should be noted that the computational

time of Method 1a is less than double that of Method 2 and thus it is more computational effi-

cient.

To conclude this subsection, we point out that while the computation time reported in Table 1

appears to be quite long for some cases, it should be emphasized that these are the total running

time for the entire experiment. In the above experiment, we need to compute the whole hedging

process {h∗0, h∗1, . . . , h∗τ−1} for each of theN = 2, 000 simulated paths of mortality rate process,

because we need to evaluate the distribution of hedging error based on the N = 2, 000 time-0

values of hedged and unhedged pension liability. For a practical application, however, we do

not need to simulate these N = 2, 000 paths of mortality rate process, and instead we only need

to compute a single value of h∗t based on the information observable at time t. Indeed, it only

takes about 7 seconds to compute h∗0 even for Method 3, the most computational demanding

one among the five methods we considered in the experiment. Thus, our proposed method is

computationally convenient to implement for practical use.

25

4.4 Robustness to q-forwards’ time to maturity

In this subsection we examine the robustness of the HE of our proposed hedging strategy with

respect to the maturity of the q-forwards. In our previous baseline example considered in sub-

section 4.3, the q-forwards’ time to maturity T ∗ was set at 10 years, which is one third of our

hedging horizon τ = 30 years; and the q-forwards’ reference age was set at xf = 75 which is

approximately the average age of the underlying population during the hedging period. How-

ever, in practice, a certain type of q-forward contract may not be always liquidly traded in the

market. Therefore it is of interest to evaluate the HE of our proposed strategy if the hedging

instruments are based on different maturities and reference ages.

In this subsection we repeat our experiments using Methods 1c and 2 described in subsection

4.3, by keeping reference age fixed at xf = 75 and varying the q-forwards’ time to maturity

with T ∗ = 3, 4, ..., 20 years. Simulation results on the HE with different time to maturity are

depicted in Figure 1.

2 4 6 8 10 12 14 16 18 20

time to maturity

0.87

0.88

0.89

0.9

0.91

0.92

0.93

Hed

ge E

ffect

iven

ess

OptimalDelta

Figure 1: HE with different time to maturity

From Figure 1 we can see that the obtained HE is quite stable with respect to the maturity

26

of q-forwards, T ∗. For our proposed optimal hedging strategy, the estimated HE lies within the

interval [91%, 92%] while the dynamic “delta” method is consistently less effective. Since we

adjust our hedging position in the hedging instrument every year based on the latest information

from the market, we would not be very concerned if the amount of hedging instrument used for

next year does not reflect mortality change in the far future. Additionally, when the q-forwards’

time to maturity is too long, for example, T ∗ = 20 years, it may become less effective as time

goes by because the mortality rate, which it is linked to, will be out of our hedging horizon.

The numerical study reveals another advantage of hedging the pension liability dynamically,

as opposed to statically. The static hedging strategy typically requires long-dated longevity in-

struments. These instruments tend to be rare, expensive and illiquid. In contrast, the dynamic

hedging strategy can easily be implemented using longevity instruments with shorter term to

maturity and because of liquidity, these securities are more appealing to investors and specula-

tors.

4.5 Robustness to q-forwards’ reference age

Next we investigate the impact of q-forwards’ reference age on the hedging performance. We

fix time to maturity at T ∗ = 10 years as in our baseline example, and respectively consider

reference ages xf = 60, 61, ..., 84. Since the underlying cohort of the pension plan liability are

aged at 60 at inception, it does not make sense if we use q-forwards with reference ages that

are lower than 60; for very high ages such as 85, it is unlikely to reflect our pension liability as

well. This explains why we consider those values as the range for the reference age xf . Results

of the HE for different reference ages are shown in Figure 2.

Figure 2 shows that the HE indeed heavily depends on q-forwards’ reference age. HE

remains above 85% within the interval of 71 ≤ xf ≤ 81, while it fluctuates drastically for

xf < 71 or xf > 81. This is consistent with our conjecture that the best hedging performance

should be obtained by using q-forwards that are linked to approximately the average age of the

pension plan cohort during our hedging horizon. The curves shown in Figure 2 are not smooth

due to the fact that age effects captured by Bx and b(i)x , i = 1, 2 in the ACF model are not

necessarily smooth functions of x. Last but not least, we can also see that our hedging method

27

60 65 70 75 80 85

reference age

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Hed

ge E

ffect

iven

ess

OptimalDelta

Figure 2: HE with different reference ages

generally outperforms the dynamic “delta” method, despite the unstable hedging performance

when the reference age xf is too large or too small.

