Equity Risk Premium: How It Is Estimated and Why Estimates Differ

An equity risk premium is the extra annual return equity buyers demand before they will hold shares instead of the safest paper a state issues. The premium prices equity risk across a whole market, before any one company's beta scales it. In this worked example the mature market half is 3.50 per cent and the country half is 1.50, so the total is 5.00, and every one of those three figures is an assumption.

Everything else in a cost of capital can be argued from a document. The equity risk premium cannot, and that single fact shapes every argument anyone will ever have about it. A yield is printed on a screen. A coupon is typed into a loan agreement and signed. A betaA measure of how far a company's equity return tends to move when the whole market's return moves by one point. Where it is measured from is a separate subject. is estimated from prices that actually traded. The premium is a claim about what investors require before they will hold equities at all, and nobody anywhere transacts in a requirement. There is no screen showing it. Every route to it is an inference, and the three serious routes infer it from three different things.

What is the premium actually the price of?

Consider a shopkeeper who keeps some money in a bank deposit and lends the rest to a cousin who is expanding a workshop. He will not lend to the cousin at the deposit rate. He wants something extra, and asked how much extra he will say something vague about it not being the same thing at all. He cannot give a number. He is nonetheless pricing risk every time he decides, and the gap between what he accepts from the bank and what he demands from the cousin is his own premium, revealed by behaviour rather than stated.

An equity risk premium is that same gap, measured across a whole market rather than across one shopkeeper. It is the extra annual return the buyers of shares in aggregate demand for holding equities instead of the safest paper the state issues. The premium is not a company's number and not an industry's number. One figure applies to the whole equity market at once, and it enters a company's cost of equity only after being scaled by that company's beta.

Scaling by beta is what makes the arithmetic feel counter-intuitive at first. A single market-wide premium of 5.00 per cent does not put 5.00 points into everybody's cost of equity. A business whose equity moves less than the market takes less of it; a business whose equity moves more takes more. Sankalp Industrial Systems Limited, an invented manufacturer, carries a levered beta of 1.25, so the 5.00 per cent arrives as 6.25 points. Multiplication, not addition.

Why can nobody just look it up?

Take the four inputs that build this manufacturer's cost of equity and its borrowing cost, and sort them by how much reading is involved. The 7.75 per cent base rateWhatever a build starts from before any risk is priced. The base rate stands here for what long-dated government paper is assumed to pay, and which paper qualifies is settled elsewhere. is a quoted yield: somebody publishes it, and two analysts reading the same screen get the same figure. The 8.00 per cent pre-tax borrowing cost is the blend of three contracted coupons, and the contracts exist as documents with signatures on them. The levered beta of 1.25 is inferred rather than read, but what it infers is something that already happened, pulled out of prices people really paid.

The premium is the only one of the four with nothing behind it to read, and it is also the one carrying the most weight in the answer. Of this manufacturer's 14.00 per cent cost of equity, 7.75 points come from the base rate and 6.25 points come from the premium after the beta has scaled it. The single input nobody can observe carries not far off half the number.

An unobservable input carrying nearly half the answer is uncomfortable, and it should be. The discomfort is why a cost of capital quoted as a bare figure with two decimals is a stronger claim than its inputs support, and why the honest form of the output is a range with the assumption behind each end of it named.

Route one: what does a historical premium measure?

The oldest route rests entirely on things that actually happened, so it is also the easiest to defend in a room full of people. Take a long run of equity market returns. Take the return on government securities over exactly the same run. Subtract the second from the first, year by year, and average the differences. The average that comes out is the amount by which equities did beat the safe alternative over that stretch of history.

The appeal is obvious. Nobody has to forecast anything. Nobody has to argue about what investors are thinking. The inputs are published series that anyone can pull, and the arithmetic is subtraction and division. The trouble is that the route is not one calculation at all: it is a chain of three choices, and each of the three moves the answer by an amount that matters.

Which three choices sit inside that one number?

Choice one is the window. A premium measured across the longest run anybody has data for and a premium measured across the last twenty-five years are different numbers, and neither window is obviously correct. The long one includes economies, industries and market structures that no longer exist in any recognisable form. The short one contains so few independent observations that the average it produces barely constrains anything. Whichever window is chosen, somebody sensible will choose the other.

