Equity Research puzzles, solved step by step

EPS grows 7.1%, and the multiple should not rise on that growth. Net income goes from 70 to 75 on flat pre-tax profit of 100. The price can reasonably rise about 7.1% at the same P/E, because each share now earns more. But the growth happens once: next year, with pre-tax profit still flat, EPS growth is zero.

Where does the 7.1% come from?

Think of a salaried employee whose tax bill falls. Take-home pay jumps once, but next year's take-home pay only grows if the salary does. A lower tax rate raises net income by the tax saved, here 5 on a base of 70, which is 7.1%, even though the business earned nothing more. Say the base out loud: people who answer 5% are measuring against pre-tax profit.

Why should the multiple stay where it is?

A P/E multiple pays for the level of earnings and for their future growth. The tax cut raises the level, so at an unchanged P/E of 20 the price rises 7.1% to match. It does not raise future growth, so a higher multiple would pay twice for the same one-off step. Re-rate to 22x on the back of 7% growth and the price would rise 17.9%, most of it paying for growth that will not repeat.

Two caveats a research analyst adds. First, ask why the rate fell: a statutory cut is durable, while a one-time credit or a shift of profit to a lower-tax region may reverse, in which case even the level change deserves a discount. Second, check whether competitors got the same cut; if they did, some of it may be passed to customers through prices. Confirm any tax rate you use against the current rules rather than memory.

About 9%: duration of 6 times a 1.5 point rise in yield. For an 8-year, 7% coupon bond at par, the true fall is 8.5%, a little less than the straight-line estimate because the price curve bends. For the equity analyst, the market now charges the company about 1.5 points more for new debt, so the cost of debt in WACC should use the new yield, not the old coupon.

How does duration turn a spread move into a price move?

Think of a fixed-rate deposit you cannot break. When new deposits start paying more, yours is worth less to anyone who might buy it from you, and the longer it has left to run, the bigger the discount. Modified duration is that sensitivity in one number: the approximate percentage price change for each 1 point change in yield. A spread widening raises the bond's yield by the same amount, so 6 x 1.5 = 9%.

Why is the true fall a little smaller than 9%?

The price-yield curve is convex: it flattens as yields rise. Duration draws a straight tangent at today's yield, so for large moves it overstates price falls and understates price rises. Here the tangent says 91.0 and the bond is actually worth 91.5. For small moves the gap is negligible; at 150 basis points it is about 0.5 points, worth one sentence.

Now the equity view. The cost of debt in a WACC is what the company would pay to borrow today, so a 150 basis point widening raises it by about 1.5 points, whatever the coupon on existing bonds. With debt at 30% of capital and a 25% tax rate, that alone adds about 0.34 points to WACC. The larger message is the signal: credit investors are pricing more risk, and equity sits below the bonds, so the analyst should ask what the bond market has seen.

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.

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.

Rs 112. For every four shares worth Rs 120, a holder pays Rs 80 for one new share. The five shares then hold Rs 480 of old value plus Rs 80 of new cash, Rs 560 in all, so each is worth Rs 560 divided by 5, which is Rs 112. The right to buy one new share is worth Rs 112 minus Rs 80, or Rs 32.

Why does the share price fall when nobody has lost anything?

Picture four friends who each own a Rs 120 share in a shared tiffin business. A fifth friend joins by paying only Rs 80. The business is now worth Rs 560 and there are five equal owners, so each stake is worth Rs 112. After a rights issue the old and new shares are identical, so the price has to settle at the weighted average of the old value and the new cash per share. The fall from Rs 120 to Rs 112 is not a loss to existing holders, because they were offered the cheap share too.

How do you prove an existing holder is no worse off?

Take a holder of four shares. If she subscribes, she had Rs 480 of shares and Rs 80 of cash; now she has five shares at Rs 112, still Rs 560. If she sells her right instead, she keeps four shares worth Rs 448 and receives about Rs 32 for the right, still Rs 480. Her wealth is unchanged either way; only a holder who ignores the right loses its value.

The analyst's follow-through: because the issue is priced below market, it carries a bonus element. Historical EPS is restated by dividing by the factor 120 over 112, or 1.0714, so per-share figures before and after the issue compare like with like.

LTV is Rs 1,500, LTV to CAC is 1.67x, and payback is 15 months per retained user, about 22 months for the cohort. Lifetime value is Rs 60 divided by 4% churn. CAC over monthly contribution, Rs 900 over Rs 60, gives the usual 15 month payback. Counting users who leave along the way, the cohort's cumulative contribution only reaches Rs 900 after about 22.4 months.

Why does churn set the lifetime value?

Think of a leaking water tank. If 4% of the water drains out every hour, a litre poured in stays, on average, 25 hours. With a constant monthly churn rate, the average customer life is one divided by churn, so lifetime value is monthly contribution divided by churn. Rs 60 over 0.04 is Rs 1,500. Halve churn to 2% and LTV doubles to Rs 3,000; that is why subscription analysts watch churn more closely than price.

Which payback number do you give the interviewer?

