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Dynamic Longevity Hedging Residual Risk Transferring
Dynamic Longevity Hedging in the Presence ofPopulation Basis Risk: A Feasibility Analysisfrom Technical and Economic Perspectives
Kenneth Q. Zhou and Johnny S.-H. Li
University of Waterloo
September 3, 2014
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Outline
Figure 1: The outline of the proposed dynamic hedging strategy.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Overview
1 Dynamic Longevity HedgingConstructing the dynamic hedging strategyA two-population example
2 Residual Risk TransferringConstructing the customized surplus swapA multi-population example
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the dynamic hedging strategy
Model and Approximation
The augmented common factor model (Li and Lee, 2005):
ln(m(i)x ,t) = a
(i)x + BxKt+1 + b
(i)x k
(i)t+1 + e
(i)x ,t , i = 1, 2, . . .
where
Kt follows a random walk:
Kt = c + Kt−1 + εt ;
k(i)t follows an AR(1) process:
k(i)t = c(i) + φ(i)k
(i)t−1 + ε
(i)t .
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the dynamic hedging strategy
The approximation method - an extension of the work of Cairns(2011):
f(i)x ,t (T ,Kt , k
(i)t ) = Φ−1(p
(i)x ,t(T ,Kt , k
(i)t ))
≈ D(i)x ,t,0(T ) + D
(i)x ,t,1(T ) · (Kt − K̂t)
+ D(i)x ,t,2(T ) · (k
(i)t − k̂
(i)t )
where
Φ−1 is the probit function;
p(i)x,t(T ,Kt , k
(i)t ) is the time-t spot
survival probability for T years;
K̂t = E(Kt |K0);
k̂(i)t = E(k
(i)t |k
(i)0 );
D(i)x,t,0(T ) = f
(i)x,t (T , K̂t , k̂
(i)t );
D(i)x,t,1(T ) =
∂f(i)x,t (T ,Kt ,k̂
(i)t )
∂Kt
∣∣∣∣Kt=K̂t
;
D(i)x,t,2(T ) =
∂f(i)x,t (T ,K̂t ,k
(i)t )
∂k(i)t
∣∣∣∣k
(i)t =k̂
(i)t
.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the dynamic hedging strategy
Evaluation of the Required Quantities
To construct the dynamic hedging strategy, the followingquantities are calculated using the approximation method:
The time-t present value of the pension liabilities.
The time-t present value of the hedging instruments.
The first derivative with respect to Kt of the time-t presentvalue of the pension liabilities, ∆liab
Kt.
The first derivative with respect to Kt of the time-t presentvalue of the hedging instruments, ∆hedge
Kt.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the dynamic hedging strategy
Dynamic Hedging
To determine the optimal hedge ratio, ht , the first derivativesof the pension liabilities and the hedging instruments withrespect to Kt are matched:
∆liabKt
= ht · ∆hedgeKt
.
At time t, ht units of the hedging instruments are purchasedto hedge the uncertainty of the pension liabilities from time tto t + 1.
At time t + 1, the hedging instruments purchased at time tare closed out to compensate for the shortfall of the pensionplan from the longevity risk.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the dynamic hedging strategy
Hedge Effectiveness
The hedge effectiveness at time t is calculated as:
HEt = 1 − Var(Ht − Lt)
Var(Lt)
where
Lt is the time-0 present value of the realized liabilities fromtime 0 to t and the future liabilities after time t;
Ht is the time-0 present value of the payoffs of the hedginginstruments up to time t with an initial reserve of H0 = L0.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A two-population example
A Two-Population Example
Two populations: Continuous Mortality Investigation (CMI)and England and Wales (EW)
Liability: 30 years of $1 pension liabilities subject to themortality experience of a male individual aged 60 in year 2005from the CMI population.
Hedging instrument: 10-year age-75 q-forward contractslinked to the EW population, which are unlimitedly availableand liquidly traded.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A two-population example
0 5 10 15 20 25 30
14.3
14.4
14.5
14.6
14.7
14.8
14.9
15CMI: Liability
Time
L t
Figure 2: The time-0 present value of the liabilities over time.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A two-population example
0 5 10 15 20 250.015
0.02
0.025
0.03
0.035
0.04
Time
Rat
e
EW: q−forward
Figure 3: The 10-year age-75 q-forward rate over time.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A two-population example
0 5 10 15 20 25 300
20
40
60
80
100
120
140
160
Time
h t
No Population Basis Risk
0 5 10 15 20 25 300
20
40
60
80
100
120
140
160
Time
h t
Population Basis Risk
Figure 4: The optimal hedge ratio over time.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A two-population example
0 5 10 15 20 25 30−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time
H t − L
t
No Population Basis RiskHE30 = 0.9996
0 5 10 15 20 25 30−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time
H t − L
t
Population Basis RiskHE30 = 0.9187
Figure 5: The hedge result over time.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the customized surplus swap
Motivations
Population basis risk exists if standardized index-linkedhedging instruments are used.
The degree of the population basis risk varies according to thepopulations involved in the hedge.
