Application: The Valuation of Longevity Swaps

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Table 4-5. MAPE of Logarithm of Mortality Projection in 1984-2008 (Unit: %)

Model England and Wales France Italy

Original RH-Normal 4.9344 5.1999 7.2268

Original RH-JD 4.9343 5.1915 7.2822

Original RH-VG 4.9393 5.1000 7.9326

Original RH-NIG 4.9319 5.1891 7.2983

VG-Normal 4.8072 5.1719 7.0612 VG-JD 4.8064 5.1619 7.1135 VG-VG 4.807 5.0595 6.9687

VG-NIG

4.8063

5.1598 7.1293

Note: Original RH-Normal is the same as M2 of Cairns et al. (2009). The X-Y model corresponds to an X error term in the RH model and to Y distributions for the time and cohort effects.

4.3 Application: The Valuation of Longevity Swaps

In this section, we first price a longevity swap. Using the mortality data of England and Wales from 1900 to 2008, we then re-fit the RH model to attain the fair swap premium of the longevity swap for both the original RH model (M2) and the best projection model. Finally, we provide the VaR and CTE of the longevity swaps.

Pricing Longevity Swaps

The traditional method of transferring longevity risk in a pension plan or an annuity book is to sell the liability through an insurance or reinsurance contract, known as pension buyouts. These tactics have attracted increasing attention since 2006, especially in the United Kingdom. However, such transactions involve the transfer of all risks, including longevity and investment risk. To transfer longevity risk only to capital markets, Blake and Burrows (2001) first advocate the use of longevity bonds, whose coupon payments depend on the proportion of the population surviving to particular ages. Bauer (2006) and Barbarin (2008) also apply the Heath-Jarrow-Morton methodology (see Heath et al., 1992) to price longevity bonds.

The EIB/BNP longevity bond was the first securitization instrument designed to

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transfer longevity risk but ultimately was withdrawn. The lack of success in issuing longevity bonds led to new securitization instruments, such as longevity swaps,¹² which were pioneered in capital markets by J.P. Morgan and Canada Life in July 2008.

As Blake et al. (2012) show, 16 publicly announced longevity swaps were executed between 2007 and 2012 in the United Kingdom. In this context, the valuation of longevity swaps represents an important research topic for developing capital market solutions for longevity risk.

Longevity swaps have been widely explored in prior literature (Dawson, 2002;

Lin and Cox, 2005; Dowd et al., 2006; Dawson et al., 2010; Biffis et al., 2011; Wang and Yang, 2012). Dowd et al. (2006) introduce the mechanism for transferring longevity risk; this instrument involves exchanging actual pension payments for a series of pre-agreed fixed payments. On each payment date, the fixed-rate payer (e.g., pension plan) receives from the hedge supplier a random mortality-dependent payment and, in return, makes a fixed payment to the hedge supplier. Dowd et al.

(2006) demonstrate that the hedge is almost perfect when the reference index is based on the survivor experience of the insurer’s annuity book. If the expected reference indices and insurers’ own survivor experiences are highly correlated, the longevity swap can still hedge the insurer against a considerable amount of the aggregate longevity risk it faces. In this article, following the vanilla longevity swap structure analyzed by Dowd et al. (2006) and Dawson et al. (2010), we discuss a T-year bespoke longevity swap linked to a benchmark cohort of a given initial age for the England and Wales mortality data.¹³

For a given time horizon T, we consider a filtered probability space        

12 For the recent development of longevity-linked securities, see Blake et al. (2012) and reference therein.

13 To bear no basis risk, the variable payments in bespoke longevity swaps are designed to match precisely the mortality experience of each individual hedger.

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enlarged filtration ℋ^࣡ where ℋ is the filtration related to risk factors and ࣡ is the complement, such that  is a stopping time on . Conditional on the path followed by the mortality rates, the t-year survival probability that a 65-year-old person in calendar year 2008+t reaches age 65+t is of the form:

 

^xp



^{0 65 ,2008}



( ) _T e ^t _{s s}

S t P  t ^  



m _{ }ds . (4-17) We assume that the mortality rates are constant within certain age and time windows but may vary from one window to the next. Specifically, given any integer age and calendar year , we presume that

To transfer longevity risk, on each of the payment dates t, the fixed-rate payer pays the notional principal multiplied by a prespecified fixed proportion (1) ( )H t to the floating-rate payer and receives the notional principal multiplied by S t( ), where H t( ) is anticipated by using the best estimate of the underlying mortality model, and  is the swap premium that would be set so that the initial value of the swap is zero for each party.

