Nominal and Effective Annual Rates: Why Frequency Changes the Answer

A quoted annual rate settles nothing on its own until the frequency with which it lands is also known. Charge 7.60 per cent twelve times and the borrower ends the year 7.8704 per cent poorer; charge 7.80 per cent four times and the figure is 8.0311 per cent. The two quotes sit 0.20 of a percentage point apart. The two costs sit 0.16 apart.

All of that rests on one familiar thing: interest charged inside a year lands on an amount that earlier interest has already lifted. A rate stated per year but charged four times is really four charges of a quarter of it, and the second of those four is applied to a slightly bigger balance than the first. The quoted figure is what the contract says. The effective figure is the cost that arithmetic imposes on the borrower. The more often a charge lands, the further apart those two drift, and the drift is completely predictable once one line can be written.

The three borrowings of Sankalp Industrial Systems Limited, an invented listed maker of industrial valves and precision castings, carry every figure worked below, all of them read at Year 0. Each of the three rates is Sankalp's own contracted rate rather than an observation of what anything costs to borrow anywhere.

What is a quoted annual rate actually saying?

Consider how a shopkeeper describes a moneylender. The rate comes out as two per cent, and the sentence stops there. Two per cent per what? Per month, it turns out, and two per cent a month is a very different animal from two per cent a year. The trade has a convention, so everybody in that conversation knows which one is meant. Written contracts have conventions too. The convention is that a rate is stated as an annual figure even when nobody intends to charge it once a year.

So a quoted annual rate is a label with two halves. The first half is a number. The second half is a frequency. The frequency sits somewhere else in the document, often in a schedule, sometimes in a single clause about rests. The number on its own is accurate and incomplete at the same time. A number that is simply wrong is a far easier thing to handle. Nothing about the 7.60 per cent on Sankalp's working capital facilityA borrowing a company can draw on and repay as it needs, secured on receivables and inventory, and usually renewed every year. is untrue. The contract charges a twelfth of it every month, so the figure divided by twelve is exactly 7.60, and the lender has said so.

A label cannot stand next to another label and be compared with it. Two quoted rates are two different objects wearing the same coat if their frequencies differ. The danger in a quoted rate is never that it lies. The danger is that it invites a comparison it cannot support.

What does charging four times a year actually do to the amount owed?

Sankalp's first borrowing is a secured rupee term loanA borrowing drawn once and repaid on a fixed schedule, rather than one a company dips into and out of. of Rs 3,00,00,00,000, quoted at 7.80 per cent a year and charged quarterly, with a bullet maturityA repayment shape in which nothing comes off the principal along the way and the whole amount falls due on one day at the end. at the end of Year 5. Quarterly charging means the periodic rate is 7.80 divided by four, being 1.95 per cent, and that 1.95 per cent is applied four times.

Walk it. The first charge is 1.95 per cent of Rs 3,00,00,00,000, or Rs 5,85,00,000. If nothing is paid across, the amount the second charge sees is Rs 3,05,85,00,000, so the second charge is bigger than the first without the rate moving at all. The third sees a bigger amount again, and so does the fourth. Four identical rates applied to four different balances do not produce four identical charges, and the whole of the effective rate story is contained in that sentence.

Two notes on that table before the point it makes. The four charges are each rounded to the rupee for printing, and the four printed lines add to Rs 24,09,33,913 while the same walk carried at full precision lands on Rs 24,09,33,912. One rupee, and it is the rounding rather than a disagreement. Second, the loan is a bullet, so nothing is paid across during the year and each charge really does sit on the balance the last one left.

Now the point. A single charge of 7.80 per cent at the end of the year would have cost Rs 23,40,00,000. The four quarterly charges cost Rs 69,33,912 more. Nobody added a fee. The rate never moved. The Rs 69,33,912 is the price of the calendar alone, and stated as a rate it is what separates 7.80 per cent from 8.0311 per cent.

Nominal vs Effective Annual Rate: which of the two is a cost?

The four steps need not be walked every time. One expression does that walk, and it is the only formula anywhere here.

