Financial Analysis puzzles, solved step by step

The cost of equity is 13.6% at debt-to-equity of 0.5, and 15.4% at 1.0. Relevering puts your own debt back onto the peer's business risk: 0.8 times (1 plus 0.75 times 0.5) gives a beta of 1.10, and 7% plus 1.10 times 6% is 13.6%. At 1.0 the beta is 1.40, so the cost of equity is 15.4%.

Why unlever the peer's beta and then relever it?

Two friends buy identical flats. One pays cash; the other borrows 60% of the price. When flat prices move 10%, the cash buyer's wealth moves 10%, while the borrower's stake moves far more, because the loan does not shrink with the price. A peer's observed beta mixes two things, the risk of its business and the risk its own debt adds, so you strip out its debt before borrowing its business risk. Then you add back your own debt, because your shareholders carry your leverage, not the peer's.

This is the Hamada relation, which assumes debt carries no market risk of its own. Then the capital asset pricing model does the rest: cost of equity is the risk-free rate plus beta times the equity risk premium, so 7% plus 1.10 times 6% is 13.6%. At debt-to-equity of 1.0, beta becomes 0.8 times 1.75, or 1.40, and the cost of equity is 15.4%.

Does more debt make the company more expensive to fund overall?

Not by this alone. Equity gets dearer as debt rises, but debt is cheaper than equity and its interest is tax-deductible, so the weighted cost of capital can fall even while the cost of equity climbs. Where it turns is where lenders start charging more and distress becomes real, which this formula does not capture. Say that limit out loud: the straight line only holds while debt is safe.

Its beta is zero, yet its expected return is minus 8%. Beta measures movement with the market, and a slot machine's payouts are random and unrelated to the market. CAPM would give a zero-beta asset the risk-free rate, say 7% a year, so the machine falls short by at least 15 points. There is no contradiction: CAPM prices assets bought as investments, and a slot machine is bought as entertainment.

How can something so risky have a beta of zero?

Think of an umbrella seller and an ice cream seller in the same town. Each has a volatile income, but whether it rains has nothing to do with the stock market. BetaHow much an asset tends to move when the market moves, measured as its covariance with the market divided by the variance of the market. measures how an asset moves with the market, not how much it moves, so a gamble driven by a random number generator has a beta of zero. The slot machine is about as volatile as anything in a town, but all of that risk is the kind a diversified owner can spread away, and in this case the casino does exactly that across thousands of players.

Why does CAPM not give it the risk-free rate?

CAPM says expected return equals the risk-free rate plus beta times the equity risk premium. With beta of zero that is just the risk-free rate, 7% a year in this example. The machine instead returns minus 8% on every stake, and a stake lasts seconds, so over a year of play the gap is far wider than the 15 points the two headline numbers suggest. In CAPM language that is a large negative alphaThe return an asset earns above or below what its beta implies under CAPM..

So is CAPM wrong?

No, it is answering a different question. CAPM describes the prices of assets that diversified investors hold to earn a return, where anyone could sell an overpriced asset short. Nobody plays a slot machine for return; players pay 8% of each stake for entertainment, the way a cinema ticket has a negative return. And you cannot short a single slot machine to collect the edge. The only way to take the other side is to own the casino, which needs licences, buildings and capital, and the casino's return on that capital is what an investor would compare with CAPM. Say that and the interviewer hears that you know where a model applies, not just its formula.

Equity falls from Rs 2,400 crore to Rs 800 crore, a 66.7% drop on a 20% fall in EV. Debt is 7 x 800 = Rs 5,600 crore. At 10x, EV is Rs 8,000 crore; at 8x it is Rs 6,400 crore. Debt keeps its claim, so the equity slice absorbs the whole Rs 1,600 crore. At 7x leverage, equity moves 3.3 times as fast as EV, and a 7x multiple wipes it out.

Why does equity fall so much further than the business?

Think of a flat bought for Rs 1 crore with a Rs 70 lakh home loan. If the flat's price drops 20% to Rs 80 lakh, the bank still wants its Rs 70 lakh, so the owner's stake falls from Rs 30 lakh to Rs 10 lakh, a two-thirds loss. Debt is a fixed claim in rupees, so any change in enterprise value lands entirely on the equity that sits on top. Here debt is 7 x 800 = Rs 5,600 crore. At 10x, EV is Rs 8,000 crore and equity is Rs 2,400 crore, only 30% of the value.

So what does 7x leverage tell you about sensitivity?

