Financial Analysis puzzles, solved step by step

Puzzles: 100
Traced to a firm: 47
Topics: 13
Hard: 30

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006A product's price rises 25%, its volume falls 4% and the currency it is sold in adds 10% when translated back. Without paper, what is the combined change in revenue?Mental mathsCoreLeveraged financePrivate equity

Try it first

Say a number before you work it out properly.

Show the worked solution

Revenue rises 32%. Growth rates multiply, they do not add. Pair the price and volume factors first because they cancel neatly: 1.25 times 0.96 is exactly 1.20. Then 1.20 times 1.10 is 1.32. Adding 25, minus 4 and 10 gives 31%, which misses the cross terms, the growth earned on growth, worth one point here.

Why do the three changes multiply instead of add?

Think of a tea stall that sells 100 cups at Rs 10. Raise the price to Rs 12.50 and it now takes Rs 1,250 if it still sells 100 cups. If it then sells 4% fewer cups, those 4% are lost at the new, higher price, not the old one. Each change acts on revenue that the earlier changes have already moved, so the factors multiply. Revenue is price times volume times the exchange rate, and a product of three factors changes by the product of their changes.

The relationship

1+g = (1.25)(0.96)(1.10) = 1.20 \times 1.10 = 1.32

1.25	price factor, a 25% rise
0.96	volume factor, a 4% fall
1.10	currency factor, a 10% gain on translation
g	the combined revenue growth

What it says in wordsTurn every change into a factor, multiply the factors, and subtract one.

How do you do it in your head without fumbling?

Choose the order. Multiplication does not care about order, but your head does. 0.96 is 24 over 25 and 1.25 is 5 over 4, so their product is 120 over 100, exactly 1.20, and the hard step disappears. Ten per cent of 1.20 is 0.12, so the last step is 1.32. Pair the factors the other way and you are stuck with 1.25 times 1.10, then 1.375 times 0.96, which is the same answer by a much longer road.

Pair first	First product	Then multiply by	Ease in your head
Price x volume	1.2000	1.10	Easy: 24/25 x 5/4 is exactly 1.2
Price x currency	1.3750	0.96	Harder: 1.375 less 4% of it
Volume x currency	1.0560	1.25	Harder: 1.056 plus a quarter of it

All three routes give 1.32. The first is the one to say out loud, because every step is a round number.

Revenue of 100 moves to 125, then 120, then 132, so the combined change is 32%. Adding the rates gives 31%, because it leaves out the cross terms, which net to plus one point: 2.5 from price on the currency gain, less 1.0, 0.4 and 0.1 from the other pairings.

Why does the one-point gap matter on a real desk?

On a Rs 2,000 crore revenue line, one point is Rs 20 crore, which is a material miss in a budget bridge. The gap between adding and multiplying grows with the size of the changes, so it is small for 2% moves and large for 25% moves. When a board pack splits revenue growth into price, volume and currency, the cross terms have to sit somewhere, and an analyst should say where they put them rather than leave the bridge one point short.

Where candidates lose it

Saying 31% is the fast loss. It is close enough to sound right, and the interviewer asked precisely because the cross terms are what separates someone who knows growth compounds from someone who adds percentages.

The second loss is getting 32% by a slow route, multiplying 1.25 by 1.10 first and grinding through 1.375 times 0.96. Spend one second choosing the pairing that cancels; the interviewer is watching that choice as much as the answer.

What the interviewer asks next

Price falls 25% and volume rises 25%. Is revenue up, down or flat?
In local currency terms, how much did revenue grow?
How would you show the 32% split into price, volume and currency in a revenue bridge so it adds up?