Choice two is the averaging method, and this one catches people who have never thought about it. The arithmetic mean of a run of annual excess returns is always higher than the geometric mean of the same run, and the gap between them widens with the volatilityHow widely a series of returns swings around its own average from one period to the next. Wide swings mean the two ways of averaging drift further apart. of the series. Equity returns swing a lot, so on equity data the gap is not a rounding difference. The arithmetic mean answers a question about a typical single year: how large has one year's excess return been on average. The geometric mean answers a question about a whole holding period: at what rate did somebody who stayed in the entire time actually compound. Both are correct answers. The two means answer different questions, and a premium quoted without saying which one it is has not yet said what it means.

Choice three is which government security the comparison was made against, and it is the choice that reaches forward into the rest of the build. A premium measured against short treasury billsGovernment paper maturing within a year. A bill is sold below face value and repaid at face value, so the difference stands in for interest. The safest paper is identified under the risk-free rate. is larger than one measured against long government bonds, for the plain reason that bills earned less over the same period. Same equity returns, lower comparison, bigger gap. Nothing has been discovered; the measuring stick was changed.

Does a longer window fix it?

The instinct is to reach for more data. If twenty-five years is too few, take a hundred. The instinct is right in direction and much weaker in effect than people expect, and the reason is worth carrying around.

Equity returns are volatile enough that the standard errorA measure of how much an average would move around if the sample could be collected again. A large one says the average at hand is a loose reading rather than a sharp one. around an average excess return stays large relative to the average itself, even across very long runs. Adding decades shrinks it, but slowly, and the series never becomes long enough for the arithmetic to become tight. The honest reading is that a historical premium is a wide estimate that happens to be printed as a narrow one.

None of that is an argument against the route. The width of that estimate is the argument against pretending the route delivers precision. Somebody who quotes a historical premium to two decimals is making a claim the underlying data cannot support, and the correct response is not to reject the figure but to ask what a different setting of the three choices would have given.

Route two: how does the implied premium run the valuation backwards?

The second route throws the history away entirely. The implied route does not ask what equities earned, but what buyers are currently being paid, given what they are currently paying.

Here is the move. A valuation normally starts with a discount rate, applies it to a forecast of cash flows and produces a value. Run that machinery in reverse. Take today's level of a market indexA single running number built from a basket of listed shares, used as a stand-in for the whole market. The idea of a market portfolio and why it matters are set out separately. as the value, take a forecast of the cash the constituents will hand back to their owners, being dividends together with buybacks, and solve for the discount rate that makes the two sides equal. Subtract the base rate from that solved rate and what remains is the premium the market is pricing at this moment. Aswath Damodaran publishes a premium on exactly this basis, at pages.stern.nyu.edu, and the argument for the route is his.

Two properties follow immediately. The route uses a forecast and today's price rather than any past, so it is forward-looking. And it moves as prices move. A sharp fall in the index, holding the cash flow forecast still, raises the implied premium the same day. A price of risk should behave in exactly that way.

Now the part that keeps the route honest. An implied premium depends entirely on the cash flow forecast fed into it, and above all on the growth rate assumed beyond the explicit forecast. Nudge that growth assumption and the solved rate moves with it, and so does the premium left over after the base rate is removed. The route has not escaped assumption; it has swapped a choice about which slice of history to use for a choice about how fast the cash handed back to owners will grow. A growth assumption is genuinely different and often more comfortable to defend, but an assumption is what it remains, and two careful people running the identical route on the identical day will still disagree if their growth inputs differ.

Route three: what does a survey actually measure?

The third route does the simplest possible thing. It asks. Put the question to fund managers, or to chief financial officers, or to finance academics, collect the answers and report the middle one.

Being sniffy about a survey is easy, and a mistake. The survey route is the only one of the three that measures what people actually believe, and when the task at hand is explaining a decision somebody already made rather than valuing a business, belief is precisely the right input. If a board approved a project at a hurdle rate, the premium sitting inside that hurdle rate is the one the board believed in, not the one an index implies.

The objections are well known and each is specific. The answer depends heavily on who was asked, and the three groups above do not give the same number as each other. The dispersionHow far apart the individual answers in a set sit. Two sets can share the same middle answer and still disagree about almost everything. across respondents inside any one group is wide, so the middle answer conceals a great deal of disagreement. And the answers drift with recent returns: the same survey taken after a good stretch reports a higher number than one taken after a bad stretch, even though nothing about the long-run riskiness of holding equities has changed in between.

