My Math Forum Value of an annuity in any given month

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 September 6th, 2011, 09:15 AM #1 Newbie   Joined: Sep 2011 Posts: 4 Thanks: 0 Value of an annuity in any given month I have an annuity where $2,000 is being invested at the beginning of every year for 20 years and the interest is compounded monthly. I want to find out for any given month what the balance of the annuity is. Is there a formula that will allow this? Example: I invest$2,000 at the start of every year for 20 years. The annual interest rate over the life of the annuity is 9%, compounded monthly. I want to be able to find out the balance at month 15, 30, 45, 60, etc. Thank you very much in advance for any help you can offer. It is much appreciated!
 September 6th, 2011, 09:49 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Value of an annuity in any given month You can use this in reverse, with some care: http://en.wikipedia.org/wiki/Mortgage_c ... nt_formula I'm sure you can find the formula written "forward" if you search for it.
 September 6th, 2011, 09:53 AM #3 Newbie   Joined: Sep 2011 Posts: 4 Thanks: 0 Re: Value of an annuity in any given month This is a great start. I'll give it a try. Thanks!
 September 6th, 2011, 10:05 AM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Value of an annuity in any given month The formula can also be derived as a geometric series, but that may be more difficult. (It's probably the easiest way for me, but ymmv.)
 September 7th, 2011, 12:39 PM #5 Newbie   Joined: Sep 2011 Posts: 4 Thanks: 0 Re: Value of an annuity in any given month I came across this link and the "Calculating the Future Value of an Annuity Due" sections seems to outline what I am looking to do. The only issue is that I do not know how to modify the formula to account for different payment and compounding periods (annual payments, monthly compounding). Every formula I find assumes the payment and compounding periods are the same. http://www.investopedia.com/articles...#ixzz1XIgPHZ6Y I took your geometric series suggestion and found the link below. At the very end under the "A Common Application" section they given an example of how to calculate the annuity value assuming the deposit cycle and the compounding cycle are the same. http://fym.la.asu.edu/~tturner/MAT_1..._Sequences.htm How can I modify these formulas to account for different payment and compounding periods?
September 7th, 2011, 01:43 PM   #6
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Re: Value of an annuity in any given month

Quote:
 Originally Posted by Business School Guy How can I modify these formulas to account for different payment and compounding periods?
You're letting a fixed (integer) number of compounding periods pass between investing, right? Then just find the equivalent interest over that whole period and you can use the effective interest rate instead. If over 10 years you go from $P to$P * k, then the interest rate per ten-year period is k-1.

 September 8th, 2011, 07:33 PM #7 Senior Member   Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: Value of an annuity in any given month The problem with the future value annuity equation is that this situation isn't the type of annuity for which this equation applies. It only works when the payments match the compounding periods, so you can't use it, nor am I aware of a way to "adjust" it to work. My degree's in accounting and we did a fair amount of time value of money, but I've only had one finance class (and no banking/investment classes or anything of the sort), so there may be some special way to do this that I'm not aware of. We did problems like this, but we used a combination of equations, and just good old addition/subtraction. If there were another equation, not sure why we wouldn't have learned it, unless the math behind it was considered too advanced. The following is how I would solve something like this, and if it were personal for me, I'd stick it into Excel and make some type of input section so that I could take it to any time period. The equation for just a future value (no payments) is: $\text{FV= PV(1 + i)^n}$ I noticed on the one site they quote i to be the "interest rate." Actually, i is the interest per compounding period, which is an important point to note. That is, i in this case is .09/12, or .0075. This equation does not account for making more payments. It's a lump sum equation; that is, you stick $2000 in at the beginning of the year, and what's it worth after n periods, but without adding any more payments to it. So I would use this to get the value of the thing after one year, since you'd be changing it after a year: $\text{FV= 2000 (1 + \frac{.09}{12})^{12}}$ $\text{FV= 2187.61}$ That gives you the value after the first year. At that point, you're adding another$2000 to it. So I would just add it: $\text{2187.61 + 2000= 4187.61}$ Now you have a value for the beginning of the second year. And you can repeat this with your new number: $\text{FV= 4187.61 (1 + \frac{.09}{12})^{12}}$ $\text{FV= 4580.44}$ Repeat ad nauseum. If you want to know what it is after, say, 15 months, just adjust the n in the second one: $\text{FV= 4187.61 (1 + \frac{.09}{12})^3}$ Because I can handle Excel fairly well, I'm sure I could find a way to put it in there. I'm also sure there's some fancy dancy equation that could take care of this, without manually adding the new payment each year. As I said, this is how we would do problems like this, and perhaps the math to do it as one equation was beyond our level. Perhaps someone here can take what I've done and turn it into one nice equation. It's also possible that CRGreathouse's method would work, but I don't see how. The effective rate stays the same on an annual basis. But I don't know how you can do this between years, like at 15 months. Perhaps if he used these numbers as example to demonstrate that?
September 9th, 2011, 07:55 AM   #8
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Re: Value of an annuity in any given month

