Equity Research puzzles, solved step by step

About 6.7%. Revenue is price times volume, so the growth factors multiply: 1.12 = 1.05 x (1 + v). Dividing, 1.12 / 1.05 = 1.0667, so volume grew about 6.67%. Subtracting 5 from 12 gives 7%, which overstates volume by the small cross term, 0.33 points.

Why divide instead of subtract?

Think of a tea stall that raises the price of a cup and also sells more cups. The takings grow for both reasons, and the extra cups are also sold at the higher price, which is a little extra neither change produces alone. Revenue is price times volume, so growth factors multiply, and the way to back one out is to divide. 1.12 divided by 1.05 is 1.0667.

When does the shortcut of subtracting matter?

The error from subtracting is the corner of the rectangle, price growth times volume growth. At single-digit rates the corner is small, about 0.33 points here, but it grows quickly with the rates, and in high-inflation markets it becomes the whole story. With 30% price rises and 40% revenue growth, subtracting says 10% volume growth; dividing says 7.7%.

For an analyst this split is the first question on any results call: how much of the growth was price and how much was volume. Volume growth says whether the company is winning customers; price growth says whether it has pricing power or is passing on costs. The limitation: reported price growth often mixes true price changes with shifts in product mix, so ask how the company defines it.

Rs 19,920 crore, and 100 crore make a billion. USD 2.4 billion at Rs 83 is Rs 199.2 billion. A crore is ten million, 10 to the 7, and a billion is 10 to the 9, so a billion rupees is 100 crore. Rs 199.2 billion times 100 is Rs 19,920 crore, just under a fifth of a lakh crore.

Why do people get the crore conversion wrong?

Think of two rulers marked in different units, one in inches and one in centimetres. Reading a length off one and writing it in the other is easy once you line them up, and error-prone if you do it from memory. Indian and Western number names are two rulers on the same powers of ten. The Indian system groups by lakh and crore, 10 to the 5 and 10 to the 7, while the Western system groups by thousands, so the two only line up at a thousand and at a billion, which equals 100 crore.

What is the fastest safe route in the room?

Do the currency first, then the units. 2.4 x 83 is 199.2, so Rs 199.2 billion. Multiply billions by 100 to get crore, and divide crore by 1,00,000 to get lakh crore; never convert through million unless you have to. That gives Rs 19,920 crore, or about 0.2 lakh crore. If you prefer the Western route: Rs 199,200 million divided by 10 million per crore gives the same 19,920.

In an Indian research role this comes up every day, because company filings report in crore or lakh while global peers and many investors think in millions and billions. The exchange rate here is the one given in the question; for real work, use the rate on the date of the numbers you are converting and state it.

About 18 million a year, on these assumptions. Take 300 million households, half owning a two-wheeler, at 1.1 each: a fleet of about 165 million. Replacing each every 11 years gives 15 million a year. Ownership rising one point a year adds 3 million first-time buyers. Check the total against published industry sales before relying on it.

Where does demand for new vehicles come from?

Think of a housing society's parking lot. Each year a few old scooters are swapped for new ones, and a few families who never had one buy their first. New vehicle sales are replacement of the existing fleet plus first-time buyers, and splitting the two is what makes the estimate defensible, because each has its own driver. Replacement depends on fleet size and vehicle life; first-time buying depends on how fast ownership spreads.

How do you build each branch, and which assumption matters most?

Start with households: roughly 1,400 million people at a little under five a household gives about 300 million, stated as an assumption. Half own a two-wheeler, and owning households average 1.1, so the fleet is about 165 million. If a vehicle lasts 11 years, about one in eleven is replaced each year, 15 million, which makes replacement most of the market. First-time demand is ownership rising one point a year on 300 million households, 3 million.

Test the most sensitive input out loud. A 9-year life instead of 11 lifts replacement to about 18 million; a 13-year life cuts it to about 13 million. That range, about 6 million, is twice the whole first-time branch, which tells you where to look first. Then say you would check the total against the industry body's published annual sales rather than quoting a figure from memory.

Exactly where it started. Rs 100 rises 25% to Rs 125. The 20% fall is taken on Rs 125, which is Rs 25, bringing it back to Rs 100. As factors, 1.25 x 0.80 = 1.00. The rupee gain and the rupee loss are the same Rs 25; they look different as percentages because they are measured on different bases.

Why does a 20% fall cancel a 25% gain?

