When you enter “calculator” into Google, they provide you with one. Might just be in Chrome, but it’s handy.
I can’t imagine the scenario where doing one’s taxes without any sort of calculator would be advantageous.
watchwolf49’s comment on books brings back the memories. Things like CRC Math tables for compound interest based on different interest rates, etc. Now throw in a sliderule and we’ve got a full nostalgia trip.
Why would you do this? Why would you even think of doing this? :eek: 
:dubious:
If you’re wanting to figure in your head, the rule of 72 is frequently useful. This rule relates the interest rate to the number of years it takes for the original amount to double. Those two multiplied together is always (for reasonable values of interest rates) very near to 72.
At 4% interest, your money doubles in 18 years (4 x 18 = 72).
At 6%, it takes 12 years.
At 8%, it takes 9 years.
At 9%, it takes 8 years.
At 12%, it takes 6 years.
A few years ago, a co-worker was assigned the task by his grad school prof to prove this, and he got stuck, so he asked me. I’ve been out of college 25 years at the time but I was able to do it. I was kinda proud of that.
If you really want to do this by hand, without the use of log tables, you can at least use a Taylor series approximation.
ln(1+x) is well approximated by x - x^2/2 for small x. Your 5.7% annual rate is 0.475% monthly, so you’re trying to compute:
ln(1 + 0.00475)
Which is well approximated by:
0.00475 - 0.00475^2/2 = 0.004739
Multiply by 120 for:
0.56868
Now we need an approximation for the exponential function. Again, we can use a Taylor series, though unfortunately we need more terms. Working up to x^3, and with some rearrangement of terms, we get:
e^x =~ x(x(x/3 + 1)/2 + 1) + 1
That isn’t too many multiplies. You end up with 1.761, which isn’t too far from the real value of 1.7659. You could add another term to the e^x approximation with:
e^x =~ x(x(x(x/4 + 1)/3 + 1)/2 + 1) + 1
If anyone’s interested in the derivation:
As with my comment above, we can use a Taylor series approximation, but this time with just one term:
ln(1 + x) =~ x
We’re trying to find n (given a percentage p) such that:
(1 + p/100)^n = 2
Put in terms of n:
ln((1 + p/100)^n) = ln(2)
n*ln(1 + p/100) = ln(2)
n = ln(2) / ln(1 + p/100)
Substitute the approximation:
n = ln(2) / (p/100)
n = ln(2)*100 / p
n = 69.3 / p
So it’s really the “rule of 69.3”. However, that’s kind of an annoying number. 72 has a lot of divisors, and so it’s easy to work out in your head for 2%, 3%, 4%, etc. It also turns out that nudging the value up by a little bit cancels out some of the error for typical interest rates (the approximation works best for small numbers, so it’ll be closer to 69.3 on the low end but in the 70s for higher rates). Occasionally I’ll hear the “rule of 70”, which is indeed better for low rates but not higher ones, and doesn’t have so many easy divisors.
…when compared to alternatives for the same future year. Start one year later, just end one year later and it’s the same amount.
I’m under the impression that’s because 69.3 would require instantaneous compounding. The longer the compounding period, the higher number you should use.
But it’s got 5, 7, and 10. 72 doesn’t have those, so if it’s a 10-year or 5-year plan, 70 fits the bill nicely.
It’s more about low interest rates–ones where the ln(1+x)=~x approximation holds. Though this could be seen as the same as continuous compounding if you look at the limit as the number of periods goes to infinity (and the per-period interest rate goes to 0).
You can see what’s going on if you graph ln(2)*x/ln(1 + x/100)) (try this online calculator). It starts at 69.3 for 0% interest and gets to 72 at roughly 8% (quick check: 1.08^9 = 1.999).
At any rate, it’s just a quick mental approximation, so the percent difference between 69.3 or 70 or 72 isn’t too important. They’re all in the right ballpark.
If you want to do compounding, from scratch, the simplest method is to use a spreadsheet. Column for date, column for amount, column for interest earned.
Many people have access to Excel or similar. Use the fixed reference (like $A$1) method to point to a fixed cell for interest rate, and you can experiment with changing the interest rate and see month over month value changes.
Of course if you are an Excel expert there are several financial functions that do it in one step.
Another quick rule of thumb is the rule of 70. Divide 70 by annual interest rate to get roughly how many years to double an amount. (I.e. 5% interest, will double in 70/5=14 years. 10%, 7 years. 3%, 23 years…)
First some fatherly advice. You should be putting the maximum allowable savings into whatever kind of retirement account available to you. If your job “pays well” then you can afford to shave off some expenses somewhere. There are people who make less than you and do OK–just live like they do and you’ll save a bundle.
Suppose you can get a return of 5.7%, as in your example. This year you start putting $1000 a year into an IRA for 10 years, then stop. In year 11, the year after you stop, your twin in a parallel universe, after partying for 10 years, starts to put $1000 a year in his. It’s not until year 35 when you are 63 that he has as much money as you saved up. You contributed $10,000, but he contributed $25,000.
You can’t afford not to invest young.
Also, you don’t pay taxes on an IRA until you take the money out. Taxes are not really what you should be worrying about here.
Nobody does this. Nobody. It’s error-prone and time-consuming.
Unless you have a Roth, of course. (And withdraw the money after 59 1/2).
(Although you might not count it as paying taxes on an IRA so much as using after-tax income to fund it.)
nm.