# Compounding formula



## Steve64 (Jun 28, 2016)

Can someone show me the formula for calculating a daily rate of return .
if I’m expecting a rate of return of 10% in 1 year for instance , how much would the principle grow day by day to equal the 10% by day 365?

thanks in advance


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## Steve64 (Jun 28, 2016)

Let me clarify a little bit, what I would like to do is know that on day 100 of the year the principle plus the daily interest would have earned x amount (based on an expected annual return of 10%) and on day 200 of the year the principle + the daily compounded interest would have a value of ”y”


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## Topo (Aug 31, 2019)

My guess:

x=(0.10/365)^100
y=P(1+0.10/365)^200
P is principal.


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## Steve64 (Jun 28, 2016)

thanks for the reply, sorry for the stupid question but what function is ^


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## like_to_retire (Oct 9, 2016)

Raised to the power of.............

For example 5^2 is 5 raised to the power of 2 (or 5 squared in this case).

ltr


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## Topo (Aug 31, 2019)

I used it to mean "to the power of." I don't know if that is the standard notation for this function.


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## Steve64 (Jun 28, 2016)

Topo said:


> I used it to mean "to the power of." I don't know if that is the standard notation for this function.


still unsure what to do with the 200 after I get to that part of your formula. So I multiply the result by 200 or divide it by 200 for instance? Sorry ut I’m still u sure


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## Steve64 (Jun 28, 2016)

Opps, “sorry but I’m still unsure”


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## Topo (Aug 31, 2019)

First raise what is in the parenthesis (~1.000273)to the power of 200 and then multiply by P.


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## Topo (Aug 31, 2019)

Upon further reflection, I think it should be that 365 is divided into 0.10 and not 10 to arrive at the daily return, since it is a 10% annual return. The above posts have been edited to reflect this.


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## Steve64 (Jun 28, 2016)

Topo said:


> Upon further reflection, I think it should be that 365 is divided into 0.10 and not 10 to arrive at the daily return, since it is a 10% annual return. The above posts have been edited to reflect this.


Tks Topo


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## MrBlackhill (Jun 10, 2020)

Dividing 10% by 365 leads to an approximation, but I think most banks use that approximation. Maybe they use the approximation when it's in their favour and otherwise calculate the true daily interest.

10% yearly divided by 365 equals about 0.0274% daily, which will lead to a bit more than 10% yearly once compounded (10.5156%). The exact daily interest would be 0.0261% for a 10% yearly interest.

You can use this calculator : Compound Interest & APY Calculator


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## agent99 (Sep 11, 2013)

I will have a try.
Let SV = starting value (assume $100), FV = final value($110), r=interest rate, d=days

FV=SV*(1+r)^d or 110=100*(1+r)^365 . (eqn 1)
Rearrange that and you get (1+r)^365 = 110/100=1.1
then (1+r)=1.1^(1/365) = 1.000261158
so r= 0.000261158. This is your daily interest rate (0.0261158%)

To calculate value on any day, using eqn 1:
For 100 days
FV=100*(1+0.000261158)^100 = 100x1.026456306 = $102.6456306.

To test, use 365 days
FV=100*(1.000261158)^365 = $110

So, your formula for Future value d days from start:
FV=SV*1.000261158^d


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## crooked beat (Jan 19, 2011)

This might be helpful Effective interest rate - Wikipedia


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## Steve64 (Jun 28, 2016)

MrBlackhill said:


> Dividing 10% by 365 leads to an approximation, but I think most banks use that approximation. Maybe they use the approximation when it's in their favour and otherwise calculate the true daily interest.
> 
> 10% yearly divided by 365 equals about 0.0274% daily, which will lead to a bit more than 10% yearly once compounded (10.5156%). The exact daily interest would be 0.0261% for a 10% yearly interest.
> 
> ...


thanks this is very helpful!


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## Steve64 (Jun 28, 2016)

agent99 said:


> I will have a try.
> Let SV = starting value (assume $100), FV = final value($110), r=interest rate, d=days
> 
> FV=SV*(1+r)^d or 110=100*(1+r)^365 . (eqn 1)
> ...


thanks, the calculator in the previous example is exactly what i'm looking for.


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## agent99 (Sep 11, 2013)

Steve64 said:


> thanks, the calculator in the previous example is exactly what i'm looking for.


Great! - Certainly easier to use than using my Math, but is does give slightly inaccurate results. Could be a factor if amounts are large.


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## MrBlackhill (Jun 10, 2020)

agent99 said:


> Great! - Certainly easier to use than using my Math, but is does give slightly inaccurate results. Could be a factor if amounts are large.


I think banks uses the same inaccurate calculation because it's on their favour since loans have higher interests rate than savings accounts.


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## MrBlackhill (Jun 10, 2020)

I just calculated my effective rate for my mortgage and I can confirm that banks uses the approximation.

Mortgages in Canada are compounded semi-annually with the simple method. I have an annual rate of 2.99% but my effective rate is 3.012%.

