Investment Banking puzzles, solved step by step

Day 29. The coverage doubles every day, so it halves when you step back one day. Full on day 30 means half on day 29, a quarter on day 28 and one thirty-second on day 25. Most of the growth in a doubling process arrives right at the end.

Why is day 15 wrong?

Day 15 would be right if the pond gained the same area every day, like a tank filling from a tap. Doubling is different: it is money compounding at 100% a day. Under doubling, each step back in time halves the amount, so half full is always exactly one step before full. On day 15 the pond is only one part in about 32,768 covered.

Why would a banking interviewer ask this?

It is a test of whether you think in growth rates or in straight lines. The same instinct that answers day 15 also overestimates how much of a compounding investment's value arrives early. Say that link in one sentence after you answer: it shows you know why the question is on the table.

About 11.1%. A price of 100 falls 10% to 90. Getting back to 100 needs a rise of 10 on a base of 90, which is 10 divided by 90, or 11.1%. The general rule: to recover a loss of L, you need L divided by (1 minus L), which grows much faster than the loss itself.

Why is the recovery bigger than the fall?

A 10% discount on a 100 rupee shirt takes it to 90. If the shop then marks it up by 10%, the tag reads 99, not 100, because the markup is taken on 90. Percentages are always measured against a base, and after a fall the base is smaller, so the same rupee gain is a bigger percentage.

What do you add to sound like a banker rather than a calculator?

Give the number, then the pattern. Large drawdowns are hard to climb out of: a fund that loses 50% has to double just to break even. That one sentence shows you see why lenders and investors care about downside, which is the real reason the question is asked.

Net income falls Rs 7.5, cash rises Rs 2.5, inventory falls Rs 10 and retained earnings fall Rs 7.5. The write-down is an expense, so pre-tax income falls 10 and, after the Rs 2.5 tax saving, net income falls 7.5. No cash left the business, so the cash flow statement adds the 10 back and cash ends Rs 2.5 higher. Assets fall 7.5 and equity falls 7.5, so it balances.

Why does a loss leave the company with more cash?

A shopkeeper finds a carton of biscuits past their date. They cost Rs 10 and are now worth nothing. No money changes hands today; the cash went out when the biscuits were bought. What changes is that the loss lowers this year's taxable profit. A write-down is a non-cash loss, so its only cash effect is the tax it saves, and cash rises by 25% of Rs 10. Inventory is carried at the lower of cost and net realisable valueWhat the stock can be sold for, less the costs of selling it., which is why the carrying amount is cut when goods lose value.

How does each statement move?

Income statement: the write-down usually sits inside cost of goods sold, so pre-tax income falls 10, tax falls 2.5 at 25%, and net income falls 7.5. Cash flow statement: start from net income of minus 7.5, add back the 10 because no cash left, and operating cash flow is plus 2.5. Balance sheet: inventory is down 10 and cash is up 2.5, so total assets are down 7.5, and retained earnings are down 7.5. The check is minus 10 plus 2.5 on the asset side equalling minus 7.5 in equity.

What assumption should you say out loud?

That the write-down is deductible for tax now, as the question states. Whether a tax system allows the deduction when the stock is written down or only when it is sold depends on its rules, which you would confirm. If the deduction comes later, cash does not move this year and the company records a deferred tax asset of Rs 2.5 instead, with the balance sheet still balancing. Offering that variant in one sentence shows you understand why the cash moved in the first place.

About 8 years: 72 divided by 9 is 8.0, and the exact answer is 8.04 years. The exact doubling time is the log of 2 divided by the log of 1.09. The rule is a shortcut built for moderate rates: it is almost exact around 8%, slightly long at low rates and increasingly short at high ones. At 40% it says 1.8 years against an exact 2.06.

Why does 72 work at all?

Think of a sapling that grows 9% taller each year; the question is how many of those steps multiply up to 2. The exact answer uses logarithms: years equal ln 2 divided by ln(1 + r). For small r, ln(1 + r) is close to r, and ln 2 is 0.693, so the exact rule is close to 69.3 divided by the rate in per cent. 72 replaces 69.3 because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because ln(1 + r) sits below r at the rates people actually meet, which pushes the true constant up. At 9%, ln 1.09 is 0.0862, and 0.693 / 0.0862 is 8.04.

Where does the rule break down?

At high rates. ln(1 + r) falls further below r as r grows, so the true doubling time is longer than 72 divided by r: at 20% the rule says 3.6 years against 3.80, and at 40% it says 1.8 against 2.06, an error of 12.6%. At very low rates it errs the other way: 36 years against 35.0 at 2%. The rule is within about 1% of the exact answer only between roughly 6% and 10%, and should be adjusted outside that band.

