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Algebra 2 · A.SSE.B.3C · F.LE.A.2

Compound interest and compounding periods

A bank quotes one yearly rate but adds the interest several times a year. How much does adding it more often really change what you end with?

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Exponential growth and decayGrades 9 to 10
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The number e and the natural exponential functionAlgebra 2
Where you are 6 in Exponential functions
Functions13
Exponential functions6
Exponential growth and decay6
Compound interest and compounding periods
Graphs of exponential functionsThe number e and the natural exponential functionTransforming an exponential graphWriting an exponential function from a table or graph

Why this exists

With simple interest, interest is paid only on the money first put in. Most savings accounts and most loans do something else: each time interest is worked out, it is added to the balance, and from then on it earns interest of its own. That is compound interest, and over whole years it is the exponential growth you already know, with the factor 1 + r for a yearly rate r.

But banks rarely add interest once a year. A savings account may add it every month, a credit card every month or every day, a bond twice a year. The advertised rate is still a yearly one, such as 5.4\% a year compounded monthly. So two questions arise that the once-a-year picture cannot answer. What does “5.4\% a year, compounded monthly” actually do to a balance? And is 5.4\% compounded monthly worth more or less than 5.5\% compounded once a year? This lesson answers both, with one formula and one comparison number.

Definition

Compound interest is interest that is added to the balance at the end of each period, so that in later periods it earns interest too. Each such adding is a compounding, and the time between two of them is a compounding period.

A rate quoted as r a year compounded n times a year means: the year is split into n equal periods, and at the end of each one the balance grows by the fraction \dfrac{r}{n} of itself. The yearly figure r is called the nominal rate (from the Latin for “name”: it is the rate in name). The common choices are

CompoundedPeriods a year, nRate per period
annually (once a year)1r
semi-annually (twice a year)2\dfrac{r}{2}
quarterly4\dfrac{r}{4}
monthly12\dfrac{r}{12}
daily (taking a year as 365 days)365\dfrac{r}{365}

The effective annual rate r_{\text{eff}} is the percent by which the balance really grows in one whole year, after all n compoundings. It is the single number that lets two accounts with different n be compared.

Intuition

Take 2{,}700 dollars at 5.4\% a year. Compounded once a year, nothing happens for twelve months, and then 5.4\% of 2{,}700, which is 145.80 dollars, is added at once: the balance becomes 2{,}845.80 dollars.

Compounded quarterly, the rate per quarter is \dfrac{5.4\%}{4} = 1.35\%, and interest is added four times. Amounts are to the nearest cent.

End ofInterest addedBalance
quarter 12{,}700 \times 0.0135 = 36.452{,}736.45
quarter 22{,}736.45 \times 0.0135 \approx 36.942{,}773.39
quarter 32{,}773.39 \times 0.0135 \approx 37.442{,}810.83
quarter 42{,}810.83 \times 0.0135 \approx 37.952{,}848.78

Four times 36.45 would be exactly 145.80, the yearly interest. The quarterly account does better, by 2.98 dollars, only because each quarter’s interest is taken of a balance that already holds the earlier quarters’ interest. The extra is interest on interest earned inside the year.

0
3
6
9
12
\text{months}
\$2{,}700
\$2{,}750
\$2{,}800
\$2{,}850
\text{quarterly: } \$2{,}848.78
\text{once a year: } \$2{,}845.80
Figure 1The first year of 2{,}700 dollars at 5.4\%. Quarterly interest arrives in four smaller steps, and each step is taken on a slightly larger balance.

Two things follow from the picture. More periods mean more of this interest on interest within the year, so the balance after a year rises as n rises. And the gain is small, because the interest earned in the first part of a year is small compared with the balance, and interest on a small amount for part of a year is smaller still. Splitting the year finer and finer keeps helping, but by less and less.

Results

The compound interest formula. A principal P at a nominal rate r a year, compounded n times a year, has after t years the balance

A(t) = P\left(1 + \frac{r}{n}\right)^{nt}

Here 1 + \dfrac{r}{n} is the growth factor for one period and nt is the number of periods. With n = 1 it is the yearly model A(t) = P(1 + r)^t.

The effective annual rate. One year is n periods, so the yearly growth factor is \left(1 + \dfrac{r}{n}\right)^n and

r_{\text{eff}} = \left(1 + \frac{r}{n}\right)^{n} - 1

The balance can then be written A(t) = P\,(1 + r_{\text{eff}})^t: compounding n times a year at the nominal rate r is the same as compounding once a year at the rate r_{\text{eff}}.

