Financial Analysis puzzles, solved step by step

Puzzles: 100
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005A dairy cow gives 10 litres of milk a day for 300 days a year. Milk sells at Rs 40 a litre and the cow costs Rs 70,000 a year to keep. It will produce for 6 more years and then be sold to another farm for Rs 20,000. At a 12% required return, what would you pay for the cow?Compounding and time valueCoreRothschild & CoNew York · 2026
Try it first
Before discounting, what is the cow's net cash each year?
Show the worked solution
About Rs 2.16 lakh. The cow earns Rs 1,20,000 of milk a year and costs Rs 70,000, leaving Rs 50,000 of net cash for six years. At 12% that stream is worth about Rs 2,05,570, and the Rs 20,000 sale in year 6 adds about Rs 10,133. Paying more than about Rs 2,16,000 means earning less than 12% on the purchase.
How do you value something that is not a company?
The same way you value a company. A shop, a flat you rent out and a cow are all machines that turn money in now into money out later. Anything that produces cash can be valued as the present value of the cash it will hand you, after the costs of keeping it running. So the first job is to find the net cash each year, then the life, then the value left at the end.
Net cash: 10 litres a day for 300 days is 3,000 litres, worth Rs 1,20,000 at Rs 40. Upkeep of Rs 70,000 leaves Rs 50,000 a year. The six years and the Rs 20,000 sale at the end complete the cash flows.
The cow's Rs 50,000 of yearly net cash is worth Rs 44,643 today in year 1 but only Rs 25,332 in year 6, and the Rs 20,000 sale is worth Rs 10,133. Added up, the cow is worth about Rs 2.16 lakh at 12%.
The relationship
$V = 50{,}000 \times \frac{1 - 1.12^{-6}}{0.12} + \frac{20{,}000}{1.12^{6}} = 50{,}000 \times 4.111 + 10,133 \approx 215,703$
50,000 net cash each year, Rs
4.111 the six-year annuity factor at 12%
20,000 sale value at the end of year 6, Rs
What it says in wordsValue is the six equal yearly cash flows discounted as an annuity, plus the final sale discounted six years.
What would make you pay less?
Every assumption in this answer is a risk to price: yield falling with age, illness, milk price, feed cost. If milk yield drops 10% a year, net cash falls fast because upkeep does not fall with it, and the value drops sharply. A higher required return for a risky asset does the same. The interviewer who asks you to value an animal is checking that you name the cash, the life, the end value and the risk, in that order, and do not stop at the milk bill.
Where candidates lose it
The common slip is discounting Rs 1,20,000 of milk sales instead of Rs 50,000 of net cash, which more than doubles the answer. Revenue is not what the owner keeps.
The second is forgetting the end value, or adding it undiscounted. Rs 20,000 in six years is worth about Rs 10,000 today at 12%, and saying so shows you treat every cash flow the same way.
What the interviewer asks next
Milk yield falls 10% a year from year 2. What is the cow worth now?
What discount rate would you use, and why might it be higher than for a bond?
How would you value a pet dog, which produces no cash?
Asked at Rothschild & Co, Generalist, New York, 2026 (Wall Street Oasis): How would you value your favorite animal? What is your personal beta?
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?

057A toll road concession will pay Rs 50 crore a year forever, with the first payment at the end of year 4. At a 10% discount rate, what is it worth today?Compounding and time valueCoreProject financeCorporate finance

Try it first

What is the concession worth today?

Show the worked solution

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.

The perpetuity formula values the year 4 onward payments at Rs 500 crore at year 3, one year before the first payment, and discounting three years at 10% brings that to Rs 375.66 crore today; discounting four years gives a wrong Rs 341.5 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.

The relationship

PV = \frac{50 / 0.10}{1.10^{3}} = \frac{500}{1.331} \approx 375.66 \qquad PV = 500 - 50 \times \frac{1 - 1.10^{-3}}{0.10} = 500 - 124.34

50 / 0.10	the perpetuity's value at year 3, one year before the first payment
1.10^3	three years of discounting at 10%
124.34	the value today of the three payments years 1 to 3 that this stream does not have

What it says in wordsEither discount the year 3 value three years, or take a perpetuity from year 1 and remove the three missing payments; both give about Rs 375.66 crore.

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%.

Where candidates lose it

The common slip is discounting four years because the first payment arrives in year 4. The formula has already placed its value a year before the first payment, so a fourth year of discounting counts that year twice and understates the value by about 9%.

