Is it possible to work out approximately what $10,000 compounding at say 11% per annum would be worth after 30 years using the rule of 72 or any other rules without resorting to a calculator?
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The above captioned table contains the factors for compound interest for differing periods 1 to 30 at a compound rate of 11% for $1.
Without the use of a financial calculator, I can read off the answer to your question from the table as being the factor of 22.892 x $10 000 = $228 920.
So to answer your question, yes one could estimate the answers to Future Value compound interest type problems without a financial calculator but it would presuppose that you could commit a table of factors to memory to achieve that feat. Obviously as the compound interest model is premised on the geometric progression, it requires one to use exponential functions in arriving at the correct answer to Future Values of a given lump sum investment algebraically.
Easiest method to solve is obviously using a financial calculator which nowadays one can download onto their cell phones as an app.
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- Wikipediaās article on Rule 72 has a very interesting table (Rule Selection section) on which the base number is the most accurate to use for the estimate.
- Itās not exactly the same, but relevant, this mathematical article published by a guy who documents how his father taught him the āPersian folk method of calculating interestā to calculate approximate monthly payments. For those who donāt want to follow the link, itās essentially a Taylor polynomial.
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Thanks Fred for the reference. I guess my question is can the Rule of 72 or any other rules derive the 22.892 or a reasonable approximation without the financial table? We know that at 11% money will double at approx every 6.5 years, at 6.5, 13, 19.5, 26, 32.5 By the time it has reached year 26 it will have grown 16 times, at year 32.5 it will be 32 times. The table shows a factor of 23 times after 30 years which is between 16 and 32 times. Can these rules be manipulated to work out this rate of growth?
Thanks so much for sharing SH. Iām going to experiment with this but it looks good!
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Of course. 72 / 11 is roughly 6.5.
30 / 6.5 = 4.6.
In other words, the sum doubles 4.6 times. We now need to work out 2 ^ 4.6.
2 ^ 4 = 16 and 2 ^ 0.6 = 1.5. See my other posts how to work out 2 ^ 0.6 (or ask me).
16 * 1.5 = 24.
The sum gets multiplied by a factor of 24 times.
$10,000 becomes $240,000.
On a calculator, the factor is closer to 23 (22.9).
The reason is that ā72ā is used for small percentages.
For 11%, 72 needs to be smaller, around 66 even.
See sovereign_headspaceā first link for a formula how to get to this correction from 72 to 66.
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Thanks again Kinma, if one can use the Rule of 72 or 70 or 69 or any of the other variants to calculate Future value that makes it a powerful tool in mental calculation. I am fascinated by the 2 ^ 4.6 calculation and Iāll explore it further, does it involve logs?
It would seem that almost anything can be calculated mentally providing one takes time to learn a few rules and useful numbers. While accepting that mental arithmetic wonāt give spot on precision results, approximations are mostly fine for my purposes.
Iāve been playing around with some scenarios and wonder how the Rule of 72 could be used to estimate the growth factor. Letās say that an amount has grown from $1000 to $10000 over a 25 year period, in other words it has grown to 10 times its original value (this could be a monetary amount, population etc). How does the rule give us the compounding factor?
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Iām stuck here at 2^0.6, thought 2^0.5 x 2^0.1 and then memorising 2^0.1 but it feels awkward.
Will need Kinma or others to help here Iām afraid.
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There are different ways of working 2 ^ 0.6 out.
Since the topic is the rule of 72, letās use that.
Using this rule, ask yourself, what percentage will double a sum in 10 years?
Clearly this is 72 divided by 10 years or 7.2.
Therefore; 2 ^ 0.1 = 1.072.
2 ^ 0.5 is the square root of 2 or 1.41.
If 2^0.1 is 7.2%, then add 7.2% to 1.41 to get 1.51.
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Easier than you might think.
A factor of 10 means it doubled 3.3 times (1/log 2).
72 times 3.3 is 237.6.
237.6 / 25 = 9.5.
The compounding factor is 9.5%.
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Thanks Kinma, let me ponder this. What are the steps if weād been trying to work out 2^0.35?
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Again, thanks for responding, that does seem easy. Can I confirm with you where does the log 2 come from?
Easy.
72 * 0.35 = 36 * .7 = 25.2. The 25.2 is a percentage, so the number is 1.252.
This is somewhat inaccurate. A calculator gives 1.275.
The reason is that 72 works for small interests. Less than 10%.
35% is not small, so we need a bigger number than 72.
We can use the correction formulas. However; they work for percentages up to about 20% and 35% is still too big for them.
What I like to do is first get close to the answer in a different way and then use 72 only for the correction. Realize that 0.35 is close to 1/3.