Leverage of 7x EBITDA means the debt alone is worth 7x EBITDA. Whenever the valuation multiple sits close to the leverage multiple, equity is a thin slice of the gap between them, and small changes in the multiple swing it hard. At 10x the gap is 3 turns; each turn of multiple is Rs 800 crore, a third of the equity. Software companies can carry this because their cash flows recur and their multiples are high, but the same 7x on a business valued at 8x leaves one turn of cushion. Say the limitation: the sum assumes debt is worth its face value, while in distress the debt itself would trade below par and soak up some of the loss.

EPS falls from Rs 10.00 to Rs 9.125; the buyback breaks even at a P/E of 14.8x. After-tax interest is 400 x 9% x 75% = Rs 27 crore, so net income drops to Rs 73 crore over 8 crore shares. The shares cost 20x earnings, a 5% earnings yield, but the debt costs 6.75% after tax. EPS rises only if the P/E is below 1 / 0.0675 = 14.8x.

What decides whether a buyback raises or lowers EPS?

Think of borrowing at 9% to buy a shop that pays you 5% of its price in profit each year. Your income falls, even though you now own more. Buying your own shares is buying their earnings: it raises EPS only if the earnings yield on the shares, one over the P/E, beats the after-tax cost of the money used. At Rs 200 a share and EPS of Rs 10, the shares yield 5%. The debt costs 9% x (1 - 25%), which is 6.75%. You pay 6.75% to buy a 5% stream, so EPS falls.

Does a lower EPS mean the buyback destroys value?

Not by itself. EPS is an accounting test, and it ignores what happens to risk. Replacing equity with debt raises the risk borne by every remaining share, so a fall in EPS can sit alongside an unchanged or higher share price if the debt tax shield is worth more than the extra risk. Equally, a rise in EPS at a low P/E is not proof of value created. Say both sides: the EPS answer is Rs 9.125, the breakeven is 14.8x, and the value question turns on whether the shares were bought below what they are worth and on what the extra leverage does to the cost of equity.

The company's own cash repaid Rs 400 crore of debt, and every rupee repaid moved a rupee of the unchanged Rs 1,000 crore value from lenders to the fund. Equity went in at Rs 400 crore and came out at Rs 800 crore: a MOIC of 2.0x and, over five years, an IRR of about 14.9%. No growth and no change in multiple were needed.

Where did the gain come from if the price did not move?

Think of a flat bought for Rs 1 crore with a Rs 60 lakh home loan. The rent covers the loan payments and brings the loan down to Rs 20 lakh over five years. You sell the flat for exactly Rs 1 crore, yet your stake has grown from Rs 40 lakh to Rs 80 lakh. Enterprise value is a pie shared between lenders and owners, and each rupee of debt repaid from the business's own cash moves a rupee of that pie to the owners. The price never had to rise.

The return arithmetic follows. MOIC, the multiple on invested capital, is Rs 800 crore out over Rs 400 crore in, 2.0x. IRR is the yearly rate that turns 400 into 800 over five years: 2 to the power one fifth, less 1, which is 14.87%. The rule of 72 gives a quick check: doubling in five years is roughly 72 / 5, about 14.4%.

Is that skill or just leverage?

Run the same business with no debt. The fund pays Rs 1,000 crore of equity, the company's cash flow piles up as at least Rs 400 crore of cash, and at exit the fund receives Rs 1,000 crore plus that cash, about Rs 1,400 crore. That is 1.4x and about 7.0% a year, before adding the interest it never paid. Debt did not create the Rs 400 crore; the business earned it. Debt let a smaller cheque claim all of it, which lifts the return from 1.4x to 2.0x.

Leverage cuts both ways, and saying so is the limit worth stating. Had the company sold for only Rs 500 crore, the levered fund would get back Rs 300 crore on Rs 400 crore, 0.75x, a loss, while the unlevered owner would hold 0.9x. The same debt that doubled the good outcome deepened the bad one.

What does the interviewer listen for?

Name the three ways a buyout makes money: EBITDA growth, a higher exit multiple, and debt paydown. This deal used only the third, which is the point of the question. Then add time: the same 2.0x earned over three years instead of five is about 26% a year, which is why funds care about IRR and not only the multiple. A finishing touch is noting that the Rs 400 crore repaid is cash after interest, so the business had to earn more than Rs 400 crore to do it.

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.

A's lenders are in trouble and B's are merely uncomfortable; both sets of shareholders take a far bigger hit than revenue. A's EBIT swings from Rs 100 crore to minus Rs 40 crore and cannot cover Rs 50 crore of interest. B's falls to Rs 60 crore and covers it 1.2 times. Profit before tax falls 280% at A and 80% at B: operating and financial leverage multiply.

