Share the (real) ways you do financial arithmetic mentally

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nisiprius
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Share the (real) ways you do financial arithmetic mentally

Post by nisiprius »

I think it is a really good idea to get in the habit of doing financial arithmetic mentally, getting estimates and using round numbers. The goal in this thread is not to display dazzling nifty tricks, but to be as honest as possible in sharing the actual mental steps we use to tackle real-world financial number estimation.

For example, and the point is NOT to show some best or recommended way, but to show that I am NOT a mental math whiz, but nevertheless have a sort of laborious almost-like-counting-on-fingers way to slog through. For me, the key is to find a series of steps that are simple, natural, and practiced enough that I'm likely to get each one right.

How many dollars a year is the difference in expenses on a $300,000 investment in two ETFs with expense ratios of 0.16% and 0.09%?

How I do it in my head:

The difference is 0.07%.
I have a little trouble with getting decimal points correctly placed, so I start with 7% of $100,000 because that's easy for me.
7% of 100 of anything is 7 of that thing.
7% of $100 thousand is $7 thousand, $7,000.
0.07% is 1/100 of that, so it's $70.
$300,000 is three times as much as $100,000, so multiply by 3.
3 x $70 = $210.
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Re: Share the (real) ways you do financial arithmetic mental

Post by momar »

That's also what I do.

For tips, it's easy enough to move the decimal over one and then add half.
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Re: Share the (real) ways you do financial arithmetic mental

Post by sscritic »

I own a calculator, and I will not pretend that I don't. But if forced to get an estimate, I would start with .09 being almost .10 (this is how estimates are constructed). .1% of 300,000 is 1/10 of 1% of 300,000 or 1/10 0f 3000 or 300. .16 is between .2 and .15 so we are looking at between 600 (twice the .1 number) and 450 (1.5 times the .1 number). Our answer is between 150 and 300, and since I just need a quick estimate, I am not going to waste my time trying to be accurate (accurate and estimate don't go together), so let's split the difference and say $225.
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Re: Share the (real) ways you do financial arithmetic mental

Post by The Wizard »

This also comes into play with some people when using foreign currency.
A Euro was about $1.35 during a recent trip. My GF initially made a table to figure how much 10 or 20 Euro are in dollars, but found it simpler just to ask me most times.
After several days, I just started "thinking" in Euros, mainly when looking at dinner menus...
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Re: Share the (real) ways you do financial arithmetic mental

Post by sscritic »

To test your method, whatever it is, try it on $368,462 rather than $300,000.
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Re: Share the (real) ways you do financial arithmetic mental

Post by Random Walker »

I think in terms of monthly living expenses. So when I think about financial goals, I start with some odd numbers like 120k or 1.2M (multiples of 12). Then when I work backwards and divide by 12 months per year, I get some pretty round numbers to ballpark with.

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Re: Share the (real) ways you do financial arithmetic mental

Post by The Wizard »

In Nisi's example (0.07% of $300K), I'd do it differently.
I'd start with 0.01% and express that as 0.0001 rather than percentage.
Hence 4 decimal places.
Hence $300,000 --> $30,000 --> $3,000 --> $300. --> $30.
Then multiply by 7 --> $210.
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Re: Share the (real) ways you do financial arithmetic mental

Post by The Wizard »

sscritic wrote:To test your method, whatever it is, try it on $368,462 rather than $300,000.
Good point.
Similar to above, move decimal point 4 places and get $36.84, but round it to $37.
Then multiply by 7 which is hard to do, so do 7 x $40 instead.
So something a bit less than $280...
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Re: Share the (real) ways you do financial arithmetic mental

Post by VictoriaF »

The Wizard wrote:In Nisi's example (0.07% of $300K), I'd do it differently.
I'd start with 0.01% and express that as 0.0001 rather than percentage.
Hence 4 decimal places.
Hence $300,000 --> $30,000 --> $3,000 --> $300. --> $30.
Then multiply by 7 --> $210.
I do something similar, but I don't count decimal places in my mind. I do it step by step as follows:
1. 1% of $300k is $3k
2. 0.1% is $300
3. 0.01% is $30
4. 0.07% is $210.

Then I check myself in several way. One check is to repeat my steps and watch out for missteps. Alternatively, I may try to get the result in another way. For example, instead of 0.07% assume that it's 0.1%. 0.1% of $300k is $300. $210 for 0.07% looks right.

Another check is the plausibility of the result, i.e., whether it make sense. If it seems too high or too low, look for more information.

