![Page 1: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/1.jpg)
The Pricing of Tranched Longevity Bonds
Samuel Wills and Prof. Michael Sherris
School of Actuarial StudiesAustralian School of Business
The University of New South Wales.
9th November 2007
![Page 2: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/2.jpg)
Outline
1. Longevity and Mortality Risk2. Risk Management Strategies3. Longevity Risk Securitization4. Models for Mortality5. The Proposed Mortality Model6. The Longevity Bond7. The Pricing Model8. Data and Assumptions9. Results10. Conclusion
![Page 3: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/3.jpg)
1. Longevity and Mortality Risk
Survival Functions for Italian Male Populations (1881-1992) (Pitacco, 1992)
Rectagularization Expansion
An Increasing Exposure:• Longevity is improving with greater variability.
• OECD Male 60-64 Labour Participation:
- 60-90% (1970s) to 20-50% (today).
• Shift to DC Superannuation.
• Australian Super Industry:
-$912b assets (June 2006),
-2/3 DC or Hybrids.
• Australian Life Annuities:
-$4.3b assets (June 2006).
• Supply/demand constraints (Purcal, 2006).
• Reinsurance:
- Longevity is “toxic” (Wadsworth, 2005).
![Page 4: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/4.jpg)
2. Risk Management Strategies 1. Avoidance
– Participating Annuities
– Reverse Mortgages
3. Transfer– Reinsurance– Bulk Purchase
Annuities– Securitization
2. Retention
– Capital Reserves
– Contingent Capital
4. Hedging– Natural Hedges– Survivor Bonds– Mortality Swaps– Longevity Options and
Futures
![Page 5: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/5.jpg)
• Vehicle for risk transfer: CDOs in the late 1980s.
• Insurance-Linked Securitization – USD 5.6b issued in 2006 (Lane and Beckwith, 2007).
– Insurance-Linked Bonds
– Industry Loss Warranties
– Sidecars
• Mortality Bond Issues (Vita I-III, Tartan, Osiris, 2003-2007)
• Survivor Bond Issues (BNP Paribas/EIB, 2004).
• Benefits:– Improved capacity for risk transfer. Tranching broadens appeal to investors.
– Issuer can tailor issue to manage basis risk vs. moral hazard / info. asymmetry.
– Diversification benefits for investors.
3. Longevity Risk Securitization Securitization Background
![Page 6: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/6.jpg)
( , ) , , ( , ) , , ( , ) td t x t x t x dt t x t x dB
a. Lee Carter (1992) Model and Extensions:
,ln[ ( , )] x x t x tm x t a b k all all
1 and 0x tx t
b k where
- Derived from finance theory, see Vasicek (1977), Cox et al (1985).
b. Dahl (2004) Model and Extensions:
( , ) ( , ) ( , ) ( , ) ( , ) ( , ) td t x t t x t x t x t dt t x t x t dB
- With a specific form based on Cox et al (1985):
4. Models for Mortality
c. Forward Rate Models:- Model the dynamics of the forward mortality surface.
- Based on work by Heath, Jarrow and Morton (1992).
![Page 7: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/7.jpg)
4. Models for Mortality
- This method has been subject to criticism (Cairns et al, 2006; and Bauer and Russ, 2006) as it does not incorporate varying λ over age and time.
a. Lee Carter (1992) Model and Extensions: - Pricing employs the Wang (1996, 2000, 2002) transform that shifts the survival curve using a fixed ‘price of risk’, λ:
Denuit, Devolder and Goderniaux (2007) using stochastic mortality, following Lin and Cox (2005) (deterministic).
![Page 8: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/8.jpg)
5. The Proposed Modeli) A Multivariate Mortality Process
- This falls within the Dahl (2004) family of models.
- To incorporate dependence, we introduce a M.V. random vector dW(t), length N:
- For lives at time t, initially aged x, the mortality rate μ(x,t) is given by:
- Where dZ(t) is a random vector of independent B.M. of length N; and Δ is a N x N matrix of constants, such that:
Note: the dimension of dZ(t) can be reduced using PCA.
