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

SLIDE 1 Ma rk et V alue

f

Life Insuran e Contra ts under Sto hasti Interest Rates and Default Risk Ca role Berna rd Olivier Le Courtois F ranois Quitta rd-Pinon Universit y

f

Ly

n

1 E.M. Ly

n

SLIDE 2 A Sho rt A tua rial Bibliography ...

➠

Briys and de V a renne [1993, 1997℄

➠

Grosen and Jrgensen [1997, 2000, 2002℄

➠

T ansk anen and Lukk a rinen [2003℄

➠

Jrgensen [2004℄

SLIDE 3 ...Complemented b y some Majo r Referen es from Finan e

➠

F

rtet

[1943℄

➠

Merton [1974℄

➠

Heath, Ja rro w and Mo rton [1992℄

➠

Longsta and S hw a rtz [1995℄

➠

Collin-Dufresne and Goldstein [2001℄

SLIDE 4 Capital Stru ture

f

the Insuran e Company Assets Liabilities

A0 E0 = (1 − α)A0 L0 = αA0

The

life-insuran e

mpany

has no debt. E0 = initial equit y value L0 = initial investment

f

the p

li yholders

who all p

ssess

the same

ntra t.

SLIDE 5 Simplied Des ription

f

the Contra t The p

li yholders

investment L0 yields the minimum gua ranteed rate rg at

ntra t

expiry T .

➠

In ase

f

No-default : AT ≥ L0 ergT P

li yholders

re eive the gua ranteed amount at T :

Lg

T = L0 ergT

➠

In ase

f

Default : AT < Lg

T

(Company Insolven y) P

li yholders

re eive AT. Equit yholders re eive nothing.

SLIDE 6 A P a rti ipating P

li y

P

li yholders

a re given a

ntra tual

pa rt δ

f

the b enets

f

the

mpany

when its assets at maturit y a re su iently high :

AT > Lg

T

α

where

α < 1.

Assuming no p rio r bankrupt y , p

li yholders

re eive at maturit y T :

ΘL(T) =

                  

AT

si AT < Lg

T

Lg

T

si Lg

T ≤ AT ≤ Lg

T

α

Lg

T + δ(αAT − Lg T )

si AT > Lg

T

α

SLIDE 7 Company Ea rly Default The rm pursues its a tivities until T if :

∀t ∈ [0, T[ , At > λL0ergt Bt

where λ is xed. Note that the ase λ > 1 is in favour

f

the p

li yholders.

Let τ b e the default time

τ = inf{t ∈ [0, T] / At < Bt}

In ase

f

p rio r insolven y , p

li yholders

re eive :

ΘL(τ) =

    

L0ergτ

si

λ ≥ 1 λL0ergτ

si

λ < 1 = min(λ, 1)L0ergτ

SLIDE 8 Contra t Ma rk et V alue Denoting b y Q the risk-neutral p robabilit y measure, the p ri e

f
ur

life insuran e

ntra t

writes at t < τ :

VL(t) = Et

Q

e− T

t rsds

Lg

T + δ(αAT − Lg T)+ − (Lg T − AT)+

1τ≥T + e− τ

t rsds min(λ, 1)Lg

τ 1τ<T

This
ntra t

an b e split up into four simpler sub

ntra ts

:

VL =

GF +
BO −
PO +

LR

GF

: the nal gua rantee

BO

: the "b

nus
ption"

whi h is the pa rti ipating lause

PO

: the default put

n

whi h p

li yholders

a re sho rt.

LR

: the rebate paid in ase

f

ea rly default.

SLIDE 9 Assets Dynami s and Interest Rate Mo delling Exp

nential

V

latilit

y fo r the Zero-Coup

ns

: σP(t, T) = ν

a

1 − e−a(T −t)

The dynami s under Q

f

the sho rt interest rate r and the Zero- oup

n P(t, T)

a re :

drt = a(θ − rt)dt + νdZQ

1 (t)

and

dP(t, T) P(t, T) = rtdt − σP(t, T)dZQ

1 (t)

The assets follo w : dAt

At = rtdt+σdZQ(t)

where ZQ and ZQ

1

a re

rrelated Q -Bro

wnian motions. (dZQ.dZQ

1 = ρdt

).

