Where did you get 15% as the savings mark to aim for?
It is just a simple rule-of-thumb that I came up with. It just relies on the concept that you want to retire with roughly the same available yearly money as you had before retirement. That you don't want to live like a king at 64 and a pauper at 65 (or visa-versa). I've been spouting it off here for well over a decade if you want to search "15%" and my name to see more discussion about it. But below is how to derive it.
1) If your wages are W, and you save R as a percent of your wages each year, then you get to live on W*(1-R) to pay for your daily expenses. For example, W=$100,000 and R=15%, then you can live on $100,000 * (1-0.15) = $85,000 a year. Then you pay for taxes, expenses, fun, etc. out of that $85,000. The other R*W = $15,000 gets invested for the future each year.
2) If you save R*W every year, then after Y years, you will have (R*W/i)*[(1+i)^(Y+1)-1] in your retirement account when you retire. Here 'i' is the geometric average of your yearly investment returns. I'll save you the gory math that gets that formula.
3) A rule-of-thumb is that you can withdraw about x=5% of your retirement account each year. Yes, there are debates if you should be extra conservative and go with 4%, or more realistically you can usually be fine at 6%.
https://www.morningstar.com/news/marketwatch/20231117283/goodbye-4-rule-hello-6-rule So, you can withdraw (R*W*x/i)*[(1+i)^(Y+1)-1] each year to spend in retirement on taxes, expenses, fun, etc.
So, I put all those ideas together. What if the money you had when working was the same as when you retire? That way your standard of living is the same before and after. In that case, the amount you get to live on while working from equation (1) is equal to the amount you can withdraw after retirement from equation (3). Then solve for R, the percentage that you need to save each year for retirement. Saving you the gory math details:
R = (i / x) / [(1 + i)^(Y + 1)+(i / x) - 1]
Notice that R does not depend at all on how much you earn (W). Thus it is income independent. It works for the poor, the average bloke, and the wealthy. For the same reason, R is not dependent on inflation if you assume your wage goes up with the rate of inflation (which, yes is a big assumption, but I don't want to make this post even harder to follow).
Plug in any numbers you choose. Here is an example. Suppose someone earned an average 7.5% return (a fairly normal return to expect on investments). Suppose that person works and saves properly for Y = 30 years (not everyone can invest every year or work consistently every possible year, so 30 years of investing seems reasonable). Suppose that person withdraws x = 5% per year. Then:
R = (0.075 / 0.05) / [(1 + 0.075)^(30 + 1)+(0.075 / 0.05) - 1] = 15.1%. Google math link:
https://www.google.com/search?q=(0....AQD4AQL4AQHiAwQYACBBiAYB&sclient=gws-wiz-serp
Fiddle with the numbers, i, x, and Y all you want. The end result will change a bit, but not much. Unless you only save for a very short time (like
@repoman0 explained) or expect unusual investment returns, the end result is usually in the 10% to 20% range, with ~15% being the result in many cases.