The Risk-Free Rate: What Qualifies and Why the Choice Matters

The base rate in a cost of capital is what a security with no default risk and no reinvestment risk over the horizon being valued would pay. In practice one borrower fits: a government borrowing in its own currency at a matching maturity. For Sankalp Industrial Systems Limited, an invented maker of industrial valves, this worked example assumes 7.75 per cent rather than a current market level.

Begin somewhere that has nothing to do with valves, spreadsheets or a cost of capital. A household has put money aside for a child who will start a degree in ten years, and the fees fall due then and not before. There are two ways to hold that money. One is a single deposit locked for the whole ten years at a rate agreed today. The other is a one year deposit, rolled over nine times, at whatever rate is going in each of those years. Both are with an issuer nobody in the house expects to fail. Only one of them tells the household today what it will have on the day the fees arrive.

Ask which of the two is the safer choice and most people say the one year deposit. A year is shorter, and shorter feels safer. The instinct is exactly backwards for the question being asked. The household is not asking what happens over the next year. The household is asking what it will have in ten years, and the rolled deposit answers that question with nine unknowns in it. A rate is only certain over the period it is locked in for, so certainty about the issuer and certainty over the horizon are two separate questions, and a baseline has to answer both.

Those two questions are the whole of a base rate. Applied to a discount rate rather than to school fees, and worked through on one company with the arithmetic shown, they settle which security can serve as a baseline and which cannot.

What is this number actually being asked to do?

Look at how a cost of equity is put together and the answer becomes obvious. A cost of equity is a base rate plus a premium, and the premium is compensation for bearing something: the ordinary risk of holding shares rather than something safer, and, where the cash flows arise somewhere with more of it, the extra risk of the jurisdiction. Both of those are premiums for bearing a risk. A premium is a comparison. A comparison is only meaningful if there is something it is a premium over.

The base rate is that something, and it therefore has to be genuinely free of the things everything else is being paid for. If the baseline itself carries a little bit of credit risk, then the premium measured on top of it is no longer the price of credit risk. The premium becomes the price of credit risk minus whatever was already sitting in the baseline, and nobody can say how much that was. The measurement stops meaning what it says it means. The objection is arithmetic rather than philosophical, and the two conditions below are strict for exactly that reason.

Two conditions follow directly, and both of them are practical.

The two conditions have very little to do with each other. The first is about who is on the other side of the promise. The second is about how long the promise runs. A security can be perfect on one and hopeless on the other, and failing either one is enough to disqualify it. The independence of the two conditions is the single most useful thing to hold on to. Almost every mistake with a base rate comes from satisfying one condition thoroughly and forgetting the other exists.

Why must the issuer be able to produce what it promised?

Take the first condition on its own. Default riskThe chance that the issuer does not pay what it promised. is the chance that the promise is not kept, and the practical question is what would have to happen for it to be broken. For an ordinary borrower, the answer is a bad enough run of business. Cash does not arrive, the account is empty on the day, and the payment is missed. Nothing exotic is required.

Now ask the same question of a government borrowing in the currency it itself issues. Ask what would have to happen for that promise to be broken. The government would have to be unable to produce units of a currency it produces. The obligation and the ability to create the means of settling it sit with the same body. A government borrowing in its own currency can always meet the promise as written, and that is the whole of why such a borrowing serves as the baseline. The claim is about the mechanics of settlement rather than a compliment to any government.

Say the claim carefully. In careless hands the sentence gets shortened into something false. The claim is not that such a borrowing is a good thing to hold, or that the money will be worth what the holder hoped, or that the issuer is well run. The claim is narrow and mechanical: the amount stated will be produced. The stated amount is all condition one asks for, and it is all condition one gets.

The instrument that satisfies both conditions for a long-lived business is therefore a long-dated government securityA government borrowing repayable many years out, in its own currency., and its yield is the base rate. For Sankalp Industrial Systems Limited this worked example assumes that yield is 7.75 per cent. The figure is an assumption of the example rather than a reading from any market on any date.

What do the two conditions actually disqualify?

Three candidates fall away, and each falls away for its own reason. Naming them one at a time is worth more than waving at them together.

A government borrowing in a currency it does not issue fails condition one, and it fails it in the same way an ordinary borrower does. The obligation is in units the issuer cannot create. The issuer has to obtain them, by earning them or borrowing them, and either route can run dry. The distinction is not academic. A market charges a different rate on the two kinds of borrowing by the same government, and the currency is the only reason it does. If the currency made no difference, there would be nothing to price.

