Basic annual compounding

[nevaco]

I'm not a student anymore, I'm just trying to do some personal finance planning/budgeting.

I want to know how long I have to wait until my investiment reaches a certain goal. Besides the principal amount and annual interest rate, there are monthly additions to the investiment.

I know that the general formula is like this:

[ P(1+r/n)^nt ] + [ PMT * (((1 + r/n)^nt - 1) / (r/n)) ]

Where:

A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
PMT = the monthly payment
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

so assuming the an annual compounding for the interests, but monthly additions I come up with

[ P(1+r)^t ] + [ PMT * (((1 + r/12)^12t - 1) / (r/12)) ]

Now what I would really like is to solve this on 't'.

The steps I did so far where:

[ -P(1+r)^t ] / (r/12) = PMT * (1+r/12)^12t -1

1 = PMT (1+r/12)^12t + (12P (1+r)^t) / r

but from this point I'm stuck... I can't remember how to do it. Any help is appreciated.

 

[stapel]

I want to know how long I have to wait until my investiment reaches a certain goal. Besides the principal amount and annual interest rate, there are monthly additions to the investiment.

I know that the general formula is like this:

. . . . .\(\displaystyle FV\, =\, \left[\, P\, \times\, \left(1\, +\, \dfrac{r}{n}\right)^{nt}\, \right]\, +\, \left[\, PMT\, \times\, \left(\, \dfrac{\left(1\, +\, \dfrac{r}{n}\right)^{nt}\, -\, 1}{\left(\dfrac{r}{n}\right)} \, \right) \, \right]\)

It can be hard to get nested parentheses to mean what you want. I've typeset, in the above, what I'm assuming is the correct expression for the future value FV. If I've made a mistake, kindly please reply with corrections.

By the way: I haven't checked your formula. You can confirm (or correct) from sources online (such as here).

Where:

A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
PMT = the monthly payment
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

so assuming the an annual compounding for the interests, but monthly additions I come up with

. . . . .\(\displaystyle \left[\, P\, (1\, +\, r)^t\, \right]\, +\, \left[\, PMT\, \times\, \left(\, \dfrac{\left(1\, +\, \dfrac{r}{12}\right)^{12t}\, -\, 1}{\left(\dfrac{r}{12}\right)}\, \right) \, \right]\)

Why is the interest rate "r" not divided by 12 in the first term, but is divided by 12 in the second term? What is the difference? (Isn't the interest, though "annual", awarded to the account's balance on a monthly basis?)

Now what I would really like is to solve this [for] 't'.

The steps I did so far where:

. . . . .\(\displaystyle \dfrac{-P\, (1\, +\, r)^t}{\left(\dfrac{r}{12}\right)}\, =\, PMT\, \times\, \left(\left(1\, +\, \dfrac{r}{12}\right)^{12t}\, -\, 1\right)\)

I'm not sure how you got to this point, since you'd started with something that had no "equals" sign in it. From whence did the "equals" in the last line above come? To what had you set the previous expression equal? Also, how did the denominator of the second term (in the previous expression) convert to being the denominator of the term on the left-hand side (of the equation; what had previously been the first term in the previous expression)?

Thank you!

 

[Ishuda]

I'm not a student anymore, I'm just trying to do some personal finance planning/budgeting.

I want to know how long I have to wait until my investiment reaches a certain goal. Besides the principal amount and annual interest rate, there are monthly additions to the investiment.

I know that the general formula is like this:

[ P(1+r/n)^nt ] + [ PMT * (((1 + r/n)^nt - 1) / (r/n)) ]

Where:

A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
PMT = the monthly payment
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

so assuming the an annual compounding for the interests, but monthly additions ...

As indicated by stapel, you are really talking about two interest rates. For the initial payment P, n=1 and t=5. However, for the PMT part of the equation, you must use an interest rate r' such that you get an effective annual interest rate of r unless you are changing the interest rate to r/12 compounded monthly.

As an example, assume an annual interest rate r of 12% paid annually, a PMT=1000 and consider what the first additional payment would be at the end of five years [yeah, I know your problem only has a time of 59 months until the end of the initial 5 years]. If it were paid annually, you would have about $1762.34 at the end of five years. However, if it were an interest rate of r'=1% compounded monthly (=r/12) you would have 1000*1.01⁶⁰ ~ $1816.70 or about $54.36 more. If instead you had changed the r' to
r' = 1.12^(1/12) - 1 ~ 0.94888%
then you would have 1000*(1.0094888)⁶⁰ ~ $1762.34 which is what you should have for 12% compounded yearly.

BTW: The last part of that equation is actually a sum for the months you invested the additional payment. The first additional payment only earns interest for 59 months, the second additional payment for 58 months, the third for 57 months, etc.

 

[nevaco]

Thanks so much for all the inputs so far.

Why don't you stick to annual deposits/compounding
for now...as with your other post...
Once you got that straight, you can look at different
compounding frequencies...which really doesn't seem
to matter much in your case, according to your other post...

Thanks for your reply on the other topic. I already replied there. I started with annual compounding because I didn't know how to do monthly compounding. I discover that after posting that other topic.
As I will anyway add it on a monthly basis it would be nice if I could measure it with monthly compounding.

By the way: I haven't checked your formula. You can confirm (or correct) from sources online (such as here).

That's exactly what I meant.

Why is the interest rate "r" not divided by 12 in the first term, but is divided by 12 in the second term? What is the difference? (Isn't the interest, though "annual", awarded to the account's balance on a monthly basis?)

Perhaps the formula does not represent exactly what I want: I would want the interest (any interest) to be awarded annually, but to take into account monthly additions. Is it clear? (I think the formula considers that interest of monthly additions is paid monthly... that's not exactly what I want)

I'm not sure how you got to this point, since you'd started with something that had no "equals" sign in it. From whence did the "equals" in the last line above come? To what had you set the previous expression equal? Also, how did the denominator of the second term (in the previous expression) convert to being the denominator of the term on the left-hand side (of the equation; what had previously been the first term in the previous expression)?

well, now I see I totally forgot FV (future value) and my steps were totally wrong, but anyway, my problem is much the same:

by replacing n as I did before (which may not be totally correct)

FV = [ P(1+r)^t ] + [ PMT * (((1 + r/12)^12t - 1) / (r/12)) ]

FV - [ P(1+r)^t ] = [ PMT * (((1 + r/12)^12t - 1) / (r/12)) ]

(r/12) * ( FV - [ P(1+r)^t ] ) = PMT * ((1 + r/12)^12t - 1)

r/12 * FV - r/12 * [ P(1+r)^t ] = PMT * (1 + r/12)^12t - PMT

PMT * (1 + r/12)^12t + r/12 * [ P(1+r)^t ] = r/12 * FV + PMT

but now what?

ps:

I know that x^t = c <=> t = Log c / Log x,
but I'm almost sure that x^t + x^12t = c does not correspond to t = Log c / (12 Log x + Log x)