Financial Analysis puzzles, solved step by step

The cost of equity is 13.6% at debt-to-equity of 0.5, and 15.4% at 1.0. Relevering puts your own debt back onto the peer's business risk: 0.8 times (1 plus 0.75 times 0.5) gives a beta of 1.10, and 7% plus 1.10 times 6% is 13.6%. At 1.0 the beta is 1.40, so the cost of equity is 15.4%.

Why unlever the peer's beta and then relever it?

Two friends buy identical flats. One pays cash; the other borrows 60% of the price. When flat prices move 10%, the cash buyer's wealth moves 10%, while the borrower's stake moves far more, because the loan does not shrink with the price. A peer's observed beta mixes two things, the risk of its business and the risk its own debt adds, so you strip out its debt before borrowing its business risk. Then you add back your own debt, because your shareholders carry your leverage, not the peer's.

This is the Hamada relation, which assumes debt carries no market risk of its own. Then the capital asset pricing model does the rest: cost of equity is the risk-free rate plus beta times the equity risk premium, so 7% plus 1.10 times 6% is 13.6%. At debt-to-equity of 1.0, beta becomes 0.8 times 1.75, or 1.40, and the cost of equity is 15.4%.

Does more debt make the company more expensive to fund overall?

Not by this alone. Equity gets dearer as debt rises, but debt is cheaper than equity and its interest is tax-deductible, so the weighted cost of capital can fall even while the cost of equity climbs. Where it turns is where lenders start charging more and distress becomes real, which this formula does not capture. Say that limit out loud: the straight line only holds while debt is safe.

About Rs 2.16 lakh. The cow earns Rs 1,20,000 of milk a year and costs Rs 70,000, leaving Rs 50,000 of net cash for six years. At 12% that stream is worth about Rs 2,05,570, and the Rs 20,000 sale in year 6 adds about Rs 10,133. Paying more than about Rs 2,16,000 means earning less than 12% on the purchase.

How do you value something that is not a company?

The same way you value a company. A shop, a flat you rent out and a cow are all machines that turn money in now into money out later. Anything that produces cash can be valued as the present value of the cash it will hand you, after the costs of keeping it running. So the first job is to find the net cash each year, then the life, then the value left at the end.

Net cash: 10 litres a day for 300 days is 3,000 litres, worth Rs 1,20,000 at Rs 40. Upkeep of Rs 70,000 leaves Rs 50,000 a year. The six years and the Rs 20,000 sale at the end complete the cash flows.

What would make you pay less?

Every assumption in this answer is a risk to price: yield falling with age, illness, milk price, feed cost. If milk yield drops 10% a year, net cash falls fast because upkeep does not fall with it, and the value drops sharply. A higher required return for a risky asset does the same. The interviewer who asks you to value an animal is checking that you name the cash, the life, the end value and the risk, in that order, and do not stop at the milk bill.

About 2,000 sit-down cafés, as a range of roughly 1,000 to 4,000. Take about 1.3 crore people, 70% of them aged 15 to 64, and assume 20% of those visit a café twice a week. That is about 5.2 lakh cups a day; at 250 cups a café a day, about 2,080 cafés. A supply check, 150 commercial clusters with 10 cafés each plus 500 neighbourhood ones, gives about 2,000.

What exactly are you counting?

Say the definition before any number. In Bengaluru the answer changes several times over depending on whether small standing filter coffee counters count, so define a coffee shop as a sit-down café selling espresso-style coffee and say you are leaving the counters out. Then state the population as an assumption, about 1.3 crore for the metro area, and tell the interviewer you would confirm it. In a market sizing nobody is marking the population figure; they are marking whether your chain of reasoning holds.

How do you build it from demand?

Think of a single café first: how many cups does it need to sell to pay its rent? Then ask how many cups the city drinks. Of 1.3 crore people, about 70% are aged 15 to 64, so 91 lakh. Assume 20% of them use cafés, 18.2 lakh people, at two cups a week each: 36.4 lakh cups a week, or about 5.2 lakh a day. A café open 12 hours selling around 20 cups an hour serves about 250 cups a day, so the city supports about 2,080 cafés.

How do you check it, and which assumption matters most?

