Time-Consistent and Market-Consistent Actuarial Valuations

Transcription

1 Time-Consistent and Market-Consistent Actuarial Valuations Antoon Pelsser 1 Mitja Stadje 2 1 Maastricht University & Kleynen Consultants & Netspar a.pelsser@maastrichtuniversity.nl 2 Tilburg University & Netspar 23 May 2013 ASTIN Conference Den Haag A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 1 / 34

2 Introduction Motivation Standard actuarial premium principles usually consider static premium calculation: What is price today of insurance contract with payoff at time T? Actuarial premium principles typically ignore financial markets Financial pricing considers dynamic pricing problem: How does price evolve over time until time T? Financial pricing typically ignores unhedgeable risks Examples: Pricing very long-dated cash flows T years Pricing long-dated options T > 5 years Pricing pension & insurance liabilities Pricing employee stock-options A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 2 / 34

3 Introduction Ambitions In this joint paper with Mitja Stadje: [Pelsser and Stadje, 2013] we want to combine 1 Time-Consistent pricing operators, see [Jobert and Rogers, 2008] 2 Market-Consistent pricing operators, see [Malamud et al., 2008] We will be interested in continuous-time limits of these discrete algorithms for different actuarial premium principles: 1 Variance Principle 2 Mean Value Principle 3 Standard-Deviation Principle 4 Cost-of-Capital Principle A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 3 / 34

4 Introduction Content of This Talk 1 Pure Insurance Risk Diffusion Model for Insurance Risk Variance Principle ( exponential indiff. pricing) Standard-Dev. Principle ( E[] under new measure) Cost-of-Capital Principle ( St.Dev price) Davis Price, see [Davis, 1997] St.Dev is small perturbation of Variance price 2 Financial & Insurance Risk Diffusion Model for Financial Risk Market-Consistent Pricing Variance Principle Numerical Illustration 3 Conclusions A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 4 / 34

5 Pure Insurance Risk Model for Insurance Risk Pure Insurance Model Consider unhedgeable insurance process y: dy = a(t, y) dt + b(t, y) dw To keep math simple, concentrate on diffusion setting Discretisation scheme as binomial tree: { +b t with prob. 1 y(t + t) = y(t) + a t + 2 b t with prob. 1 2 A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 5 / 34

6 Pure Insurance Risk Time Consistency Time Consistency Time Consistent price π(t, y) satisfies property π[f (y(t )) t, y] = π[π[f (y(t )) s, y(s)] t, y] t < s < T Price of today of holding claim until T is the same as buying claim half-way at time s for price π(s, y(s)) Semi-group property Similar idea as tower property of conditional expectation A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 6 / 34

7 Pure Insurance Risk Variance Principle Variance Principle Actuarial Variance Principle Π v : Π v t[f (y(t ))] = E t [f (y(t ))] αvar t[f (y(t ))] α is Absolute Risk Aversion Apply Π v to one binomial time-step to obtain price π v : π v( t, y(t) ) = E t [π v( t + t, y(t + t) ) ] + Note: we omit discounting for now 1 2 αvar t[π v( t + t, y(t + t) ) ] A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 7 / 34

8 Pure Insurance Risk Variance Principle Pricing PDE Assume π v (t, y) admits Taylor approximation in y Evaluate Var.Princ. for binomial step & take limit for t 0 Same as derivation of Feynman-Kaç, but for E[] αvar[] This leads to pde for π v : π v t + aπ v y b2 π v yy α(bπv y ) 2 = 0 Note, non-linear term = local unhedgeable variance b 2 (π v y ) 2 Find general solution to this non-linear pde via log-transform: π v (t, y) = 1 α ln E t [e αf (y(t )) y(t) = y ]. Exponential indifference price, see [Henderson, 2002] or [Musiela and Zariphopoulou, 2004] A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 8 / 34

