Equity Research puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 19
- Topics
- 11
- Hard
- 30
083A stock has a market loading of 1.1, a size loading of 0.3 and a value loading of minus 0.2. The risk-free rate is 7%, and the factor premiums are 6% for the market, 2% for size and 3% for value. What cost of equity does a Fama-French three-factor model give?State StreetCambridge · 2019
Try it first
What does the negative value loading do to the cost of equity?
Show the worked solution
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.
Starting from a 7.0% risk-free rate, the market adds 6.6 points, size adds 0.6 and a negative value loading takes 0.6 away, so the three-factor cost of equity is 13.6%. The relationshipr_f the risk-free rate, 7% beta_M, beta_S, beta_V the stock's loadings on the market, size and value factors MRP, SMB, HML the premiums for the market, small minus big, and high minus low book to market What it says in wordsRequired return is the risk-free rate plus, for each factor, how exposed the stock is times what that exposure earns.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.
Where candidates lose it
The usual slip is treating every factor as additive risk and adding 0.6 for value, which gives 14.8%. The sign of the loading matters as much as its size.
The second loss is stopping at 13.6% without noticing it equals CAPM. Pointing out that size and value cancel here, and would not in general, shows you understand the model rather than the arithmetic.
What the interviewer asks next
- What value loading would make the three-factor answer 1 point higher than CAPM?
- Why might a small, cheap stock have a higher cost of equity than CAPM suggests?
- How would you estimate the loadings for an Indian stock?
Asked at State Street, Investment Banking, Cambridge, 2019 (Wall Street Oasis):
some basic market knowledge, such as factor model (Fama French), portfolio optimization, risk analysis
087Monty Hall: there are three doors and a prize behind one. You pick a door. The host, who knows where the prize is, opens a different door with no prize and offers you a switch. Why is the intuitive answer of one half wrong, and what is the chance that switching wins?Squarepoint CapitalLondon · 2026
Try it first
Two doors are left. What is the chance that switching wins?
Show the worked solution
Switching wins two times in three. Your first pick is right with probability one third, and nothing the host does changes that. In the other two thirds of cases the prize is behind one of the two doors you did not pick, and the host, who must open an empty door, is forced to leave the prize door shut. One half is wrong because it treats the host's reveal as random when it is not.
Why does one half feel right?
Two closed doors, one prize: the picture in front of you looks like a coin toss, and intuition prices the picture rather than the process that produced it. One half would be right only if the two remaining doors were equally likely before the reveal and the reveal told you nothing about either of them. Neither is true. Your door was chosen blind; the other door survived a filter run by someone who knew the answer.
Starting from a pick of door 1, the prize is behind door 1, 2 or 3 with one third chance each, and in the two branches where the first pick is wrong the host is forced to open the only empty door, so switching wins in two of three equally likely branches. What exactly does the host's knowledge change?
Think of a shopkeeper who knows which of three mangoes is ripe. You pick one; she then removes an unripe one from the two you did not pick, always careful never to remove the ripe one. She has concentrated the two thirds chance that the ripe mango was among the other two onto the single one she left. Your own pick is untouched at one third, because she would have been able to remove an unripe mango whatever you picked. The reveal carries information about the other door, not about yours.
When would one half actually be the right answer?
This is the part the interviewer is asking for when they ask why intuition fails. If the host did not know and opened one of the other doors at random, and it just happened to be empty, then switching would win only one half of the time. A random reveal that could have shown the prize rules out the branches where it would have, and what is left is symmetric. So the answer depends on the host's rule, not on the doors. Scale it up to make the point land: with 100 doors, you pick one, and a knowing host opens 98 empty ones; nobody thinks the last closed door is a coin toss.
The analyst's version is reading a signal from someone who knows more than you do. A management team that chooses which numbers to disclose is running a filter, and what survives the filter is not a random sample.
Where candidates lose it
Saying one half is the obvious loss. The subtler one is saying two thirds and then being unable to explain why, which is exactly what this interviewer asked for: they want the reason intuition fails, not the memorised answer.
Have the three-branch tree ready and the ignorant-host variant as the contrast. Showing that the answer changes when the host's rule changes proves you understand the mechanism.
What the interviewer asks next
- The host opens a door at random and it happens to be empty. Now what is the chance switching wins?
- With four doors and a host who opens one empty door, what does switching win?
- Where in reading company disclosures do you meet the same kind of filtered evidence?
Asked at Squarepoint Capital, Quant Research Intern Interview, London, 2026 (Wall Street Oasis):
notably I was asked why the 'intuitive answer' was not true rather than just what the correct answer was, related to the Monty Hall problem