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Slides nantes

  1. 1. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Pricing insurance linked securities : interplay between finance and insurance Arthur Charpentier http ://perso.univ-rennes1.fr/arthur.charpentier/ Atelier Finance & Risque Universit´e de Nantes, Avril 2008 1
  2. 2. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. survey of literature • Fundamental asset pricing theorem, in finance, Cox & Ross (JFE, 1976), Harrison & Kreps (JET, 1979), Harrison & Pliska (SPA, 1981, 1983). Recent general survey – Dana & Jeanblanc-Picqu´e (1998). March´es financiers en temps continu : valorisation et ´equilibre. ´Economica. – Duffie (2001). Dynamic Asset Pricing Theory. Princeton University Press. – Bingham & Kiesel (2004). Risk neutral valuation. Springer Verlag • Premium calculation, in insurance. – B¨uhlmann (1970) Mathematical Methods in Risk Theory. Springer Verlag. – Goovaerts, de Vylder & Haezendonck (1984). Premium Calculation in Insurance. Springer Verlag. – Denuit & Charpentier (2004). Math´ematiques de l’assurance non-vie, tome 1. ´Economica. 2
  3. 3. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. survey of literature • Price of uncertain quantities, in economics of uncertainty, von Neumann & Morgenstern (1944), Yaari (E, 1987). Recent general survey – Quiggin (1993). Generalized expected utility theory : the rank-dependent model. Kluwer Academic Publishers. – Gollier (2001). The Economics of Risk and Time. MIT Press. • Bentoglio & Betbeze (2005). L’Etat et l’assurance des risques nouveaux. La Documentation Fran¸caise. 3
  4. 4. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Agenda A short introduction to insurance risks • Catastrophe and (very) large risks • Mortality risks, from short term pandemic to long term risk Insurance linked securities • Insurance linked securities • Catastrophe or mortality bonds Financial versus insurance pricing • Insurance : from pure premium to other techniques • Finance : from complete to incomplete markets Pricing Insurance linked • Distorted premium • Indifference utility 4
  5. 5. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Agenda A short introduction to insurance risks • Catastrophe and (very) large risks • Mortality risks, from short term pandemic to long term risk Insurance linked securities • Insurance linked securities • Catastrophe or mortality bonds Financial versus insurance pricing • Insurance : from pure premium to other techniques • Finance : from complete to incomplete markets Pricing Insurance linked • Distorted premium • Indifference utility 5
  6. 6. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. from mass risk to large risks insurance is “the contribution of the many to the misfortune of the few”. 1. judicially, an insurance contract can be valid only if claim occurrence satisfy some randomness property, 2. the “game rule” (using the expression from Berliner (Prentice-Hall, 1982), i.e. legal framework) should remain stable in time, 3. the possible maximum loss should not be huge, with respect to the insurer’s solvency, 4. the average cost should be identifiable and quantifiable, 5. risks could be pooled so that the law of large numbers can be used (independent and identically distributed, i.e. the portfolio should be homogeneous), 6. there should be no moral hazard, and no adverse selection, 7. there must exist an insurance market, in the sense that demand and supply should meet, and a price (equilibrium price) should arise. 6
  7. 7. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. risk premium and regulatory capital (points 4 and 5) Within an homogeneous portfolios (Xi identically distributed), sufficiently large (n → ∞), X1 + ... + Xn n → E(X). If the variance is finite, we can also derive a confidence interval (solvency requirement), i.e. if the Xi’s are independent, n i=1 Xi ∈   nE(X) ± 1.96 √ nVar(X) risk based capital need    with probability 95%. High variance, small portfolio, or nonindependence implies more volatility, and therefore more capital requirement. 7
  8. 8. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. independent risks, large portfolio (e.g. car insurance) q q independent risks, 10,000 insured q q Fig. 1 – A portfolio of n = 10, 000 insured, p = 1/10. 8
  9. 9. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. independent risks, large portfolio (e.g. car insurance) q q independent risks, 10,000 insured, p=1/10 900 950 1000 1050 1100 1150 1200 0.0000.0020.0040.0060.0080.0100.012 casindépendant,p=1/10,n=10,000 distribution de la charge totale, N((np,, np((1 −− p)))) RUIN (1% SCENARIO) RISK−BASED CAPITAL NEED +7% PREMIUM 969 Fig. 2 – A portfolio of n = 10, 000 insured, p = 1/10. 9
