Equity Research puzzles, solved step by step

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.

The probability is 0.875. The sum of two uniform numbers has a triangular density, rising from 0 to a peak at 1 and falling back to 0 at 2. The region above 1.5 is a small triangle with base 0.5 and height 0.5, area 0.125. So the chance the sum is below 1.5 is 1 minus 0.125, which is 0.875.

Why is the sum not spread evenly from 0 to 2?

Think of rolling two dice: a total of 7 turns up far more often than 2 or 12, because many pairs make 7 and only one pair makes 2. Continuous numbers behave the same way. Adding two flat distributions gives a triangle, because middle values can be reached in many ways and extreme values in very few. Combining two densities like this is called convolutionThe operation that gives the distribution of a sum of independent random quantities by sliding one density across the other and adding up the overlap..

How do you get 0.875 quickly?

Use geometry rather than integrals. Every pair of draws is a point in the unit square, all points equally likely, so probability is area. The line x + y = 1.5 cuts off a corner triangle whose legs are each 0.5 long, so its area is half of 0.5 times 0.5, which is 0.125. Everything else, 0.875, is where the sum is below 1.5. The triangle density on the right shows the same tail from the other side.

The general rule for a threshold t between 1 and 2 is 1 minus half of (2 minus t) squared; for t below 1 it is half of t squared. The research version is diversification of two independent return sources: the average of two is less likely to be extreme than either alone, which is the whole argument for combining them.

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.

135 over 211, about 64%. Let P(i) be the chance of reaching Rs 5 from Rs i. Each P(i) is 0.6 times P(i+1) plus 0.4 times P(i-1), with P(0) = 0 and P(5) = 1. The solution is P(i) = (1 - (2/3)^i) / (1 - (2/3)^5), and at i = 2 that is (5/9) divided by (211/243), or 135/211. A fair coin would give 40%.

How do you set the problem up without drawing every path?

Think of a cricket team chasing a small target with few wickets in hand. What matters is not the path of each ball but where the game stands now: runs needed and wickets left. The only thing that matters at any moment is how much money you hold, so write one unknown for each amount and one equation linking it to its neighbours. From Rs i you move to Rs i+1 with chance 0.6 and to Rs i-1 with chance 0.4, so P(i) = 0.6 P(i+1) + 0.4 P(i-1). The two ends are fixed: P(0) = 0 because you are broke, and P(5) = 1 because you have won.

How do you check 64% without trusting the formula?

Solve the chain by hand, starting from the bottom. Write P(1) = a. Then P(2) = (a - 0.4 x 0) / 0.6 = a/0.6 = 1.667a, and each later step follows the same rule until P(5) = 211/81 x a. Setting P(5) = 1 gives a = 81/211, or 38.4%, and P(2) = 135/211. Two routes landing on the same fraction is the check an interviewer wants to hear. It also tells you the shape: the edge matters most in the middle, where the fair line and the edge curve are furthest apart.

Say the desk version in one line. A small, repeatable edge does not make ruin impossible when you start thin: from Rs 2 you still go broke about 36 times in 100. More capital, not a bigger edge, is what pushes the ruin chance towards zero.

Stake 20% of capital each time; at 40% your money slowly shrinks. The Kelly fraction for an even-money bet is the win chance minus the loss chance, 0.6 minus 0.4. At 20% the typical path grows about 2.0% a bet. At 40% each bet still has a positive expected gain, but growth is about -0.24% a bet, below zero, because big losses compound harder than big wins.

Why is the bet with the highest expected value not the one that grows fastest?

Imagine a shopkeeper who puts half the till into stock every morning. A good day lifts the till by half; a bad day cuts it in half. One of each leaves 1.5 x 0.5 = 0.75 of where she started, even though the good and bad days were the same size. Wealth compounds, so what matters over many bets is the average of the logarithm of each outcome, not the average outcome. Expected value per bet rises in a straight line with the stake; growth rises, peaks and then falls.

What do the numbers look like at each stake?

