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Work out what a cash flow will be worth in future

CFA® Level I · Quantitative Methods · The Time Value of Money in Finance · QM.1.a

The idea

The future value of a single cash flow is FV = PV x (1 + r)^n, where r is the interest rate per period and n the number of periods. Compounding is what makes the relationship exponential rather than linear: interest earns interest. When compounding happens more often than annually, divide the stated annual rate by the number of periods per year and multiply the number of years by that same figure. Two habits prevent most errors here: keep the rate and the period on the same time basis, and draw a timeline before touching a formula.

Formulas

Future value of a single cash flow

FV = PV × (1 + r)ⁿ

PV
the amount invested today
r
the rate per compounding period, not per year
n
the number of compounding periods, not years

r and n must be on the same footing. A 6% annual rate compounded quarterly is r = 1.5% and n = 4 per year, not r = 6% and n = 1.

A practice question

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

$1,000 is invested at 6% compounded annually. What is it worth after two years?

  1. $1,123.60 — correct
  2. $1,120.00Why not: Simple interest: 1,000 × (1 + 0.06 × 2). It pays 6% on the original 1,000 twice and never on the first year's interest.
  3. $1,060.00Why not: One year of growth, 1,000 × 1.06. The second year was never applied.

1,000 × 1.06² = 1,123.60. The extra $3.60 over simple interest is the second year's interest on the first year's interest.

A harder one

An investor deposits $5,000 at 8% compounded quarterly for three years. Which rate and period count apply?

  1. 8% for 3 periodsWhy not: The stated annual rate used unchanged. Quarterly compounding means the rate must be divided by 4 and the periods multiplied by 4.
  2. 2% for 12 periods — correct
  3. 2% for 3 periodsWhy not: The rate was converted to quarterly but the periods were left in years, so nine of the twelve quarters are lost.

Divide the annual rate by periods per year (8% / 4 = 2%) and multiply years by the same figure (3 × 4 = 12). Rate and period must share a time basis.

The hardest tier

A stated annual rate of 6% compounded monthly has an effective annual rate closest to:

  1. 6.00%Why not: The stated rate, quoted straight back. It only equals the effective rate when compounding is annual.
  2. 6.09%Why not: Semi-annual compounding: 1.03² − 1. Monthly compounding is twelve periods of 0.5%, not two of 3%.
  3. 6.17% — correct

(1 + 0.06/12)¹² − 1 = 6.17%. 6.09% is the semi-annual figure, which is why the compounding frequency has to be read carefully.

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