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

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Any Reported at a firm Standard

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038A stock trades at Rs 500 and reports results tomorrow. You think there is a 60% chance it moves to Rs 560 and a 40% chance it moves to Rs 440. What is the expected price, and what does it say about today's price?Expected value and decisionsWarm upSell-side equity researchHedge fund long/short
Try it first
What is the expected price after results?
Show the worked solution
The expected price is Rs 512, 2.4% above today. Weight each outcome by its chance: 0.6 x 560 plus 0.4 x 440 gives 336 plus 176. Today's Rs 500 sits exactly halfway between Rs 440 and Rs 560, so the market is pricing roughly a 50% chance of good results. Your 60% view is what separates Rs 512 from Rs 500, and that probability is what you would have to defend.
Why is the expected price not the likely price?
A cricket fan who thinks her team wins 60% of the time does not expect the team to win exactly 0.6 of a match; she expects the win most often and a loss sometimes. Expected value is the probability-weighted average of every outcome, and it can be a price that never actually trades. Here the stock will be at Rs 560 or Rs 440 tomorrow, never at Rs 512, yet Rs 512 is the right number to compare with today's price.
Weighting Rs 560 by 0.6 and Rs 440 by 0.4 gives an expected price of Rs 512, while today's Rs 500 sits halfway between the outcomes and implies a 50% chance of good results.
What does today's price tell you about the market's view?
Run the calculation backwards. If Rs 500 is the market's expected price, the chance q of the good outcome solves 560q + 440(1 - q) = 500, so q = 60 / 120 = 50%. The interesting number is not Rs 512 but the gap between your 60% and the market's 50%: that gap is the whole of your view. An analyst would next ask what evidence justifies seeing more upside than the market does.
The relationship
$E[P] = 0.6 \times 560 + 0.4 \times 440 = 512 \qquad q = \frac{500 - 440}{560 - 440} = 50\%$
E[P] the expected price after results
q the chance of good results that makes today's price fair
What it says in wordsWeight the outcomes by your probabilities to get your expected price, and solve for the probability that makes today's price the expected one.
Say the limitations. Two outcomes are a simplification of a whole spread of possible moves, and a 2.4% expected gain on one event is small next to the Rs 60 swing either way. The expected value is a way to state a view precisely, not a reason on its own to act on it.
Where candidates lose it
The common slip is answering Rs 560 because it is the more likely outcome. The expected value averages both branches.
The second loss is stopping at Rs 512. The interviewer wants the implied probability too: today's price already carries a view, and saying 50% shows you know your edge is the difference between two probabilities, not a price.
What the interviewer asks next
What probability of good results would make Rs 500 fair if the upside were Rs 580?
Options on the stock imply a move of plus or minus 12%. Is that consistent with your tree?
How would you size a position when the expected gain is 2.4% but the swing is 12%?
088Two stock pitches. Pitch A returns 5x your money with a 10% probability and zero otherwise. Pitch B returns 1.5x with a 60% probability and 0.5x otherwise. Which has the higher expected multiple?Expected value and decisionsWarm upBuy-side equity researchLong-only asset management
Try it first
Which pitch has the higher expected multiple of your money?
Show the worked solution
Pitch B, at 1.1x against 0.5x for Pitch A. Weight each outcome by its probability. A gives 10% of 5x plus 90% of nothing, which is 0.5x: on average it loses half the money. B gives 60% of 1.5x plus 40% of 0.5x, which is 0.9 plus 0.2, 1.1x. A would need better than a one in five chance of the 5x just to break even.
Why does the big number not win?
A lottery ticket that costs Rs 100 and pays Rs 500 one time in ten is a bad ticket, however good Rs 500 sounds. An expected value multiplies each outcome by its probability, so a large payoff is shrunk by a small chance before it counts. For A, 5x shrinks to 0.5x. For B, a modest win that happens more often than not, plus a partial loss that still returns half the money, adds up to 1.1x.
Pitch A's 5x outcome has only a 10% chance and the other 90% returns nothing, so it expects 0.5x, while Pitch B's 60% chance of 1.5x and 40% chance of 0.5x expect 1.1x, above the line where you get your money back.
The relationship
$E[A] = 0.1 \times 5 + 0.9 \times 0 = 0.5\times \qquad E[B] = 0.6 \times 1.5 + 0.4 \times 0.5 = 1.1\times$
E[A], E[B] the expected multiple of money for each pitch
0.1, 0.9, 0.6, 0.4 the probabilities of each outcome
5, 0, 1.5, 0.5 the multiples of money in each outcome
What it says in wordsWeight every outcome by its probability and add: that is what you get on average per rupee put in.
What would make Pitch A worth taking?
Work backwards from breakeven. A returns your money on average only if the chance of the 5x is one in five, 20%. So the real question about A is not how big the upside is but whether you can defend a probability above 20%, double what the pitch claims. That is how a buy-side analyst would push back on a story built around one dramatic outcome: ask for the odds, then check them.
Is the higher expected value the whole answer?
No, and saying so earns the extra point. B still loses half the money 40% of the time, so the expected value tells you which pitch to prefer, not how much to put in. Position size depends on how bad the bad outcome is and how often it comes. The limitation of the question is that it hands you the probabilities. In practice those are the hardest number to estimate, and a pitch with a vivid upside tends to come with an optimistic probability attached.
Where candidates lose it
Candidates are pulled to A by the 5x and justify it with language about asymmetric upside. The interviewer is checking whether you multiply by the probability before you get excited.
The second loss is stopping at the expected value. Mention that B still halves your money 40% of the time, so the choice of pitch and the size of the position are separate questions.
What the interviewer asks next
What probability of the 5x makes A as attractive as B?
If you could hold both, each with half your money, what is the expected multiple and the chance of losing money?
Why might a fund still take a small position in something like Pitch A?

E[P]	the expected price after results
q	the chance of good results that makes today's price fair

E[A], E[B]	the expected multiple of money for each pitch
0.1, 0.9, 0.6, 0.4	the probabilities of each outcome
5, 0, 1.5, 0.5	the multiples of money in each outcome

Try it first

Why is the expected price not the likely price?

What does today's price tell you about the market's view?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the big number not win?

What would make Pitch A worth taking?

Is the higher expected value the whole answer?

Where candidates lose it

What the interviewer asks next