Victoria
Last edited by VictoriaF on Tue Nov 26, 2013 9:34 am, edited 1 time in total.
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Re: Share the (real) ways you do financial arithmetic mental

Post by Kosmo »

My method:

The difference between 0.16% and 0.09% is 0.07%.
A % sign takes away 2 zeros. And the decimal is 2 places in front of the 7 so that takes away another 2 zeros for a total of 4 zeros.
So take 4 zeros off the initial value of $300,000 leaving 30, and replace the 0.07% with 7.
7*3 = 21. Plus the 1 remaining zero gives $210. (Or just as easily: 7*30 = $210)

Having nice round numbers works easy, but this method works the same for arbitrary number by just moving the decimal around and then rounding.
sscritic wrote:To test your method, whatever it is, try it on $368,462 rather than $300,000.
7*36.8 is close enough to 7*40 = $280, so I'd guess about $265. My calculator says $257.92, so I'm off by < $10.
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Re: Share the (real) ways you do financial arithmetic mental

Post by gkaplan »

VictoriaF wrote:
The Wizard wrote:In Nisi's example (0.07% of $300K), I'd do it differently.
I'd start with 0.01% and express that as 0.0001 rather than percentage.
Hence 4 decimal places.
Hence $300,000 --> $30,000 --> $3,000 --> $300. --> $30.
Then multiply by 7 --> $210.
I do something similar, but I don't count decimal places in my mind. I do it step by step as follows:
1. 1% of $300k is $3k
2. 0.1% is $300
3. 0.01% is $30
4. 0.07% is $210.

Then I check myself in several way. One check is to repeat my steps and watch out for missteps. Another check is the plausibility of the result, i.e., whether it make sense. If it seems too high or too low, look for more information.

Victoria
That's exactly how I do it, Victoria, although I don't know if I always do the check.
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Re: Share the (real) ways you do financial arithmetic mental

Post by The Wizard »

I think a lot of non-STEM types get tripped up with fractional PERCENTS.
Which is why they sometimes have percent keys on simple calculators.
I think I have a tendency to go directly to decimals or even exponential notation,
10^-9 is a lot quicker to comprehend, for instance, than either 0.000000001 or (heaven help us) 0.0000001%.
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Re: Share the (real) ways you do financial arithmetic mental

Post by VictoriaF »

The Wizard wrote:I think a lot of non-STEM types get tripped up with fractional PERCENTS.
Which is why they sometimes have percent keys on simple calculators.
I think I have a tendency to go directly to decimals or even exponential notation,
10^-9 is a lot quicker to comprehend, for instance, than either 0.000000001 or (heaven help us) 0.0000001%.
I agree. I much prefer to see powers than zeros. And I don't appreciate comparisons such as "it would take $100 bills head-to-toe to encircle the equator 17 times." I never thought about this as the difference between STEM majors and others, but you are probably right.

Victoria
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Re: Share the (real) ways you do financial arithmetic mental

Post by Clive »

try it on $368,462 rather than $300,000
I work the numbers first and then figure out the precision.

0.07% becomes just 7
$368,462 becomes just 37

mentally multiplying those two
7x7=49
7x3=21
210 + 49 = 259

Once I've got that 259 numbers in mind I figure out the precision - something like :
if it were 1% of 300.000 it would be 3000
So for 0.1% it would be 300
0.07% is a bit less than that so = $259
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Re: Share the (real) ways you do financial arithmetic mental

Post by mhc »

When I was a child, my dad taught me the rule of 72. The rate of return times 100 (7% would be merely 7) times the number of years compounding equals 72.

For example, if you earn 8% it takes 9 years to double your money. 8x9=72

I use the rule of 72 and linear approximation to calculate how much I will have saved for retirement all the time. It mentally goes like this:

I have $100k. I save $10k a year. My return will be 6%. I know I will double my money every 12 years (6x12=72). In 12 years I will have added an additional $120k. Half of that is $60k. So, $100k + $60k=$160k. Then double it. I will have $320k in 12 years. I know it is on the low side of the actual number, so it is a conservative estimate. Excel says it would actually be $369,919.06. 13% error. I can live with that for making predictions 12 years out.
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Re: Share the (real) ways you do financial arithmetic mental

Post by Clive »

11 times table for 2 digit numbers can be simplified to the first digit, sum of two digits, last digit

i.e. 35 x 11 =

first digit = 3
sum of two digits = 3+5 = 8
last digit = 5
answer 385

You do however have to revise the first digit if the sum of the two digits is greater than 10 i.e. carry the extra 1 e.g. 57 x 11 = 5 first digit, 5+7 middle digit = 12 so add one onto the first digit (making it 6), last digit = 7 so 627.

There are other tricks such as to determine if a number is wholly divisible by three, sum the individual digits and if that is divisible by three then the larger number is divisible by three.

