Calculating daily compound interest with increasing APY

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Lipton

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Say I have saved amount of money (e.g. $1) that earns a dividends which increases at a steady rate per day.
e.g. First day 0.01 $ earned per 1$ dollar saved, 2nd day $0.02 earned per $1 saved, 3rd $0.03 earned per $1 saved.

How can I calculate the total amount earned from the beginning until now using 1 formula without just totalling up the amount earned per day?
 
Lipton said:
Say I have saved amount of money (e.g. $1) that earns a dividends which increases at a steady rate per day.
e.g. First day 0.01 $ earned per 1$ dollar saved, 2nd day $0.02 earned per $1 saved, 3rd $0.03 earned per $1 saved.

How can I calculate the total amount earned from the beginning until now using 1 formula without just totalling up the amount earned per day?
Click to expand...
Do you know the formula for compound interest?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
I can find the formula for compound interest online, but its based on a constant interest rate. My version is that the APY rate would increase slightly every day at a constant rate.
This is for a personal programming task I am working on, I didn't want to post code as it is a math forum and not everyone would know the programming language I am using.

Basically the interest per day can be simply calculated by:

(start rate + (rate increase per day * days elapsed)) * $amount

but I'm trying to find a single formula to efficiently calculate the total interest since the first day without looping through and totalling the calculations every day.
 
Beer induced query follows.
Lipton said:
Say I have saved amount of money (e.g. $1) that earns a dividends which increases at a steady rate per day.
e.g. First day 0.01 $ earned per 1$ dollar saved, 2nd day $0.02 earned per $1 saved, 3rd $0.03 earned per $1 saved.

How can I calculate the total amount earned from the beginning until now using 1 formula without just totalling up the amount earned per day?
Click to expand...
Could you post a solid example for a period of say 10 days?
 
jonah2.0 said:
Beer induced query follows.

Could you post a solid example for a period of say 10 days?
Click to expand...
These values in this example will always be constant:
- initial amount $100
- start rate 0.01
- increase per day 0.001

First 10 days:
(0.01 + (0.001 * 0)) *100 = 1
(0.01 + (0.001 * 1)) *100 = 1.1
(0.01 + (0.001 * 2)) *100 = 1.2
(0.01 + (0.001 * 3)) *100 = 1.3
(0.01 + (0.001 * 4)) *100 = 1.4
(0.01 + (0.001 * 5)) *100 = 1.5
(0.01 + (0.001 * 6)) *100 = 1.6
(0.01 + (0.001 * 7)) *100 = 1.7
(0.01 + (0.001 * 8)) *100 = 1.8
(0.01 + (0.001 * 9)) *100 = 1.9

So the total earned after 10 days would be 14.5.
How could I express this as a formula, so I could calculate any arbitrary day e.g. day 45.
 
The formulae for compound interest will not apply to this more complicated arrangement.
Interest on successive days is 1%, 2%, 3% ........ So the amount is multiplied, in succession, by 1.01, then 1.02, 1.03 ....... What type of series do 1.01, 1.02, 1.03 ........ form? What process do you apply to these terms to calculate the final amount?
 
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Lipton said:
These values in this example will always be constant:
- initial amount $100
- start rate 0.01
- increase per day 0.001

First 10 days:
(0.01 + (0.001 * 0)) *100 = 1
(0.01 + (0.001 * 1)) *100 = 1.1
(0.01 + (0.001 * 2)) *100 = 1.2
(0.01 + (0.001 * 3)) *100 = 1.3
(0.01 + (0.001 * 4)) *100 = 1.4
(0.01 + (0.001 * 5)) *100 = 1.5
(0.01 + (0.001 * 6)) *100 = 1.6
(0.01 + (0.001 * 7)) *100 = 1.7
(0.01 + (0.001 * 8)) *100 = 1.8
(0.01 + (0.001 * 9)) *100 = 1.9

So the total earned after 10 days would be 14.5.
How could I express this as a formula, so I could calculate any arbitrary day e.g. day 45.
Click to expand...
In your example above, the APY is changing, but the interest is NOT compounding - i.e.

the interest earned in previous period is NOT added to the principle to be used for calculating interest for next period.

Thus, it is a variation of "simple interest" calculation.
 
The mathematical answer is, for a month of 30 days, 465%.

[math]\dfrac{1}{100} * \dfrac{30 * 31}{2} = 4.65 = 465 \%.[/math]
Good luck with that investment. Available through a guy named Madoff by any chance?
 
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Subhotosh Khan said:
In your example above, the APY is changing, but the interest is NOT compounding - i.e.

the interest earned in previous period is NOT added to the principle to be used for calculating interest for next period.

Thus, it is a variation of "simple interest" calculation.
Click to expand...
Yes that is true, the initial investment that earns interest per day will stay the same. As you can see I can calculate the simple interest for each individual day no problem

I need to see if it is possible to express the amount of interest earned up until any day in the future using these variables:

-investment amount
-start rate
-rate increase per day
-number of days invested
 
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JeffM said:
The mathematical answer is, for a month of 30 days, 465%.

[math]\dfrac{1}{100} * \dfrac{30 * 31}{2} = 4.65 = 465 \%.[/math]
Good luck with that investment. Available through a guy named Madoff by any chance?
Click to expand...
Yeh it does sound like a ponzi if it was a real investment.
Your answer may work for the example values I gave. But I need an algebraic expression that would work if the investment amount, start rate, rate increase per day are different.
 
Lipton said:
These values in this example will always be constant:
- initial amount $100
- start rate 0.01
- increase per day 0.001

First 10 days:
(0.01 + (0.001 * 0)) *100 = 1
(0.01 + (0.001 * 1)) *100 = 1.1
(0.01 + (0.001 * 2)) *100 = 1.2
(0.01 + (0.001 * 3)) *100 = 1.3
(0.01 + (0.001 * 4)) *100 = 1.4
(0.01 + (0.001 * 5)) *100 = 1.5
(0.01 + (0.001 * 6)) *100 = 1.6
(0.01 + (0.001 * 7)) *100 = 1.7
(0.01 + (0.001 * 8)) *100 = 1.8
(0.01 + (0.001 * 9)) *100 = 1.9

So the total earned after 10 days would be 14.5.
How could I express this as a formula, so I could calculate any arbitrary day e.g. day 45.
Click to expand...
- initial amount = P

- start rate = I

- increase per day = α

Total amount after n days = P * \(\displaystyle \sum_{k=0}^{n}\left(I+k*α\right)\)​

Replace the "summation" with appropriate equation from "arithmetic progression series"

continue....
 
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