Investment Banking puzzles, solved step by step

488 small cubes are painted and 512 are not. The unpainted ones form a hidden core with one layer peeled off every side, so the core is 8 x 8 x 8, which is 512. The whole cube is 1,000, so 1,000 minus 512 leaves 488 with at least one painted face.

Why count the cubes you cannot see?

Picture a mango with its skin peeled. Measuring the skin piece by piece is fiddly; measuring the fruit left inside and subtracting from the whole is easy. The unpainted cubes form one clean block, so counting them and subtracting is faster and harder to get wrong than adding up the painted surface. Peeling one layer from both sides of each dimension takes 10 down to 8.

How do you check 488 the other way?

Count the surface directly, taking care not to double count. There are 8 corners, 12 edges each carrying 8 cubes once the corners are removed, and 6 faces each carrying an 8 x 8 centre. That gives 8 plus 96 plus 384, which is 488. Two methods agreeing is exactly what the interviewer wants to hear.

Fill the 3-litre bottle, pour it into the 4, fill the 3 again and pour until the 4 is full: 2 litres are left in the 3-litre bottle. That is four moves. Because 3 and 4 share no common factor, fills and pours can build every whole number of litres, so any total from 1 to 7 is possible across the two bottles.

What does each move actually measure?

Think of paying for something with only Rs 3 and Rs 4 coins. You can pay Rs 2 by handing over two Rs 3 coins and taking a Rs 4 coin back as change. The bottles work the same way. Every fill adds a full bottle and every pour into a full bottle sets one bottle's worth aside, so the volumes you can make are sums and differences of 3s and 4s. Two litres is 3 plus 3 minus 4: you draw two small bottles from the tap, the big bottle swallows 4 of those 6 litres, and 2 are left.

Is there a second route, and which volumes are out of reach?

Start from the big bottle and it still works, only slower. Fill the 4, pour into the 3 to leave 1, empty the 3, move the 1 litre across, fill the 4 again and top up the 3, which takes 2 litres and leaves 2 in the big bottle. That is six moves, 4 plus 4 minus 3 minus 3. Saying both routes and choosing the shorter one shows the interviewer you searched the problem rather than remembered it. Write each state as a pair, small bottle first, so neither of you loses track.

For the second half of the question, use BezoutA result in number theory: whole-number combinations of two numbers produce exactly the multiples of their highest common factor, and nothing else.'s rule: two bottle sizes can measure exactly the multiples of their highest common factor. The highest common factor of 3 and 4 is 1, so every whole number of litres can be built, limited only by capacity: up to 4 in one bottle and 7 across both. Contrast a 4 and a 6: their common factor is 2, so 3 litres is impossible however long you pour.

About Rs 92.4 per Rs 100 of face value. Discount the five Rs 8 coupons and the Rs 100 repayment at 10%: the coupons are worth Rs 30.33 today and the repayment Rs 62.09. It sits below par because it pays Rs 2 a year less than a new bond, and a buyer wants those five years of shortfall, worth Rs 7.58 today, taken off the price.

How do you price it from first principles?

Imagine you let a flat on a five-year lease at Rs 8,000 a month, and every similar flat now fetches Rs 10,000. Someone buying your lease will pay less for it, by roughly the rent you miss over the remaining term. A bond is the same lease on money. A bond's price is every future payment discounted at the yield the market now demands, so when that yield rises above the coupon, the price has to fall below par. At 10%, each Rs 8 coupon and the final Rs 100 are worth less today the further away they sit.

Is there a shortcut that skips most of the discounting?

Yes, and it is the one to use out loud. Price equals par minus the yearly coupon shortfall times the annuity factor: 100 minus 2 x 3.79 is 92.42. The annuity factorThe present value of Rs 1 a year for a set number of years. At 10% for five years it is about 3.79. is the only number that needs any work, and at 10% for five years it is close to 3.8. The same rule runs the other way: a bond paying above the market yield trades above par by the extra coupon times the annuity factor.

Two limits are worth one sentence each. The yield must come from bonds of the same maturity and credit quality, which is what similar bonds means; in today's market you would read it off comparable issues. Real bonds also pay coupons twice a year in many markets and trade with accrued interest, which shifts the answer by small amounts. For an interview answer, annual discounting is the expected level.