SLIDE 10 De o rrelation Let us no w

nsider

a Bro wnian motion ZQ

2

indep endent from

ZQ

1

. The Bro wnian motion ZQ an b e exp ressed as

dZQ(t) = ρdZQ

1 (t) +

1 − ρ2dZQ

2 (t)

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 :

dAt At = rtdt + σ

ρdZQ

1 (t) +

1 − ρ2dZQ

2 (t)

SLIDE 11 F

rw

a rd-Neutral Exp ressions Let QT b e the T

fo

rw a rd-neutral measure. F rom Girsanov theo rem, ZQT

1

and ZQT

2

a re indep endent QT

Bro

wnian motions.

dZQT

1 = dZQ

1 + σP (t, T)dt , dZQT 2

= dZQ

2

Under QT the p ri es P(t, T) and At follo w the sto hasti dif- ferential equations :

dP(t, T) P(t, T) = (rt + σ2

P(t, T))dt − σP (t, T)dZQT 1

and

dAt At = (rt − σρσP (t, T))dt + σ

ρdZQT

1 +

1 − ρ2dZQT

2

SLIDE 12 Contra t V aluation at t = 0 After hanging the p robabilit y measure, w e have in the F

rw

a rd-Neutral universe :

VL(0) = P(0, T) ( GF + BO − PO + LR )

where

                      

GF = Lg

T (1 − E1)

BO = αδ(E7 − E2) − δLg

T(E8 − E3)

PO = Lg

T (E9 − E4) − E10 + E5

LR = min(λ, 1)L0 E6

SLIDE 13 with the follo wing quantities that remain to b e

mputed

:

E1 = QT [τ < T] E6 = EQT [ergτ1τ<T] E2 = EQT

   AT1

AT >

Lg T α

, τ<T



  

E7 = EQT

 AT1

AT >

Lg T α

 

E3 = QT

AT > Lg

T

α , τ < T

E8 = QT
AT > Lg

T

α

E4 = QT[AT < Lg

T , τ < T]

E9 = QT[AT < Lg

T]

E5 = EQT

AT1AT<Lg

T 1τ<T

E10 = EQT
AT1AT<Lg

T

SLIDE 14 Metho dology : Longsta and S hw a rtz App ro ximation Problem : W e need to kno w the la w

f τ

, rst passage time

f

the assets b ey

nd

the default-triggering ba rrier.

➠

Longsta and S hw a rtz (1995) use F

rtet's

result to ap- p ro ximate the densit y

f τ

in a p roblem simila r to

urs.

➠

Collin-Dufresne and Goldstein (2001) give a

rre tion

to the p revious metho d to tak e p rop erly into a ount the sto hasti feature

f

the interest rates.

SLIDE 15 Rationale Let us rememb er the p rop er exp ression fo r τ

τ = inf{t ∈ [0, T] / At < λL0ergt}

Idea : App ro ximate the densit y

f τ

at time t under QT as a pie ewise

nstant

fun tion.

The

interval [0, T] is sub divided into nT subp erio ds.

The

interest rate is dis retized b et w een rmin and rmax into

nr

intervals.

tj = jδt

and ri = rmin + iδr a re the dis retized values

f

time and interest rate.

SLIDE 16 The p robabilit y

f

the event τ ∈ [tj, tj+1] with r ∈ [ri, ri+1] exp resses as : q(i, j) . Collin-Dufresne and Goldstein give a re ursive fo rmula fo r these p robabilities :

q( i, 1 ) =

nr

u=0

q( u, 1 ) Ψ( ri, t1 | ru, t1 )

One w

uld

rst

mpute q( i, 1 )

fo r ea h i , and then q(i, j) re ursively fo r j ≥ 2 using :

q(i, j) = Φ( ri, tj ) −

j−1

v=1

nr

u=0

q( u, v ) Ψ( ri, tj | ru, tv )

where Φ and Ψ a re

mpletely

kno wn.

SLIDE 17 Exp ressions

f Φ

and Ψ

L (lt|Fs, rt) = Gauss

µ( rt, ls, rs ), Σ2( rt, ls, rs )
let N

b e the umulative fon tion

f

the Gauss(0, 1) la w, then :

Φ( rt, t ) = fr( rt, t| l0, r0, 0) N

  h − µ( rt, l0, r0 )

Σ2( rt, l0, r0 )

  

Ψ( rt, t | rs, s ) = fr( rt, t | ls = h, rs, s) N

  h − µ( rt, ls = h, rs )

Σ2( rt, ls = h, rs )

  

where :

fr( rt, t | ls = h, rs, s) = 1 √ 2πv e−(rt−m)2

2v

, m = E[rt|rs] , v = V

a r[rt|rs]

SLIDE 18 Empiri al Densit y and F

rtet's

App ro ximate Densit y

1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 3.5 x 10

−3

Empirical Density nr=10 and nT=50 nr=50 and nT=200

SLIDE 19 Computation

f

the Ei dep ending

n τ

No w, ea h Ei an b e

mputed

easily even if it dep ends

n τ

. W e detail ho w to valuate E2 fo r instan e ; its exa t exp ression is :

E2 = EQT

   AT1

AT >

Lg T α

, τ<T



  

Then, using

nditional

la ws w e

btain

:

E2 = ergT

T

ds

+∞

−∞

drs g(rs, s)EQT

elT1{lT>ln(

L0 α )} | ls = h, rs, s, τ = s

As L (lt|Fs, rt) = Gauss
µ, Σ2

and as w e kno w the transition densit y

f r

: fr w e an

mpute E2

dis retizing the integrals.