9 Pure Insurance Risk Variance Principle Include Discounting We should include discounting into our pricing Absolute Risk Aversion α is not unit-free, but has unit 1/e This conveniently compensates the unit (e) 2 of Var[]... Therefore, α-today is different than α-tomorrow Relative Risk Aversion γ is unit-free Express ARA relative to benchmark wealth X 0 e rt Explicit notation: α γ/x 0 e rt leads to pde: πt v + aπy v b2 πyy v + 1 γ 2 X 0 e rt (bπv y ) 2 rπ v = 0 π v (t, y) = X 0e rt [ ] γ X ln E e 0 e rt f (y(t )) y(t) = y γ Note: express all prices in discounted terms A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 9 / 34

10 Pure Insurance Risk Variance Principle Backward Stochastic Differential Equations Pricing PDE: πt v + aπy v b2 πyy v + 1 γ 2 X 0 e rt (bπv y ) 2 rπ v = 0 This non-linear PDE, represents the solution to a so-called BSDE for the triplet of processes (y t, Y t, Z t ) dy t = a(t, y t ) dt + b(t, y t ) dw t dy t = g(t, y t, Y t, Z t ) dt + Z t dw t Y T = f ( y(t ) ), with generator g(t, y, Y, Z) = 1 2 X 0 e Z 2 ry. rt Recent literature studies uniqueness & existence of solutions to BSDE s, see [El Karoui et al., 1997] Via BSDE s we can study time-consistent pricing operators in a much more general stochastic setting. But we will not pursue this here. A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 10 / 34 γ

11 Pure Insurance Risk Mean Value Principle Mean Value Principle Generalise to Mean Value Principle Π m t [f (y(t ))] = v 1 (E t [v(f (y(t )))]) for any function v() which is a convex and increasing Exponential pricing is special case with v(x) = e αx Do Taylor-expansion & limit t 0: π mf t + aπ mf y b2 πyy mf + 1 v (π mf ) 2 v (π mf ) (bπmf y ) 2 = 0 Note: π mf (t, y) := π m (t, y)/e rt is price expressed in discounted terms Interpretation as generalised Variance Principle with local risk aversion term: v ()/v () A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 11 / 34

12 Pure Insurance Risk Standard-Deviation Principle Standard-Deviation Principle Actuarial Standard-Deviation Principle: Π s t[f (y(t ))] = E t [f (y(t ))] + β Var t [f (y(t ))]. Pay attention to time-scales : Expectation scales with t St.Dev. scales with t Thus, we should take β t to get well-defined limit Note: β has unit 1/ time A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 12 / 34

13 Pure Insurance Risk Standard-Deviation Principle Pricing PDE Do Taylor-expansion & limit t 0: πt s + aπy s b2 πyy s + β (bπy s ) 2 rπ s = 0 Again, non-linear pde. But if π s is monotone in y then π s t + (a ± βb)π s y b2 π s yy rπ s = 0 π s (t, y) = E S t [f (y(t )) y(t) = y] Upwind drift-adjustment into direction of risk A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 13 / 34

14 Pure Insurance Risk Cost-of-Capital Principle Cost-of-Capital Principle Cost-of-Capital principle, popular by practitioners Used in QIS5-study conducted by EIOPA Idea: hold buffer-capital against unhedgeable risks. Borrow from shareholders by giving excess return δ Define buffer via Value-at-Risk measure: ] Π c t[f (y(t ))] = E t [f (y(t ))] + δvar q,t [f (y(t )) E t [f (y(t ))]. A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 14 / 34

15 Pure Insurance Risk Cost-of-Capital Principle Scaling & PDE Again, pay attention to time-scaling : First, scale VaR back to per annum basis with 1/ t Then, δ is like interest rate, so multiply with t Net scaling: δ t/ t = δ t. Limit: for small t the VaR behaves as Φ 1 (q) St.Dev. Hence, limiting pde is same as π s but with β = Φ 1 (q)δ. Conclusion: In the limit for t 0, CoC pricing is the same as st.dev. pricing (for a diffusion process!) A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 15 / 34

