Financial Analysis puzzles, solved step by step

About 1.49% a month, or 19.4% a year effective. The real price is the Rs 57,000 cash price, so choosing the EMI means borrowing Rs 57,000 and repaying Rs 60,000 in six monthly instalments. The rate that makes six payments of Rs 10,000 worth exactly Rs 57,000 today is 1.49% a month. The forgone Rs 3,000 discount is the interest, charged on a balance that shrinks every month.

Where is the interest, if the EMI says no-cost?

A friend offers to sell you his bike for Rs 57,000 today, or for Rs 60,000 paid over six months. Nobody calls the second option interest-free; it is a loan of Rs 57,000 with Rs 3,000 of interest. A shop doing the same thing is no different. When a cash discount is available, the discount you give up by paying in instalments is the interest on the loan. The 'no-cost' label means the instalments add up to the sticker price, not that money has no time value.

So the question becomes: at what monthly rate is a stream of six Rs 10,000 payments, starting in one month, worth Rs 57,000 today? That is the internal rate of return of the loan.

How do you solve it without a spreadsheet?

Guess and check. The annuity factor must be 57,000 / 10,000 = 5.70. At 1% a month, six payments are worth 5.80 times one payment; at 2% they are worth 5.60. Halfway gives about 1.5%, and 1.5% gives 5.697, a whisker under 5.70, so the answer is just below: 1.49%. Then compound, do not multiply: 1.0149 to the power 12 is 1.194, so the effective annual rate is 19.4%, against 17.8% if you simply multiply by 12.

A second check uses the average balance. The loan starts at Rs 57,000 and is paid down to zero over six months; the balance at the start of each month averages about Rs 33,700. Rs 3,000 of interest on about Rs 33,700 over six months is 8.9%, or about 17.8% a year simple, which matches 1.49% x 12.

What changes the answer in real life?

Timing matters a great deal. If the first instalment is due today rather than in a month, you really borrow only Rs 47,000 and repay Rs 50,000 over five months, and the hidden rate jumps to 2.10% a month, about 28% a year. Any processing fee, or tax charged on the interest component, raises the true cost further, so always ask what you pay on day one. Check the specific charges and the current tax treatment with the lender. And if there was never a cash discount to forgo, the EMI really is close to free for the buyer, because the retailer, not the buyer, is paying the lender.

Year one: profit falls Rs 15 crore, inventory falls Rs 20 crore, cash rises Rs 5 crore from tax saved. Year two: gross margin is flattered. The scrapped stock sits on the books at zero, so the Rs 5 crore sale is pure gross profit, lifting margin from 30.0% to 30.9% on these figures. The write-down moved profit between years, because it wrote off more than the stock turned out to be worth.

How does a write-down move through the three statements?

Suppose you bought Rs 20,000 of winter jackets for your shop two years ago, and they are now unsellable. Admitting it does not take any money out of your till today; the money left when you bought them. An inventory write-down is a non-cash expense that recognises a loss of value already suffered, so its only cash effect is the tax it saves.

Check the balance sheet: assets are down Rs 15 crore, Rs 20 crore of stock gone and Rs 5 crore of cash added, and retained earnings are down by the same Rs 15 crore. It balances, which is the test an interviewer listens for.

Why does next year's gross margin look better than normal?

The stock now has a book value of zero. When it is sold for Rs 5 crore, revenue rises Rs 5 crore and cost of goods sold rises by nothing, because there is no cost left to expense. On a normal Rs 400 crore of sales at 30%, the write-down year shows 25.0% and the scrap year 30.9%. Writing down more than the stock is finally worth pulls a loss into this year and pushes an equal gain into the next.

Over both years nothing is lost or gained by the timing. Pre-tax, the total is minus Rs 20 crore plus Rs 5 crore, or minus Rs 15 crore: the true economic loss on stock that cost Rs 20 crore and fetched Rs 5 crore. Accounting rules generally require inventory to be carried at the lower of cost and net realisable value. If the company expected Rs 5 crore of scrap value, the honest write-down was Rs 15 crore, and the sale would have produced no gain at all.

What does an analyst do with this?