4527 = sum of 4+5+2+7 = 18 and as 18 is divisible by three, so also is 4527 (=1509)
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Re: Share the (real) ways you do financial arithmetic mental

Post by VictoriaF »

I distinguish between two applications of mental arithmetic. One application is to calculate precise numbers by applying some numerical methods. For example, 99 x99 = (100 - 1)^2 = 10,000 - 200 + 1 = 9801.

The other application is to quickly evaluate the order of magnitude. The rule of 72 helps to estimate when some investment will double.

Even less precise estimates help me with long term planning. For example, I have $X in a tax-deferred account, and I am planning to spend $0.3X in taxes to convert it to Roth (where $X itself is rounded). I may need less than $0.3X if I don't have a state income tax, and I may need more than $0.3X if my account appreciates significantly, but $0.3X is a good planning number. I don't want it to be any more precise than that; extra digits will upset my intuition into thinking that the number is more precise than it really is.

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Re: Share the (real) ways you do financial arithmetic mental

Post by TareNeko »

The Wizard wrote:
sscritic wrote:To test your method, whatever it is, try it on $368,462 rather than $300,000.
Good point.
Similar to above, move decimal point 4 places and get $36.84, but round it to $37.
Then multiply by 7 which is hard to do, so do 7 x $40 instead.
So something a bit less than $280...
In these kind of situations, one last step will help to get you close to the answer:

You want to know what 7 x $37 is, so you calculate 7 x $40 = $280.
Then, calculate 7 x $3 =$21.
Now subtract $280 - $21 = $259.
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Re: Share the (real) ways you do financial arithmetic mental

Post by House Blend »

When getting a rough idea of monthly payments on a (30 year fixed) mortgage, an old rule of thumb is that the first year is nearly all interest, so the payment is

(loan amount) x APR / 12.

Worked much better back in the days of 12% loans, rather than 3%. (Try it and see.)
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Re: Share the (real) ways you do financial arithmetic mental

Post by MathWizard »

sscritic wrote:To test your method, whatever it is, try it on $368,462 rather than $300,000.
I always try to transform the problem into an easier one, by some approximation, and then fix up
the approximation as many times as I need to get better estimates.
I worked this out on my own, but later found out it is called iterative refinement.

You migt find all this convoluted, but the idea is not to worry about slight inaccuracies, they
get corrected later, but to concentrtae on numbers you can keep in your head.

.07% of $368,462 in my head?

First estimate:

I recognize 368,462 as close to 1/3 of $1.1 Million

.07% of $1.1 million is .7% of $0.11M
or .7% of $110K =
7% of 11,000
= 7 times 110
=770.

Now I earlier tripled the original $368,462, so $770 is too high by a factor of 3,
dividing by 3 gives me $256.67.

This is the initial estimate. Which is often good enough to know I am in the ballpark.

Seond estimate (first refinement):

If I want to make this now more accuate, I recognize that my initial approximation
of 368,462 as 1/3 of $1.1M was low, since 366,667 is 1/3 of $1.1M, when rounded to the nearest dollar,

So my final answer was to low by a factor of (368,462-366,667)/366,667 which is about 1,800/360,000 or 1/200,
so I should adjust the 256.67 up by
1/2 of 2.5667 which is about
$1.28, so
256.67+1.28 is $257.95.

Thrid estimate (second refinemnet):

I could do this once again recognizing that the correction that I applied was slighly high, because 366,667 is not the same as
360,000. 6,667 out of 360,000 is 1111* 6 parts of 360,000 or 1111* 1 part in 60,000 = 1111/60,000 = 1.111 * 1/6th of 1/10 or 1 part in 60.

So my correction of 1.28 was high by a little over 2 cents.

So the third, more accurate estimate is

$257.95 - $.02 = $257.93



This is quite close to the actual $257.92

Is this the way you approach this problem , sscritic?
Last edited by MathWizard on Tue Nov 26, 2013 11:04 am, edited 1 time in total.
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Re: Share the (real) ways you do financial arithmetic mental

Post by sschullo »

nisiprius wrote: How many dollars a year is the difference in expenses on a $300,000 investment in two ETFs with expense ratios of 0.16% and 0.09%?
Estimates and round numbers, of course, but perhaps I missed your point and just responding to your example, anybody here would choose .09% ETF.
I don't care about the difference, that's my thinking. Just look at the cheaper cost. And .16% is extremely cheap too compared to the my personal 403b fixed annuity history and managed funds costing well over 1.25%.
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Re: Share the (real) ways you do financial arithmetic mental

Post by sscritic »

MathWizard wrote:
sscritic wrote:To test your method, whatever it is, try it on $368,462 rather than $300,000.
I always try to transform the problem into an easier one, by some approximation, and then fix up
the approximation as many times as I need to get better estimates.
I worked this out on my own, but later found out it is called iterative refinement.

