Equity Research puzzles, solved step by step

The cost of equity rises from 12% to 16% and WACC stays at 12%. With debt equal to equity, shareholders carry the same business risk on half the capital, so their required return rises by the spread between the unlevered rate and the debt rate, 4 points, times debt over equity. Half at 16% and half at 8% is still 12%. Without taxes, cheap debt only moves risk around.

Why can cheap debt not lower the cost of capital on its own?

Think of two friends buying a food truck. If one lends at a fixed rate and the other takes whatever is left after paying the loan, the truck's takings are no less risky; the owner just now carries all of the ups and downs on a smaller stake. Borrowing does not change the business, so it cannot change the total return the business must earn for all its funders; it only shifts risk from lenders to shareholders.

How do you get the 16%?

The Modigliani and MillerThe 1958 result, from Franco Modigliani and Merton Miller, that in a world without taxes or distress costs the value of a firm does not depend on how it is financed. relation gives it directly. The cost of equity is the unlevered rate plus the gap between the unlevered rate and the debt rate, scaled by debt over equity. Here that is 12% plus (12% minus 8%) times 1, which is 16%. Check with WACC: half at 16% plus half at 8% is 12%, exactly the unlevered rate.

Now add back what the puzzle removed. With tax, interest is deductible, so debt does lower WACC a little; at high debt, the cost of debt itself rises and distress costs appear. That is why an analyst who sees WACC fall sharply as a model adds debt should check whether the cost of equity was left unchanged. In practice this shows up as re-levering beta: the equity beta must rise when leverage rises.

About 9%: duration of 6 times a 1.5 point rise in yield. For an 8-year, 7% coupon bond at par, the true fall is 8.5%, a little less than the straight-line estimate because the price curve bends. For the equity analyst, the market now charges the company about 1.5 points more for new debt, so the cost of debt in WACC should use the new yield, not the old coupon.

How does duration turn a spread move into a price move?

Think of a fixed-rate deposit you cannot break. When new deposits start paying more, yours is worth less to anyone who might buy it from you, and the longer it has left to run, the bigger the discount. Modified duration is that sensitivity in one number: the approximate percentage price change for each 1 point change in yield. A spread widening raises the bond's yield by the same amount, so 6 x 1.5 = 9%.

Why is the true fall a little smaller than 9%?

The price-yield curve is convex: it flattens as yields rise. Duration draws a straight tangent at today's yield, so for large moves it overstates price falls and understates price rises. Here the tangent says 91.0 and the bond is actually worth 91.5. For small moves the gap is negligible; at 150 basis points it is about 0.5 points, worth one sentence.

Now the equity view. The cost of debt in a WACC is what the company would pay to borrow today, so a 150 basis point widening raises it by about 1.5 points, whatever the coupon on existing bonds. With debt at 30% of capital and a 25% tax rate, that alone adds about 0.34 points to WACC. The larger message is the signal: credit investors are pricing more risk, and equity sits below the bonds, so the analyst should ask what the bond market has seen.

About 11.7%. Scale the dollar rate by relative inflation: (1 + 9%) x (1 + 5%) / (1 + 2.5%) = 1.1166, so the rupee cost of equity is 11.66%. Adding the 2.5 point gap gives 11.5%, close but slightly low. The conversion keeps the value the same in both currencies, because rupee cash flows grow faster by exactly the same factor.

Why must the rate change when the business has not?

Think of a salary quoted two ways. A Rs 10 lakh salary rising with 5% inflation and the same salary quoted in dollars rising with 2.5% inflation are one job. If you discount the faster-rising rupee salary at the slower dollar rate, you make the same job look more valuable. A discount rate carries the inflation of its currency, so moving between currencies means moving the rate by the inflation gap, in the same way the cash flows move.

How do you prove the value is the same in both currencies?

Take a cash flow worth 100 in today's money, received in year five. In dollars it grows at 2.5% inflation and is discounted at 9%: 1.025 to the fifth over 1.09 to the fifth gives a discount factor of 0.7353. In rupees it grows at 5% and is discounted at 11.66%: the factor is 0.7353. The two factors match to the fourth decimal in every year, which is the proof that the conversion is right.

