Hailiang Yang, Ph.D., ASA, HonFIA - Member | SOA · 2010-01-29 · Hailiang Yang, Ph.D., ASA,...

Hailiang Yang, Ph.D., ASA, HonFIA

Session 3

Equity linked insurance products: Introduction and valuation

Equity linked insurance products: Introduction and valuation

Hailiang YangDepartment of Statistics & Actuarial Science

The University of Hong KongHong Kong

Joint work withHans U. Gerber and Elias S. W. Shiu

Support from SoA CAE grant is acknowledged.

Motivation

To value guarantees and options

in Variable Annuities

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S&P500

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S&P500 MFC ING AXA HIG

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ING

ING announced in December 2011 that it would take a charge of about € 1 billion to close its U.S. variable annuity business It stopped selling variable annuities in 2009.

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Financial Post, January 29, 2010

“When markets collapsed in September 2008, Manulife's net exposure of guarantees from segregated fund products was [Canadian] $ 72 billion ...

Financial Post, January 29, 2010

“When markets collapsed in September 2008, Manulife's net exposure of guarantees from segregated fund products was [Canadian] $ 72 billion ... The company's capital levels … sank because of the massive stock portfolio associated with the variable annuities”

In 2004, Manulife merged with John Hancock, creating the largest life insurer in Canada, the second largest in the USA, and the fifth largest in the world.


In 2004, Manulife decided remove the hedging on the equity positions that it held in its variable annuity business.


In 2004, Manulife decided remove the hedging on the equity positions that it held in its variable annuity business. For several years, the decision boosted Manulife’s profits. Then …

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S&P500 AXA

“Much of the downturn in AXA's profit outlook comes from variable-annuities hedging.

“Much of the downturn in AXA's profit outlook comes from variable-annuities hedging. The costs of hedging the annuities – all from its U.S. business – ballooned from 64 million euros in the first half of the year to 123 million euros in the third quarter -- and as much as 450 million euros in the fourth quarter.”

The Wall Street Journal “MarketWatch”, 25 November 2008

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“SNL Financial reported an industry net loss of $900.3 million, down from a net income of $8.9 billion in second-quarter 2009. Primarily variable annuity writers were affected … The largest insurers to see an increase in reserves were The Hartford, with an increase in $2.7 billion …”

TheStreet, Sep 29, 2010

“CIGNA Corporation, on September 3, 2002,

“CIGNA Corporation, on September 3, 2002, announced that it would record an after-tax charge of $720 million to strengthen reserves relating to reinsurance contracts on variable annuity death-benefit guarantees.

“CIGNA Corporation, on September 3, 2002, announced that it would record an after-tax charge of $720 million to strengthen reserves relating to reinsurance contracts on variable annuity death-benefit guarantees. Originally, CIGNA Corporation had only set aside $300 million against these policies and did not use capital markets to hedge the embedded options.

“CIGNA Corporation, on September 3, 2002, announced that it would record an after-tax charge of $720 million to strengthen reserves relating to reinsurance contracts on variable annuity death-benefit guarantees. Originally, CIGNA Corporation had only set aside $300 million against these policies and did not use capital markets to hedge the embedded options. In early September 2002, it realized, and then announced, that it had mis-estimated the risk and was forced to add the extra $720 million to its actuarial reserves”

The Wall Street Journal, Sept. 5, 2002

What are Variable Annuities (VA)?


Variable Annuities

= Investment Funds (Mutual Funds)

+


Variable Annuities


+

Rider(s) : Guaranteed Minimum BenefitsI. GMDB

II. GMAB

III. GMIB

IV. GMWB


Variable Annuities


+

Rider(s) : Guaranteed Minimum BenefitsI. GMDB Death

II. GMAB Accumulation

III. GMIB Income

IV. GMWB Withdrawal

For a life age x, let Tx denote the time-until-death random variable.

For a life age x, let Tx denote the time-until-death random variable.Let S(t) denote the price of a stock or stock fund at time t, t .

For a life age x, let Tx denote the time-until-death random variable.Let S(t) denote the price of a stock or stock fund at time t, tExample of an equity-linked death benefit:

For a life age x, let Tx denote the time-until-death random variable. Let S(t) denote the price of a stock or stock fund at

Example of an equity-linked death benefit: Guaranteed minimum death benefit

Max[S(Tx), K],where K is a guaranteed amount, e.g., K = 0.9×S(0).



