Financial Analysis puzzles, solved step by step

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.

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.

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.