A common adjustment for high rates adds one to the 72 for every three points of rate above 8%. At 20% that gives 76 divided by 20, 3.80 years, and at 40% about 82.7 divided by 40, 2.07 years, both within a hundredth or two of the exact figures. For deal work, where target returns of 20% to 30% are common, that adjustment is worth knowing.

115.6. Notice that 34% of 170 is the same as 17% of 340: halving one number and doubling the other leaves a product unchanged. So the sum is 17% of 340 twice, which is 34% of 340. That is 30% of 340, which is 102, plus 4% of 340, which is 13.6, giving 115.6.

What is the trick hiding in the numbers?

Two plots of land, one 34 metres by 17 and one 17 metres by 34, have the same area; turning a rectangle on its side does not change it. Percentages behave the same way, because x% of y is x times y divided by 100. The percentage and the base can trade places, so 34% of 170 equals 17% of 340: each is 17 x 340 / 100, which is 57.8. Once you see that, the question is one product, not two.

How do you do 34% of 340 in your head?

Split it into easy pieces: 30% of 340 is 102 and 4% of 340 is 13.6, so the total is 115.6. Or notice that 34% of 340 is 34 x 3.4, and 34 x 34 is 1,156, so the answer is 115.6. Spotting a structure first and calculating second is the habit the interviewer is checking, and it usually turns two awkward products into one easy one.

How do you check it before you say it?

Estimate first: 17% is a little more than a sixth, and a sixth of 340 is about 57, so each term is near 57 and the sum near 115. Then check each term directly: 10% of 340 is 34 and 7% is 23.8, so 17% is 57.8; 30% of 170 is 51 and 4% is 6.8, so 34% of 170 is 57.8. The two terms match, which confirms the swap.

Enterprise value stays the same and equity value rises by Rs 10 crore. Shareholders own the extra cash, so their equity is worth Rs 10 crore more. Enterprise value is equity plus debt minus cash: equity is up 10 and cash is up 10, so EV does not move. EV measures the operating business, and picking money up off the street does not change what that business earns. Ignoring tax on the windfall keeps the numbers clean.

What is enterprise value actually measuring?

Think of buying a shop whose till holds Rs 10,000. You would pay for the business plus the cash in the till, and the cash is worth exactly its face value to you, no more and no less. Enterprise value prices the operating business on its own, so cash, which is worth its face value to whoever holds it, is taken out of the bridge. Equity value is what shareholders own, and they own both the business and the cash.

How does the bridge move?

Take a company with equity worth Rs 500 crore, debt of Rs 300 crore and cash of Rs 50 crore, so enterprise value is Rs 750 crore. Find Rs 10 crore: cash becomes Rs 60 crore and equity Rs 510 crore, and EV is 510 plus 300 minus 60, still Rs 750 crore. Nothing in the operating business changed, so multiples of EBITDAEarnings before interest, tax, depreciation and amortisation: a rough measure of the cash profit the operating business produces. or revenue do not change either.

What assumptions should you say out loud?

Two. First, tax: the windfall is probably taxable income, so at a 25% rate equity and cash each rise by Rs 7.5 crore rather than Rs 10 crore. Whatever amount sticks, cash and equity move together and enterprise value stays put. Second, the cash sits idle. If the company used it to repay debt, EV would still not change, but the split between lenders and shareholders would. Naming both shows the interviewer you know which line each event touches.

66 handshakes. Each of the 12 bankers shakes hands with the other 11, which gives 12 x 11 = 132, but every handshake has two people in it and so has been counted twice. Half of 132 is 66. The same number falls out of adding 11 + 10 + 9 and so on down to 1, or from 12 choose 2.

Why do you halve the count?

Think of counting the phone calls in a group where everyone rang everyone else once. If each person reports the calls they took part in, every call shows up in two reports. Each handshake involves two people, so counting from each person's side counts every handshake exactly twice, and the true number is half. Twelve people times eleven others is 132 ends of handshakes, which is 66 handshakes.

How do you check 66 another way?

Let people arrive one at a time. The second person shakes 1 hand, the third 2, and so on to the twelfth, who shakes 11, and adding 1 through 11 gives 66. Pair the terms to add them quickly: 1 and 11, 2 and 10, and so on make five pairs of 12 plus a 6 in the middle, 66. Two routes to the same number is the check interviewers like to hear.

Where does this count show up on a desk?

Anywhere pairs matter. The number of pairs grows roughly with the square of the group: double the group to 24 and the handshakes rise to 276, more than four times as many. That is why a 12-stock portfolio has 66 pairwise correlationsMeasures of how closely two things move together, from minus 1 to plus 1. to estimate, and why a deal with many parties needs far more conversations than its headcount suggests.

