Financial Analysis puzzles, solved step by step

Stop on a 5 or 6 after the first roll, on 4 or more after the second, and take whatever the third gives. The game is worth 14/3, about 4.67. Solve it from the end: a last roll is worth 3.5, so with two rolls left you keep anything above 3.5, which makes two rolls worth 4.25. With three rolls left you keep only what beats 4.25.

Why do you start from the last roll?

Think of house hunting with three viewings booked and a rule that you must take the last flat if you get that far. You cannot judge the first flat until you know what walking away from it is worth, and that depends on the viewings still to come. A stop or continue decision is only as good as your value for continuing, so you price the last stage first and carry that value backwards. The method is called backward inductionSolving a sequence of decisions from the final step back to the first, so each earlier choice is made knowing what the later ones are worth., and it is how an option to wait is valued in finance too.

On the third roll there is no choice: you get the face, and a fair die averages (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. That 3.5 is the price of walking away from the second roll. So on the second roll you keep a 4, 5 or 6, each of which beats 3.5, and re-roll a 1, 2 or 3. Half the time you keep an average of 5; half the time you collect 3.5. Two rolls are worth 0.5 x 5 + 0.5 x 3.5 = 4.25.

What changes when you hold three rolls?

The bar goes up. With three rolls in hand, walking away from the first roll is worth 4.25, so a 4 is no longer good enough: only a 5 or a 6 beats it. Two faces in six you keep, averaging 5.5; four faces in six you roll on and collect 4.25. That is (2/6) x 5.5 + (4/6) x 4.25 = 1.83 + 2.83 = 4.67.

What do you add to show you see the pattern?

Two observations. First, the bar rises with the number of chances left. A candidate who applies one rule, keep 4 or more, on every roll gets 4.625 instead of 4.667: a small loss that shows the continuation value was never priced. Second, each extra roll is worth less than the one before: the second roll adds 0.75, the third only 0.42, and a fourth would add 0.28. An extra option is worth less when the options you already hold are good. The limit to say out loud: this strategy maximises the average, which is right for a player who plays many times; someone playing once who needs at least 4 would play differently.

200 km. The car takes 100 / 50 = 2 hours to reach B, and the bird flies without stopping for exactly those 2 hours at 100 km/h. Distance is speed times time, so the bird covers 200 km. How many trips it makes and where each turn happens do not matter; only the time in the air does.

Why is the zigzag the wrong thing to count?

Picture a dog running back and forth between you and your front gate while you walk home. You could chart every lap, or you could notice that the dog runs for exactly as long as you walk. When something moves at a constant speed for a known time, its distance is speed times time, and the route it takes is irrelevant. The puzzle is built to pull you into the laps.

Here the clock belongs to the car. It has 100 km to cover at 50 km/h, so the whole episode lasts 2 hours. The bird flies at 100 km/h throughout, so it flies 200 km. That is the whole answer, and it takes one sentence.

How do you check it the long way if the interviewer asks?

Sum the laps. On the first leg the bird and the car close at 150 km/h across 100 km, so they meet after 40 minutes, 33.3 km from A. The bird has flown 66.7 km and flies the same back to B, a round trip of 133.3 km. By then the car is 33.3 km from B, so the next round trip is 44.4 km. Each round trip is one third of the one before, so the laps form a geometric series that sums to 133.3 / (1 - 1/3) = 200 km.

Why would a finance interviewer ask this?

It tests whether you look for the quantity that controls a problem before you start calculating. Most analysis questions hide one controlling number under a lot of detail, and the skill is finding it first. In a model review that might be the single assumption the valuation hangs on; in a variance analysis it might be one volume line. Give the two hour answer first, then offer the series as a check. Speed plus a second method is exactly what the interviewer is listening for.

The reported yield is 5.0%; with stock pay treated as a cost it is 3.1%. Market value is 10 crore shares at Rs 800, Rs 8,000 crore. Rs 400 crore over that is 5.0%; Rs 250 crore is 3.1%. You do not need a future share price: deduct stock pay as if it were cash salary, because whatever the price, the shares handed to staff are worth Rs 150 crore.

Why is stock pay a real cost when no cash leaves?

Think of a family shop that pays its manager with a slice of the business instead of a salary. The till looks fuller, but the family now owns less of the shop. Stock-based compensation is wages paid in ownership rather than cash, and adding it back to free cash flow counts the saving without counting the slice given away. For this company, Rs 150 crore of the Rs 400 crore reported, 37.5%, exists only because staff were paid in shares. Put the other way, the reported yield is 60% higher than the yield an owner actually earns.

How does stock pay reach equity value if you do not know the future share price?

