Nominal vs Real Discount Rate: Matching the Rate to the Cash Flow

A nominal rate contains expected inflation; a real rate has it stripped out. Fisher's relation divides rather than subtracts: 1.12 over 1.05 less one is 6.6667 per cent, not the 7.00 per cent subtraction gives. Discount nominal cash at the nominal rate or real cash at the real rate, and both give the same Rs 29,25,00,00,000 on this worked example's assumptions. Mix them and the answer breaks.

Almost everybody has met a fixed deposit, and almost nobody has been asked the second question about one. So a fixed deposit is the place to start. A saver puts Rs 1,00,000 away for a year and the bank states the rate as 12.00 per cent. A year later the balance is Rs 1,12,000. The arithmetic is not in dispute and nobody has been cheated. Then the second question. Is the saver better off?

The rate alone cannot answer that. The rate counts rupees, and the saver lives on what rupees buy. Suppose the school fee, the cylinder, the bag of rice and the bus pass that together cost Rs 1,00,000 last year cost Rs 1,05,000 now. The Rs 1,12,000 buys 1.066667 of last year's basket. The saver is better off, but by 6.6667 per cent rather than by 12.00 per cent, and the difference between those two numbers is the whole of the distinction that follows.

Notice what has just happened. The moment is more interesting than it looks. Two people can look at the same deposit, the same year and the same bank, and both say something completely true. One says the return was 12.00 per cent. The other says the return was 6.6667 per cent. The two speakers are not disagreeing about the facts; they are counting in two different units, and finance has a name for each of them.

What is the difference between a nominal and a real discount rate?

One counts rupees and the other counts what rupees buy, and everything below follows from that single sentence. A nominal rateA rate that includes expected inflation. is stated in money as money is actually printed and paid: it takes no view about whether prices moved. A real rateA rate with expected inflation taken out. is the same return restated in constant purchasing power, with the change in prices taken back out of it.

The deposit above is the whole idea in miniature. In rupees the stake went from Rs 1,00,000 to Rs 1,12,000. The rise is 12.00 per cent, and the rise stated that way is the nominal statement. In baskets the stake went from one basket to 1.066667 baskets. The rise is 6.6667 per cent, and the rise stated that way is the real statement. Nothing about the deposit changed between those two sentences. Only the unit of account did.

The arithmetic does not change when the numbers get bigger, so the same carries straight across to a company. Sankalp Industrial Systems Limited, invented, is a listed manufacturer of industrial valves, precision castings and the aftermarket parts and service that go with them. The worked example assumes a discount rate of 12.00 per cent for the company and expected inflation of 5.00 per cent. Both figures are assumed rather than measured, and a different price assumption would move every rupee figure that follows.

The two assumed numbers are enough to produce every figure that follows. The 12.00 per cent is a nominal rate: it is what somebody would want in printed rupees for putting money into this business for a year. Taking the assumed 5.00 per cent price rise back out of it gives the real rate. The real rate is what the same somebody would want in purchasing power. The nominal rate and the real rate are not two different rates for two different situations; they are one rate written in two different units, exactly like the deposit above.

The same is true on the cash side, and this is where the trouble usually starts. A nominal cash flowA cash flow stated in the rupees of the year it arrives in. is stated in the rupees of the year it actually turns up in, so a Year 6 figure is in Year 6 rupees and a Year 12 figure is in Year 12 rupees, and the two are not the same unit even though both are called rupees. A real cash flowA cash flow restated in the rupees of one fixed year. is restated into the rupees of one fixed year, so every figure in the column is in the same unit and can be compared straight down without doing anything else to it.

Why is taking inflation out a division rather than a subtraction?

Because prices do not add themselves to a return, they scale it. Scaling rather than adding is the entire argument, and once that has landed the subtraction is never written again unknowingly. Back to the deposit. The rupees were multiplied by 1.12. The price of the basket was multiplied by 1.05. A multiplication is undone by a division and never by a subtraction, so the number of baskets is 1.12 divided by 1.05.

Irving Fisher wrote that relation down, and it is his name that travels with it. The Fisher relationThe rule that one plus the nominal rate equals one plus the real rate times one plus inflation. says that one plus the nominal rate equals one plus the real rate multiplied by one plus expected inflation. Rearranged to give the real rate, it says: divide one plus the nominal rate by one plus expected inflation, then take one away.