Roughly 2,000 to 2,500 taxis, inside a defensible range of about 1,000 to 4,000. Assume 300,000 people in Central at lunch, 2% of them taking a taxi trip that touches the district in the peak hour, and each trip spending 15 minutes inside it: 1,500 cabs occupied at any moment. Add half as many again cruising or queuing empty and you reach about 2,250.

Why count trips and minutes rather than cabs?

Picture a busy restaurant. If 60 diners arrive an hour and each stays half an hour, about 30 seats are taken at any moment, whatever is on the menu. That rule, known as Little's lawThe number of items in a system on average equals the rate they arrive multiplied by how long each one stays., turns a flow into a count. The taxis in Central at one moment equal the trips touching Central per hour multiplied by the hours each trip spends there, plus the cabs that are empty. You cannot see cabs from an interview room, but you can reason about people and journeys.

Where do the numbers come from, and which one moves the answer most?

Every number is an assumption you state, not a fact you recall. Central is a dense office district, so assume 250,000 office workers plus 50,000 visitors and shoppers at lunch. Most walk or take the train; assume 2% take a taxi trip that starts or ends in Central in the busiest hour, which is 6,000 trips. A cab picking up, crawling through lunchtime traffic and dropping off spends perhaps 15 minutes in the district per trip. The minutes-per-trip assumption moves the answer as much as the head count does, so say it slowly and invite challenge on it. Empty cabs cruising or waiting at ranks add about half again.

How do you sanity-check it from the supply side?

Come at it from the fleet. Suppose the city has about 18,000 licensed taxis, a number you would state as an assumption and check against the current licence count, and the busiest business district holds 10% to 15% of them at peak. That gives 1,800 to 2,700. Two methods landing in the same range is worth more than either number on its own. If they had disagreed by a factor of three, you would say which assumption you trust less and why.

Net income falls Rs 60, cash falls Rs 10, debt rises Rs 50 and retained earnings fall Rs 60. The full Rs 100 is expensed, so pre-tax income drops 100 and, after 40% tax, net income drops 60. The cash flow statement adds back the Rs 50 of PIK as a non-cash expense. On the balance sheet the PIK accrues to the debt, and assets of minus 10 equal liabilities and equity of plus 50 minus 60.

What does paid in kind mean in plain terms?

Suppose you owe a friend Rs 100 of interest, pay Rs 50 in cash and write an IOU for the other Rs 50. Your wallet is Rs 50 lighter, and you now owe Rs 50 more than before. Paid in kindInterest settled by adding it to the loan principal instead of paying cash. Often shortened to PIK. interest works exactly like that IOU. PIK interest is a real expense paid with more debt instead of cash, so it reduces profit while adding to the loan. That one sentence tells you where every number in the walk will land.

Why does cash fall by only Rs 10?

Because tax is worked out on the whole expense, not just the cash part. Assume both halves of the interest are tax deductible, which is the usual set-up for this question; real deductibility rules vary by country and are often capped, so say the assumption. The Rs 100 expense saves Rs 40 of tax, and the only cash going out is the Rs 50 of cash interest, so cash falls Rs 50 minus Rs 40, which is Rs 10. On the cash flow statement the same Rs 10 appears as net income of minus 60 plus the Rs 50 PIK add-back.

How do you prove the balance sheet balances?

Say it in one breath. Assets fall 10 through cash. Liabilities rise 50 because the PIK interest is added to the principal of the debt. Equity falls 60 through retained earnings. Minus 10 on the asset side equals plus 50 minus 60 on the other side, so the balance sheet balances. Saying the check out loud is what turns a recited answer into a reasoned one, and it catches the slip of forgetting where the PIK went.

EV/Revenue is 2.0x and price to sales is 1.8x. Take Rs 100 of revenue. A 20% margin gives EBITDA of Rs 20, and at 10x the enterprise value is Rs 200, two times revenue. Net debt equals one year of EBITDA, Rs 20, so equity value is Rs 200 minus Rs 20, which is Rs 180, or 1.8 times revenue.

How do multiples convert into one another?