The distribution function of S t( ) under the real-world (physical) probability measure P is

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Wang (2000) proposes a distortion operator to change the probability measure from the real-world probability measure P to an equivalent martingale measure Q, with the following transformation:¹⁴ normal distribution function. Therefore, as shown by Denuit et al. (2007), the expectation value of S t( ) under the equivalent martingale measure Q is defined as



^{( )}



₀¹



^{1 ( )}



₀¹



¹



¹

^

^{( )}

^  

Q t t

E S t







F y dy

^ 



  ^

F y



 dy

. (4-22) Let be the total annuities issued to an initial population that consists of persons aged 65 years who also are alive in 2008. Under the equivalent martingale measure Q, the fair value of a pay-fixed longevity swap at issue year 2009, denoted by

LS

₀, can be calculated as

where is the risk-free rate. We also consider the term structure of the interest rate in our valuation framework. Let denote the price of a zero-coupon bond issued at time t that pays $1 at time T, . With the assumption that mortality rates and financial risk are independent, the fair value of a pay-fixed longevity swap takes the form:

14 The Wang transform represents only one possible choice among several incomplete market pricing methods. For example, Biffis et al. (2010) provide the equivalent changes of measures that preserve the structure of the LC model and the tractability of the doubly stochastic setup. The specification of both a real-world and an equivalent martingale measure raises the issue of whether the doubly stochastic setting applies under the two measures. For more details, please refer to the Proposition 3.2 in Biffis et al. (2010).

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The fair swap premium , which is set when the initial value of the swap equals zero, is given by

the Monte Carlo algorithm to compute the expected value of the t-year survival probability under the equivalent martingale measure Q in Appendix D.

Numerical Analysis

To simulate the mortality rates, we first re-fit the RH model with four distributions—normal, JD, VG, and NIG—to the mortality data of England and Wales from 1900 to 2008 in Table 4-6. Similar to the results based on the 1900–1983 period, the best model for England and Wales is still the VG model. Consequently, we use the best prediction models presented in Table 4-5 to simulate mortality rates, which is the VG-NIG model for England and Wales.

Table 4-6. Goodness-of-fit Measures for the Number of Deaths in 1900-2008

Model England and Wales

LLF AIC BIC

Normal -41760 42094 43111

JD -41616 41953 42980

VG

-41524 41860 42883

NIG -41526 41862 42886

In this section, we provide a numerical example of the longevity swaps based on a cohort of 65-year-old persons in calendar year 2008. The initial term structure is obtained from the U.S. Department of the Treasury.¹⁵ We also assume that M = 1.

       

15 See http://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/TextView.aspx?

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Figure 4-1 depicts the swap premium curve by varying the level of the risk-adjusted parameter  . The fair swap premium is higher for a longer duration swap, because long-duration contracts are usually more expensive for covering longevity risk. The lower  implies higher survival probabilities, so the fair swap premiums should be bigger for lower  . In addition, the fair swap premiums of the RH model (the M2 model of Cairns et al., 2009) are higher than those of the best prediction model, which means that the fixed-rate payer (longevity risk hedger) can pay lower swap premium, according to the best prediction model.

Figure 4-1. Swap Premium Curves for Distinct Level of Risk-Adjusted Parameter



Table 4-7 reveals the fair swap premiums with time to maturity equal to 25 years when is -0.1, -0.15, and -0.2, with parallel shifts upward of 0%, 2%, and 4% in the yield curve. From Table 4-7, we see that the lower the and the interest rate are, the higher is the fair swap premium. Similarly, the fair swap premiums of the RH        

data=yieldYear& year=2008. The 1-year, 2-year, 3-year, 5-year, 7-year, 10-year, 20-year, and 30-year yield rates are 0.37%, 0.76%, 1%, 1.55%, 1.87%, 2.25%, 3.05%, and 2.69% on December 31, 2008, respectively. We use the linear interpolation to obtain other yield rates. 