Also, loans are compounded monthly. That means if you take a 10 000$ loan at 5% which you pay monthly for 60 months, your effective rate will be 5.116%.


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## agent99 (Sep 11, 2013)

The question Steve asked in post #1 was about an investment that would yield exactly 10% in one year. He wanted to know what it would be worth on any day before maturity. No room for approximations if yield had to be 10%. I took it as a simple math question.


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## andrewf (Mar 1, 2010)

For n days, isn't the answer (1+i)^(n/365)? So for i=10% and n = 100 days, it simplifies to (1.1)^(100/365)=~1.0265, or a 2.65% return? Casting back to my actuarial science classes (though I think I learned it first in high school)...

Really, there are a few different ways time value of money are calculated that give you nearly the same answer. I don't think your average retail investor needs to worry about the precise amount.


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## MrBlackhill (Jun 10, 2020)

agent99 said:


> The question Steve asked in post #1 was about an investment that would yield exactly 10% in one year. He wanted to know what it would be worth on any day before maturity. No room for approximations if yield had to be 10%. I took it as a simple math question.


Good point.

@Steve64 if you still want to use an online calculator you would have to input 9.5323% as the annual interest rate that leads to exactly 10% after 365 days for financial products.



andrewf said:


> For n days, isn't the answer (1+i)^(n/365)? So for i=10% and n = 100 days, it simplifies to (1.1)^(100/365)=~1.0265, or a 2.65% return? Casting back to my actuarial science classes (though I think I learned it first in high school)...
> 
> Really, there are a few different ways time value of money are calculated that give you nearly the same answer. I don't think your average retail investor needs to worry about the precise amount.


Yes.

Just to clarify :


Topo provided the formula for financial products which have an effective annual rate different to their nominal annual rate depending on compounding frequency
crooked beat provided a link explaining that difference
I provided a calculator but most online calculators are using the formulas for financial products, I also mentioned the effective rate and what would be the daily rate for an exact 10%
agent99 & andrewf provided the formula for daily compounding that would lead to exactly 10% after 365 days of compounding

Since I guess you won't be getting 10% from a financial product, I guess you are setting a target of 10% from investments in the market, so agent99 & andrewf formula makes more sense in that case. I am not sure why you are looking to calculate the daily return and the returns after 100 and 200 days since you will certainly be off target due to volatility, but maybe you just want to check periodically if you are above or below target. It could make sense in low-volatility investments, but again there's no low-volatility investment yielding 10%...


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## Topo (Aug 31, 2019)

The result depends on exactly how the 10% rate is compounded. It makes sense to assume it is compounded daily since OP mentions "100 days" and "200 days", but one could calculate returns for those periods even if the 10% is compounded annually.


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## Steve64 (Jun 28, 2016)

MrBlackhill said:


> Good point.
> 
> @Steve64 if you still want to use an online calculator you would have to input 9.5323% as the annual interest rate that leads to exactly 10% after 365 days for financial products.
> 
> ...


I see how this is evolving into something more complicated than i originally expected... ha. i really appreciate the thought everyone is putting into this.
What i intend to do is use 10% / 365 days = 0.0274% so on a $100k principle the "daily growth " would be $27.40. Creating a spreadsheet with $100k growing by $27.40 each day for 365 days will give me the final growth of 10% ($10k) I was looking for. I'm going to use this as a base line to compare the actual daily #'s to. I think you can see what i'm trying to do with this description - so if you see any problems with the logic (or math) please let me know.

thanks again everyone.


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## MrBlackhill (Jun 10, 2020)

Steve64 said:


> I see how this is evolving into something more complicated than i originally expected... ha. i really appreciate the thought everyone is putting into this.
> What i intend to do is use 10% / 365 days = 0.0274% so on a $100k principle the "daily growth " would be $27.40. Creating a spreadsheet with $100k growing by $27.40 each day for 365 days will give me the final growth of 10% ($10k) I was looking for. I'm going to use this as a base line to compare the actual daily #'s to. I think you can see what i'm trying to do with this description - so if you see any problems with the logic (or math) please let me know.


Ok, so if you want $100k to be worth exactly $110k (+10%) after 365 days, then the daily increase has to be 0.0261157876% and not 0.0274%. That means $26.12 after the first day.

But as you may know, that doesn't mean a daily growth of $26.12. That's only for the first day because then your investment will be worth $100 026.12 and the interests on that new value is a little bit more than $26.12 and so on. That's the power of compounding.


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## MrBlackhill (Jun 10, 2020)

If you want to do a spreadsheet, here's a few calculation examples


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## Steve64 (Jun 28, 2016)

thanks Guys, i will rethink my daily increment adjustment. I was looking for a single value but this may not serve my needs as well as your suggestions.


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## agent99 (Sep 11, 2013)

I think this thread has run it's course.

One question, would be why would anyone want to know the value of an annual investment yielding 10% on a daily basis? Second question, would be where do you find one of those?

Mind you, some brokerages quote daily values for GICs with 2-5 year terms. Not sure how they do the calculation.


