Equity Research puzzles, solved step by step

Today: cash up Rs 50 crore, deferred revenue up Rs 50 crore, and no change to the income statement. Operating cash flow rises by 50 through the change in deferred revenue. On delivery, revenue of 50 and tax of 12.5 give net income of 37.5; deferred revenue falls by 50, so operating cash flow that year is minus 12.5, the tax paid, and equity rises 37.5.

Why is there no revenue on the day the cash arrives?

Think of a gym that sells a year's membership in January. The money is in the bank on day one, but the gym has not yet provided a single workout; if it shut down in February, it would owe most of that money back. Revenue is recorded when the service is delivered, so cash received in advance is a debt of service owed to the customer, and it sits on the balance sheet as deferred revenue.

On payment day the balance sheet grows on both sides: cash up 50 and deferred revenueCash received for goods or services not yet delivered. It is a liability until the company delivers, when it moves to revenue. up 50. The cash flow statement starts from net income of zero and adds the 50 rise in the liability, so operating cash flow is plus 50. The income statement is untouched.

What happens on the day the service is delivered?

Now the company has earned it. Revenue rises 50, tax at 25% is 12.5, net income rises 37.5. The cash flow statement starts at 37.5 and subtracts the 50 fall in deferred revenue, which leaves operating cash flow of minus 12.5: the profit was paid for in cash a year earlier, so the only cash that moves on delivery is the tax. The balance sheet balances: cash down 12.5 on one side, deferred revenue down 50 and retained earnings up 37.5 on the other.

For a research analyst this is the pattern behind subscription and advance-booking businesses: cash flow runs ahead of profit while bookings grow, and falls behind when they shrink. A growing deferred revenue balance flatters operating cash flow, so read the two together. The tax assumption matters too: some tax systems tax advances when received, so confirm the rule that applies before modelling it.

Total revenue grows 12%, from Rs 1,000 crore to Rs 1,120 crore. Existing stores add 4%, Rs 40 crore. The 20 new stores sell at Rs 8 crore a year but trade for only half the year, adding Rs 80 crore, 8 points. Split the two when you present it, because they tell different stories about the business.

Why split growth into old stores and new stores?

Think of a restaurant owner who opens a second branch. Total takings rise, but that says nothing about whether the first branch is doing better. Total growth mixes two different things: how the existing stores are trading, which is like-for-like growthSales growth from stores open for the whole of both periods, so new openings and closures do not distort it. Also called same store sales growth., and how many stores were added. A retailer can grow 12% on openings while every existing store shrinks.

How do you work the new stores' contribution?

Three adjustments, one at a time. Twenty stores at a mature rate of Rs 10 crore would be Rs 200 crore. They run at 80%, so Rs 160 crore a year. They trade for half the year, so Rs 80 crore this year. Part-year openings and ramp-up each shrink a new store's first-year contribution, which is why store count growth overstates revenue growth in the year of opening.

The follow-up an analyst should volunteer: next year, even with zero like-for-like growth and no openings, the 20 stores trade a full year and add another Rs 80 crore. That annualisation is growth already in the bag, and it is worth saying separately from the growth the business still has to earn.

20% for four years, narrowly: 2.07x against 2.01x. The rule of 72 says 20% doubles in about 3.6 years and 15% in about 4.8, so both paths just pass double, the 20% path by more. Check exactly: 1.2 squared is 1.44 and 1.44 squared is 2.0736; 1.15 squared is 1.3225, squared again is about 1.749, times 1.15 is 2.0114.

How do you get close without multiplying anything?

Think of two savers racing to double their money. The rule of 72A shortcut: dividing 72 by a growth rate in per cent gives roughly the number of years it takes to double. tells you when each one doubles, and whoever has more time left after doubling is ahead. At 20%, 72 over 20 is 3.6 years, leaving 0.4 years of 20% growth. At 15%, 72 over 15 is 4.8 years, leaving 0.2 years of 15%. The 20% saver has twice as long left at a higher rate, so 20% for four years should win.

Why check the edge exactly, and how?

The rule of 72 is rough at high rates: 20% actually doubles in 3.8 years, not 3.6, and 15% in 4.96, not 4.8. When a shortcut says the answer is close, the gap it shows can be smaller than the shortcut's own error, so you check the edge with exact arithmetic. Here it is easy: square 1.2 twice to get 2.0736, and for 1.15 square twice and multiply once more to get 2.0114. The margin is about 3%, real but narrow.