More periods, more growth, by shrinking amounts. For a positive rate r, a larger n always gives a larger effective rate, so r_{\text{eff}} \geq r with equality only when n = 1. The increases shrink as n grows: for 5.4\% a year, going from once a year to quarterly adds about 0.110 percentage points, from quarterly to monthly about 0.025, and from monthly to daily about 0.012.

5.40\%
5.45\%
5.50\%
5.55\%
n = 1
5.400\%
n = 4
5.510\%
n = 12
5.536\%
n = 365
5.548\%
Figure 2More periods always help, by less each time. The vertical axis starts at 5.38\%, so the differences are drawn much larger than they are.
Why it is true

The formula. Adding \dfrac{r}{n} of the balance to the balance is multiplying it by 1 + \dfrac{r}{n}, by the distributive property: B + \dfrac{r}{n}B = B\left(1 + \dfrac{r}{n}\right). Each period does this to whatever the balance is at that moment, so, just as with two percent changes in a row, the factors multiply. In t years there are n periods a year for t years, nt periods in all, so the principal is multiplied by the same factor nt times: A(t) = P\left(1 + \dfrac{r}{n}\right)^{nt}.

The effective rate. After one year the factor is \left(1 + \dfrac{r}{n}\right)^n, and a growth factor 1 + q means a growth of q, so the yearly growth is that factor minus 1. By the power of a power rule,

\left(1 + \frac{r}{n}\right)^{nt} = \left(\left(1 + \frac{r}{n}\right)^{n}\right)^{t}

so A(t) = P(1 + r_{\text{eff}})^t.

Why more periods give more. Start with two periods. Expanding,

\left(1 + \frac{r}{2}\right)^2 = 1 + r + \frac{r^2}{4}

The 1 + r is exactly the once-a-year factor, and \dfrac{r^2}{4} is the extra: the first half’s interest, \dfrac{r}{2} of the principal, earning \dfrac{r}{2} in the second half. For a positive r it is positive, so two periods beat one.

For general n, the binomial theorem writes \left(1 + \dfrac{r}{n}\right)^n as a sum of terms, and the term with r^k is

\begin{aligned} \binom{n}{k}\frac{r^k}{n^k} &= \frac{n(n-1)\cdots(n-k+1)}{k!\,n^k}\,r^k \\ &= \frac{r^k}{k!}\cdot\frac{n}{n}\cdot\frac{n-1}{n}\cdots\frac{n-k+1}{n} \\ &= \frac{r^k}{k!}\left(1 - \tfrac{1}{n}\right)\cdots\left(1 - \tfrac{k-1}{n}\right) \end{aligned}

The first two terms are 1 and r whatever n is. Every bracket 1 - \dfrac{j}{n} grows when n grows, so with r > 0 every later term grows too, and a larger n also brings extra positive terms on the end. So the whole sum grows with n: more periods, more growth. The same formula shows why the gains shrink: each bracket is already close to 1 once n is large, so raising n further can only nudge it.

Worked examples

Example 1

The same deposit, four ways

2{,}700 dollars is deposited at 5.4\% a year for 6 years. Find the balance, to the nearest cent, when interest is compounded once a year, quarterly, monthly and daily (a year of 365 days). Find each effective annual rate too.

Solution

In every case P = 2700, r = 0.054 and t = 6; only n changes, and with it the rate per period \dfrac{r}{n} and the number of periods nt.

nPeriods ntBalancer_{\text{eff}}
163{,}701.755.400\%
4243{,}725.065.510\%
12723{,}730.445.536\%
3652{,}1903{,}733.065.548\%

The factors per period are 1.054, 1.0135, 1.0045 and 1 + \dfrac{0.054}{365}, and each balance is 2700\left(1 + \dfrac{r}{n}\right)^{nt}. For instance, monthly: \dfrac{0.054}{12} = 0.0045 and 12 \times 6 = 72 periods, so the balance is 2700(1.0045)^{72} \approx 2700 \times 1.381643 \approx 3730.44 dollars, and the effective rate is 1.0045^{12} - 1 \approx 0.05536. For the daily row, keep \dfrac{0.054}{365} as a fraction in the calculator rather than rounding it, since a rounding error in the factor is raised to the power 2{,}190.

The interest earned runs from 1{,}001.75 dollars (once a year) to 1{,}033.06 dollars (daily), a difference of 31.31 dollars. Moving from once a year to quarterly gains 23.31 dollars; moving from monthly to daily gains only 2.62 dollars. For comparison, simple interest would earn 2700 \times 0.054 \times 6 = 874.80 dollars.