Max[S(Tx), K],where K is a guaranteed amount, e.g., K = 0.9×S(0).Because Max[S(Tx), K] = S(Tx) + Max[0, K - S(Tx)]



Max[S(Tx), K],where K is a guaranteed amount, e.g., K = 0.9×S(0).Because Max[S(Tx), K] = S(Tx) + Max[0, K - S(Tx)],the guarantee is provided by the payoff of a life-contingent put option,

Max[0, K - S(Tx)] = [K - S(Tx)]+

S0 t

Let M (t) Max {S( )}, running maximum

Example 2:High water mark (or lookback)

Death benefit = MS(Tx)

S0 t




Example 3: Death benefit

= Max[S(Tx), 0.9 MS(Tx)]

S0 t





= Max[S(Tx), 0.9 MS(Tx)]= S(Tx) + Max[0, 0.9 MS(Tx) S(Tx)].

S0 t





= Max[S(Tx), 0.9 MS(Tx)]= S(Tx) + Max[0, 0.9 MS(Tx) S(Tx)].

So we are to value the life-contingent fractional lookback put option payoff

[0.9 MS(Tx) S(Tx)]+

S0 t


Example 4: Lapses & Surrenders

Example 4: Lapses & SurrendersAs the stock price rises, put options become lessvaluable

Example 4: Lapses & SurrendersAs the stock price rises, put options become lessvaluable; hence policies may lapse.

Example 4: Lapses & SurrendersAs the stock price rises, put options become lessvaluable; hence policies may lapse. Instead of

[K S(Tx)]+,


[K S(Tx)]+,consider the up-and-out put option payoff

I[MS(Tx) < H]×[K S(Tx)]+

where I[.] is the indicator function and is a “barrier.”




where I[.] is the indicator function and is a “barrier.”xT

x S x: Find E[e b(S(T ),M (T ))]Problem





for DB functions b(s, m) = (K – s)+

b(s, m) = mb(s, m) = (0.9×m – s)+

b(s, m) = I(m < H)×(K – s)+

xTx S x: Find E[e b(S(T ),M (T ))]Problem





for DB functions b(s, m) = (K – s)+

b(s, m) = mb(s, m) = (0.9×m – s)+

b(s, m) = I(m < H)×(K – s)+

The expectation is taken with respect to an appropriate probability distribution.

xTx S x: Find E[e b(S(T ),M (T ))]Problem

By conditioning on the time-until-death r.v. Tx ,x

x

Tx S x

Tx S x xE

E[e b(S(T ),M (T ))]

E[ e b(S(T ),M (T )) T ][ ]|

By conditioning on the time-until-death r.v. Tx ,x

x

Tx S x

Tx S x xE

E[e b(S(T ),M (T ))]

E[ e b(S(T ),M (T )) T ][ ]|

x

tS x T

0

e b(S(t),M (t))E[ ] T =t f (t) dt|

By conditioning on the time-until-death r.v. Tx ,

if Tx is independent of {S(t)}.

x

x

Tx S x

Tx S x xE

E[e b(S(T ),M (T ))]

E[ e b(S(T ),M (T )) T ][ ]|

x

tS x T

0

e b(S(t),M (t))E[ ] T =t f (t) dt|

x

tS T

0

e b(S(t),M (t)) fE[ (t) dt]

By conditioning on the time-until-death r.v. Tx ,

if Tx is independent of {S(t)}.

x

x

Tx S x

Tx S x xE

E[e b(S(T ),M (T ))]

E[ e b(S(T ),M (T )) T ][ ]|

x

tS x T

0

e b(S(t),M (t))E[ ] T =t f (t) dt|

x

tS T

0

e b(S(t),M (t)) fE[ (t) dt]

xT t x x tIn actuarial notation: f (t) = p

So we want to calculate

x

tS T

0

E[e b(S(t),M (t))]f (t)dt.