017A vendor offers a game: a fair coin is tossed until it shows heads. If heads comes on the first toss you get Rs 100, on the second toss Rs 200, and the prize doubles with every toss after that. The game costs Rs 1,000 to play, but the vendor has only Rs 10 lakh to pay out. What is the game really worth, and would you play?Probability and expected valueCoreEquity capital marketsConsulting-style case
Try it first
With the vendor's Rs 10 lakh limit, what is the game worth on average?
Show the worked solution
About Rs 761, so at Rs 1,000 the game is not worth playing. Each possible toss adds Rs 50 of expected value: a prize that doubles times a chance that halves. Prizes fit under the Rs 10 lakh cap for 14 tosses, which is Rs 700, and all longer runs are paid Rs 10 lakh, which adds only about Rs 61. The infinite value of the textbook game rests entirely on prizes no vendor can pay.
Why does every toss add exactly Rs 50?
Imagine a lottery stall where a ticket wins Rs 200 one time in four, or Rs 400 one time in eight. Each prize is worth the same on average, Rs 50, because the prize doubles exactly as the chance halves. In this game, heads first appearing on toss k has probability one over 2 to the k, and pays 100 times 2 to the k minus 1, so every possible toss contributes Rs 50 of expected value. With no limit there are infinitely many tosses, so the expected value is infinite. That is the famous St Petersburg paradox, and it is why nobody sensible pays a fortune to play.
The relationship
$E = \sum_{k=1}^{\infty} \frac{1}{2^{k}}\,\min(100 \cdot 2^{k-1},\ 10^{6}) = 14 \times 50 + \frac{10^{6}}{2^{14}} \approx 761$
k the toss on which the first head appears
1/2^k the chance the first head comes on toss k
100 x 2^(k-1) the promised prize
10^6 the vendor's limit, Rs 10 lakh
What it says in wordsFourteen tosses each add Rs 50; every longer run pays the capped Rs 10 lakh, and the chance of reaching toss 15 is one in 16,384.
What does the Rs 10 lakh limit do to the value?
The prize on toss 14 is Rs 100 times 2 to the 13, which is Rs 8,19,200; on toss 15 it would be Rs 16,38,400, more than the vendor holds. So only 14 tosses pay in full, worth Rs 700, and every longer run pays Rs 10 lakh, which happens one time in 16,384 and adds about Rs 61. The game is worth about Rs 761. Paying Rs 1,000 means losing about Rs 239 a game on average.
Each of the first 14 tosses adds Rs 50 of expected value, reaching Rs 700, and the capped tail adds only about Rs 61, so the game is worth about Rs 761. Without the cap the value would keep rising Rs 50 a toss, but the capped game never reaches the Rs 1,000 price.
How rich would the vendor need to be for Rs 1,000 to be fair?
Value grows very slowly with the vendor's wealth, because each doubling of the bankroll adds just one more Rs 50 toss. A vendor holding Rs 1,000 crore would make the game worth only about Rs 1,425, and the game reaches Rs 1,000 only if the vendor can pay about Rs 2.6 crore. Say the limit too: even a fair price ignores how much risk you can stomach, since almost every game pays Rs 100 or Rs 200. That is why economists use this game to show people value money by its usefulness to them, not by its face amount.
Where candidates lose it
The common loss is reciting "infinite expected value" and stopping, or worse, saying you would pay any price. The interviewer added the vendor's limit precisely to see whether you notice that infinite value depends on payouts that cannot happen.
The second loss is getting the cap wrong by assuming the tail adds a lot. Work out the last toss that pays in full, count Rs 50 per toss, and add the small capped tail; it takes thirty seconds.
What the interviewer asks next
What would you pay if the vendor could pay out Rs 100 crore?
The vendor offers to play the game ten times in a row for Rs 7,000. Does that change your answer?
How does this game relate to valuing a company with a tiny chance of an enormous outcome?
021You invest Rs 10,000 in a fund when its NAV is Rs 100, and another Rs 10,000 when the NAV is Rs 50. What is your average cost per unit, and why is it not Rs 75?Percentages and averagesCoreCorporate FP&ABusiness finance
Try it first
What is your average cost per unit?
Show the worked solution
About Rs 66.67 a unit, not Rs 75. The first Rs 10,000 buys 100 units at Rs 100 and the second buys 200 units at Rs 50, so Rs 20,000 buys 300 units. Rs 75 is the average of the two prices, which would be your cost only if you had bought equal numbers of units. Equal rupee amounts buy more units when the price is low, so the average cost is the harmonic mean of the prices.
Why does the cheap purchase count for more?