![Page 9: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/9.jpg)
5. The Proposed Modeli) A Multivariate Mortality Process- The covariance matrix of dW(t), Σ, has each element:
such that
- This gives the Cholesky decomposition of Σ.
ii) Incorporating Age-Dependence- Using PCA, decompose Σ into its eigenvectors (V), and eigenvalues (diagonal matrix T):
- Simulations of dW(t) can be generated with the same dependence properties:
![Page 10: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/10.jpg)
6. The Longevity Bond
Principal (FV) + Loss Leg (LL)
Premium (P)
Benefit
Issuer
Annuitants
A l(x,t)Mezzanine
Junior
Senior
Collateral Account
Investors:
SPVPremium Leg (PL)
+ FV
- Both the PL and the LL are based on the percentage cumulative losses incurred on an underlying annuity portfolio:
- Where the loss on the portfolio in each period is:
The Proposed Longevity Bond Structure
![Page 11: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/11.jpg)
6. The Longevity Bond- The total variance of the number of lives alive at time t, initially aged x is given by:
- The first term gives the binomial variability in the portfolio given a fixed tpx (the focus of Lin and Cox, 2005).
- The second is the variability due to changes in the mortality rate, which accounts for almost all of the portfolio variance:
Variability in tpx accounts for almost all the variability in l(x,t).
![Page 12: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/12.jpg)
6. The Longevity Bond
- Tranche losses are allocated by the cumulative loss on the portfolio. From this we can find the cumulative tranche loss:
Tranching
where
- The tranche loss as a percentage of its prescribed principal is given by:
- The assumed tranche thresholds are:
Portfolio cumulative loss simulations.
![Page 13: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/13.jpg)
7. The Pricing Model- The premium on tranche j, P*
j, is set to equate the cashflows on the premium leg (PLj), and the loss leg (LLj):
such that
where:
- B(0,t) is the price of a ZCB.
- TCLj(t) is the tranche % cum. loss at time t.
- Premiums need to be set under a risk-adjusted Q mortality measure. Using the Cameron-Martin-Girsanov Theorem:
and for all ages:
where Δλ(t) is a ‘risk adjustment’ that can differ for each age and time.
- and the risk adjusted mortality process is:
![Page 14: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/14.jpg)
7. The Pricing Model- However, the choice of Q, and thus Δλ(t) is not unique (like IR derivatives). It thus needs to be calibrated to market prices.
- These are approximated using an empirical model proposed by Lane (2000), fit to the price of 2007 mortality bond issues using non-linear least squares:
- To facilitate calibration with limited data, simplifying assumptions are made on the risk adjustment:
So that for each x and t:
- λ* is chosen so that: - As a result,
![Page 15: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/15.jpg)
8. Data and AssumptionsData- Australian Population Mortality Data, ages 50-99, 1971-2004. Human Mortality Database (www.mortality.org)
- Australian Gov’t Treasury Bill and Note rates: maturity 1-12 years. Bloomberg 24/09/2007.
- Market insurance-linked security data: 2007 issues. Drawn from Lane and Beckwith (2007).
Assumptions
Mortality process parameter estimates.
- dW(t) is modeled under 3 assumptions of age dependence:
1. Perfect age independence.
2. Observed age dependence using PCA.
3. Perfect age dependence.
![Page 16: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/16.jpg)
9. ResultsThe Mortality Model
A 20 year projection of male expected mortality (linear and log scales).
95% confidence intervals for projected male mortality, under 3 age-dependence assumptions.
![Page 17: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/17.jpg)
9. ResultsThe Mortality Model
Fitted residuals (left) and descriptive statistics (above).
Asymptotic variance/covariance matrix for MLE estimates.
- Analysis of Fit:
Pearson’s chi-square statistic.
![Page 18: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/18.jpg)
9. ResultsThe Longevity Bond
Portfolio expected cumulative loss and 95% bounds.
Tranche expected cumulative loss and 95% bounds under 3 age-dependence assumptions.
![Page 19: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/19.jpg)
Tranche cumulative losses, disaggregated by age.
9. ResultsThe Longevity Bond
![Page 20: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/20.jpg)
The Pricing Model9. Results
Model I: Tranche Premiums (column)and Risk Adjustments (line)
0
500
1000
1500
2000
2500
3000
Tranche 1 Tranche 2 Tranche 3
Pre
miu
ms
(bp
s)
0
1
2
3
4
5
6
Ris
k A
dju
stm
ent
Indep
PCA
Dep
Indep
PCA
Dep
- Calibrated tranche premiums and associated ‘prices of risk’ λ. Consistent with risk averse investors.
