Longevity and pension funds Paris 3-4 February 2011 Ragnar Norberg - PowerPoint PPT Presentation

Longevity and pension funds Paris 3-4 February 2011 Ragnar Norberg ISFA Universit´ e Lyon 1 Email: ragnar.norberg@univ-lyon1.fr Homepage: http://isfa.univ-lyon1.fr/ ∼ norberg/ MANAGEMENT OF MORTALITY AND LONGEVITY RISK IN LIFE INSURANCE AND PENSIONS: RISK SHARING VS MARKET OPERATIONS

State of economic-demographic environment at time 푡 is 푌 ( 푡 ) ( 푌 ( 푡 )) 푡 ∈ [0 ,푇 ] generates filtration F 푌 = ( ℱ 푌 푡 ) 푡 ∈ [0 ,푇 ] Assume 푌 Markov chain with state space 풴 = { 0 , 1 , . . . , 퐽 푌 } and intensity matrix Λ = ( 휆 푒푓 ), 푒, 푓 ∈ 풴 , 푌 (0) = 0. 푇 -year insurance policy (or portfolio of policies) issued at time 0. State of policy at time 푡 is 푍 ( 푡 ). ( 푍 ( 푡 )) 푡 ∈ [0 ,푇 ] generates filtration F 푍 = ( ℱ 푍 푡 ) 푡 ∈ [0 ,푇 ] . Conditional on ℱ 푌 푇 , 푍 is Markov chain, state space 풵 = { 0 , 1 , . . . , 퐽 푍 } , intensity matrix M 푌 ( 푡 ) ( 푡 ) = ( 휇 푌 ( 푡 ) ,푗푘 ( 푡 )), 푗, 푘 ∈ 풵 , 푍 (0) = 0. Condi- tional transition probabilities P ( 푠, 푡 ) = ( 푝 푗푘 ( 푠, 푡 )) 푝 푗푘 ( 푠, 푡 ) = P [ 푍 ( 푡 ) = 푘 ∣ 푍 ( 푠 ) = 푗, ℱ 푌 푇 ] = P [ 푍 ( 푡 ) = 푘 ∣ 푍 ( 푠 ) = 푗, ℱ 푌 푡 ] are solutions to forward differential equations 푑 P ( 푠, 푡 ) = P ( 푠, 푡 ) M 푌 ( 푡 ) ( 푡 ) 푑푡 , P ( 푠, 푠 ) = I 2

Thus, ( 푌, 푍 ) is a Markov chain with state space 풴 × 풵 and intensities ⎧ 휆 푒푓 , 푒 ∕ = 푓, 푗 = 푘 ,  ⎨ 휅 푒푗,푓푘 ( 푡 ) = 휇 푒,푗푘 ( 푡 ) , 푒 = 푓, 푗 ∕ = 푘 ,  0 , 푒 ∕ = 푓, 푗 ∕ = 푘 . ⎩ Indicator processes 퐼 푌 푒 ( 푡 ) = 1[ 푌 ( 푡 ) = 푒 ] Counting processes 푁 푌 푒푓 ( 푡 ) = ♯ { 휏 ; 푌 ( 휏 − ) = 푒, 푌 ( 휏 ) = 푓, 휏 ∈ (0 , 푡 ] } Observe that 푑퐼 푌 푑푁 푌 푑푁 푌 퐼 푌 ∑ ∑ 푒 ( 푡 ) = 푓푒 ( 푡 ) − 푒푓 ( 푡 ) , 푒 (0) = 훿 0 푒 푓 ; 푓 ∕ = 푒 푓 ; 푓 ∕ = 푒 Similar for policy process 퐼 푍 푗 ( 푡 ) and 푁 푍 푗푘 ( 푡 ) ( 푌, 푍 ) generates filtration F = ( ℱ 푡 ) 푡 ∈ [0 ,푇 ] , ℱ 푡 = ℱ 푌 푡 ∨ ℱ 푍 푡 . 3