4.6 Robustness to model risk

4.6.1 The Cairns-Blake-Dowd (CBD) model

In this subsection, we study the impact of model risk, i.e., the risk coming from misidentifying

the actual stochastic longevity model. In order to do so, we introduce the two-population Cairns-

Blake-Dowd (CBD) model defined as follows:

logit(q(i)x,t) := ln

(q(i)x,t

1− q(i)x,t

)= κc1,t + κc2,t(x− x̄) + κ

(i)1,t + κ

(i)2,t(x− x̄) + ε

(i)x,t, (25)

where i = H,R, are referred as two populations; x̄ denotes the average age over the sample

age range [xa, xb] (defined to be [60,89] in this example); κc1,t and κc2,t are time-varying factors

28

which are common to both two populations; κ(i)1,t and κ(i)2,t are time-varying factors applied to the

specific population i; ε(i)x,t captures all the remaining variations and the error terms for different

populations, ages and times are assumed to be independent.

Further, the common factors κc1,t and κc2,t are modeled by a bivariate random walk with drift,

i.e., ∀t = 1, 2, 3, ..., κc1,t = µc1 + κc1,t−1 + ηc1,t,

κc2,t = µc2 + κc2,t−1 + ηc2,t,

where ηc1,t and ηc2,t are constantly correlated, i.e., the correlation coefficient ρ(ηc1,t, ηc2,t) is a

constant number that does not change with time t. Moreover, the population specific factors

κ(i)1,t and κ(i)2,t are modeled by two correlated AR(1) model, ∀t = 1, 2, 3, ...

κ(i)1,t = µ

(i)1 + φ

(i)1 κ

(i)1,t−1 + η

(i)1,t,

κ(i)2,t = µ

(i)2 + φ

(i)2 κ

(i)2,t−1 + η

(i)2,t,

where η(i)1,t and η(i)2,t are also constantly correlated.

4.6.2 Hedging results with model risk

With model risk involved, we would expect a lower HE when applying our proposed hedging

scheme due to the additional error from the model misspecification. To demonstrate this we

investigate model risk by comparing results from the four following cases.

• Case 1: the “true” model is CBD model, but the calculation of hedging strategy is based

on ACF model (referred as “assumption” model in the rest of this section).

• Case 2: the “true” model is ACF model, but the “assumption” model is CBD model.

• Case 3: the “true” model is CBD model, and the “assumption” model is CBD model.

• Case 4: the “true” model is ACF model, and the “assumption” model is ACF model. This

is the same as baseline results.

29

“True”/“assumption” model mean variance min max 95% C.I.

Case 1 (CBD/ACF) 0.6437 1.7231× 10−4 0.5828 0.6942 (0.6174,0.6687)Case 2 (ACF/CBD) 0.8712 3.2003× 10−5 0.8416 0.8952 (0.8597,0.8819)Case 3 (CBD/CBD) 0.6492 1.7890× 10−4 0.5839 0.7029 (0.6222,0.6747)

Case 4 (ACF/ACF, baseline) 0.9213 1.2142× 10−5 0.9035 0.9341 (0.9143,0.9279)

Table 2: HE results with different “true” and “assumption” models

We take the following numerical procedure for each of the above four cases:

Step 1. Simulate N = 2, 000 mortality sample paths using the “true” model.

Step 2. Based on each simulated path, calibrate model parameters for the “assumption” model.

If the “true” model and the “assumption” model are identical, skip this step.

Step 3. Calculate optimal hedging strategy for each time point on each simulated path based on

the calibrated model parameters of the “assumption” model.

Step 4. Calculate time-0 value of both hedged and unhedged position based on simulated paths

and hedging strategy obtained in step 3.

Step 5. Apply a bootstrap procedure with resampling size of Nb = 100, 000 to obtain an estima-

tion for the distribution of the HE.