Say the limitation. This converts the currency, nothing else. If the dollar rate was built for a US listed peer, it may not carry any premium for country risk, and whether to add one is a separate judgement you should state and defend, not a number to fold silently into the conversion.

A levered beta of 1.10. Relever with levered beta = unlevered beta x (1 + (1 - tax rate) x debt/equity). That is 0.8 x (1 + 0.75 x 0.5) = 0.8 x 1.375 = 1.10. The 0.8 is the risk of the business itself; the extra 0.3 is the financial risk shareholders take on because lenders are paid first.

Why does debt raise the equity's beta?

Think of two families with the same salary. One has a home loan EMI to pay first each month; the other has none. A 10% pay cut hurts the family with the EMI far more, because the EMI does not shrink. Lenders are paid a fixed amount first, so the same swing in the business moves the shareholders' leftover by more, and beta measures exactly that swing. The business risk is the 0.8; debt stacks financial risk on top.

Where do candidates go wrong with the inputs?

Two places. Debt to equity uses market values where you can, and it is debt over equity, not debt over total capital: 0.5 debt to equity is one third debt in the capital structure. Using 0.33 by mistake gives 0.8 x (1 + 0.75 x 0.33) = 1.00. And the tax rate is the one that applies to this company's interest deduction; confirm the current rate rather than assuming it.

Say the limitation. The formula assumes the debt itself carries no market risk and that the debt level stays constant. For a heavily indebted company, debt starts to behave like equity and the simple formula overstates the levered beta.

About 14%. In a growing perpetuity the price is next year's dividend divided by (r - g). The gap goes from 12% - 6% = 6 points to 13% - 6% = 7 points, so the price is multiplied by 6/7, a fall of 14.3%. With a Rs 6 dividend the price goes from Rs 100 to Rs 85.7. A stock growing only 2%, priced at the same Rs 100, falls just 9.1%.

Why does one percentage point matter so much?

Think of a shop you could buy whose rent rises every year. What you would pay depends on the return you want minus the rate at which the rent grows: the faster the rent grows, the less of your required return has to come from today's income. Price depends on the gap r minus g, not on r itself, so when growth is high and the gap is narrow, a one point move in r is a large change in the gap. Here the gap goes from 6 points to 7, a rise of a sixth, so the price falls to six sevenths.

Which stocks are most exposed to a rise in rates?

Compare a slow grower priced the same way. A stock with a Rs 10 dividend growing at 2% is also worth Rs 100 at 12%; at 13% it is worth Rs 90.9, a fall of 9.1%. The more growth is built into a price, the further in the future its cash arrives, and the more the price moves when the discount rate moves. The durationThe sensitivity of a price to its discount rate, roughly the percentage price change for a one point change in the rate. of a growing perpetuity is about 1/(r - g): 16.7 for the fast grower and 10 for the slow one. The duration shortcut predicts a 16.7% fall for the fast grower; the actual 14.3% is smaller because the curve bends.

What are the limits of this answer?

The perpetuity assumes 6% growth forever and a discount rate that shifts cleanly by one point with nothing else changing. Real companies do not grow steadily forever, and a rise in rates often arrives with a change in growth expectations too. Treat 14.3% as the sensitivity of the valuation to its discount rate, not as a forecast of the share price. The ranking survives the simplification: high-growth, long-dated cash flows are the most rate sensitive.

The real rate is about 5.7%, and discounting real cash flows at the nominal rate understates value. Real and nominal rates are linked by multiplying: 1.11 / 1.05 = 1.0571, so the real rate is 5.71%, a little below the 6% you get by subtracting. A forecast in today's prices has no inflation in it, so an 11% rate removes inflation twice. Rs 100 in year 10 is worth Rs 57.4 at the matched rate but Rs 35.2 at the mismatched one.

How are real and nominal rates linked?

Your salary rises 11% in a year when prices rise 5%. You can buy more, but not 6% more: you have 1.11 rupees for every 1.05 rupees of goods you used to buy, which is 5.71% more stuff. A real rate is the nominal rate with inflation divided out, not subtracted, so 1 plus the real rate equals 1.11 over 1.05. At low rates the two methods differ by a fraction of a point; at high inflation the gap matters.

Why does the mismatch understate value?