Why do identical profits hide very different risks?

Take two tea stalls that each clear Rs 10,000 a month. One rents a shop for Rs 60,000 a month and buys cheap; the other rents a cart for Rs 10,000 and pays more per cup. In a good month they look the same. In a bad month the shop's rent is still due while the cart's costs fall with its sales. Fixed costs do not shrink when revenue does, so the higher the fixed share of costs, the more of every lost rupee of revenue comes straight out of profit. That sensitivity is operating leverage.

Work out each company's variable cost. A spends Rs 900 crore to make Rs 100 crore of EBIT; Rs 600 crore of that is fixed, so Rs 300 crore, or 30% of revenue, is variable. B's Rs 900 crore is Rs 100 crore fixed and Rs 800 crore variable, 80% of revenue. So a lost rupee of revenue costs A 70 paise of EBIT and B only 20 paise. Revenue falls by Rs 200 crore: A loses Rs 140 crore of EBIT, B loses Rs 40 crore.

What does each lender actually face?

Each lender is owed Rs 50 crore of interest a year, 10% on Rs 500 crore. B's lender is paid out of operating profit with Rs 10 crore to spare: thin, and a rating analyst would note coverage dropping from 2.0x to 1.2x, but no default. A's lender is not paid from operations at all. A must find Rs 90 crore of cash, Rs 50 crore of interest plus the Rs 40 crore operating loss, from its cash balance, an asset sale or new borrowing, so its lender's safety now depends on liquidity, not earnings. A's break-even revenue to cover interest is Rs 929 crore, just 7% below today; B's is Rs 750 crore, 25% below.

Why do shareholders and lenders feel operating leverage differently?

Run revenue up 20% as well as down. A's profit before tax jumps from Rs 50 crore to Rs 190 crore and B's to Rs 90 crore. A's shareholders get the steep line both ways, and their loss is capped at the value of their shares. The lender gets the same Rs 50 crore in the good year and the base year, and loses only in the bad year, so operating leverage hands the upside to equity and the downside to debt. That is why credit analysts ask about the fixed cost base before they ask about margins, and why lenders to high fixed cost businesses ask for lower debt or tighter covenants.

The limitation: real fixed costs are not perfectly fixed. A could cut overheads, defer maintenance or renegotiate rent within a few quarters, so the first year is the dangerous one. Say that, and the answer moves from a formula to a judgement.

The zero falls more: about 8.8% against about 6.4% for the coupon bond. Price sensitivity follows duration, the present-value-weighted average time to each cash flow. The zero's only cash flow is at year 10, so its duration is 10 years. The coupon bond returns part of its value early, so its duration is 7.25 years. Modified duration predicts falls of 9.26% and 6.71%; convexity makes the actual falls slightly smaller.

Why does the timing of cash flows decide the sensitivity?

Two friends each owe you Rs 1,000 in total. One will repay it all in ten years; the other pays Rs 80 a year and the rest at the end. If prices start rising faster and money loses value faster, which IOU loses more value? The one where every rupee is ten years away. The second friend's early payments are already in your hands and can be reinvested at the new higher rates. A bond's price sensitivity to yield depends on when its value arrives on average, not on its final maturity date. That average, weighted by present value, is the Macaulay duration.

How big is each fall?

Use modified duration, Macaulay duration divided by one plus the yield, as the percentage price change per point of yield. For the zero, 10 / 1.08 = 9.26; for the coupon bond, 7.25 / 1.08 = 6.71. So a one point rise should cut prices by about 9.3% and 6.7%. Repricing exactly: the zero goes from 46.32 to 42.24, down 8.80%; the coupon bond goes from par, 100.00, to 93.58, down 6.42%.

Why are the actual falls smaller than duration predicts?

Duration is the slope of the price curve at today's yield, a straight-line estimate. The real curve bows outward, so as yields rise each further step hurts a little less, and as yields fall each step helps a little more. That curvature, convexity, makes a plain bond fall less than duration predicts when yields rise and gain more than predicted when they fall. For a one point move the gap is small, under half a point here; for larger moves, add the convexity term or simply reprice the bond.

Two practical notes. Most Indian government bonds pay coupons twice a year, which shortens the duration slightly but leaves the answer unchanged. And the gap between the bonds widens with maturity and narrows as coupons fall: a coupon bond's duration always sits below its maturity, while a zero's always equals it.