Portfolio Management puzzles, solved step by step

Puzzles
100
Traced to a firm
31
Topics
13
Hard
30
Topic
All topicsStatistics and forecasting9Portfolio risk maths10Logic brainteasers7Behavioural and decision traps7Probability and expected value8Bond maths10Valuation riddles8Performance measurement8Private and real asset maths8Funds, ETFs and implementation7Compounding and fee drag7Market sizing and estimation6Currency and global returns5
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 043You have n cars, each fuelled to drive exactly 1,000 miles, and fuel can be moved from one car to another along the way. Tanks cannot be overfilled. How far can one car get, and how does that distance grow as n becomes very large?Logic brainteasersHardMillennium ManagementLondon · 2024

    Try it first

    With four cars, how far can one car get?

    Show the worked solution

    1,000 x (1 + 1/2 + 1/3 + ... + 1/n) miles, which grows without limit but only like the logarithm of n. Drive all n cars 1,000/n miles; together they have burned one full tank, so one car can refill the rest and be abandoned. Repeat with n minus 1 cars for 1,000/(n minus 1) miles, and so on. Four cars reach 2,083 miles, 100 cars about 5,187, and the distance tracks 1,000 x (ln n + 0.577).

    When should a car drop out of the convoy?

    Think of friends sharing water on a long walk, where every bottle is full at the start and nobody can carry more than one. The moment the group has drunk exactly one bottle's worth, one friend can pour the rest of theirs into everyone else's bottles and head home. A car should drop out the instant the convoy has burned exactly one tank in total, because that is the first moment its remaining fuel exactly fills the others. With n cars that happens after 1,000/n miles, since n cars burn fuel n times as fast as one.

    Each extra car adds a shorter leg: 1000/n, then 1000/(n-1), and so on1000/4 = 2501000/3 = 3331000/2 = 5001000/1 = 1,000Car 4tops up, left emptyCar 3tops up, left emptyCar 2tops up, left emptyCar 12,083 miles1,0003,0005,0001255075100Number of carsMiles reachable10 cars: 2,929100 cars: 5,187
    Four cars travel 250, 333, 500 and 1,000 miles in successive legs as one car at a time tops up the rest and drops out, reaching 2,083 miles. The distance with n cars keeps growing but ever more slowly, from 2,929 miles with 10 cars to 5,187 with 100.

    Why does the distance grow only like a logarithm?

    The total is 1,000 times the harmonic sum 1 + 1/2 + ... + 1/n. Each extra car adds the shortest leg of the journey, 1,000/n miles, so the gains shrink as the convoy grows, and the harmonic sum rises like ln n plus about 0.577. It never stops growing, so any distance is reachable in principle, but slowly: 10 cars reach about 2,929 miles, and getting past 5,000 miles takes 83 cars. Doubling the fleet adds only about 1,000 x ln 2, roughly 693 miles.

    The relationship
    D(n)=1000∑k=1n1k≈1000 (ln⁡n+0.577)D(n) = 1000\sum_{k=1}^{n}\frac{1}{k} \approx 1000\,(\ln n + 0.577)
    nthe number of cars at the start
    1000/kthe leg driven while k cars remain
    0.577the Euler-Mascheroni constant
    What it says in wordsEach leg is one tank shared among the cars still running, and the legs add up to a harmonic series.

    Why a hedge fund asks it: the structure is the same as scaling a strategy. Each extra unit of capital or effort buys a smaller increment, and a candidate who sees the diminishing returns and names the rate of decay is showing the instinct the desk wants. Also be ready to argue optimality in one sentence: any plan that drops a car earlier wastes fuel it cannot hand over, and dropping later wastes the fuel spent carrying a car that is no longer needed.

    Where candidates lose it

    The quick wrong answers are n times 1,000, as if all the fuel could be pooled into one car, or a flat 1,000 because tanks cannot be overfilled. Both skip the key idea that the convoy itself consumes fuel while carrying the reserve.

    Work n equals 2 out loud first: drive 500, pour the rest of car 2 into car 1, and drive 1,000 more, for 1,500. Then generalise. The interviewer wants the harmonic series and the words grows like log n.

    What the interviewer asks next

    • Roughly how many cars do you need to travel 10,000 miles?
    • What changes if cars can come back to a depot and cache fuel along the road?
    • Where do you see diminishing returns of this shape in portfolio construction?

    Asked at Millennium Management, Investments, London, 2024 (Wall Street Oasis): Suppose you have n cars, each fueled so that they can drive for 1000 miles.