Why do three careful people get three different answers?

One sentence carries the whole argument. The three routes disagree because they are answering three different questions, and their disagreement is information about the market rather than evidence that two of them are broken.

Read them side by side and the point becomes almost obvious. The historical route asks what equities earned. The answer is a statement about a stretch of the past, and it is as good as that stretch is representative. The implied route asks what buyers are being paid today, given what they are paying. The answer is a statement about now, and it is as good as the forecast behind it. The survey route asks what people say they expect. The answer is a statement about belief, and it is as good as the sample and as fresh as the date it was taken.

Three questions, three objects, three answers. Nobody is wrong. Somebody who reports all three and notes that the implied figure sits well below the historical one has said something real about how the market is currently pricing, and has said it in a way that no single number could.

Which premium goes with which base rate?

The pairing rule is the one rule here with no judgement in it at all. The rule is a single question with two branches and no third option.

A premium and a base rate must be measured against the same security. If the premium was measured against short treasury bills, the base rate is the bill rate. If the premium was measured against long government bonds, the base rate is the long bond rate. The premium was defined as a gap relative to something specific; it can only be added back onto that same something.

In this worked example the 3.50 per cent mature market premium is stated as being measured against a long-dated government security, and the 7.75 per cent base rate belongs to that same long-dated paper. Leaving the pairing to be assumed is exactly how the failure below happens, so the pairing is stated outright.

There is a second failure in this territory and it looks far more respectable than the first. Faced with a historical figure, an implied figure and a survey figure, somebody takes the middle one and calls it balanced. The three arguments behind those figures were answers to different questions, so averaging the three routes produces a number that none of the three can defend. The disciplined move is the plain one: pick a route, name it, state the date, and show the reader what a different choice would have done to the answer.

How much of a valuation rests on which premium was chosen?

Abstract disagreement is easy to nod along with. Traced through to the answer, it stops being abstract very quickly.

Follow a single point of mature market premium through this manufacturer's build. One extra point of premium raises the total premium by one point. The levered beta of 1.25 scales that total, so the cost of equity rises by 1.25 points. Equity is three quarters of the capital at market, so the cost of equity is weighted at 75.0 per cent and the weighted average rises by 0.9375 points. One point of argument about an unobservable input is 93.75 basis points of the rate that discounts every future rupee the business produces.

Set that against the thing this manufacturer's model is actually being asked to explain. The discounted cash flow produces an enterprise value of Rs 2,128.14 crore. The traded enterprise value, built back out of the share price, is Rs 2,240.00 crore. Holding the growth assumption still, closing that gap requires the weighted average to be 11.67 per cent rather than 12.00. The two rates sit 33 basis points apart. Working backwards through the same chain, 33 basis points of rate is roughly 35 basis points of mature market premium: a premium of about 3.15 per cent in place of 3.50 would have made the model reproduce the traded figure exactly. Both of those last two figures are derived from a rate the record carries rounded to two decimals, so read them as close rather than as exact.

One point of disagreement about the premium is 2.84 times the size of the entire gap between what the model says and what the market says. That is the number to carry away. The input nobody can look up moves the answer by nearly three times as much as the discrepancy everybody argues about.

What does the whole range of defensible premiums do to the rate?

Serious estimates of a mature market premium do not span an unlimited range, but they span a wide enough one that the choice is not a detail. Take a span from 2.50 to 5.50 per cent, hold every other input in this manufacturer's build still, and watch what happens to the answer.

Because only one input is moving, the relationship is a straight line, and the slope of that line is the 0.9375 traced through in the block above. Every 0.10 added to the premium adds 9.375 basis points to the weighted average. Across the whole span the rate moves 2.8125 points, from 11.06 to 13.88 per cent. The span produces nearly three points of discount rate out of moving one number that nobody can observe.

The ladder below is the same arithmetic without the control, so a reader who never touches the slider still has the argument in front of them. The last column is worth reading down rather than across: seven defensible discount rates for one unchanged business.

All figures in per cent. The highlighted row is this worked example's locked build. Half a point of premium is half of the 93.75 traced through above, so between any two adjacent rows the weighted average moves by 46.88 basis points.

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