A nominal discount rate assumes the cash flows it meets include inflation, and discounting removes it. If the forecast was built in today's prices, the inflation was never added, so the nominal rate removes it anyway and the value falls by about 39% on a year-10 cash flow. The reverse mistake, inflated cash flows at a real rate, overstates value by just as much in the other direction. Both errors grow with the length of the forecast, which is why they hurt long-lived assets most.

Which approach does an analyst use?

Either, as long as it is consistent. Most equity models are nominal, because reported accounts, debt costs and tax are all in current rupees. Real models are common for long-lived assets such as infrastructure, where prices are linked to inflation. One trap sits inside real models: tax depreciation is fixed in rupees at the purchase price, so in a real model its value must be deflated or it is overstated. Check the terminal growth rate too: a 6% nominal growth rate is only about 1% in real terms at 5% inflation.

13.6%. Start from the 7% risk-free rate and add each factor's loading times its premium: market 1.1 times 6% is 6.6 points, size 0.3 times 2% is 0.6, and value minus 0.2 times 3% is minus 0.6. So 7 plus 6.6 plus 0.6 minus 0.6 gives 13.6%. Here the size and value terms cancel, so the answer matches a plain CAPM with the same beta, which is a coincidence of these numbers.

How does a factor model build a required return?

Think of a taxi fare: a flag-down charge, then so much per kilometre, then so much per minute of waiting. Each meter runs at its own rate, and the fare is the sum. A factor model prices a stock the same way: the risk-free rate is the flag-down charge, and each factor adds how much the stock is exposed to it, the loading, times what that exposure pays, the premium. The {term('Fama-French three-factor model', 'An asset pricing model from Eugene Fama and Kenneth French that explains stock returns with three factors: the market, company size, and value against growth.')} uses three meters: the market, small against large companies, and cheap against expensive stocks.

Why does a negative loading subtract?

A negative value loading means the stock tends to do well when cheap stocks do badly: it behaves like a growth stock. The model says investors are paid a premium for holding value exposure, so a stock with the opposite exposure is priced to earn less, and its cost of equity falls. Push the value loading to minus 0.5 and the answer drops to 12.7%. Say the sign out loud; it is exactly what the interviewer is listening for.

Why does the answer equal a plain CAPM here, and should you trust it?

With the same market beta, CAPM gives 7 plus 1.1 times 6, which is 13.6%. The size and value terms happen to cancel in this question, so the two models agree only by coincidence. In practice the market loading from a three-factor regression is usually different from the CAPM beta, because the other factors absorb some of the movement. The bigger limit is the inputs: factor premiums are estimated from history, vary by market and period, and should be treated as assumptions to be stated, not facts to be quoted.

About 11.8%. Weight each source of money by its share and use debt after tax. Equity contributes 70% of 14%, 9.8 points. Debt costs 9% before tax but only 9% times 75%, 6.75%, after it, so it contributes 30% of 6.75%, about 2.0 points. The total is 11.825%. Using debt before tax would overstate WACC at 12.5%.

Why is WACC a weighted average rather than a simple one?

A household that pays for a flat with 70% of its own savings and a 30% home loan has a blended cost of money closer to what its savings could have earned than to the loan rate, because most of the money is savings. WACC weights each source by its share of the funding, measured at market value, because that is the mix the company's investments must pay for. A simple average of 14% and 9% would give debt far more say than 30% of the money deserves.

Why does the tax rate touch only the debt?

Interest is deducted before tax is worked out; dividends are not. So every Rs 100 of interest cuts the tax bill by Rs 25, and debt at 9% really costs the company 9% times 75%, which is 6.75%. The tax shieldThe reduction in tax a company gets because interest is deductible: interest times the tax rate. is why WACC uses the after-tax cost of debt, and why the equity cost stays as it is. Confirm the applicable tax rate for the company before using a headline figure.

What would make you distrust this number?

The weights and the inputs both move. Weights should be market values, not book values, and they should reflect the mix the company will hold over the forecast, not a single year-end snapshot. Adding debt does not lower WACC forever: as borrowing rises, lenders charge more and shareholders demand more for the extra risk. And the tax shield only exists if the company has profits to shield. A loss-making company's debt costs the full 9%, which would lift WACC to 12.5%.