The other slip is subtracting the three missing payments at face value, 500 less 150. Payments in years 1 to 3 are worth less than Rs 150 crore today, so the shortcut takes off too much and lands at Rs 350 crore.

What the interviewer asks next

The payments grow at 3% a year after year 4. What is the concession worth today?
The concession ends after 30 payments. How much value is lost against the perpetuity?
In a DCF with a five-year forecast, which year's cash flow goes into the terminal value, and how many years do you discount it?

093A perpetuity pays Rs 100 a year forever, discounted at 10%. What share of its present value comes from the first 10 years of payments? From the first 30?Compounding and time valueCoreEquity researchCorporate finance
Try it first
Roughly what share of the value comes from the first 10 years?
Show the worked solution
About 61% from the first 10 years and 94% from the first 30. The perpetuity is worth Rs 100 / 0.10 = Rs 1,000. Payments beyond year N are themselves a perpetuity, starting later, so their share is 1/1.1^N: 38.6% beyond year 10 and 5.7% beyond year 30. Add growth and the far years matter much more: at 5% growth only 37% of the value arrives in the first decade.
Is there a quick way to split a perpetuity by year?
Suppose you are promised a pension of Rs 100 a year forever. Starting ten years from now, what you are owed is again a Rs 100 perpetuity, worth Rs 1,000 at that date; you just have to wait ten years for it. The value of the payments after year N is the full perpetuity value, discounted back N years, so its share of the total is 1/(1 + r)^N and the first N years take the rest. At 10% that is 1/1.1^10 = 0.386 beyond year 10 and 1/1.1^30 = 0.057 beyond year 30.
The relationship
$\text{Share in first } N \text{ years} = 1 - \left(\frac{1+g}{1+r}\right)^{N} \qquad 1 - 1.1^{-10} = 61.4\% \qquad 1 - 1.1^{-30} = 94.3\%$
r the discount rate, 10%
g the growth rate of the payment, zero here
N the number of years counted
What it says in wordsThe share of value received in the first N years is one minus the discount factor for N years, adjusted for growth.
In rupees, the first ten years are worth Rs 614 of the Rs 1,000. Half the value arrives in the first 7.3 years, because 1.1 to the power 7.3 is 2. A useful mental marker: at 10%, value halves every seven years or so.
Discounted at 10%, a flat perpetuity delivers 61% of its value in the first 10 years and 94% in 30, but if the payment grows 5% a year only 37% arrives in the first decade, which is why a DCF's terminal value dominates.
Why does this matter for a DCF?
A DCF forecasts a company for perhaps ten years and puts everything after into a terminal value. With no growth, the terminal value would already be 39% of the total. Companies are usually assumed to grow, and growth pushes value further out: at 5% growth and a 10% discount rate, the gap between them shrinks to about 4.8% a year, half the value arrives only after 14.9 years, and the terminal value is 63% of the total. Most of a DCF's value sits beyond the explicit forecast, so the terminal growth rate and discount rate deserve more scrutiny than the detail of year three.
The limitation runs the other way too. The terminal value depends on the gap between the discount rate and growth, so a small change in either swings it sharply. Present a DCF with the terminal value's share stated and a sensitivity table beside it, rather than a single number.
Where candidates lose it
The intuitive answer is tiny, because ten years is nothing against forever. It ignores discounting: a payment in year 50 is worth under one hundredth of its face value today. The first decade carries 61% of the value with no growth.
The opposite loss comes in the follow-up. Candidates who learn the 61% conclude that the terminal value is a minor detail. Add realistic growth and the far years matter far more: at 5% growth only 37% of the value sits in the first ten years.
What the interviewer asks next
At what discount rate would the first 10 years carry 75% of a flat perpetuity's value?
If a DCF's terminal value is 80% of the enterprise value, what does that tell you about where the risk in the valuation sits?
How does the share change if the discount rate falls from 10% to 8% with 5% growth?

50,000	net cash each year, Rs
4.111	the six-year annuity factor at 12%
20,000	sale value at the end of year 6, Rs

g_i	growth in year i
product	multiply the yearly growth factors, do not add the rates

Try it first

How do you value something that is not a company?

What would make you pay less?

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

Where in time does the perpetuity formula put its value?

How do you check it a second way?

Why does this matter beyond the puzzle?

Where candidates lose it

What the interviewer asks next

Try it first

Is there a quick way to split a perpetuity by year?

Why does this matter for a DCF?

Where candidates lose it

What the interviewer asks next