Finance-Theory · Unit 3 · Video 9 · Interactive Practice

Why Banks Compound Interest: APR vs the Effective Annual Rate

IKey Formulas

Formula	Name	What it does
$\dfrac{r}{n}$	Rate per period	Splits an APR $r$ across the $n$ crediting dates in a year: $r = 6\%$ with $n = 12$ is $0.5\%$ a month
$r_{\text{EAR}} \equiv \left(1 + \dfrac{r}{n}\right)^{n} - 1$	Effective annual rate	What a year actually delivers once each period's interest earns interest for the rest of the year
$(1 + r_{6})^{2} = 1.10 \;\Longrightarrow\; r_{6} = \sqrt{1.10} - 1 \approx 4.881\%$	The withdrawal-proof half-year rate	The rate that leaves a depositor with exactly $10\%$ however often the money is withdrawn and redeposited — the rate the compounding convention replaces
$r_{\text{EAR}} = \lim\limits_{n \to \infty}\left(1 + \dfrac{r}{n}\right)^{n} - 1 = e^{r} - 1$	Continuous compounding	The ceiling no frequency gets past: at $r = 10\%$ , $e^{0.10} - 1 \approx 10.517\%$

Key Insight: An APR fixes the rate per period, $r/n$ , not the year's return. Pro-rating an annual rate by calendar time would let a depositor withdraw and redeposit for more than the posted rate; compounding is the convention that closes that hole by writing the schedule of $n$ periods into the account. It pays the depositor, who lends to the bank, and costs the borrower, who pays interest on interest for exactly the same reason.

IIGaming a Pro-Rated Rate

Can a depositor beat a posted $10\%$ by withdrawing at a break point and redepositing the same minute?

Bank's rule

💡 Banks do not leave the choice open: they credit interest $n$ times a year at $r/n$ , which writes a fixed schedule of $n$ stays into the account so nobody has to visit the branch to collect them.

IIIMore Frequent Compounding, and Where It Stops

Each step up in frequency adds less than the step before, and the additions stop at $e^{r} - 1$ .

Quote Frequency 1

IVAn APR Is Incomplete Without Its Frequency

The highest of four quoted APRs pays least over the year, and the lowest pays nearly the most.

💡 On a deposit account the effective rate is disclosed as the annual percentage yield, which by law reflects the compounding frequency; on a loan only the APR must be disclosed, so the borrower's true annual rate is the one left unstated.

VQuiz Questions

Problem 1 · APR to EAR

Given: a savings account quotes an APR of $r = 12\%$ compounded quarterly — find its effective annual rate.

✅ Correct!

(1.03)^{4} = 1.1255088

, so the year returns

12.5509\%

: the

0.55

percentage points above the APR are the first three quarters' interest earning interest.

❌ That is the APR. It is the quoted rate with compounding left out — the first quarter's interest is credited in March and earns interest for the remaining nine months.

❌ That is $r/n$ .

3\%

is the rate for one quarter; four such quarters compound to more than

12\%

, not to

3\%

❌ Close, but that is $n = 12$ .

(1 + 0.12/12)^{12} - 1 = 12.6825\%

is monthly compounding; quarterly means four periods a year.

Show solution

Quarterly compounding means $n = 4$ , so each quarter pays the per-period rate

\frac{r}{n} = \frac{12\%}{4} = 3\%

A dollar left alone for the year is multiplied once per quarter:

r_{\text{EAR}} = \left(1 + \frac{0.12}{4}\right)^{4} - 1 = (1.03)^{4} - 1 = 1.1255088 - 1 = 12.5509\%

So $12.55\%$ to two decimals. Checking the size of the gap: $12\%$ simple interest on $1,000 is $120.00, while quarterly compounding gives $125.51 — the extra $5.51 is interest earned on the first three quarterly credits.

Problem 2 · What Pro-Rating Costs the Bank

Given: a bank posts $12\%$ a year and pro-rates it, paying $6\%$ for a six-month stay. A depositor puts in $5,000, withdraws the whole balance at six months and redeposits it the same minute — find how much more than the posted $5,600 she holds at year end.

✅ Correct! The second half-year pays

6\%

on $5,300, not on $5,000: the extra $18.00 is

6\%

of the $300 of first-half interest.

❌ The principal changed. After six months she redeposits $5,300, so the second

6\%

is charged against a larger balance than the first.

❌ That doubles the interest on interest. Only the $300 earned in the first half-year earns a second

6\%

, and

6\%

of $300 is $18.00.

❌ That is a whole half-year of interest on the principal. The $300 is what the first six months pay; the question asks only for the part that is interest earning interest.

