Equity Research puzzles, solved step by step

Puzzles: 100
Traced to a firm: 19
Topics: 11
Hard: 30

Topic

All topics Probability and brainteasers12 Expected value and decisions8 Market sizing and estimation12 Returns and compounding9 Valuation riddles12 Three statement riddles10 EPS and share count9 Cost of capital and rates8 Growth, mix and unit economics8 Mental maths6 Data and reasoning traps6

Level

Any Warm up Core Hard

Source

Any Reported at a firm Standard

Showing 1–2 of 2 · filtered from 100Clear filters

044Comparable companies give you an unlevered beta of 0.8. Your company has debt to equity of 0.5 and a 25% tax rate. What levered beta do you use for its cost of equity?Cost of capital and ratesCoreSell-side equity researchBuy-side equity research
Try it first
Before the formula: which direction and roughly how far?
Show the worked solution
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.
The relationship
$\beta_L = \beta_U\,\bigl[1 + (1 - t)\tfrac{D}{E}\bigr] = 0.8 \times (1 + 0.75 \times 0.5) = 1.1$
beta_U unlevered beta, the risk of the business with no debt
t the tax rate, 25%
D/E debt to equity, 0.5
What it says in wordsLevered beta is the business's beta scaled up by debt to equity, with the tax shield softening the debt's effect.
The business risk of 0.80 stays the same at every debt level, and debt adds financial risk on top: 0.30 at debt to equity of 0.5, for a levered beta of 1.10, and 0.60 at 1.0.
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.
Where candidates lose it
The fast wrong answers are 0.8, treating beta as fixed, and 1.2, relevering without the tax shield. The interviewer wants the formula said out loud with the numbers in it.
The quieter loss is mixing up debt to equity with debt to capital, which moves the answer from 1.10 to 1.00 and is hard to spot once it is buried in a model.
What the interviewer asks next
Go the other way: a peer has levered beta 1.3, D/E 0.8 and a 25% tax rate. What is its unlevered beta?
Why unlever peer betas before averaging them?
With a risk-free rate of 7% and an equity risk premium of 6%, what cost of equity does a beta of 1.1 give?
069Your nominal cost of capital is 11% and expected inflation is 5%. What is the real discount rate, and what goes wrong if you discount a forecast built in today's prices at the nominal 11%?Cost of capital and ratesCoreSell-side equity researchResearch KPO and GCC
Try it first
Discounting a forecast in today's prices at the nominal 11% makes the value...
Show the worked solution
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.
The relationship
$1 + r_{real} = \frac{1 + r_{nom}}{1 + \pi} = \frac{1.11}{1.05} = 1.0571$
r_nom the nominal cost of capital, 11%
\pi expected inflation, 5%
r_real the real discount rate, the growth in buying power demanded
What it says in wordsOne plus the real rate is one plus the nominal rate divided by one plus inflation.
Rs 100 of year-10 cash in today's prices is worth Rs 57.4 whether you pair real cash with the real rate or nominal cash with the nominal rate, but mixing them gives Rs 35.2 or Rs 93.4, wrong in both directions.
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.
Where candidates lose it
The question is not really about 5.7% against 6%. The trap is the pairing: candidates happily build a forecast in today's prices and then pick up the company's nominal cost of capital from a different sheet, which quietly takes a third or more off the value of distant cash flows.
The secondary slip is subtracting inflation. It is fine as a quick estimate at low rates; say that it is an approximation and give the exact figure.
What the interviewer asks next
Nominal rates are 30% and inflation is 25%. What is the real rate by subtraction and exactly? (5% against 4%)
Why do tax depreciation shields cause trouble in a real-terms model?
Your terminal growth is 6% nominal. What is that in real terms?

beta_U	unlevered beta, the risk of the business with no debt
t	the tax rate, 25%
D/E	debt to equity, 0.5

r_nom	the nominal cost of capital, 11%
\pi	expected inflation, 5%
r_real	the real discount rate, the growth in buying power demanded

Try it first

Why does debt raise the equity's beta?

Where do candidates go wrong with the inputs?

Where candidates lose it

What the interviewer asks next

Try it first

How are real and nominal rates linked?

Why does the mismatch understate value?

Which approach does an analyst use?

Where candidates lose it

What the interviewer asks next