Equity Research puzzles, solved step by step

EPS rose to Rs 16.1 and the P/E rose from 20x to 27.9x, so earnings explain about 59% of the gain and re-rating the rest. The price multiplied by 2.25. EPS multiplied by 1.1⁵ = 1.611, and the multiple by 1.397; the two multiply to 2.25. A year, that is 17.6% total: 10% from earnings and 6.9% from the multiple.

Why split a return into earnings and multiple at all?

A house bought for Rs 50 lakh sells for Rs 1 crore. Part of the gain came from the rent rising, part from buyers being willing to pay more years of rent for the same flat. A share price is EPS times P/E, so any change in price is a change in earnings multiplied by a change in the multiple. The split matters because earnings growth can continue while the business grows, but a multiple cannot rise for ever.

Does the rupee split depend on which step you take first?

Yes, and that is worth saying before the interviewer does. Because the effects multiply, the cross term goes to whichever leg is taken second, so a rupee split is a convention, while the log split of 59% and 41% is not. The analyst's conclusion: 41% of this return came from investors paying more for each rupee of profit. If the multiple drifts back to 20x, a holder earns only the EPS growth, and the next five years look very different from the last.

EPS and net worth fall; cash and net debt to EBITDA do not move. Assuming no tax deduction for the charge, reported profit falls by Rs 200 crore, taking EPS from Rs 5.00 to Rs 3.00 on 100 crore shares, and net worth falls by Rs 200 crore. No cash leaves, and EBITDA sits above the charge, so a covenant at 2.5x net debt to EBITDA is untouched. A covenant set on debt to equity would move.

Why does a Rs 200 crore loss leave cash untouched?

Imagine you paid Rs 20 lakh for a car three years ago and a valuer now says it is worth Rs 12 lakh. You are poorer on paper, but your bank balance did not change today; the money left when you bought it. A goodwill impairment admits that an acquisition was worth less than was paid, and the cash for that acquisition left the business when the deal closed. The goodwillThe part of an acquisition price above the fair value of the net assets bought, carried as an asset on the balance sheet. is written down, reported profit takes the charge and equity falls with it.

Which covenant does move, and why does the difference matter?

A covenant built on EBITDA is blind to the charge. A covenant built on net worth or on debt to equity is not: gross debt of Rs 1,300 crore against equity of Rs 2,000 crore is 0.65x, and after the write-down it is 0.72x. So the same accounting entry can be harmless under one loan and a breach under another. An analyst reads the covenant definitions before saying which.

Two more things to say. Most analysts strip the charge out of adjusted EPS, which stays at Rs 5.00, because it says nothing about next year's earnings. And whether the charge saves tax depends on local rules, which usually do not allow a deduction for goodwill impairment; confirm before assuming either way. The real signal is about management: the acquisition is now expected to earn less than was paid for it.

A levered beta of 1.10. Relever with levered beta = unlevered beta x (1 + (1 - tax rate) x debt/equity). That is 0.8 x (1 + 0.75 x 0.5) = 0.8 x 1.375 = 1.10. The 0.8 is the risk of the business itself; the extra 0.3 is the financial risk shareholders take on because lenders are paid first.

Why does debt raise the equity's beta?

Think of two families with the same salary. One has a home loan EMI to pay first each month; the other has none. A 10% pay cut hurts the family with the EMI far more, because the EMI does not shrink. Lenders are paid a fixed amount first, so the same swing in the business moves the shareholders' leftover by more, and beta measures exactly that swing. The business risk is the 0.8; debt stacks financial risk on top.

Where do candidates go wrong with the inputs?

Two places. Debt to equity uses market values where you can, and it is debt over equity, not debt over total capital: 0.5 debt to equity is one third debt in the capital structure. Using 0.33 by mistake gives 0.8 x (1 + 0.75 x 0.33) = 1.00. And the tax rate is the one that applies to this company's interest deduction; confirm the current rate rather than assuming it.

Say the limitation. The formula assumes the debt itself carries no market risk and that the debt level stays constant. For a heavily indebted company, debt starts to behave like equity and the simple formula overstates the levered beta.

0.648. At least two means exactly two or all three. Exactly two can happen three ways, each with chance 0.6 x 0.6 x 0.4 = 0.144, so 0.432 together. All three beat with 0.6 cubed, 0.216. Adding them gives 0.648. The complement checks it: none beat 0.064 plus exactly one 0.288 is 0.352, and 1 minus 0.352 is 0.648.

How do you avoid missing an outcome?

Think of three friends each deciding whether to come to dinner. At least two coming covers four different guest lists: each of the three possible pairs, and all three. List the outcomes by how many succeed, count the ways each count can happen, and multiply by the chance of one such way. With three stocks there are only eight results, so write them down; with more, the count of ways is a binomial coefficientThe number of ways to choose k items out of n, written n choose k; here 3 choose 2 is 3..

Is independence a fair assumption for three stocks you cover?