Think of filling your scooter with Rs 500 of petrol every week. In a week when petrol is cheap, Rs 500 buys more litres; when it is dear, fewer. At the end of the month the litres you own lean towards the cheap weeks. A fixed rupee amount buys more units when the price is low, so the low price carries more weight in your average cost than the high one. Here Rs 10,000 buys 100 units at Rs 100 but 200 units at Rs 50, so the Rs 50 price carries two thirds of the weight.
The relationship
$\text{average cost} = \frac{\text{total spent}}{\text{total units}} = \frac{20{,}000}{100 + 200} = 66.67 = \frac{2}{1/100 + 1/50}$
total spent Rs 20,000 across both purchases
total units 100 units at NAV 100 plus 200 at NAV 50
2 / (1/100 + 1/50) the harmonic mean of the two prices
What it says in wordsAverage cost is money spent over units bought, which for equal rupee amounts is the harmonic mean of the prices.
Rs 10,000 buys 100 units at a NAV of 100 and 200 units at a NAV of 50, so Rs 20,000 buys 300 units at an average cost of Rs 66.67. That harmonic mean sits below both the geometric mean of 70.7 and the simple average price of 75.
When is Rs 75 the right answer?
When you buy equal numbers of units rather than equal rupee amounts. Buy 100 units at Rs 100 and 100 units at Rs 50 and you spend Rs 15,000 on 200 units, which is exactly Rs 75 a unit. So the right average depends on what you held fixed: units fixed gives the simple average of prices, money fixed gives the harmonic mean. This is the arithmetic behind the idea of rupee cost averagingInvesting the same rupee amount at regular intervals, so that more units are bought when prices are low and fewer when they are high..
What you hold fixed Units bought Spent, Rs Average cost, Rs
Rs 10,000 each time 300 20,000 66.67
100 units each time 200 15,000 75.00
The same two prices give two different average costs, depending on whether the rupee amount or the number of units was held constant.
Does the lower average cost mean you made money?
No, and saying so is the mark of a careful answer. A lower average cost is arithmetic about what you paid, not a return. If the NAV stays at Rs 50, your 300 units are worth Rs 15,000 against Rs 20,000 invested, a 25% loss. Your break-even NAV is Rs 66.67, not Rs 75, which is the useful fact. The effect also needs prices to vary: at a constant price every average is the same.
Where candidates lose it
The common loss is answering Rs 75 by averaging the two prices. It ignores that the purchases bought different numbers of units, and it is the same slip people make when they average two speeds for a journey.
The second loss is going too far the other way and describing the lower average cost as a gain. Give Rs 66.67, say break-even is at that NAV, and stop short of claiming anything about returns.
What the interviewer asks next
You invest Rs 10,000 at NAV 100, then 50, then 100 again. What is your average cost?
What NAV do you need to break even on the two purchases?
Why is the harmonic mean of two different positive numbers always below their simple average?
025A supermarket has annual cost of goods sold of Rs 3,650 crore. It pays suppliers in 45 days, holds 20 days of stock, and is paid in cash at the till. If its sales fall 10%, how much cash leaves the business through working capital?Working capital and cash riddlesCoreCorporate FP&ATreasury
Try it first
Sales fall 10% with payment and stock days unchanged. What happens to cash from working capital?
Show the worked solution
About Rs 25 crore of cash leaves. Cost of goods sold is Rs 10 crore a day, so payables are 45 days, Rs 450 crore, and stock is 20 days, Rs 200 crore. With no receivables, net working capital is minus Rs 250 crore: suppliers fund the business. A 10% fall cuts daily cost to Rs 9 crore, payables to Rs 405 crore and stock to Rs 180 crore, so net working capital rises to minus Rs 225 crore. That Rs 25 crore is cash consumed.
How can a business be funded by its suppliers?
Think of a school canteen where parents pay for the term in advance and the canteen pays its vegetable seller at the end of each month. The canteen holds other people's money for weeks. A supermarket sells for cash at the till, keeps stock for 20 days and pays suppliers after 45, so it collects from customers about 25 days before it pays for the goods. Its net working capital is negative: suppliers are lending it money, free of interest, all year.
The relationship
$\text{NWC} = \underbrace{10 \times 20}_{\text{stock}} - \underbrace{10 \times 45}_{\text{payables}} = 200 - 450 = -250$
10 cost of goods sold per day, Rs 3,650 crore over 365
20 days of stock held
45 days taken to pay suppliers
What it says in wordsNet working capital is stock less payables when customers pay at the till; here suppliers fund Rs 250 crore.
Before the fall, payables of Rs 450 crore fund stock of Rs 200 crore, leaving net working capital of minus Rs 250 crore. After a 10% fall, payables of Rs 405 crore and stock of Rs 180 crore leave minus Rs 225 crore, so Rs 25 crore of cash has left the business.
