Ma rk et V alue of Life Insurane Contrats under Sto hasti - PowerPoint PPT Presentation

Ma rk et V alue of Life Insuran e Contra ts under Sto hasti Interest Rates and Default Risk Ca role Berna rd Olivier Le Courtois F ran�ois Quitta rd-Pinon Universit y of Ly on 1 E.M. Ly on

A Sho rt A tua rial Bibliography ... Briys and de V a renne [1993, 1997℄ Grosen and J�rgensen [1997, 2000, 2002℄ T ansk anen and Lukk a rinen [2003℄ ➠ J�rgensen [2004℄ ➠ ➠ ➠

...Complemented b y some Majo r Referen es from Finan e F o rtet [1943℄ Merton [1974℄ Heath, Ja rro w and Mo rton [1992℄ ➠ Longsta� and S hw a rtz [1995℄ ➠ Collin-Dufresne and Goldstein [2001℄ ➠ ➠ ➠

Capital Stru ture of the Insuran e Company Assets Liabilities A 0 E 0 = (1 − α ) A 0 � The life-insuran e ompany has no debt. L 0 = αA 0 � E 0 = initial equit y value � L 0 = initial investment of the p oli yholders who all p ossess the same ontra t.

Simpli�ed Des ription of the Contra t The p oli yholders investment L 0 yields the minimum gua ranteed rate r g at ontra t expiry T . In ase of No-default : A T ≥ L 0 e r g T P oli yholders re eive the gua ranteed amount at T : ➠ In ase of Default : A T < L g (Company Insolven y) P oli yholders re eive A T . Equit yholders re eive nothing . L g T = L 0 e r g T ➠ T

A P a rti ipating P oli y P oli yholders a re given a ontra tual pa rt δ of the b ene- �ts of the ompany when its assets at maturit y a re su� iently high : where Assuming no p rio r bankrupt y , p oli yholders re eive at matu- rit y T : A T > L g T α < 1 . si A T < L g α si L g   A T   T si A T > L g       T ≤ A T ≤ L g L g T Θ L ( T ) = T α         L g T + δ ( αA T − L g  T T ) α

Company Ea rly Default The �rm pursues its a tivities until T if : where λ is �xed. Note that the ase λ > 1 is in favour of the p oli yholders. A t > λL 0 e r g t � B t Let τ b e the default time ∀ t ∈ [0 , T [ , In ase of p rio r insolven y , p oli yholders re eive : si τ = inf { t ∈ [0 , T ] / A t < B t } si  L 0 e r g τ  λ ≥ 1  = min( λ, 1) L 0 e r g τ Θ L ( τ ) =   λL 0 e r g τ λ < 1

Contra t Ma rk et V alue Denoting b y Q the risk-neutral p robabilit y measure, the p ri e of our life insuran e ontra t writes at t < τ : � e − � T This ontra t an b e split up into four simpler sub ontra ts : t r s ds � T − A T ) + � T ) + − ( L g L g T + δ ( αA T − L g V L ( t ) = E t 1 τ ≥ T Q � + e − � τ t r s ds min( λ, 1) L g τ 1 τ<T � : the �nal gua rantee � : the "b onus option" whi h is the pa rti ipating lause � : the default put on whi h p oli yholders a re sho rt. � � PO + � � V L = GF + BO − LR � � : the rebate paid in ase of ea rly default. � GF � BO � PO LR

Assets Dynami s and Interest Rate Mo delling Exp onential V olatilit y fo r the Zero-Coup ons : σ P ( t, T ) = ν The dynami s under Q of the sho rt interest rate r and the Zero- oup on P ( t, T ) a re : � 1 − e − a ( T − t ) � a and = a ( θ − r t ) dt + νdZ Q dr t 1 ( t ) The assets follo w : dA t where Z Q and Z Q a re dP ( t, T ) P ( t, T ) = r t dt − σ P ( t, T ) dZ Q o rrelated Q -Bro wnian motions. ( dZ Q .dZ Q 1 ( t ) ). A t = r t dt + σdZ Q ( t ) 1 1 = ρdt

De o rrelation Let us no w onsider a Bro wnian motion Z Q indep endent from . The Bro wnian motion Z Q an b e exp ressed as 2 Z Q 1 In this w a y w e de o rrelate the interest rate risk from the �rm � assets risk. The assets dynami s then writes : dZ Q ( t ) = ρdZ Q 1 − ρ 2 dZ Q 1 ( t ) + 2 ( t ) � � � dA t ρdZ Q 1 − ρ 2 dZ Q = r t dt + σ 1 ( t ) + 2 ( t ) A t

F o rw a rd-Neutral Exp ressions Let Q T b e the T -fo rw a rd-neutral measure. F rom Girsanov theo rem, Z Q T and Z Q T a re indep endent Q T -Bro wnian mo- tions. 1 2 Under Q T the p ri es P ( t, T ) and A t follo w the sto hasti dif- ferential equations : dZ Q T 1 + σ P ( t, T ) dt , dZ Q T = dZ Q = dZ Q 1 2 2 and dP ( t, T ) P ( t, T )) dt − σ P ( t, T ) dZ Q T P ( t, T ) = ( r t + σ 2 1 � � � dA t ρdZ Q T 1 − ρ 2 dZ Q T = ( r t − σρσ P ( t, T )) dt + σ + 1 2 A t