The subtraction that most people use instead is an approximation of this, and it is worth being precise about why it is only an approximation. Expanded, the Fisher relation says that one plus the nominal rate is one plus inflation plus the real rate plus inflation multiplied by the real rate. Cancelling the ones leaves an exact identity: the nominal rate equals inflation plus the real rate plus the product of the two, and the subtraction shortcut is simply the version that throws that last product away.

Put the numbers in. The assumed inflation is 5.00 per cent and the real rate is 6.6667 per cent, so the product of the two is 0.05 multiplied by 0.066667. The product is 0.003333, or 0.3333 of a percentage point. Add the three together: 5.0000 plus 6.6667 plus 0.3333 is exactly 12.0000. Nothing is left over and nothing is fudged. The shortcut's whole error is that one product, and it is worth 33 basis points here, a basis pointOne hundredth of a percentage point. being one hundredth of a percentage point.

One more application of the same division, and then this idea is finished. The worked example also assumes a risk-free rate of 7.75 per cent for the invented company. The 7.75 per cent is a nominal figure, like every other rate above. Dividing 1.0775 by 1.05 and taking one away gives 2.6190 per cent. The 2.6190 per cent is the same assumed risk-free rate written in purchasing power. How that 7.75 per cent, and the 12.00 per cent built on top of it, are arrived at in the first place is set out under the build of a discount rate. The division above only takes them apart into their two units.

How much does the subtraction shortcut actually cost?

Thirty-three basis points sounds like nothing, and on a rate it very nearly is. A rate does not sit on its own, though. A rate sits in the denominator of a stream that never ends, and a denominator is a place where small errors stop being small.

The invented company's model supplies the figures for the working below. After Year 5, the forecast assumes the business settles into a steady state and produces free cash flow of Rs 2,04,75,00,000 in Year 6, growing at an assumed 5.00 per cent a year for ever. How that Rs 2,04,75,00,000 is arrived at, and how a terminal figure is made consistent with the reinvestment it assumes, are covered separately and are taken here as given. The Year 6 cash flow is discounted twice below, once in each of the two units.

Restate that Year 6 cash flow in Year 5 rupees: Rs 2,04,75,00,000 divided by 1.05 is Rs 1,95,00,00,000. Value that real stream properly, at the Fisher real rate of 6.6667 per cent, and it is worth Rs 29,25,00,00,000. Now value exactly the same stream using the subtracted 7.00 per cent instead. Rs 1,95,00,00,000 divided by 0.07 is Rs 27,85,71,42,857, and the working takes four seconds and looks entirely respectable.

The two answers are Rs 1,39,28,57,143 apart, or 4.76 per cent of the correct figure, and one arithmetic habit is all that separates them. The figures above are rounded to the rupee from the exact division; on an answer of this size the last two digits carry no information whatever, and the load-bearing part of Rs 27,85,71,42,857 is the leading Rs 27.86 crore-scale magnitude rather than the tail.

When is the shortcut harmless, and when is it not?

A feeling fails at exactly the wrong moment, so state it as a rule. The shortcut's error is the product of inflation and the real rate, so it is small when either of them is small and it is largest when they are both middling. The shape of that product gives a usable test that needs no calculator: multiply the two rates in question, in per cent, and the answer is the error in basis points. Five multiplied by 6.6667 is 33.33, and that is the gap in basis points, exactly.

Run at the ends, that test shows something most readers get backwards. Almost everybody assumes the subtraction gets worse and worse as inflation rises, without limit. It does not. At an assumed 2.00 per cent inflation the real rate is 9.8039 per cent and the error is 19.61 basis points. At 5.00 per cent it is 33.33. At 8.00 per cent the real rate has fallen to 3.7037 per cent, so the product falls too, and the error is back down to 29.63. At 11.00 per cent the real rate is only 0.9009 per cent and the error is 9.91 basis points, smaller than it was at 2.00 per cent.

On a fixed 12.00 per cent nominal rate the shortcut's error peaks at 33.99 basis points when assumed inflation is 5.83 per cent, and it falls away on both sides of that. There is a neat reason for the peak sitting where it does. The error is a product of two numbers that trade off against each other, and a product like that is largest when the two are equal. At 5.83 per cent assumed inflation the Fisher real rate is also 5.83 per cent. Equality is what makes that the widest point rather than any other.

So when is the shortcut safe? When the answer only has to be indicative and the numbers are small. Working out roughly what a one-year deposit did in purchasing power over a cup of tea, the subtraction is fine and nobody will be harmed by twenty basis points. The moment the rate is going to be divided into something, and especially the moment it is going into the denominator of a stream with no end date, the shortcut stops being an approximation and starts being a defect that produces a nine-figure difference.