Why does shrinking cost this business cash?
Run it as two movements. Stock falls by Rs 20 crore, which frees cash, but supplier credit falls by Rs 45 crore, which the supermarket has to pay out, so the net is Rs 25 crore out. Each day of lower sales means paying suppliers for last month's larger purchases while this month's till receipts are smaller. The mirror image is why such businesses love growth: a 10% rise in sales would release about Rs 25 crore.
What makes the real number worse?
The answer assumes the days stay fixed, and in a downturn they often do not. Suppliers who see sales fall may shorten credit; if payable days drop from 45 to 40, payables fall to Rs 360 crore and the cash outflow grows to about Rs 70 crore. On top of the working capital effect, lower sales also cut profit. Say both, then name the lesson a lender draws: a business funded by its suppliers can look cash-rich while it grows and turn cash-hungry quickly when it shrinks.
Where candidates lose it
The common loss is saying cash comes in, because less stock is needed. That counts only the asset side and forgets that supplier credit shrinks faster, since payables are more than twice the size of stock.
The second loss is saying nothing changes because the days are unchanged. Days are ratios; the rupee balances scale with sales, and the cash moves with the rupees.
What the interviewer asks next
If sales grew 10% instead, how much cash would working capital release?
How would the answer change if 20% of sales were on credit with 30-day terms?
Why might a supermarket's suppliers accept 45-day terms?
027A product sells at a 30% contribution margin. You raise the price by 5% and lose 8% of your volume. Is total contribution better or worse than before, and by how much?Pricing, costing and unit economicsCoreCorporate FP&ACost accounting
Try it first
Gut call: you gained 5% on price and lost 8% on volume. Where does total contribution land?
Show the worked solution
Better, by about 7.3%. Take a price of 100 with variable cost 70, so each unit contributes 30. After the rise the price is 105 and each unit contributes 35. Selling 92 units instead of 100 gives 92 x 35 = 3,220 against 3,000 before. Volume could fall as far as 14.3% before the price rise stopped paying.
Why does 5% on price beat 8% on volume?
Picture a tea stall that sells a cup for 10 rupees with 7 rupees of milk, leaves and sugar in it. Raise the price to 10.50 and the stall's profit per cup goes from 3 to 3.50, a sixth more. A price rise falls straight to contribution, so its effect is measured against the margin, not against the price. The volume loss, by contrast, removes whole cups and their contribution at the same rate as the lost units, 8%.
Raising price from 100 to 105 lifts contribution per unit from 30 to 35, a 16.7% gain, so 92 units at 35 earn 3,220 against 3,000 for 100 units at 30, and volume would need to fall 14.3% to erase the gain.
How much volume could you lose before the price rise hurts?
Set new contribution equal to old: units times 35 must equal 100 times 30, so units can fall to 3,000 over 35, which is 85.7. The breakeven volume loss for a price rise is the price change divided by the new margin: 5 over 35, or 14.3%. That is why thin-margin businesses gain so much from price and lose so much from discounts: at a 30% margin, a 5% discount needs volume to rise by 5 over 25, a full 20%, just to stand still.
The relationship
$\Delta V_{\text{breakeven}} = -\frac{\Delta p}{m + \Delta p} = -\frac{5\%}{30\% + 5\%} = -14.3\%$
Delta p the price change as a share of the old price, 5%
m the contribution margin before the change, 30%
What it says in wordsVolume can shrink by the price change divided by the new margin before the rise stops paying.
Say the limit too. The sum treats variable cost per unit as fixed and ignores any fixed costs that could be cut when volume falls, and it says nothing about customers who leave and do not come back next year.
Where candidates lose it
The usual loss is adding the two percentages: plus 5 and minus 8 is minus 3, so the move looks bad. That treats the price change as if it acted on revenue, when it lands entirely on the contribution slice.
The second loss is working in revenue instead of contribution. Revenue does fall slightly, 105 x 92 is 9,660 against 10,000, and a candidate who stops there gets the answer backwards.
What the interviewer asks next
At the same 30% margin, how much must volume rise to justify a 10% price cut?
How does the answer change if the margin is 70%, as in software?
Which fixed costs would you look at if volume did fall 8%?
030A company's net debt is 3x EBITDA and its cost of debt is 10%. What is its EBITDA interest cover? What does cover become if leverage rises to 5x and the rate to 12%?Ratio and margin riddlesCoreRating agenciesBank credit
Try it first