Treat both effects as one event. Strip the write-down out of year one's gross margin and the scrap gain out of year two's, and you see the business's real run-rate margin of 30%. A large write-down followed by margins above the old normal is a pattern worth asking management about, because over-provisioning creates a reserve that can be released into later profits. That is not proof of anything; obsolete stock genuinely varies in what it fetches. It is a question to ask.

The chance of at least one default is 18.3%, and the expected number of defaults is 0.2. Work from the complement: each loan survives with probability 98%, so all ten survive with probability 0.98^10 = 81.7%, and at least one default is the rest. The expected number is simply 10 x 2% = 0.2, which holds whether or not the loans are independent. Independence matters for the first answer, not the second.

Why work from the chance that nothing happens?

Ask what the chance is that at least one of ten friends is late for dinner, each being late one time in fifty. Counting the ways someone could be late is messy: one late, two late, any combination. Counting the single way nobody is late is easy: everyone on time. 'At least one' is one minus 'none', and 'none' for independent events is just the single probabilities multiplied. Each loan survives with probability 0.98, so all ten survive with probability 0.98^10.

To do 0.98^10 in your head, use the shortcut that (1 minus x) to the power n is close to 1 minus nx plus a correction of n(n minus 1)/2 times x squared: 1 minus 0.20 plus 45 x 0.0004 = 0.818. The exact figure is 0.8171, so the chance of at least one default is 18.29%. A second check is the Poisson shortcut, 1 minus e to the minus 0.2, which gives 18.1%.

Why is the expected number so much simpler?

Expected values add, always. Each loan contributes 0.02 expected defaults, and ten loans contribute 0.2. The expected number of defaults is 0.2 regardless of how the loans are connected, because the expectation of a sum is the sum of the expectations even when the events are correlated. The distribution behind it is lopsided: 81.7% of the time nothing happens, 16.7% of the time exactly one loan fails, and two or more fail 1.6% of the time.

What does correlation change?

Take the extreme. If all ten borrowers are suppliers to one factory and fail together or survive together, there is a 2% chance that all ten default and a 98% chance that none does. Expected defaults are still 0.2. But the chance of at least one default falls to 2%, and the chance of losing the entire book jumps from practically zero to 2%. Correlation does not change the average loss; it moves probability from many small losses into rare large ones, which is the loss a lender cannot survive. That is why credit portfolio work spends its effort on concentration and correlation rather than on the average default rate, and why a portfolio of ten loans to one industry is riskier than its expected loss suggests.

Cost of goods sold must have grown about 44%. Inventory days are inventory divided by daily cost of goods sold. For days to fall by a sixth while inventory rises a fifth, the denominator must grow by 1.2 x 60 / 50 = 1.44 times. So the larger stock is serving a much larger business, and is leaner relative to it. Had days stayed at 60, the distributor would be holding Rs 28.8 crore more stock in this example.

How can a bigger stock be leaner?

A kirana shop keeps Rs 60,000 of stock and sells Rs 1,000 a day at cost: two months of cover. A year later it keeps Rs 72,000 and sells Rs 1,440 a day: fifty days of cover. The shelf is fuller, yet each rupee of stock waits less time to be sold. Inventory days measure how long stock waits, not how much there is, so a larger rupee stock can still be leaner if what it serves grew faster.

Put rupees on it. Last year: COGS of Rs 730 crore is Rs 2.0 crore a day, and Rs 120 crore of stock is 60 days of it. This year inventory is Rs 144 crore. For that to be 50 days, daily COGS must be Rs 2.88 crore, an annual Rs 1,051.2 crore, up 44%.

What is the cash meaning of the change?

Compare the stock with what it would have been at the old turn. At 60 days, this year's cost of goods sold would need Rs 172.8 crore of inventory. The distributor holds Rs 144 crore. The faster turn means Rs 28.8 crore less cash tied up in stock than the business would otherwise need, even though the balance sheet shows inventory up Rs 24 crore. An analyst who reads only the rupee change calls this a working capital problem; the ratio says it is a working capital improvement.

What would make you distrust the 50 days?

