• Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers

## 1.3.4 Present Value of an Annuity

Valuation of an annuity - methodology

A shortcut is available to compute the present value of an annuity -- a stream of annual flows. You can use the attached present value annuity table we described before:

 PV = SUM [i=1 to n] 1/(1 + i)n present value of \$1 deposited at the end of n periods discounted at i percent

From the present value table, you will notice that receiving \$1 each year for 25 years assuming a 12% discount rate has a present value of \$7.84. You should be indifferent whether you are paid the 25-year stream of \$1 payments or a present lump-sum payment of \$7.84 (which you can invest at 12% to replicate the same 25-year stream).

Or, to compute an annuity's present value, you can use a formula (imbedded in the attached present value annuity table):

 PV = Pymtn * [(1+i)n - 1] / (1+i)n * i PV the present value (or initial principal) Pymtn the payment made at the end of each of n periods i the discount rate for each period (assumed equal throughout) n the number of periods

Loan amortization

The seller of an annuity contract exchanges a current payment from the buyer for the promise to provide a stream of payments in the future. Seen in this way, a loan to be repaid through a series of future payments is a kind of annuity. The lender makes a current payment to the borrower in exchange for the borrower's promise to make a stream of future payments.

 A curious note: It's interesting that mortgage lenders in this country, which lend by advertising the annual interest rate, compute monthly payments by using an average monthly discount rate - not a compounded rate. For example, an annual interest rate of 5.68% can be replicated by compounding at a monthly rate of .4614% - which is lower than the average monthly rate of .4733%. Mortgage lenders get the use of monthly payments, but at a higher rate than the advertised annual rate suggests! This adds up to a discrepancy of hundreds of millions of dollars between their advertised annual rates and the actual monthly rates they charge.

Example: Loan amortization

Suppose you want to borrow money to buy a house. You are considering a 15-year or a 30-year loan. The lender offers different interest rates, reflecting the differences in risks of shorter-term and longer-term lending.

• 15-year loan: annual rate of 6.25% (compounded monthly, with 180 equal monthly payments)
• 30-year loan: annual rate of 6.75% (compounded monthly, with 360 monthly payments).

If you borrow \$150,000, what would your monthly payments be for each loan? (More>>)

 Annuities in legal contexts Valuation of annuities is often an issue in legal contexts, such as in computing the taxes on a state lottery jackpot, or valuing a pension in an equitable distribution, or valuing an annuity contract for estate tax purposes. Valuation often turns on legal policy. Consider the case of John Hance, who owned an annuity contract that provided for payment during his lifetime and then to his widow for her life. John died on February 15, and his wife Mabel died on May 15. What is the value of John's annuity contract for estate tax purposes? the value on the date of John's death of the payments to be received by Mabel on the basis of her life expectancy, discounted to reflect a reasonable rate of return - \$44,633 the value of similar annuity contracts issued on the date of John's death to a female applicant of the same age as Mabel - \$121,905 the value of payments received by Mabel between John's death and hers - \$5,007 the value of the annuity contract one year after John's death, a valuation date the estate may choose under IRC S 2032, which because of Mabel's death made the annuities worthless - \$0 The Tax Court's answer: \$5,007. The Tax Court interpreted IRC § 2032, which permits the estate to make "adjustment for any difference in its value as of [one year after death] not due to mere lapse of time," to take into consideration Mabel's death. Although the annuity had become "totally worthless" at the one year mark because of her death, it did have value "due to mere lapse of time" during the period she had "merely remained alive" after John's death and before hers. See Estate of Hance v. Commissioner, 18 TC 499 (1952), acquiesced, 1953-1 CB 4. Example: Hitting the Jackpot In 1992, Bronx residents Edison and Salvadora Blanco won \$10 million in the New York State Lotto. Under the lottery rules, Mr. Blanco was entitled to one initial payment of \$476,100.00 and 20 subsequent annual payments of \$476,195.00. To fund the installment payout obligation, the Lottery purchased an annuity for \$4,724,421.26. (More>>) Another jackpot Gribauskas won a \$16 million state lottery - to be paid in 20 annual installments of \$800,000. The right to installment payments could not be transferred or accelerated. Two years later, Gribauskas died. What was the value of the prize to his estate? Actuarial tables in IRC Sec 7520 placed the value at \$3.5 million. The estate discounted the prize because of its restrictions to \$2.6 million. Should the prize be valued using actuarial tables or reduced to reflect the lack of transferability? Gribauskas, -- F.3d -- (2d Cir 2003). Shakelford v. US, 262 F.3d 1028 (9th Cir. 2001) (Need not use tables if another valuation method is more realistic and reasonable). Student paper Don Lee's paper discusses how North Carolina courts value a pension during an equitable distribution proceeding. Example: Equitable Distribution You represent Wilma in a divorce. She is married to Harold, who is retired and has a pension that entitles him to a monthly benefit of \$900.00 on the date of separation. Harold has a life expectancy of twenty years. The pension company is well-established, and you believe a reasonable rate of return would be 3.0% above 20-year Treasury bonds—or 9% compounded monthly. If the entire pension was earned during marriage, what is its current value as a marital asset? (More>>)
 1.3.3 Present Value of Cash Flow Streams 1.3.5 Present Value of a Perpetuity