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## MrBlackhill (Jun 10, 2020)

Steve64 said:


> thanks Guys, i will rethink my daily increment adjustment. I was looking for a single value but this may not serve my needs as well as your suggestions.


A single value for daily increase is not a compounding interest, it's a simple interest.

Compounded interests means they grow at an exponential rate (blue curve), which means each day the interests will be greater than the previous day. That's because compounding means you multiply by a value every day instead of adding a value every day. If you use a single additive value for the daily interests, then the increase is linear (red line).










If you really want to add a single value on a daily basis but still be compounding, then you would have to work in a logarithmic scale, but I don't think you want to go there... In a logarithmic scale, your initial value of $100k would become a value of 5 and the single value additive daily interests would be 0.0001340461687. Logarithms convert multiplication into addition, that's how cool they are...


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## andrewf (Mar 1, 2010)

By its very nature, compounding means you don't have a fixed daily increase. You can get a simple daily rate of return that is equivalent...

I can't help but think this is getting massively over-engineered for the intended purpose (some simple benchmarking, I imagine).


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## Steve64 (Jun 28, 2016)

MrBlackhill said:


> Ok, so if you want $100k to be worth exactly $110k (+10%) after 365 days, then the daily increase has to be 0.0261157876% and not 0.0274%. That means $26.12 after the first day.
> 
> But as you may know, that doesn't mean a daily growth of $26.12. That's only for the first day because then your investment will be worth $100 026.12 and the interests on that new value is a little bit more than $26.12 and so on. That's the power of compounding.
> 
> View attachment 20610


It took a while for me to get my head around what you are saying above MrBlackhill but i think i have the jest.... it seems to me the formula you used to get the 0.0261157876% is missing? Can you share this with me please? i'd like to know how to arrive at the daily interest amt for any return rate i choose (10%, 5% .... etc). It's surprising to me how fast this gets complicated - though i get the compounding effect is what's causing this.

Thanks in advance


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## Steve64 (Jun 28, 2016)

MrBlackhill said:


> A single value for daily increase is not a compounding interest, it's a simple interest.
> 
> Compounded interests means they grow at an exponential rate (blue curve), which means each day the interests will be greater than the previous day. That's because compounding means you multiply by a value every day instead of adding a value every day. If you use a single additive value for the daily interests, then the increase is linear (red line).
> 
> ...


I understand what you are saying about the Blue and Red curves. For my purpose though i want to know what i would add to the principle (each day) to arrive at the $110,000 value . I'm not sure i need to have the principle compounded (for my purpose) I want to know that on day 5 the principle + interest would be "x" based on an annual rate of 10%, on day 200 the principle + interest would be "x" based on an annual rate of 10% - know what i mean?

Another way to say this is if you gave me 100,000 and i told you i would give you back 110,000 in 1 year that would reflect a 10% ROI (return on investment), If you said you wanted it back on day 100 i would give you back $100,000 plus the interest you accumulated based on a 365 day return of 10,000. This is where I get confused. This is where having a static number i plug in each day for a grand total of $10,000 after 365 days would work as opposed to a variable number that also equals $10,000 after 365 days. If you're red line ended up exactly where your blue line ends up (10,000) that would mean the red line (simple interest) will always be a higher value than the compounded interest - though both ending up at the same value on day 365. I understand this. Are you able to show me a simple way to calculate the daily "static" number for simple interest to arrive at 110,000 after 365 days?


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## agent99 (Sep 11, 2013)

Steve64 said:


> It took a while for me to get my head around what you are saying above MrBlackhill but i think i have the jest.... it seems to me the formula you used to get the 0.0261157876% is missing?


I guess you did not read my initial post. One of the first things I did, was provide a formula that would allow you calculate the interest rate.



> V=SV*(1+r)^d or 110=100*(1+r)^365 . (eqn 1)
> Rearrange that and you get (1+r)^365 = 110/100=1.1
> then (1+r)=1.1^(1/365) = 1.000261158
> so r= 0.000261158. This is your daily interest rate (0.0261158%)


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## MrBlackhill (Jun 10, 2020)

Steve64 said:


> Are you able to show me a simple way to calculate the daily "static" number for simple interest to arrive at 110,000 after 365 days?


A simple way to calculate the daily "static" number for simple interest to arrive at 110,000 after 365 days starting at 100,000?

Sure, that would be $27.40 daily for simple interests (linear).

Take 110,000 - 100,000 = 10,000
Then 10,000 / 365 = 27.40

But that's not compounding.

Or maybe you want to do yearly compounding interests but daily simple interests. That means on year 2, when you have 110,000, your goal for 10% growth would be 121,000 and the daily simple interest would be $30.14. And on year 3, when you have 121,000, your goal for 10% growth would be 133,100 and the daily simple interest would be $33.15. The maximum tracking error would be 0.1136% of the total amount, in favour of the simple interests.


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## Steve64 (Jun 28, 2016)

These posts are causing me to think hard about the Linear vs Compound calculations and what exactly i'm trying to do here. Thanks for the perspectives everyone!


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