Think of a shop that marks a Rs 100 item up to Rs 125, then offers 20% off the new price. The customer pays Rs 100: the discount is taken on Rs 125, so it is worth Rs 25, the same as the mark-up. A percentage change is always measured against the level just before it, so a fall taken on a higher base removes more rupees per point than the rise added.

What is the general rule?

Multiply the growth factors, never add the percentages. 1.25 x 0.80 is exactly 1.00. To undo a rise of r you need a fall of r divided by (1 plus r): 0.25 / 1.25 = 20%; to undo a fall of r you need a rise of r divided by (1 minus r). That asymmetry is why a 50% loss needs a 100% gain to recover, while a 50% gain is undone by a 33% loss.

For an analyst this matters when reading reported returns. A fund that was up 25% last year and down 20% this year shows an average annual return of 2.5% but has made nothing. The average of percentage returns is not the return an investor earned; the compound return, here 0%, is.

A is cheaper on PEG, 1.2 against 1.9, but PEG ignores how long the growth lasts and what it costs. PEG divides the P/E by the growth rate, so it rewards fast growth whatever its duration. If A's 25% lasts only three years, today's price is 15.4 times year-three earnings for A against 11.9 times for B, and A is the dearer stock.

What does PEG measure?

Think of two rented flats, one expensive in a fast-improving area and one cheap in a quiet one. Comparing rent per square foot is not enough; you would also ask how quickly each area is improving. PEG does that for stocks. It divides the P/E by the growth rate, so it expresses price per unit of growth: A pays 1.2 times per point of growth, B pays 1.88. A PEG near 1 is often used as a rough marker of fair value, which is a convention, not a law.

What does PEG leave out?

Two things. First, duration: a growth rate is a speed, and PEG ignores how long the speed lasts. If A grows 25% for three years and then slows to B's pace, you are paying 15.4 times its year-three earnings for a company then growing like B, which costs 11.9 times. Second, the cost of growth: a company that must reinvest most of its earnings to grow is worth less than one that grows with little capital, and PEG cannot see the difference.

How a research analyst would use it: as a quick screen across a sector, then replaced by something that sees duration and reinvestment, such as a DCF or a return on capital comparison. The limitation in one line: PEG compares price to the speed of growth, not to its length or its cost.

The expected price is Rs 512, 2.4% above today. Weight each outcome by its chance: 0.6 x 560 plus 0.4 x 440 gives 336 plus 176. Today's Rs 500 sits exactly halfway between Rs 440 and Rs 560, so the market is pricing roughly a 50% chance of good results. Your 60% view is what separates Rs 512 from Rs 500, and that probability is what you would have to defend.

Why is the expected price not the likely price?

A cricket fan who thinks her team wins 60% of the time does not expect the team to win exactly 0.6 of a match; she expects the win most often and a loss sometimes. Expected value is the probability-weighted average of every outcome, and it can be a price that never actually trades. Here the stock will be at Rs 560 or Rs 440 tomorrow, never at Rs 512, yet Rs 512 is the right number to compare with today's price.

What does today's price tell you about the market's view?

Run the calculation backwards. If Rs 500 is the market's expected price, the chance q of the good outcome solves 560q + 440(1 - q) = 500, so q = 60 / 120 = 50%. The interesting number is not Rs 512 but the gap between your 60% and the market's 50%: that gap is the whole of your view. An analyst would next ask what evidence justifies seeing more upside than the market does.

Say the limitations. Two outcomes are a simplification of a whole spread of possible moves, and a 2.4% expected gain on one event is small next to the Rs 60 swing either way. The expected value is a way to state a view precisely, not a reason on its own to act on it.

The price falls to about Rs 400, EPS halves, and the P/E is unchanged. A 1:1 bonus gives one free share for each share held, so the share count doubles while the business, its profit and its value do not change. With EPS of, say, Rs 40, EPS becomes Rs 20 and the price Rs 400, leaving the P/E at 20x and the market value at Rs 8,000 crore.

If shareholders get free shares, why are they not richer?

Cut a pizza into eight slices instead of four and nobody gets more pizza. A bonus issue changes how many slices the company is cut into, not the size of the company, so each slice is worth proportionally less. A holder of 10 shares at Rs 800 had Rs 8,000; after the bonus she has 20 shares at Rs 400, still Rs 8,000. Nothing was paid and nothing was received.

What happens in the accounts?

The company moves an amount equal to the face value of the new shares out of reserves and into share capital. Net worth is unchanged, because money only moves between two lines inside equity. No cash moves, profit is untouched, and the dividend per share usually halves unless the board chooses to keep it, which would then be a real increase in payout.

