Financial Analysis puzzles, solved step by step

WACC is 7.25%. With no growth, a P/E of 10 means an earnings yield of 1 / 10 = 10%, which stands in for the cost of equity. Debt costs 6% before tax and 6% x (1 - 25%) = 4.5% after. Weighted half and half: 0.5 x 10% + 0.5 x 4.5% = 5% + 2.25% = 7.25%.

Why can the P/E stand in for the cost of equity?

Suppose you buy a shop for ten years of its profit, and the profit never grows and is all paid to you. Each year you collect a tenth of what you paid: a 10% return. With no growth and all earnings paid out, the price is earnings divided by the cost of equity, so the earnings yield, one over the P/E, is the return shareholders require. At 10x that is 10%.

The tax step matters because interest is deducted before tax is calculated. Every Rs 100 of interest saves Rs 25 of tax, so lenders cost the company only Rs 75 of every Rs 100 they receive. Dividends earn no such deduction, so equity gets no adjustment.

When does the shortcut break?

When earnings grow. The dividend growth relation gives P/E = payout / (k - g), so with growth the same 10x implies a higher cost of equity. At a 70% payout and 4% growth, k = 0.7 / 10 + 4% = 11%, and WACC becomes 7.75%. With growth, the earnings yield understates the cost of equity, because part of the shareholder's return comes from growth rather than from today's earnings. That is why the question says no growth.

What would an interviewer probe next?

Two things. First, weights: they must be market values, not book values, because WACC is the return investors require on what their claims are worth today. Second, leverage: the 10x P/E belongs to shares in a company already carrying 50% debt, so the 10% already includes the extra risk that debt places on equity. Unlever it before applying it to a company with a different capital structure. The tax shield is only real if the company has profits to deduct interest from, which is the limit to name.

In the year it happens, the capex cut: it adds the full Rs 10 crore to free cash flow, while more revenue or lower cost of goods sold adds Rs 7.5 crore after 25% tax. Capex is a direct cash item. If the change repeats every year, lower capex means lower depreciation and more tax, so all three converge on Rs 7.5 crore a year, and an exit multiple on EBITDA favours revenue and cost.

Why does capex hit cash harder than revenue or cost?

If you earn Rs 10,000 more, you keep what is left after income tax. If you decide not to buy a Rs 10,000 laptop, you keep all Rs 10,000, because buying it was never a deduction in that year. Revenue and cost changes reach free cash flow after tax; a capex change reaches it in full, because capex is subtracted below the tax line. So with a 25% tax rate, Rs 10 crore more revenue or Rs 10 crore less cost of goods sold adds Rs 7.5 crore, and Rs 10 crore less capex adds Rs 10 crore.

One assumption sits under the revenue answer: that the extra Rs 10 crore comes with no extra cost and no extra working capital. In practice more revenue usually needs more receivables and stock, which takes a little more away. Say so in one clause.

What changes when the change repeats every year?

Capex becomes depreciation. Spend Rs 10 crore less every year on assets depreciated over five years, and depreciation falls by Rs 2 crore more each year until it is Rs 10 crore lower, which raises tax by Rs 2.5 crore. In steady state a recurring capex cut adds Rs 10 crore of cash less Rs 2.5 crore of lost tax shield, Rs 7.5 crore a year: exactly the same as the revenue and cost changes. The capex cut's edge is timing, worth something in present value, but not a permanent advantage.

Even a one-off cut keeps only part of its edge. Discounted at 10%, Rs 10 crore saved next year is worth Rs 9.09 crore, but the depreciation shield given up over the following five years is worth Rs 1.72 crore, leaving Rs 7.37 crore against Rs 6.82 crore for a one-year revenue gain. Still ahead, by much less than Rs 2.5 crore.

How does the terminal value change the ranking?

If the terminal value uses an exit multiple of EBITDA, a permanent Rs 10 crore of extra revenue or lower cost raises terminal EBITDA by Rs 10 crore and the terminal value by Rs 80 crore at 8x, while a capex cut does not touch EBITDA and adds nothing through the multiple. With an exit multiple, recurring revenue and cost changes beat capex by a wide margin; with a perpetuity growth terminal, all three are equal in steady state. Name which terminal method you are assuming before you rank them. And add the business limit: capex cut without consequence is rare, because the assets usually drive future revenue.

36 ways for 10, 45 for 11, and (n minus 1)(n minus 2) / 2 in general. Write 10 as a row of ten ones. Three positive parts means two dividers in two different gaps between the ones, and there are nine gaps, so the count is C(9, 2) = 36. For 11 there are ten gaps: C(10, 2) = 45. For any n it is C(n minus 1, 2). If order did not matter, 10 has only 8 splits, and if zero were allowed it would have 66, so say which version you are answering.

How do you turn 'three numbers add to 10' into something you can count?

Ten sweets in a row, three children, each must get at least one. You hand them out by cutting the row in two places, and every pair of cuts is one way to share. Each ordered triple is one choice of two distinct gaps out of the nine between ten objects, so counting triples is the same as counting pairs of gaps: 9 choose 2. That is 9 x 8 / 2 = 36. The same picture gives 11 at once: eleven sweets have ten gaps, and 10 x 9 / 2 = 45.

Check the small cases before trusting the formula. A total of 3 can only be 1 + 1 + 1, one way, and C(2, 2) = 1. A total of 4 gives 1 + 1 + 2 in three orders, and C(3, 2) = 3. A total of 5 gives six, C(4, 2). The counts 1, 3, 6, 10, 15 are the triangular numbers, which is what (n minus 1)(n minus 2) / 2 produces.

What changes if the interviewer meant something else?

Three variants are common, and the right first move is to ask or to state your reading. If zero is allowed, the dividers may share a gap or sit at the ends, and the count becomes C(n + 2, 2), which is 66 for 10. If order does not matter, you are counting partitions of 10 into three parts: 1+1+8, 1+2+7, 1+3+6, 1+4+5, 2+2+6, 2+3+5, 2+4+4, 3+3+4, 8 in all, and there is no neat formula, so you list them from the smallest part upwards. Say which version you are answering, then answer it; the count is easy once the question is fixed, and most lost marks here come from solving a different question than the one asked. If negative numbers were allowed the count would be infinite, which is worth one sentence to show you noticed.

Why would a trading desk ask this?

Counting outcomes is the floor under every probability question, and the follow-up about 11 tests whether you hold a method or a memorised answer. The natural next step is three dice: how many of the 216 rolls add to 10? Start from 36 ordered triples and remove those with a part above 6: an 8 with two 1s gives 3 orders, a 7 with 1 and 2 gives 6, so 36 minus 9 is 27, and the chance is 27/216, 12.5%. The limitation of the gap method is exactly that it has no upper bound on any part, so a capped problem needs the subtraction.