To focus on model risk, in this example we fix the q-forward contract with reference age

xf = 75 and time to maturity T ∗ = 10, to be consistent with the parameter values in our

baseline experiments. Results of the HE under these four cases are shown in Table 2.

Table 2 shows that the HE by adopting our proposed hedging strategy is quite stable even

when the “assumption” model deviates from the “true” model. Comparing Case 2 with baseline

result Case 4, we can see that the average HE decreases from 92.13% to 87.12% if the “as-

sumption” model is misidentified as the CBD model. All the percentiles also decrease slightly

and variance of the HE increases substantially, from 1.2142 × 10−5 to 3.2003 × 10−5. When

the model is misspecified, additional hedging error occurs due to the discrepancy between the

“true” model and the “assumption” model, and the hedging strategy becomes less accurate,

however, the HE we obtain in this example should still be considered as satisfactory in general.

30

When the “true” model is the CBD model (Cases 1 and 3), we observe a similar pattern.

If the underlying model is correctly specified, the HE is slightly better in terms of average and

percentiles, but the difference is quite small. It further implies that our hedging strategy is not

sensitive to model risk.

It is worth mentioning that the difference in HE achieved between “true” model of CBD

cases (Cases 1 and 3) and “true” model of ACF cases (Cases 2 and 4) is not caused by the

hedging strategy, but the implied population basis risk of different models. In this paper cor-

relations between mortality rates of different populations are modeled by common factors, and

those model parameters we use in this example are estimated by a least square procedure. As a

result, correlation between these two populations under different mortality model assumptions

are not necessarily the same or even close, even though model parameters are calibrated from

the same data set. Table 2 shows that in this example there is much larger population basis risk

if future mortality rates follow the two-population CBD model calibrated from the dataset.

4.6.3 Correlation coefficients under different models

In what follows, we provide some explanation about the seemingly large difference in basis

risk we observed under different “true” model assumptions in section 4.6.2, by comparing the

correlation coefficients between some representative death rates of two longevity models: the

ACF model and the CBD model. To be consistent with our original hedging problem assump-

tions, we choose q(H)59+t,t and q(R)

75,t+9, t = 1, 2, 3, ..., 30, as the representative death rates. Based

on each of the ACF model and the CBD model calibrated from the same data set, we simulate

N = 10, 000 future mortality paths and calculate correlation coefficient ρ(q(H)59+t,t, q

(R)75,t+9) for

t = 1, 2, 3, ..., 30, and numerical results are shown in Figure 3.

Figure 3 shows that the ACF model and the CBD model have different patterns to model

the correlation between death rates q(H)59+t,t and q

(R)75,t+9 of two populations. When t is small,

correlation coefficients ρ(q(H)59+t,t, q

(R)75,t+9) for ACF are generally larger, however, as t increases,

ρ(q(H)59+t,t, q

(R)75,t+9) for CBD model surpasses that from ACF model till the end of our hedging

horizon T = 30 years. Because the pension plan liability is more dependent on mortality rates

in early years of the hedging horizon, intuitively it explains why there is much higher basis risk

when the underlying model is the CBD model. A more comprehensive and rigorous analysis on

31

0 5 10 15 20 25 30

Time

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Cor

rela

tion

coef

ficie

nt

CBDACF

Figure 3: Change with time to maturity

the modeling issue of multi-population longevity models is beyond the scope of this paper and

worth more thorough investigations in the future.

5 Conclusion

In this paper we study the optimal dynamic hedging strategy for pension plans that are ex-

posed to longevity risk and basis risk. Under the commonly used longevity models such as

the Augmented Common Factor model and the Cairns-Blake-Dowd model, we derive a semi

closed-form hedging plan for the pension plan sponsor which minimizes the variance of time-0

value of the total future liabilities. Further with a Monte Carlo simulation procedure and an ap-

proximation formula for forward mortality rates, we show that numerical calculation of optimal

hedging strategy only requires moderate computational time and therefore it offers an effective

solution for practical applications.

As the theoretically best strategy under the variance criterion, our proposed hedging scheme

is also shown to outperform the “delta” strategy in terms of both effectiveness and stability.