Three checks. Year-end timing: inventory is a snapshot, and a stock-light last week of March can flatter the ratio; average inventory over the year is a better numerator. The denominator: some analysts compute days on revenue, not COGS, and mixing the two across years breaks the comparison. And mix: if the growth came from a fast-moving new product line, the old lines may be turning as slowly as ever. The ratio summarises; it does not explain.

Depreciation and amortisation rose from Rs 60 crore to Rs 90 crore, up Rs 30 crore or half again. D&A is exactly the gap between EBITDA and EBIT: 180 minus 120 last year, 200 minus 110 this year. The usual causes are a burst of capex now being depreciated, an acquisition bringing amortisation of intangibles, or an accounting change, such as lease accounting, that moves a cash cost out of operating expenses and into depreciation.

How do you find D&A from two margins?

If your take-home pay rises but your savings fall, something between the two lines got bigger: rent, EMI, school fees. You find it by looking at the gap, not the lines. EBITDA and EBIT differ by exactly one thing, depreciation and amortisation, so when the two margins move in opposite directions, D&A has grown by the sum of the two moves. Here, 2 points up and 1 point down is 3 points of Rs 1,000 crore of revenue, or Rs 30 crore.

Convert to rupees to be sure. EBITDA is Rs 180 crore last year and Rs 200 crore this year; EBIT is Rs 120 crore and Rs 110 crore. D&A is Rs 60 crore, then Rs 90 crore. Revenue is flat, so D&A went from 6% to 9% of revenue.

What could explain a 50% jump in D&A on flat revenue?

Three stories, each with a different meaning. First, a capex burst: a new plant or system has come into use and is being depreciated, but has not yet lifted revenue. That is a timing story, and the question is when revenue arrives. Second, an acquisition: buying a business brings customer lists and brands onto the balance sheet, and amortising them raises D&A without any new cash spend. Third, an accounting change. Under lease accounting standards such as Ind AS 116, rent stops being an operating expense and becomes depreciation plus interest, which raises EBITDA and D&A together without changing the cash the company pays.

The lease story fits the EBITDA rise neatly, since rent leaving operating expenses lifts EBITDA directly. But EBIT usually edges up under that change too, because part of the old rent now sits in interest, below EBIT. EBIT falling here suggests at least some genuine new depreciation as well.

Which margin should an analyst trust?

Neither on its own. EBITDA ignores the cost of the assets that produce the profit, so a company can raise its EBITDA margin simply by buying capacity or by reclassifying rent. EBIT charges for those assets, but through accounting depreciation that may not match their real wear. The honest check is cash: EBITDA less capex less lease payments, compared across both years, tells you whether the business really earns more. Ask management which of the three stories is true before praising the higher EBITDA margin.

Basic EPS is Rs 30.0; diluted EPS is Rs 26.88, about 10% lower. The if-converted method assumes conversion: add back the after-tax interest saved, Rs 30 crore x 0.75 = Rs 22.5 crore, and add 2 crore shares. That gives Rs 322.5 crore over 12 crore shares. On actual conversion, debt falls Rs 500 crore and equity rises Rs 500 crore, interest expense stops, and no cash changes hands.

What does the if-converted method assume?

Imagine a friend lent your shop Rs 5 lakh and can choose, any time, to swap the loan for a share of the business. To be honest with your other partners about what each share is worth, you would show the numbers as if the friend has already swapped: no more interest to pay, but one more partner splitting the profit. Diluted EPS shows earnings per share as if every convertible security that would reduce EPS had already converted, adding back the interest it would save and adding the shares it would create.

The interest is Rs 500 crore x 6% = Rs 30 crore a year. It was tax-deductible, so removing it raises net income by only Rs 30 crore x (1 minus 25%) = Rs 22.5 crore. Conversion creates Rs 500 crore / Rs 250 = 2 crore shares. Diluted EPS is (300 + 22.5) / (10 + 2) = Rs 26.875, which rounds to Rs 26.88.

How do you know whether a convertible dilutes?

