Simple Compound Interest Calculation Question + [bonus question]

[NickNick]

On a similar note, this is confusing me:

Initial Investment: $1000
Yearly addition: $1000 for 20 years
Interest Rate: 5%

If interest compounded yearly = $37,372.55
If interest compounded monthly = $37,108.17

http://www.moneychimp.com/calculator...calculator.htm

Why is the final amount less if the interest is compounded more regularly?

Thanks a lot! - Sorry if this is a basic oversight!

[Garland]

[ QUOTE ]
On a similar note, this is confusing me:

Initial Investment: $1000
Yearly addition: $1000 for 20 years
Interest Rate: 5%

If interest compounded yearly = $37,372.55
If interest compounded monthly = $37,108.17

http://www.moneychimp.com/calculator...calculator.htm

Why is the final amount less if the interest is compounded more regularly?

Thanks a lot! - Sorry if this is a basic oversight!

[/ QUOTE ]

It took my mind a while to wrap around the concept when studying:
From time 0 to time 19, there are 20 payments of $1000, one for each year t = 0, t = 1, ..., t = 19. In addition, there's one more for the initial investment at time t = 0, which is also $1000. This all takes place in the span of 19 years, and not 20 like I kept thinking. The 20th payment occurs in 19 years. With that out of the way, here goes!

I could use my calculator:

Mode=BGN for Due (1st payment right away as opposed to a full period from now)

N = 20 (20 payment periods)
I/Y = 5 (5% per year)
PV = 1000 (Initial Investment)
PMT = 1000 (Payment per period)

Compute Future value = $37,372.55

That's correct.

Now, when they calculate only changing the number of compounding periods = 12. This is what they did:

Switching to periods of months instead of years.

N = 240 (20 years * 12 months)
I/Y = (0.05/12)*100 (Interest per period)
PV = 1000
Payment = 1000/12 = 83.333333 for each month

Compute Future Value = $37,108.17

So they also assume you are putting your payments in monthly. But also they are using the faulty 5% effective yearly [called i upper 1 or just “i”] and treating it like it’s convertible monthly. This is only equivalent to 4.8889 convertible monthly [called i upper 12].

We need to convert 5% convertible monthly, to its equivalent effective yearly rate.

In order to convert to get the “right” %, you have to do a little math:

[(1 + (0.05/12)^(12) – 1] = 0.05116 or 5.116%

To get per period, we have to divide by 12: 0.05116/12 = 0.004263 or 0.4263%

So now

N = 240 (20 years * 12 months)
I/Y = (0.05116)/12 * 100 = 0.4263 (Interest per period)
PV = 1000
Payment = 1000/12 = 83.333333 for each month

Compute Future Value = $37,639.42

There’s the extra interest you were looking for. And that’s assuming you’re depositing monthly, rather than yearly, so you get yet another number for that. But it’s really, really late, and I have to sleep.

Garland