Return Order Does Not Change the Final Value When Returns Compound
Summary
The document addresses an interview puzzle about applying a fixed set of positive and negative returns to an asset and asks which ordering maximizes the ending value. Each return multiplies the current asset value by one plus that return. Since multiplication is commutative, rearranging the same return factors cannot change their product or the final value.
The answer also offers an intuition for why a gain and loss of equal percentage size do not cancel in dollar terms: the loss applies to the higher value after a gain, while the gain applies to the lower value after a loss. This connects the puzzle to geometric returns and high-water marks. The document mentions a visualization based on enumerating possible paths, but gives no independent empirical evidence; the result follows from the arithmetic and assumes the return set is fixed, with no order-dependent costs, cash flows, or trading constraints.
Key ideas
- A fixed set of simple returns produces the same compounded ending value in any order.
- Each return contributes a multiplicative factor of one plus the return rate.
- Equal-sized percentage gains and losses do not offset in dollar terms because they apply to different asset values.
- The order-invariance result assumes no order-dependent costs, cash flows, or constraints.
Tags
Full text
# Trading Interview Question (Bullish, Bearish)?
# Trading Interview Question (Bullish, Bearish)?
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?
## Answer by amsh (score 5)
https://quant.stackexchange.com/a/22337
It doesn't matter since multiplication is commutative (in $\mathbb{R}$); you will always end up losing the same.
## Answer by Jacob Amos (score 3)
https://quant.stackexchange.com/a/24342
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.
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())))
windowsFonts(Euclid=windowsFont("Euclid"))
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)
```Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.