Compare what each new share brings with what each existing share earns. Each conversion share brings Rs 22.5 crore / 2 crore = Rs 11.25 of earnings, far below basic EPS of Rs 30, so including the convertible lowers EPS and it must be counted. If basic EPS were below Rs 11.25, conversion would raise EPS; that security is anti-dilutive and is left out of diluted EPS. The test is per security, so a company with several convertibles ranks them from most to least dilutive.

What happens to each statement on actual conversion?

Balance sheet: borrowings fall Rs 500 crore and equity rises Rs 500 crore, split between share capital at face value and securities premium. Total assets do not move; the right-hand side changes its mix and leverage falls sharply. Income statement: from the conversion date, the Rs 30 crore of interest stops, so profit rises Rs 22.5 crore after tax. Cash flow statement: nothing at conversion, because no cash changes hands; it is disclosed as a non-cash transaction. Afterwards, the Rs 30 crore of cash interest is no longer paid. Reported basic EPS after conversion should land close to the diluted EPS already shown, which is why analysts value on the diluted figure.

No: 20% then 10% off is 28% off, not 30%; buy two get one free is a 33.3% discount per unit. Successive discounts multiply, because the second applies to the already reduced price: 0.80 x 0.90 = 0.72, so you pay 72% of the original. Buy two get one free means paying for two units and taking three, so each unit costs two thirds of its price, a third off, but only if you wanted three.

Why do discounts not simply add?

A Rs 1,000 kurta at 20% off costs Rs 800. The cashier then takes 10% off what you are about to pay, which is Rs 800, so the second discount is worth Rs 80, not Rs 100. You pay Rs 720. Each successive discount is taken off a price that the earlier discounts have already reduced, so discounts combine by multiplying what is left: 0.80 x 0.90 = 0.72, or 28% off. The order does not matter, 10% then 20% also gives 0.72, but the base does.

The shortfall from simple addition is the product of the two discounts: 20% x 10% = 2 points, the discount on the discount. It is small here and grows with the size of the cuts: 50% then 50% off is 75% off, not 100%.

What is buy two, get one free really worth?

You pay for two units and walk out with three, so each unit costs two thirds of the list price. Buy two get one free is a 33.3% discount per unit, deeper than the flat 30%, but only on the condition that you buy three. Buy one, get the second at half price is milder: one and a half prices for two units, 25% off each. If you needed only one, the effective discount of either offer is zero.

Where does this show up at work?

Everywhere rates are applied one after another. A distributor margin taken on a price that already carries a retailer margin, a trade discount followed by a cash discount, a fee charged after another fee: all compound. An FP&A analyst who adds layered discounts to estimate net price will overstate the discount and understate revenue per unit. The same arithmetic runs in reverse for mark-ups: a 20% price rise followed by a 20% cut leaves you 4% below where you started, because 1.2 x 0.8 = 0.96.

About 61% from the first 10 years and 94% from the first 30. The perpetuity is worth Rs 100 / 0.10 = Rs 1,000. Payments beyond year N are themselves a perpetuity, starting later, so their share is 1/1.1^N: 38.6% beyond year 10 and 5.7% beyond year 30. Add growth and the far years matter much more: at 5% growth only 37% of the value arrives in the first decade.

Is there a quick way to split a perpetuity by year?

Suppose you are promised a pension of Rs 100 a year forever. Starting ten years from now, what you are owed is again a Rs 100 perpetuity, worth Rs 1,000 at that date; you just have to wait ten years for it. The value of the payments after year N is the full perpetuity value, discounted back N years, so its share of the total is 1/(1 + r)^N and the first N years take the rest. At 10% that is 1/1.1^10 = 0.386 beyond year 10 and 1/1.1^30 = 0.057 beyond year 30.

In rupees, the first ten years are worth Rs 614 of the Rs 1,000. Half the value arrives in the first 7.3 years, because 1.1 to the power 7.3 is 2. A useful mental marker: at 10%, value halves every seven years or so.

Why does this matter for a DCF?

A DCF forecasts a company for perhaps ten years and puts everything after into a terminal value. With no growth, the terminal value would already be 39% of the total. Companies are usually assumed to grow, and growth pushes value further out: at 5% growth and a 10% discount rate, the gap between them shrinks to about 4.8% a year, half the value arrives only after 14.9 years, and the terminal value is 63% of the total. Most of a DCF's value sits beyond the explicit forecast, so the terminal growth rate and discount rate deserve more scrutiny than the detail of year three.

