Risk Management and Financial Institutions. Hull John C.

show similar statistics for a woman. The probability of a 31-year-old woman dying within one year is 0.000699 (0.0699 %), the probability of a woman surviving to age 31 is 0.98486 (98.486 %), and the remaining life expectancy for a 31-year-old woman is 50.86 years.

TABLE 3.1 Mortality Table

      Source: U.S. Department of Social Security, www.ssa.gov/OACT/STATS/table4c6.html.

      The full table shows that the probability of death during the following year is a decreasing function of age for the first 10 years of life and then starts to increase. Mortality statistics for women are a little more favorable than for men. If a man is lucky enough to reach age 90, the probability of death in the next year is about 16.8 %. The full table shows this probability is about 35.4 % at age 100 and 57.6 % at age 110. For women, the corresponding probabilities are 13.1 %, 29.9 %, and 53.6 %, respectively.

      BUSINESS SNAPSHOT 3.1

      Equitable Life

      Equitable Life was a British life insurance company founded in 1762 that at its peak had 1.5 million policyholders. Starting in the 1950s, Equitable Life sold annuity products where it guaranteed that the interest rate used to calculate the size of the annuity payments would be above a certain level. (This is known as a Guaranteed Annuity Option, GAO.) The guaranteed interest rate was gradually increased in response to competitive pressures and increasing interest rates. Toward the end of 1993, interest rates started to fall. Also, life expectancies were rising so that the insurance companies had to make increasingly high provisions for future payouts on contracts. Equitable Life did not take action. Instead, it grew by selling new products. In 2000, it was forced to close its doors to new business. A report issued by Ann Abraham in July 2008 was highly critical of regulators and urged compensation for policyholders.

      An interesting aside to this is that regulators did at one point urge insurance companies that offered GAOs to hedge their exposures to an interest rate decline. As a result, many insurance companies scrambled to enter into contracts with banks that paid off if long-term interest rates declined. The banks in turn hedged their risk by buying instruments such as bonds that increased in price when rates fell. This was done on such a massive scale that the extra demand for bonds caused long-term interest rates in the UK to decline sharply (increasing losses for insurance companies on the unhedged part of their exposures). This shows that when large numbers of different companies have similar exposures, problems are created if they all decide to hedge at the same time. There are not likely to be enough investors willing to take on their risks without market prices changing.

      Some numbers in the table can be calculated from other numbers. The third column of the table shows that the probability of a man surviving to 90 is 0.16969. The probability of the man surviving to 91 is 0.14112. It follows that the probability of a man dying between his 90th and 91st birthday is 0.16969 − 0.14112 = 0.02857. Conditional on a man reaching the age of 90, the probability that he will die in the course of the following year is therefore

      This is consistent with the number given in the second column of the table.

      The probability of a man aged 90 dying in the second year (between ages 91 and 92) is the probability that he does not die in the first year multiplied by the probability that he does die in the second year. From the numbers in the second column of the table, this is

      Similarly, the probability that he dies in the third year (between ages 92 and 93) is

      Assuming that death occurs on average halfway though a year, the life expectancy of a man aged 90 is

      EXAMPLE 3.1

      Assume that interest rates for all maturities are 4 % per annum (with semiannual compounding) and premiums are paid once a year at the beginning of the year. What is an insurance company's break-even premium for $100,000 of term life insurance for a man of average health aged 90? If the term insurance lasts one year, the expected payout is 0.168352 × 100, 000 or $16,835. Assume that the payout occurs halfway through the year. (This is likely to be approximately true on average.) The premium is $16,835 discounted for six months. This is 16, 835/1.02 or $16,505.

      Suppose next that the term insurance lasts two years. In this case, the present value of expected payout in the first year is $16,505 as before. The probability that the policyholder dies during the second year is (1 − 0.168352) × 0.185486 = 0.154259 so that there is also an expected payout of 0.154259 × 100, 000 or $15,426 during the second year. Assuming this happens at time 18 months, the present value of the payout is 15, 426/(1.02³) or $14,536. The total present value of payouts is 16, 505 + 14, 536 or $31,041.

      Consider next the premium payments. The first premium is required at time zero, so we are certain that this will be paid. The probability of the second premium payment being made at the beginning of the second year is the probability that the man does not die during the first year. This is 1 − 0.168352 = 0.831648. When the premium is X dollars per year, the present value of the premium payments is

      The break-even annual premium is given by the value of X that equates the present value of the expected premium payments to the present value of the expected payout. This is the value of X that solves

      or X = 17, 251. The break-even premium payment is therefore $17,251.

      3.4 LONGEVITY AND MORTALITY RISK

      Longevity risk is the risk that advances in medical sciences and lifestyle changes will lead to people living longer. Increases in longevity adversely affect the profitability of most types of annuity contracts (because the annuity has to be paid for longer), but increases the profitability of most life insurance contracts (because the final payout is either delayed or, in the case of term insurance, less likely to happen). Life expectancy has been steadily increasing in most parts of the world. Average life expectancy of a child born in the United States in 2009 is estimated to be about 20 years higher than for a child born in 1929. Life expectancy varies from country to country.

      Mortality risk is the risk that wars, epidemics such as AIDS, or pandemics such as Spanish flu will lead to people living not as long as expected. This adversely affects the payouts on most types of life insurance contracts (because the insured amount has to be paid earlier than expected), but should increase the profitability of annuity contracts (because the annuity is not paid out for as long). In calculating the impact of mortality risk, it is important to consider the age groups within the population that are likely to be most affected by a particular event.

      To some extent, the longevity and mortality risks in the annuity business of a life insurance company offset those in its regular life insurance contracts. Actuaries must carefully assess the insurance company's net exposure under different scenarios. If the exposure is unacceptable, they may decide to enter into reinsurance contracts for some of the risks. Reinsurance is discussed later in this chapter.

      Longevity Derivatives

      A longevity derivative provides payoffs that are potentially attractive to insurance companies when they are concerned about their longevity exposure on annuity contracts and to pension funds. A typical contract is a longevity bond, also known as a survivor bond, which first traded in the late 1990s. A population group is defined and the coupon on the bond at any given time is defined as being proportional to the number of individuals in the population that are still alive.

      Who will sell such bonds to insurance companies and pension funds?