A high grade corporate borrowing fails condition one too, and readers resist this one because the failure is so small. The company is large, the borrowing is senior, nobody expects trouble. But the spreadThe difference between a borrower's rate and the base rate of the same maturity. over the government rate is not zero, and a spread that is small is still a spread. Put it in the baseline and every premium measured on top is measured from the wrong place, by an amount nobody has written down and nobody can now recover. On this invented company the point is easy to see: the blended pre-tax cost of debt is 8.00 per cent against a base rate of 7.75, so the spread the company actually pays is 25 basis pointsOne hundredth of a percentage point.. Small, real, and not nothing.

A bank rate fails for the same reason with an extra wrinkle. A rate at which banks lend to each other is a rate between borrowers who can fail, and the history of the last few decades is largely a history of people rediscovering that. Where such a rate is used as a base, it is being used because it is convenient and available, not because it satisfies condition one.

And a short government instrument passes condition one with room to spare and fails condition two. The short instrument is the trap people walk into while being careful. Maturity matching is taken on its own below.

Why is a rate that is certain for one year not certain for ten?

Back to the household and the school fees. The finance version is the same object wearing different clothes. Roll a one year deposit nine times and eight of those nine renewals happen at a rate nobody knows today. The issuer never fails, every single payment arrives, and the household still cannot say today what it will have in ten years. Reinvestment riskThe chance that money coming back early has to be put to work at an unknown rate. is the name for that, and it has nothing whatever to do with whether anybody defaults.

Condition two is a statement about the length of the cash flows being valued, not about the quality of the issuer. An instrument that passes condition one perfectly can therefore fail condition two outright. A short government instrument is about as free of default risk as anything gets. Over one year its outcome is settled. Over ten it is nine renewals of guesswork stitched together, and a valuation model that runs for ever is asking a question about a great deal more than one year.

A second reason the short instrument tempts people sounds sophisticated, and it is worth naming. The argument goes: the shortest instrument is the purest, having the least of everything else mixed into it. The argument is true and beside the point. Purity over one year is the wrong property. The property the baseline needs is certainty over the period the cash flows actually run, and reaching for the shortest instrument optimises the wrong thing with great care.

Which maturity should the rate come from?

Maturity matchingUsing a rate whose term corresponds to the length of the cash flows being discounted. is the rule, and the rule is decided by the shape of the cash flows rather than by anybody's preference. So look at the shape. Sankalp Industrial Systems Limited is forecast to produce free cash flow to the firm of Rs 98,00,00,000 in Year 1, rising by exactly Rs 18,00,00,000 a year to Rs 1,70,00,00,000 in Year 5, and then to continue in perpetuity. The stream is five explicit years and then for ever.

Discount that stream at 12.00 per cent, this worked example's cost of capital, and the value falls out as follows. The surrounding sequence builds that rate input by input. The five explicit years contribute Rs 4,68,41,43,564 of present value. The terminal value contributes Rs 16,59,72,35,530. The total is Rs 21,28,13,79,094, of which the terminal value is 77.99 per cent.

Nearly seventy eight per cent of this business's value arrives after Year 5, so the stream being discounted is long-duration and the rate must come from a long maturity. Matching a one year yield against that is matching a rate for one year against a claim on cash flows that mostly do not arrive for decades. The mismatch is not subtle and it is not a matter of taste.

Why must the currency of the rate match the currency of the cash flows?

Currency matchingUsing a rate denominated in the same currency as the cash flows. is the second matching rule and it is the one that produces the largest errors in practice. Sankalp Industrial Systems Limited sells valves, castings and aftermarket service, prices them in rupees, pays its people in rupees and reports in rupees. Its cash flows are rupee cash flows. The base rate therefore has to be a rupee rate, and the reason is not tidiness.

A nominal rateA rate that includes expected inflation. carries an inflation expectation inside it. The inflation expectation is what makes the rate nominal. Every economy has its own, so every currency has its own, and a rate in one currency has a different one baked in from a rate in another. Now pair a rate carrying one economy's inflation expectation with cash flows that were grown at a different economy's. The two halves of the calculation now disagree about how fast prices rise, and that disagreement runs the same way in every single year of the forecast and then for ever in the terminal value.

One property makes the error dangerous rather than merely annoying: the error is systematic, not noisy. A noisy error is sometimes too high and sometimes too low, and a longer forecast averages it away. A systematic error has a sign, keeps it, and a longer forecast makes it worse. Here is that shown rather than asserted: take this model, with its 12.00 per cent rate and its 5.00 per cent terminal growth, and suppose the mismatch is worth just one point of rate. The bias in each year is the ratio of the two discount factors, and the bias in the terminal value is larger still because the growth rate is subtracted from a smaller number.