Contra t V aluation at t = 0 After hanging the p robabilit y measure, w e have in the F o rw a rd-Neutral universe : where V L (0) = P (0 , T ) ( GF + BO − PO + LR )  = L g GF T (1 − E 1 )         = αδ ( E 7 − E 2 ) − δL g  T ( E 8 − E 3 )  BO   = L g   T ( E 9 − E 4 ) − E 10 + E 5 PO         LR = min( λ, 1) L 0 E 6

with the follo wing quantities that remain to b e omputed : E 6 = E Q T [ e r g τ 1 τ<T ] E 1 = Q T [ τ < T ]         A T 1 � �   A T 1  E 2 = E Q T E 7 = E Q T Lg Lg  T T A T > , τ<T A T > α α � � � � A T > L g A T > L g T T E 3 = Q T E 8 = Q T α , τ < T α E 4 = Q T [ A T < L g E 9 = Q T [ A T < L g T , τ < T ] T ] � � � � E 5 = E Q T A T 1 A T <L g E 10 = E Q T A T 1 A T <L g T 1 τ<T T

Metho dology : Longsta� and S hw a rtz App ro ximation Problem : W e need to kno w the la w of τ , �rst passage time of the assets b ey ond the default-triggering ba rrier. Longsta� and S hw a rtz (1995) use F o rtet's result to ap- p ro ximate the densit y of τ in a p roblem simila r to ours. Collin-Dufresne and Goldstein (2001) give a o rre tion to ➠ the p revious metho d to tak e p rop erly into a ount the sto hasti feature of the interest rates. ➠

Rationale Let us rememb er the p rop er exp ression fo r τ Idea : App ro ximate the densit y of τ at time t under Q T as a pie ewise onstant fun tion. τ = inf { t ∈ [0 , T ] / A t < λL 0 e r g t } � The interval [0 , T ] is sub divided into n T subp erio ds. � The interest rate is dis retized b et w een r min and r max into intervals. and r i = r min + iδ r a re the dis retized values of time and interest rate. n r t j = jδ t

The p robabilit y of the event τ ∈ [ t j , t j +1 ] with r ∈ [ r i , r i +1 ] exp resses as : q ( i , j ) . Collin-Dufresne and Goldstein give a re ursive fo rmula fo r these p robabilities : One w ould �rst ompute q ( i, 1 ) fo r ea h i , and then q ( i, j ) re ursively fo r j ≥ 2 using : n r � q ( i, 1 ) = q ( u, 1 ) Ψ( r i , t 1 | r u , t 1 ) u =0 where Φ and Ψ a re ompletely kno wn. j − 1 n r � � q ( i, j ) = Φ( r i , t j ) − q ( u, v ) Ψ( r i , t j | r u , t v ) v =1 u =0

Exp ressions of Φ and Ψ let N b e the umulative fon tion of the G auss (0 , 1) la w, then : � � µ ( r t , l s , r s ) , Σ 2 ( r t , l s , r s ) L ( l t |F s , r t ) = G auss    h − µ ( r t , l 0 , r 0 )   Φ( r t , t ) = f r ( r t , t | l 0 , r 0 , 0) N �  Σ 2 ( r t , l 0 , r 0 ) where :    h − µ ( r t , l s = h, r s )   a r [ r t | r s ] Ψ( r t , t | r s , s ) = f r ( r t , t | l s = h, r s , s ) N �  Σ 2 ( r t , l s = h, r s ) 2 πv e − ( rt − m )2 1 √ f r ( r t , t | l s = h, r s , s ) = , m = E [ r t | r s ] , v = V 2 v

Empiri al Densit y and F o rtet's App ro ximate Densit y −3 3.5 x 10 Empirical Density n r =10 and n T =50 3 n r =50 and n T =200 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10

Computation of the E i dep ending on τ No w, ea h E i an b e omputed easily even if it dep ends on τ . W e detail ho w to valuate E 2 fo r instan e ; its exa t exp ression is :   Then, using onditional la ws w e obtain :    A T 1 �  �  E 2 = E Q T Lg  T A T > , τ<T α As L ( l t |F s , r t ) = G auss and as w e kno w the transition � � densit y of r : f r T + ∞ w e an ompute E 2 dis retizing the integrals. � � E 2 = e r g T e l T 1 { l T > ln ( ds dr s g ( r s , s ) E Q T α ) } | l s = h, r s , s, τ = s L 0 0 −∞ � µ, Σ 2 �

Let X b e a random va riable with la w N ( m, σ 2 ) , w e denote � � � m + σ 2 − ln( a ) � then admits the simpler exp ression : m + σ 2 Φ 1 ( m ; σ ; a ) = E [ e X 1 e X >a ] = exp N 2 σ E 2 � � � T � + ∞ � + ∞ The extended F o rtet's app ro ximation of E 2 writes : Σ s,T ; L 0 µ s,T ; � E 2 = e r g T dr s g ( r s , s ) dr T f r ( r T | r s , s, l s ) Φ 1 � ds α −∞ −∞ 0 � � n T n r n r � � � Σ t j ,T ; L 0 µ t j ,T ; � E 2 = e r g T � δ r f r ( r k | r i , t j , l t j ) Φ 1 q ( i, j ) α j =1 i =0 k =0

Numeri al Analysis W e set our pa rameter range a o rding as : 100 0.4 0.008 0.06 0.03 - 0.02 0.1 10 0.8 0.7 A 0 a ν θ r 0 ρ σ T λ α Contra t Maturit y : 10 y ea rs L 0 = αA 0 = 70