  10. 10. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. independent risks, large portfolio (e.g. car insurance) q q independent risks, 10,000 insured, p=1/10 900 950 1000 1050 1100 1150 1200 0.0000.0020.0040.0060.0080.0100.012 casindépendant,p=1/10,n=10,000 distribution de la charge totale, N((np,, np((1 −− p)))) RUIN (1% SCENARIO) RISK−BASED CAPITAL NEED +7% PREMIUM 986 Fig. 3 – A portfolio of n = 10, 000 insured, p = 1/10. 10
  11. 11. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. independent risks, small portfolio (e.g. fire insurance) q q independent risks, 400 insured q q Fig. 4 – A portfolio of n = 400 insured, p = 1/10. 11
  12. 12. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. independent risks, small portfolio (e.g. fire insurance) q q independent risks, 400 insured, p=1/10 30 40 50 60 70 0.000.010.020.030.040.050.06 casindépendant,p=1/10,n=400 distribution de la charge totale, N((np,, np((1 −− p)))) RUIN (1% SCENARIO) RISK−BASED CAPITAL NEED +35% PREMIUM 39 Fig. 5 – A portfolio of n = 400 insured, p = 1/10. 12
  13. 13. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. independent risks, small portfolio (e.g. fire insurance) q q independent risks, 400 insured, p=1/10 30 40 50 60 70 0.000.010.020.030.040.050.06 casindépendant,p=1/10,n=400 distribution de la charge totale, N((np,, np((1 −− p)))) RUIN (1% SCENARIO) RISK−BASED CAPITAL NEED +35% PREMIUM 48 Fig. 6 – A portfolio of n = 400 insured, p = 1/10. 13
  14. 14. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. nonindependent risks, large portfolio (e.g. earthquake) q q independent risks, 10,000 insured q q Fig. 7 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent. 14
  15. 15. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. nonindependent risks, large portfolio (e.g. earthquake) q q non−independent risks, 10,000 insured, p=1/10 1000 1500 2000 2500 0.0000.0020.0040.0060.0080.0100.012 distribution de la charge totale nonindependantcase,p=1/10,n=10,000 RUIN (1% SCENARIO) RISK−BASED CAPITAL NEED +105% PREMIUM 897 Fig. 8 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent. 15
  16. 16. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. nonindependent risks, large portfolio (e.g. earthquake) q q non−independent risks, 10,000 insured, p=1/10 1000 1500 2000 2500 0.0000.0020.0040.0060.0080.0100.012 distribution de la charge totale nonindependantcase,p=1/10,n=10,000 RUIN (1% SCENARIO) RISK−BASED CAPITAL NEED +105% PREMIUM 2013 Fig. 9 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent. 16
  17. 17. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. some stylized facts about natural disasters “climatic risk in numerous branches of industry is more important than the risk of interest rates or foreign exchange risk” (AXA 2004, quoted in Ceres (2004)). Fig. 10 – Major natural catastrophes (from Munich Re (2006).) 17
  18. 18. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Some stylized facts : natural catastrophes Includes hurricanes, tornados, winterstorms, earthquakes, tsunamis, hail, drought, floods... Date Loss event Region Overall losses Insured losses Fatalities 25.8.2005 Hurricane Katrina USA 125,000 61,000 1,322 23.8.1992 Hurricane Andrew USA 26,500 17,000 62 17.1.1994 Earthquake Northridge USA 44,000 15,300 61 21.9.2004 Hurricane Ivan USA, Caribbean 23,000 13,000 125 19.10.2005 Hurricane Wilma Mexico, USA 20,000 12,400 42 20.9.2005 Hurricane Rita USA 16,000 12,000 10 11.8.2004 Hurricane Charley USA, Caribbean 18,000 8,000 36 26.9.1991 Typhoon Mireille Japan 10,000 7,000 62 9.9.2004 Hurricane Frances USA, Caribbean 12,000 6,000 39 26.12.1999 Winter storm Lothar Europe 11,500 5,900 110 Tab. 1 – The 10 most expensive natural catastrophes, 1950-2005 (from Munich Re (2006)). 18
  19. 19. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. basics on extreme value theory When modeling large claims (industrial fire, business interruption,...) : extreme value theory framework is necessary. The Pareto distribution appears naturally when modeling observations over a given threshold, F(x) = P(X ≤ x) = 1 − x x0 b , where x0 = exp(−a/b) Then equivalently log(1 − F(x)) ∼ a + b log x, i.e. for all i = 1, ..., n, log(1 − Fn(Xi)) ∼ a + b · log Xi. Remark : if −b ≥ 1, then EP(X) = ∞, the pure premium is infinite. The estimation of b is a crucial issue (see Zajdenweber (JRI, 1998) or from Charpentier (BFA, 2005).) 19