Usually not, and saying so is the part that sounds like an analyst. Stocks in one sector share demand, input costs and the same consensus-setting habits, so beats tend to cluster: when one beats, the others are more likely to. Clustering fattens both ends. The chance of all three beating and of none beating both rise above 0.216 and 0.064, and whether at least two beat can move either way, so the independent answer is a starting point, not a forecast.

About Rs 69 crore of revenue and Rs 2.2 crore of gross profit a year, on stated assumptions. Assume 12,000 vehicles a day pass on the pump's side, about 500 an hour. About 384 stop, filling 20,232 litres a day, most of it diesel for trucks. At assumed pump prices near Rs 90 to Rs 100 a litre that is Rs 18.8 lakh a day; at an assumed dealer margin of Rs 3 a litre the pump keeps about Rs 2.2 crore a year.

Where does the estimate start?

Think of a roadside dhaba: its takings depend less on how many vehicles pass than on which ones stop and how much each group eats. Start from the traffic on the pump's side of the road, split it by vehicle type, and apply a stop rate and litres per fill to each, because the three types differ by a factor of forty in what they buy. Every input below is an assumption for the method; fuel prices and dealer margins change and should be checked before use.

How do you turn litres into revenue and profit, and check the answer?

Revenue is litres times the pump price: petrol for two-wheelers and cars at an assumed Rs 100, diesel for trucks at Rs 90, which gives Rs 18.8 lakh a day and Rs 68.6 crore a year. The dealer does not keep the pump price; it earns a commission per litre, so gross profit is litres times margin, about Rs 2.2 crore a year at Rs 3 a litre. Sanity check the stops: 384 a day is about 16 an hour, one fill every four minutes, which a pump with a few nozzles handles easily. The weakest assumption is the truck stop rate, since fleet operators choose pumps by contract and credit terms, not by chance.

A implies a 20% margin and B 16.7%; B is cheaper for the margin you get. EV/sales divided by EV/EBITDA is EBITDA over sales, so A is 3 / 15 and B is 2 / 12. A pays 1.5 times as much per rupee of sales, but its margin is only 1.2 times B's. The rest of A's premium is a higher price per rupee of EBITDA, 15x against 12x, which has to be earned by faster growth or better quality.

How do two multiples reveal a margin?

If a flat costs Rs 60 lakh, which is 20 years of rent, and 3 times the owner's salary, you can work out that the rent is a fifteenth of the salary without seeing either. Two multiples on the same numerator divide to give the ratio of their denominators, so EV/sales over EV/EBITDA is EBITDA over sales, the margin. The enterprise value cancels, and the market has told you what margin it is capitalising.

So which is cheaper?

Split A's premium. A trades at 1.5 times B's sales multiple, and that decomposes exactly into 1.20 times the margin and 1.25 times the price per rupee of EBITDA. The margin explains part of the premium; the rest is the market paying 15x rather than 12x for each rupee of profit. On EBITDA, B is cheaper by a fifth. That does not make B the better stock: if A grows faster, converts more EBITDA to cash or carries less risk, the higher multiple may be deserved. The next question to ask is which of those it is.

Say the limitation: the implied margin is only as good as the EBITDA in the multiple. If one company's EBITDA is a forecast and the other's is last year's, or one capitalises costs the other expenses, the comparison is off before you start.

About 6.9% a month, and a 95% one-month value at risk of about Rs 1.13 crore. Variance adds across independent days, so volatility scales with the square root of time: 1.5% x the square root of 21, which is 4.58, gives 6.87%. The one-tailed 95% point of a normal distribution is 1.645 standard deviations, so a bad month costs 1.645 x 6.87% = 11.3% of Rs 10 crore, about Rs 1.13 crore.

Why does volatility grow with the square root of time?

Think of someone taking random steps left or right along a lane. After 100 steps they are not 100 steps from where they began; many steps cancelled, and the typical distance is about 10 steps, the square root of 100. Daily returns behave the same way when one day does not predict the next. Variances add across days, so the standard deviation, which is the square root of the variance, grows with the square root of the number of days.

How do you turn a volatility into a rupee value at risk?

Value at riskA loss threshold that a position should exceed only with a stated small probability over a stated period, such as 5% over one month. answers one question: how much would a bad month cost, where bad means the worst month in twenty. Under a normal distribution 5% of outcomes fall below the mean minus 1.645 standard deviations, so the 95% one-month VaR is 1.645 x 6.87% = 11.3% of the position. On Rs 10 crore that is about Rs 1.13 crore. Assume a zero expected return for the month; over one month the drift is small against the volatility.

What does the number leave out?

Two things, and saying them separates an analyst from a calculator. The square root rule needs independent days; when bad days cluster, as they do in a sell-off, real monthly risk is higher than 6.9%. And VaR marks the edge of the tail, not what sits inside it: it says nothing about how bad the worst 5% of months can get. Real returns have fatter tails than the normal curve, so a one-in-twenty month can cost well over Rs 1.13 crore.