You migt find all this convoluted, but the idea is not to worry about slight inaccuracies, they
get corrected later, but to concentrtae on numbers you can keep in your head.

.07% of $368,462 in my head?

First estimate:

I recognize 368,462 as close to 1/3 of $1.1 Million

.07% of $1.1 million is .7% of $0.11M
or .7% of $110K =
7% of 11,000
= 7 times 110
=770.

Now I earlier tripled the original $368,462, so $770 is too high by a factor of 3,
dividing by 3 gives me $256.67.

This is the initial estimate. Which is often good enough to know I am in the ballpark.

Seond estimate (first refinement):

If I want to make this now more accuate, I recognize that my initial approximation
of 368,462 as 1/3 of $1.1M was low, since 366,667 is 1/3 of $1.1M, when rounded to the nearest dollar,

So my final answer was to low by a factor of (368,462-366,667)/366,667 which is about 1,800/360,000 or 1/200,
so I should adjust the 256.67 up by
1/2 of 2.5667 which is about
$1.28, so
256.67+1.28 is $257.95.

Thrid estimate (second refinemnet):

I could do this once again recognizing that the correction that I applied was slighly high, because 366,667 is not the same as
360,000. 6,667 out of 360,000 is 1111* 6 parts of 360,000 or 1111* 1 part in 60,000 = 1111/60,000 = 1.111 * 1/6th of 1/10 or 1 part in 60.

So my correction of 1.28 was high by a little over 2 cents.

So the third, more accurate estimate is

$257.95 - $.02 = $257.93



This is quite close to the actual $257.92

Is this the way you approach this problem , sscritic?
No, because I don't have $368,462 in a single fund, nor do I have $300,000. I also have no interest in the exact answer. Actually, I prefer a different question. I want to know both fees. If the difference is $257.93, is that $2347.26 vs $2695.19 (excuse me if my addition is not precise) or $23.12 vs $281.05 (same caveat). Or rather, is it $300 above $2000 or $300 above $20?
I think it is a really good idea to get in the habit of doing financial arithmetic mentally, getting estimates and using round numbers.
Last edited by sscritic on Tue Nov 26, 2013 11:21 am, edited 1 time in total.
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Re: Share the (real) ways you do financial arithmetic mental

Post by 2beachcombers »

sschullo wrote:
nisiprius wrote: How many dollars a year is the difference in expenses on a $300,000 investment in two ETFs with expense ratios of 0.16% and 0.09%?
Estimates and round numbers, of course, but perhaps I missed your point and just responding to your example, anybody here would choose .09% ETF.
I don't care about the difference, that's my thinking. Just look at the cheaper cost. And .16% is extremely cheap too compared to the my personal 403b fixed annuity history and managed funds costing well over 1.25%.
I also did this for a year. Then I got an education about actual performance and content of etfs.

So I switched from VBR to IJS to get a better small value content and went from 10 to 30bp.
300k * (30-10) = 600$. I did this in my head by taking actual 277556 in roth ; rounding to 300 and multipying by 2.
by 2 cause 300 is missing 3 zeros and 20 is adding 4 zeros. 20- less a zero is 2.

Now I hope over the next 20 years IJS outperforms VBR by 600/300000 or 6/3000 = 0.2% :sharebeer
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Re: Share the (real) ways you do financial arithmetic mental

Post by MathWizard »

sscritic wrote:[ I also have no interest in the exact answer. Actually, I prefer a different question. I want to know both fees. If the difference is $257.93, is that $2347.26 vs $2695.19 (excuse me if my addition is not precise) or $23.12 vs $281.05 (same caveat). Or rather, is it $300 above $2000 or $300 above $20?
Fair enough. I actually find doing such problems in my head keeps me sharp, and is actually fun.

Some people like creating poetry or sculpture, I like math.

Oh and by the way: It's good to have you back. I've always liked your
posts.
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Re: Share the (real) ways you do financial arithmetic mental

Post by mlipps »

I've always used little tricks like this, mostly taught to me by my mom, who is the fastest mental arithmetic calculator I've ever known. I will never forget in 1st grade when I told my teacher that to add 9 to something I added 10 then subtracted one. She told me I was right, but not to talk about it because I would confuse the other students. :oops: No wonder people use calculators for absolutely everything anymore.
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Re: Share the (real) ways you do financial arithmetic mental

Post by sscritic »

My grandfather worked for National Cash Register, but he never needed one for himself. He used to add columns (as in twelve or fifteen numbers) of three digit numbers in his head, in what to me was a very short time. There were no shortcuts, just right answers. It always amazed me (and yes, I would slowly add up the same figures, doing it the way I learned it: ones column, carry, twos column, carry, threes column, carry).
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Re: Share the (real) ways you do financial arithmetic mental