Example 2

What a credit card balance becomes

Ifeoluwa has a credit card balance of 1{,}350 dollars at 21\% a year compounded monthly. If she makes no payments and no new purchases for 2 years, what does she owe, and how much of that is interest? What is the card’s effective annual rate?

Solution

A loan grows the same way as a deposit: the interest is added to what is owed. The rate per month is \dfrac{0.21}{12} = 0.0175, and 2 years is 24 months, so

\begin{aligned} A(2) &= 1350(1.0175)^{24} \\ &\approx 1350 \times 1.516443 \\ &\approx 2047.20 \end{aligned}

She would owe 2{,}047.20 dollars, of which 2{,}047.20 - 1{,}350 = 697.20 dollars is interest. Simple interest at 21\% for 2 years would have been 1350 \times 0.21 \times 2 = 567 dollars; the extra 130.20 dollars is interest charged on interest. The effective rate is 1.0175^{12} - 1 \approx 0.2314, about 23.14\% a year, not 21\%.

Example 3

Comparing two offers

Bank One offers 4.5\% a year compounded monthly. Bank Two offers 4.6\% a year compounded once a year. Which pays more?

Solution

The nominal rates cannot be compared directly, because they are paid out on different schedules. Turn each into an effective annual rate. For Bank One the monthly rate is \dfrac{0.045}{12} = 0.00375, so

r_{\text{eff}} = 1.00375^{12} - 1 \approx 0.04594

about 4.594\%. Bank Two compounds once a year, so its effective rate is its nominal rate, 4.6\%. Bank Two pays slightly more: monthly compounding lifts 4.5\% to 4.594\%, which is not enough to reach 4.6\%. On 10{,}000 dollars for a year the two give 10{,}459.40 and 10{,}460 dollars. Since both effective rates hold every year, Bank Two stays ahead for any whole number of years.

Example 4

How much to deposit now

A family wants 10{,}000 dollars in 5 years. An account pays 3.9\% a year compounded monthly. How much must they deposit now, to the nearest cent?

Solution

This time A(5) = 10000 is known and P is not. The rate per month is \dfrac{0.039}{12} = 0.00325 and there are 60 months, so

10000 = P(1.00325)^{60}

Dividing both sides by (1.00325)^{60} \approx 1.214927, which is allowed because it is not zero,

P = \frac{10000}{(1.00325)^{60}} \approx 8230.95

They must deposit about 8{,}230.95 dollars. Check: 8230.95 \times 1.214927 \approx 10000.00.

Example 5

Reading and rewriting an expression

(a) An account’s balance after t years is A(t) = 2000(1.0035)^{12t}. Read off the principal, the number of compoundings a year, the nominal rate and the effective rate. (b) A different account grows by an effective 8\% a year, B(t) = P(1.08)^t. Rewrite the factor as a monthly one, and give the nominal rate compounded monthly that matches it.

Solution

(a) Match the expression with P\left(1 + \dfrac{r}{n}\right)^{nt}. The principal is 2{,}000 dollars. The exponent 12t says n = 12: monthly. The factor per period is 1.0035, so \dfrac{r}{12} = 0.0035 and r = 0.042, a nominal 4.2\% a year. The effective rate is 1.0035^{12} - 1 \approx 0.04282, about 4.28\%.

(b) By the power of a power rule, 1.08^t = \left(1.08^{1/12}\right)^{12t}, and 1.08^{1/12} \approx 1.006434. So

B(t) \approx P(1.006434)^{12t}

which is growth of about 0.6434\% a month. Twelve times that monthly rate is 12 \times 0.006434 \approx 0.0772: a nominal 7.72\% a year compounded monthly gives the same balances as 8\% compounded once a year.

Example 6

When does the balance first pass a target?

5{,}000 dollars is deposited at 8\% a year compounded quarterly. After how many whole quarters does the balance first exceed 7{,}000 dollars?

Solution

The rate per quarter is \dfrac{0.08}{4} = 0.02, so after q quarters the balance is 5000(1.02)^q. Interest is only added at the end of a quarter, so the balance changes in whole quarters, and we want the first whole q with 5000(1.02)^q > 7000, that is (1.02)^q > 1.4. The powers of 1.02 increase, so try values near where the product passes 1.4:

\begin{aligned} 5000(1.02)^{16} &\approx 6863.93 \\ 5000(1.02)^{17} &\approx 7001.21 \end{aligned}

After 16 quarters the balance is still below 7{,}000 dollars, and after 17 it is above. So the balance first exceeds 7{,}000 dollars after 17 quarters, which is 4 years and 3 months. Solving such an equation exactly, without trying values, needs logarithms.