32

Additionally, extensive numerical results show that the hedging performance achieved by our

hedging plan is robust to the hedging instrument we use and the underlying longevity model we

choose. One useful discovery is that for dynamic longevity hedging plans, it is unnecessary to

choose those long-maturity longevity instruments that tend to be more costly and less liquid, in

order to achieve a satisfactory hedging performance.

33

References and Notes

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survival functions and Dynamic hedging. Insurance: Mathematics and Economics, 49,

438-453.

[4] Cairns, A.J.G., 2013. Robust hedging of Longevity Risk. Journal of Risk and Insurance,

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[11] Li, J.S.H. and Luo, A., 2012. Key q-Duration: A Framework for Hedging Longevity Risk,

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Extension of the Lee-Carter Method. Demography, 42, 575-594.

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Benefit Plan Considering Longevity Basis Risk. North American Actuarial Journal 18(1),

68-86.

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ities: The effectiveness of static hedging using longevity bonds and derivatives. Insurance:

Mathematics and Economics, 49(1), 100-114.

[17] Rosa, C.D., Luciano, E. and Regis, L., 2017. Basis risk in static versus dynamic longevity-

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35

[21] Zhou, K.Q. and Li, J.S.H., 2016. Dynamic Longevity Hedging in the Presence of Pop-

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[23] Zhou, R., Li, J.S.H. and Tan, K.S., 2013 “Pricing Standardized Mortality Securitizations:

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Appendix

A.1 Approximations for forward survival rates in Section 4.2

By Zhou and Li (2016), forward survival rates under the ACF assumption are approximated by

p(R)x,u (T,Kt, k

(R)t ) ≈ Φ

−E(V(R)u |Ft)√

Var(V(R)u |Ft)

, ∀u > t, (A.26)

where Φ(·) denotes the cumulative distribution function of a standard normal distribution, and

other components of the formula are defined as below:

E(V (R)u |Ft) = −D(R)

x,u,0(T )−D(R)x,u,1(T )(E(Ku|Ft)− E(Ku|F0))

−D(R)x,u,2(T )(E(k(R)

u |Ft)− E(k(R)u |F0)),

Var(V (R)u |Ft) = 1 + (D

(R)x,u,1(T ))2Var(Ku|Ft) + (D

(R)x,u,2(T ))2Var(k(R)

u |Ft)

+2D(R)x,u,1(T )D

(R)x,u,2(T )Cov(Ku, k

(R)u |Ft),

E(Ku|Ft) = Kt −K0 − Ct,

E(k(R)u |Ft) = (φ

(R)1 )u((φ

(R)1 )−tk

(R)t − k(R)

0 ) +(φ

(R)1 )u(1− φ(R)

1 )−t)

1− φ(R)1

φ(R)0 ,

36

Var(Ku|Ft) = σ2K(u− t),

Var(k(R)u |Ft) =

1− (φ(R)1 )2(u−t)

1− (φ(R)1 )2

σ2k,i,

Cov(Ku, k(R)u |Ft) = 0,

f(R)x,t (T, k

(R)t ) = Φ−1(p

(R)x,t (T, k

(R)t )),

D(R)x,t,0(T ) = f

(R)x,t (T,E(Kt|F0),E(k

(R)t |F0)),

D(R)x,t,1(T ) =

∂f(R)x,t (T,Kt,E(k

(R)t |F0))

∂Kt

∣∣∣∣∣Kt=E(Kt|F0)

,

D(R)x,t,2(T ) =

∂f(R)x,t (T,E(Kt|F0), k

(R)t )

∂k(R)t

∣∣∣∣∣k(R)t =E(k

(R)t |F0)

.

For proof, see Zhou and Li (2016).