Leverage rises from 3x to 5x and the rate from 10% to 12%. What happens to cover?
Show the worked solution
Cover is 3.3x, falling to 1.7x. Take EBITDA of 100. Net debt of 300 at 10% costs 30 of interest, so EBITDA covers it 3.33 times. At 5x and 12%, net debt is 500 and interest is 60, so cover is 1.67x. Cover is one over leverage times the rate, and 0.30 doubling to 0.60 halves it.
Why is there no need for an actual EBITDA figure?
Think of a household whose loan EMIs are some multiple of monthly salary. Whether the salary is 50,000 or 5 lakh, the share of salary that goes on interest depends only on how many months of salary it borrowed and at what rate. Interest cover is EBITDA over interest, and interest is leverage times EBITDA times the rate, so EBITDA cancels and cover is one over leverage times the rate. Pick EBITDA of 100 only to make the arithmetic visible.
The relationship
$\text{Cover} = \frac{\text{EBITDA}}{L \cdot \text{EBITDA} \cdot r} = \frac{1}{L\,r} = \frac{1}{3 \times 0.10} = 3.3\times$
L net debt as a multiple of EBITDA
r the average cost of debt
What it says in wordsInterest cover is the reciprocal of leverage multiplied by the interest rate.
With EBITDA of 100, net debt of 300 at 10% costs 30 of interest and gives cover of 3.3x, while net debt of 500 at 12% costs 60 and gives cover of 1.7x, so the two moves together cut cover exactly in half.
Why do leverage and rates tend to move together, and what does 1.7x mean?
A lender charges more to a borrower who owes more, so the second scenario is not a coincidence; it is how a stretched balance sheet usually looks. Because leverage and rate multiply, a company that adds debt while rates rise loses cover much faster than either change alone suggests. Cover of 1.7x also flatters the position: EBITDA is before tax, capex and working capital, so once maintenance capex is paid the cash left to service interest can be well under 1.7 times the bill. Say that limitation; credit analysts look at cover after capex for exactly this reason.
Where candidates lose it
Candidates compute each change separately and add them: leverage up two-thirds, rate up a fifth, so cover falls by something like 50 to 90% depending on how they combine it. Say the formula first, cover equals one over L times r, and the answer is a single division.
The second loss is treating 1.7x as comfortable because it is above one. The interviewer wants to hear that EBITDA is not cash available for interest.
What the interviewer asks next
At 5x leverage, what rate would bring cover down to 1.0x?
How would you compute cover after maintenance capex, and why is it lower?
Which covenant would a lender set on this company, and at what level would you expect trouble?
036A business has a 10% operating margin. Next year revenue grows 20% and costs grow 15%. What is the new margin?Percentages and averagesCoreCorporate FP&ABusiness finance
Try it first
Quick call: where does the margin land?
Show the worked solution
The new margin is 13.75%. Take revenue of 100, so cost is 90 and profit 10. Revenue grows to 120 and cost to 90 x 1.15 = 103.5, leaving profit of 16.5. The margin is 16.5 over 120, which is 13.75%. Profit itself grows 65%, because a 5 point gap between growth rates is large next to a 10% margin.
Why is the answer not simply 15%?
Think of a tea stall that takes in Rs 100 a day and spends Rs 90. If takings rise 20% and spending rises 15%, the extra takings are Rs 20 but the extra spending is Rs 13.50, because 15% is taken on 90, not on 100. Growth rates act on their own bases, so a margin moves by the gap in rupees, not by the gap in percentage points. Work in rupees on a base of 100 and the trap disappears: profit goes from 10 to 16.5 on revenue of 120.
On revenue of 100, profit of 10 gains 20 from revenue growth and loses 13.5 to cost growth, ending at 16.5 on revenue of 120, a margin of 13.75% rather than the 15% you get by adding the growth gap to the margin.
The relationship
$m_1 = 1 - (1-m_0)\frac{1+g_c}{1+g_r} = 1 - 0.90 \times \frac{1.15}{1.20} = 13.75\%$
m0, m1 the old and new margins
g_r, g_c revenue growth and cost growth
1 - m0 cost as a share of revenue before the change
What it says in wordsThe new cost share is the old cost share scaled by cost growth over revenue growth, and the margin is what is left.
Why does a thin margin make the move so large?
Profit is the small gap between two big numbers, so a modest difference in their growth rates is a big change to the gap. The thinner the starting margin, the more operating leverage a growth gap carries: here profit grows 65% on 20% revenue growth. Run the same 20% and 15% from a 30% margin and the new margin is 32.9%, a smaller relative move, because cost is a smaller share of revenue. Say the limit too: the formula assumes all cost grows at 15%. In practice fixed costs grow slowly and variable costs track volume, so you would split the cost line before trusting the answer.