So first work out the third root of 2. This is about 1.26.
Now for the correction.
We have calculated 2^ {0.33333\ldots} We need 2^{0.35}.
The difference: 0.35 - 0.3333\ldots = 0.0166666\ldots = 1/60
72 / 60 = 1.2.
So add 1.2\% of 1.26 \approx 1.5 to 1.26 to get 1.275.
I said a factor of 10 means it doubled 3.3 times.
If 1 doubles 3 times we get 8. This is 2^3.
If 1 doubles 4 times we get 16. This is 2^4.
If we want to know how many doublings we need to get to 10, the answer is bigger than 3 and lower than 4.
Letās try 3.5.
8 times the square root of 2 \approx 11.3. 10 is lower, so the answer is lower than 3.5. etc.
We need to work out x such that 2^x = 10.
This is \log_2 10 or 1/log 2 \approx 1/{0.301} \approx 3.3.
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Thanks for clarifying Kinma, the individual steps are valuable in understanding your thought processes, itās also important to understand the limitations of the 72 rule. Having said that though this rule is still pretty impressive, Iām still learning about it, I needed to work out how much 8% compounded annually would be in the next 4 years. Applying the Rule of 72, 72/8 gives me 9 which tells me that an amount would double in 9 years. The calculation then becomes 2^0.44 which is the subject of another thread that I started. This is a more difficult calculation. I know that the answer is 1.08^4 which gives us 1.36, in other words at this interest rate and for 4 years we have a 36% increase. I can estimate the latter through base 10 logarithms quicker than I can calculate 2^0.44 at this stage however the Rule of 72 can give a very quick estimate when the numbers allow.
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1.08^4 is not very difficult to calculate.
First, square 1.08.
(a+b)^2 = a^2 +2a +b^2.
Take a = 1 & b= 0.08. Alternatively, take a = 100 & b= 8.
Letās visit the terms one by one:
a^2 = 1
2ab = 0.16
b^2 = 0.0064
Add them together: 1.1664.
Now square 1.1664 to get 1.08^4.
a = 1 & b= 0.1664.
a^2 = 1
2ab = 0.3328
b^2 is a little more āinvolvedā.
The amount of time you need to get an answer depends on the precision you want.
Letās do a quick and dirty calculation: just average 16^2 and 17^2 .
This gives an answer quickly close to 0.165^2 & 0.165 is close to 0.1664.
16^2 = 256 & 17^2 = 289. Difference is 33, half is 16.5, so the average is 256 + 16.5 = 272.5. Maybe drop the last 0.5.
Now add 33|28 + 2|72.
I hope you immediately realize that 28+72 = 1|00.
The calculation now becomes 33|00 + 3|00.
Answer is 36|00.
Lastly add the one: 1.36
Pretty close to the actual answer of 1.36048896.
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Thanks very much Kinma, you have shown that Compound interest questions can indeed be done mentally which I didnāt think was possible. Iām wondering even though I canāt see how, whether annuity type calculations can be estimated, this will be another thread though.
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While the Rule of 72 is a handy tool for quick estimations, itās not perfectly accurate, especially for higher interest rates like 11%. For a more precise calculation without a calculator, you might want to consider the Rule of 115 or Rule of 114. These rules take into account higher interest rates and longer periods, offering better estimates. If youāre interested in exploring further, you can find detailed explanations and formulas for these rules in financial literature or online resources.
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This is not entirely accurate.
The 114 or 115 rule is used for calculations where you triple your money.
I wrote about the rule of 72 - also for higher interest rates - here:
Back to the rule of 115.
An 10 {\%} interest will triple your money in 115/10 = 11.5 years.
Indeed: 1.1^{11.5} \approx 3. It is 2.9923... if you want to be exact, but thatās a rounding issue.
An 5 {\%} interest will triple your money in 115/5 = 23 years.
Indeed: 1.05^{23} \approx 3. It is 3.07... if you want to be exact, but thatās a rounding issue.
Longer periods do not affect the rule.
Letās take the rule of 72 as an example.
An 0.5 {\%} interest will double your money in 72/0.5 = 144 years.
Indeed: 1.005^{144} \approx 2. It is 2.0507... if you want to be exact, but thatās a rounding issue.
Also for the rule of 115:
An 0.5 {\%} interest will triple your money in 115/0.5 = 230 years.
Indeed: 1.005^{230} \approx 3. It is 3.1419... if you want to be exact, but thatās a rounding issue.
Can you give examples of how you take into account higher interest rates and/or periods?
This is a mental calculation forum, so we like to see how you do these things mentally.
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Thanks Kinma for clarifying. It helps enormously to see the proofs through the calculations.
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