Calculating same payments in with different rates

[jai012]

Could someone help me with this question:
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What is the amount of money that I need to deposit every year in order to have $200k in 20 years if the interest rate for the first 10 years is 8% and for the last 10 years 12%?

Note: The first yearly deposit will happen in a year (we are now in T0). All the 20 deposits are of the same amount (equal payments).
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Can someone help me with a full walkthrough of the answer?

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Ans : $3,197.86

 

[tkhunny]

As always, draw a map!

P is the level payment

T1: P
T2: P
etc...
T10: P
T11: P
etc...
T20: P

There's your map.

Now, add interest accumulation.

i = 0.08, q = 1+i = 1.08
j = 0.12, r = 1+j = 1.12

It's often easiest to start from the back.

T20: P
T19: Pr
T18: Pr^2
T17: Pr^3
continue...
T10: is a little tricky. Think really hard!

Once you have 20 deposits with 20 interest rate accumulations, just add them all up. It's not very difficult if you are VERY familiar with the summation of Finite Geometric Series.

 

[Mr. Bland]

Welcome to Free Math Help!

In researching this problem, I found a formula that gives the correct result, but I don't know how it works. Hopefully others with more knowledge on the subject can spell it out for me!

The formula comes from this page, which is a JavaScript interest calculator. It documents various formulas such as compound interest, annual percentage rate, and this one: principal with interest and recurring payments:

[MATH]total = payment\left(\left(1 + \frac{interest}{compounds}\right)^{years * compounds} - 1\right)\frac{compounds}{interest}[/MATH]​

Specifically, this calculates the total amount contributed by recurring payments on top of the interest to the initial principal. In other words, the total principal is calculated like this:

[MATH]total = initial * \left(1 + \frac{interest}{compounds}\right)^{years * compounds} + payment * \left(\left(1 + \frac{interest}{compounds}\right)^{years * compounds} - 1\right) * \frac{compounds}{interest}[/MATH]​

I recognize [MATH]\left(1 + \frac{interest}{compounds}\right)^{years*compounds} - 1[/MATH] as the formula for the compounded interest rate (let's call it [MATH]foo[/MATH]), so the above formula reduces to this:

[MATH]total = initial * (1 + foo) + payment * foo * \frac{compounds}{interest}[/MATH]​

What I don't get is, why does it work to multiply the recurring payment amount by the compounded interest rate and the reciprocal [MATH]\frac{compounds}{interest}[/MATH]? It's easy enough to remember, and I'm glad I stumbled on it, but I don't understand how the meaning is associated with the formula...