Where candidates lose it
The fast wrong answer is 15%: 10% plus the 5 point gap between the growth rates. It treats both rates as if they applied to the same base, when cost growth applies to 90 and revenue growth to 100.
The second loss is stopping at the margin. The interviewer often wants the profit growth too, and 65% on 20% revenue growth is the sentence that shows you understand operating leverage.
What the interviewer asks next
What cost growth would keep the margin at exactly 10%?
If half the cost base is fixed and does not grow, what is the new margin?
Revenue falls 10% and costs fall 5%. What happens to a 10% margin?
042A company sells land with a book value of Rs 40 crore for Rs 100 crore and pays 20% tax on the gain. Walk it through the three statements, and explain why the gain is subtracted in cash from operations.Accounting flow riddlesCoreBig FourCorporate FP&A
Try it first
What does cash from operations show for this transaction?
Show the worked solution
Net income rises Rs 48 crore, cash rises Rs 88 crore, and the balance sheet balances at plus Rs 48 crore. The gain of Rs 60 crore less Rs 12 crore tax gives net income of Rs 48 crore. Cash from operations starts at 48 and subtracts the Rs 60 crore gain, leaving minus 12; investing shows the full Rs 100 crore. Cash is up Rs 88 crore, land down Rs 40 crore.
Why subtract a gain you actually made?
Think of selling your old scooter for Rs 50,000 when you had it on your books at Rs 20,000. You did receive Rs 50,000 in cash, all of it, on the day of the sale. If your diary records a Rs 30,000 profit under daily earnings and also Rs 50,000 under scooter sold, you have counted Rs 30,000 twice. The cash flow statement puts the full sale proceeds in investing, so the gain sitting inside net income has to be taken out of operations, or part of the sale would be counted twice.
Cash from operations starts at net income of Rs 48 crore and backs out the Rs 60 crore gain, leaving minus Rs 12 crore of tax, while investing carries the full Rs 100 crore of proceeds, so cash rises Rs 88 crore and the balance sheet balances at plus Rs 48 crore.
How does each statement move, line by line?
Income statement: a gain on sale of Rs 60 crore, tax of 20% on it, Rs 12 crore, so net income is up Rs 48 crore. Cash flow statement: start from net income of 48, subtract the non-operating gain of 60, and cash from operations is minus 12; investing shows plus 100; cash is up 88. Balance sheet: cash up 88, land down 40, so assets are up 48; retained earnings up 48. The check that matters is that the change in assets equals the change in equity, Rs 48 crore each side.
State the assumption: the tax is paid in cash in the same year. If it were deferred, operations would show zero, a tax liability would sit on the balance sheet and cash would be up the full Rs 100 crore. Say also why the tax stays in operations: accounting rules generally classify income taxes as operating unless they are specifically tied to an investing activity, and practice varies, so confirm the treatment the company uses.
Where candidates lose it
The trap is putting the Rs 100 crore in operations, or putting it in investing and forgetting to back out the gain. The second version leaves cash up Rs 148 crore, and the balance sheet will not balance.
Candidates also put the gain on the cash flow statement as an add-back, the way they treat depreciation. Depreciation is a non-cash expense, so it is added back. A gain is a non-operating income, so it is subtracted. Say which one it is before you pick the sign.
What the interviewer asks next
What changes if the land is sold for Rs 30 crore, a loss?
How would the answer change if the tax is deferred to next year?
Where would the sale of a machine at book value show up?
044Revenue went from Rs 100 crore to Rs 200 crore in five years and management calls it 20% annual growth. What is the true compound annual growth rate? And what is it for yearly growth of 50%, minus 20%, 30%, 10% and 5%?Compounding and time valueCoreCorporate FP&AEquity research
Try it first
Doubling in five years is what compound rate?
Show the worked solution
The true CAGR is 14.9%, and the second series compounds at 12.5%, not the 15% its average suggests. Doubling in five years means (200/100) to the power one-fifth, less one: 14.87%. For the yearly series, multiply the growth factors: 1.5 x 0.8 x 1.3 x 1.1 x 1.05 = 1.8018, whose fifth root gives 12.50%. A simple average ignores both compounding and volatility.
Why is doubling in five years not 20% a year?
Think of a savings account that pays interest on interest. To double your money in five years you need less than 20% a year, because each year's growth is earned on a larger balance. Total growth divided by the number of years ignores compounding, so it always overstates the annual rate for a gain. The rate that compounds 100 to 200 in five years is the fifth root of 2 less one, 14.87%. Compounding at the claimed 20% would have produced 248.8.