  20. 20. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. goodness of fit of the Pareto distribution ! " # $ % &! !'!#!(!"!&! !o#!lo# &areto +lot, -urri0ane losses )oga-it01 o3 t0e loss a1ount )oga-it01o39u1ulate:;-o<a<ilites =>'?@ slo;e> ! &A"'B =>"'?@ slo;e> !!A%$# 0 20 40 60 80 1000.51.01.52.0 Hill estimator of the tail index Percentage of bservations exceeding the threshold Tailindex,with95%confidenceinterval Fig. 11 – Pareto modeling of hurricanes losses (Pielke & Landsea (WF, 1998)). 20
  21. 21. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. longevity and mortality risks Year Age 0 20 40 60 80 100 5e−055e−045e−035e−02 Age 1899 1948 1997 1900 1920 1940 1960 1980 2000 5e−042e−035e−032e−025e−02 Age 60 years old 40 years old 20 years old Fig. 12 – Mortality rate surface (function of age and year). 21
  22. 22. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. longevity and mortality risks Main problem= forecasting future mortality rates, i.e. m(x, t) = # deaths during calendar year t aged x last birthday average population during calendar year t aged x last birthday cf. LifeMetrics (JPMorgan), – Lee & Carter (JASA, 1992), log m(x, t) = β (1) x + β (2) x κ (2) t , – Renshaw & Haberman (IME, 2006), log m(x, t) = β (1) x + β (2) x κ (2) t + β (3) x γ (3) t−x, – Currie (2006), log m(x, t) = β (1) x + κ (2) t + γ (3) t−x, 1950 1960 1970 1980 1990 2000 −15−10−50510 Lee Carter model, total France (fit701) Year (y) Kappa(LeeCarter) 55 60 65 70 75 80 85 −4.5−4.0−3.5−3.0−2.5−2.0 Lee Carter model, total France (fit701) Age (x) Beta1(LeeCarter) 55 60 65 70 75 80 85 0.0300.0320.0340.036 Lee Carter model, total France (fit701) Age (x) Beta2(LeeCarter) 22
  23. 23. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Agenda A short introduction to insurance risks • Catastrophe and (very) large risks • Mortality risks, from short term pandemic to long term risk Insurance linked securities • Insurance linked securities • Catastrophe or mortality bonds Financial versus insurance pricing • Insurance : from pure premium to other techniques • Finance : from complete to incomplete markets Pricing Insurance linked • Distorted premium • Indifference utility 23
  24. 24. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. insurance linked securities, from insurance to finance Finn & Lane (1995) “there are no right price of insurance, there is simply the transacted market price which is high enough to bring forth sellers and low enough to induce buyers”. traditional indemni industry parametr ilw reinsurance securitiza loss securitiza derivatives securitization traditional indemnity indust parametric ilw reinsurance securitization loss securitization derivatives securitiza 24
  25. 25. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. (re)insurance principle q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 02468101214 Reinsurance Excess−of−Loss Indemnity Time (exposure period) Insurancelosses q q INSURANCE COMPANY REINSURANCE COMPANY PREMIUM (2) Fig. 13 – (re)insurance excess-of-loss principle. 25
  26. 26. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. (re)insurance principle q q q q q q q q q q q q q q q q q q q q q q q 0.2 0.4 0.6 0.8 02468101214 Reinsurance Excess−of−Loss Indemnity Time (exposure period) Insurancelosses q q INSURANCE COMPANY REINSURANCE COMPANY PREMIUM (2)INDEMNITY (5) Fig. 14 – (re)insurance excess-of-loss principle. 26
  27. 27. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. catastrophe or mortality bonds q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 02468101214 Parametric cat bond Time (exposure period) Insurancelosses q q INSURANCE COMPANY SPECIAL PURPOSE VEHICLE PREMIUM (2) FINANCIAL INVESTORS CAT BOND NOMINAL (100) Fig. 15 – catastrophe or mortality bonds principle. 27
  28. 28. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. catastrophe or mortality bonds q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 02468101214 Parametric cat bond Time (exposure period) Insurancelosses q q INSURANCE COMPANY SPECIAL PURPOSE VEHICLE PREMIUM (2) FINANCIAL INVESTORS CAT BOND NOMINAL (100) NOMINAL + RISKY COUPON (100+5+2) Fig. 16 – catastrophe or mortality bonds principle. 28
  29. 29. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. catastrophe or mortality bonds q q q q q q q q q q q q q q q q q q q q q q q 0.2 0.4 0.6 0.8 02468101214 Parametric cat bond Time (exposure period) Insurancelosses q q INSURANCE COMPANY SPECIAL PURPOSE VEHICLE PREMIUM (2) FINANCIAL INVESTORS CAT BOND NOMINAL (100) NOMINAL + RISKY COUPON (100+5−9) INDEMNITY (9) Fig. 17 – catastrophe or mortality bonds principle. 29
