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

1. Actuarial Approach to Pricing Exotic Options Elias S. W. Shiu Department of Statistics & Actuarial Science The University of Iowa Iowa City Iowa, U.S.A. M.A.R.C. June 13, 2011

2. Correct title: Valuing GMDB in Variable Annuities Without Tears

3. Correct title: Valuing GMDB in Variable Annuities Without Tears Introduction to “Omega or Chocolate?”

4. Manulife Financial Corporation

5. 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”

6. 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.

7. 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 …

8. Hartford Financial Services Group

9. Hartford Financial Services Group (ITT Hartford)

10. “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,

11. “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

12. “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 annuities and asset management” BestWire Apr 24, 2009

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

14. “CIGNA Corporation, on September 3, 2002,

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

16. “CIGNA Corporation, on September 3, 2002, announced that it would record an after-tax charge of $720 millionto 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.

17. “CIGNA Corporation, on September 3, 2002, announced that it would record an after-tax charge of $720 millionto 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

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

20. Annuities

21. Annuities Deferred Annuities -- Accumulation Immediate Annuities -- Payout

22. Annuities Deferred Annuities -- Accumulation Immediate Annuities -- Payout Fixed Annuities Variable Annuities

23. Annuities Deferred Annuities -- Accumulation Immediate Annuities -- Payout Fixed Annuities Variable Annuities Equity-Indexed Annuities

24. The U.S. Annual Annuity Sales from 1996-2006 Source: A 2007 Report from LIMRA (Life Insurance & Market Research Association) $Billion 231.0 218.4 214.6 210.1 210.5 190.5 185.5 164.1 131.9 124.2 112.4

25. What are Variable Annuities (VA)?

26. What are Variable Annuities (VA)? Variable Annuities = Investment Funds (Mutual Funds) +

27. What are Variable Annuities (VA)? Variable Annuities = Investment Funds (Mutual Funds) + Rider(s) : Guaranteed Minimum Benefits • GMDB • GMAB • GMIB • GMWB

28. What are Variable Annuities (VA)? Variable Annuities = Investment Funds (Mutual Funds) + Rider(s) : Guaranteed Minimum Benefits • GMDB Death • GMAB Accumulation • GMIB Income • GMWB Withdrawal

29. The basic Variable Annuity product: Allows policyholders to invest proceeds into a variety of investment funds. Deposits grow on a tax-deferred basis.

30. 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.

31. Historical Notes Variable annuities were first offered in 1952.

32. Historical Notes Variable 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.

33. Historical Notes Variable 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.

34. Valuing Guaranteed Minimum Death Benefits (GMDB) Without Tears

35. 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.

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

37. 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 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.

38. 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)]]

39. 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-dtMax(S(t), K) | T(x) = t]fT(x)(t) dt,

40. 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-dtMax(S(t), K) | T(x) = t]fT(x)(t) dt, = E[e-dtMax(S(t), K)] fT(x)(t) dt if T(x)is independent of {S(t)}.

41. So we want to calculate

42. So we want to calculate If

43. So we want to calculate If then

44. So we want to calculate If then

45. We know how to do the approximation with {Tj} being exponential random variables.

46. We know how to do the approximation with {Tj} being exponential random variables. Thus, the problem valuing GMDB is to determine E[e-dTp(S(T))], where T is an exponential random variable independent of the stock-price process {S(t)}.

47. We know how to do the approximation with {Tj} being exponential random variables. Thus, the problem valuing GMDB is to determine E[e-dTp(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.

48. In fact, the expectation E[e-dTp(S(T), Max{S(t); 0 ≤ t ≤ T})],

49. In fact, the expectation E[e-dTp(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

50. In fact, the expectation E[e-dTp(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