CFA Program Level I

Present value, annuities and annuity due: CFA Level I TVM

The present value of an amount due in n periods is the amount times 1 / (1 + i)^n, where i is the rate per period. The value of a series of cash flows is the sum of each one discounted for its own date. An annuity due pays at the start of each period, so it is worth the ordinary annuity times (1 + i).

One amount

Treasury's auction rules (31 CFR Part 356, Appendix B) write the present value of 1 due at the end of n periods as v^n = 1 / (1 + i/2)^n, where i is the annual yield with semiannual compounding and n counts half-years. The general form is PV = FV / (1 + rate per period)^(number of periods).

Example: a zero-coupon bond pays 100 in 5 years at a yield of 4.00% with semiannual compounding. Rate per period = 2%, periods = 10. PV = 100 / 1.02^10 = 82.03. Using 4% for 5 periods instead would give 82.19, which is wrong.

Annuities and annuity due

The same Treasury appendix gives the present value of 1 per period for n periods as a(n) = (1 - v^n) / i, where i is the rate per period. This is v + v^2 + ... + v^n, an ordinary annuity with payments at the end of each period.

Example: 1,000 at the end of each of the next 3 years at 5%. v^3 = 1 / 1.05^3 = 0.863838. a(3) = (1 - 0.863838) / 0.05 = 2.72325. PV = 2,723.25.

If the payments come at the start of each year, the first one is paid today and each is one period earlier, so PV = 2,723.25 x 1.05 = 2,859.41. Equivalently, 1,000 + 1,000 / 1.05 + 1,000 / 1.05^2 = 1,000 + 952.38 + 907.03 = 2,859.41.

Uneven cash flows

When the amounts differ, discount each one for its own date and add. A share is worth the sum of its expected future cash flows discounted at a rate that reflects their risk, which is the starting point of every dividend discount model.

Example: dividends of 3 at the end of years 1 and 2 and a final payment of 50 at the end of year 3, at 10%. PV = 3 / 1.10 + 3 / 1.10^2 + 50 / 1.10^3 = 2.7273 + 2.4793 + 37.5657 = 42.77.

Where the exam catches people

Believing a semiannual-pay yield can be applied per year: divide the annual rate by 2 and double the number of periods.
Believing an annuity due is discounted the same way as an ordinary annuity: each payment is one period earlier, so multiply the ordinary annuity value by (1 + i).
Believing the first payment of an annuity due is discounted for one year: it is paid today, so its discount factor is 1.
Believing uneven cash flows can be valued with one annuity factor: each amount is discounted separately for its own date.

Written from CFA Institute's documentation

Try 3 questions on this

Pick an answer to see why each option is right or wrong.

Under a contract, Larkfield Ltd. pays USD20 million at the beginning of each of the next three years, the first payment today. Year-end discount factors at a 4% discount rate are 0.9615 for year 1, 0.9246 for year 2 and 0.8890 for year 3. The present value of the three payments is closest to:
Right. In the game that’s Renown for your Quantitative Methods building.
Not that one: B is right. In the game your Quantitative Methods building would lose a little Renown, and this question would come back later.
Your answerA.USD55.50 millionUSD55.50 million discounts each payment as if it fell at the end of its year. The first payment is made today and is not discounted; the second and third fall one and two years out, so the factors are 1.000, 0.9615 and 0.9246.The answerB.USD57.72 millionA payment at the beginning of year t falls at the same date as the end of year t − 1, so its discount factor is the year-end factor for year t − 1, and 1.000 for year 1.Your answerC.USD60.00 millionUSD60.00 million is the sum of the payments with no discounting. Only the first payment is made today; the two later ones are worth less than USD20 million each in present value terms.
Show the full explanation and doc quotes
PV = 20 × (1.000 + 0.9615 + 0.9246) = 20 × 2.8861 = USD57.72 million. Equivalently, the year-end value of USD55.50 million times 1.04 is USD57.72 million.
Written from www.whitehouse.gov/wp-content/uploads/2023/11/CircularA-94.pdf
A zero-coupon government note will pay 100 at maturity in exactly 10 years. Its yield is 6.40%, a nominal annual rate based on semiannual compounding. The note's present value per 100 of face value is closest to:
Right. In the game that’s Renown for your Quantitative Methods building.
Not that one: A is right. In the game your Quantitative Methods building would lose a little Renown, and this question would come back later.
The answerA.53.26The present value of 1 due at the end of n periods is v^n = 1 / (1 + i/2)^n, where i is the nominal annual yield based on semiannual payments and n counts semiannual periods.Your answerB.53.7853.78 discounts at 6.40% once a year for 10 years: 100 / 1.064^10. The yield is stated on a semiannual basis, so the note is discounted at 3.20% per half-year for 20 half-years, which gives the lower 53.26.Your answerC.72.9872.98 uses the 3.20% half-year rate but only 10 periods: 100 / 1.032^10. Ten years hold 20 semiannual periods, so the factor is 1 / 1.032^20 and the value is 53.26.
Show the full explanation and doc quotes
Here i/2 = 0.032 and n = 20: v^n = 1 / 1.032^20 = 1 / 1.8776 = 0.5326, so the present value is 100 × 0.5326 = 53.26.
Written from www.treasurydirect.gov/files/laws-and-regulations/auction-regulations-uoc/31-cfr-part-356.pdf
Wrenfield Holdings is winding down. An analyst expects it to pay dividends of USD4.00 per share at the end of each of the next two years and a final liquidating dividend of USD30.00 per share at the end of year 3, after which the shares will have no value. With a required return of 8%, the value of one share today is closest to:
Right. In the game that’s Renown for your Quantitative Methods building.
Not that one: A is right. In the game your Quantitative Methods building would lose a little Renown, and this question would come back later.
The answerA.USD30.95A share is worth the present value of all the cash flows expected to reach its holders: V = 4.00 / 1.08 + 4.00 / 1.08² + 30.00 / 1.08³ = 3.7037 + 3.4294 + 23.8150 = USD30.95.Your answerB.USD32.85USD32.85 discounts the USD30.00 for two years. It is paid at the end of year 3, so it is discounted for three: 30.00 / 1.08³ = 23.82, not 30.00 / 1.08² = 25.72. Each cash flow is discounted for the number of years until it is received.Your answerC.USD35.19USD35.19 discounts all three payments by a single year. Payments further away are worth less today; with each discounted for its own number of years, the value is 3.70 + 3.43 + 23.82 = USD30.95.
Written from www.newyorkfed.org/medialibrary/media/research/epr/2015/2015_epr_equity-risk-premium.pdf