  30. 30. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. comparing (re)insurance and financial markets From AIG (1997), q q q q q q q q q q q World Equity market, $15,000 billion World governement bond market, $9,000 billion US non−life insurance and reinsurance capital $200 billion Hurricane Andrew $20 billion Hurricane in Florida $100 billion (potential) =⇒ capital markets have significantly greater capacity then (re)insurance markets 30
  31. 31. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. insurance linked securities, from insurance to finance Mortality/mortality risk : • deviation of the trend (e.g. consequence of obesity) • one off deviation (e.g. avian flu, Spanish influenza) Natural catastrophe risk : • wind related (e.g. hurricanes in Florida or winter storms in Europe) • flood related (e.g. Paris) • soil related (e.g. earthquakes in California or Japan, or volcanic eruptions) Man-made disasters : • terrorism (e.g. 9/11) • institutional investors : pension funds, (re)insurance ? • financial markets ? • governments ? 31
  32. 32. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Agenda A short introduction to insurance risks • Catastrophe and (very) large risks • Mortality risks, from short term pandemic to long term risk Insurance linked securities • Insurance linked securities • Catastrophe or mortality bonds Financial versus insurance pricing • Insurance : from pure premium to other techniques • Finance : from complete to incomplete markets Pricing Insurance linked • Distorted premium • Indifference utility 32
  33. 33. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. the pure premium as a technical benchmark Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century proposed to evaluate the “produit scalaire des probabilit´es et des gains”, < p, x >= n i=1 pixi = n i=1 P(X = xi) · xi = EP(X), based on the “r`egle des parties”. For Qu´etelet, the expected value was, in the context of insurance, the price that guarantees a financial equilibrium. From this idea, we consider in insurance the pure premium as EP(X). As in Cournot (1843), “l’esp´erance math´ematique est donc le juste prix des chances” (or the “fair price” mentioned in Feller (AS, 1953)). Problem : Saint Peterburg’s paradox, i.e. infinite mean risks (cf. natural catastrophes) 33
  34. 34. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. the pure premium as a technical benchmark For a positive random variable X, recall that EP(X) = ∞ 0 P(X > x)dx. q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Expected value Loss value, X Probabilitylevel,P Fig. 18 – Expected value EP(X) = xdFX(x) = P(X > x)dx. 34
  35. 35. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. from pure premium to expected utility principle Ru(X) = u(x)dP = P(u(X) > x))dx where u : [0, ∞) → [0, ∞) is a utility function. Example with an exponential utility, u(x) = [1 − e−αx ]/α, Ru(X) = 1 α log EP(eαX ) , i.e. the entropic risk measure. See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern (PUP, 1994), Rochet (E, 1994)... etc. 35
  36. 36. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Distortion of values versus distortion of probabilities q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Expected utility (power utility function) Loss value, X Probabilitylevel,P Fig. 19 – Expected utility u(x)dFX(x). 36
  37. 37. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Distortion of values versus distortion of probabilities q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Expected utility (power utility function) Loss value, X Probabilitylevel,P Fig. 20 – Expected utility u(x)dFX(x). 37
  38. 38. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. from pure premium to distorted premiums (Wang) Rg(X) = xdg ◦ P = g(P(X > x))dx where g : [0, 1] → [0, 1] is a distorted function. Example • if g(x) = I(X ≥ 1 − α) Rg(X) = V aR(X, α), • if g(x) = min{x/(1 − α), 1} Rg(X) = TV aR(X, α) (also called expected shortfall), Rg(X) = EP(X|X > V aR(X, α)). See D’Alembert (1754), Schmeidler (PAMS, 1986, E, 1989), Yaari (E, 1987), Denneberg (KAP, 1994)... etc. Remark : Rg(X) will be denoted Eg ◦ P. But it is not an expected value since Q = g ◦ P is not a probability measure. 38