16 Pure Insurance Risk Davis Price Davis Price The variance price π v is hard to calculate, the st.dev. price π s is easy to calculate Can we make a connection between these two concepts? Yes, we can! using small perturbation expansion Consider existing insurance portfolio with price π v (t, y), now add small position with price επ D (t, y). Subst. into pde: (πt v + επt D ) + a(πy v + επy D ) b2 (πyy v + επyy) D + 1 γ ( 2 X 0 e rt b2 (πy v ) 2 + 2επy v πy D + ε 2 (πy D ) 2) r(π v + επ D ) = 0 π v () solves the pde, cancel π v -terms A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 16 / 34

17 Pure Insurance Risk Davis Price Pricing PDE Simplify pde, and divide by ε: πt D + aπy D b2 πyy D + 1 γ ( 2 X 0 e rt b2 2πy v πy D + ε(πy D ) 2) rπ D = 0 Approximation: ignore small ε-term ( πt D + a + γ X 0 e rt b2 π v y ) π D y b2 π D yy + rπ D = 0 π D (t, y) = E D t [f (y(t )) y(t) = y] Davis price π D is defined only relative to existing price π v of insurance portfolio Note, drift-adjustment of st.dev. price scales with b A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 17 / 34

18 Financial & Insurance Risk Financial & Insurance Risk Investigate environment with financial risk that can be traded (and hedged!) in financial market and non-traded insurance risk Model financial risk as [Black and Scholes, 1973] economy. Model return process x t = ln S t under real-world measure P: dx = ( µ(t, x) 1 2 σ2 (t, x) ) dt + σ(t, x) dw f Binomial time-step: { x(t + t) = x(t) + (µ 1 +σ t with P-prob. 1 2 σ2 ) t + 2 σ t with P-prob. 1 2 A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 18 / 34

19 Financial & Insurance Risk No-arbitrage pricing BS-economy is arbitrage-free and complete unique martingale measure Q. No-arbitrage pricing operator for financial derivative F (x(t )): π Q (t, x) = e r(t t) E Q t [F (x(t ))] Binomial step for x under measure Q: x(t + t) = x(t) + (µ 1 2 σ2 ) t+ +σ ( t with Q-prob µ r ) σ t σ ( t with Q-prob µ r ) σ t Quantity (µ r)/σ is Radon-Nikodym exponent of dq/dp Quantity (µ r)/σ is also known as market-price of financial risk. A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 19 / 34

20 Financial & Insurance Risk Quadrinomial Tree Joint discretisation for processes x and y using quadrinomial tree with correlation ρ under measure P: State: y + y y y ( ) ( ) 1 + ρ 1 ρ x + x 4 4 ( ) ( ) 1 ρ 1 + ρ x x 4 4 Positive correlation increases probability of joint ++ or co-movement A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 20 / 34

21 Financial & Insurance Risk Market-Consistent Pricing Market-Consistent Pricing We are looking for market-consistent pricing operators, see e.g. [Malamud et al., 2008] Definition A pricing operator π() is market-consistent if for any financial derivative F (x(t )) and any other claim G(t, x, y) we have π F +G (t, x, y) = e r(t t) E Q t [F (x(t ))] + π G (t, x, y). Observation: generalised notion of translation invariance for all financial risks A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 21 / 34

22 Financial & Insurance Risk MC Variance Pricing MC Variance Pricing Intuition: construct MC pricing in two steps using conditional expectations. For full results we refer to [Pelsser and Stadje, 2013]. First: condition on financial risk & use actuarial pricing for pure insurance risk π v (t + t x±) := E[π v (t + t) x±]+ 1 γ 2 X 0 e r(t+ t) Var[πv (t + t) x±] E[π v (t + t) x+] = Var[π v (t + t) x+] = E[π v (t + t) x ] = Var[π v (t + t) x ] = ( 1+ρ 2 ( 1 ρ 2 4 ( 1 ρ 2 ( 1 ρ 2 4 ) ( ) π++ v + 1 ρ 2 ) (π v ++ π+ v ) 2 ) ( ) π + v + 1+ρ 2 ) (π v + π v ) 2. For ρ = 1 or ρ = 1 no unhedgeable risk left Var = 0 π v + π v A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 22 / 34

