Compounding More Than Once a Year

Once a year

Adding interest to a balance, so that the interest earns interest in turn, is called compounding. Compounded once a year at r% a year, a principal P is multiplied by $1+\frac{r}{100}$1 + r/100 once for every year, so after n years it is $A=P{\left(1+\frac{r}{100}\right)}^n$A = P(1 + r/100)ⁿ.

Many accounts and loans compound more often than that: every six months, every quarter, every month or every day. The time between two additions of interest is called a compounding period.

Every quarter

A rate of 8% a year compounded quarterly adds interest four times a year. Each quarter does not pay the full 8%: the year's rate is shared among the four quarters, so each quarter pays 8% ÷ 4 = 2% of the balance at that time.

Over 3 years there are 4 × 3 = 12 quarters, so the balance is multiplied by 1.02 twelve times. Compounded once a year instead, it would be multiplied by 1.08 three times.

Three years at 8% a year. Compounded yearly, 8% is paid 3 times; compounded quarterly, 2% is paid 12 times.

The formula

In general, a rate of r% a year compounded k times a year pays $\frac{r}{k}$r/k percent in each period, and there are k periods in each year, so k × n periods in n years. The amount after n years is $A=P{\left(1+\frac{r}{\left(100k\right)}\right)}^{kn}$A = P(1 + r/(100k))^(kn). With k = 1 this is the yearly formula again.

For $1000 at 8% for 3 years, compounded quarterly, k = 4 gives $A=1000\times 1.02^{12}=1268.24$A = 1000 × 1.02¹² = 1268.24 dollars, to the nearest cent. Compounded yearly, the same $1000 becomes $1000\times 1.08^3=1259.71$1000 × 1.08³ = 1259.71 dollars. Quarterly compounding pays $8.53 more.

Side by side: 6% a year and 0.5% a month

Take $1000 for one year at 6% a year. Compounded yearly, the year pays 6% once: 1000 × 1.06 = 1060 dollars. Compounded monthly, each month pays 6% ÷ 12 = 0.5%, twelve times: $1000\times 1.005^{12}=1061.68$1000 × 1.005¹² = 1061.68 dollars, to the nearest cent.

The monthly account pays $1.68 more. Its first month of interest is 0.5% of $1000, which is $5, and from the second month on that $5 earns interest too. Each month's interest is worked out on a balance that already holds the interest of the months before, while the yearly account adds nothing until the year ends.

$1000 for one year at 6% a year. Paid once, the rate gives $1060. Paid as 0.5% twelve times, it gives $1000\times 1.005^{12}=1061.68$1000 × 1.005¹² = 1061.68 dollars.

The effective annual rate

To compare offers that compound at different times, turn each into the rate that, paid once a year, would give the same growth in a year. That is the effective annual rate. The stated rate, such as 6% a year compounded monthly, is called the nominal rate.

Monthly at 6%, a year multiplies the balance by $1.005^{12}=1.061678$1.005¹² = 1.061678, to 6 decimal places. Take away the 1 that stands for the balance itself: 1.061678 − 1 = 0.061678, so the effective annual rate is 6.17%, to 2 decimal places. For 8% compounded quarterly, $1.02^4=1.082432$1.02⁴ = 1.082432, so the effective annual rate is 8.24%.

More often pays more, by less and less

The more often the interest is compounded, the more it pays, but each step up adds less. For $1000 at 8% a year for 3 years, compounding yearly gives $1259.71, quarterly $1268.24, monthly $1270.24 and daily, 365 times a year, $1271.22. Quarterly adds $8.53 to yearly; monthly adds only $2.00 more, and daily only $0.98 more.

$1000 at 8% a year for 3 years, compounded 1, 4, 12 and 365 times a year. Each step pays more, by $8.53, then $2.00, then $0.98.

A look ahead: compounding all the time

The gain from compounding more often does not grow without limit. On $10,000 at 6% for one year, compounding yearly pays $600. Compounding k times a year pays more than that by $9.00 when k = 2, by $12.08 when k = 3, by $13.64 when k = 4, by $15.20 when k = 6 and by $16.78 when k = 12.

As k grows without limit, the year's multiplier ${\left(1+\frac{0.06}{k}\right)}^k$(1 + 0.06/k)^k closes in on $e^{0.06}=1.061837$e^0.06 = 1.061837, to 6 decimal places, where $e\approx 2.718$e ≈ 2.718 is the base of the natural logarithm. So $10,000 at 6% can never grow past $10,618.37 in a year, however often the interest is added. Interest added at every instant is called continuous compounding. The Natural Logarithm finds e itself in the same way, from a rate of 100%: ${\left(1+\frac{1}{k}\right)}^k$(1 + 1/k)^k closes in on e.

The extra interest in a year on $10,000 at 6%, over the $600 that yearly compounding pays, for k = 1, 2, 3, 4, 6 and 12 compoundings a year: $0, $9.00, $12.08, $13.64, $15.20 and $16.78. The points climb by less and less toward the line at $18.37, which they never reach.

The usual mistakes

Not dividing the rate. 8% compounded quarterly pays 2% a quarter. Paying the full 8% twelve times gives $1000\times 1.08^{12}=2518.17$1000 × 1.08¹² = 2518.17 dollars, which is what 12 years at 8% a year would give, not 3.

Not multiplying the number of periods. Quarterly for 3 years is 12 periods, not 3. $1000\times 1.02^3=1061.21$1000 × 1.02³ = 1061.21 dollars covers only the first 9 months.

Reading the nominal rate as the growth. 0.5% a month is 6% a year nominal, but the balance grows by 6.17% in a year, because the monthly interest is compounded.