Read the numbers off it. Year 1 is overstated by 0.90 per cent, Year 2 by 1.81, Year 3 by 2.73, Year 4 by 3.65 and Year 5 by 4.59. Not one of them has a different sign from any other. The terminal value holds 77.99 per cent of the answer, and it is overstated by 22.02 per cent. Length is not the friend of this error. Length is what feeds it.

The rule that falls out of the two matching conditions is one line long. The rate must match the currency and the horizon of the cash flows, and nothing else about it is negotiable. If a valuation has to be presented in another currency, the honest route is to convert the cash flows first and then rebuild the entire rate in that currency from its own base rate and its own premiums. No component is carried across. Half a rate from one economy and half from another is not a compromise; it is two models glued together at the seam.

A compact check makes the abstraction concrete. Inflation in this worked example is 5.00 per cent alongside a nominal rate of 12.00 per cent. Take the inflation out properly and the real rate is 1.12 divided by 1.05 less one, or exactly 6.6667 per cent. Take it out by subtraction, the shortcut everybody reaches for, and the answer is 7.00 per cent. The shortcut is 33 basis points too high, and the whole of that gap is the compounding the subtraction ignores. The relationship between nominal rates, real rates and inflation is covered separately. The 33 point gap is what shows that a nominal rate has an inflation expectation genuinely inside it rather than sitting alongside it.

How is the choice actually made, in what order?

Put the two matching rules and the default condition into a sequence and something useful happens: the order itself prevents the two errors that actually occur. Ask which currency first. No later choice of maturity can repair that decision. Ask how long the cash flows run second. The answer fixes the maturity. Ask about the issuer third, by which point the question is narrow enough to have one answer.

Answer those three questions in that order and the choice of rate is settled before anybody looks at a single yield. The discipline is exactly what stops a rate being chosen because it happened to be the one on the screen.

Why use one rate rather than the whole curve?

Here is an objection a careful reader raises at about this point, and it is a good one. A government does not borrow at one rate. A government borrows at many rates across many maturities, and the set of those yields is a yield curveThe set of yields on the same issuer's securities across different maturities.. Strictly, then, each year of cash flow should be discounted at the rate for its own maturity: Year 1 at the one year rate, Year 5 at the five year rate, and so on outward.

The objection is correct, and almost nobody discounts that way. Using a single long rate for the whole model is a convention rather than the right answer. A convention is worth keeping when what it buys is worth more than what it costs, so both sides deserve to be stated.

The convention costs precision, and only where the curve is steep. If short rates sit well below long ones, a single long rate discounts the near cash flows at a rate that is too high and the far ones at a rate that is too low. Where the curve is nearly flat the whole objection evaporates. There is only one rate to speak of anyway.

The convention buys two things that matter more than they sound. The first is comparability. Two models built on one rate each can be set beside each other and argued about. Two models built on two different sets of thirty rates cannot. The second is arguability. A reader can look at a single number, disagree with it, and say by how much. Bury the rate in a term structure and the reader has nothing to push against. In a subject whose whole purpose is to make assumptions visible, that is a real loss.

What does the base rate constrain further down the model?

The base rate does one more job that nobody advertises, and it catches models out constantly. The base rate puts a ceiling on the growth rate a model may assume for ever.

The reasoning is not financial at all; it is arithmetic about compounding. If a business is assumed to grow at some nominal rate for ever, and that rate sits above the nominal rate on a default-free security in the same currency, then the business is being assumed to compound faster than the economy it sits inside, permanently. Run that forward far enough and the business is the economy, then larger than it. No forecast means to claim that. Most forecasts that do claim it have simply never checked. The nominal base rate is a usable ceiling for a growth rate assumed to run for ever, and checking a terminal growth assumption against it takes about four seconds.

On this worked example the check passes comfortably. Terminal growthThe rate at which cash flows are assumed to grow for ever after the explicit forecast. is 5.00 per cent against a base rate of 7.75, leaving 2.75 points of headroom. The headroom is a constraint being met rather than a formality being observed.

Two figures in this worked example are the same, and the match is not a typing error. Assumed inflation is 5.00 per cent and terminal growth is also 5.00 per cent, so the model assumes no real growth at all after Year 5. What that assumption implies for the terminal value is covered separately.

How much of this company's cost of equity is the base rate?