The analyst's housekeeping: restate past EPS, dividends per share and price charts for the bonus, otherwise the history shows a false halving. Say the limitation too. In practice the price can drift from the exact half, because a lower share price can widen the pool of buyers, but that is a market effect, not a change in value.

About 20.7%. Set the market at 100. The company's sales go from 20 to 22.4 and the market from 100 to 108, so its share is 22.4 over 108, or 20.74%. The gain is only 0.74 of a point, because share moves with relative growth: 1.12 divided by 1.08, about 3.7% more share, on a 20% base.

Why does growing 4 points faster add less than 1 point of share?

A student who scored 20 out of 100 last term and improves her marks by 12% while the class average improves by 8% has done a little better relative to the class, not a lot. Market share is a ratio, so it changes by the ratio of the two growth rates, 1.12 over 1.08, applied to the share you started with. A 3.7% relative gain on a 20% share is about 0.74 of a point.

How do you use this the other way round in a model?

Analysts often forecast a company as market growth plus a share assumption. If you forecast 12% growth in an 8% market, you are assuming a share gain of about 0.74 points a year, and five years of that takes 20% to about 24.0%. Saying that out loud tests whether the forecast is believable: which competitor is giving up that share, and why? The limitation: this assumes the market figure and the company's sales are measured the same way, which is often not true when companies report by segment.

Both, if you say it properly: the margin rose 3 percentage points, which is a 25% relative rise. The difference between two percentages is measured in percentage points, here 15 minus 12, or 300 basis points. The relative change is 3 divided by 12, which is 25%. On flat sales of Rs 1,000 crore, that is EBITDA rising from Rs 120 crore to Rs 150 crore, 25% more.

Why are both numbers right and one of them misleading?

If a bank's loan rate moves from 8% to 9%, nobody says the rate rose 1%; they say it rose one percentage point, even though the interest bill rose 12.5%. A percentage point is the difference between two percentages; a per cent change is that difference measured against the starting value. Saying margin improved 3% leaves the reader guessing whether the margin went to 15% or to 12.36%, which is 3% more than 12%.

When does the 25% matter more than the 3 points?

When you are forecasting profit. At flat sales, a margin that rises by a quarter of itself lifts EBITDA by a quarter, so the 25% is what flows into earnings and valuation. On Rs 1,000 crore of sales, EBITDA moves from Rs 120 crore to Rs 150 crore. The same 3 points on a 30% margin would be only a 10% rise in EBITDA, which is why the same points of margin matter much more for a thin-margin business.

In the room, give the answer in one line: 3 percentage points, 300 basis points, a 25% relative improvement. Then add the one caution that shows judgement: check whether sales were flat, because a margin can rise while EBITDA falls if revenue shrinks.

About 6 years to double and about 18 years to grow eightfold. The rule of 72 says the doubling time is roughly 72 divided by the percentage rate, so 72 / 12 = 6 years; the exact figure is 6.1 years. Eightfold is three doublings, 2 x 2 x 2, so it takes three doubling times, about 18 years, and exactly 18.3. It is not eight times six.

Why does 72 divided by the rate work?

Think of a savings pot that earns 12% a year and never has anything taken out. Each year's interest earns interest the next year, so the pot grows faster in rupees every year while growing at the same percentage. The time to double depends only on the rate, and for everyday rates it is close to 72 divided by the rate in percent. The exact answer is the log of 2 over the log of 1.12, 6.12 years. The number 72 is used because it sits near the exact constant and divides neatly by 2, 3, 4, 6, 8, 9 and 12.

Why is eightfold 18 years and not 48?

Because growth multiplies. Eight is 2 x 2 x 2, so eightfold is three doublings, and each doubling takes the same six years whatever the size of the pot. The same counting gives the other useful anchors: fourfold is two doublings, 12 years; a thousandfold is about ten doublings, since 2 to the power 10 is 1,024, so about 60 years. Tenfold is a bit more than three doublings, exactly 20.3 years.

Where does an analyst use this?

Anywhere a growth rate needs to be turned into a sense of scale, fast. A company promising 24% growth is promising to double every three years; a fund charging 2% a year in fees gives up about a third of the pot over twenty years; inflation at 6% halves the value of money in about 12 years. Turning rates into doubling times is the quickest way to check whether a claim is plausible before any spreadsheet is opened.