Post by IlliniDave »

.16% of 300,000 =480 minus .09% of 300,000 = 270 = 210. I can just sorta do the multiplication in my head...some subconscious version of 3e5*16e-4=48e1=480, etc., I suppose. Works as long as the numbers are somewhat round.
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Re: Share the (real) ways you do financial arithmetic mental

Post by Kosmo »

MathWizard wrote:always try to transform the problem into an easier one, by some approximation, and then fix up
the approximation as many times as I need to get better estimates.
I'd say this process takes you way beyond the range of estimating. Understanding this method is important, but when you're doing it all in your head on the fly, it's overkill. By the time you apply multiple refinements to account for the approximations, you could have multiplied out on paper the exact solution.

After reading the responses, it seems the best tools to have a firm grasp on rounding, powers of 10, and the commutative property of multiplication and addition. The rest is just details.
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Re: Share the (real) ways you do financial arithmetic mental

Post by BaldTom »

Erhan touched on the distributive law of mathematics. A*(B+C) =A*B +A*C.

I use this for a lot of rough estimates. The components do not have to be positive or larger than one.

First clear up the decimals.
368000 x .0007 = 368 x.7 =
368*(1 -.3) = 368 - 3*36.8 =
368 - 1.10 =
2.58

Many paths to the same result. These can be used to check our work. Eighty percent, for example, can be reached by:
Subtract a tenth twice,
subtract a fifth,
or multiply a fifth by four.

I had a lot of trouble doing temperature conversions until I realized 9/5ths of a Celsius degree is a doubling, less 10%.

This was an excellent question. I've been doing things like this so automatically, it took some time to sort it out enough to respond. THANKS!
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Re: Share the (real) ways you do financial arithmetic mental

Post by Bonnan »

I start out similar to Victoria's but at .1% ($300) I merely take .7's of 300 to get $210
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Re: Share the (real) ways you do financial arithmetic mental

Post by mlipps »

Kosmo wrote:
MathWizard wrote:always try to transform the problem into an easier one, by some approximation, and then fix up
the approximation as many times as I need to get better estimates.
I'd say this process takes you way beyond the range of estimating. Understanding this method is important, but when you're doing it all in your head on the fly, it's overkill. By the time you apply multiple refinements to account for the approximations, you could have multiplied out on paper the exact solution.

After reading the responses, it seems the best tools to have a firm grasp on rounding, powers of 10, and the commutative property of multiplication and addition. The rest is just details.
Yes but personally, I'm always doing this kind of math in the shower! When I try to write out the multiplication on the steamy shower door, the numbers evaporate before I get through the whole problem...So estimating & refining all in my head works best.
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Re: Share the (real) ways you do financial arithmetic mental

Post by Epsilon Delta »

mhc wrote:When I was a child, my dad taught me the rule of 72. The rate of return times 100 (7% would be merely 7) times the number of years compounding equals 72.

For example, if you earn 8% it takes 9 years to double your money. 8x9=72
You may also like the rule of 112. It works the same as the rule rule of 72 except the result is a triple.

For example, if you earn 8% it takes 14 years to triple your money. 8x14=112.

Conveniently 3 ~ 2 * sqrt(2) so this fills the holes in time series, with the rule of 72 you can work out how long it takes to multiply by 2, 4, 8, 16 etc. Using the rule of 112 allows you to figure out how long it takes to multiply by 2,3,4,6,8,9,12,15,16 ... .
mhc wrote: I use the rule of 72 and linear approximation to calculate how much I will have saved for retirement all the time. It mentally goes like this:

I have $100k. I save $10k a year. My return will be 6%. I know I will double my money every 12 years (6x12=72). In 12 years I will have added an additional $120k. Half of that is $60k. So, $100k + $60k=$160k. Then double it. I will have $320k in 12 years. I know it is on the low side of the actual number, so it is a conservative estimate. Excel says it would actually be $369,919.06. 13% error. I can live with that for making predictions 12 years out.
Another approach is that if you have $10k/6% ~ $170 it will throw off 10k a year. So "borrow" $170k, invest it for 12 years to get 2*$170k, pay the loan back and your left with $170k. So saving $10k per year for 12 years at 6% gives you $170k. Your $100k also doubles to $200k. Result $370k,
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Re: Share the (real) ways you do financial arithmetic mental

Post by 2beachcombers »

MathWizard wrote:
sscritic wrote:[ I also have no interest in the exact answer. Actually, I prefer a different question. I want to know both fees. If the difference is $257.93, is that $2347.26 vs $2695.19 (excuse me if my addition is not precise) or $23.12 vs $281.05 (same caveat). Or rather, is it $300 above $2000 or $300 above $20?
Fair enough. I actually find doing such problems in my head keeps me sharp, and is actually fun.