SLIDE 20 Let X b e a random va riable with la w N (m, σ2) , w e denote

Φ1(m; σ; a) = E[eX1eX>a] = exp

m + σ2

2

N

m + σ2 − ln(a)

σ

E2

then admits the simpler exp ression :

E2 = ergT

T

ds

+∞

−∞

drs g(rs, s)

+∞

−∞

drT fr(rT | rs, s, ls) Φ1

µs,T;

Σs,T; L0 α

The

extended F

rtet's

app ro ximation

f E2

writes :

E2 = ergT

nT

j=1

nr

i=0

nr

k=0

δrfr(rk | ri, tj, ltj) Φ1

µtj,T;

Σtj,T; L0 α

q(i, j)

SLIDE 21 Numeri al Analysis W e set

ur

pa rameter range a o rding as :

A0 a ν θ r0 ρ σ T λ α

100 0.4 0.008 0.06 0.03

0.02

0.1 10 0.8 0.7

L0 = αA0 = 70

Contra t Maturit y : 10 y ea rs

SLIDE 22 Numeri al Results Extended F

rtet

GF BO PO LR Contra t Time

nT = 200, nr = 50

28.11 89.05 0.09 1.27 60.9967 2 min

nT = 500, nr = 50

28.11 89.03 0.09 1.29 69.9996 10 min Monte-Ca rlo GF BO PO LR Contra t Time

step = 1/12

28.10 89.28 0.14 1.30 70.1108 15 min

step = 1/52

28.11 89.14 0.13 1.31 70.0451 1h20

step = 1/365

28.14 89.07 0.13 1.30 70.0201 1 jour

SLIDE 23 Contra t F air V alue Denition : The initial investment

f

p

li yholders L0 = αA0

must b e equal to the

ntra t

ma rk et value at t = 0 . The P a rameters : rg : minimum gua ranteed interest rate δ : pa rti ipating b enets annot b e xed a rbitra rily : they

b

ey regulato ry

nstraints,

and need to b e set in su h a w a y as to mak e the

ntra t

fair b et w een the insurer and the p

li yholder.

SLIDE 24 Contra t V alue w.r.t. δ the pa rti ipating

e ient

0.83 0.85 0.87 0.89 0.91 0.93 0.95 68.5 69 69.5 70 70.5 71

δ

Contract Value

89.8%

SLIDE 25 Contra t V alue w.r.t. rg Minimum Gua ranteed Rate

0.01 0.02 0.03 0.04 67 68 69 70 71 72 73 rg Contract Value

0.026

SLIDE 26 Ho w to x the P a rameters

f

a F air Contra t ? W e use a ro

t

sea r h algo rithm

n

the follo wing equation to nd the fair value

f

a pa rameter, eteris pa ribus :

L0 = { Contra t

V alue at t = 0}

SLIDE 27 The

ntra t

value at t = 0 dep ends

n

:

the

initial stru ture

f

the

mpany

: A0 , α , the interest rate pa rameters and the

rrelation ρ

b et w een the assets and the interest rate,

the
ntra t

maturit y T , the ba rrier level λ ,

some

pa rameters σ , δ and rg that w e will study mo re in details in the follo wing.

SLIDE 28

δ

with resp e t to σ

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.7 0.75 0.8 0.85 0.9 0.95 1 σ δ rg=1% rg=2,5% rg=4%

SLIDE 29

rg

with resp e t to σ

0.05 0.1 0.15 0.2 0.25 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 σ Guaranteed Rate rg δ=50% δ=70% δ=90%

SLIDE 30 Con lusion

➠

A study

f

relevan e in the

ntext
f

the new IAS and IFRS Standa rds

➠

A new metho d to p ri e standa rd life insuran e gua rantees (gua ranteed apital and minimum rate with pa rti ipating b

nuses

when interest rates a re sto hasti and the p

ssible

default

f

the

mpany

is tak en into a ount)

➠

The next step is to p ri e supplementa ry

ptions

t ypi al to life insuran e

ntra ts

(surrender and

nversion
ptions,

apital paid up

n

death and not at a xed time making ne essa ry the use

f

mo rtalit y tables and so

n)