Think of a corner shop priced at 10 years of profit, where profit is a fifth of takings. Priced against takings, it costs two years' worth: 10 times a fifth. An enterprise value multiple converts into another by the ratio of their denominators, so EV/Revenue equals EV/EBITDA times the EBITDA margin. Working per Rs 100 of revenue keeps every step visible: EBITDA Rs 20, enterprise value Rs 200.

Why is price to sales lower than EV to revenue here?

Price to sales is an equity multiple: the market value of the shares over revenue. To reach it from enterprise value you cross the equity bridgeThe steps from enterprise value to equity value: subtract debt and similar claims, add cash.. Equity value is enterprise value less net debt, so an equity multiple sits below the enterprise multiple when a company has net debt, and above it when it holds net cash. Net debt of one year's EBITDA is Rs 20 per Rs 100 of revenue, which takes Rs 200 down to Rs 180, or 1.8x.

The limitation to state is why bankers prefer one family over the other. Revenue belongs to every funder of the business, so comparing it with enterprise value is consistent; comparing it with equity value alone mixes a whole-business line with one funder's slice. Two companies with identical operations but different debt will show the same EV/Revenue and different price to sales ratios.

About Rs 50. Prices rising 6% a year compound to 1.06 to the power of 12, which is 2.01, so a basket costing Rs 100 today costs about Rs 201 in year 12. Rs 100 then buys what Rs 49.70 buys now. The rule of 72 gets there in one line: 72 divided by 6 is 12 years for prices to double.

Why does the answer land so close to half?

Think of a plate of biryani that costs Rs 100 today. If its price rises 6% every year, each rise is taken on the previous year's higher price, so the rises themselves grow: Rs 6 in the first year, close to Rs 12 by the twelfth. Inflation compounds like interest, so over 12 years at 6% prices roughly double, and the same Rs 100 note buys roughly half as much. To state a future rupee in today's money, you divide by the growth in prices, 1.06 to the 12th.

How do you get 1.06 to the 12th without a calculator?

Square your way up. 1.06 squared is 1.1236; squared again gives 1.2625 for four years; squared again gives 1.594 for eight years. Twelve years is eight plus four, so multiply: 1.594 x 1.262 is 2.012. Repeated squaring turns eleven multiplications into four, and every step can be said out loud. Then Rs 100 divided by 2.012 is Rs 49.70, and the rule of 72 has already told you to expect about 2.

What are the two wrong answers, and why are they tempting?

The straight-line answer, Rs 28, takes Rs 6 off every year as if prices rose by the same rupee amount each time. The subtler one, Rs 48, shrinks the money by 6% a year, 0.94 to the 12th. Inflation raises prices by 6%, which cuts buying power by 6 divided by 106, about 5.7% a year, not by a full 6%. The gap between Rs 48 and Rs 50 is small here and widens with the rate and the years.

Why a deal team cares: a return in rupees is not a return in buying power. Money growing at an assumed 12% a year while prices rise 6% grows in real terms by 1.12 divided by 1.06, minus one, which is 5.66% a year, a little under the 6% you get by simple subtraction. Over a long holding period that gap compounds too.

Enterprise value is unchanged, equity value rises by Rs 100 crore and EV/EBITDA stays the same. Equity value rises because Rs 100 crore of new shares now exist, and debt falls by the same Rs 100 crore. Enterprise value is equity plus net debt, so the two moves cancel. EBITDA sits above the interest line and is untouched, so the multiple cannot change either.

Why does enterprise value not move?

Think of a house worth Rs 1 crore with a Rs 40 lakh mortgage. If you pay down Rs 10 lakh of the loan with your savings, your stake rises to Rs 70 lakh and the loan falls to Rs 30 lakh, but the house is worth exactly what it was. Enterprise value measures the business, and swapping one kind of claim for another changes how that value is split, not how big it is. At first order, issuing shares to repay debt is exactly that swap.

Then what does change?

Everything below the interest line. Take EBITDA of Rs 100 crore, D&A of Rs 20 crore, Rs 400 crore of debt at 10% and a 25% tax rate, with 60 crore shares at Rs 10. Interest falls from Rs 40 crore to Rs 30 crore, so net income rises from Rs 30 crore to Rs 37.5 crore, and with equity now worth Rs 700 crore, P/E falls from 20.0x to 18.7x. Equity multiples such as P/E move with capital structure; enterprise multiples such as EV/EBITDA do not, which is why bankers compare companies on enterprise multiples.