Boundaries

Dividing by n applies only to a nominal rate. \dfrac{r}{n} is the rate per period only when r is quoted as a yearly rate to be split into n equal parts. Some accounts quote the effective rate instead (often called the annual percentage yield). An effective rate already includes the compounding, so it goes straight into P(1 + r_{\text{eff}})^t and is never divided by n.

t is in years, and real accounts move in whole periods. The formula’s exponent nt counts periods only when t is in years. A bank adds interest only at the end of a period, so between two compoundings the balance stays at its last value; putting a t that is not a whole number of periods into the formula gives a smooth model, not the account’s real balance at that moment.

The rate is fixed and nothing is added or taken out. The formula assumes one rate for the whole time and no deposits, withdrawals or payments. Accounts with regular deposits, and loans with regular payments, need more than this one formula.

Rounding. A bank rounds to the cent at every compounding, while the formula carries full precision, so the two can differ by a cent or so over a long time. The formula’s per-period factor must also not be rounded before raising it to a large power.

More periods do not mean unlimited growth. The balance rises with n, but the rises shrink, and the next lesson shows they settle toward a limit. The argument above that more periods help used r > 0; for a shrinking quantity it needs another look.

Where this goes next

The values \left(1 + \dfrac{r}{n}\right)^n climb with n but by less each time, and it is natural to ask what happens when the year is split into infinitely many periods, so that interest is added at every instant. That question leads to a special number and to continuous compounding in the number e and the natural exponential function. Rewriting a factor for a different period, as in Example 5, is the same move as in growth and decay factors, and finding the time to reach a target exactly, without trying values, is a job for logarithms. Compound interest is also the engine inside everyday percent in money situations: savings, loans and credit cards.

Exercises

The formula

  1. 1{,}500 dollars is deposited at 4\% a year compounded quarterly. Find the balance after 3 years.
  2. 900 dollars is deposited at 6\% a year compounded monthly. Find the balance after 4 years.
  3. 4{,}200 dollars is invested at 3.6\% a year compounded semi-annually. Find its value after 10 years.
  4. 650 dollars is saved at 5\% a year compounded daily (a year of 365 days). Find the balance after 2 years.
  5. Write A(t) for 3{,}000 dollars at 2.4\% a year compounded monthly, and find A(8).
  6. 12{,}000 dollars is invested at 4.4\% a year compounded quarterly for 5 years. How much interest does it earn?

Comparing frequencies

  1. 5{,}000 dollars is deposited for 1 year at 6\% a year. Find the balance compounded once a year, semi-annually, quarterly and monthly, and the gain from each step to the next.
  2. Find the effective annual rate of 9\% a year compounded monthly.
  3. Find the effective annual rate of 5.2\% a year compounded quarterly.
  4. Which pays more: 4.25\% a year compounded monthly, or 4.3\% a year compounded once a year? Justify with effective rates.

Loans, deposits and targets

  1. How much must be deposited now at 4\% a year compounded quarterly to have 8{,}000 dollars in 6 years?
  2. How much must be invested now at 5.5\% a year compounded monthly to have 25{,}000 dollars in 15 years?
  3. A credit card charges 18\% a year compounded monthly. A balance of 2{,}000 dollars is left with no payments or purchases for 1 year. What is owed, and what is the effective annual rate?
  4. 3{,}000 dollars is deposited at 6\% a year compounded quarterly. After how many whole quarters does the balance first exceed 3{,}600 dollars? Show the two values you compare.

Reading and rewriting

  1. An account’s balance after t years is A(t) = 1200(1.0025)^{12t}. Give the principal, how often interest is compounded, the nominal rate and the effective rate, and find A(10).
  2. A balance grows by an effective 7\% a year, A(t) = P(1.07)^t. Find the equivalent monthly factor to four decimal places and the nominal rate compounded monthly that matches it.
  3. Compare \displaystyle 2000\left(1 + \frac{0.05}{4}\right)^{4t} and 2000(1.05)^t at t = 10. Which is larger, and by how much?

