Present Value of An Annuity

simplequestion · #1 $Old$ 10-07-2008, 02:02 PM

Hello. I have been banging my head on the wall with this problem and would appreciate some help.
I know the general formula for calculating the present value of an annuity (due or ordinary) can be found here:
Present Value of an Annuity and I completely understand everything on that page which is assuming that interest is compounded annually.
Where I run into trouble is to change the assumption that interest is compounded, say, daily. I would think it is as easy as saying i = rate/365 and n = number_of_years * 365. However, using those numbers the formula does not work out to a realistic answer.
Can somebody help? Thanks!

TKHunny · #2 $Old$ 10-08-2008, 11:33 AM

You must understand the compounding or you cannot work such problems. Sometimes you can just divide the rate and get the right thing. Have you a specific problem you are workign on? Let's see where you are wandering off.

simplequestion · #3 $Old$ 10-09-2008, 07:08 AM

I am trying to figure out the formula that is used for places like this:

How much do I need to fund my retirement?

How much do I need to fund my retirement?

Which I understand use the present value of an annuity formula I referenced in my first post but I am not sure what they used for "i" and "n" in the formula. Thanks for any help you can provide.

TKHunny · #4 $Old$ 10-09-2008, 07:25 AM

Sorry, I'm not clicking on any external links.

Formulas work the way they are coded. You must read and understand the instructions. If there is a specific instruction with which you are struggling, feel free to present it here for explanation.

It makes a difference how you intend to accumulate your deposits and you your investment credits interest. Definitions can be very specific. You must hunt them down if you really want the right answer. There are not any shortcuts unless a close-enough answer is sufficient.

simplequestion · #5 $Old$ 10-09-2008, 07:50 AM

I can appreciate not wanting to click on any external links. Maybe I can explain it well enough here.

We have the formula for figuring out the present value of an annuity here:
PVoa = PMT [(1 - (1 / (1 + i)n)) / i]

I think it is straight forward when the interest is comounded annually and the withdrawals are done on an annual basis. Where I am getting confused is what values to use for "i" and "n" if the interest is compounded daily and the withdrawals are done on a monthly or semi-annual basis. What numbers do we plug in for "i" and "n" in that case?

Thanks again.

TKHunny · #6 $Old$ 10-09-2008, 11:34 AM

Now we're talkin'. The notation is a little rough, but I think I have it.

You must be careful with the interest rates, but even a good guess will get you in the ballpark. Generally, you are given an annual rate. It should be called "APR" or "Nominal". If this is so, there is no problem using a variant of this formula.

Here's the general version:

$\frac{1 - \left(\frac{1}{1+\frac{i}{m}}\right)^{n*m}}{\frac{i}{m}}$

n = Number of Years
m = Number of Periods per year
i = Annual Nominal Interest Rate

It's an easy enough cut-and-past task. I'll just type in a bunch...

Annual: m = 1 -- After a little algebra, this should look familiar.

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

Semi-Annual: m = 2

$\frac{1 - \left(\frac{1}{1+\frac{i}{2}}\right)^{n*2}}{\frac{i}{2}}$

Quarterly: m = 4

$\frac{1 - \left(\frac{1}{1+\frac{i}{4}}\right)^{n*4}}{\frac{i}{4}}$

Monthly: m = 12

$\frac{1 - \left(\frac{1}{1+\frac{i}{12}}\right)^{n*12}}{\frac{i}{12}}$

Weekly: m = 52

$\frac{1 - \left(\frac{1}{1+\frac{i}{52}}\right)^{n*52}}{\frac{i}{52}}$

Daily: m = 365 (You may want 360 or 366, sometimes.)

$\frac{1 - \left(\frac{1}{1+\frac{i}{365}}\right)^{n*365}}{\frac{i}{365}}$