EPS also rises, from Rs 0.50 to Rs 0.54. The reason is a comparison of two costs: the new shares were sold at an earnings yieldNet income divided by equity value, the inverse of the P/E. It is what each rupee of equity earns. of 5%, and the debt they retired cost 7.5% after tax. Replacing a 7.5% cost with a 5% cost lifts earnings per share.

What could make enterprise value move after all?

Three second-order effects, worth one sentence each. Fees on the share issue leak value out to advisers. Less debt means a smaller interest tax shield, which in Modigliani and Miller's analysis with taxes lowers value slightly. Against that, lower leverage reduces the risk and cost of financial distress. In an interview, give the first-order answer, unchanged, and then name the effects that could nudge it.

531,441. Rewrite 9 to the 6th as the square of 9 cubed. 9 cubed is 729. Then square 729 by splitting it into 700 plus 29: 700 squared is 490,000, twice 700 times 29 is 40,600, and 29 squared is 841. They add to 531,441. As a check, it should be a little over half a million, because 0.9 to the 6th is about 0.53.

How do you keep every step in your head?

Think of carrying a heavy box up six floors: you take it in two flights of three with a rest between, not six lifts in a row. Rewriting a power as the square of a smaller power cuts the work to one easy cube and one square you can split. 9 squared is 81, and 81 times 9 is 729, so 9 cubed is 729. Then 9 to the 6th is 729 squared, and squaring a number near a round one is the most practised move in mental maths.

How do you check it before saying it?

Three quick checks. An even power of 9 ends in 1, because 81 does, and 531,441 ends in 1. Any power of 9 is divisible by 9, so its digits must sum to a multiple of 9: 5 + 3 + 1 + 4 + 4 + 1 is 18. And the size is right: 9 is 10% below 10, and six 10% trims leave about 53% of a million. A digit check and a size check together catch nearly every slip in the cross term, which is where most mental errors happen.

A second route, if you prefer it: 9 to the 6th is also 81 cubed. 81 times 81 is 6,561, and 6,561 times 81 is 6,561 times 80 plus 6,561, which is 524,880 plus 6,561, giving 531,441 again. Saying one route and confirming with the other is worth more than speed.

It adds about Rs 18.5 crore, or 4.9%, taking the value from Rs 379.1 crore to Rs 397.6 crore. The mid-year convention assumes each year's cash arrives evenly through the year, so on average halfway through rather than at the end. Every cash flow is then discounted for half a year less, which multiplies each present value by 1.10 to the power of one half, 1.0488.

Why would cash arrive in the middle of the year?

A shop does not collect its year's takings in one lump on the last day of March; money comes in every day. A DCF that discounts each year's cash flow as if it all arrived on the final day is too harsh. The mid-year convention treats each year's cash flow as arriving, on average, halfway through the year, so it is discounted for t minus one half years instead of t. For a business with steady receipts that is the more accurate assumption, and it is the default in many banks' models.

Why is the uplift exactly the square root of 1.10 minus one?

Because the shift is the same for every cash flow. Moving a payment six months earlier multiplies its present value by 1.10 to the power of one half, whatever year it sits in. Since every cash flow gets the same uplift, the whole value rises by the same 4.88%, and neither the size nor the pattern of the cash flows changes that. Half the discount rate, 5%, is a fair first approximation to say out loud; at a 20% rate the exact uplift is 9.5%, against the 10% the approximation would give, so the shortcut weakens as rates rise.

Where should you not apply it?

Two cautions. A terminal valueThe value of all cash flows beyond the forecast years, usually from a growing perpetuity or an exit multiple. built on a perpetuity of flows is usually discounted on the same mid-year basis, but one built on an exit multiple is not, because a sale happens at a point in time, the end of the final year. Shifting an exit-multiple terminal value half a year earlier overstates the answer, and an interviewer who asks about the convention is often fishing for exactly that. For seasonal businesses that collect most cash late in the year, the convention itself flatters the value.