Equity Research puzzles, solved step by step

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.

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.

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.