Population basis risk cannot be diversified within a pensionplan, but may be diversifiable across different pension plans.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the customized surplus swap
The Customized Surplus Swap
The swap is a cash exchange agreement between a pensionplan and a reinsurer.
It transfers population basis risk and sampling risk from oneparty to another.
The swap exchanges the economic surplus of a pension planafter the implementation of a dynamic longevity hedgingstrategy.
It eventually offloads the residual risk from the pension planto the counterparty.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the customized surplus swap
The Surplus of a Pension Plan
The total liability of the pension plan at time t is
Lt = CLt + FLt ,
where CLt and FLt are the time-t values of the current andfuture liabilities, respectively.
The hedging portfolio for the pension plan at time t is
Ht = (1 + r)(Ht−1 − CLt−1) + Pt ,
where Pt is the payoff of the hedging instruments at time t,and H0 = L0 is the initial reserve.
Finally, the surplus of the pension plan at time t is defined as
SPt = Ht − Lt .
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
Constructing the customized surplus swap
The Cash Flow of the Swap
The goal is to have SPt = 0. Hence, the cash flow of the swap attime t is defined as CFt = −SPt . It follows that
CFt = Lt − (1 + r)(Ht−1 − CLt−1) − Pt .
If the swap is set up every year, then
CFt = Lt − (1 + r)FLt−1 − Pt .
Both Lt and FLt−1 can be determined by whichever valuationmethods agreed to by the two parties. The payoff of the hedginginstruments will be determined by the market price at time t andthe number of holdings at time t − 1.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A multi-population example
A Multi-Population Example
20 national populations.
Liability: 30 years of $1 pension liabilities subject to themortality experience of a male individual aged 60 in year 2008from each population.
Hedging instrument: 10-year age-75 q-forward contractslinked to the EW population, which are unlimitedly availableand liquidly traded.
Sampling risk: a binomial frequency model with an assumedpopulation size of 10,000 for each population.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A multi-population example
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
EWHE30=0.9668
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
SWEHE30=0.7247
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
FRAHE30=0.9054
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
BELHE30=0.9024
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
NLDHE30=0.68
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
CHEHE30=0.9289
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
CANHE30=0.853
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
NZLHE30=0.9479
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
PRTHE30=0.8504
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
ITAHE30=0.9511
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
NORHE30=0.6416
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
AUSHE30=0.7832
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
ESPHE30=0.7815
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
DEUWHE30=0.6855
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
AUTHE30=0.8528
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
USAHE30=0.7402
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
FINHE30=0.7053
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
LUXHE30=0.6467
0 15 30−1
−0.5
0
0.5
1
TimeH t −
L t
ISLHE30=0.5682
0 15 30−1
−0.5
0
0.5
1
Time
H t − L t
SCOHE30=0.4404
Figure 6: The hedge result over time for the 20 populations.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A multi-population example
0 15 30−2
−1
0
1
2
Time
SPt
EW
0 15 30−2
−1
0
1
2
Time
SPt
SWE
0 15 30−2
−1
0
1
2
Time
SPt
FRA
0 15 30−2
−1
0
1
2
Time
SPt
BEL
0 15 30−2
−1
0
1
2
Time
SPt
NLD
0 15 30−2
−1
0
1
2
Time
SPt
CHE
0 15 30−2
−1
0
1
2
Time
SPt
CAN
0 15 30−2
−1
0
1
2
Time
SPt
NZL
0 15 30−2
−1
0
1
2
Time
SPt
PRT
0 15 30−2
−1
0
1
2
Time
SPt
ITA
0 15 30−2
−1
0
1
2
Time
SPt
NOR
0 15 30−2
−1
0
1
2
Time
SPt
AUS
0 15 30−2
−1
0
1
2
Time
SPt
ESP
0 15 30−2
−1
0
1
2
Time
SPt
DEUW
0 15 30−2
−1
0
1
2
Time
SPt
AUT
0 15 30−2
−1
0
1
2
Time
SPt
USA
0 15 30−2
−1
0
1
2
Time
SPt
FIN
0 15 30−2
−1
0
1
2
Time
SPt
LUX
0 15 30−2
−1
0
1
2
TimeSP
t
ISL
0 15 30−2
−1
0
1
2
Time
SPt
SCO
Figure 7: The surplus over time for the 20 populations.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A multi-population example
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
EW
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
SWE
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
FRA
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
BEL
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
NLD
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
CHE
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
CAN
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
NZL
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
PRT
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
ITA
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
NOR
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
AUS
0 15 30−0.2
−0.1
0
0.1
0.2
TimeCF
t
ESP
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
DEUW
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
AUT
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
USA
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
FIN
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
LUX
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
ISL
0 15 30−0.2
−0.1
0
0.1
0.2
Time
CFt
SCO
Figure 8: The cash flow over time for the 20 populations.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging
Dynamic Longevity Hedging Residual Risk Transferring
A multi-population example
0 5 10 15 20 25 30−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2Mean Cash Flow
Time
CF t
Figure 9: The mean cash flow of the 20 populations.
Kenneth Q. Zhou and Johnny S.-H. Li University of Waterloo
Dynamic Longevity Hedging