Equity Research puzzles, solved step by step

About Rs 50,000 crore a year on these assumptions, with the two routes landing within 5% of each other. Bottom up, urban repaints give Rs 35,000 crore, rural repaints Rs 9,000 crore and new homes Rs 7,200 crore, Rs 51,200 crore in all. Top down, 140 crore people spending Rs 350 a head gives Rs 49,000 crore. Every input is an assumption to be checked against published data.

Where do you start the bottom-up build?

Start from what gets painted, not from who sells paint. A home is repainted every few years, and each repaint uses a number of litres set by the wall area. So the market is homes, divided by the repaint cycle, times litres per job, times the price of a litre, plus the new homes painted for the first time. Split urban and rural before you multiply anything, because they differ on every input: bigger walls, shorter cycles and costlier emulsions in cities; smaller homes, longer cycles and cheaper finishes in villages.

State each number as an assumption and move on. Assume 140 crore people in about 30 crore households, a third urban. Urban homes are repainted every five years, with 3,000 square feet of wall and ceiling; rural homes every eight years, with 1,200 square feet. A litre covers about 60 square feet for a two-coat finish, and a litre costs a blended Rs 350 in cities and Rs 180 in villages. About 60 lakh new homes a year take 40 litres each at Rs 300. None of these is a published figure; each is a round number you can defend in one sentence and replace later. Confirm household counts against the latest census or survey data.

How does a top-down check work, and why does it help?

Go the other way with a number you can feel. A family of four that repaints a flat for about Rs 7,000 of paint every five years spends Rs 1,400 a year, which is Rs 350 a head. Across 140 crore people that is Rs 49,000 crore. Two routes built from different inputs that land within a few percent of each other are worth more than one route worked to the last rupee. If they had landed a factor of two apart, you would know one assumption was wrong and go looking for it.

Then name your weakest input. Urban repaints are about 68% of the answer, so the urban cycle carries the most risk: at six years instead of five, the urban figure falls to Rs 29,167 crore. An analyst would close by saying which number to check first, and where: the decorative revenue that listed paint makers publish in their annual reports is the natural cross-check.

Rs 2.16 lakh on price alone and Rs 2.84 lakh on total return. Price compounds at 8% a year: 1.08 to the tenth is 2.16. Reinvested dividends lift the yearly return to 11%, and 1.11 to the tenth is 2.84. The gap of about Rs 68,050 is more than the Rs 30,000 that ten years of 3% seems to promise, because the dividends compound too.

Why is the answer not simply 8% growth plus 3% times ten?

Think of a bank fixed deposit with two options: interest paid out every quarter, or interest added to the deposit. The cumulative option ends with more money because each quarter's interest starts earning interest. A reinvested dividend is the cumulative option: it buys more shares, and those shares rise in price and pay dividends of their own. Adding 3% a year for ten years treats the dividend like the payout option on a fixed base, which undercounts twice: the base keeps rising and the reinvested money keeps compounding.

Where does the gap come from, piece by piece?

Split it into two layers. Dividends taken in cash are 3% of a price that grows 8% a year, so they start at Rs 3,000 and end near Rs 6,000, adding to about Rs 43,460 over ten years. Reinvesting them adds a further Rs 24,590, the return earned on dividends already received. So the Rs 68,050 gap is roughly two thirds the dividends themselves and one third compounding on them, and the second layer grows fastest in the later years.

What does this change about how you compare stocks?

Compare on total return, always. A high-yield stock with slow price growth can beat a faster grower on the number an investor keeps, and a price chart alone hides that. Index providers publish total return versions of their indices for exactly this reason. The limits are worth one sentence: tax on dividends leaks some of the gap, and reinvesting assumes you can buy at a fair price each year.

About 9.7% a year, forever. For a bank, justified price to book is ROE minus growth over cost of equity minus growth. Setting 2.5 equal to (18% minus g) over (13% minus g) gives g of 9.67%. With no growth the bank would deserve 1.38x, so about 45% of today's price is paying for growth, which needs 54% of profits retained every year.

Why does price to book depend on ROE against cost of equity?

