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1 Valuation of life annuities A life annuity is a financial contract in the form of an insurance policy in which a seller (issuer), a life insurance company, makes a series of future payments to a buyer (annuitant) in exchange for the immediate payment of a lump sum (single premium annuity) or a series of regular payments (level premium annuity), prior to the onset of the annuity. Recall that a premium in an insurance contract is also called consideration in insurance contracts. The payment stream from the insurer to the annuitant has an unknown duration based principally upon the date of death of the annuitant. At this point the contract will terminate and the remainder of the fund accumulated is forfeited unless there are other annuitants or beneficiaries in the contract. Thus a life annuity is a form of longevity insurance, where the uncertainty of an individual's lifespan is transferred from the individual to the insurer, which reduces its own uncertainty by pooling many clients. Annuities can be purchased to provide an income during retirement, or originate from a structured settlement of a personal injury lawsuit. A structured settlement is a financial or insurance arrangement, defined by Internal Revenue Code as periodic payments; a claimant accepts to resolve a personal injury tort claim or to compromise a statutory periodic payment obligation. In Canada, structured settlements were first utilized after settlement for children affected by Thalidomide. Structured settlement cases became more popular in the United States during the 1970s as an alternative to lump sum settlements. The increased popularity was also due to several rulings by the IRS and an increase in personal injury awards. The IRS rulings changed policies such that if the Internal Revenue Code requirements were met then claimants could have federal income tax waived. Annuities can also be purchased as individual policies, as group policies, and are also used in settling obligations other than structured settlements, for example, pension plans. Annuities have two phases: - Accumulation phase, when the policyholder pays premiums and accumulates them with interest and other gains (or losses) earned in the account, - Payout phase, when the insurance company makes payments to the insured/insureds. The methodologies used in valuation of reserves during those two phases are different. During the payout phase the calculation amounts to finding the actuarial present value of future benefits at the interest rate and mortality table set by law (or by actuary s judgment). Note that during the payout period death is the only decrement. The general approach for calculation of reserves is to take the worst case scenario approach: find the greatest possible excess of present value of guaranteed benefits over the present value of the required premiums, guaranteed to be paid through the same period of time as the corresponding guaranteed benefit. This method is a standard for fixed annuities, while for variable annuities there are additional issues related to the variability of the value of the account and the resulting uncertainty of any guaranteed benefit payment, if such payment is in any way different than the value of the account (and some form of guarantee is always required for the contract to be an insurance policy). We will cover the methodology used for fixed annuities reserving, and discuss For educational use by Illinois State University students only, do not redistribute

2 some issues that come up for variables annuities. There are additional readings on variable annuities also linked at the class site, and principles based reserving is especially applicable to variable annuities. Commissioners Annuity Reserve Valuation Method Fixed deferred annuity in the accumulation period is subject to the Commissioners Annuity Reserve Valuation Method. This method calls for accumulation of all required premium (simplest in the case of a single premium deferred annuity) at guaranteed rates to every policy year-end, and then discounting every year-end guaranteed accumulated value at the valuation rate back to the valuation date. Reserve is established as the largest of all so calculated present values. There are two key principles in CARVM: The first of the two main principles CARVM is the concept of greatest present value. And it is the greatest present value of guaranteed benefits not potential benefits. Important: CARVM uses only year-end guaranteed values, but in New York, all possible future accumulated values must be considered. Exercise 1. Consider a single premium deferred annuity with a single premium of $10,000 and no front-end load, or any other charges at time 0. The policy guarantees the interest rate of 10% for the first five years, and 4% thereafter. The policy has a surrender charge of 7% during the first year, and then the surrender charge declines by 1% every year, until it becomes 1% in the seventh year, and there is no surrender charge after that. The valuation interest rate is 8%. The policy has a death benefit equal to its cash surrender value. Calculate the policy reserves at the beginning of each policy year for the first ten years. The guaranteed accumulated values and cash surrender values at the end of the first ten policy years are: End of Year Guaranteed Accumulated Values Cash Surrender Values 1 $11, $10, $12, $11, $13, $12, $14, $14, $16, $15, $16, $16, $17, $17, $18, $18, $18, $18, $19, $19, Note also that there is a guaranteed cash surrender value of $9,300 as of time 0, which must be also taken into account in the consideration of the reserve, only because it exists as a cash value and the reserve can never be lower than the current cash surrender value, For educational use by Illinois State University students only, do not redistribute