Count from the other side: supply. Picture the tech parks, malls and busy high streets, say 150 of them, with about 10 cafés each, plus about 500 standalone neighbourhood cafés. That is about 2,000, close to the demand answer. Then name the swing factor. The 20% share of café users is the softest input: halve it and the answer falls to about 1,040. Say the range out loud, roughly 1,000 to 4,000, because a single precise number from soft inputs sounds less credible, not more.

Roughly 21,800 to 26,200. Scale the 16% annual volatility by the square root of the time, not the time itself: four months is a third of a year, the square root of a third is about 0.58, and 16% times 0.58 is about 9.2%. That is about 2,217 points either side of 24,000, a band that holds the close about two times in three.

Why can you not just say where the index will close?

You cannot, and saying so is part of a good answer. The honest reply to a where-will-it-close question is a central point and a range, with the range doing most of the work. Today's level is the natural centre over a few months, since the expected drift is small next to the spread. The interviewer reported here asked a sales and trading candidate exactly this, and what they are testing is whether you can turn a volatility number into a range in your head.

Why does volatility scale with the square root of time?

Picture someone walking along a lane, taking each step forward or back at random. After four steps they are not usually four steps away; good and bad steps partly cancel, and the typical distance is about two steps, the square root of four. Price moves behave the same way: variance adds up with time, so the standard deviation grows with the square root of time. A third of a year is not a third of the annual volatility; it is the square root of a third, about 58% of it.

How do you check it, and what are you leaving out?

Check by another route. Monthly volatility is 16% over the square root of 12, about 4.6%, and four months is the square root of 4, which is 2, times that: about 9.2%. Same answer. Two standard deviations, about 19,600 to 28,400, covers roughly 95% of outcomes, and naming that wider band shows you know a one standard deviation range will be wrong a third of the time. Say the limits too: prices compound, so the upper side is slightly wider, about 21,900 to 26,300 on a log basis; the drift is ignored; and real index returns have fatter tails than a bell curve.

Net income falls Rs 60 crore, cash falls Rs 10 crore and debt rises Rs 50 crore. All Rs 100 crore of interest is an expense, so pre-tax profit drops 100 and tax drops 40. On the cash flow statement, add back the Rs 50 crore paid in kind because no cash left, so cash from operations falls 10. On the balance sheet, cash is down 10, debt up 50 and retained earnings down 60: both sides fall 10.

What does paid in kind actually mean?

Think of a credit card bill where you pay half the interest and the bank adds the other half to what you owe. You have still been charged the full interest; you just have not paid all of it in cash. PIK interestPaid-in-kind interest: interest that is settled by adding it to the loan principal instead of paying cash, so the debt grows each period. is a real expense that is settled by growing the loan instead of draining the bank account. That single sentence tells you where each half goes: the expense hits profit in full, and the PIK half turns up as more debt.

How does it move through each statement?

Income statement: interest of Rs 100 crore cuts pre-tax profit by 100. Assuming all of it is deductible, tax falls by Rs 40 crore, so net income falls Rs 60 crore. Cash flow statement: start from net income, down 60, and add back the Rs 50 crore of PIK as a non-cash charge. Cash from operations falls by only Rs 10 crore, because the tax saving on the whole Rs 100 crore nearly covers the Rs 50 crore of cash interest. Balance sheet: cash down 10 on the assets side; debt up 50 and retained earnings down 60 on the other side. Both sides fall by 10.

What should you add after the walk-through?

Two points earn credit. First, the tax assumption: some tax systems limit how much interest a company can deduct, and the rules on PIK can differ, so say you have assumed full deductibility and would confirm the current rule. If the PIK half were not deductible, net income would fall Rs 80 crore and cash Rs 30 crore. Second, PIK compounds: next year's interest is charged on a loan Rs 50 crore larger, which is why lenders price it higher and why it shows up in leveraged buyouts where cash is tight early on.

About Rs 157.5 crore. Start from potential rent of Rs 20 crore and take off 10% vacancy to reach effective gross income of Rs 18 crore. Operating expenses at 30% of that are Rs 5.4 crore, leaving net operating income of Rs 12.6 crore. Divide NOI by the 8% cap rate: Rs 12.6 crore over 0.08 is Rs 157.5 crore, which is 12.5 times NOI.