Ifx jjT

j

f (t) f (t)c ,

x

tS T

0



If

thenx jjT

j

f (t) f (t)c ,

x

jj

tS T

0

tS

0j

E[e b(S(t), M (t))]f (t)dt

= E[e b(S(t), M (t))]fc (t)dt

x

tS T

0



If

thenx jjT

j

f (t) f (t)c ,

x

jj

tS T

0

tS

0j

E[e b(S(t), M (t))]f (t)dt

= E[e b(S(t), M (t))]fc (t)dt

x

tS T

0


jSj j j

j

= E[e b(S( ), M ( ))].c

The time-until-death probability density function can be approximated by linear combinations of exponential density functions

j

jx j j jj

tT

j

f (t) f (t) = ecc

The time-until-death probability density function can be approximated by linear combinations of exponential density functions

Thus, our valuation problem becomes finding

E[e b(S( ), MS( ))],

where is an exponential random variable

independent of {S(t)}.

j

jx j j jj

tT

j

f (t) f (t) = ecc

E[e b(S( ), MS( ))]

= E[E[e- b(S( ), MS( ))| ]]

= E[ , ] t

E[e b(S( ), MS( ))]

= E[E[e- b(S( ), MS( ))| ]]

= E[ , ] t

= E[ , ]( + ) ( )t = E[b(S( ), MS( ))],

where exp( + ).


where is an exponential random variable with

mean

E[ ( , (


where is an exponential random variable with

mean

S(t) can be written as

S(t) = S(0) ( )

E[ ( , (

Let M(t) = max

Let M(t) = maxE[ ( , (E[ ( 0) ( ) , (0) ( ))]E[ ( , (( , ) , ( , ) ,

Let M(t) = max[ ( , ([ ( 0) ( ) , (0) ( ))][ ( , (( , ) , ( , ) ,where , ( , ) is the joint density function of and .

Black-Scholes Model S(t) = S(0)e t + Z(t) , t ,

where {Z(t)} is a standard Brownian motion.

Black-Scholes Model S(t) = S(0)e t + Z(t) , t ,

where {Z(t)} is a standard Brownian motion.

is an independent exponential r.v. with mean 1/ .

( is an independent exponential r.v. with mean

1/( + ))

A key result

< 0 and > 0 solutions of the quadratic equation2 + + = 0

A key result

< 0 and > 0 solutions of the quadratic equation2 + + = 0 , , = m x

= , ,=

= , ,== , , ( , ) ( , ), > 0,, > 0,with =

Our examples areb(s, m) = (K – s)+

b(s, m) = mb(s, m) = (0.9×m – s)+

b(s, m) = I(m < H)×(K – s)+

, = ; High water mark lookback payoffE 0 = (0) 0

, = ; High water mark lookback payoffE 0 = (0) 0ThereforeE 0 0 = 0

, = +; put option Out of the money put option

, = +; put option Out of the money put optionE 0 + ( ( ) 0 ) = ( ) [ ]

Proof of the key result

[ ( )] = ( ) = ( )( )where < 0 and > 0 are the solutions of2 + + = 0

Because and M( ) are independentM( ) ( ) has the same distribution as – m( )


[ ( )] = = ( )] [ ] = ( )( )


[ ( )] = = ( )] [ ] = ( )( )where < 0 and > 0 are the solutions of2 + + = 0

Because 0 and m 0, we have ( )] = and E[ ] = .

Because 0 and m 0, we have ( )] = and E[ ] = .

So

( ) ( ) = , 0and

m( ) ( ) = 0.

Then the joint density is

( ), ( ) ( , ( )

for max , 0where < 0 and > 0 are solutions of 2 + + = 0

Models with regime-switching

• This part is based on a joint paper with Phillip Yam and Chi Chung Siu: “Valuing Equity-Linked Death Benefits in a Regime-Switching Framework”, ASTIN Bulletin, To appear.

Index price model

• Assume that • S(t)= S(0) ( )• where X(t) = ( ) ds + ( ) ds

is a finite state Markov chain.

• Properties of the regime-switching model.

Bad news

• The joint distribution of ( )cannot be obtained

Good news

• The Laplace transformation of andM( ) can be obtained easily,

• Numerical method can be used to solve the problem

Hailiang Yang, Ph.D., ASA, HonFIA - Member | SOA · 2010-01-29 · Hailiang Yang, Ph.D., ASA,...

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