Finance Formulas

Time Value of Money

core

Effective Annual Rate (EAR)

Builds onFuture Value of a Single Sum — if this page feels steep, start there.

EAR=(1+rm)m−1EAR = \left(1 + \frac{r}{m}\right)^{m} - 1

Reading the notation

EAREAR
effective annual rate — what your money truly grows by in one year
rr
the stated (nominal) annual rate the bank quotes, as a decimal
mm
how many times per year interest is credited (12 = monthly, 4 = quarterly)
rm\frac{r}{m}
the slice of the annual rate you actually earn each crediting period
(1+rm)m(1+\frac{r}{m})^m
grow by that slice m times in the year — compounding each time
− 1-\,1
subtracting 1 turns the growth factor back into a rate (1.083 becomes 8.3%)

Play with it

Compounding frequency moves you between these curves: annual is the red path, continuous is the ceiling. EAR is how far up you actually sit.

1×2×5×0y5y10y15y20yyearsgrowth of $1continuous 5.0×compound (annual) 4.7×simple 2.6×

2.6×

Simple @ 20y

4.7×

Compound @ 20y

2.1×

Interest on interest

The gap between the straight line and the curves is interest earning interest. Continuous compounding is the ceiling — barely above annual at low rates, decisive at high ones.

Why it must be true

A "stated" (nominal) annual rate rr compounded mm times a year actually pays r/mr/m per sub-period, mm times. Because each sub-period earns interest on the previous ones, the year's true growth is (1+r/m)m(1 + r/m)^m — more than 1+r1 + r.

The EAR converts any compounding convention into one honest number: the rate that, compounded once a year, produces the same growth. More frequent compounding ⇒ higher EAR, approaching er−1e^{r} - 1 in the continuous limit.

The derivation

Each sub-period the balance is multiplied by (1+rm)\left(1 + \frac{r}{m}\right). Over one year there are mm sub-periods:

growth over one year=(1+rm)m\text{growth over one year} = \left(1 + \frac{r}{m}\right)^{m}

The effective annual rate is defined as the single annual rate producing the same growth:

1+EAR=(1+rm)m⇒EAR=(1+rm)m−11 + EAR = \left(1 + \frac{r}{m}\right)^{m} \quad\Rightarrow\quad EAR = \left(1 + \frac{r}{m}\right)^{m} - 1

When to reach for it

Two quotes compound at different frequencies and must be compared, or a stated (nominal) rate needs converting into what the money actually earns in a year.

Listen for

“stated / nominal / quoted annual rate”“compounded monthly, credited daily”“APR vs APY”“which offer is better”“effective annual basis”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

  • ≈

    EAR is always a whisker above the stated rate — bracket it: r<EAR<er−1r < EAR < e^r - 1. For 12% monthly: between 12% and 12.75%.

  • ≈

    Quick estimate: EAR≈r+r22EAR \approx r + \frac{r^2}{2} for monthly-or-better compounding. At 8%: 8% + 0.32% ≈ 8.32% (true monthly: 8.30%).

  • ≈

    The more frequent the compounding, the bigger the bump — but the bump shrinks fast: monthly→daily adds almost nothing. If two choices are near-identical, the higher one is likelier for higher m.

Traps in applying it

  • ✗Compounding the whole stated rate m times instead of splitting it into r/m first.
  • ✗Answering with the EAR when the problem needs the per-period rate (or vice versa).
  • ✗Ignoring day-count conventions — 365 vs 360-day years shift the answer.

Limits & criticisms

EAR assumes every sub-period's interest is reinvested at the same rate — fine for a savings account, shakier for instruments where coupons can't be reinvested on identical terms. And unlike a regulatory APR, the pure formula ignores fees, which often matter more than compounding frequency.

Where it came from

Compounding-frequency games are old enough that Jacob Bernoulli's famous 1683 problem — what happens to interest as you compound more and more often — was posed about exactly this. For centuries lenders quoted rates in whatever convention flattered them, which is why the EAR became a legal concept: the US Truth in Lending Act (1968) and its APY disclosures exist to force quotes onto this one honest scale.

In practice you meet it whenever two offers compound differently: deposit accounts, credit cards, money-market yields. It is also the bridge to continuous compounding — push mm to infinity and the EAR converges to er−1e^r - 1.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

True annual growth

EAR=(1+rm)m−1EAR = \left(1+\frac{r}{m}\right)^m - 1

Converts any quoting convention into one honest annual rate, making offers with different compounding directly comparable.

Drill this face →

Stated rate behind an EAR

r=m[(1+EAR)1/m−1]r = m\left[(1+EAR)^{1/m} - 1\right]

Read backwards: take the m-th root to find the per-period rate, then annualize by m. This is how a contract's nominal rate is recovered from disclosed effective terms.

Drill this face →

On the BA II Plus

Worked example: A loan carries a stated annual rate of 10.5%, compounded semiannually. What effective annual rate is the borrower really paying?

  1. 1.[2ND] [ICONV]interest-conversion worksheet
  2. 2.10.5 [ENTER]NOM = the stated annual rate, in percent
  3. 3.[↓] [↓]scroll to C/Y
  4. 4.2 [ENTER]C/Y = compounding periods per year
  5. 5.[↑] [CPT]EFF = the effective annual rate

→ 10.78%

Where it leads

Master this and the following come almost for free:

Continuous Compounding
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