Compounding frequency: monthly, quarterly, semi-annual

◈ 6 cards

The same 6 % is three different numbers — divide the rate by p and multiply the years by p, both at once, and rank the results.

The rate is annual; the period is not

Interest rates are quoted per year unless stated otherwise, but a bank may credit interest monthly, quarterly or semi-annually. When it does, the compounding period is shorter than a year, and every TVM function needs two adjustments made together:

$rate per period = \frac{r}{p} periods in total = n \times p$

where $p$ is the number of compounding periods in a year (12, 4 or 2). The closed form becomes

$F V = P V \times (1 + \frac{r}{p})^{n p}$

Make both adjustments or neither. The half-adjusted answer — an annual rate with monthly periods — is not slightly wrong; it is absurd, and Lesson 1.3 showed how to spot it.

Worked example — Tomas's $3,000, three ways

Tomas puts $3,000 into a savings product paying 6 % for 5 years. The product comes in three versions, differing only in how often interest is credited.

Monthly ( $p = 12$ ): rate 0.06/12 = 0.005, nper 5 × 12 = 60. =FV(0.005,60,0,-3000) = 4,046.55.
Quarterly ( $p = 4$ ): rate 0.06/4 = 0.015, nper 5 × 4 = 20. =FV(0.015,20,0,-3000) = 4,040.57.
Annual ( $p = 1$ ): rate 0.06, nper 5. =FV(0.06,5,0,-3000) = 4,014.68.

More frequent compounding gives more money — by a little. Monthly beats annual by 31.87 on 3,000 over five years, about 1 % of the deposit. The ordering is what matters on a multiple-choice stem; the size of the gap is what tells you your arithmetic is sane.

Add semi-annual ( $p = 2$ ): rate 0.03, nper 10. =FV(0.03,10,0,-3000) = 4,031.75 — between quarterly and annual, exactly where it should sit.

The half-adjusted answer

Type =FV(0.06,60,0,-3000) — the annual rate with the monthly period count — and Excel returns 98,963.07. Three thousand dollars does not become ninety-nine thousand in five years at 6 %; that would be 6 % a month. The other half-adjustment, =FV(0.005,5,0,-3000), returns 3,075.75 — five months of interest dressed up as five years. Either number should stop you before you write it down.

Four answers, one right

For $3,000 at 6 % for 5 years compounded monthly: $4,047 (right); $4,015 (annual compounding — forgot to adjust either); $98,963 (annual rate, monthly nper); $3,900 (simple interest, =3000*(1+0.06*5)). Know which mistake produced each wrong one, and you can tell them apart on sight.