There are two consistent routes. Treat stock pay as a cash expense in the DCF, leave it out of the add-backs, and divide the equity value by today's diluted share count. Or keep the add-back and forecast every share that will be issued in future, which needs the future share price you do not know. The first route avoids the circle, because the rupee value handed to staff is fixed by the grant, Rs 150 crore a year, whatever the price turns out to be. Only the number of shares depends on the price: at Rs 400 the grant is 37.5 lakh shares, at Rs 800 it is 18.75 lakh, at Rs 1,600 about 9.4 lakh. Owners give up Rs 150 crore of value each time.

What is the one line that shows you understand it?

The gap between the two yields, 5.0% less 3.1%, is 1.9%, exactly the share of the company given to employees each year. An investor who takes the 5.0% at face value is being paid partly in their own dilution. The limit to say: stock pay retains staff and may cost less than the cash salary it replaces, so it is a real expense rather than a waste. The only point is that it must be counted once, as a cost.

About Rs 156 crore on cash flow, but a buyer will not pay much above Rs 150 crore, the cost of building the same plant. Revenue is Rs 100 crore and cash costs Rs 70 crore. Taking depreciation equal to the Rs 5 crore of capex, tax is 25% of Rs 25 crore, Rs 6.25 crore, leaving Rs 18.75 crore a year: over 12% that is Rs 156.25 crore. Replacement cost caps it.

How do you get from units to a value?

Valuing a factory is the same sum as valuing a flat you rent out: the rent left after upkeep and tax, divided by the return you need. A business with no growth is worth its steady yearly free cash flow divided by the discount rate. All the work is in getting the yearly cash right, and the two lines people forget are tax and the capex needed just to stand still.

Why does the rebuild cost set the ceiling?

Suppose someone asks Rs 25 lakh for a used car when the same model costs Rs 20 lakh new at the showroom. You walk to the showroom. No sensible buyer pays much more for an asset than it would cost to build an identical one, so replacement cost caps the price even when the cash flows say more. Here the gap is small, Rs 156.25 crore against Rs 150 crore, so a defensible answer sits close to Rs 150 crore.

The cap is loose in three ways, and naming them is what separates a good answer. Building takes time, perhaps two years with no cash coming in, so a working factory earns a premium for the lost years. Land, permits and trained staff may be hard to copy. And if cash flow value stays well above rebuild cost, rivals build plants, supply rises and prices fall, which pulls the cash flows back down. That last force is why the two numbers tend to converge over time.

What if the numbers pointed the other way?

If the cash flow value were Rs 100 crore against a Rs 150 crore rebuild cost, the factory would be worth about Rs 100 crore. Replacement cost is a ceiling, not a floor: nobody pays Rs 150 crore for a plant whose cash flows justify only Rs 100 crore. The floor is what the land and machinery would fetch if sold off, which is a third number worth asking for before you commit.

Revenue up 10% adds the most, Rs 4 crore, against Rs 3 crore for the COGS cut and Rs 0.75 crore for EBITDA up 5%. New revenue brings only its gross margin: 10% of Rs 40 crore of gross profit. The COGS cut saves 5% of Rs 60 crore. The two levers tie when 10% of gross profit equals 5% of COGS, at a gross margin of 33.3%; below that, cutting COGS wins.

Why is a revenue increase worth less than it sounds?

A tea stall that sells 10% more cups also buys 10% more milk and tea leaves. When costs move with sales, extra revenue adds only its gross margin to profit, not the whole rupee. Here each extra rupee of sales brings 40 paise of gross profit, so Rs 10 crore of new revenue adds Rs 4 crore. The Rs 25 crore of other costs is assumed fixed; if part of it rose with sales too, the revenue lever would shrink further.

The COGS cut is simpler: 5% of Rs 60 crore is Rs 3 crore straight to EBITDA. EBITDA up 5% is the decoy: 5% of a Rs 15 crore base is only Rs 0.75 crore, because a percentage of a small number is a small number.

At what gross margin does the answer flip?

Write both gains per rupee of revenue. The revenue lever adds 10% times the gross margin; the COGS lever adds 5% times the cost ratio, which is one minus the gross margin. They tie when 0.10 x GM = 0.05 x (1 - GM), which gives a gross margin of one third. Above 33.3%, growing sales does more; below it, as in grocery or commodity processing, the cost cut does more. At the tie each lever adds Rs 3.33 crore on Rs 100 crore of revenue.

What does the interviewer want to hear beyond the number?

Ask for the margin structure before answering, because the right lever depends on it. Then add the practical view: a 5% cost cut is often more within management's control than 10% more sales, which may need price cuts or marketing spend that eat into the gain. Finally, EBITDA up 5% can never beat the revenue lever, since EBITDA can never exceed gross profit; it beats the COGS cut only when EBITDA is larger than COGS, as in some software businesses. If half the other costs were variable, the revenue lever would fall to Rs 2.75 crore and the COGS cut would win.