Picture a shop that earns Rs 18 a year on every Rs 100 its owner has put in, when the owner could earn Rs 13 elsewhere for the same risk. Nobody would sell that shop for Rs 100. A bank's book value is the owners' capital, so a bank earning more on it than shareholders demand is worth more than its book, and one earning less is worth less. With no growth, every Rs 100 of book produces Rs 18 a year forever, worth 18 over 13, or {pb0_80:.2f} times book.

How do you get the growth out of the price?

Growth needs capital. A bank growing its book at g must retain g over ROE of its profits, so the dividend on each Rs 100 of book is ROE minus g, not ROE. Put that dividend into the Gordon growth modelA valuation of a stream of dividends growing at a constant rate forever: next year dividend divided by cost of equity minus growth. and the price to book falls out. Set the formula equal to 2.5 and solve: 2.5 times (13% minus g) equals 18% minus g, so 1.5g equals 14.5%, and g is 9.67%.

Is 9.7% forever believable?

That is the question the interviewer really wants answered. A reverse valuation is only useful if you then judge the implied number. Growing book at 9.7% while paying out 46% of profits requires the 18% ROE to hold for decades, with asset quality intact. Notice how steep the curve is near the answer: a point more of growth, or a point less of cost of equity, moves the justified multiple a long way, which is why small changes in rate expectations swing bank valuations. The limitation is the model itself: one growth rate forever is a simplification, and a two-stage version is fairer to a bank growing fast today.

EBITDA rises by the full Rs 12 crore, EBIT by about Rs 4.3 crore, net debt by a lease liability of about Rs 77 crore, and EV/EBITDA falls from 10.0x to 9.40x. Rent leaves operating costs and returns as depreciation of Rs 7.7 crore and interest of Rs 6.9 crore. The cash paid is identical, so the only safe comparison is between companies measured on the same basis.

Why does the rent move out of EBITDA at all?

Think of the difference between renting a flat and buying it with a home loan. The owner pays an EMI that is partly interest and partly repayment, and the flat sits on the family balance sheet as an asset with a loan against it. The lease standard treats a long lease the same way: the right to use the store becomes an asset, the promise to pay rent becomes a debt, and the rent itself disappears from operating costs. Discount ten payments of Rs 12 crore at an assumed borrowing rate of 9% and the lease liabilityThe present value of the lease payments still to be made, carried on the balance sheet like a loan. is Rs 77.0 crore.

What replaces the rent in the income statement?

Two lines, both below EBITDA. The right-of-use assetThe asset recognised for the right to use a leased item, depreciated over the lease term. of Rs 77.0 crore is depreciated over ten years, Rs 7.7 crore a year, and the liability accrues interest at 9%, Rs 6.9 crore in year one. So EBITDA gains the whole Rs 12 crore, EBIT gains only Rs 4.3 crore, and year-one pre-tax profit actually falls by Rs 2.6 crore, because interest is heaviest while the liability is largest. Over the ten years the charges add up to Rs 120 crore, the same as the rent; only the timing moves.

Why does this matter when you compare two retailers?

Because the multiple now depends on whether a company rents or owns its stores. A retailer that leases every store will look cheaper on EV/EBITDA and more levered on net debt to EBITDA than an identical one that owns its stores, unless you put both on the same basis. Here leverage rises from 3.33x to 3.85x while the multiple falls. Say which basis you are using, and either include leases for everyone or strip them out for everyone. The standard's exemptions for short-term and low-value leases also vary in use, so check the notes before comparing.

Diluted EPS is about Rs 1.90, against basic EPS of Rs 2.00. Assume the bond converts. Earnings rise by the after-tax interest no longer paid: Rs 300 crore at 8% is Rs 24 crore, Rs 18 crore after tax. Shares rise by 15 crore. Rs 218 crore over 115 crore shares is Rs 1.90, a dilution of 5.2%. Check first that the bond dilutes at all.

Why do earnings go up when the bond converts?

Imagine a friend who lent you money and agrees to take a share of your business instead of repayment. From that day you stop paying them interest, so your profit rises, but you now split it with one more owner. A convertible works the same way: conversion removes the coupon and adds the shares, and the if-converted methodThe standard way to include a convertible in diluted EPS: assume it converts at the start of the year, add back its after-tax interest and add the new shares. captures both. The add-back is after tax, because the interest was saving tax: Rs 24 crore of interest cost only Rs 18 crore of profit.