Fund B ends higher, at Rs 121 against Rs 117. Fund A goes to 130 and then loses 10% of 130, ending at 117. Fund B compounds 10% twice to 121. Both average 10% a year, but A's compound growth rate is only about 8.2% because its returns swing. Volatility drags on compound returns: roughly half the variance comes off the average each year.

Why does the same average give different endings?

Think of a salary that rises 30% one year and is cut 10% the next, against one that rises 10% twice. The cut lands on the higher salary, so it takes away more rupees than the same percentage would have earlier. Returns multiply rather than add, so a loss after a gain is taken from a bigger base, and an uneven path ends below a smooth one with the same average. Rs 100 grows to 130 and then gives back Rs 13; the steady fund never gives anything back.

How big is the drag, and can you estimate it in your head?

The growth rate that matters is the geometric averageThe constant yearly return that would turn the starting amount into the ending amount over the same period.. Fund A's is the square root of 1.17 minus 1, about 8.2%, against B's 10%, even though both arithmetic averages are 10%. A quick estimate: subtract half the variance. A's returns sit 20 points either side of 10%, so half of 0.2 squared is 2 points, and 10% minus 2% is about 8%, close to the exact 8.2%.

Why would an interviewer care?

Because fund reports often quote average returns, and investors live on compound ones. Two funds with the same average return can leave an investor with very different money, and the more volatile one leaves less. The same arithmetic explains why a fund that rises 50% and then falls 50% is down 25%, and why leverage that doubles volatility can lower long-run growth even while it raises the average.

Only the balance sheet changes. Property, plant and equipment rises by Rs 100 crore and accounts payable rises by Rs 100 crore, so both sides grow by the same amount. No cash has moved, so the cash flow statement is untouched, and no time has passed for depreciation, so the income statement is untouched too. Cash and capex appear in 60 days, when the supplier is paid.

Why does a purchase on credit touch neither cash nor profit?

Think of buying a refrigerator on a shop's 60-day credit. The fridge is in your kitchen today and you owe the shop, but your bank balance has not moved and nothing has come out of this month's budget. A credit purchase adds an asset and a debt of the same size, so the balance sheet grows on both sides and nothing else moves. The machine will be used for years, so its cost is not an expense on the day it arrives. It reaches the income statement slowly, through depreciation, over the years it is used.

What happens over the next 60 days and beyond?

Day 60 is when the cash flow statement wakes up. The company pays the supplier: cash falls by 100 and payables fall by 100. The Rs 100 crore appears as capital expenditure in investing cash flow in the year it is paid, not the year the machine arrived. The notes to the accounts usually flag the year-end purchase as a non-cash investing item, so a reader is not surprised. From the following year, depreciation starts: on a 10-year straight line, Rs 10 crore a year comes off pre-tax profit, is added back in operating cash flow, and lowers the machine's book value.

Close by saying the balance check out loud. Assets up 100, liabilities up 100: the sheet balances, and that one sentence tells the interviewer you walk the statements in a fixed order rather than guessing. Interviewers use small timing questions like this one to see whether you separate when something is owned, when it is paid for and when it is expensed.

The WACC is 11.4%. Equity is 60% of the funding at 14%, contributing 8.4 points. Debt is 40% at 10% before tax, but interest is tax-deductible, so its after-tax cost is 10% x (1 minus 25%) = 7.5%, contributing 3.0 points. Add them: 8.4 + 3.0 = 11.4%. Leaving out the tax shield gives 12.4%.

Why is it a weighted average and not a plain one?

A family buying a flat with 60% from savings and 40% from a home loan pays a blended cost that leans towards the bigger source. Each source of money counts in proportion to how much of the funding it provides. Here equity provides 60 rupees in every 100, so its 14% carries more weight than debt's rate. A plain average of 14% and 10% would be 12.0%, which pretends the two sources are the same size.

Why does debt go in after tax?

Interest is deducted before profit is taxed, so every 10 rupees of interest cuts the tax bill by 2.50 rupees at a 25% rate. The tax saved pays a quarter of the interest, so the company's true cost of debt is 7.5%, not 10%. That saving is the tax shieldThe tax a company avoids because interest is deducted from profit before tax is worked out.. Equity has no such shield, because dividends are paid out of profit after tax, which is why the adjustment sits on the debt term only.

One more sentence wins the point. The weights should be market values, not the book values on the balance sheet, because investors demand a return on what their stake is worth today. Then say what the number is for: 11.4% is the rate at which you would discount the company's unlevered free cash flows in a DCF.