Actuarial Approach to Pricing Exotic Options Elias S. W. Shiu

Actuarial Approach to Pricing Exotic Options

Elias S. W. ShiuDepartment of Statistics & Actuarial Science

The University of IowaIowa City

Iowa, U.S.A.

M.A.R.C. June 13, 2011

Correct title:Valuing GMDB in Variable Annuities

Without Tears

Correct title:Valuing GMDB in Variable Annuities

Without Tears

Introduction to“Omega or Chocolate?”

Manulife Financial Corporation

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 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. For several years, the decision boosted Manulife’s profits. Then …

Hartford Financial Services Group

Hartford Financial Services Group(ITT Hartford)

“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,

“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, Prudential, with an increase in $964 million, Jackson National Life, with an increase in $1.8 billion, ING, with an increase in $1.5 billion …”

TheStreet, Sep 29, 2010

“A.M. Best Co. has downgraded the financial strength rating [of ING] to A (Excellent) from A+ (Superior) … The rating actions primarily reflect the significant decline in ING’s 2008 global insurance and banking operating results … most pronounced within variable annuitiesand asset management”

BestWire Apr 24, 2009

“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 thirdquarter -- and as much as 450 million euros in the fourth quarter.”

The Wall Street Journal “MarketWatch”, 25 November 2008

“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

18

Income of U.S. Life InsurersSource: Data from Life Insurers’ Fact Book (US$ million)

Year Life Insurance Premiums

Annuity Consid-erations

Health Insurance Premiums

1950 6,249 939 1,0011960 11,998 1,341 4,0261970 21,679 3,721 11,3671980 40,829 22,429 29,3661986 66,213 83,712 44,1531990 76,692 129,064 58,2542000 130,616 306,693 105,6192005 142,261 277,117 118,2672008 147,182 328,135 165,034

Annuities

AnnuitiesDeferred Annuities -- AccumulationImmediate Annuities -- Payout


Fixed AnnuitiesVariable Annuities


Fixed AnnuitiesVariable AnnuitiesEquity-Indexed Annuities

73.586.0

99.0122.0

137.7114.0 116.0

128.5 134.0 137.5160.0

38.938.2

32.9

42.1

52.871.5

98.6 81.684.4 73

71

0.0

50.0

100.0

150.0

200.0

250.0

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Variable Annuity Sales Fixed and Equity Indexed Annuity Sales

The U.S. Annual Annuity Sales from 1996-2006

112.4124.2

131.9

164.1

190.5185.5

214.6210.1

218.4210.5

231.0

$BillionSource: A 2007 Report from LIMRA (Life Insurance & Market Research Association)

What are Variable Annuities (VA)?


Variable Annuities = Investment Funds (Mutual Funds) +


Variable Annuities = Investment Funds (Mutual Funds) + Rider(s) : Guaranteed Minimum Benefits

I. GMDBII. GMABIII. GMIBIV. GMWB


Variable Annuities = Investment Funds (Mutual Funds) + Rider(s) : Guaranteed Minimum Benefits

I. GMDB DeathII. GMAB AccumulationIII. GMIB IncomeIV. GMWB Withdrawal

The basic Variable Annuity product:

Allows policyholders to invest proceeds into a variety of investment funds.

Deposits grow on a tax-deferred basis.

The basic Variable Annuity product:

Allows policyholders to invest proceeds into a variety of investment funds.

Deposits grow on a tax-deferred basis. There a 10% federal-tax penalty on withdrawals before age 59½. Withdrawals after 59½ are taxed as income (not capital gains).

Monthly asset management, expense, and risk charges are deducted by the insurer, usually as a (fixed) % of the prevailing account balance.

Sold mostly to ages 50+ and frequently rolled over within the VA market after 5 to 7 years. Roll-overs, referred to as 1035 exchanges, represents a significant portion of “new” sales.

Historical NotesVariable annuities were first offered in 1952.

Historical NotesVariable annuities were first offered in 1952.In 1918, Andrew Carnegie and his Carnegie Foundation for the Advancement of Teaching created the Teachers Insurance and Annuity Association of America (TIAA), a fully-funded system of pensions for professors.

Historical NotesVariable annuities were first offered in 1952.In 1918, Andrew Carnegie and his Carnegie Foundation for the Advancement of Teaching created the Teachers Insurance and Annuity Association of America (TIAA), a fully-funded system of pensions for professors.Recognizing the need for its participants to invest in equities in order to diversify their retirement funds, TIAA created the College Retirement Equities Fund (CREF) in 1952.

Valuing Guaranteed Minimum Death Benefits (GMDB) Without Tears


Consider a Variable Annuity (VA) policy issued to a life aged x.Let S(t) be the investment fund value at time t.Let T(x) denote the time-until-death random variable.