When would you leave the bond out altogether?

When including it would raise EPS. Work out what the bond costs per new share: Rs 18 crore of after-tax interest over 15 crore shares is Rs 1.20 a share. If that figure is below basic EPS, conversion dilutes and the bond goes in; if it is above, the bond is antidilutiveA security whose assumed conversion would increase EPS. Accounting standards exclude it from diluted EPS. and is left out. Here Rs 1.20 is below Rs 2.00, so it goes in. If the same bond converted into only 5 crore shares, its cost per share would be Rs 3.60 and it would be excluded.

An analyst uses diluted EPS for valuation because a convertible that is in the money will convert, and the market prices the stock on the larger share count. The limit is that diluted EPS is a snapshot: it counts only securities that dilute at today's numbers, and a rise in the share price can bring more of them in.

Revenue is Rs 219 in year three, and Rs 100 of it, 46%, comes from new customers. The three layers are the new batch at 100, the year-two batch at 70 and the year-one batch at 49. Growth falls from 70% to 28.8% even though the company signs just as many customers, because revenue creeps towards a ceiling of Rs 333, which is new spend divided by the share lost each year.

Why do you build revenue layer by layer?

Picture a gym that signs 100 members every January, where 30% of any class drops out each year. Its member count is not 100 a year times the years open; it is this year's class plus what is left of every earlier class. Revenue built from customers is a stack of cohortsA group of customers who joined in the same period, tracked together over time., each decaying at the retention rate, and the total is the sum of the layers. Writing the layers down stops you from projecting last year's growth rate forward, which is the mistake the question is set up to catch.

Why does growth slow when the company is signing just as many customers?

Because the loss grows with the base. In year three the company loses 30% of 170, which is 51, and adds 100, so net growth is 49. Revenue stops growing when the spend lost each year equals the spend added, which here is at Rs 333: 100 divided by 0.3. Retention sets that ceiling. At 80% retention, year-three revenue would be Rs 244 and the ceiling Rs 500; at 60% the ceiling is Rs 250.

What does an analyst do with the 46%?

It tells you how much of the business must be won again every year. When nearly half of revenue comes from customers signed in the last twelve months, sales hiring and marketing spend are carrying the top line, and any slowdown in new wins shows up in revenue almost at once. By year five the share from new customers falls to 36% as the older layers pile up. The limitation of the model is equal batches: a company that grows its sign-ups each year will show faster growth, and one that spends more per customer in year two, through upsell, can see a batch grow rather than decay.

About 1.54 standard deviations too high. Each forecast is right on average, but choosing the highest selects for the biggest positive error. The expected maximum of ten independent draws from a normal distribution is 1.54 standard deviations above the mean. If each analyst's error has a spread of Rs 2 around a true EPS of Rs 50, the top forecast averages Rs 53.08.

Why does choosing the highest create bias when nobody is biased?

Weigh yourself on ten bathroom scales, each accurate on average but each a little off, and write down only the heaviest reading. You will always look heavier than you are. The bias is not in any one estimate; it comes from the choosing, because the maximum picks whichever estimate had the largest positive error. Auction theory calls the same effect the winner's curseThe tendency for the winning bid in an auction to overestimate the value of the prize, because the highest estimate of many noisy ones is usually too high.: the bidder with the highest estimate wins, and usually overpaid.

How do you get to 1.5 without tables?

Ask where the maximum has an even chance of landing. All ten estimates must fall below a level for the maximum to fall below it, so you need the level where the chance for one estimate, raised to the tenth power, is one half. That single-estimate chance is 0.5 to the power one tenth, about 0.933, which a normal table puts at 1.50 standard deviations. The mean of the maximum is a touch higher, 1.54, because the distribution of the maximum has a longer right tail. Either number is a fine answer in the room if you show the route.

Where does this bite an analyst?

Anywhere the top of a list is chosen after the fact. The most bullish forecast in the consensus, the best of ten back-tested strategies and the top-ranked fund of the year all carry a selection premium that will not repeat. The effect grows with the list: the highest of five is 1.16 standard deviations high, of twenty about 1.87. The fix is to shrink the chosen number back towards the average in proportion to how noisy the estimates are. The limit of this answer is the independence assumption: analysts who talk to the same management team share errors, and correlated errors shrink the gap.