1

If this pattern continues what will the balance be in January 2017? I noticed every month there's a higher percentage increase but don't know how to calculate. Thank you!

january- 25.29$

Feb- 39.69$

March- 58.36$

April- 86.91$

Jason
  • 11
  • Why do you imagine that there is a pattern here? It seems like it's going up by something around $50%$ a month (good investment, by the way). But otherwise... – lulu Apr 13 '16 at 00:54
  • It is not true that every month the percentage increase goes up. From january to february the increase is $56.94%$. From february to march the increase is $47.04%$ – callculus42 Apr 13 '16 at 01:03
  • 1
    How do you figure there's a higher percentage increase every month? From Jan->Feb is a 56.94% increase, from Feb->March is a 47.04% increase, from March->April is a 48.92% increase. If we assume the account experiences a month-to-month increase of the mean of these values, it seems you'd end up with around $$3544$, but the error margin on this computation would be quite large. A back-of-the-envelope computation suggests the account balance would lie somewhere in the range $[2316.24, 5353.05]$. – Nicholas Stull Apr 13 '16 at 01:04
  • In a situation such as this, a conservative estimate of $$2300$ seems most appropriate. – Nicholas Stull Apr 13 '16 at 01:09
  • @Nicholas Stull: what do you mean by "a back-of-the-envelope computation " ? – KonKan Apr 13 '16 at 01:10
  • Do you have any additional information ? – callculus42 Apr 13 '16 at 01:19
  • 1
    @KonKan, it's a rough computation where I made some semi-reasonable assumptions about the percentage growths that would be sustained. I took the percent increases, computed their mean $\mu$ and standard deviation $\sigma$. The lower bound was computed by assuming that the mean monthly rate of return would be $\mu-\sigma$, and the upper bound was computed by assuming that the mean monthly rate of return would be $\mu+\sigma$. – Nicholas Stull Apr 13 '16 at 01:24
  • By the way, I say "semi-reasonable assumptions", because a standard deviation and mean aren't really that useful if you only have 3 data points to judge from. – Nicholas Stull Apr 13 '16 at 01:27
  • @Nicholas Stull: nice treat! you should post that as an answer. – KonKan Apr 13 '16 at 01:30

1 Answers1

0

From January to February is a 56.94% increase, from February to March is a 47.04% increase, and from March to April is a 48.92% increase, so you are not seeing an increasing rate of return from month to month. Indeed, your greatest rate of return is the first month, with the lowest rate of return being the third month.

This leads to a mean rate of return of $\mu=50.96667\%$, with a standard deviation of $\sigma=5.25777\%$. If we assume that you begin with a principal of $\$25.29$ in January of 2016, then a back-of-the-envelope calculation suggests that you would attain a final balance in January of 2017 of between $\$2316.24$ and $\$5353.05$.

The biggest problem with a computation like this is sustainable growth vs. anomalous growth. We are forced to make assumptions and perform a "back-of-the-envelope calculation", where we make some semi-reasonable assumptions (noting that mean and standard deviation aren't all that useful if you only have 3 data points to work with). That said:

The "middle-of-the-road" figure assumes that the rate of return is sustainable at an average of $\mu$, yielding an estimated final balance of $\$3544.23$. The conservative estimate of $\$2316.24$ assumes that your sustained rate of return averages out to $\mu-\sigma\approx 45.7089\%$ (i.e., one standard deviation below the mean), while the high-end estimate of $\$5353.05$ assumes that your sustained rate of return averages out to $\mu+\sigma \approx 56.2244\%$.

In this kind of situation, it is definitely better to act on the conservative estimate, and perhaps even regard this rate of return as a bit too optimistic.

Let's assume, for a moment, that you just start with the April figure and we go with the same sort of computation. A sustained rate of return of $\mu$ yields a final balance of $\$3539.97$, while a sustained rate of return of $\mu-\sigma$ yields a final balance of $\$2573.04$, and a sustained rate of return of $\mu+\sigma$ yields a final balance of $\$4822.91$.

The disparity in values comes from the fact that changes of the rate of return have a larger effect on larger balances, and the first computation is basically making assumptions of catch-up returns, so the safest value is probably to go with the conservative estimate of $\$2573.04$, or perhaps even the slightly more conservative value of $\$2500$.

To be honest, from a fiscal perspective, I'd be extremely hesitant to act on either of these conservative estimates, because they are both making assumptions that are not necessarily realistic for sustainable growth. We could conceivably allow for similar calculations with sustained rates of return of between $\mu-3\sigma$ and $\mu+3\sigma$ (step size of $\sigma$), which would yield wildly spread results:

$\$1311.22$ for $\mu-3\sigma$, $\$1848.42$ for $\mu-2\sigma$, $\$2573.04$ for $\mu-\sigma$, $\$3539.97$ for $\mu$, $\$4822.91$ for $\mu+\sigma$, $\$6503.64$ for $\mu+2\sigma$ and $\$8686.18$ for $\mu+3\sigma$

In short, such figures must be taken for what they are: rough estimates based on assumptions on return rate that may or not be sustainable in the long run, especially since the rate of return won't be constant month-to-month.

Nicholas Stull
  • 3,014