Reasoning

  1. Expand \displaystyle \left(1 + \frac{r}{2}\right)^2 and explain, in terms of money, what each of the three terms stands for.
  2. Someone says that with enough compounding periods, 10\% a year can turn 1 dollar into as much as you like in one year. Find \displaystyle \left(1 + \frac{0.1}{n}\right)^n for n = 1, 12, 365 and 1{,}000{,}000, and use the values to argue against the claim.
  3. Explain why two accounts should be compared by effective annual rates, not by nominal rates.
  4. Show that 6\% a year compounded semi-annually has exactly the same effect as 6.09\% a year compounded once a year.
Answers
  1. 1{,}690.24 dollars. 1500(1.01)^{12} \approx 1690.24, with \displaystyle \frac{0.04}{4} = 0.01 and 12 quarters.
  2. 1{,}143.44 dollars. 900(1.005)^{48} \approx 1143.44, with \displaystyle \frac{0.06}{12} = 0.005 and 48 months.
  3. 6{,}000.74 dollars. 4200(1.018)^{20} \approx 6000.74, with \displaystyle \frac{0.036}{2} = 0.018 and 20 half-years.
  4. 718.36 dollars. \displaystyle 650\left(1 + \frac{0.05}{365}\right)^{730} \approx 718.36.
  5. A(t) = 3000(1.002)^{12t}, since \displaystyle \frac{0.024}{12} = 0.002; A(8) = 3000(1.002)^{96} \approx 3634.31 dollars.
  6. 2{,}934.97 dollars. The balance is 12000(1.011)^{20} \approx 14934.97, and 14934.97 - 12000 = 2934.97.
  7. Once a year 5{,}300.00; semi-annually 5000(1.03)^2 = 5304.50; quarterly 5000(1.015)^4 \approx 5306.82; monthly 5000(1.005)^{12} \approx 5308.39. The gains are 4.50, then 2.32, then 1.57 dollars: each step helps, by less each time.
  8. About 9.38\%. (1.0075)^{12} - 1 \approx 0.0938.
  9. About 5.30\%. (1.013)^4 - 1 \approx 0.0530.
  10. 4.25\% compounded monthly. Its effective rate is \displaystyle \left(1 + \frac{0.0425}{12}\right)^{12} - 1 \approx 0.04334, about 4.33\%, which beats 4.3\%.
  11. 6{,}300.53 dollars. P = \dfrac{8000}{(1.01)^{24}} \approx 6300.53.
  12. 10{,}976.54 dollars. \displaystyle P = \dfrac{25000}{\left(1 + \frac{0.055}{12}\right)^{180}} \approx 10976.54.
  13. 2{,}391.24 dollars owed. 2000(1.015)^{12} \approx 2391.24; the effective rate is 1.015^{12} - 1 \approx 0.1956, about 19.56\%.
  14. After 13 quarters (3 years and 3 months). 3000(1.015)^{12} \approx 3586.85, still below 3{,}600, and 3000(1.015)^{13} \approx 3640.66, above it.
  15. Principal 1{,}200 dollars, compounded monthly (n = 12), nominal rate 12 \times 0.0025 = 0.03, so 3\%; effective rate 1.0025^{12} - 1 \approx 0.0304, about 3.04\%. A(10) = 1200(1.0025)^{120} \approx 1619.22 dollars.
  16. 1.07^{1/12} \approx 1.0057, about 0.565\% a month. The nominal rate is 12 \times 0.0056541 \approx 0.0678, about 6.78\% a year compounded monthly.
  17. The quarterly one, by 29.45 dollars. 2000(1.0125)^{40} \approx 3287.24 and 2000(1.05)^{10} \approx 3257.79.
  18. \displaystyle 1 + r + \frac{r^2}{4}. The 1 is the principal, r is a full year’s interest at the nominal rate, and \displaystyle \frac{r^2}{4} is the first half-year’s interest, \displaystyle \frac{r}{2} of the principal, earning interest at \displaystyle \frac{r}{2} in the second half-year: interest on interest.
  19. The values are 1.1, about 1.104713, about 1.105156 and about 1.105171. They rise, but by less and less: from 365 periods to a million the gain is under two thousandths of a cent. The values settle near 1.1052 rather than growing without end, so one dollar cannot become as much as you like.
  20. A nominal rate leaves out how often interest is added, so two accounts with the same nominal rate can grow at different speeds, and a lower nominal rate can win, as in Exercise 10. The effective rate is the actual growth over a year with all compounding included, so the account with the larger effective rate has the larger balance after every whole year.
  21. Semi-annually the yearly factor is \displaystyle \left(1 + \frac{0.06}{2}\right)^2 = 1.03^2 = 1.0609, exactly the yearly factor of 6.09\% compounded once a year. Equal yearly factors give equal balances after any whole number of years, since both are P(1.0609)^t.
kg1128 · updated 2026-10-19 · CC-BY-SA-4.0