  39. 39. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Distortion of values versus distortion of probabilities q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Distorted premium beta distortion function) Loss value, X Probabilitylevel,P Fig. 21 – Distorted probabilities g(P(X > x))dx. 39
  40. 40. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Distortion of values versus distortion of probabilities q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Distorted premium beta distortion function) Loss value, X Probabilitylevel,P Fig. 22 – Distorted probabilities g(P(X > x))dx. 40
  41. 41. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. some particular cases a classical premiums The exponential premium or entropy measure : obtained when the agent as an exponential utility function, i.e. π such that U(ω − π) = EP(U(ω − S)), U(x) = − exp(−αx), i.e. π = 1 α log EP(eαX ). Esscher’s transform (see Esscher (SAJ, 1936), B¨uhlmann (AB, 1980)), π = EQ(X) = EP(X · eαX ) EP(eαX) , for some α > 0, i.e. dQ dP = eαX EP(eαX) . Wang’s premium (see Wang (JRI, 2000)), extending the Sharp ratio concept E(X) = ∞ 0 F(x)dx and π = ∞ 0 Φ(Φ−1 (F(x)) + λ)dx 41
  42. 42. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. pricing options in complete markets : the binomial case In complete and arbitrage free markets, the price of an option is derived using the portfolio replication principle : two assets with the same payoff (in all possible state in the world) have necessarily the same price. Consider a one-period world, risk free asset 1 → (1+r), and risky asset S0 → S1 =    Su = S0u( increase, d > 1) Sd = S0d( decrease, u < 1) The price C0 of a contingent asset, at time 0, with payoff either Cu or Cd at time 1 is the same as any asset with the same payoff. Let us consider a replicating portfolio, i.e.    α (1 + r) + βSu = Cu = max {S0u − K, 0} α (1 + r) + βSd = Cd = max {S0d − K, 0} 42
  43. 43. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. pricing options in complete markets : the binomial case The only solution of the system is β = Cu − Cd S0u − S0d and α = 1 1 + r Cu − S0u Cu − Cd S0u − S0d . C0 is the price at time 0 of that portfolio. C0 = α + βS0 = 1 1 + r (πCu + (1 − π) Cd) where π = 1 + r − d u − d (∈ [0, 1]). Hence C0 = EQ C1 1 + r where Q is the probability measure (π, 1 − π), called risk neutral probability measure. 43
  44. 44. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. financial versus actuarial pricing, a numerical example risk-free asset risky asset contingent claim 1 →    1.05 1.05 100 →    110 70 ??? →    150 10 probability 75% probability 25%    Actuarial pricing : pure premium EP(X) = 3 4 × 150 + 1 4 × 10 = 115 (since p = 75%). Financial pricing : 1 1 + r EQ(X) = 126.19 (since π = 87.5%). The payoff can be replicated as follows,    −223.81 · 1.05 + 3.5 · 110 = 150 −223.81 · 1.05 + 3.5 · 70 = 10 and thus −223.81 · 1 + 3.5 · 100 = 126.19. 44
  45. 45. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. financial versus actuarial pricing, a numerical example 0.00 0.01 0.02 0.03 0.04 0.05 0.06 115120125130135140145 Comparing binomial risks, from insurance to finance Alpha or lambda coefficients Prices ACTUARIAL PURE PREMIUM FINANCIAL PRICE (UNDER RISK NEUTRAL MEASURE) WANG DISTORTED PREMIUM ESSCHER TRANSFORM EXPONENTIAL UTILITY q Fig. 23 – Exponential utility, Esscher transform, Wang’s transform...etc. 45
  46. 46. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. risk neutral measure or deflators The idea of deflators is to consider state-space securities contingent claim 1 contingent claim 2 ??? →    1 0 ??? →    0 1 probability 75% probability 25%    Then it is possible to replicate those contingent claims    −1.667 · 1.05 + 0.025 · 110 = 1 −1.667 · 1.05 + 0.025 · 70 = 0    2.619 · 1.05 + −0.02 · 110 = 0 2.619 · 1.05 + −0.02 · 70 = 1 The market prices of the two assets are then 0.8333 and 0.119. Those prices can then be used to price any contingent claim. E.g. the final price should be 150 × 0.8333 + 10 × 0.119 = 126.19. 46
  47. 47. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. pricing options in incomplete markets The valuation of option is based on the idea of replicating portfolios. What if we cannot buy the underlying asset. risk-free asset risky asset contingent claim 1 →    1.05 1.05 100 →    110 70 ??? →    150 10 probability 75% probability 25%    Actuarial pricing : pure premium EP(X) = 3 4 × 150 + 1 4 × 10 = 115. Financial pricing : ... The payoff cannot be replicated =⇒ incomplete market Remark : this model can be extended in continuous time, with continuous prices (driven by Browian diffusion). 47