Some people like creating poetry or sculpture, I like math.

sscritic---------------Oh and by the way: It's good to have you back. I've always liked your
posts.

+10-- on top of my allstar list---sscritic, doc, livesoft, YNDAL, Rick, grabiner, etc--you guys are great
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Re: Share the (real) ways you do financial arithmetic mental

Post by asset_chaos »

0.1% of 300,000 is 300. 0.1 is close enough to 0.09 to say the cheaper fund costs a little less than 300 a year. The more expensive is 60% more than 0.1. 60% of 300 is 180. Because I slightly overestimated on the cheaper, I round up and say the difference is around 200 a year. If I need more precision than the right order of magnitude, then it's time to get out a calculator.
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Re: Share the (real) ways you do financial arithmetic mental

Post by asset_chaos »

sscritic wrote:To test your method, whatever it is, try it on $368,462 rather than $300,000.
The cheaper one, then, is about 350 and the more expensive is a little less than twice as much, say 650. For a difference of around 300.
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Re: Share the (real) ways you do financial arithmetic mental

Post by nisiprius »

sscritic wrote:My grandfather worked for National Cash Register, but he never needed one for himself. He used to add columns (as in twelve or fifteen numbers) of three digit numbers in his head, in what to me was a very short time. There were no shortcuts, just right answers. It always amazed me (and yes, I would slowly add up the same figures, doing it the way I learned it: ones column, carry, twos column, carry, threes column, carry).
It sounds as if it might be too late to ask him, but do you know whether he worked from left to right or right to left?

One of the weird things I've read is that almost uniformly, people who are skilled at that kind of mental math--accurate non-estimated four-function arithmetic--perform the operations working from left to right, i.e. from most significant to least significant.

What is even weirder is that when I try to do it myself--two digits plus two digits being about my comfortable limit--I, too, find it easier to work from left to right.

It is utterly illogical, and means that you sometimes need to perform multiple carries and adjust several digits in your head, but it seems to be true.
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Re: Share the (real) ways you do financial arithmetic mental

Post by TareNeko »

BaldTom wrote: I had a lot of trouble doing temperature conversions until I realized 9/5ths of a Celsius degree is a doubling, less 10%.

This was an excellent question. I've been doing things like this so automatically, it took some time to sort it out enough to respond. THANKS!
I use similar approach for lb / kg conversion:

kg =~ (lb * 0.5) - ( lb/10 * 0.5)

100lb --> 100*0.5 - 10*0.5 = 45kg.

Or you can think of it as subtracting 1/10th of lb*0.5 from itself.
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Re: Share the (real) ways you do financial arithmetic mental

Post by mhc »

Epsilon Delta wrote:
mhc wrote:When I was a child, my dad taught me the rule of 72. The rate of return times 100 (7% would be merely 7) times the number of years compounding equals 72.

For example, if you earn 8% it takes 9 years to double your money. 8x9=72
You may also like the rule of 112. It works the same as the rule rule of 72 except the result is a triple.

For example, if you earn 8% it takes 14 years to triple your money. 8x14=112.

Conveniently 3 ~ 2 * sqrt(2) so this fills the holes in time series, with the rule of 72 you can work out how long it takes to multiply by 2, 4, 8, 16 etc. Using the rule of 112 allows you to figure out how long it takes to multiply by 2,3,4,6,8,9,12,15,16 ... .
mhc wrote: I use the rule of 72 and linear approximation to calculate how much I will have saved for retirement all the time. It mentally goes like this:

I have $100k. I save $10k a year. My return will be 6%. I know I will double my money every 12 years (6x12=72). In 12 years I will have added an additional $120k. Half of that is $60k. So, $100k + $60k=$160k. Then double it. I will have $320k in 12 years. I know it is on the low side of the actual number, so it is a conservative estimate. Excel says it would actually be $369,919.06. 13% error. I can live with that for making predictions 12 years out.
Another approach is that if you have $10k/6% ~ $170 it will throw off 10k a year. So "borrow" $170k, invest it for 12 years to get 2*$170k, pay the loan back and your left with $170k. So saving $10k per year for 12 years at 6% gives you $170k. Your $100k also doubles to $200k. Result $370k,
Thank you. When I see my dad at Christmas, I'll have to ask him why he didn't teach me the rule of 112. Just think what I could have done with that knowledge. :D

I'll have to work through your "borrow" example and see why it works. It seems like a much better way to do the estimating. Thanks.
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Re: Share the (real) ways you do financial arithmetic mental

Post by wilpat »

sscritic wrote:My grandfather worked for National Cash Register, but he never needed one for himself. He used to add columns (as in twelve or fifteen numbers) of three digit numbers in his head, in what to me was a very short time. There were no shortcuts, just right answers. It always amazed me (and yes, I would slowly add up the same figures, doing it the way I learned it: ones column, carry, twos column, carry, threes column, carry).
I have done it this way for over 60 years, but as I get older it is more trouble to remember what was in column 1 or 2 etc.