23 Financial & Insurance Risk MC Variance Pricing Pricing PDE Second: use no-arbitrage pricing for artificial financial derivative π v (t, x, y) = e r t E Q [π v ( (t + t x±)] ( = e r t µ r ) σ t π v (t + t x+) + ( µ r ) ) σ t π v (t + t x ) Do Taylor-expansion & limit t 0: πt v + (r 1 2 σ2 )πx v + ( a ρb µ r ) σ π v y σ2 πxx v + ρσbπxy v b2 πyy v + 1 γ 2 X 0 e rt (1 ρ2 )(bπy v ) 2 rπ v = 0 Impact on x: Q-drift (r 1 2 σ2 ) Impact on y: adjusted drift ( a ρb µ r ) σ Non-linear term for locally unhedgeable variance (1 ρ 2 )(bπy v ) 2 A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 23 / 34

24 Financial & Insurance Risk MC Variance Pricing Pure Insurance Payoff Unfortunately, we cannot solve the non-linear pde in general Special case: consider pure insurance payoff (and constant ), then no (explicit) dependence on x MPR µ r σ πt v + ( a ρb µ r ) σ π v y b2 πyy v + 1 γ 2 X 0 e rt (1 ρ2 )(bπy v ) 2 rπ v = 0 Note that for ρ 0 the pde is different from the pure insurance pde Via correlation with financial market we can still hedge part of the insurance risk Note: incomplete market no martingale representation, therefore delta-hedge is not π v x In fact: hold ρbπ v y /σ in x as hedge Economic explanation for drift-adjustment in y, a kind of quanto-adjustment A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 24 / 34

25 Financial & Insurance Risk MC Variance Pricing Pure Insurance Payoff (2) General solution via log-transform: π v (t, y) = X 0e rt P γ(1 ρ 2 ln E ) Measure P induces drift-adjusted process for y [ e ] γ(1 ρ 2 ) X 0 e rt f (y(t )) y(t) = y See also, [Henderson, 2002] and [Musiela and Zariphopoulou, 2004] who derived this solution in the context of exponential indifference pricing We can generalise to Mean Value Principle for any convex function v() A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 25 / 34

26 Financial & Insurance Risk MC Variance Pricing Numerical Illustration Consider unit-linked insurance contract with payoff: y(t )S(T ) = y(t )e x(t ) Numerical calculation in quadrinomial tree with 5 time-steps of 1 year Naive hedge is to hold y(t) units of share S(t) In fact: hold π v x + ρbπ v y /σ as hedge MC Variance hedge also builds additional reserve as buffer against unhedgeable risk Payoff at T=5 Fin Delta t= Share Price Insurance Risk "y" Share Price Insurance Risk "y" A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 26 / 34

27 Financial & Insurance Risk MC Variance Pricing Numerical Illustration (2) Investigate impact of correlation ρ Compare ρ = 0.50 (left) and ρ = 0 (right) Fin Delta t=4 Fin Delta t= Share Price Insurance Risk "y" Share Price Insurance Risk "y" Positive correlation leads to higher delta, as this also hedges part of insurance risk: hold π v x + ρbπ v y /σ as hedge Price for ρ = 0.00 at t = 0 is e26.75 Price for ρ = 0.50 at t = 0 is e18.79, due to less unhedgeable risk Price for ρ = 0.99 at t = 0 is e1.98, due to drift-adjustment A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 27 / 34