Why a research interviewer asks it: growth comparisons of this kind turn up every day, a company growing earnings faster for fewer years against one growing slower for longer. Saying the approximation first and then checking it shows the habit they want, and the limitation is worth one line: the answer flips if the 15% path runs for six years.

The revenue mix shifted towards the lower-margin segment. Say segment A earns 30% on 80 of revenue and B earns 10% on 20: the group earns 26%. Next year A earns 32% on 40 and B earns 12% on 60. Both improved by 2 points, but the group earns 20 on 100, 20%, because most revenue now sits in the 12% business.

How can an average fall when every part of it rises?

Think of a class where both the science and arts sections improve their average marks, yet the school's overall average falls, because far more students joined the lower-scoring section this year. An overall average depends on the weights as well as the parts, so a big enough shift in weights can reverse the direction of the average. Statisticians call this Simpson's paradoxThe pattern where a trend that holds in every group reverses when the groups are combined, because the group sizes changed..

How do you show the numbers work?

Check the group line each year. Year 1: 80 x 30% plus 20 x 10% is 24 plus 2, or 26 on 100. Year 2: 40 x 32% plus 60 x 12% is 12.8 plus 7.2, or 20 on 100. The whole fall comes from moving 40 of revenue out of a 30% business and into a 10% one, which costs 8 points of group margin; the segment improvements claw back only 2.

In a research note this is the difference between a margin miss that signals trouble and one that simply reflects mix. Split the change into a mix effect and a rate effect before judging management. The limitation: the split depends on which year's weights you use, so state the convention.

The reroll right is worth Rs 75. A single roll is worth Rs 350 on average. With the reroll, keep 4, 5 and 6, which average Rs 500, and reroll 1, 2 and 3 for an expected Rs 350. Half the time you get Rs 500 and half the time Rs 350, so the game is worth Rs 425, Rs 75 more than without the right.

What is the rule for keeping or rerolling?

Think of a job offer in hand while you wait on another interview. You take the offer if it beats what you expect the other process to give you, not if it beats your dream job. Keep any result worth more than the expected value of the alternative; here the alternative is a fresh roll worth Rs 350. So 4, 5 and 6 are kept, and 1, 2 and 3 are rerolled.

How do you value the right itself?

Value the game with the right, then subtract the game without it. With the right: half the time you hold 4, 5 or 6, averaging Rs 500; half the time you reroll and expect Rs 350. That is Rs 425. Without it the game is worth Rs 350. The right is worth the difference, Rs 75, and all of it comes from the three bad faces being replaced.

This is an option, and a research interviewer asks it to see whether you value flexibility correctly. The right has value only because you can refuse it when the first roll is good. Its value is the gain in the bad states, Rs 250, 150 and 50 on faces 1, 2 and 3, averaged over six faces: Rs 75. The same logic prices an option to expand a project or to delay an investment.

About Rs 271 crore a year, on these assumptions. A weekday carries 240,000 boardings: six peak hours at 25,000, four shoulder hours at 12,000 and seven quiet hours at 6,000. Weekends and holidays run at 60% of a weekday, giving 77.5 million trips a year. At an average fare of Rs 35, that is about Rs 271 crore.

How do you estimate a weekday's ridership?

Think about any busy road near an office district: jammed from 8 to 11 and from 5 to 8, calm in between. A metro line has the same shape. Ridership is concentrated in two peaks, so build the day hour by hour; a flat hourly average taken from the peak overstates it badly. Assume the line runs 6 am to 11 pm, 17 hours, with 25,000 boardings in each of six peak hours, 12,000 in four shoulder hours and 6,000 in seven quiet hours. That gives 240,000.

How do you check it and turn it into revenue?

Check from the stations. A line of 25 stations at 240,000 boardings a day is about 9,600 per station, which is plausible for a mix of busy interchanges and quiet suburban stops. Two routes that land close together give you licence to use the number; if they did not, the gap would tell you which assumption to test. Then annualise: 260 weekdays at 240,000 and 105 weekend and holiday days at 60% of that give 77.52 million trips.