EPS rises 13.75%, from Rs 5.00 to about Rs 5.69. The cash was earning 6% before tax, Rs 60 crore, or Rs 45 crore after tax, so net income falls to Rs 455 crore. Rs 1,000 crore at Rs 50 retires 20 crore shares, leaving 80 crore. Rs 455 crore over 80 crore shares is Rs 5.69. The buyback helps because the earnings yield on the shares, 10%, beats the 4.5% after-tax yield on the cash.

Why does a buyback change EPS at all?

Four friends own a small shop and split its profit. One sells out to the other three, paid from the shop's savings account. Profit dips a little, because the savings no longer earn interest, but it is now split three ways instead of four. A buyback lowers earnings by the lost return on the cash and lowers the share count by the shares retired; EPS rises when the second effect is larger. Here the share count falls 20% and net income falls only 9%.

When does a buyback lift EPS?

Compare two yields. The earnings yieldEPS divided by the share price: the earnings each rupee spent on the shares buys. It is one over the P/E. is Rs 5 over Rs 50, 10%: what each rupee spent on shares buys in earnings. The after-tax yield on cash is 6% x (1 - 25%), 4.5%: what each rupee was earning where it sat. A buyback lifts EPS whenever the earnings yield beats the after-tax yield on the cash spent, which is the same as a P/E below one over that yield. Here the break-even P/E is 22.2x, so the same buyback would leave EPS flat at a share price of about Rs 111.

Does higher EPS mean each share is worth more?

Not by itself. EPS rose 13.75%, but the company also handed out Rs 1,000 crore of cash, and a buyback at a fair price leaves each remaining share worth the same. Before, the equity was worth Rs 5,000 crore over 100 crore shares; after, Rs 4,000 crore over 80 crore shares, still Rs 50 each. Holders own a bigger slice of a company that has less cash and so carries a little more risk, which is why the P/E often slips when EPS is lifted this way.

About Rs 61 crore. A perpetuity of Rs 10 crore at 10% is worth Rs 100 crore. A perpetuity that starts only after year 10 is worth that Rs 100 crore discounted back ten years; 1.1 to the power 10 is about 2.59, giving Rs 38.6 crore. The ten-year stream is the difference, Rs 100 crore less Rs 38.6 crore, about Rs 61.4 crore, or roughly six years of undiscounted cash.

Why subtract two perpetuities?

Think of a shop lease that pays you rent for ten years. It is the same as a lease that pays you forever, minus a lease that starts in year eleven and pays you forever after. A ten-year annuity equals a perpetuity starting now less a perpetuity starting in year eleven, and a perpetuity is one division: payment over rate. That turns an awkward ten-term sum into two numbers you can hold in your head: Rs 100 crore, and the same Rs 100 crore pushed ten years into the future.

How do you get 1.1 to the power 10 in your head?

Square, don't multiply. 1.1 squared is 1.21; squared again is about 1.46 for four years; squared again is about 2.14 for eight years; one more 1.21 for years nine and ten gives about 2.59. Three squarings and one multiplication get you to 1.1^10 = 2.59, and 100 divided by 2.59 is about 38.6. A second route checks it: the rule of 72 says 10% doubles money in about 7.2 years, so ten years is a little more than one doubling, around 2.6.

What sanity checks can you say out loud?

Bracket it first. Every payment is worth between Rs 3.86 crore, the year ten payment, and Rs 9.09 crore, the year one payment, so the total sits between Rs 38.6 crore and Rs 90.9 crore. A tighter check treats all ten payments as arriving at the midpoint, year 5.5: Rs 100 crore over 1.1^5.5 is about Rs 59 crore, close to 61.4. The midpoint shortcut always comes out a little low, because discounting curves, which is itself a useful thing to say.

Pull again straight away: you survive with probability 3/4, against 2/3 after a fresh spin. The four empty chambers sit together in a row and the cylinder advances one chamber per pull. The click you heard came from one of those four, each equally likely. Only one of them, the last empty chamber before the bullets, is followed by a bullet, so the chance of dying without a spin is 1/4. A fresh spin puts you back at two bullets in six.

Why does the first click carry information?

Picture a street with four dark houses in a row, then two lit ones, laid out in a circle. You are told you are standing at a dark house and that you will walk to the next door. Three of the four dark houses have another dark house next door. The click tells you that you are somewhere in the run of four empty chambers, and because the bullets sit together, most of that run is followed by another empty chamber. A fresh spin throws that information away.

Would the answer change if the bullets were not next to each other?

Yes, and saying so shows you understand the mechanism rather than the trick. Put the bullets in chambers 1 and 4. The empty chambers are 2, 3, 5 and 6, and two of them, 3 and 6, sit in front of a bullet. With the bullets apart, pulling straight away survives only 50% of the time, so a fresh spin at 66.7% becomes the better choice. The adjacency is what makes the empty chambers cluster, and the clustering is what makes the click informative in your favour.

Why would an investing firm ask this?

Because it is a small version of the question every analyst faces after new information: does what just happened change the odds of what comes next? An outcome you have already seen is data about the state of the world, and ignoring it in favour of the unconditional average is a common and costly habit. Say that link in one line after the numbers.