Three keystrokes, in order, and they never change. Divide the quote by the number of charges. Add one and raise it to that number. Take one away. Every effective rate worked below, including the ones that look surprising later on, comes out of those three keystrokes and nothing else.

Run it on the term loan. 0.078 divided by 4 is 0.0195. One plus that is 1.0195. Raised to the fourth power it is 1.08031130. Take one away and 0.08031130 is left, or 8.0311 per cent. The 8.0311 per cent is the Rs 24,09,33,912 from the walk above expressed as a rate, and the two agree: they are the same arithmetic written two ways.

Run it on Sankalp's second borrowing. The second borrowing is Rs 2,00,00,00,000 of listed unsecured non-convertible debenturesListed borrowings that pay a stated coupon and repay their face amount at maturity, with no route into shares at any point. quoted at 8.50 per cent and paid half-yearly. Half of 8.50 is 4.25. One plus that is 1.0425. Squared it is 1.08680625. Take one away and the year costs 8.6806 per cent.

How do the three borrowings read when both numbers are put side by side?

Sankalp holds three borrowings at Year 0, and each of them is charged on a different rhythm. Each borrowing was negotiated with a different counterparty at a different time for a different purpose, and nobody harmonised the schedules afterwards. Mixed rhythms are what a real capital structure looks like.

The last column is the one that surprises people. Read it first. The basis pointRate work is counted in hundredths of one per cent, and each of those hundredths is one of these. A quarter point is twenty-five of them. gains run the wrong way round from the rates. The cheapest quote gains the most, 27.04, and the dearest quote gains the least, 18.06. Frequency is doing that work rather than size: twelve charges compound more than two, and it does not matter at all which quote they are compounding.

And now the headline of the whole worked case: the order does not change. Ranked on the quoted rates the three run tranche 3, tranche 1, tranche 2. Ranked on the effective rates they run tranche 3, tranche 1, tranche 2. Nobody overtakes anybody. A dramatic reversal would require numbers other than these, and what actually happens here is more useful than a reversal would be.

The gap between the two cheapest borrowings narrows instead. Tranche 3 and tranche 1 are quoted 20 basis points apart. Tranche 3 picked up 27.04 basis points and tranche 1 only 23.11, so on an effective basis the two sit 16.07 basis points apart. The difference between those two gains, 3.93 basis points, is what got eaten. Frequency has taken about a fifth of the quoted gap away without touching the ranking.

How much does frequency add at each charging pattern?

Almost everybody gets this wrong in both directions at once, underestimating what monthly charging does and then badly overestimating what daily charging does on top of it.

The default setting reproduces Sankalp's working capital facility exactly. Monthly charging on a 7.60 per cent quote costs 7.8704 per cent. On Rs 1,00,00,00,000 drawn that is Rs 7,87,04,026 for the year, against the Rs 7,60,00,000 the quote alone would suggest. Here is the whole ladder as static figures, for a reader who never touches the control.

The six rows above are the whole of the frequency story, and the interesting thing about them is not the first row or the last but the spacing between them.

Why are the additions not equal, and where do they stop growing?

Line the five gains up: 14.44, 21.94, 27.04, 29.03, 29.54. Every step buys less than the step before it. Going from one charge a year to two buys 14.44 basis points. Going from fifty-two to three hundred and sixty-five buys half a basis point. The first step is worth more than every step from monthly onwards put together. A lender switching from annual to half-yearly rests is doing something material; a lender switching from weekly to daily is doing almost nothing.

The reason is that compounding needs a gap to work in. Two charges give the first one six months to sit before the second lands on it. Three hundred and sixty-five charges give the first one a single day. Cut the interval finely enough and there is nothing left for the next charge to feed on, so the total stops climbing. The total does not merely slow down. It walks into a hard ceiling at 7.8963 per cent, the limit as the interval shrinks towards nothing altogether. Daily charging is already within 0.09 of a basis point of that ceiling.

Can a lower quoted rate ever cost more than a higher one?