Consider a Variable Annuity (VA) policy issued to a life aged x.Let S(t) be the investment fund value at time t.Let T(x) denote the time-until-death random variable.Consider the payoff

Max(S(T(x)), K),where K is the guaranteed amount.


Consider a Variable Annuity (VA) policy issued to a life aged x.Let S(t) be the investment fund value at time t.Let T(x) denote the time-until-death random variable.Consider the payoff

Max(S(T(x)), K),where K is the guaranteed amount.One way to value this GMDB is to calculate

E[e-dT(x) Max(S(T(x)), K)],where the expectation is taken with respect to an appropriate probability measure and d is a valuation force of interest.

By conditioning on the time of death,

E[e-dT(x) Max(S(T(x)), K)]

= E[E[e-dT(x) Max(S(T(x)), K) | T(x)]]




= E[e-dt Max(S(t), K) | T(x) = t] fT(x)(t) dt,0




= E[e-dt Max(S(t), K) | T(x) = t] fT(x)(t) dt,

= E[e-dt Max(S(t), K)] fT(x)(t) dt

if T(x) is independent of {S(t)}.

0

0

So we want to calculatet

T(x)0

E[e (S(t))]f (t)dt.

- d

So we want to calculate

IfjT(x) j

jTf (t) f (tc ) ,

tT(x)

0

E[e (S(t))]f (t)dt.

- d


If

thenjT(x) j

jTf (t) f (tc ) ,

j

tT(x)

0

tj

0T

jc

E[e (S(t))]f (t)dt

= E[e (S(t))]f (t)dt

- d

- d

tT(x)

0

E[e (S(t))]f (t)dt.

- d


If

thenjT(x) j

jTf (t) f (tc ) ,

j

tT(x)

0

tj

0T

jc

E[e (S(t))]f (t)dt

= E[e (S(t))]f (t)dt

- d

- d

tT(x)

0

E[e (S(t))]f (t)dt.

- d

jTj j

j= E[e (S(c ]T )) .- d

We know how to do the approximation

with {Tj} being exponential random variables.

jT(x) jj

Tf (t) f (tc ) ,


with {Tj} being exponential random variables. Thus,

the problem valuing GMDB is to determine

E[e-dT (S(T))],

where T is an exponential random variable

independent of the stock-price process {S(t)}.

jT(x) jj

Tf (t) f (tc ) ,


with {Tj} being exponential random variables. Thus,

the problem valuing GMDB is to determine

E[e-dT (S(T))],

where T is an exponential random variable

independent of the stock-price process {S(t)}. It turns

out that this is an elementary calculus exercise.

jT(x) jj

Tf (t) f (tc ) ,

In fact, the expectation

E[e-dT (S(T), Max{S(t); 0 ≤ t ≤ T})],


E[e-dT (S(T), Max{S(t); 0 ≤ t ≤ T})],

where {S(t)} is a geometric Brownian motion and T is

an independent exponential r. v., is


E[e-dT (S(T), Max{S(t); 0 ≤ t ≤ T})],


an independent exponential r. v., ism

x m x ( )m2

0

2 S(0)e S(0)e e dx( e) m., d

- - -

-


E[e-dT (S(T), Max{S(t); 0 ≤ t ≤ T})],


an independent exponential r. v., is

This will be the not funny part of this afternoon’s

“Omega or Chocolate?” talk.

mx m x ( )m

20

2 S(0)e S(0)e e dx( e) m., d

- - -

-

Let X = {X(t)} be a stochastic process and gt(X) be a

real-valued functional of the process up to time t. For

example,

gt(X) = (S(0)eX(t), Max{S(0)eX(s); 0 ≤ s ≤ t}).

Let X = {X(t)} be a stochastic process and gt(X) be a

real-valued functional of the process up to time t. For

example,

gt(X) = (S(0)eX(t), Max{S(0)eX(s); 0 ≤ s ≤ t}).

The process X is stopped at time T, an independent

positive random variable. We are interested in the

expectation

E[e-dT gT(X)].

Factorization Formula: If T is exponential with

mean 1/, then

E[e-dT gT(X)] =


mean 1/, then

E[e-dT gT(X)] = E[e-dT] × E[gT*(X)],

where T* is an exponential random variable with

mean 1/(+d) and independent of X.


mean 1/, then




Notes: (i) E[e-dT] = /( + d).


mean 1/, then




Notes: (i) E[e-dT] = /( + d).

(ii) The condition d > 0 can be weakened to d > -.

Time for Lunch?

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Actuarial Approach to Pricing Exotic Options Elias S. W. Shiu