FINANCIAL MARKET: Assume there exists arbitrage-free market for environmental risk: Equivalent martingale measure ˜ P under which 푌 is a Markov chain with transition rate matrix ˜ Λ = (˜ 휆 푒푓 ). The compensated environment-related counting processes 푑푀 푌 푒푓 ( 푡 ) = 푑푁 푌 푒푓 ( 푡 ) − 퐼 푌 푒 ( 푡 ) ˜ 휆 푒푓 푑푡 are the driving ( F 푌 , ˜ P )-martingales. Locally risk free asset: Market money account with price process ∫ 푡 0 푟 ( 푠 ) 푑푠 푆 0 ( 푡 ) = 푒 yields interest at fixed rate 푟 푒 in state 푒 : 퐼 푌 ∑ 푟 ( 푡 ) = 푟 푌 ( 푡 ) = 푒 ( 푡 ) 푟 푒 푒 4

Risky assets, fundamental or derivatives, with unpredictable price jumps. Generic 푈 - claim with a single pay-off 퐻 ∈ ℱ 푌 푈 at fixed time 푈 . Price at 푒 − ∫ 푈 [ � ] 푡 푟 ( 푠 ) 푑푠 퐻 time 푡 ≤ 푈 is 푆 ( 푡 ) = ˜ � ℱ 푌 � . Discounted price process E � 푡 푆 ( 푡 ) = 푆 − 1 푆 − 1 � [ � ℱ 푌 ] 0 ( 푡 ) 푆 ( 푡 ) = ˜ ˜ 0 ( 푈 ) 퐻 E � 푡 is ( F 푌 , ˜ P )-martingale: 휉 푒푓 ( 푡 ) 푑푀 푌 ˜ 푑 ˜ ∑ 푆 ( 푡 ) = 푒푓 ( 푡 ) , 푒 ∕ = 푓 휉 푒푓 are F 푌 -predictable. Work with “inflated” coefficients, where the ˜ 휉 푒푓 ( 푡 ) := 푆 0 ( 푡 ) ˜ 휉 푒푓 ( 푡 ) : 푆 ( 푡 ) = 푆 − 1 휉 푒푓 ( 푡 ) 푑푀 푌 푑 ˜ ∑ 0 ( 푡 ) 푒푓 ( 푡 ) . 푒 ∕ = 푓 5

Analysis for identifying the coefficients 휉 푒푓 . Consider fairly general 푈 -claim 퐻 = 푒 − ∫ 푈 0 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ℎ 푌 ( 푈 ) , ( 푔 푒 ( 푡 )) 푡 ∈ [0 ,푈 ] and ℎ 푒 , 푒 = 0 , . . . , 퐽 푌 are non-random scalars. − ∫ 푈 ( ) � [ ] 푑푠 ℎ 푌 ( 푈 ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) � � ℱ 푌 ˜ ˜ 0 푆 ( 푡 ) = E 푒 � 푡 � − ∫ 푡 − ∫ 푈 ( ) ( ) � [ ] 푑푠 ˜ 푑푠 ℎ 푌 ( 푈 ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) � � ℱ 푌 0 푡 = 푒 E 푒 � 푡 � − ∫ 푡 ( ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ∑ 퐼 푌 0 = 푒 ( 푡 ) 푣 푒 ( 푡 ) 푒 푒 − ∫ 푈 ( ) [ � ] 푑푠 ℎ 푌 ( 푈 ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) � 푣 푒 ( 푡 ) = ˜ 푡 � 푌 ( 푡 ) = 푒 E 푒 � � 6

Ito: − ∫ 푡 ( ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ( ) 퐼 푌 푑 ˜ ∑ 0 푆 ( 푡 ) = − 푒 푟 ( 푡 ) + 푔 푌 ( 푡 ) ( 푡 ) 푑푡 푒 ( 푡 ) 푣 푒 ( 푡 ) 푒 − ∫ 푡 ( ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ∑ 퐼 푌 0 + 푒 푒 ( 푡 ) 푑푣 푒 ( 푡 ) 푒 − ∫ 푡 ( ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ∑ ( 푣 푓 ( 푡 ) − 푣 푒 ( 푡 )) 푑푁 푌 0 + 푒 푒푓 ( 푡 ) 푒 ∕ = 푓 − ∫ 푡 ( ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ∑ 퐼 푌 0 = 푒 ( 푡 ) × 푒 푒 ⎛ ⎞ ( ) ∑ ˜ ⎝ − ( 푟 푒 + 푔 푒 ( 푡 )) 푣 푒 ( 푡 ) 푑푡 + 푑푣 푒 ( 푡 ) + 휆 푒푓 푣 푓 ( 푡 ) − 푣 푒 ( 푡 ) 푑푡 ⎠ 푓 ; 푓 ∕ = 푒 − ∫ 푡 ( ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ∑ ( ) 푑푀 푌 0 + 푒 푣 푓 ( 푡 ) − 푣 푒 ( 푡 ) 푒푓 ( 푡 ) 푒 ∕ = 푓 7

