Financial Analysis puzzles, solved step by step
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- Hard
- 30
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?Leveraged 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 relationship1.25 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?
048Quickly, without a calculator: what is 1/7 as a decimal, what is 5/7 of 910, and what is 3/7 as a percentage to two decimal places?Private equityConsulting-style case
Try it first
Which is 3/7 to two decimals?
Show the worked solution
1/7 is 0.142857 repeating, 5/7 of 910 is 650, and 3/7 is 42.86%. 910 divides cleanly by 7 to give 130, and five of those is 650. For any seventh, the six digits 142857 repeat in the same order; only the starting digit changes. 3/7 is about 0.43, so it starts at the 4: 0.428571, or 42.86%.
Why should sevenths be memorable when they look messy?
Think of a clock face with six numbers on it instead of twelve. Whatever hour you start from, you read the same numbers in the same order. Sevenths work like that: every k/7 is the cycle 1, 4, 2, 8, 5, 7 read round the ring from a different starting digit. You only need to learn one string, 142857, and a way to pick the start: k/7 is roughly k x 0.14, so 2/7 starts at 2 (0.285714), 3/7 at 4 (0.428571) and 6/7 at 8 (0.857142).
The six digits 142857 repeat in every seventh, so 3/7 reads 0.428571 starting from the 4 and is 42.86%, while 5/7 of 910 is simply 910 divided by 7, which is 130, times 5, or 650. When should you divide first and when should you use the decimal?
For 5/7 of 910, check whether 7 divides 910: 7 x 130 is 910, so divide first and multiply second, and the answer is a clean 650. When the number you are taking a fraction of is a multiple of the denominator, divide first; reach for the decimal only when it is not. For 5/7 of 1,000, there is no clean division, so use the cycle: 5/7 starts at the 7, 0.714285, and the answer is about 714.3. A check that works on both: 5/7 is a little over 70%, and 70% of 910 is 637, so 650 sits in the right place.
Interviewers use sevenths because the decimals look random and catch people who have only learned halves, quarters and eighths. The cycle gives you six decimal places in a second, which is more precision than you will ever need to say out loud.
Where candidates lose it
Candidates try long division for 3/7, get 0.43 after a pause, and then guess the second decimal. The cycle gives the full 0.428571 at once, so 42.86% is not a guess.
For 5/7 of 910, the slip is converting 5/7 to 0.714 first and multiplying, which invites a rounding error. Notice that 910 is a multiple of 7 and the answer is exact.
What the interviewer asks next
- What is 4/7 as a percentage to two decimals?
- What is 6/7 of 1,400?
- Why do sevenths repeat in a six-digit cycle?
071Without a calculator, estimate the square root of 50 to two decimal places. Then annualise a daily volatility of 1.2% over 250 trading days.Private equityTreasury
Try it first
What is the annualised volatility of 1.2% a day over 250 trading days?
Show the worked solution
Root 50 is about 7.07, and 1.2% a day is about 19% a year. Root 49 is 7, and near a known square the curve is nearly straight, so add the step divided by twice the root: 7 + 1 / 14 = 7.0714 against a true 7.0711. Volatility scales with the square root of time, so multiply by root 250, about 15.81: 1.2 x 15.81 = 19.0%.
How do you get root 50 without a calculator?
Think of walking up a gentle hill: one more step forward lifts you by the slope at the point you are standing, and for a single step the hill may as well be straight. Near a square you know, the root curve is nearly straight, so the root of 49 plus 1 is 7 plus 1 divided by the slope's denominator, twice 7. That gives 7 + 1/14 = 7.0714; the true value is 7.0711, an error of 0.0004. The slope of the root curve at a point is one over twice the root, which is why the denominator is 14 and not 7.
The relationshipa the nearest perfect square you know, 49 d the step from that square to the number you want, 1 2 root a twice the known root, the slope's denominator, 14 What it says in wordsStart from a square you know and add the step divided by twice the known root; the smaller the step relative to the square, the better the estimate.The tangent at 49 puts root 50 at 7.07 and the tangent at 256 puts root 250 at 15.81, and the same curve, read as volatility against days, carries 1.2% a day to 19.0% a year while straight-line scaling would claim 300%. Why does volatility scale with the square root of time?
Toss a coin 100 times and count heads minus tails. The lead is almost never 100; it is typically about 10, the root of 100, because the tosses partly cancel. Independent moves add in variance, not in standard deviation, so the spread after n days is the daily spread times root n. Daily variance is 1.2 squared, 1.44; over 250 days it is 360; the root of 360 is 19.0. Root 250 by the tangent at 256 is 16 - 6/32 = 15.812, close enough to the true 15.811.
Horizon Trading days Root of days Volatility 1 day 1 1.00 1.20% 1 week 5 2.24 2.68% 1 month 21 4.58 5.50% 1 quarter 63 7.94 9.52% 1 year 250 15.81 18.97% The same 1.2% daily figure becomes 2.68% over a week, 5.50% over a month and 18.97% over a year, each the daily number times the root of the days, never the days themselves. When does the root-of-time rule mislead?
The rule assumes each day's move is independent of the last and drawn from the same distribution. Trending markets, where moves follow moves, make the true spread wider than root n suggests; mean-reverting ones make it narrower. Volatility also clusters, so a 1.2% daily figure measured in a calm month understates a stormy one. The day count is a convention: 250, 252 and 365 are all in use, and annualised volatilityThe standard deviation of returns restated to a one-year horizon, usually by multiplying a daily figure by the square root of the number of trading days. quoted on each will differ, so state the one you used. The limit to say: the rule converts a horizon, it does not forecast the year.
Where candidates lose it
The common loss on the second part is multiplying by 250 and announcing 300%, a number that should sound wrong before it is spoken: no index moves three times its value in a typical year.
On the first part, candidates either guess 7.5, halfway between 7 and 8 although 50 is barely past 49, or divide the step by 7 instead of 14 and get 7.14. Say the method aloud: twice the root in the denominator.
What the interviewer asks next
- Estimate root 30 using the same method. Why is the error larger than for root 50?
- A fund reports 24% annualised volatility. What is its daily volatility, and what weekly move would be a two-standard-deviation event?
- A ten-day risk horizon is often scaled from a one-day figure by root 10. What assumption does that make, and when would you distrust it?