CM1 · explained

Nominal rates payable p times a year, and the loan that costs more than it says

A rate of twelve per cent charged monthly is not a rate of twelve per cent. It is 12.6825 per cent, and the gap is the whole reason the subject distinguishes a nominal rate from an effective one. The same arithmetic shows what a flat-rate loan really charges.

CM1 · Actuarial Mathematics · 8 min read

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Convert a nominal rate payable p times a year into an effective annual rate and back, and find the effective rate a flat-rate instalment loan is really charging

What a nominal rate actually is

A nominal rate payable p times a year, written i(p), is not a rate at all in the sense that matters. It is an instruction: divide me by p and apply the result p times.

So "12% payable monthly" means a monthly rate of 0.12 ÷ 12 = 1%, applied twelve times. The 12% itself is never charged to anybody; it is shorthand for the 1%.

The effective annual rate is the rate that answers the only question a borrower or depositor cares about: by how much has the money grown over a year? Compounding the 1% twelve times gives it:

(1 + i(p) ÷ p)p = 1 + i

That single equation is the whole topic. Everything below is reading it in one direction or the other.

The mechanism: the same money, two labels

Because interest charged in January is itself earning interest from February onwards, applying 1% twelve times beats applying 12% once. The effective rate is therefore always higher than the nominal rate, for any p above 1, and the gap widens as p rises.

Turn that round and you have the fact that makes the distinction worth money. For a given real cost, charging more often lets you quote a smaller nominal number. Two lenders can advertise different nominal rates and charge the identical amount. The worked example shows 12% payable monthly and 12.3040% payable half-yearly costing exactly the same.

So a nominal rate is only meaningful together with its p. "12% a year" with no frequency attached does not state a price, and comparing two nominal rates with different frequencies is not a comparison.

Where this bites in India: flat rate against reducing balance

A consumer loan is often quoted as a flat rate: interest is calculated on the original amount for the whole term, added on, and the total divided into equal instalments.

That is not a nominal rate, and it is not an effective rate. It is simple interest on a balance that is in fact falling, and it overstates how long the lender's money is actually out. A borrower repaying monthly has handed back roughly half the principal by halfway, so charging the full term's interest on the full amount charges for money that was returned long ago.

The effective rate is found the way every rate in this subject is found — by setting the present value of the instalments equal to the amount borrowed and solving. The worked example does it exactly, as a quadratic, and the answer is well above the quoted figure.

A rough rule worth carrying: a flat rate on a loan repaid in equal instalments works out close to twice the flat figure as an effective rate. Treat it as a smell test, not a calculation — the example gives 10.8554% on an 8% flat quote, and the multiple depends on the term and the frequency.

Reading the equation backwards

Questions ask for the conversion in both directions, and the backwards one is the more awkward:

i(p) = p × [ (1 + i)1/p − 1 ]

Two sanity checks that take seconds and catch most slips. The effective rate must exceed the nominal one for p above 1. And raising p must lower the nominal rate for a fixed effective rate. If either fails, a p has gone in upside down.

Pushing p higher and higher does not run away: the nominal rates fall towards a limit, and that limit is the force of interest, which has its own page. Nothing here needs it, but it is why the sequence of nominal rates is bounded below rather than heading to zero.

Questions students ask

Is the advertised rate on a savings account nominal or effective?

Both are advertised, which is exactly the problem. Where a deposit quotes a rate and a compounding frequency, that rate is nominal and the frequency tells you to divide. Where it quotes an annual equivalent or annual effective figure, that is the effective rate and no conversion is needed. The only safe habit is to ask which one a quoted number is before using it.

Why is a flat-rate quotation allowed at all?

Because it is simple to compute and simple to explain, and because an instalment is what a borrower actually budgets for. It is not dishonest in itself — the instalments are stated plainly. What is misleading is the rate, because it invites comparison with reducing-balance rates that are not on the same footing.

Does a bigger p always help the lender?

For a fixed nominal rate, yes: charging more often raises the effective rate. For a fixed effective rate it makes no difference to either party, which is the point of the comparison in the worked example. The frequency only matters once you hold one of the two rates fixed, and which one is held fixed is what a question is really telling you.