Why does a cap rate apply to net income and not to rent?

Think of a flat you rent out for Rs 30,000 a month. Some months it sits empty, and the society charges, repairs and property tax come out of your pocket. What you would pay for the flat depends on what is left, not on the rent written in the agreement. A cap rate is the yield a buyer wants on net operating income, the cash the building throws off after vacancy and running costs but before any loan payments. Applying it to gross rent values money the owner never receives.

How do you walk from potential rent to value?

Three steps, in order. Gross potential rent is what the building would earn fully let: Rs 20 crore. Take off vacancy first, because operating expenses here are a share of the income actually collected, not of the potential. 10% vacancy leaves effective gross incomeRent the building actually collects after vacancy and bad debts, before operating expenses. of Rs 18 crore. Expenses at 30% of 18 are Rs 5.4 crore, which leaves NOI of Rs 12.6 crore. Then divide by the cap rate.

What makes the exit cap rate the number to argue about?

Value is very sensitive to the cap rate: one point lower, at 7%, the building is worth Rs 180 crore; one point higher, at 9%, Rs 140 crore. That 1 point swing moves value by about Rs 40 crore on a Rs 157.5 crore building. Analysts usually set the exit cap rate a little above today's rate, because the building will be older when it is sold. Say the limit as well: a single cap rate assumes NOI is stable, so a building with large leases expiring soon needs a cash flow model, not one division.

EBITDA is wrong: it should be Rs 80 crore, not Rs 90 crore. Check every subtotal. Revenue less cost of goods sold is 200, so gross profit is right. Gross profit less opex is 80, not 90. EBITDA less D&A is 70, not the 60 shown. EBITDA is the only number in both failed checks, and setting it to 80 makes every line reconcile, including EBIT at 60.

Where do you start when one number in a page is wrong?

Think of a shop's daily cash sheet where the till, the card machine and the total do not agree. You do not stare at the biggest number; you re-add each subtotal and see which ones break. A P&L is a chain of subtractions, so every subtotal can be tested against the lines above it and the lines below it. Write the three identities out loud: gross profit is revenue less cost of goods sold; EBITDA is gross profit less operating expenses; EBIT is EBITDA less D&A.

Why is EBITDA the culprit and not opex or EBIT?

Two checks fail, and they share exactly one line. The wrong number is the one whose single correction fixes every failed check at once. Suppose opex were wrong instead: setting it to 110 makes EBITDA of 90 look right, but 90 less 20 still is not 60, so a second error would be needed. Suppose EBIT were wrong: 70 would fix the bottom check but leave 200 less 120 against 90. Only EBITDA at 80 repairs both. The puzzle says exactly one number is wrong, so that settles it.

What should you say beyond the answer?

State the assumption that made the puzzle solvable: operating expenses here exclude D&A, so EBITDA is gross profit less opex. If opex included depreciation, the chain would read differently. On the job, this test is the first thing a reviewer runs on any summary table, because subtotal errors usually come from a hard-coded number that did not update when the line above changed. Say where you would look next: the cell that feeds EBITDA, and whether margins quoted elsewhere in the pack used the wrong 90. At 90, the EBITDA margin reads 18% instead of 16%.

531,441. Split the power: 9 to the 6th is 9 cubed, squared, and 9 cubed is 729. Square 729 as (730 minus 1) squared: 730 squared is 532,900, less twice 730, which is 1,460, plus 1, gives 531,441. Check it: an even power of 9 ends in 1, the digits add to 18, and 0.9 to the 6th is about 0.53, so the size is right.

How do you break a big power into steps you can hold in your head?

Think of carrying a heavy suitcase up six floors: you rest on the landings. Split the exponent into pieces you know, 6 = 3 x 2, so 9 to the 6th is 9 cubed, squared. Most people know 9 squared is 81 and 9 cubed is 729. That turns the question into one square, 729 times 729, which is still awkward until you move it next to a round number.

Why square 730 instead of 729?

Because 730 squared is easy: 73 squared is 5,329, so 730 squared is 532,900. The identity (a minus 1) squared equals a squared minus 2a plus 1 lets you square any number just below a round one with two easy corrections. Twice 730 is 1,460. So 532,900 less 1,460 is 531,440, and adding 1 gives 531,441. Each step is small enough to say out loud while you do it.

