# Help me figure out a finance equation

October 26, 2011 6:13 PM Subscribe

Please help me with a compound interest question (specifically with IRAs)

I'm writing an article about IRAs and when to make contributions, and I'm trying to compare someone who contributes $5,000 every year, right at the start of the year, versus someone who waits until April 15 of the next year to make a contribution of $5,000 for the previous year.

I understand how to compare someone who makes contributions on January 1 against someone who makes contributions on December 31 of the same year, but what do I do for the person who waits until April 15 of the next year?

I would like to know the actual math, not just the answer. But for illustrative purposes, let's assume each guy contributes $5,000 per year from age 25 to 67 and has a 10% interest rate (yeah, I wish) that compounds monthly. What would the difference be? What about if it only compounds annually?

I'm writing an article about IRAs and when to make contributions, and I'm trying to compare someone who contributes $5,000 every year, right at the start of the year, versus someone who waits until April 15 of the next year to make a contribution of $5,000 for the previous year.

I understand how to compare someone who makes contributions on January 1 against someone who makes contributions on December 31 of the same year, but what do I do for the person who waits until April 15 of the next year?

I would like to know the actual math, not just the answer. But for illustrative purposes, let's assume each guy contributes $5,000 per year from age 25 to 67 and has a 10% interest rate (yeah, I wish) that compounds monthly. What would the difference be? What about if it only compounds annually?

Assuming constant returns and the same number of payment, the only difference is that you will have the amount of money you otherwise would have one year, four months, and 15 days earlier.

All you're doing is moving each cash flow forward by some amount of time, assuming you're making the same number of payments.

posted by cupcake1337 at 6:35 PM on October 26, 2011 [1 favorite]

All you're doing is moving each cash flow forward by some amount of time, assuming you're making the same number of payments.

posted by cupcake1337 at 6:35 PM on October 26, 2011 [1 favorite]

Surely the difference is only a one-off difference the first year, because the first guy puts his first $5000 in a lot earlier than the other guy. That difference will compound, but it is equivalent to putting into a compound interest calculator a one-off payment of $5000 for the number of months that is different between the two, seeing what the final total is, and then putting that total into the calculator for the number of years left until age 67. The later contributions are irrelevant when working out the difference, I think.

(IANAMG: I am not a maths geek.)

posted by lollusc at 6:36 PM on October 26, 2011

(IANAMG: I am not a maths geek.)

posted by lollusc at 6:36 PM on October 26, 2011

On (not) previewing: I was assuming the number of payments was not the same, unlike cupcake1337. The answer will be different if the delayed guy pays in his final payment after turning 67.

posted by lollusc at 6:38 PM on October 26, 2011

posted by lollusc at 6:38 PM on October 26, 2011

Your question is awkward, because I don't know of any assets that can be held in an IRA that actually do have annual or monthly interest. All assets are continually priced such that they include any accrued interest, since otherwise you could continually make money by buying assets right before they pay interest, then immediately selling them. This is why you can't "buy a dividend" and make money, nor can you buy a bond right before an interest payment and make money. As a result, all assets that I can think of are best modeled with continually accruing interest.

In that case, the person that invests on 1 January for the 43 years between ages 25 and 67 accrues 43 years of interest. The person that invests on 31 December accrues 42 years of interest (they miss out on the first year). The person that invests on 15 April of the next year accrues 41.71 years of interest (since they miss out on the first 1.29 years).

The difference in net worth between the three people is the difference in month accrued in 43 years vs 42 years vs 41.71 years. However, all that assumes that the people just stop investing on 1 January of the year they turn 68, which is probably also not likely.

posted by saeculorum at 6:40 PM on October 26, 2011 [1 favorite]

In that case, the person that invests on 1 January for the 43 years between ages 25 and 67 accrues 43 years of interest. The person that invests on 31 December accrues 42 years of interest (they miss out on the first year). The person that invests on 15 April of the next year accrues 41.71 years of interest (since they miss out on the first 1.29 years).

The difference in net worth between the three people is the difference in month accrued in 43 years vs 42 years vs 41.71 years. However, all that assumes that the people just stop investing on 1 January of the year they turn 68, which is probably also not likely.

posted by saeculorum at 6:40 PM on October 26, 2011 [1 favorite]

"I" am planning on giving people advice to invest in January (if they can afford it) rather than waiting until the following April. I say "I" because I'm not really providing that advice. I'm writing on assignment.