- In the absence of a closed form, sensitivities of λ to the inputs into the Lane (2000) model are provided based on mortality rates under observed age dependence.
![Page 21: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/21.jpg)
9. Results
Implications of Results
- Mortality can effectively be modelled as a dynamic, multi-age process.
- Tranched longevity bonds provide an effective vehicle for managing longevity risk.
- Dynamic mortality models are well suited to pricing longevity-linked securities.
Further Research
- Calibration of the risk-adjusted mortality process.
- Application of the proposed mortality model to a broader range of ages
- Alternative definitions for portfolio loss, eg. changes in future obligations on the annuity portfolio (Sherris and Wills, 2007).
![Page 22: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/22.jpg)
10. ConclusionThe Mortality Model- The first time that the Dahl (2004) framework has successfully fit changes in mortality by age and time simultaneously. Importance of age-dependence. Implications for modelling mortality-linked securities on multi-age portfolios.
- First time that the Dahl family has been considered in an Australian context.
The Longevity Bond
The Pricing Model
- The first consideration of a longevity-linked security on multiple ages.
- The first detailed analysis of the impact of tranching, under a range of age dependence assumptions.
- Mortality model is sufficiently flexible to allow the ‘price of risk’ to vary by age and time. Incorporate range of investor sentiments.
- Calibrated price of risk consistent with risk averse investor with non-linear risk/return tradeoff.
![Page 23: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/23.jpg)
References- Bauer, D., and Russ, J., 2006, Pricing Longevity Bonds using Implied Survival Probabilities. (Available at http://www.mortalityrisk.org/Papers/Models.html)
- Cairns, A.J.G., Blake, D., and Dowd, K., 2006a, Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk, ASTIN Bulletin, 36, 79-120
- Cox, J., Ingersoll, J., and Ross, S., 1985, A Theory of the Term-Structure of Interest Rates, Econometrica, 53, 385-408.
- Cox, S.H., and Lin, Y., 2007, Natural Hedging of Life and Annuity Mortality Risks, North American Actuarial Journal, 11(3), 1-15.
- Denuit, M., Devolder, P., and Goderniaux, A.-C., 2007, Securitization of Longevity Risk: Pricing Survivor Bonds with Wang Transform in the Lee-Carter Framework, The Journal of Risk and Insurance, 74(1), 87-113.
- Dahl, M., 2004, Stochastic Mortality in Life Insurance: Market Reserves and Mortality-Linked Insurance Contracts, Insurance: Mathematics and Economics, 35, 113-136.
- Heath, D., Jarrow, R., and Morton, A., 1992, Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation, Econometrica, 60, 77-105.
- Lane, M.N., and Beckwith, R., 2007, That was the Year that was! The 2007 Review of the Insurance Securitization Market, Lane Financial L.L.C. (Available at http://lanefinancialllc.com/)
- Lee, R.D., and Carter, L.R., 1992, Modeling and Forecasting U.S. Mortality, Journal of the American Statistical Association, 87, 659-675.
- Pitacco, E., 2002, Longevity Risk in Living Benefits, Presented at the Third Annual CeRP Conference, Turin.
-Purcal, S., 2006, Supply Challenges to the Provision of Annuities, Working paper, University of New South Wales, Sydney.
- Sherris, M., and Wills, S., 2007, Financial Innovation and the Hedging of Longevity Risk, Presented at the Third International Longevity Risk and Capital Market Solutions Symposium, Taipei.
- Vasicek, O., 1977, An Equilibrium Characterisation of the Term Structure, Journal of Financial Economics, 5, 177-188.
- Wang, S.S., 1996, Premium Calculation by Transforming the Layer Premium Density, ASTIN Bulletin, 26, 71-92.
- Wang, S.S., 2000, A Class of Distortion Operations for Pricing Financial and Insurance Risks, Journal of Risk and Insurance, 67, 15-36.
- Wang, S.S., 2002, A Universal Framework for Pricing Financial and Insurance Risks, ASTIN Bulletin, 32, 213-234.
![Page 24: The Pricing of Tranched Longevity Bonds Samuel Wills and Prof. Michael Sherris School of Actuarial Studies Australian School of Business The University](https://reader036.vdocuments.net/reader036/viewer/2022062714/56649d3a5503460f94a14b88/html5/thumbnails/24.jpg)
Questions and Comments