Say which input you trust least: the average fare, since metro fares usually rise with distance and a line's trip-length mix is hard to guess. A research analyst would also note that fare revenue is only part of a metro's income, with advertising and property often material, and would check the operator's published ridership rather than rely on the estimate.

About 30% of the final wealth. At 12% a rupee grows to 9.65 in 20 years; at 10% after fees it grows to 6.73. The difference, 2.92, is 30.3% of the gross result. A fee that looks like one sixth of the annual return takes close to a third of the ending wealth, because the fee compounds too.

Why is the answer so much bigger than 2%?

Think of a leaking water tank that loses a small share of its contents every day. The leak looks trivial against the day's inflow, but it runs every day on the whole tank, and over a year it drains a large share of what would have collected. A percentage fee is charged on the whole balance every year, and every rupee it removes also loses the growth it would have earned for the rest of the period.

How do you work it quickly in the room?

Use the ratio rather than the two big numbers. The net path grows at 1.10 against 1.12, so each year it keeps 1.10 / 1.12 = 98.2% of the gross path. Over 20 years that ratio compounds to 0.982 to the 20th, about 0.70, so the fee takes about 30%. A mental shortcut: 20 years at roughly 1.8% a year is about 36% by simple addition, and compounding pulls it back to about 30%.

The same arithmetic applies to any recurring drag: fund expenses, trading costs, a tax on annual gains. It is also why a long-horizon investor compares costs in basis points. The limitation: the calculation assumes the gross return is the same with and without the fee; a manager who earns the fee through better returns changes the comparison.

About 7.8% a year, forever. In a growing perpetuity the price equals next year's cash flow, Rs 30 x (1 + g), divided by 11% minus g. Setting that equal to Rs 1,000 and solving gives g = (110 - 30) / 1,030 = 7.77%. The quick version: an 11% required return minus a 3% cash yield leaves about 8% for growth.

Why solve for growth instead of assuming it?

Think of a shop listed for sale at a price that looks high for its current takings. Rather than argue about the price, ask what sales growth would justify it; then the argument becomes whether that growth is believable. A price is a bundle of assumptions, and solving for the growth it needs turns a vague sense that a stock is expensive into one number you can test. This is the idea behind a reverse DCFA valuation run backwards: start from the market price and solve for the growth or margins the price implies..

How do you solve it cleanly?

Write the growing perpetuity with next year's cash flow on top: 1,000 = 30 x (1 + g) / (0.11 - g). Multiply out: 110 - 1,000g = 30 + 30g, so 80 = 1,030g and g = 7.77%. The shortcut, cost of equity minus cash yield, gives 8% and is close because the yield is small; the exact answer is a little lower because the Rs 30 grows before it is paid.

Then judge it. Growing at 7.8% forever means growing faster than most economies can for ever, which is a demanding assumption. Look at how steep the curve is near the answer: at 7% the stock would be worth Rs 803, at 8.5% Rs 1,302. Small changes in the growth belief move the value a lot, which is why a single-stage model is a sense check here, not a valuation.

Working capital absorbed Rs 130 crore, more than all of the profit. Start at net income of 100 and add depreciation of 20. Receivables up 80 and inventory up 60 each tie up cash; payables up 10 frees some. Operating cash flow is minus 10. Capex of 20 takes the change in cash to minus 30, matching the fall.

How can a profitable company lose cash?

Think of a wholesaler who sells a lot this month, but on 90 days' credit, and stocks up for the festive season. The books show a profit, yet the bank balance falls, because the sales have not been collected and the new stock has been paid for. Profit counts sales when they are made; cash counts them when they are collected, and growing receivables and inventory are the gap between the two.

Which way does each working capital line push cash?

Read the balance sheet change and ask whether cash went out or came in. A rise in an asset such as receivables or inventory uses cash; a rise in a liability such as payables provides it. Receivables and inventory together took Rs 140 crore and payables returned Rs 10 crore, so working capital absorbed Rs 130 crore against Rs 120 crore of profit plus depreciation. That makes operating cash flow minus 10.

What a research analyst does next: compare the rise in receivables to sales growth. If receivables grew much faster than revenue, customers are paying more slowly or sales were pushed through on easy terms, and that is a question for management. The limitation: a fast-growing company can have this pattern for healthy reasons, so the answer needs the trend, not one year.

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.