Drift term vanishes, which gives constructive PDE-s: ⎡ ⎤ ( ) ˜ ⎦ 푑푡 , ∑ 푑푣 푒 ( 푡 ) = ⎣ ( 푟 푒 + 푔 푒 ( 푡 )) 푣 푒 ( 푡 ) − 휆 푒푓 푣 푓 ( 푡 ) − 푣 푒 ( 푡 ) 푒 ∈ 풴 , 푓 ; 푓 ∕ = 푒 with side conditions 푣 푒 ( 푈 ) = ℎ 푒 , 푒 ∈ 풴 . Martingale dynamics reduces to − ∫ 푡 ( ) 푟 ( 푠 ) + 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ∑ ( ) 푑푀 푌 푑 ˜ 0 푆 ( 푡 ) = 푣 푓 ( 푡 ) − 푣 푒 ( 푡 ) 푒푓 ( 푡 ) 푒 푒 ∕ = 푓 = 푆 − 1 휉 푒푓 ( 푡 ) 푑푀 푌 ∑ 0 ( 푡 ) 푒푓 ( 푡 ) 푒 ∕ = 푓 휉 푒푓 ( 푡 ) = 푒 − ∫ 푡 0 푔 푌 ( 푠 ) ( 푠 ) 푑푠 ( ) 푣 푓 ( 푡 ) − 푣 푒 ( 푡 ) 8

INSURANCE PAYMENTS: Benefits less premiums generate payment function 퐵 : 퐼 푍 푏 푗푘 ( 푡 ) 푑푁 푍 ∑ ∑ 푑퐵 ( 푡 ) = 푗 ( 푡 ) 푑퐵 푗 ( 푡 ) + 푗푘 ( 푡 ) 푗 ∈풵 푗,푘 ∈풵 ; 푗 ∕ = 푘 Policy terminates at a finite time 푇 . Market reserve at time 푡 ≤ 푇 is [∫ 푇 푒 − ∫ 휏 � ] 푡 푟 ( 푠 ) 푑푠 푑퐵 ( 휏 ) � ˜ 푉 ( 푡 ) = � ℱ 푡 E � � 푡 퐼 푌 푒 ( 푡 ) 퐼 푍 ∑ ∑ = 푗 ( 푡 ) 푉 푒푗 ( 푡 ) , 푒 푗 where (Markov assumptions) state-wise reserves are [∫ 푇 푒 − ∫ 휏 � ] 푡 푟 ( 푠 ) 푑푠 푑퐵 ( 휏 ) � 푉 푒푗 ( 푡 ) = ˜ E � 푌 ( 푡 ) = 푒, 푍 ( 푡 ) = 푗 � � 푡 9

Sums at risk: 휂 푒,푗푘 ( 푡 ) = 푏 푗푘 ( 푡 ) + 푉 푒푘 ( 푡 ) − 푉 푒푗 ( 푡 ) , 푒 ∈ 풴 , 푗 ∕ = 푘 ∈ 풵 , and 휂 푒푓,푗 ( 푡 ) = 푉 푓푗 ( 푡 ) − 푉 푒푗 ( 푡 ) , 푒 ∕ = 푓 ∈ 풴 , 푗 ∈ 풵 . The 푉 푒푗 ( 푡 ) are solutions to the ODE-s 휂 푒푓,푗 ( 푡 )˜ ∑ ∑ 푑푉 푒푗 ( 푡 ) = 푉 푒푗 ( 푡 ) 푟 푒 푑푡 − 푑퐵 푗 ( 푡 ) − 휂 푒,푗푘 ( 푡 ) 휇 푒,푗푘 ( 푡 ) 푑푡 휆 푒푓 푑푡 − 푓 ∕ = 푒 푘 ∕ = 푗 푉 푒푗 ( 푇 − ) = ∆ 퐵 푗 ( 푇 ) Market value of total cash-flow at time 푡 , discounted at time 0, is [∫ 푇 ∫ 푡 � ] � 0 − 푆 − 1 0 − 푆 − 1 0 ( 휏 ) 푑퐵 ( 휏 ) + 푆 − 1 푀 ( 푡 ) = ˜ E 0 ( 휏 ) 푑퐵 ( 휏 ) � ℱ 푡 = 0 ( 푡 ) 푉 ( 푡 ) � � 10