Show solution

Each six-month stay is paid the pro-rated rate $12\%/2 = 6\%$ , so the balance is multiplied twice:

\text{\textdollar}5{,}000 \times 1.06 = \text{\textdollar}5{,}300 \qquad \text{\textdollar}5{,}300 \times 1.06 = \text{\textdollar}5{,}618.00

Against the posted $12\%$ , which would give $5,600.00:

\text{\textdollar}5{,}618.00 - \text{\textdollar}5{,}600.00 = \text{\textdollar}18.00 = 6\% \times \text{\textdollar}300

Her realised annual return is $(1.06)^{2} - 1 = 12.36\%$ , not $12\%$ . The bank closes the gap either by paying the withdrawal-proof half-year rate $\sqrt{1.12} - 1 \approx 5.830\%$ , or — the convention it actually uses — by quoting the $12\%$ as compounded semiannually and owning the $12.36\%$ .

Problem 3 · Comparing Two Quotes

Given: Bank P quotes a $6.00\%$ APR compounded monthly and Bank Q quotes a $5.95\%$ APR compounded daily — find which account pays more over a year, and the winner's effective annual rate.

Which account pays more over a year?

What is the winner's effective annual rate?

✅ Correct! Bank P returns

(1.005)^{12} - 1 = 6.1678\%

against Bank Q's

6.1301\%

: the extra

0.05

points of APR are worth more than the jump from monthly to daily crediting.

❌ Frequency is the smaller effect here. Going from monthly to daily at a

6\%

APR is worth only

0.0153

percentage points, while Bank P's APR is

0.05

points higher to begin with.

❌ They differ by about four hundredths of a point. That is

0.0377

percentage points — $3.77 a year on a $10,000 balance.

❌ That is Bank Q's rate.

6.1301\%

is the daily-compounded

5.95\%

; the question asks for the winner's, and Bank P's

(1.005)^{12} - 1

is higher.

❌ Check the exponent. Monthly compounding means twelve periods of

0.06/12 = 0.5\%

, not two periods of

3\%

and not the APR itself.

Show solution

Each quote has to be converted to the one number that is comparable, the effective annual rate.

Bank P — $r = 6.00\%$ , $n = 12$ , so $r/n = 0.5\%$ a month:

r_{\text{EAR}} = (1.005)^{12} - 1 = 1.0616778 - 1 = 6.1678\%

Bank Q — $r = 5.95\%$ , $n = 365$ , so $r/n \approx 0.0163\%$ a day:

r_{\text{EAR}} = \left(1 + \frac{0.0595}{365}\right)^{365} - 1 = 1.0613006 - 1 = 6.1301\%

Bank P wins by $0.0377$ percentage points — $3.77 a year on $10,000. Daily compounding is genuinely worth something, but at these rates it is worth only about $0.015$ percentage points more than monthly, which a $0.05$ -point difference in APR easily outweighs. Neither ranking can be read off the APRs alone.

Problem 4 · The Ceiling

Given: a short-term lender quotes an APR of $r = 18\%$ and computes interest continuously, so $n \to \infty$ — find the effective annual rate, and how daily compounding of the same APR would compare.

What is the effective annual rate?

Compounded daily instead, the same $18\%$ APR would give an effective annual rate that is

✅ Correct!

e^{0.18} - 1 = 19.7217\%

, and daily compounding reaches

19.7164\%

— short of the limit by about five thousandths of a point, since

(1 + r/n)^{n}

rises to

e^{r}

from below.

❌ That is the APR. It is the stated rate; continuous compounding credits interest at every instant, so the year delivers more.

❌ That is monthly compounding.

(1 + 0.18/12)^{12} - 1 = 19.5618\%

; letting

n

run to infinity adds another

0.16

points.

❌ Subtract the principal.

e^{0.18} = 1.1972

is the growth factor for one dollar; the rate is that factor minus

1

❌ Not quite.

❌ $e^{r} - 1$ is a ceiling. Every finite

n

falls short of it, and no compounding frequency can push the effective rate past it or below the APR.

Show solution

Compounding at every instant is the limit of the convention as $n$ grows without bound:

r_{\text{EAR}} = \lim_{n \to \infty}\left(1 + \frac{0.18}{n}\right)^{n} - 1 = e^{0.18} - 1 = 1.1972174 - 1 = 19.7217\%

Daily compounding is a finite $n$ , so it lands just underneath:

\left(1 + \frac{0.18}{365}\right)^{365} - 1 = 19.7164\%

The gap is $0.0053$ percentage points — about five cents a year on $1,000 borrowed ($1,197.22 continuously against $1,197.16 daily). Monthly compounding gives $19.5618\%$ , weekly $19.6845\%$ : the sequence climbs toward $e^{r} - 1$ and never reaches it, which is why more frequent compounding always adds something but never more than that.

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