Compounding at 14.9% takes revenue from 100 to 200 in five years while the claimed 20% would have reached 249, and yearly growth that averages 15% only takes 100 to 180.2, a compound rate of 12.5%.
The relationship
$\text{CAGR} = \Big(\prod_{i=1}^{n}(1+g_i)\Big)^{1/n} - 1 = (1.8018)^{1/5} - 1 = 12.5\%$
g_i growth in year i
product multiply the yearly growth factors, do not add the rates
What it says in wordsMultiply the growth factors, take the n-th root, and subtract one.
Why does an uneven path lose so much to its average?
A fall hurts more than an equal rise helps, because the rise then starts from a smaller base: 100 up 50% is 150, down 20% is 120, not 130. The gap between the average and the compound rate grows with how much the yearly rates vary; roughly, CAGR is the average less half the variance of the rates. Here the rates have a variance of 0.056, so the shortcut gives 15% less 2.8 points, about 12.2%, close to the exact 12.5%. Use this when a pitch quotes average growth: ask for the start and end values and compute the compound rate yourself.
Where candidates lose it
The trap is dividing the total gain by the years: 100% over five is 20%. It is the number management wants you to repeat, and it overstates growth by about five points a year.
For the uneven series, candidates add the rates and divide by five. Multiply the factors instead, or at least say that volatility drags the compound rate below the average.
What the interviewer asks next
What is the CAGR if revenue triples in ten years?
A fund gains 100% then loses 50%. What is its two-year CAGR?
Why do analysts quote CAGR rather than average growth for revenue?
046Sales grow by exactly Rs 10 lakh every month. A team forecasts next month's sales as the average of the last three months. By how much does the forecast miss, and in which direction?Data and statistics intuitionCoreCorporate FP&ABusiness finance
Try it first
The trend is steady at Rs 10 lakh a month. How far off is the three-month-average forecast?
Show the worked solution
It under-forecasts by Rs 20 lakh every month. On a straight-line trend, the average of the last three months equals the middle month, so the average lags the latest month by one month. The forecast is for the following month, one more month ahead. Two months of Rs 10 lakh growth is Rs 20 lakh. For a window of n months, the miss is the monthly growth times (n + 1) / 2.
Why does an average lag behind a trend?
Think of judging how fast a child is growing by averaging her height over the last three birthdays. The average describes her at the middle birthday, not today, and certainly not next year. A moving average describes the middle of its window, so on a rising trend it always sits below the latest value and further below the next one. With months 4, 5 and 6 at 130, 140 and 150, the average is 140, the month 5 figure, while month 7 will be 160.
Sales rise Rs 10 lakh a month while the three-month average forecast runs parallel but always Rs 20 lakh below, because the average sits at the middle month and the forecast is two months ahead of it.
The relationship
$\text{Miss} = b \times \frac{n+1}{2} = 10 \times \frac{3+1}{2} = 20$
b the trend: growth per month, Rs 10 lakh
n the number of months in the average
(n + 1)/2 months between the centre of the window and the month forecast
What it says in wordsOn a straight trend, a moving average forecast misses by the monthly growth times half the window plus one.
What does that mean for the window you choose?
A longer window smooths noise better but lags more: a 12-month average on the same trend would miss by 10 x 6.5 = Rs 65 lakh. Moving averages trade noise against lag, so they suit flat, noisy series and fail on trending ones. The fix on a trend is to model the trend itself: add the slope back, or use a method that tracks level and slope separately, such as Holt's linear exponential smoothing. Say the limit both ways: if sales are seasonal, a three-month average also mixes high and low months, and if the trend turns, any trend-adjusted method overshoots for a while.
Where candidates lose it
The common answer is Rs 10 lakh: the forecast uses last month's level, so it is one month behind. That forgets that the average sits at the middle of the window, not at the latest month.
The other loss is saying the error is random. On a steady trend it is a bias: the same Rs 20 lakh in the same direction every month, which is exactly what a forecast review should catch.
What the interviewer asks next
What is the miss for a six-month moving average on the same trend?
How would you adjust the moving average to remove the bias?
What happens to the forecast error in the month after the trend stops?

