Rates: nominal, effective and continuous

A rate is meaningless without its compounding convention. "7%" is not a number until somebody says per what, how often.

This chapter is short and it prevents a specific, common error: comparing two products quoted under different conventions and concluding the wrong one is better.

Nominal

The nominal annual rate is the quoted one, and it is a convention rather than a measurement. A nominal 12% compounded monthly means 1% a month — the annual figure is the monthly rate multiplied by twelve, which deliberately ignores that the months compound.

$\text{periodic rate} = \frac{r_{\text{nominal}}}{m}$

where $m$ is the number of compounding periods a year.

Effective

The effective annual rate is what you actually end the year with. Grow by the periodic rate $m$ times:

$(1 + r_{\text{eff}}) = \left(1 + \frac{r_{\text{nominal}}}{m}\right)^{m}$

$r_{\text{eff}} = \left(1 + \frac{r_{\text{nominal}}}{m}\right)^{m} - 1$

At 12% nominal:

Compounding	$m$	Effective
Annually	1	12.000%
Half-yearly	2	12.360%
Quarterly	4	12.551%
Monthly	12	12.683%
Daily	365	12.747%

The effective rate is always at least the nominal rate, and equal only when $m = 1$. More frequent compounding is strictly better for a depositor and strictly worse for a borrower.

Why this matters in India specifically

The RBI's deposit directions require interest on domestic term deposits to be calculated at quarterly or shorter rests. So a fixed deposit advertised at 7% pays an effective

$\left(1 + \frac{0.07}{4}\right)^{4} - 1 = 7.186\%$

That is not a trick; it is the rule, and it is why the "annualised yield" column on a bank's deposit page is higher than the rate column beside it. The two numbers describe the same deposit.

The error to avoid is the reverse one: a loan quoted at a monthly rate is more expensive than twelve times that rate suggests, for exactly the same arithmetic.

Continuous compounding

Let $m$ grow without bound. The limit

$\lim_{m \to \infty}\left(1 + \frac{r}{m}\right)^{m} = e^{r}$

is one of the definitions of $e$. Continuous compounding is that limit, and the table above is already converging on it: at 12%, $e^{0.12} - 1 = 12.750\%$, barely above the daily figure.

So continuous compounding is not an exotic product anyone sells. It is a mathematical convenience, and the convenience is real:

• Growth over $t$ years is $e^{rt}$, so rates add across periods instead of multiplying. Two years at 5% then 7% is $e^{0.05}e^{0.07} = e^{0.12}$.
• The inverse is a logarithm, which makes the algebra of option pricing tractable. Black-Scholes is written in continuous rates for this reason and no other.
• Log returns, which chapter 7 uses, are the returns implied by continuous compounding.

Converting between the two conventions:

$r_{\text{continuous}} = \ln(1 + r_{\text{eff}}) \qquad r_{\text{eff}} = e^{r_{\text{continuous}}} - 1$

The rule that prevents the error

Convert everything to effective annual before comparing anything. Not because effective is the "true" rate in some philosophical sense, but because a comparison between two different conventions is not a comparison at all.

A worked case: 7.2% quarterly gives $(1 + 0.018)^4 - 1 = 7.397\%$, against 7.3% annual at 7.300%. The quarterly deposit wins — by 0.097 percentage points, which on ₹5,00,000 is about ₹485 a year. Real, but worth knowing before spending an afternoon on it.

The point

A rate is not a number until its compounding frequency is stated. The effective annual rate is what you actually earn, it is always at least the nominal rate, and India's deposit rules mean quoted FD rates systematically understate it. Continuous compounding is the limit of ever-more-frequent compounding, worth knowing because it makes rates add rather than multiply — which is why option pricing is written in it.