I recently had a trading interview, and they asked this question. However, I had no idea how to answer it, and I was wondering if you could help me undersatnd it.

Say you have a set of returns applied to an asset value, such as: {5%, -5%, 10%, -10%, 15%, -15%}. What sequence of returns would maximize the final asset value?

I wasn't sure, because wouldn't you end up with the same amount at the end? aka gaining 5% and losing 5% would put you less than you began with, how would you go about this?

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    $\begingroup$ did you get the job? $\endgroup$ – vonjd Dec 15 '15 at 11:25
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    $\begingroup$ when you say combination of returns do you mean permutation of returns? $\endgroup$ – noob2 Dec 15 '15 at 13:35
  • $\begingroup$ It seems like a strange question: Are these expected values only? Are these deterministic or stochastic? Are these returns on different assets or the same asset? If they are deterministic, why would you ever choose anything else except 10%? If they are expected values, what is their variance? Given their mean and variance, what is the utility function? Are you risk neutral/risk averse? $\endgroup$ – ChinG Dec 15 '15 at 22:19
  • $\begingroup$ The first question was look for the answer provided by @amsh, there was probably a follow-up question asking if the price would be exactly the same as the initial price. $\endgroup$ – SRKX Dec 16 '15 at 1:07

It doesn't matter since multiplication is commutative (in $\mathbb{R}$); you will always end up losing the same.

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    $\begingroup$ For convenience, the meaning of "multiplication is commutative (in $\mathbb{R}$)" is that multiplication of real numbers can be re-ordered without changing the result. Notice that this is not the case for matrices for example. $\endgroup$ – user25064 Dec 15 '15 at 13:04

While amsh's answer definitely gives you what you need for interview purposes, for any visual learners like me, here's what running through all possible paths ends up looking like.

enter image description here

Another thing I might consider bringing up in an interview context would be the intuitive reasoning behind why it works out this way. In other words, just explaining that a 15% loss on \$130 has a larger gross impact than a 15% gain on \$70. Then you can talk about geometric returns, high water marks, etc. to demonstrate that you understand the financial/business relevance of what's going on mathematically.


For reference, the R code I threw together to make the graph above:

r = c(5, -5, 10, -10, 15, -15) * 0.01
x0 = 100

k = length(r)
n = factorial(k)
r = sort(r)
permList = combinat::permn(r)
results = matrix(NA, n, k+1)
results[,1] = x0
if (n != length(permList)) stop("factorial(length(r)) != length([permutations])")
for (i in 1:n)
    for (j in 2:(k+1))
        results[i,j] = results[i,j-1] + results[i,j-1]*permList[[i]][j-1]

if (!("Euclid" %in% names(windowsFonts())))
par(family="Euclid", mar=c(2.5,2.5,0.5,0.5), mgp=c(1.5,0.5,0), cex=0.75)
matplot(t(results), pch=19, type='o', col=rgb(1:n/n,0,n:1/n,n/(k*n)), xlab='', ylab='', main='')
grid(lty=1, col=rgb(0,0,0,0.2))
abline(h=x0, lty=2)
title(xlab=expression(t), ylab=expression(X[t]), font=3)

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