Financial Analysis puzzles, solved step by step

Output doubles in about 7.2 months and reaches eight times in about 21.6 months, by the rule of 72. Divide 72 by the growth rate in per cent: 72 over 10 is 7.2 months per doubling, and eight times is three doublings. The exact answer, the log of 2 over the log of 1.1, is 7.27 months, or 21.8 months for eight times, so the rule is within a fifth of a month.

Why does eight times take three doublings and not 70 months?

Think of a rumour that doubles the number of people who know it every week. One becomes two, then four, then eight in three weeks, not eight weeks. Compounding growth multiplies, so the question to ask is how many doublings fit in, not how many 10% steps add up to 700%. Two times two times two is eight, so eight times today's output is three doublings away. Dividing 700% by 10% a month treats each month's growth as if it were earned only on the starting level.

Where does the rule of 72 work, and where does it slip?

The exact doubling time uses the log of 1 plus the rate, which is a little less than the rate itself, and 72 is chosen to compensate for typical rates. The rule is within about 1% for rates from 5% to 10% a period and drifts at the extremes: at 2% it overstates, and at 50% it understates. At 10% the gap is only 0.07 months. A check in whole months helps too: 1.1 to the 7th is 1.95 and to the 8th is 2.14, so output passes double during month 8.

Say the limit in context. Output growing 10% a month for two years is a startup or a ramp-up, not a steady state; a plant hits capacity, demand or cash constraints long before it has grown nearly tenfold. The interviewer wants the arithmetic, then one sentence of business sense.

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.

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.

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%.

About 1.49% a month, or 19.4% a year effective. The real price is the Rs 57,000 cash price, so choosing the EMI means borrowing Rs 57,000 and repaying Rs 60,000 in six monthly instalments. The rate that makes six payments of Rs 10,000 worth exactly Rs 57,000 today is 1.49% a month. The forgone Rs 3,000 discount is the interest, charged on a balance that shrinks every month.

Where is the interest, if the EMI says no-cost?

A friend offers to sell you his bike for Rs 57,000 today, or for Rs 60,000 paid over six months. Nobody calls the second option interest-free; it is a loan of Rs 57,000 with Rs 3,000 of interest. A shop doing the same thing is no different. When a cash discount is available, the discount you give up by paying in instalments is the interest on the loan. The 'no-cost' label means the instalments add up to the sticker price, not that money has no time value.

So the question becomes: at what monthly rate is a stream of six Rs 10,000 payments, starting in one month, worth Rs 57,000 today? That is the internal rate of return of the loan.

How do you solve it without a spreadsheet?

Guess and check. The annuity factor must be 57,000 / 10,000 = 5.70. At 1% a month, six payments are worth 5.80 times one payment; at 2% they are worth 5.60. Halfway gives about 1.5%, and 1.5% gives 5.697, a whisker under 5.70, so the answer is just below: 1.49%. Then compound, do not multiply: 1.0149 to the power 12 is 1.194, so the effective annual rate is 19.4%, against 17.8% if you simply multiply by 12.

A second check uses the average balance. The loan starts at Rs 57,000 and is paid down to zero over six months; the balance at the start of each month averages about Rs 33,700. Rs 3,000 of interest on about Rs 33,700 over six months is 8.9%, or about 17.8% a year simple, which matches 1.49% x 12.

What changes the answer in real life?

Timing matters a great deal. If the first instalment is due today rather than in a month, you really borrow only Rs 47,000 and repay Rs 50,000 over five months, and the hidden rate jumps to 2.10% a month, about 28% a year. Any processing fee, or tax charged on the interest component, raises the true cost further, so always ask what you pay on day one. Check the specific charges and the current tax treatment with the lender. And if there was never a cash discount to forgo, the EMI really is close to free for the buyer, because the retailer, not the buyer, is paying the lender.

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.

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.

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.