Coherent Pricing of Life Settlements Under Asymmetric Information

Transcription

1 Longevity Seven, Frankfurt Coherent Pricing of Life Settlements Under Asymmetric Information Nan Zhu Georgia State University Daniel Bauer Georgia State University

2 Page 2 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Introduction One-Period Model for Life Settlements The Extended Framework Application Conclusion

3 Page 3 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Introduction Introduction One-Period Model for Life Settlements The Extended Framework Application Conclusion

4 Page 4 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Introduction Background & Literature Review The life settlement market: Abiding investment opportunity Senior insureds w/ below average health "Viatical settlements" (1980s) Securitization in the capital market (Chen et al. (2011), Stone and Zissu (2006)) Limited number of contracts idiosyncratic risk factors Recent market investigations: Expected returns 8-12% from a policy-by-policy basis (Gatzert (2010)) Open-end life settlement funds returned 4.8% (Braun et al. (2011)) Bad quality of underlying life expectancy estimates? Systematic biases should be swiftly corrected Unsystematic errors cannot explain aggregate underperformance Rating agencies declined rating these "death bonds" due to "unique risks"

5 Page 5 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Introduction Main Findings Different view points based on adverse selection One-period expected utility model offer price in competitive market With symmetric information on health condition With asymmetric information on health condition Adjustment of pricing scheme clientele effects (Hoy and Polborn (2000), Villeneuve (2003)) Extended framework applicable pricing formulas Frailty model heterogeneity in life tables Life-time utility evaluation threshold set for settling Generalizations option to settle in later periods Numerical examples Impact of asymmetric information varies Extreme cases: no effect or market breakdown

6 Page 6 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu One-Period Model for Life Settlements Introduction One-Period Model for Life Settlements The Extended Framework Application Conclusion

7 Page 7 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu One-Period Model for Life Settlements Symmetric Information Simple one-period expected utility model Representative policyholder One-period term-life insurance: F No future contingent premiums, zero cash surrender value Condition: p (survival probability to the end of the period) u( ) and v( ): utilities from life insurance benefits Under a competitive secondary life market: OP sym (p) = (1 p) F 1+R U r = p u(0) + (1 p) v(f) U s = p u((1 p)f) + (1 p) v((1 p)f ) Settle if and only if U s U r

8 Page 8 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu One-Period Model for Life Settlements Asymmetric Information p: estimate of p from third party f (p p) Without considering policyholder s behavior: OP a ( p) = E[(1 p) p] F 1+R Not economically rational! With considering policyholder s behavior: U r = p u(0) + (1 p) v(f) U s (p, OP) = p u(op (1 + R)) + (1 p) v(op (1 + R)) U s (p, OP) U r (p) 0 v(f) v(op (1+R)) p u(op (1+R)) u(0)+v(f) v(op (1+R)) = p (OP) { OP e ( p) = } 1 arg max x p (x) ((1 p)f x(1 + R))f (p p)dp = 0 Average Clientele Risk? Time point: settling vs. purchasing the policy Derived price: independent settlement vs. level premium

9 Page 9 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu One-Period Model for Life Settlements Implication Proposition With asymmetric information with respect to p, the rational expectation offer price, OP e ( p), will be smaller than OP a ( p), for all estimates p. Proof. It is sufficient to show that 1 ((1 p)f (1 + R) OP a )f (p p)dp 0 F p (OP a ) 1 p (OP a ) ((1 p) E[(1 p) p])f (p p)dp 0 E[p p, p p (OP a )] E[p p] 0 Value of the policy is far underestimated reject Offer price exceeds intrinsic value settle Explanation for the discrepancy between expected and realized returns! Coherent pricing should take policyholder s decision into account

10 Page 10 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu The Extended Framework Introduction One-Period Model for Life Settlements The Extended Framework Application Conclusion

11 Page 11 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu The Extended Framework Heterogenous Life Tables Extended framework: Multi-period environment Whole-life policy, annual premium P, death benefit F Heterogeneity w.r.t. individual mortality rates (Vaupel et al. (1979), Hoermann and Russ (2008)) Current frailty models: fail to connect average of (heterogeneous) individual tables to population table E j [ τ p x j (T )] = τ p x (T ), τ, and τ p x j (T ) [0, 1], τ, j We propose: τ p x j (T ) = τ p x (T ) + A j min{ τ p x (T ), 1 τ p x (T )}e γ(τ 1), s.t. A j [ 1, 1], and E[A j ] = 0.