I find that there are 2 signs of approaching old age "Loss of memory and - uh - I forget the other one!"
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Re: Share the (real) ways you do financial arithmetic mental

Post by Trikkur »

nisiprius wrote:
sscritic wrote:My grandfather worked for National Cash Register, but he never needed one for himself. He used to add columns (as in twelve or fifteen numbers) of three digit numbers in his head, in what to me was a very short time. There were no shortcuts, just right answers. It always amazed me (and yes, I would slowly add up the same figures, doing it the way I learned it: ones column, carry, twos column, carry, threes column, carry).
It sounds as if it might be too late to ask him, but do you know whether he worked from left to right or right to left?

One of the weird things I've read is that almost uniformly, people who are skilled at that kind of mental math--accurate non-estimated four-function arithmetic--perform the operations working from left to right, i.e. from most significant to least significant.

What is even weirder is that when I try to do it myself--two digits plus two digits being about my comfortable limit--I, too, find it easier to work from left to right.

It is utterly illogical, and means that you sometimes need to perform multiple carries and adjust several digits in your head, but it seems to be true.
I've always been fairly good with big number operations in my head. I work left to right because when you work right to left you have to remember all of the numbers in reverse order for the correct answer. When you work left to right when adding or multiplying large digits, you are thinking out the correct answer immediately. It is difficult to explain exactly, but I'll do my best.

3435
+
4936

I'm going to try to type every thought process that would have to go into this during mental math.
Left-Right
4+3 is 7
9+4 is 13 - since this is over 10, enlarge the last number -> 83
3+3 is 6 -> 836
5+6 is 11 - enlarge last -> 8371

Right-Left
5+6 is 11
3+3 is 6+1 is 7
4+9 is 13
3+4 is 7+1 is 8
Now you have to remember them in opposite order. 8, then 3, then 7, then 1.

This is easier when I'm typing it because the numbers are right there, but it is much more difficult to recall them in your head after doing various math operations on random sets of numbers.

This becomes even more obvious in multiplication when you aren't just simply carrying over 1s.
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Re: Share the (real) ways you do financial arithmetic mental

Post by MathWizard »

Epsilon Delta wrote:
mhc wrote:When I was a child, my dad taught me the rule of 72. The rate of return times 100 (7% would be merely 7) times the number of years compounding equals 72.

For example, if you earn 8% it takes 9 years to double your money. 8x9=72
You may also like the rule of 112. It works the same as the rule rule of 72 except the result is a triple.

For example, if you earn 8% it takes 14 years to triple your money. 8x14=112.

Conveniently 3 ~ 2 * sqrt(2) so this fills the holes in time series, with the rule of 72 you can work out how long it takes to multiply by 2, 4, 8, 16 etc. Using the rule of 112 allows you to figure out how long it takes to multiply by 2,3,4,6,8,9,12,15,16 ... .
mhc wrote: I use the rule of 72 and linear approximation to calculate how much I will have saved for retirement all the time. It mentally goes like this:

I have $100k. I save $10k a year. My return will be 6%. I know I will double my money every 12 years (6x12=72). In 12 years I will have added an additional $120k. Half of that is $60k. So, $100k + $60k=$160k. Then double it. I will have $320k in 12 years. I know it is on the low side of the actual number, so it is a conservative estimate. Excel says it would actually be $369,919.06. 13% error. I can live with that for making predictions 12 years out.
Another approach is that if you have $10k/6% ~ $170 it will throw off 10k a year. So "borrow" $170k, invest it for 12 years to get 2*$170k, pay the loan back and your left with $170k. So saving $10k per year for 12 years at 6% gives you $170k. Your $100k also doubles to $200k. Result $370k,
I always used sqrt(2) for half the duration computed using the rule of 72 when continuously adding the same contribution..
So $10K a year means 120K over 12 years with an average duration of 6 years, so I estimated
120K*sqrt(2) which is about 120K*1.414 = 141.4K+28.28K = 169.68K
The original $100K doubles, so this add up to 200K+169.68K = $369.68K.
Still an estimation.

I have no real basis for using sqrt(2), but it seems to work. I think it is just a close approx for interest rates in the 3 to 18 range,
like the rule of 72 is.

Maybe grabiner has an explanation.
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Re: Share the (real) ways you do financial arithmetic mental

Post by umfundi »

My father had a book,

The Trachtenberg Speed System of Mathematics, with all kind of tricks.