The limitation runs the other way too. The terminal value depends on the gap between the discount rate and growth, so a small change in either swings it sharply. Present a DCF with the terminal value's share stated and a sensitivity table beside it, rather than a single number.

One coffee buyer in three, 33.3%, must also buy a food item. Each cup loses Rs 20: Rs 140 to serve against Rs 120 of price. Each food item adds Rs 60 of contribution. So one food sale covers the loss on three coffees, and the break-even attach rate is 20 / 60. Two checks matter: only food bought because of the coffee counts, and Rs 140 must be the true extra cost of a cup, not a share of rent.

How do you price a product that loses money on purpose?

A shopkeeper who sells milk below cost does it because people who come in for milk also pick up bread and biscuits. The milk is a door, and its price is justified by what walks through it. A loss leader pays its way when the profit on what customers also buy covers the loss on the leader, so the number to manage is the attach rate. Here, each coffee loses Rs 20 and each food item earns Rs 60.

What makes the one-in-three figure misleading?

Two things. First, only incremental food counts. If half the people buying food would have come in for lunch anyway, their food was not brought in by the cheap coffee, and the coffee cannot claim it. The attach rate that matters is the share of coffee buyers who buy food because they came for coffee, which is lower than the attach rate on the till receipts. Measuring it needs a comparison: a price test in a few outlets, or the behaviour of customers before and after the coffee price changed.

Second, ask what is inside the Rs 140. If it includes a share of rent, staff and the coffee machine, those costs are there whether or not one more cup is sold. Suppose the extra cost of a cup, beans, milk and the cup itself, is Rs 70. Then each coffee contributes Rs 50 at the margin and needs no food to pay its way; it only looks like a loss after fixed costs are spread across it. Decisions about one more sale use contribution; fully loaded cost answers a different question, whether the whole cafe covers its overheads.

What would you track after launch?

Attach rate by time of day, average food contribution per coffee buyer, and the number of new customers the coffee brings in. A cafe that hits 40% attach in the morning and 15% in the afternoon may want the cheap coffee only before noon. The limitation is that habit takes time: a promotion judged on its first two weeks can look worse than it will in month three.

Roughly Rs 4.4 crore a year, on the assumptions below. Five shows a day, 200 seats and 30% average occupancy give about 1.1 lakh admissions a year. At an average ticket of Rs 250 that is Rs 2.74 crore; food and beverage at Rs 130 a head adds Rs 1.42 crore; on-screen advertising adds perhaps Rs 0.2 crore. Every input is an assumption to state and then test.

What is the structure before any number?

Estimating what a hall earns is like estimating a restaurant: count the covers, then what each cover spends, then any income that does not depend on covers. Estimate each revenue stream separately: tickets and food both hang off admissions, while advertising depends on the screen, not the audience on the day. So the first job is admissions: shows a day times seats times occupancy times days.

Assume five shows a day on a 200-seat screen, open every day. Occupancy is the soft input: a weekend evening may be nearly full while a Tuesday matinee is mostly empty. Build it rather than guess it: if weekdays run at 20% and weekends at 55%, the weighted average is (5 x 20% + 2 x 55%) / 7, about 30%.

How do you know the number is sensible?

Check it three ways. Per day: Rs 4.4 crore over 365 days is about Rs 1.19 lakh, which is 300 admissions spending about Rs 400 each in total, plausible for a screen with five shows. Per seat: about Rs 2.2 lakh a year. And the mix: food at about a third of revenue is the kind of share multiplex operators discuss, and it is high-margin money. The softest input is occupancy: every 10 points on it moves revenue by about Rs 1.4 crore, because it drives tickets and food together.

Say what revenue is not. Gross ticket revenue includes entertainment taxes in some form, and a large share of net ticket income goes to the film's distributor. Confirm current tax rates and typical revenue-sharing terms before using the figure for anything beyond a size estimate. Food, by contrast, the operator keeps in full, which is why its share matters to profit far more than its share of revenue suggests.