1 5 10 15 20 25

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model are higher than those of the best prediction model, even when the yield curve moves up in parallel. Longevity risk hedgers can use the best prediction model to price longevity swaps and pay lower swap premiums.

Table 4-7. Swap Premiums for Different Interest Rates (Units: bps)

Yield Rates Model λ = -0.1 λ = -0.15 λ = -0.2

Original yield curve RH 7.15 19.07 30.91

Best 6.08 18.42 30.67

Parallel shift up of 2% RH 6.11 16.32 26.48

Best 5.15 15.73 26.24

Parallel shift up of 4% RH 5.21 13.94 22.63

Best 4.36 13.40 22.38

Note: Time to maturity is 25 years.

As market conditions change (e.g., mortality patterns, a parallel shift in yield curve), the marking-to-market (MTM) procedure could mean that the longevity swap switching status in the hedger’s balance sheet falls between that of an asset and that of a liability. Assume that  is -0.1 and the maturation time is 25 years, as in our baseline case. The initial swap premiums are 7.15 bps and 6.08 bps for the RH and best prediction models in the baseline case, respectively. In Table 4-8, applying Equation (4-24), we report the impacts of market condition changes (a parallel shift in yield curve and different risk-adjustment parameters  ) on the MTM profits or losses of the longevity swaps. When the yield curve moves up in parallel, ceteris paribus, the fair value of the longevity swap decreases, which means that a parallel shift up in the yield curve leads to a loss for the fixed-rate payer (hedger). In addition, a lower level of the risk-adjustment parameter results in a higher expected value of survival probability (higher mortality improvement), which in turn leads to a higher value of the longevity swap. Because the U.S. Fed reiterated its plan to keep its key short-term interest rate near zero until at least late 2014, it may be not favorable for the hedgers

‧

(e.g., pension funds, annuity providers) to hedge their exposure to longevity risk through longevity swaps in this low interest rate environment. However, as shown in Table 4-8, the risk-adjustment parameter has a larger impact than the parallel shift up in the yield curve on the fair value of the longevity swap. Consequently, as life expectancy increases dramatically in developed countries, it is reasonable to find the recent surge in transactions in longevity swaps.

Table 4-8. The MTM Values of Longevity Swaps

Model Yield Rates λ

Note: Assume that λ is -0.1 and maturation time is 25 years in the baseline case.

From the standpoint of the pay-fixed payer of a longevity swap, the unexpected loss at time t is of the form:

 

( ) (1 ) ( ) ( )

L t



M



 H t



S t

, t1,...,T. (4-26) The present value of the total unexpected loss, denoted as PVL, is given by

1

Figure 4-2 depicts the pdf of PVL for the RH model and the best prediction model of England and Wales mortality data; it also marks the areas for the other three subplots in the upper left-hand panel. We find that the pdf of PVL for the best prediction model possesses leptokurticity and a high tip. In addition, Table 4-9 presents the VaR and CTE of the PVL with maturation times of up to 25 years. It is

‧

clear that, compared with the RH model, the best prediction model has higher VaR and CTE. Because shorter-duration contracts cover less longevity risk, the VaR and CTE values are smaller for shorter duration longevity swaps. The differences of RH and the best prediction model are larger for longer durations. Therefore, the loss distribution of longevity swaps is centralized and heavy-tailed, especially for longer duration contracts. It is critical to have a good mortality model to calculate accurate loss distributions.

Figure 4-2. Probability Density Functions of Present Value of the Losses (

 0.1

,

T 25

)

Present Value of the losses pdf

Present Value of the losses pdf

Present Value of the losses pdf

Present Value of the losses pdf

RH model Best model

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Table 4-9. The VaR and CTE of the Losses for Different Maturation Times (λ = -0.1)

Time to Maturity Model VaR95 VaR99 CTE95 CTE99

10 RH 0.0800 0.1188 0.1047 0.1380

Best 0.0847 0.1355 0.1171 0.1656

15 RH 0.2136 0.3061 0.2713 0.3596

Best 0.2254 0.3452 0.3002 0.4176

20 RH 0.4137 0.5940 0.5256 0.6951

Best 0.4355 0.6590 0.5757 0.7926

25 RH 0.6720 0.9563 0.8497 1.1165

Best 0.7057 1.0526 0.9233 1.2528

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