Which pairings of cash flow and rate are legal, and which are not?

Two of the four, and that is the whole of the consistency rule. Nominal cash flows may be discounted at a nominal rate. Real cash flows may be discounted at a real rate. Neither of the other two combinations is permissible. The bar is not a convention or a house style: each mismatched combination divides a quantity measured in one unit by a rate measured in another, and the result means nothing at all.

A household version of the rule makes it impossible to forget, and the household version is worth carrying. Suppose a household is putting money aside for a school fee that is Rs 1,00,000 today and will rise with prices. Planning in the rupees of each future year means letting the fee grow and comparing it against a deposit rate quoted in the same printed rupees. Planning instead in today's rupees, holding the fee at a flat Rs 1,00,000, means comparing it against a rate with the price rise already taken out. One of each either terrifies or reassures, and neither reaction has anything to do with the household's actual position.

Does the choice of basis change the answer at all?

Not by a rupee. The belief that real is somehow the careful option and nominal the aggressive one is extremely common and completely wrong, so the proof is worth doing slowly. Both routes below value the same terminal stream of the invented company. One does it entirely in printed rupees; the other does it entirely in Year 5 rupees. The two routes agree exactly.

Both use the same formula, the growing perpetuity that Gordon set out in Dividends, Earnings and Stock Prices in the Review of Economics and Statistics in 1959. A growing perpetuityA stream that never stops and rises at a constant rate each period. is a stream that never stops and rises at a constant rate each period, and its value is next period's cash flow divided by the rate less the growth rate.

Route one, everything nominal. The Year 6 cash flow is Rs 2,04,75,00,000, the growth is 5.00 per cent nominal and the rate is 12.00 per cent nominal. The gap between rate and growth is 0.07, and Rs 2,04,75,00,000 divided by 0.07 is Rs 29,25,00,00,000.

Route two, everything real. Restate the cash flow in Year 5 rupees: Rs 2,04,75,00,000 divided by 1.05 is Rs 1,95,00,00,000. The stream is assumed to grow at exactly the rate prices are assumed to rise, so its real growth is 1.05 divided by 1.05 less one, or 0.00 per cent. The real rate is 6.6667 per cent. The gap between rate and growth is 0.066667, and Rs 1,95,00,00,000 divided by 0.066667 is Rs 29,25,00,00,000.

The two routes agree to the rupee, and that agreement is the whole proof that neither basis is more conservative than the other. The choice between them is a choice about which set of assumptions is easier to defend out loud, not a choice about how high or low the answer comes out.

The same agreement holds on the five explicit forecast years, and running it there is the more useful check because the numbers are not round. The five free cash flow figures of Sankalp Industrial Systems Limited are nominal: Rs 98,00,00,000, Rs 1,16,00,00,000, Rs 1,34,00,00,000, Rs 1,52,00,00,000 and Rs 1,70,00,00,000. Discounted at the nominal 12.00 per cent with year-end discounting, they come to Rs 4,68,41,43,564.

Now do the same five years the real way. Restate each nominal figure into Year 0 rupees by dividing by 1.05 raised to that year's power, then discount at the Fisher real rate of 6.6667 per cent. The real discount factor turns out to be unusually clean: one divided by one plus one fifteenth is exactly 0.9375, so the factor for year n is 0.9375 raised to the power n and every figure in the column is exact.

Read the last column and then read it again. The nominal route produces that same column. Every line matches, and the total matches: Rs 4,68,41,43,564 either way. The middle two columns are completely different from anything the nominal route ever computes, and they cancel each other out exactly. The Fisher relation says that cancellation must happen. The present values are computed on the unrounded values and printed rounded, and the restated cash flows in the third column are printed rounded to the rupee from an exact division, so the third column will not multiply back into the fourth by hand at the last digit.

What happens when the two bases are mixed?

Something much larger than most people expect in one direction, and something quietly plausible in the other. The quiet one is the more dangerous of the two. Both mistakes travel through the invented company's terminal stream, so take them in turn on that number.

Mismatch one is nominal cash at a real rate. The Year 6 cash flow of Rs 2,04,75,00,000 is nominal and it is left growing at its nominal 5.00 per cent, but the rate applied to it is the real 6.6667 per cent. The gap between rate and growth collapses from 7.00 percentage points to 1.6667, and Rs 2,04,75,00,000 divided by 0.016667 is Rs 1,22,85,00,00,000. The mismatched answer is exactly 4.20 times the correct one, and the whole of the damage comes from a denominator that shrank to less than a quarter of what it should have been.