A worked example

Two invented examples, built for this page. Every figure is made up; the first is chosen so the arithmetic is exact and the second so it can be solved in closed form rather than by trial.

  1. Part 1: 12% payable monthly

    The monthly rate is 0.12 ÷ 12 = 0.01, so one rupee grows over a year to

    1.0112 = 1.126825

    The effective annual rate is therefore 12.6825%, not 12%. On a borrowing of ₹5,00,000 held for a year that is a difference of

    5,00,000 × (0.126825 − 0.12) = 5,00,000 × 0.006825 = ₹3,412.50

    Check the conversion backwards from the printed effective rate: 1.1268251/12 = 1.01, and 12 × 0.01 = 0.12, the 12% we began with.

  2. Part 1 continued: the same cost, a different label

    What nominal rate payable half-yearly costs the borrower exactly as much? Set p = 2 and read the equation backwards, using the printed 1.126825:

    i(2) = 2 × (1.1268251/2 − 1) = 2 × (1.061520 − 1) = 0.123040

    So 12.3040% payable half-yearly and 12% payable monthly are the same price. Check it forwards from the printed figures: (1 + 0.123040 ÷ 2)2 = 1.0615202 = 1.126825, which is Part 1's accumulation factor.

    Read the two together and the lesson is plain. The lender charging monthly advertises the smaller number for the same cost. A borrower choosing between "12% monthly" and "12.3% half-yearly" on the face of the quotations would pick the one that is very slightly worse.

  3. Part 2: what an "8% flat" loan really charges

    An invented consumer loan. A dealer in Nagpur offers ₹1,00,000 for one year at "8% flat", repaid in two equal half-yearly instalments.

    The instalments come straight from the flat-rate rule:

    interest = 1,00,000 × 8% = ₹8,000  →  total ₹1,08,000  →  two instalments of ₹54,000

    Now find the rate those instalments actually imply. Let v be the half-yearly discount factor. The loan is worth what it repays:

    54,000 (v + v2) = 1,00,000  →  v2 + v − 1.8518519 = 0

    where 1.8518519 = 1,00,000 ÷ 54,000. This is an ordinary quadratic, so no trial and error is needed:

    v = [ −1 + √(1 + 4 × 1.8518519) ] ÷ 2 = (−1 + √8.4074076) ÷ 2 = (−1 + 2.8995530) ÷ 2 = 0.9497765

    Check the root before using it, by discounting the two instalments with the printed v:

    InstalmentFactorPresent value
    ₹54,000 at 6 monthsv = 0.9497765₹51,287.93
    ₹54,000 at 12 monthsv2 = 0.9020754₹48,712.07
    Total₹1,00,000.00

    It returns the amount borrowed exactly, so the root is right.

    The half-yearly rate of interest is therefore

    j = 1 ÷ 0.9497765 − 1 = 1.0528793 − 1 = 5.2879%

    and the effective annual rate, compounding it twice:

    1.0528792 − 1 = 1.108554 − 1 = 10.8554%

    So "8% flat" is 10.8554% effective — about 1.36 times the quoted figure on a one-year loan with two instalments. On a longer loan repaid monthly the multiple is larger, which is why the rough rule in the body says roughly double rather than giving a number.

  4. Part 2 continued: why the quoted rate was wrong, in one line

    The flat rule charged a full year's interest on the full ₹1,00,000. But half the principal was returned at six months, so on average the lender's money was out for less than a year. Charging ₹8,000 for that shorter use is a higher rate than 8%, and the arithmetic above says how much higher.

    A quick confirmation of the direction, from the figures already printed: the first instalment's present value is ₹51,287.93, so more than half the loan is repaid at the six-month point. Interest for a full year on money that was more than half returned at halfway cannot be an 8% annual rate, and nothing about the calculation above should have come out below 8%.

Where this fits in CM1

This page explains one idea from CM1 Actuarial Mathematics. The full CM1 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM1 topics explained

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.