How do you check it before you say it?

Three checks take seconds. The last digit of an even power of 9 is always 1, the digit sum of any power of 9 is divisible by 9, and 9 to the 6th is 0.9 to the 6th times a million. Here the digits 5, 3, 1, 4, 4, 1 add to 18, and 0.9 to the 6th is about 0.53, so the answer must be about 530 thousand. That last check catches the common slip of being off by a factor of ten. A second route also helps: 81 squared is 6,561, and 6,561 times 81 is 524,880 plus 6,561, again 531,441.

17 minutes. 1 and 2 cross (2 minutes) and 1 brings the torch back (1). Then 5 and 10 cross together (10), and 2 brings the torch back (2). Finally 1 and 2 cross again (2). The obvious plan, where the 1-minute person escorts everyone, takes 19 minutes because the two slowest walkers cost 15 minutes on separate trips but only 10 together.

Why does the obvious plan lose two minutes?

The escort plan sounds efficient: the fastest person walks each one across and runs back with the torch. It costs 2, 1, 5, 1 and 10, which is 19 minutes. The waste is that the 5 and the 10 cross on separate trips, so you pay both their times, 15 minutes, when one shared trip would cost only 10. Think of two slow friends who will hold up any group they walk with; the cheapest thing is to let them hold up the same group, once.

How do you make sure the torch is on the right side?

If 5 and 10 cross first, one of them has to bring the torch back, which costs at least 5 more minutes. So before the slow pair goes, a fast walker must already be waiting on the far side to carry the torch back. That is why 1 and 2 go first and 1 returns: it parks the 2 on the far side. After the slow pair crosses, the 2 brings the torch back, and the fast pair finishes together.

How do you show 17 cannot be beaten?

There must be three forward trips and two returns. The 10 must cross on some forward trip, costing 10. If the 5 crossed on a different forward trip, the forward trips alone would cost at least 10 + 5 + 2, which is 17, and the two returns add at least 2 more, so 5 and 10 must share a trip. The slow pair cannot go first, or one of them walks the torch back. So 1 and 2 go first, costing 2, and one of them returns. After the slow pair crosses, the only fast walker on the far side is whichever of 1 and 2 stayed, so the two returns cost 1 + 2 between them, and the last trip, 1 and 2, costs 2. That gives at least 10 + 2 + 2 + 1 + 2 = 17. A brute-force search over every schedule confirms 17.

Equity falls from Rs 2,400 crore to Rs 800 crore, a 66.7% drop on a 20% fall in EV. Debt is 7 x 800 = Rs 5,600 crore. At 10x, EV is Rs 8,000 crore; at 8x it is Rs 6,400 crore. Debt keeps its claim, so the equity slice absorbs the whole Rs 1,600 crore. At 7x leverage, equity moves 3.3 times as fast as EV, and a 7x multiple wipes it out.

Why does equity fall so much further than the business?

Think of a flat bought for Rs 1 crore with a Rs 70 lakh home loan. If the flat's price drops 20% to Rs 80 lakh, the bank still wants its Rs 70 lakh, so the owner's stake falls from Rs 30 lakh to Rs 10 lakh, a two-thirds loss. Debt is a fixed claim in rupees, so any change in enterprise value lands entirely on the equity that sits on top. Here debt is 7 x 800 = Rs 5,600 crore. At 10x, EV is Rs 8,000 crore and equity is Rs 2,400 crore, only 30% of the value.

So what does 7x leverage tell you about sensitivity?

Leverage of 7x EBITDA means the debt alone is worth 7x EBITDA. Whenever the valuation multiple sits close to the leverage multiple, equity is a thin slice of the gap between them, and small changes in the multiple swing it hard. At 10x the gap is 3 turns; each turn of multiple is Rs 800 crore, a third of the equity. Software companies can carry this because their cash flows recur and their multiples are high, but the same 7x on a business valued at 8x leaves one turn of cushion. Say the limitation: the sum assumes debt is worth its face value, while in distress the debt itself would trade below par and soak up some of the loss.