3 or cash value (CV). Now we calculate present values of future guaranteed cash surrender values as of each year-end, with present values established as of times: 0 years, 1 year, 2 years, 3 years, etc., up to the beginning of the tenth year, i.e., time 9 years. Those values are: Time PV Y1 CV PV Y2 CV PV Y3 CV PV Y4 CV PV Y5 CV PV Y6 CV PV Y7 CV PV Y8 CV PV Y9 CV PV Y10 CV 0 $9, $9, $10, $10, $10, $10, $10, $9, $9, $9, $10, $10, $10, $11, $11, $11, $10, $10, $10, $9, $11, $11, $12, $12, $12, $11, $11, $10, $10, $12, $13, $13, $13, $12, $12, $11, $11, $14, $14, $14, $13, $13, $12, $12, $15, $15, $14, $14, $13, $13, $16, $15, $15, $14, $14, $17, $16, $16, $15, $18, $17, $16, $18, $18, The amounts in bold italics represent reserves at times 0, 1, 2, etc., at times stated in the first column, one reserve per row. Thus reserves as of the year beginning, at times stated below, are (note also cash surrender values are given for comparison): Policy year Time Reserve Cash Surrender Value 1 0 $10, $9, $11, $10, $12, $11, $13, $12, $14, $14, $15, $15, $16, $16, $17, $17, $18, $18, $18, $18, Note also that if the guaranteed rates of return are lower than the valuation rate, future values become irrelevant, and reserve is produced by the earliest possible value discounted. On the other hand, early years have guaranteed credited rates higher than the valuation rate, and this favors later years, but one must also take into account the effect of the surrender charge. Let us also note that New York law requires all future times to be considered, which means that instead of year-end values, we must consider every possible time, but only next year beginning, just after surrender charge is lowered, affects the outcome. This type of calculation is commonly called continuous CARVM. Note that the reserve at time 0 is $10,632.03, even though the company receives only $10,000 in single premium, and most likely had marketing and issuance expenses. This means that company s own capital will have to be invested in this product. Additionally, the company will have some form of capital requirements. Assume, for example, that the company functions in a dream world in which it has no marketing and issuance expenses, but it has a requirement of holding capital equal to 3% of its assets. This means that if we write x for the required capital, the company must have x of assets at times 0 For educational use by Illinois State University students only, do not redistribute

4 (remember that Assets = Surplus + Reserve, and Surplus is just an insurance term for x capital), so that = 0.03, and x x = So the company has to put up $ in capital to issue this policy, and make up the deficiency caused by the excess of reserve required over premium received of $10, $10, = $ Thus, effectively, the company has to invest $ $ = $ of economic capital (i.e., real money, not just an accounting entry), or additional capital, into the issuance of this policy. In a calculation like this, if the premium is paid annually, only the required premiums must be considered, and only guaranteed cash surrender values must be taken into account. Exercise 2. Consider a deferred annuity with required annual premium of $1,000, which charges a front-end load of 5% of premium, plus a one-time charge of $25 for every premium received. The policy pays a guaranteed interest rate of 10% in years 1 through 5, and 4% thereafter. The surrender charge is 5% the first five years, 4% the sixth year, 3% the seventh year, 2% the eighth year, 1% the ninth year, and 0% thereafter. The valuation interest rate is 8.75%. There is a death benefit, equal to the cash surrender value. Calculate the policy reserve at the beginning of each of the first ten policy years. The guaranteed accumulated values and cash surrender values at the end of the first ten policy years are listed below, but cash values are calculated in an instant after that, at the beginning of the following policy year: End of Year Guaranteed Accumulated Values Cash Surrender Values 0 $ $ $1, $ $2, $2, $3, $3, $4, $4, $6, $5, $7, $7, $8, $8, $9, $9, $11, $11, $12, $12, Note that the cash surrender value of $ at time 0 is the floor for any reserve calculated. Now we calculate present values of future guaranteed cash surrender values as For educational use by Illinois State University students only, do not redistribute