And actually, as someone just mentioned in a comment, doing the calculation with an average annual return of 10% makes more since than using compound interest. Though, it's certainly possible to have interest bearing devices in an IRA (CDs for one).

posted by GnomeChompsky at 6:44 PM on October 26, 2011

And actually, as someone just mentioned in a comment, doing the calculation with an average annual return of 10% makes more since than using compound interest. Though, it's certainly possible to have interest bearing devices in an IRA (CDs for one).

posted by GnomeChompsky at 6:44 PM on October 26, 2011

I am not an investment expert of any kind, but I think your assignment has you giving people some pretty bad advice. There is no real-world scenario in which you can say with any degree of certainty that investing in January is better than investing the following April. For example, someone who put in their 2008 lump sum in an index fund in January of that year would have lost nearly half its value by the time April 2009 rolled around, at which point the person who procrastinated could have bought at the fund's lowest price in a decade, ending up with substantially more money. That's a pretty dramatic example, but the point is that stock prices fluctuate, and so your question doesn't map onto real world conditions at all, making a simple compound interest equation basically useless as investment advice. For casual investors, most experts recommend something like dollar cost averaging, which tends to reduce risk. But even if investors choose to invest in one lump sum, there's really no rhyme or reason to determining when would be the best time in an 18 month period to drop that money in the market.

posted by decathecting at 6:54 PM on October 26, 2011

posted by decathecting at 6:54 PM on October 26, 2011

Decathecting, I actually tend to agree with you. When I first got the assignment, I thought of it as a lesson in compounding interest. Then I realized (looking at answers here) that most people don't put IRA contributions into an interest bearing account. That said, I think I'm going to turn the assignment into a lesson on compound interest anyway. So the initial question still stands (and is relevant).

posted by GnomeChompsky at 6:59 PM on October 26, 2011

posted by GnomeChompsky at 6:59 PM on October 26, 2011

Seconding cupcake1337 and saeculorum. If you have a defined term (e.g., 25-65 years old), investing the same amount of money earlier in that term should leave you with more money than if you invest it later (assuming a positive rate of return).

But whether the deposits happen in January or 15 months later will not make a huge difference.

posted by pollex at 7:00 PM on October 26, 2011

But whether the deposits happen in January or 15 months later will not make a huge difference.

posted by pollex at 7:00 PM on October 26, 2011

Here's some math, since you asked to see the math

Let A(t) be the amount in the account at time t (in years).

If the 10% is applied once every year, then after one year, the account will have 10% more than it started with. Call A0 the starting amount and A1 the amount after one year. A1 = A0(1 + .1) After two years - it's A2 = A1(1 + .1) and so on, but remember that A1 = A0(1 + 0.1) If we plugged that in to the A2 formula, we'd get A2 = A0(1 + 0.1)(1 + 0.1). So more generally, we could write:

A(t) = A0(1 + .10)^(t) even more generally, if we replace the .10 with r (the annual rate), we get:

A(t) = A0(1 + r)^(t)

That formula is a special case of an even more general formula. The one we just derived assumes that the interest is being compounded once every year. If it were every month instead, we could derive a similar formula, but we need to know what

Using a similar process, the reasoning goes like this:

After one month, we have

A(1month) = A0(1 + (.10/12))

After two months:

A(2months) = A1(1 + (.10/12))

Substituting as above, we can see that

A(2months) = A0(1 + (.10/12))(1 + (.10/12))

And we can see a similar pattern emerging. The only slight adjustment we'll have to make is accounting for the fact that our time variable

A(t) = A0(1 + (r/n))^nt

Since in both your scenarios,

posted by chndrcks at 7:28 PM on October 26, 2011 [1 favorite]

Let A(t) be the amount in the account at time t (in years).