Give both and say which is which. The common convention is CAC divided by monthly contribution, 15 months, which describes a user who stays. A cohort of acquired users earns less than that each month because some have left, so the money spent on the cohort comes back later, after about 22.4 months. By month 15 only 54% of the cohort is still paying.

Then give a view. A ratio of 1.67x is thin: the business keeps Rs 600 per user over the user's life before any overhead, and a common rule of thumb looks for about three times. The two levers are churn and acquisition cost, and the numbers show churn is the stronger one.

About 33%. Take a price of 100 and a cost of 60, so profit is 40 a unit. Cut the price by 10 and profit falls to 30 a unit, because the cost does not move. To keep gross profit the same you need 40 divided by 30 as many units, 1.33 times, so volume must rise by a third. The formula is cut divided by (margin minus cut).

Why is the volume needed so much bigger than the cut?

A tea stall sells a cup for Rs 10 that costs Rs 6 to make, keeping Rs 4. Knock Re 1 off the price and the stall keeps Rs 3, a quarter less on every cup. A price cut comes straight out of the margin, because costs do not fall with price, so the percentage fall in profit per unit is the cut divided by the margin. Here a 10% price cut removes a quarter of the 40% margin, and volume has to make up the whole quarter.

How does the answer change with the margin?

This is why analysts treat price cuts in thin-margin businesses with suspicion. A retailer on a 20% gross margin that cuts prices 10% needs twice the volume just to stand still, while a software company on 80% needs only 14% more. Say the limitation too: the puzzle holds unit cost fixed. If volume brings purchasing discounts or spreads fixed factory costs, the bar is lower.

About 9.6% a year, not 14%. The 14% average only covers funds that survived, and funds usually close because they did badly. Weight both groups by their count: 60 funds at 14% and 40 at 3% gives 0.6 x 14% plus 0.4 x 3%, or 9.6%. A database that drops closed funds overstates the average investor's experience by 4.4 points a year.

Why does the survivor average mislead?

Ask the toppers of a coaching class how hard the entrance exam was, and they will say it was manageable. The students who failed have gone home and nobody asked them. Survivorship bias is judging a group by the members still standing, when the ones that dropped out left because their results were poor. A fund database that lists only live funds is the coaching class that only asks its toppers.

How big is the gap once it compounds?

Over ten years, 14% a year turns Rs 100 into Rs 371, while 9.6% turns it into Rs 250. A gap of 4.4 points a year looks modest, but it is the difference between multiplying money 3.7 times and 2.5 times. That is why performance studies that ignore closed and merged funds tend to make active management look better than it was.

Say the limitation too. Averaging annual returns across funds of different lives is a simplification; a careful study would weight by assets and by years in existence. The direction of the bias does not change, though: leaving out the losers always flatters the average.

6 tosses. Call S the expected tosses from the start and H the expected tosses once you have one head. From the start, one toss either gives a head (move to H) or a tail (stay at S): S = 1 + H/2 + S/2. From one head, one toss either finishes or sends you back: H = 1 + S/2. Substituting, S = 6 and H = 4.

Why is the answer not four?

Think of climbing two steps on a slippery stair where any slip sends you back to the bottom. Reaching the second step is not just two climbs; every slip costs you what you had gained. The chance of heads-heads on a single pair is one in four, but a tail after a head wipes out your progress, so the expected wait is longer than four. The clean way to count the cost of those slips is to name each position you can be in.

How do you check 6 another way?

Count sequences directly. The first HH lands on toss n in exactly F(n-1) of the 2ⁿ equally likely sequences, where F is the Fibonacci sequence: 1 way on toss 2, 1 on toss 3, 2 on toss 4, 3 on toss 5. Summing n times F(n-1) over 2ⁿ across all n gives 6.0000, the same 6. The same series shows the spread: you finish within six tosses 67% of the time, and the other third of runs can go on for a long while, which is what pulls the average up to 6.

Roughly 607 million 5G phones worth about US$ 351 billion, on stated assumptions. Assume 4 billion smartphones in use, replaced every 3 years, so about 1.33 billion are sold in 2022. Assume all premium phones, 60% of mid-range and 15% of entry phones are 5G, a blended 45.5%, which gives 607 million units. At a blended price near $579, the value is about US$ 351 billion.

Where do you start: people, phones or networks?

Think of estimating how many pairs of school shoes a town buys. You do not count shops; you count children and ask how often a pair wears out. Annual phone sales are the phones in use divided by how many years a phone lasts, so start from the installed base and the replacement cycle, then split the year's sales by price band. Say every number as an assumption: an installed base of 4 billion and a 3 year cycle are round figures for the method, to be checked against an industry tracker before they go into a note.

How do you show the interviewer you know where the estimate is weak?

Run the sensitivity on the widest step. A cycle of 2.5 years gives 728 million 5G units and 3.5 years gives 520 million, a swing larger than any plausible error in the price assumptions. Then sanity check the output: 45% of phones sold being 5G in 2022 should sit sensibly against what you know about network rollouts, and a blended price near $579 says premium phones carry most of the value. The limitation: first-time buyers are folded into the cycle, which slightly understates sales in markets still adding users.