Derivatives Foundation puzzles, solved step by step

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  1. 066A company is worth a uniformly random amount between Rs 0 and Rs 100 crore to its owner, who knows the exact value. It is worth 1.5 times that amount to you. You make one take-it-or-leave-it offer, and the owner accepts if your offer exceeds the value. What should you offer?Games and logicHardHedge fundsMarket making

    Try it first

    The company is worth 50% more to you than to the owner. What do you bid?

    Show the worked solution

    Offer nothing. Every positive offer loses money on average. If you offer b and the owner accepts, you learn the value is below b, so its average is b/2, worth 1.5 x b/2 = 0.75b to you. You pay b, so each accepted deal loses 0.25b, and the deal is accepted a fraction b/100 of the time. Expected profit is minus 0.0025 b squared, negative for every b above zero. The 1.5 multiplier is not enough to overcome what acceptance tells you.

    Why is the owner saying yes bad news for you?

    A friend sells you their old scooter for any price you name, but only if your price beats what they privately think it is worth. If they take Rs 20,000 instantly, you have just learned the scooter is worth less than that to someone who knows it well. Acceptance is information: it tells you the true value lies below your bid, so the only companies you ever buy are the ones worth less than you paid, and the ones worth more walk away. Averaging over all possible values, as if you bought every company, is the mistake; you must average only over the values at which the owner says yes.

    Acceptance is bad news: whatever you bid, you pay more than it is worth to you on average02550751000-10-20your offer, Rs croreexpected profit, Rs crorebest offer: 0, profit 0offer 50: -6.25offer 100: -25profit = - 0.25 x offer x (offer / 100)Say you offer 60Accepted only if value < 60so value is uniform on 0 to 60average value given a yes: 30worth 1.5 x 30 = 45 to youyou paid 60loss when accepted: 15accepted 60% of the timeexpected profit: - 9Same sign at every offer
    Expected profit from an offer b is minus a quarter of b times the acceptance chance b/100, a parabola that is zero at b = 0 and falls to minus 25 crore at b = 100, so no positive offer earns anything and the best move is not to bid; an offer of 60, for instance, buys a company worth 45 to you on average and loses 9 in expectation.

    How do you set up the expected profit cleanly?

    Condition on acceptance, then multiply by its probability. Given a bid b that is accepted, the value V is uniform on 0 to b, so E[V | accepted] = b/2, the company is worth 1.5 x b/2 = 0.75b to you, and the profit on an accepted deal is 0.75b - b = - 0.25b. Acceptance happens with probability b/100, so expected profit is - 0.25b x b/100 = - b squared over 400. At b = 50 that is minus 6.25 crore; at b = 100 it is minus 25 crore. The derivative is negative everywhere above zero, so the maximum is at b = 0.

    The relationship
    E[π(b)]=b100(1.5⋅b2−b)=−b2400<0for all b>0E[\pi(b)] = \frac{b}{100}\left(1.5\cdot\frac{b}{2} - b\right) = -\frac{b^2}{400} < 0 \quad \text{for all } b > 0
    byour offer in Rs crore
    b/100the chance the owner accepts, since the value is uniform on 0 to 100
    1.5 x b/2what the company is worth to you on average once you know the value is below b
    What it says in wordsThe expected profit from any offer is a negative multiple of the offer squared, so the best offer is zero.

    At what multiplier does a bid start to make sense?

    Replace 1.5 with a general m. Given acceptance, the company is worth m x b/2 to you against the b you pay, so the deal breaks even when m/2 = 1, that is m = 2. Unless the company is worth more than twice as much to you as to the owner, acceptance always costs you, and at exactly double every bid is a wash. That is the winner's curse in its purest form: the party with less information loses whenever the informed party decides whether to trade. The limitation is the uniform prior and the single offer; with a floor on the value, or a negotiation that reveals information, positive bids can be profitable.

    Where candidates lose it

    Candidates bid around 50 or 75 by averaging over the whole range of values, forgetting that they only buy when the owner accepts. The interviewer wants you to say that acceptance is information before you touch any arithmetic.

    The second loss is getting zero and not being able to say what would change it. Give the general condition: the multiplier must exceed two for any positive bid to pay.

    What the interviewer asks next

    • At what multiplier does a positive bid first break even, and what is the best bid at a multiplier of 3?
    • The value is uniform on Rs 50 to Rs 100 crore instead. Does a positive bid make sense now?
    • How is this the same problem as a market maker being hit only when they are wrong?
    • You get two offers rather than one, and the owner rejects the first. Does that change anything?