The company's own cash repaid Rs 400 crore of debt, and every rupee repaid moved a rupee of the unchanged Rs 1,000 crore value from lenders to the fund. Equity went in at Rs 400 crore and came out at Rs 800 crore: a MOIC of 2.0x and, over five years, an IRR of about 14.9%. No growth and no change in multiple were needed.

Where did the gain come from if the price did not move?

Think of a flat bought for Rs 1 crore with a Rs 60 lakh home loan. The rent covers the loan payments and brings the loan down to Rs 20 lakh over five years. You sell the flat for exactly Rs 1 crore, yet your stake has grown from Rs 40 lakh to Rs 80 lakh. Enterprise value is a pie shared between lenders and owners, and each rupee of debt repaid from the business's own cash moves a rupee of that pie to the owners. The price never had to rise.

The return arithmetic follows. MOIC, the multiple on invested capital, is Rs 800 crore out over Rs 400 crore in, 2.0x. IRR is the yearly rate that turns 400 into 800 over five years: 2 to the power one fifth, less 1, which is 14.87%. The rule of 72 gives a quick check: doubling in five years is roughly 72 / 5, about 14.4%.

Is that skill or just leverage?

Run the same business with no debt. The fund pays Rs 1,000 crore of equity, the company's cash flow piles up as at least Rs 400 crore of cash, and at exit the fund receives Rs 1,000 crore plus that cash, about Rs 1,400 crore. That is 1.4x and about 7.0% a year, before adding the interest it never paid. Debt did not create the Rs 400 crore; the business earned it. Debt let a smaller cheque claim all of it, which lifts the return from 1.4x to 2.0x.

Leverage cuts both ways, and saying so is the limit worth stating. Had the company sold for only Rs 500 crore, the levered fund would get back Rs 300 crore on Rs 400 crore, 0.75x, a loss, while the unlevered owner would hold 0.9x. The same debt that doubled the good outcome deepened the bad one.

What does the interviewer listen for?

Name the three ways a buyout makes money: EBITDA growth, a higher exit multiple, and debt paydown. This deal used only the third, which is the point of the question. Then add time: the same 2.0x earned over three years instead of five is about 26% a year, which is why funds care about IRR and not only the multiple. A finishing touch is noting that the Rs 400 crore repaid is cash after interest, so the business had to earn more than Rs 400 crore to do it.

About Rs 375.7 crore. The perpetuity formula, 50 / 0.10 = Rs 500 crore, gives a value one period before the first payment, which here is the end of year 3. Discount that three years: 500 / 1.1 cubed = 500 / 1.331 = Rs 375.66 crore. Discounting four years because the cash starts in year 4 is the classic slip and gives Rs 341.5 crore.

Where in time does the perpetuity formula put its value?

Picture someone who will retire in four years and has been promised a pension from that day on. On the eve of the first payment, the pension is worth a neat round sum; four years out, that round sum itself is still in the future. Cash flow divided by the rate gives the value one period before the first payment, never on the day of it. The standard formula assumes the first payment comes one year from the valuation date. If the first payment is at year 4, the formula is standing at year 3.

So value the stream at year 3 and bring that single lump back to today. At year 3 the concession is worth 50 / 0.10 = Rs 500 crore. Three years of discounting at 10% means dividing by 1.1 x 1.1 x 1.1 = 1.331, which leaves Rs 375.66 crore.

How do you check it a second way?

Value a perpetuity that starts next year and subtract the three payments this one does not have. A full perpetuity from year 1 is worth Rs 500 crore. The missing payments in years 1 to 3 are a three-year annuity worth 50 x 2.487 = Rs 124.34 crore. Rs 500 crore less Rs 124.34 crore is Rs 375.66 crore, the same answer by a route that cannot misplace the year. The tempting shortcut, 500 less 150 = 350, fails because the missing payments are worth less than their face.

Why does this matter beyond the puzzle?

It is the same step as the terminal value in a DCF. A growing perpetuity built on year 6 cash flow gives a value at year 5, so it is discounted five years, not six. Discounting one year too many cuts the value by the full discount rate, here about 9%, and in a DCF the terminal value is often most of the answer. The limit to say aloud: real concessions end. If this one ran for 30 years from year 4, it would be worth Rs 354.1 crore, so the 'forever' after year 33 adds only Rs 21.5 crore. Distant cash matters little at 10%.

1,596 and 4,891. Both pairs sit equally either side of a round number, so each product is that round number squared less the gap squared. 38 x 42 = 40 squared less 2 squared = 1,600 - 4 = 1,596. 67 x 73 = 70 squared less 3 squared = 4,900 - 9 = 4,891. Two easy steps replace four partial products.

Why does a pair either side of a round number multiply so neatly?

Picture a square courtyard 40 tiles by 40. Rebuild it 42 tiles wide but only 38 deep. You lose a row of 40 tiles along the bottom and gain a column of 38 down the side. The column is two tiles shorter than the row, and the missing piece is a 2 by 2 corner. Any two numbers equally either side of a middle number multiply to the middle squared less the gap squared, because (a - b)(a + b) = a squared less b squared.