Mismatch two is real cash at a nominal rate, and it runs the other way. The real Rs 1,95,00,00,000 with its 0.00 per cent real growth is discounted at the nominal 12.00 per cent, and Rs 1,95,00,00,000 divided by 0.12 is Rs 16,25,00,00,000. The mismatched answer is 55.56 per cent of the right one. Nearly half the value is gone. A figure that is a bit lower than expected reads as caution rather than as an error, so this is the version nobody catches.

What does 5.00 per cent terminal growth mean when inflation is 5.00 per cent?

The model assumes the business does not grow at all in real terms after Year 5, and that sentence is the payload of the whole distinction. The reading is not a trick and not a criticism. The two assumptions simply say so once they are put next to each other, and most readers of most models have never put them next to each other.

Work through it. Terminal growthThe rate at which cash is assumed to grow forever after the last forecast year. for Sankalp Industrial Systems Limited is assumed at 5.00 per cent a year, in nominal rupees, forever after Year 5. The worked example also assumes prices rise at 5.00 per cent a year. Divide one by the other, exactly as Fisher's relation requires: 1.05 divided by 1.05 is one, and less one is nil. The real growth rate assumed for this business after Year 5 is 0.00 per cent, forever.

The sentence is easy to misread in the alarming direction, so say what it does and does not mean. Zero real growth does not mean the business shrinks. The business does not stop selling anything, and the rupee figures do not stop rising. The rupee figures rise at 5.00 per cent a year, for ever. Zero real growth means the business is assumed to sell the same physical quantity of valves and castings and service at prices that move with everything else, so its volumes never grow again and it never takes another share of its market. The assumption is a substantive claim about the business, and it deserves to be argued about rather than hidden inside a growth cell.

Now the part that gives the sentence its weight. In the invented company's discounted cash flow, the terminal value is 77.99 per cent of the whole answer. The five explicit years, the ones with the forecast revenue and the forecast margins and the schedule of capital spending, do 22.01 per cent of the work. Everything else rests on a stream nobody forecast in detail, growing at a rate that this worked example's own inflation assumption reduces to nothing in real terms.

There is a cross-check available on the same fact, and it is worth naming because it shows how far this assumption sits from what a market price would imply. The peer median exit multiple used elsewhere in this worked example is 7.8 times Year 5 earnings before interest, tax, depreciation and amortisation, and exiting at that multiple would imply nominal terminal growth of 6.58 per cent rather than 5.00. Put that through Fisher's division against the assumed 5.00 per cent inflation and it is 1.5048 per cent of real growth. So the difference between the model's assumption and that multiple is the difference between no real growth at all and about one and a half points of it, forever. Which of those is right is a judgement, and the division above settles only the units.

What does this go wrong on, and what does the mistake look like?

Who actually uses this, and what do they use it for?

Which neighbouring subjects does this touch, and where does the inflation basis stop?

The inflation basis holds one distinction and works it in full, and stops there. Five subjects sit close enough to be confused with it, and each is set out briefly below along with where it is treated.

Where the 12.00 per cent itself comes from

The build behind the rate, being a cost of equity, an after-tax cost of debt and the weights that combine them, is covered separately. Above, the 12.00 per cent is taken as given and one thing is done to it: the assumed price rise is taken out.

The other meaning of the word nominal

A rate can also be called nominal when it is quoted for a year but charged more often than once a year. The second meaning is a question about frequency and has nothing whatever to do with prices. The invented company's term loan is quoted at 7.80 per cent and charged quarterly, and putting that on an annual basis is a different operation entirely. Compounding frequency is covered separately, and the two ideas share a word and nothing else.

What inflation is and how it is measured

Actual inflation is a separate subject. The 5.00 per cent used throughout is this worked example's own assumption about an invented company, not a measured price change, a policy rate or an index level, and not a forecast of prices.

How the terminal cash flow was arrived at

The Rs 2,04,75,00,000 of Year 6 comes from a build that makes the terminal stream consistent with the reinvestment it assumes, an argument associated with Aswath Damodaran's published valuation material. The build is covered separately and is taken as given above, restated only in a second unit.

Whether the zero real growth assumption is a good one

Not settled here, and not settleable by arithmetic. Reading 5.00 per cent nominal growth against 5.00 per cent assumed inflation and getting nil is a calculation. Deciding whether an industrial valve business should be assumed to grow in real terms after Year 5 is a judgement about the business, and it belongs to somebody who knows the business rather than to a treatment of units.