12 Page 12 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu The Extended Framework Policyholders Decision Making Value function when retaining: ω x VT r (W ω x 0, j) = max τ 1p j x (T ) β τ 1 u(c τ P)+ ( τ 1 p j x (T ) τ p j x (T )) β τ v(w τ +F ), c τ τ=1 τ=1 s.t. W τ = (W τ 1 c τ ) Value function when settling: 1, τ = 1,..., ω x. p(τ 1, 1) ω x VT s (W ω x 0, OP, j) = max τ 1p j x (T ) β τ 1 u(c τ ) + ( τ 1 p j x (T ) τ p j x (T )) β τ v(w τ ), c τ τ=1 τ=1 s.t. W 1 = (W 0 c 1 + OP) 1 p(0, 1), and 1 W τ = (W τ 1 c τ ), τ = 2,..., ω x. p(τ 1, 1) Threshold set (settling preferred to retaining): Ω(OP) = {A j : V s T (W 0, OP, j) V r T (W 0, j)}.

13 Page 13 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu The Extended Framework Pricing Formula & Generalization With symmetric information: ω x OP sym (j) = τ=1 [ ( τ 1p x j (T ) τ p x j (T )) F (1 + R) τ τ 1p x j (T ) ] P (1 + R) τ 1 With asymmetric information: { ( ω x OP e (Ā) [ = arg max ( τ 1 p j x (T ) τ p j F x (T )) z Ω(z) (1 + R) τ τ=1 ] ) } τ 1 p j P x (T ) (1 + R) τ 1 z f (A j Ā)dA j = 0 If allowing settling in future periods: V r T (W 0, j) increases truncate Ω(OP) more significant adverse selection effects Systematic mortality risk at population level matters

14 Page 14 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Application Introduction One-Period Model for Life Settlements The Extended Framework Application Conclusion

15 Page 15 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Application Life Table Projections Population: year 1978 (age 50) year 2008 (age 80) U.S. (female) mortality data from Human Mortality Database Lee-Carter model Year 1978: mortality forecasts for premium setting (data from ) $ per $1,000 (r = 4%) Year 2008: mortality forecasts for life settlement pricing (data from ) Individual: γ = 0.1, A j +1 2 follows a Beta distribution with parameters α = β = 2 1p p p 80

16 Page 16 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Application Symmetric Case u(c) = c1 γ 1 γ, γ = (cf. Hall and Jones (2007)) v(w ) = 1+r r ( r 1+r W )1 γ 1 γ W 0 = $500, 000, F = $1, 000, 000, r = 4%, β = 1/1.04 Symmetric Information: OP sym V r T and V s T Settle only when OP sym $323, 370 R 9.95%

17 Page 17 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Application Settlement Decision By comparing value functions reservation price for each type A j Calculate OP sym with hurdle rates R at 4%, 8%, and 12% When OP sym crosses with the reservation price curve asymmetric choice from policyholders Threshold set: Ω(OP) = [A (OP), 1] Reservation and Actuarially Fair Prices A (OP)

18 Page 18 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Application Equilibrium Offer Prices Beta Distribution, R = 0.07 Beta Distribution, R = 0.08 R = 0.07: OP e = OP sym = $412, 680 (no adverse selection) R = 0.08: OP e = $367, 930 < OP sym = $378, 810 (modest effect of asymmetric information)

19 Page 19 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Application Equilibrium Offer Prices Beta Distribution, R = 0.09 Uniform Distribution, R = 0.08 R = 0.09: fatal impact from adverse selection market breaks down R = 0.08 (uniform A j ): OP e = $339, 100 (stronger adverse selection)

20 Page 20 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Conclusion Introduction One-Period Model for Life Settlements The Extended Framework Application Conclusion

21 Page 21 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Conclusion Conclusion Contributions: Effect of asymmetric information on the profit structure of life settlement company Applicable pricing formulas for life settlement transactions Explanation of the discrepancy between estimated and realized returns New angle on the financial analysis of life settlements Promote the mortality-linked capital market as a whole Future projects: Calibrate the model parameters Sensitivity tests Including option to settle later more severe impact of adverse selection

22 Page 22 Longevity Seven, Frankfurt Sep., 9th 2011 Nan Zhu Conclusion Contact Nan Zhu Georgia State University USA Thank you!