Read the two columns against each other. Every row has a positive expected gain, yet the 40% and 50% rows lose money on the path you will actually live through. Half Kelly at 10% keeps about 75% of the maximum growth with far smaller swings, which is why many desks size below full Kelly. Say the limitation too: the formula assumes you know the 60% exactly. Real edges are estimates, and overestimating one pushes you towards the overbetting side of the curve.

About Rs 19,000 crore a year on these assumptions. Take an assumed fleet of 25 crore two-wheelers, 1.8 crore in each of the last five years and 1.6 crore in each older year. All young vehicles are insured; after year five the insured share falls from 60% to 20%. That leaves 14.6 crore insured vehicles paying Rs 800 to Rs 1,500 a year, a pool near Rs 18,700 crore.

Where do you start, the vehicles or the policies?

Think of a gym. Counting everyone who ever signed up tells you little; the revenue comes from the members who still renew. Start from the fleet, but the number that decides the pool is how many vehicles are still insured, not how many are on the road. In India third-party cover is compulsory by law and new two-wheelers are sold with a multi-year third-party policy, so the young fleet is close to fully insured. Confirm the current rules before quoting them. Once that upfront cover runs out, many owners of older, cheaper bikes let it lapse.

How do you show the interviewer which assumption matters?

Run one sensitivity out loud. If ten more vehicles in every hundred older than five years renewed their cover, the pool would rise by about Rs 1,440 crore, roughly 8% of the total. That is the lever an insurer or a regulator can pull: enforcement of the third-party requirement at the roadside. The premium per vehicle matters less because it is set in narrow bands for third-party cover. The limitation to say: premiums, fleet and lapse rates here are assumptions for the method, and an analyst would replace each with a sourced number before writing it into a note.

30% (130 to 91), then 25% (120 to 90), then 10% (140 to 126). Walk the series keeping the highest close so far. Each time the price falls below that running peak, track the lowest point until a new peak is set. 120 falls to 90 before 130 sets a new high; 130 falls to 91 before 140; 140 falls to 126 and has not recovered. Divide each fall by its own peak.

Why measure from the running peak rather than the start?

Picture a savings account you check every evening. If it once showed Rs 1,30,000 and now shows Rs 91,000, the loss you feel is from the best balance you ever saw, not from what you first deposited. A drawdown is the fall from the highest value reached so far to the lowest value before that high is beaten, divided by the high. The start of the series only matters until the price first rises above it.

How do you find them in one pass?

Carry two numbers as you read left to right: the running peak and the lowest price since that peak. When the price sets a new high, close the open drawdown, record it, and reset both numbers to the new high. At the end, close whatever is still open, then sort by size and return the first n. That is the programming version of the question, and it runs in a single pass over the prices.

State the edge cases before the interviewer does. An open drawdown at the end of the series counts, even though it has not recovered. Two drops inside one drawdown episode, such as 130 to 91 and a later dip to 93 before a new high, are one drawdown, not two.

B trades lower, at 13.3x against A's 16.7x, and they match when B's debt costs 8%. A earns 80 x 0.75 = 60 on equity of 1,000. B pays 30 of interest, earns 37.5 on equity of 500. The P/Es match when the after-tax cost of debt equals A's earnings yield of 6%: r x 0.75 = 6%, so r = 8%. Cheaper debt lowers B's P/E; dearer debt raises it.

Why would the same business carry two different P/Es?

Suppose a flat earns rent of Rs 6 for every Rs 100 of its price. Buy it with half cash and half a loan at 4.5% after tax, and your cash earns more than 6%, because the borrowed half costs less than it yields. P/E divides the equity's value by the equity's earnings, and debt changes both, by different amounts. Swapping equity for debt that costs less than the equity's earnings yield lowers the P/E; swapping for dearer debt raises it.

Why is 8% the crossing point?

Swapping 500 of equity for 500 of debt removes equity that was earning its share of A's 6% earnings yield, 30 of net income, and adds an after-tax interest cost of 500 x r x 0.75. If that cost is exactly 30, net income halves along with equity and the P/E does not move; 30 over 375 is 8% before tax. The general rule: leverage cuts P/E when the after-tax cost of debt is below the earnings yield, E/P. The limitation worth saying: B's lower P/E is not a sign it is cheaper. Its equity is riskier, so investors should demand a lower multiple for the same business.