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Determine the price of an annuity as well as that of a perpetuity.

FRM® Part I · Valuation and Risk Models · Bond Yields and Return Calculations · FRM1.VR.11.e

The idea

An ordinary annuity pays a fixed amount C at the end of each of n periods. Annuity price = C × [1 − (1 + r)⁻ⁿ] ÷ r. Here r is the rate per period, matching the payment frequency. The price is the sum of n discounted payments. A perpetuity pays C forever, so the discount term vanishes. Perpetuity price = C ÷ r. An annuity due pays at the start of each period, so its value is the ordinary annuity value times (1 + r). For a deferred annuity, value it one period before the first payment and then discount that value back to today. A coupon bond is an annuity of coupons plus the discounted face value. Typical errors: using an annual rate with monthly payments, discounting a deferred annuity one period too many or too few, and applying the perpetuity formula to a stream that ends.

Formulas

Price of an ordinary annuity

Annuity = C × [1 − (1 + r)⁻ⁿ] ÷ r

C
the fixed payment at the end of each period
r
the rate per period, matching the payment frequency
n
the number of payments

Payments at the end of each period. For an annuity due multiply by (1 + r); for a deferred annuity value it one period before the first payment, then discount.

Price of a perpetuity

Perpetuity = C ÷ r

C
the fixed payment each period, for ever
r
the rate per period

First payment one period from now. Do not apply it to a stream that ends.

A practice question

From the question bank, with the reason each wrong option is wrong.

A perpetuity pays 5 each year, with the first payment one year from now. The discount rate is 4% per year. What is its price?

  1. 125.00 - correct
  2. 0.20Why not: Multiplies the payment by the rate instead of dividing.
  3. 130.00Why not: Uses a perpetuity-due formula, C x (1 + r) / r, although the first payment is a year away.
  4. 4.81Why not: Discounts a single payment one year instead of summing the whole stream.

A perpetuity pays C forever, so price = C / r = 5 / 0.04 = 125.00.

A harder one

An annuity pays 2,400 once a year for 15 years, with the first payment three years from today. The discount rate is 6% per year. What is its value today, closest to?

  1. 23,309.40Why not: Values it as an ordinary annuity starting next year, ignoring the deferral.
  2. 19,571.02Why not: Discounts the annuity value three periods; it is valued one period before the first payment, so two.
  3. 20,745.28 - correct
  4. 36,000.00Why not: Adds the payments without discounting.

Value at year 2 (one period before the first payment) = 2,400 x [1 - 1.06^-15] / 0.06 = 23,309.40. Discount two years: 23,309.40 / 1.06² = 20,745.28.

The hardest tier

Two instruments each pay 5 per year with the first payment one year away: a perpetuity and a 10-year ordinary annuity. When the discount rate falls from 4% to 3%, by how many percentage points does the perpetuity's percentage price gain exceed that of the annuity, closest to?

  1. 33.3 percentage pointsWhy not: Gives the perpetuity's gain on its own and does not subtract the annuity's.
  2. 5.2 percentage pointsWhy not: Gives the annuity's gain, not the difference between the two.
  3. 28.2 percentage points - correct
  4. 38.5 percentage pointsWhy not: Adds the two gains instead of taking the difference.

Perpetuity: 5/0.03 = 166.67 vs 5/0.04 = 125.00, a gain of 33.3%. Annuity: 42.65 vs 40.55, a gain of 5.2%. Difference = 28.2 points; the perpetuity, with all its cash flow far away, is more sensitive to the rate.

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