Greetings, Fellow Cerebrators:
I hope that you enjoyed the weekend.
Before we get started on anything new, let's answer Friday's interest problems:
1) If interest accumulates at 10% per year simple, how much will the investment be worth at the end of ten years? Simple interest is computed on the amount invested, and no interest is paid on interest accumulated. If $100,000 is invested at 10% per year simple, it simply means that an additional $10,000 will be gerated upon the account each year. In ten years, the total interest would be 10 x $10,000, or $100,000. At the end of the 10-year period, the principal of $100,000 would have grown by $100,000 of interest, for a total of $200,000.
2) If interest accumulates at 7.5% per year compounded annually, how much will the investment be worth at the end of ten years? Compound interest is computed on both the principal and the interest in an interest-bearing account. If no funds are withdrawn from the account, the amount of interest earned each year will increase. In this example, at the end of the first year, the interest will be $7,500. In the second year, the interest earned would be 107,500 x (0.075), or $8,062.50. The amount of interest added on each year will increase as the "base" amount grows. At the end of ten years, the total amount in the account would be $100,000 x(1.075)(1.075)(1.075)(1.075)(1.075)(1.075)(1.075)(1.075)(1.075)(1.075) = $206,103. The effect of the compounding at 7.5% is greater than the effect of simple interest at 10%.
3) If interest accumulates at 7.5% per year compounded annually, and the investor (my Uncle Phil) takes $5,000 out of the account at the end of each year, how much will be in the investment account at the end of ten years? This problem, not unlike Uncle Phil, has a twist to it. At the end of the first year, although we have earned $7,500 in interest, Uncle Phil takes $5,000, and leaves us with a "base" of only $102,500. At the end of the second year, we will have $102,500 x (1.075) = $110,108.50 - $5,000 (Phil's portion) = $105,108.50.
The rate of growth in the account is not 7.5% per year. It is also not 2.5% per year. we must find a way to compute the rate of growth given a) compound interest at 7.5%, and b) a deduction at the end of each year of 5% of the original principal amount. These two "forces" are competing against eachother. One is compound and the other is not. Take another day to work on this one. Is there a formula which could make it easier???
Faithfully,
Douglas Castle
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