0

I saw this question recently and am wondering about the specifics:

Alice has $\$10,000$. She saves $\$1000$ a year. The interest rate is $10\%$, and she retires $10$ years later. Which of the following leads to the biggest increase in her retirement funds?

  1. Starting with $10\%$ more money
  2. Saving $10\%$ more per year
  3. Getting $10\%$ more interest (i.e. $11\%$ per year)
  4. Saving for $10\%$ more years (i.e. $11$ years)

It's easy enough to do the calculations and find that the answer is 4 > 3 > 1 > 2.

However, I notice that this ordering is not set in stone; it is affected by the input parameters. For example, if Alice saves $\$12,000$ a year, then the ordering changes to 4 > 2 > 3 > 1. Intuitively this makes sense - when you save more money a year than the starting capital, the effect of saving $10\%$ more ought to be magnified. Similarly, if Alice had started with $\$1,000,000$, then the impact of starting with $10\%$ more money ought to be greatly magnified.

What is the threshold for these input parameters at which the ordering changes? That is, how can one calculate at what point one parameter begins to dominate over the others? It's easy enough to "experimentally" verify if the ordering has changed, but how can one derive the point of change?

Allure
  • 616
  • 1
    These calculations that you refer to doing, they use certain formulas, no doubt. You should be able to work with the formulas, with variables instead of specific numbers, to write down inequalities, and perhaps to solve them. – Gerry Myerson May 11 '22 at 05:01
  • In addition: would not be surprised when the ordering also depends on after how many years Alice retires. All calculations are totally straightforward. – Kurt G. May 11 '22 at 05:30
  • Making any progress, Allure? – Gerry Myerson May 12 '22 at 13:26
  • @GerryMyerson I don't believe there's an easy way to write the formula with annual increments. All the calculations I've done use for loops. Can you think of a simple expression for the fomula? – Allure May 12 '22 at 13:46
  • If you start with $a$ dollars, save $b$ dollars a year, if the interest rate is $c$ percent, then at the end of $d$ years, you have a sum which is a function $f(a,b,c,d)$ of those four variables. Are you saying you don't have a formula for $f$? If you do have a formula for $f$, then, if you start with $10%$ more money, you end with $f(1.1a,b,c,d)$; if you save $10%$ more, you end with $f(a,1.1b,c,d)$; if interest is $10%$ higher, you end with $f(a,b,1.1c,d)$; if you wait $10%$ longer, you end with $f(a,b,c,1.1d)$. So; do you have a formula for $f(a,b,c,d)$? – Gerry Myerson May 14 '22 at 03:14
  • Do you understand how the calculations are done in the answer at https://math.stackexchange.com/questions/1332040/saving-should-start-early ? If you know where those numbers come from, and if you know the formula for the sum of a geometric series, you should be able to get $f(a,b,c,d)$. – Gerry Myerson May 14 '22 at 03:18
  • @GerryMyerson of course I understand how to do this. The question already says what the ordering is for the numbers given. For any individual values of a, b, c and d I can calculate the result. That still does not however answer the question, which is about what values of a, b, c, d switches the relative ordering of f(1.1a, b, c, d), f(a, 1.1b, c, d), f(a, b, 1.1c, d) and f(a, b, c, 1.1d). – Allure May 14 '22 at 04:14
  • But if you have the general formula for $f(a,b,c,d)$, then you can, for instance, write down the inequality $f(1.1a,b,c,d)>f(a,1.1b,c,d)$ and try to solve it, that is, try to find the range of values of $a,b,c,d$ for which it holds. Then you can do the same for other pairs of the four perturbed formulas, and (with some luck) put together a description of the subset of $4$-space where each of the $24$ possible orderings holds. Look, it's worth a try, isn't it? – Gerry Myerson May 14 '22 at 04:22
  • So, how is it coming along? – Gerry Myerson May 15 '22 at 12:42
  • @GerryMyerson I lost interest. – Allure May 15 '22 at 13:38
  • Too bad. There might be some interesting mathematics to be learned here. ("lost interest" – was that a pun?) – Gerry Myerson May 16 '22 at 01:57
  • 2
    I’m voting to close this question because OP admits to having lost interest, and no one else has shown much interest, so it won't be missed. – Gerry Myerson May 16 '22 at 01:58

0 Answers0