The one I shortcut I find most useful is Pounds to Kilograms: Divide by 2, deduct 10% .

Kg to lbs: Double, add 10%.

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Re: Share the (real) ways you do financial arithmetic mental

Post by LadyGeek »

This thread is now in the Personal Finance (Not Investing) forum (finance math).
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Re: Share the (real) ways you do financial arithmetic mental

Post by Phineas J. Whoopee »

For mental arithmetic generally, regardless of whether the units are financial or otherwise, I too follow the break-it-down-into-fairly-easy-steps-(for-me)-then-do-that method, but without expecting myself to get the result to very tight precision. I like to use the multiplication tables they forced me to memorize up to the number twelve. For example, if I don't know the answer, I think about answers I already do know, one on each side, and say to myself "it's got to be somewhere between 121 and 144."

To boil it down to a case, I remember several years ago being at an airport bar, having a sandwich and a beer before my flight, and the bartender / waitress / entire front of house staff said, I think in gratitude to a just-departing patron so he'd come back and do it again, that he'd left a ten dollar tip on an eighteen dollar check. She loudly asked (which is why I think it was really aimed at him) "What percent is that?" Immediately I replied "more than fifty." She asked "did you have to figure that out, or are you just really good with percent?" I said "I know ten is more than half of eighteen."

I left a more customary tip a few minutes later when I departed for my gate.

My own problem with mental arithmetic is while I usually home in on a range for the mantissa, although to only one or two significant digits, I'm prone to order-of-magnitude errors.

The other thing I do is keep my trusty HP RPN calculator near.

A calculator is a device which frees us from having to focus on the arithmetic thereby enabling us to focus on the mathematics.

The math I'm reasonably good at.

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Re: Share the (real) ways you do financial arithmetic mental

Post by Grt2bOutdoors »

nisiprius wrote:I think it is a really good idea to get in the habit of doing financial arithmetic mentally, getting estimates and using round numbers. The goal in this thread is not to display dazzling nifty tricks, but to be as honest as possible in sharing the actual mental steps we use to tackle real-world financial number estimation.

For example, and the point is NOT to show some best or recommended way, but to show that I am NOT a mental math whiz, but nevertheless have a sort of laborious almost-like-counting-on-fingers way to slog through. For me, the key is to find a series of steps that are simple, natural, and practiced enough that I'm likely to get each one right.

How many dollars a year is the difference in expenses on a $300,000 investment in two ETFs with expense ratios of 0.16% and 0.09%?

How I do it in my head:

The difference is 0.07%.
I have a little trouble with getting decimal points correctly placed, so I start with 7% of $100,000 because that's easy for me.
7% of 100 of anything is 7 of that thing.
7% of $100 thousand is $7 thousand, $7,000.
0.07% is 1/100 of that, so it's $70.
$300,000 is three times as much as $100,000, so multiply by 3.
3 x $70 = $210.
The difference is 7 basis points.
$100000*.0007 is $70, $70 times 3 = $210.
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Re: Share the (real) ways you do financial arithmetic mental

Post by Grt2bOutdoors »

sscritic wrote:To test your method, whatever it is, try it on $368,462 rather than $300,000.
The difference is 7 basis points or .0007.
$100,000 times .0007 is $70.
$68,342 divided by $100,000 is .68342 times .0007 or a tad bit past 2/3's of the way, approximately $47. $210 + $47 equals $257 give or take a few pennies.
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Re: Share the (real) ways you do financial arithmetic mental

Post by BolderBoy »

Correct me if I'm wrong, but don't the typical calculators convert everything into logarithms so there is only addition and subtraction going on, then convert back to the answer?

Would it be easy to teach humans to do that from an early age? Would there be an advantage?

I find math fascinating, though I'm not good with it.
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Re: Share the (real) ways you do financial arithmetic mental

Post by Phineas J. Whoopee »

BolderBoy wrote:Correct me if I'm wrong, but don't the typical calculators convert everything into logarithms so there is only addition and subtraction going on, then convert back to the answer?

Would it be easy to teach humans to do that from an early age? Would there be an advantage?

I find math fascinating, though I'm not good with it.
Speaking for myself alone, I had so much trouble learning to spell the word rhythm that introducing the spelling, let alone the idea, of logarithms would have left me quivering under my desk.

I did think the idea of alternate bases, like 12 (shows how old I am) was pretty cool, though.

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Re: Share the (real) ways you do financial arithmetic mental

Post by umfundi »

The one easy rule is the rule of 72, for when a sum will double for a given interest rate.

Except, I have never been able to remember anything about it other than there is a rule associated with some number. :(

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