  48. 48. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. pricing options in incomplete markets • the market is not complete, and catastrophe (or mortality risk) cannot be replicated, • the guarantees are not actively traded, and thus, it is difficult to assume no-arbitrage • underlying diffusions are not driven by a geometric Brownian motion process 48
  49. 49. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Impact of WTC 9/11 on stock prices (Munich Re and SCOR) 2001 2002 30354045505560 250300350 MunichRestockprice Fig. 24 – Catastrophe event and stock prices (Munich Re and SCOR). 49
  50. 50. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Agenda A short introduction to insurance risks • Catastrophe and (very) large risks • Mortality risks, from short term pandemic to long term risk Insurance linked securities • Insurance linked securities • Catastrophe or mortality bonds Financial versus insurance pricing • Insurance : from pure premium to other techniques • Finance : from complete to incomplete markets Pricing Insurance linked • Distorted premium • Indifference utility 50
  51. 51. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. pricing insurance linked securities using distorted premium Lane (AB, 2000) proposed to fit an econometric model on yield spreads (ER, excess return), as a function of the CEL (conditional expected loss) and the PFL (probability of first loss), as a Cobb-Douglas function, ER = 0.55 · PFL0.495 · CEL0.574 . Wang (GPRI, 2004) Assume that X has a normal distribution (where µ and σ are functions of CEL and PFL), and with λ = 0.45... 51
  52. 52. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. Property Catastrophe Risk Linked Securities, 2001 0 2 4 6 8 10 12 14 16 Yield spread (%) Mosaic2A Mosaic2B HalyardRe DomesticRe ConcentricRe JunoRe ResidentialRe Kelvin1stevent Kelvin2ndevent GoldEagleA GoldEagleB NamazuRe AtlasReA AtlasReB AtlasReC SeismicLtd Lane model Wang model Empirical Fig. 25 – Cat bonds yield spreads, empirical versus models. 52
  53. 53. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. using expected utility principle to price securities Consider – a risk-free bond, – a risky traded asset, with price Xt, – a non traded insurance index, with value St We want to price the option with payoff payoff (ST − K)+. In complete market (the index is traded), we price a replicating portfolio. In incomplete market, we create a portfolio δ = (α, β) of tradable assets which should not be too far from the true payoff. See the super-replication price or risk minimization principle idea. If X is a random payoff, the classical Expected Utility based premium is obtain by solving u(ω, X) = U(ω − π) = EP(U(ω − X)). 53
  54. 54. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. using expected utility principle to price securities Consider an investor selling an option with payoff X at time T, – either he keeps the option, uδ (ω, 0) = supδ∈A EP U(ω + (δ · S)T ) , – either he sells the option, uδ (ω + π, X) = supδ∈A EP U(ω + (δ · S)T − X) . The price obtained by indifference utility is the minimum price such that the two quantities are equal, i.e. π(ω, X) = inf {π ∈ R such that uδ (ω + π, X) − uδ (ω, 0) 0} . This price is the minimal amount such that it becomes interesting for the seller to sell the option : under this threshold, the seller has a higher utility keeping the option, and not selling it. 54
  55. 55. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. conclusion • classical finance, π = EP[(S − d)+] • classical insurance, π = e−rT EQ[(ST − K)+] • large risks, expected utility approach U(ω − π) = EP(U(ω − [S − d]+)) • large risks, dual (Yaari) approach ω − π = Eg◦P(ω − S) • incomplete market, dual (Yaari) approach, i.e. Esscher • incomplete market, expected utility approach, i.e. indifference utility =⇒ a general framework for pricing of both insurance and financial risks. E.g. capital requirements of Basle II and Solvency II. 55
  56. 56. Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance. perspectives • diversification issues Consider two risks S1 and S2, π1+2 = EP[S1 + S2] = EP[S1] + EP[S2] = π1 + π2 What about other premiums ? E.g. Eg◦P[S1 + S2] ? = Eg◦P[S1] + Eg◦P[S2] For some non subadditive risk measure (e.g. VaR) π1+2>π1 + π2. • econometric and heterogeneity issues In the case of pure premium, EP(S) = EP N i=1 Yi = EP(N) frequency × EP(Y ) average cost With heterogeneity πi = EP(S|X = xi) = EP(N|X = xi) × EP(Y |X = xi) What about other premiums ? 56

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