A.2 Approximations for forward survival rates under CBD model in Sec-

tion 4.6

In order to show the following derivation more concisely, we look at one specific population

i and denote time-varying factors κc1,t, κc2,t, κ

(i)1,t and κ(i)2,t in the CBD model as κ1,t, κ2,t, κ3,t

and κ4,t, respectively. We also denote σ1 = σ(ηc1,t), σ2 = σ(ηc2,t), ρ1 = ρ(ηc1,t, ηc2,t), σ3 =

σ(η(i)1,t), σ4 = σ(η

(i)2,t), ρ2 = ρ(η

(i)1,t, η

(i)2,t) and denote the vector (κ1,t, κ2,t, κ3,t, κ4,t)

ᵀ as κt. Then

we apply the probit transformation and define f (i)x,t(T, κt) := Φ−1(p

(i)x,t(T, κt)). Therefore an

approximation based on Taylor’s theorem at κt = κ̂t := E[κt] is given by:

f(i)x,t(T, κt) = f̃

(i)x,t(T, κt)

= D(i)x,t,0(T ) +D

(i)x,t,1(T )ᵀ(κt − κ̂t) +

1

2(κt − κ̂t)ᵀD(i)

x,t,2(T )(κt − κ̂t),

where

D(i)x,t,0(T ) = f

(i)x,t(T, κ̂t),

37

D(i)x,t,1(T )j =

∂

∂κj,tf(i)x,t(T, κt)

∣∣∣∣κt=κ̂t

, j = 1, 2, 3, 4,

D(i)x,t,0(T ) is a 4× 1 vector with j-th element defined by D(i)

x,t,1(T )j,

D(i)x,t,2(T )jk =

∂2

∂κj,t∂κk,tf(i)x,t(T, κt)

∣∣∣∣κt=κ̂t

, j, k = 1, 2, 3, 4,

D(i)x,t,2(T ) is a 4× 4 matrix with jk-th element defined by D(i)

x,t,2(T )jk.

Therefore, by definition, when u > t,

p(i)x,u(T, κt) = E[p(i)x,u(T, κu)|Ft].

Applying a first-order approximation and letting Au := D(i)x,u,0(T ) + D

(i)x,u,1(T )ᵀ(κu − κ̂u),

we have:

p(i)x,u(T, κu) ≈ Φ[D(i)x,u,0(T ) +D

(i)x,u,1(T )ᵀ(κu − κ̂u)]

= Pr[Z ≤ Au|Fu] where Z is standards normal and independent of Au

= E[1{Z≤Au}|Fu].

Hence, letting Bu := Z − Au = Z −D(i)x,u,0(T )−D(i)

x,u,1(T )ᵀ(κu − κ̂u),

p(i)x,u(T, κt) ≈ E{E[1{Z≤Au}|Fu]|Ft}

= E[1{Z≤Au}|Ft]]

= Pr[Z ≤ Au|Ft]

= Pr[Bu ≤ 0|Ft]

= Φ

[−E(Bu|Ft)√Var(Bu|Ft)

],

where

E(Bu|Ft) = −D(i)x,u,0(T )−D(i)

x,u,1(T )ᵀ[E(κu)|Ft − κ̂u],

Var(Bu|Ft) = 1 + [E(κu)|Ft − κ̂u]ᵀVu[E(κu)|Ft − κ̂u],

38

where Vu is a 4× 4 covariance matrix with jk-th element defined by:

Vu,jk = Cov(κj,u, κk,u|Ft),

and

E(κj,u|Ft)− κ̂j,u = κj,t − κj,0 − µc1t, j = 1, 2,

E(κj,u|Ft)− κ̂j,u = (φ(i)j−2)

u((φ(i)j−2)

−tk(i)j−2,t − k

(i)j−2,0) +

(φ(i)j−2)

u(1− φ(i)j−2)

−t)

1− φ(i)j−2

µ(i)j−2, j = 3, 4,

Vu,jj = Var(κj,u|Ft) = σ2j (u− t), j = 1, 2,

Vu,12 = Vu,21 = Cov(κ1,u, κ2,u|Ft) = ρσ1σ2(u− t),

Vu,jj = Var(κj,u|Ft) =1− (φ

(i)j−2)

2(u−t)

1− (φ(i)j−2)

2σ2j , j = 3, 4,

Vu,34 = Vu,43 = Cov(κ3,u, κ4,u|Ft) =1− (φ

(i)1 φ

(i)2 )(u−t)

1− φ(i)1 φ

(i)2

ρσ3σ4,

Vu,13 = Vu,14 = Vu,23 = Vu,24 = Vu,31 = Vu,32 = Vu,41 = Vu,42 = 0.

39