k	the toss on which the first head appears
1/2^k	the chance the first head comes on toss k
100 x 2^(k-1)	the promised prize
10^6	the vendor's limit, Rs 10 lakh

total spent	Rs 20,000 across both purchases
total units	100 units at NAV 100 plus 200 at NAV 50
2 / (1/100 + 1/50)	the harmonic mean of the two prices

What you hold fixed	Units bought	Spent, Rs	Average cost, Rs
Rs 10,000 each time	300	20,000	66.67
100 units each time	200	15,000	75.00

10	cost of goods sold per day, Rs 3,650 crore over 365
20	days of stock held
45	days taken to pay suppliers

Try it first

Why do the three changes multiply instead of add?

How do you do it in your head without fumbling?

Why does the one-point gap matter on a real desk?

Where candidates lose it

What the interviewer asks next

Try it first

Why does every toss add exactly Rs 50?

What does the Rs 10 lakh limit do to the value?

How rich would the vendor need to be for Rs 1,000 to be fair?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the cheap purchase count for more?

When is Rs 75 the right answer?

Does the lower average cost mean you made money?

Where candidates lose it

What the interviewer asks next

Try it first

How can a business be funded by its suppliers?

Why does shrinking cost this business cash?

What makes the real number worse?

Where candidates lose it

What the interviewer asks next

Try it first

Why does 5% on price beat 8% on volume?

How much volume could you lose before the price rise hurts?

Where candidates lose it

What the interviewer asks next

Try it first

Why is there no need for an actual EBITDA figure?

Why do leverage and rates tend to move together, and what does 1.7x mean?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not simply 15%?

Why does a thin margin make the move so large?

Where candidates lose it

What the interviewer asks next

Try it first

Why subtract a gain you actually made?

How does each statement move, line by line?

Where candidates lose it

What the interviewer asks next

Try it first

Why is doubling in five years not 20% a year?

Why does an uneven path lose so much to its average?

Where candidates lose it

What the interviewer asks next

Try it first

Why does an average lag behind a trend?

What does that mean for the window you choose?

Where candidates lose it

What the interviewer asks next