5 of each year-end minus present values of future premiums paid through the date of each guaranteed cash surrender values, with present values established as of times: 0 years, 1 year, 2 years, 3 years, etc., up to the beginning of the tenth year, i.e., time 9 years. Those values are: Time PV Y1 CV PV Y2 CV PV Y3 CV PV Y4 CV PV Y5 CV PV Y6 CV PV Y7 CV PV Y8 CV PV Y9 CV PV Y10 CV 0 $(36.15) $(59.17) $(70.01) $(69.52) $(58.50) $(238.73) $(424.44) $(615.05) $(809.98) $(1,008.68) 1 $ $ $ $ $ $ $ $ $ $(91.00) 2 $2, $2, $2, $2, $1, $1, $1, $1, $ $3, $3, $3, $2, $2, $2, $2, $1, $4, $4, $4, $3, $3, $3, $3, $5, $5, $5, $5, $4, $4, $7, $6, $6, $6, $5, $8, $8, $7, $7, $9, $9, $9, $11, $10, The amounts in bold italics represent reserves at times 0, 1, 2, etc., at times stated in the first column, one reserve per row, with the exception of the first row, where the reserve is the cash value. Thus reserves as of the year beginning, at times stated below, are (note also cash surrender values are given for comparison): Policy year Time Reserve Cash Surrender Value 1 0 $ $ $ $ $2, $2, $3, $3, $4, $4, $5, $5, $7, $7, $8, $8, $9, $9, $11, $11, Note that receiving a level premium provides some relief from surplus strain we discussed in the previous example: The situation that the company issuing a policy must invest a substantial amount of its own capital in its creation. Exercise 3. Consider the following single premium deferred annuity. The policy guarantees credited interest rate of 10% in policy years 1 through 3, 6% in policy years 4 and 5, and 4% in policy years 6 and beyond. The surrender charge is 7% in the first year, and then it goes down every year by 1% until it becomes 0% in years 8 and beyond. You are given that the annuity fund value is $10,000 at the end of the second policy year. What is the For educational use by Illinois State University students only, do not redistribute

6 CARVM reserve at policy duration 2 if the valuation rate is 7%? How would those answers change if the contract were issued in the state of New York? Below we have the calculations of guaranteed fund values, cash surrender values and their present values at time 2: End of Policy Year Guaranteed fund CSV PV at time 2 2 $10, $9, $9, $11, $10, $9, $11, $11, $9, $12, $11, $9, $12, $12, $9, $13, $13, $9, $13, $13, $9, $14, $14, $9, $15, $15, $8, The largest of the entries in the last column is the standard CARVM reserve at time 2 and it equals $9,786.44, corresponding to the cash surrender value at the end of the policy year 5, when the guaranteed interest rate of 6% ends. If the valuation is done in the state of New York, then we must consider every possible time in the future. But the only substantial change in the relationship of the values occurs between the last day of each policy year and the first day of the subsequent policy year, when the surrender charge drops. If we do the same calculations as above for the first days of policy years (note the end of a policy year n is effectively the same time as the beginning of the policy year n + 1), we obtain Beginning of Policy Year Guaranteed fund CSV PV at time 2 3 $10, $9, $9, $11, $10, $9, $11, $11, $9, $12, $12, $9, $12, $12, $9, $13, $13, $9, $13, $13, $9, $14, $14, $9, $15, $15, $8, The reserve is now $9,887.33, corresponding to the guaranteed cash surrender value at time 5, at the beginning of the policy year 6. That s an increase of $9, $9, = $ over the requirement in states other than New York. For educational use by Illinois State University students only, do not redistribute

7 Exercise 4. A single premium annuity in the accumulation phase has an account value at the end of policy year 1 of $10,000. The valuation interest rate for this annuity is 5.5% annual effective. The surrender charge is 10% in policy year 1, 9% in policy year 2, and declines by 1% each year until it is 0% in policy years 11 and subsequent. The guaranteed minimum interest rate is 10% in policy year 1, 9% in policy year 2, 8% in policy years 3, 4, and 5, then 5% in policy years 6 and 7, and it is 4% in all subsequent policy years. What s the CARVM reserve at the end of policy year 1? What is the continuous CARVM reserve at the end of policy year 1? We do a pair of calculations just like in the previous problem, shown in the two tables below: End of Policy Year Guaranteed fund CSV PV at time 1 1 $10, $9, $9, $10, $9, $9, $11, $10, $9, $12, $11, $10, $13, $12, $10, $14, $13, $10, $15, $14, $10, $15, $15, $10, $16, $16, $10, $17, $16, $10, $17, $17, $10, This shows that the reserve at time 1 is $10, The continuous CARVM calculation is in the table below Beginning of Policy Year Guaranteed fund CSV PV at time 1 2 $10, $9, $9, $10, $10, $9, $11, $10, $9, $12, $11, $10, $13, $13, $10, $14, $13, $10, $15, $14, $10, $15, $15, $10, $16, $16, $10, $17, $17, $10, $17, $17, $10, The continuous CARVM reserve is $10, For educational use by Illinois State University students only, do not redistribute