Yes, and here the arithmetic stops being decorative. Take Sankalp's term loan as the thing to beat: 7.80 per cent charged quarterly, costing 8.0311 per cent for the year. Now ask a precise question. At what quoted rate would a monthly-charged loan cost exactly the same 8.0311 per cent?

The answer is the same three keystrokes run backwards. Take 1.08031130, ask for the twelfth root of it, subtract one and multiply by twelve. The answer is 7.7498 per cent. Any monthly-charged loan quoted above 7.7498 per cent is dearer over a year than a quarterly-charged loan quoted at 7.80, however comfortably it looks cheaper on the two labels.

The 7.7498 per cent is worth carrying around. It shows what frequency is actually worth here. Moving from quarterly to monthly charging is worth 5.02 basis points of quoted rate and no more. If two lenders quote within five basis points of each other and one charges monthly while the other charges quarterly, the labels have settled nothing and the arithmetic has to be done. If they quote thirty basis points apart, the labels have settled everything and the arithmetic is a formality.

Which is why Sankalp's own two borrowings do not cross. Tranche 3 is quoted at 7.60, and 7.60 sits 14.98 basis points below the 7.7498 crossover. The distance frequency can cover is about a third of that. There is no charging pattern whatever, not weekly, not daily, not one charge every second, that lifts a 7.60 per cent quote up to what tranche 1 costs. The ceiling on a 7.60 quote is 7.8963 per cent and tranche 1 sits at 8.0311.

Which rate goes into a comparison, and which one goes inside a model?

Two different questions take two different answers. The two get muddled all the same.

When offers are being compared, the effective annual rate is the only figure that means anything. A cost is what it states. The quoted rate is a label, and a label belongs beside the cost as identification rather than in the comparison itself. The discipline is small and it never varies: for each offer write the quote, write how often it lands, do the one line, and compare only the outputs. A household choosing between two lenders is doing exactly what a treasury team does here, on four figures instead of forty.

Inside a valuation model the question is different. The model is not choosing between lenders. A model needs one cost of debt for the whole company, and how the three blend into one is set out under the cost of debt. What the choice of convention is worth can be shown in this case.

Sankalp has Rs 6,00,00,00,000 of gross debtEverything the company has borrowed, added up, before subtracting any of the cash it is holding. across the three tranches. Weight the three quoted rates by their balances and the blend comes to exactly 8.00 per cent. The cost of capital build for Sankalp uses that figure. Weight the three effective rates the same way and the blend comes to 8.2208 per cent. The difference is 22.08 basis points, or Rs 1,32,50,438 a year on Rs 6,00,00,00,000.

Both figures are now on the record and nothing has been re-cut. The weighted average cost of capitalWhat funding itself costs a company once both its shareholders and its lenders are taken into account, in the proportions it actually uses each of them. worked elsewhere for Sankalp rests on the quoted blend of 8.00 per cent. Which convention such a build should use is a decision made under the cost of capital rather than here.

What sits outside the arithmetic of frequency, and where are those subjects covered?

Four neighbouring subjects press closely enough on charging frequency to be worth naming, and each of them is somebody else's ground.

Where these rates came from is one. The three figures 7.80, 8.50 and 7.60 per cent stand here as contracted facts of an invented borrower, and the only question asked of them is what frequency does. Why a lender would price a secured term loan below unsecured debentures, and how those three blend into one cost of debt that feeds a weighted average cost of capital, are covered separately.

The second neighbour is the other meaning of the word nominal, and it is the one most likely to trip a reader. Here nominal means before frequency. Where the word is set against a real rate, nominal means before inflation, and the arithmetic there divides by one plus a price rise rather than compounding inside a year. Same word, two unrelated jobs. Carrying the wrong one forward leaves neither subject making sense.

The third is what a lender must quote and disclose to a borrower in India. The Reserve Bank of India sets those requirements and revises them.

The fourth is judgement about pricing. Whether 7.60 per cent is cheap or 8.6806 per cent expensive, and whether any of these borrowings is well or badly priced, is a separate question. The work here is what a year costs given a quote and a rhythm.

Sankalp Industrial Systems Limited is invented.
Educational material. Not advice on any investment, tax, budget or market position.