Financial Analysis puzzles, solved step by step

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  1. 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.

    Each year's Rs 50,000, and what it is worth today at 12%44,64350,000Year 139,86050,000Year 235,58950,000Year 331,77650,000Year 428,37150,000Year 525,332+20,000 sale10,133Year 6present value at 12%cash received that yearSum of thefilled barsRs 2.16lakh
    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=50,000×1−1.12−60.12+20,0001.126=50,000×4.111+10,133≈215,703V = 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
    50,000net cash each year, Rs
    4.111the six-year annuity factor at 12%
    20,000sale 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?

  2. 032A machine turns a Rs 10 note into a Rs 20 note, but each use takes exactly one year, costs Rs 2 of electricity, and the machine breaks after 10 uses. At a 10% interest rate, what would you pay for it today?Compounding and time valueHardGoldman SachsDallas · 2026

    Try it first

    Before you discount anything: what is the machine's real net gain per use, measured at the end of the year?

    Show the worked solution

    About Rs 43. Each use ties up a Rs 10 note for a year, which at 10% costs you Rs 11 by year end, plus Rs 2 of electricity, and pays Rs 20. That is Rs 7 a year for ten years, and Rs 7 times the ten-year annuity factor of 6.1446 is Rs 43.01. If the electricity is paid at the start of each year the net is Rs 6.80 and the value Rs 41.78.

    What is one use of the machine actually worth?

    Think of lending a friend Rs 10 for a year and getting Rs 20 back. You doubled your note, but you also went a year without it, and that year had a price: whatever the money would have earned in the bank. The machine does not create Rs 10 a year; it creates Rs 20 at year end in exchange for Rs 10 today, and Rs 10 today is worth Rs 11 at year end at 10%. Take off Rs 2 of electricity and each use leaves Rs 7, received one year after you feed it the note.

    Ten uses, Rs 7 net each, discounted back to todayOne use, valued at year end:+20 out - 10 x 1.1 note in - 2 power = 7Sum of ten present valuesRs 43.01 (power paid up front: 41.78)6.36Yr 15.79Yr 25.26Yr 34.78Yr 44.35Yr 53.95Yr 63.59Yr 73.27Yr 82.97Yr 92.70Yr 10Dashed outline: Rs 7 received at the end of each year. Filled: what it is worth today at 10%.Forget the Rs 1 a year the note could have earned and you get Rs 49.16, too much.
    Each of the ten uses leaves Rs 7 at the end of its year once the Rs 11 cost of the note and the Rs 2 of electricity are taken off the Rs 20, and the ten present values fall from Rs 6.36 to Rs 2.70 and sum to Rs 43.01.
    The relationship
    PV=(20−10(1.1)−2)×1−1.1−100.10=7×6.1446=43.01PV = \big(20 - 10(1.1) - 2\big)\times\frac{1-1.1^{-10}}{0.10} = 7 \times 6.1446 = 43.01
    10(1.1)the Rs 10 note's value at year end had you kept it at 10%
    6.1446the present value of Rs 1 a year for ten years at 10%
    What it says in wordsFind the net gain of one use at the end of its year, then value ten of them as an annuity.

    Why does the timing of the electricity change the answer?

    The question does not say when you pay for power. If you pay at the end of the year, the net is the Rs 7 above. If you pay at the start, alongside the note, the Rs 2 also costs you a year of interest, Rs 2.20 by year end, and the net drops to Rs 6.80, worth Rs 41.78. State the timing assumption out loud, because an interviewer who wrote this puzzle is listening for whether you notice that cash flows need a date. The other common answer, Rs 49.16, discounts Rs 8 a year and is wrong for a reason, not a rounding: it treats the Rs 10 note as free to borrow.

    One check makes the answer believable. You could reproduce the machine with a bank loan: borrow Rs 10 at 10%, run the machine, repay Rs 11 and the Rs 2 of power, keep Rs 7. A buyer will pay up to the present value of that stream, and no more, because the bank can supply the money at 10% anyway.

    Where candidates lose it

    The usual answer is Rs 49.16: Rs 8 a year, discounted. It misses that the note fed in each year is capital with a cost. Candidates who think of each use as a project, with an outflow today and an inflow in a year, do not make this mistake.

    The second loss is giving one number with no assumption. Say when you assume the electricity is paid, give both values if asked, and the interviewer hears someone who dates every cash flow.