The number now goes where it belongs. For Sankalp Industrial Systems Limited the cost of equity is built as the base rate plus the levered beta times the total equity risk premium: 7.75 plus 1.25 times 5.00, giving exactly 14.00 per cent. Every one of those inputs is this worked example's assumption, and how each is estimated is covered separately.

Split the 14.00 into what came from where and the result is worth sitting with. The base rate contributes 7.75 points, or 55.36 per cent of the whole. Everything else contributes 6.25 points, or 44.64 per cent. More than half of this company's cost of equity is the baseline, before a single unit of risk has been priced. The baseline carries the most of the answer and gets the least of the attention in a discussion.

Break the 6.25 down further and the imbalance sharpens. At a beta of one, the mature market premium would contribute 3.50 points and the country premium 1.50. The beta contributes the difference between a beta of one and the actual 1.25, being 0.25 times the 5.00 per cent premium, or 1.25 points. People argue about the beta hardest and longest. The most contested input in the build is the smallest slice in the picture.

What does one point of base rate do to the answer?

Take the cost of equity first. The answer there is easy and slightly surprising. Move the base rate by one point and the cost of equity moves by exactly one point. The base rate enters as a straight addition, not as something the beta multiplies, so it passes through one for one. The base rate is therefore the only input in the whole build whose effect a reader can compute without knowing the beta at all.

Now the blend, where there are two honest answers. Two different experiments sit behind them, and people run them without saying which.

Hold the cost of debt still and one point of base rate moves the weighted average by 0.75 of a point. Only the 75.0 per cent equity weight carries the change. That is the first experiment, and it is the one most sensitivity tables silently run.

Let the cost of debt move with the base rate and the answer changes. Letting the cost of debt move is the more realistic experiment. A company's borrowing cost is a spread over the same base rate rather than a number floating free of it. On this company the pre-tax cost of debt of 8.00 per cent sits 25 basis points above the base rate of 7.75, so the spread is small and stable enough to hold constant while the base moves. If the pre-tax cost of debt rises one for one with the base rate, the after-tax cost rises by 0.75 of a point at this company's own assumed effective tax rate of 25.0 per cent, and the debt side contributes 0.25 times 0.75. Add that to the equity side and the blend moves 0.75 plus 0.1875, being 93.75 basis points.

Neither figure is more correct than the other. The two figures answer different questions, and the whole of the discipline is saying which question is being answered in the same breath as the number. A sensitivity table that reports one of them without saying which has reported an ambiguity with two decimal places on it.

Why is moving the base rate not the same experiment as moving the premium?

Both inputs move the answer, so it is tempting to treat a point of base rate and a point of premium as interchangeable levers. The two levers are not interchangeable, and the difference changes what a sensitivity result means.

A point of base rate is a point of base rate. A point of base rate arrives undiluted in the cost of equity, and in the realistic version it pushes the borrowing cost along with it. The borrowing cost is defined relative to the base rate. A point of premium behaves quite differently: the beta multiplies it, so at a beta of 1.25 one point of premium becomes 1.25 points of cost of equity, and it touches the debt side not at all.

A base rate change moves both sides of the capital structure and a premium change moves only one. The two cannot be compared point for point, even though both are quoted in points. There is a second difference, and it is about where the number comes from rather than what it does. A base rate is observable as a yield, so two careful people looking at the same market on the same day will differ about which point on the curve to take and not much else. A premium is estimated, and reasonable methods disagree about it by more than a full percentage point. The input that carries more of the answer is the one there is less to argue about, and the input everybody argues about is the smaller half.

Where would a reader find a current figure?

Everything above uses 7.75 per cent, described each time as this worked example's assumption. The repetition is not throat-clearing. A yield is a live thing. A yield moves during a session, it differs between maturities, and a figure written down in one month is a plausible-looking falsehood by the next. A printed current yield would be wrong within days and would look right for years, and those two together are the worst combination a stated figure can have.

A current figure lives elsewhere. Aswath Damodaran publishes and maintains material at pages.stern.nyu.edu on estimating any cost of capital input, including a current base rate for a given currency, and that material supplies a figure to use rather than a figure to learn from. For the government securities themselves, the arrangements through which they are issued in India and their market overseen involve the Reserve Bank of India at rbi.org.in, the authority to read for what exists, on what terms and with what tenures.

The base rate is not really a number to be memorised. The base rate is a test to run on whatever number somebody hands over: which currency is it in, what maturity does it come from, can the issuer produce that currency, and does all of that match the cash flows sitting in the model. A reader who can run those four checks on somebody else's rate has the whole of it.