Quote:
 Originally Posted by Erimess It's also possible that CRGreathouse's method would work, but I don't see how. The effective rate stays the same on an annual basis. But I don't know how you can do this between years, like at 15 months. Perhaps if he used these numbers as example to demonstrate that?
Using BSG's example, the yearly interest is (1 + .09/12)^12 = 9.38...%, so use that rate and a period of 1 year with the standard formula. That only tells you what happens at the end of each year, so if you want, e.g. 15 months, use the formula for one year and then do a standard interest calculation for 3 months.

September 9th, 2011, 08:49 PM   #9
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Re: Value of an annuity in any given month

Quote:
 Originally Posted by CRGreathouse Using BSG's example, the yearly interest is (1 + .09/12)^12 = 9.38...%, so use that rate and a period of 1 year with the standard formula.
I had to play with this for about 20 minutes to figure this out. When I said using his example, I kind of meant the whole thing. I know how to do the effective interest rate. I wasn't even sure which equation you were referring to.

Future value annuity equation:

$\text{FV=Pmt \left(\frac{(1+i)^n-1}i\right)}$

Then plug in the effective interest rate (the 9.38%) where the i is. This can only work if you do it for a full year. Also use it for the i in the denominator as well:

$\text{FV=2000\,\left(\frac{(1+.093^2-1}{.0938}\right)}" />

I'm going to jump straight to 2 periods, the value of which was 4580.44 as we saw.

If you work that out, it comes out to 4187.60. You still have to do the annuity due adjustment, still using the effective interest rate: 4187.60 (1.093 = 4580.40.

If you wanted the value in the middle of a year, you have to remember to add the next payment on first, and then proceed how I said in the other post, with the other equation for however many months you want.

I also worked it with 3 periods to check it. That comes out to 6580.40 * 1.0938 for the adjustment = 7197.64.

Etc.

Very interesting. I wonder why we never learned this in finance. I don't know how the annuity equations are derived, but don't need to know, so I wouldn't consider that the math was beyond us. I'll have to play with that and figure out how to use it for various combinations of compounding and payments.

I still don't get the k - 1 thing. k in finance stands for interest rate, so that's like saying 9% - 1, or 9.38% - 1. Don't know where that fits in.

 September 10th, 2011, 04:18 PM #10 Newbie   Joined: Sep 2011 Posts: 4 Thanks: 0 Re: Value of an annuity in any given month CRG and Erimess, I was thinking about this in the shower and I thought of the exact same approach that you guys are coming up with. I am glad to see that you guys thought it would work also! Basic formula outline will be something like this: IF Months Since Start of Extra Payments <= 12 THEN (Annual Pmt) + (Interest Earned Over X Months) ELSE IF Months Since Start of Extra Payments > 12 THEN 'To figure out the # of years passed as of month X - for FV formula Excel formula FLOOR(Months Since Start of Extra Payments/12,1) 'To figure out the # of months passed from the new year as of month X - for FV formula Months - Excel formula FLOOR(Months Since Start of Extra Payments/12,1) ELSE 0 END IF So if the month was 34 the FLOOR formula would tell me that 2 full years have passed (34/12 = 2.8333 with the number rounded down to the nearest number). 34 - (12*2) = 10 months of compounding in the current year. I could then use these figures in the FV formulas to calculate the value of the payments. I could do this very easily in Excel, but our teacher has challenged us to do this without using many of the Excel financial functions. He has also placed certain other restrictions in place, such as not making a straight monthly payment schedule and manually adding in the interest and annual payments. I am forced to take a different approach and I think this will work. I am going to try it out and I'll let you know how it goes. Thanks a ton for the help!

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