28 Financial & Insurance Risk MC Standard-Deviation Pricing MC Standard-Deviation Pricing Again, do two-step construction First: condition on financial risk & use actuarial pricing for pure insurance risk π s (t + t x±) := E[π s (t + t) x±] + δ t Var[π s (t + t) x±] Second: do no-arbitrage valuation under Q This leads to linear pricing pde (if π s (t, x, y) monotone in y): ( ) πt s + (r 1 2 σ2 )πx s + a ρb µ r σ ± δ 1 ρ 2 b πy s σ2 π s xx + ρσbπ s xy b2 π s yy rπ s = 0 Drift adjustment for y is now combination of hedge cost plus upwind risk-adjustment ±δ 1 ρ 2 b A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 28 / 34

29 Financial & Insurance Risk MC Davis Price MC Davis Price We can again consider Davis price, by small perturbation expansion This leads to pricing pde: ( ) πt D + (r 1 2 σ2 )πx D + a ρb µ r σ + γ X 0 e (1 ρ 2 )b 2 π v rt y πy D σ2 π D xx + ρσbπ D xy b2 π D yy rπ D = 0 Drift adjustment for y is now combination of hedge cost plus γ risk-adjustment X 0 e (1 ρ 2 )b 2 π v rt y Davis price defined relative to existing price π v (t, x, y) Note, st.dev. pricing depends on (1 ρ 2 ) b A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 29 / 34

30 Financial & Insurance Risk Multi-dimensional MC Variance Price Multi-dimensional MC Variance Price Vectors x of asset returns (n-vector), y of insurance risks (m-vector) dx = µ dt + Σ 1 2 dwf dy = a dt + B 1 2 dw Partitioned covariance matrix C (n + m) (n + m) P is n m matrix of financial (& insurance ) covariances Σ P C = P B The market-price of financial risks is an n-vector Σ 1 (µ r) Multi-dim pricing pde for π v : ( π πt v + r πx v + a P Σ 1 v (µ r)) y + 1 ( 2 Cij πij v ) + 1 γ ( ) 2 X 0 e rt πy v (B P Σ 1 P)πy v rπ v = 0 Note: (B P Σ 1 P) is conditional covariance matrix of y x A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 30 / 34

31 Financial & Insurance Risk Multi-dimensional MC StDev Price Multi-dimensional MC StDev Price Multi-dim pricing pde for π s : ( π πt s + r πx s + a P Σ 1 s (µ r)) y + 1 ( 2 Cij πij s ) δ π s y (B P Σ 1 P)π s y rπ s = 0 Note: unlike 1-dim case, does not simplify to linear pde Simplification only possible if (B P Σ 1 P) has rank 1 and all π s i have same sign A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 31 / 34

32 Conclusions Conclusions 1 Pure Insurance Risk Variance Principle ( exponential indiff. pricing) Standard-Dev. Principle ( E[] under new measure) Cost-of-Capital Principle ( St.Dev price) Davis Price: St.Dev. price is small perturbation of Variance price 2 Financial & Insurance Risk Market-Consistent Pricing: via two-step conditional expectations MC Variance Principle Numerical Illustration for unit-linked contract A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 32 / 34

33 References References I Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81: Davis, M. (1997). Option pricing in incomplete markets. In Dempster, M. and Pliska, S., editors, Mathematics of Derivative Securities, pages Cambridge University Press. El Karoui, N., Peng, S., and Quenez, M. (1997). Backward stochastic differential equations in finance. Mathematical finance, 7(1):1 71. Henderson, V. (2002). Valuation of claims on nontraded assets using utility maximization. Mathematical Finance, 12(4): A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 33 / 34

34 References References II Jobert, A. and Rogers, L. (2008). Valuations and dynamic convex risk measures. Mathematical Finance, 18(1):1 22. Malamud, S., Trubowitz, E., and Wüthrich, M. (2008). Market consistent pricing of insurance products. ASTIN Bulletin, 38(2): Musiela, M. and Zariphopoulou, T. (2004). An example of indifference prices under exponential preferences. Finance and Stochastics, 8(2): Pelsser, A. and Stadje, M. (2013). Time-consistent and market-consistent evaluations. Mathematical Finance. A. Pelsser (Maastricht U) TC & MC Valuations 23 May 2013 ASTIN 34 / 34