If the 10% is applied once every year, then after one year, the account will have 10% more than it started with. Call A0 the starting amount and A1 the amount after one year. A1 = A0(1 + .1) After two years - it's A2 = A1(1 + .1) and so on, but remember that A1 = A0(1 + 0.1) If we plugged that in to the A2 formula, we'd get A2 = A0(1 + 0.1)(1 + 0.1). So more generally, we could write:

A(t) = A0(1 + .10)^(t) even more generally, if we replace the .10 with r (the annual rate), we get:

A(t) = A0(1 + r)^(t)

That formula is a special case of an even more general formula. The one we just derived assumes that the interest is being compounded once every year. If it were every month instead, we could derive a similar formula, but we need to know what

**r**to use. Assuming that the 10% given is a nominal interest rate (and not an effective interest rate), we divide that r by the number of times the interest is compounded in a year (in this case 12) and call that**n**Using a similar process, the reasoning goes like this:

After one month, we have

A(1month) = A0(1 + (.10/12))

After two months:

A(2months) = A1(1 + (.10/12))

Substituting as above, we can see that

A(2months) = A0(1 + (.10/12))(1 + (.10/12))

And we can see a similar pattern emerging. The only slight adjustment we'll have to make is accounting for the fact that our time variable

**t**was years and not months. We get this general formula:A(t) = A0(1 + (r/n))^nt

Since in both your scenarios,

**A0, r,**and**n**are all the same, the only way they finish with different amounts is if**t**is different. If they take out their money at the same time, then the person who waited 16 months (1 1/3 years) has a lower t than the other person, and you can figure out the difference in final amounts by plugging in the appropriate values. If the person who waited 16 months to start also waits 16 months to finish, they should end up with the same amount.posted by chndrcks at 7:28 PM on October 26, 2011 [1 favorite]

*we divide that r by the number of times the interest is compounded in a year (in this case 12) and call that n*

Sorry, that wasn't clear.

**n**is the number of times per year the interest is compounded (

**not**r divided by the number of times)

posted by chndrcks at 7:35 PM on October 26, 2011

*one year, four months, and 15 days earlier*

One year,

*three*months, and 15 days earlier. April is the fourth month, so it won't be four months until the

*end*of April.

posted by kindall at 7:42 PM on October 26, 2011

Something you may not be considering (and might not be at all relevant to your assignment) is that the reason that some people make their IRA contribution in April of the year following tax year (IE they make the contribution for tax year 2010 in April of 2011) is because many people need to have some idea of how the contribution will affect their taxes.

Mrs. VTX and I usually figure out our taxes and then mess with the contribution amounts to figure out the ideal amount to contribute. I'd imagine that a lot of people do it in April because they forget (or otherwise procrastinate) until they do their taxes.

posted by VTX at 8:39 PM on October 26, 2011 [1 favorite]

Mrs. VTX and I usually figure out our taxes and then mess with the contribution amounts to figure out the ideal amount to contribute. I'd imagine that a lot of people do it in April because they forget (or otherwise procrastinate) until they do their taxes.

posted by VTX at 8:39 PM on October 26, 2011 [1 favorite]

Try this:

P = C (1 + r/n) nt

where

P = future value

C = initial deposit

r = interest rate (expressed as a fraction: eg. 0.06)

n = # of times per year interest is compounded

t = number of years invested

posted by Yellow at 11:01 PM on October 26, 2011

P = C (1 + r/n) nt

where

P = future value

C = initial deposit

r = interest rate (expressed as a fraction: eg. 0.06)

n = # of times per year interest is compounded

t = number of years invested

posted by Yellow at 11:01 PM on October 26, 2011

Ah, I am sorry, I just re-read your question and it requires a different formula if you are going to add money annually. I guess you can just keep changing C as you go on, but that is a lot of math. I am too tired to remember what the other formula is, though!

posted by Yellow at 11:09 PM on October 26, 2011

posted by Yellow at 11:09 PM on October 26, 2011

Add another layer here. In a real life example, wouldn't both parties get the $5,000 in assets on January 1st each year? If you are truly trying to compare the two people in economic terms, you would also need to factor in what the April 15th person is doing with their assets outside of the IRA during the 15.5 months. If you assume that either person can earn the same rate inside or outside of an IRA, then the true economic difference is only the tax savings on the interest, not the interest itself.

posted by jameslavelle3 at 4:33 AM on October 27, 2011 [1 favorite]

posted by jameslavelle3 at 4:33 AM on October 27, 2011 [1 favorite]

kindall: yes, that's right

posted by cupcake1337 at 9:46 AM on October 27, 2011

posted by cupcake1337 at 9:46 AM on October 27, 2011

This thread is closed to new comments.

posted by decathecting at 6:33 PM on October 26, 2011