So the work is two squares you already know. For 67 x 73 the middle is 70 and the gap is 3: 4,900 less 9 is 4,891. A quick check catches slips: 7 x 3 ends in 1, and so does 4,891; 8 x 2 ends in 6, and so does 1,596. Both products sit just under the square, as they must.

What do you do when the pair is not centred on a round number?

Find the middle anyway, or nudge one number. 38 x 44 has a middle of 41 and a gap of 3: 1,681 less 9 is 1,672. Or reuse what you have: 38 x 42 is 1,596, add two more 38s and get 1,672. When the sum is odd, as in 37 x 42, the middle is 39.5, which is messy; adjust instead: 38 x 42 less one 42 is 1,554. The skill is choosing the anchor you can square instantly, then correcting by a small, easy amount.

Why does a finance interviewer test this?

Interviewers use mental maths to see whether you can keep a number straight under pressure, which is what a live model review or a client call demands. They watch the method as much as the answer: a candidate who says 'that is 40 squared less 4' sounds in control. The limit is honest too: the trick only helps when a pair straddles a square you know. For 38 x 57, round and correct, or split 57 into 50 and 7, rather than forcing the pattern.

The cash never runs out. Burn reaches zero in month 17, by which point Rs 68 crore has gone, so the balance bottoms at Rs 52 crore. The burns are 8, 7.5, 7 and so on down to 0.5 in month 16: sixteen payments averaging (8 + 0.5) / 2 = Rs 4.25 crore, Rs 68 crore in all. Quoting a 15-month runway from 120 / 8 ignores the improvement.

Why is 120 / 8 the wrong runway?

A student who spends Rs 8,000 this month and cuts back by Rs 500 every month does not run through savings at Rs 8,000 a month; each month costs less than the last. A runway is cash divided by burn only when the burn is constant; when the burn changes by a fixed amount each month, the total spent is an arithmetic series, and you sum it. Here the burn falls to zero after sixteen steps, so the question is whether the sum of those sixteen burns is more or less than Rs 120 crore.

How do you sum the burn quickly?

Pair the first and last months, as the schoolboy Gauss did: 8 + 0.5 = 8.5, 7.5 + 1 = 8.5, and so on. There are sixteen burns, so eight pairs of 8.5, which is Rs 68 crore. Equivalently, sixteen months at the average burn of 4.25. Month 17's burn is zero, so nothing more is spent. Rs 120 crore less Rs 68 crore leaves Rs 52 crore at the lowest point, reached at the end of month 16.

What would you warn the founder about?

The answer rests entirely on the slope of the improvement. If the burn fell by Rs 0.25 crore a month instead of 0.5, the burn would reach zero only in month 33, and the cumulative burn would pass Rs 120 crore in month 23: the company runs out of money. Halving the pace of improvement turns a comfortable floor into a cash-out, so test the slope before trusting the floor. The second warning is that Rs 52 crore is a forecast floor, not a cushion: a lender or a board would want headroom above it for a bad quarter.

Volume must rise by a third, 33.3%, just to stand still. On a Rs 100 price, contribution is Rs 40 a unit. Cut the price 10% and variable cost stays at Rs 60, so contribution falls to Rs 30, a quarter lower. Selling 40 / 30 = 1.333 times as many units keeps total contribution flat. The general rule is the cut divided by the margin less the cut: 10 / (40 - 10).

Why does a 10% price cut need far more than 10% more volume?

A samosa seller charges Rs 20 and spends Rs 12 on ingredients, so each samosa leaves Rs 8. Cutting the price by Rs 2 does not cut the ingredient bill; it cuts the Rs 8 to Rs 6. A price cut comes entirely out of contribution, because variable cost does not fall with the price, so the volume needed to make it back is set by the margin, not the price. In the puzzle the 10% cut takes a quarter of the contribution, so volume must rise by 33.3%.

What is the general rule, and how fast does it bite?

The table shows why low-margin businesses fear discounting. At a 20% margin a 10% cut needs volume to double; at 60% it needs only 20% more. The rule also runs the other way: a 10% price rise lifts contribution to Rs 50 a unit, so volume can fall by up to 20% before total contribution drops. Price rises are usually safer than they feel and price cuts riskier than they look.

What would you check before backing the cut?

Three things. First, whether demand will really grow by a third, which depends on how sensitive buyers are to price and whether rivals match the cut. Second, capacity: a third more units may need overtime or a new shift, which turns fixed costs into step costs. Third, the existing customers who would have paid Rs 100 anyway now pay Rs 90, which is where most of the loss sits. The limit of the rule is that it holds contribution flat; it says nothing about cash tied up in the extra stock and receivables.