푀 is ( F , ˜ P )-martingale, therefore stochastic integral with respect to P )-martingales, 푀 푌 푒푓 and 푀 푍 the underlying ( F , ˜ 푗푘 given by 푑푀 푍 푗푘 ( 푡 ) = 푑푁 푍 푗푘 ( 푡 ) − 퐼 푍 퐼 푌 ∑ 푗 ( 푡 ) 푒 ( 푡 ) 휇 푒,푗푘 ( 푡 ) 푑푡 푒 11

To determine form of 푀 , apply Itˆ o: ⎛ ⎞ 푆 − 1 퐼 푍 푏 푗푘 ( 푡 ) 푑푁 푍 ⎝∑ ∑ 푑푀 ( 푡 ) = 0 ( 푡 ) 푗 ( 푡 ) 푑퐵 푗 ( 푡 ) + 푗푘 ( 푡 ) ⎠ 푗 푘 ; 푘 ∕ = 푗 − 푆 − 1 퐼 푌 푒 ( 푡 ) 퐼 푍 ∑ ∑ 0 ( 푡 ) 푟 ( 푡 ) 푑푡 푗 ( 푡 ) 푉 푒푗 ( 푡 ) 푒 푗 + 푆 − 1 퐼 푌 푒 ( 푡 ) 퐼 푍 ∑ ∑ 0 ( 푡 ) 푗 ( 푡 ) 푑푉 푒푗 ( 푡 ) 푒 푗 + 푆 − 1 퐼 푍 ( 푉 푓푗 ( 푡 ) − 푉 푒푗 ( 푡 )) 푑푁 푌 ∑ ∑ 0 ( 푡 ) 푗 ( 푡 ) 푒푓 ( 푡 ) 푗 푒 ∕ = 푓 + 푆 − 1 퐼 푌 ( 푉 푒푘 ( 푡 ) − 푉 푒푗 ( 푡 )) 푑푁 푍 ∑ ∑ 0 ( 푡 ) 푒 ( 푡 ) 푗푘 ( 푡 ) . 푒 푗 ∕ = 푘 (Left limits in the indicators 퐼 푍 푗 ( 푡 ) and 퐼 푌 푒 ( 푡 ) have been dropped.) 12

Insert 푑푁 푌 푒푓 ( 푡 ) = 푑푀 푌 푒푓 ( 푡 ) + 퐼 푌 푒 ( 푡 )˜ 휆 푒푓 푑푡 , 푑푁 푍 푗푘 ( 푡 ) = 푑푀 푍 푗푘 ( 푡 ) + 퐼 푍 푒 퐼 푌 푗 ( 푡 ) ∑ 푒 ( 푡 ) 휇 푒,푗푘 ( 푡 ) 푑푡 and identify drift terms and martingale terms, to arrive at: The martingale 푀 is given by 푀 (0) = ∆ 퐵 0 (0) + 푉 0 , 0 (0) ⎛ 푆 − 1 퐼 푌 푒 ( 푡 ) 휂 푒,푗푘 ( 푡 ) 푑푀 푍 ⎝ ∑ ∑ 푑푀 ( 푡 ) = 0 ( 푡 ) 푗푘 ( 푡 ) 푒 푗 ∕ = 푘 ⎞ 퐼 푍 푗 ( 푡 ) 휂 푒푓,푗 ( 푡 ) 푑푀 푌 ∑ ∑ + 푒푓 ( 푡 ) ⎠ 푒 ∕ = 푓 푗 The differential equations for the 푉 푒푗 are obtained by setting drift term equal to 0. 13