    What the interviewer asks next

    • What would you pay if the machine could be used once a year forever?
    • At what interest rate is the machine worth nothing?
    • If the machine could run two notes at once, what would it be worth?

    Asked at Goldman Sachs, Summer Analyst Interview, Dallas, 2026 (Wall Street Oasis): A mad scientist invents a machine that turns a standard $10 bill into a $20 bill

  3. 070One lender quotes 12% a year compounded monthly; another quotes 12.5% a year compounded annually. Which loan is cheaper, and what is the effective annual rate of each?Compounding and time valueWarm upCarlyle GroupNew York · 2015

    Try it first

    Which loan is cheaper?

    Show the worked solution

    The 12.5% annual loan is cheaper. Twelve per cent compounded monthly is an effective 12.68% a year, against 12.50%. A 12% nominal rate charged monthly means 1% a month, and interest earns interest eleven more times within the year: 1.01 to the power 12, less 1, is 12.68%. Convert every quote to an effective annual rate before comparing.

    Why is 12% a year compounded monthly more than 12%?

    Leave a credit card balance unpaid and the interest charged in January is itself charged interest in February. The bank's 'monthly rate' grows faster than the same rate charged once a year. A nominal rate tells you how interest is quoted; the effective annual rate tells you what it costs, and only effective rates can be compared. Here 12% a year becomes 1% a month, and after twelve months of compounding Rs 100 has grown to Rs 112.68.

    Same 12% headline, five different costs: convert before you compare12.0%12.4%12.8%Lender B: 12.50% compounded annually12.00%Annual12.36%Half-yearly12.55%Quarterly12.68%Monthly12.75%DailyLender A: 12% monthlyCompounding frequency of a 12% nominal rate
    The same 12% nominal rate costs anything from 12.00% to about 12.75% a year depending on how often it compounds, and at monthly compounding its 12.68% sits above the other lender's 12.50%, so the annual quote is cheaper.
    The relationship
    EAR=(1+0.1212)12−1=1.0112−1≈12.68%  >  12.50%\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 1.01^{12} - 1 \approx 12.68\% \;>\; 12.50\%
    0.12the nominal annual rate
    12compounding periods a year
    EAReffective annual rate, the true yearly cost
    What it says in wordsDivide the nominal rate by the number of periods, compound it over a year, and subtract one to get the rate you can compare.
    Compounding of 12% nominalEffective annual rate
    Annual12.00%
    Half-yearly12.36%
    Quarterly12.55%
    Monthly12.68%
    Daily12.75%
    Continuous12.75%
    More frequent compounding raises the effective rate, but with sharply diminishing steps: going from monthly to continuous adds less than a tenth of a point.

    What would the monthly lender need to quote to match?

    Run the conversion backwards: the monthly rate that compounds to 12.5% is 1.125 to the power 1/12, less 1, about 0.98% a month, or a nominal 11.84% a year. Any monthly quote above 11.84% is dearer than 12.5% annual. For a quick mental check, the extra from monthly compounding at these rates is roughly half the rate squared: 0.5 x 0.12 x 0.12 is 0.72 points, close to the true 0.68.

    What else decides which loan is really cheaper?

    The effective rate prices the money, not the whole deal. Processing fees, prepayment penalties and insurance bundled into the loan all add to the cost and must be folded in. Watch for flat-rate quotes too: a 7% flat rate on a three-year loan charges interest on the original amount even as it is repaid, which works out to an effective rate of about 13.6% a year. Confirm how any lender computes its rate before comparing; the limit of the EAR is that it compares like with like only once every cost is in it.

    Where candidates lose it

    The fast wrong answer picks 12% because it is the smaller number. It compares quotes built on different compounding, which is comparing prices in two currencies without converting.

    The other slip is the reverse: knowing that compounding adds cost but guessing the size. Monthly compounding adds about 0.68 points at 12%, not one or two points, so a 12.5% annual quote only just wins. Compute it rather than estimate it.

    What the interviewer asks next

    • What is the effective annual rate of 1.5% a month on a credit card?
    • A deposit pays 7% compounded quarterly. What is its effective annual yield?
    • Why does continuous compounding at 12% give about 12.75%, and what function produces it?

    Asked at Carlyle Group, Generalist, New York, 2015 (Wall Street Oasis): Some math brainteasers and accounting questions ranging from compounding rates to how an inventory purchase would flow