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Chapter 4.2® - Compounding Interest Homework Problem & Time Value of Money Continued - Future Value Formula, Growth of \$100 & Future Value Comparisons

From the introduction to time value of money (part 4.1), we can derive a formula, which will be the future value of \$1 invested for t periods at a rate of r per period. The equation is:

 Future Value = \$1 x (1 + r)t

The expression (1 + r)t is sometimes referred to as the future value interest factor for \$1 invested at r percent for t periods of time.

Let’s use this future value formula to derive the original \$121 at the end of the 2nd year.

 Future Value = \$1 x (1 + r)t Future Value = \$1 x (1+0.1)2 Future Value = \$1 x (1.1)2 Future Value = \$1 x (1.21) Future Value = \$1.21 x \$100 Future Value = \$121 What would be \$100 worth in 5 years, at a 10% interest rate? Future Value = \$1 x (1 + r)t Future Value = \$1 x (1+0.1)5 = 1.15 = 1.61051 Future Value = 1.61051 x \$100 = \$161.05

The table below shows the growth of \$100 each year at 10% interest. Each year, the interest earned is equal to the beginning amount multiplied by the interest rate (10%). How much of this \$161.05 is simple interest, and how much of it is compounding interest? Well \$100 x 0.10 = \$10 per year, so in 5 years, the total simple interest is 5 years x \$10 = \$50. Thus, we can calculate the compounding interest by subtracting \$161.05 - \$150 = \$11.05

 Year (A) Beginning Amount (B) Simple Interest (C) \$100 x 10% Interest Earned on Interest (D) ((B) x 10%) - C Total Interest Earned (E) (C + D) Ending Amount (F) (B + E) 1 \$100 \$10 \$0 \$10 \$110 2 \$110 \$10 \$1 \$11 \$121 3 \$121 \$10 \$2.1 \$12.1 \$133.10 4 \$133.10 \$10 \$3.31 \$13.31 \$146.41 5 \$146.41 \$10 \$4.641 \$14.641 \$161.051 The above graph shows the growth of \$100 from inception at year 1 earning simple interest of \$10 (10%) and earning \$11 in the 2nd year, thus making the new total compounded earnings to \$21 at the end of year 2. This compounding effect continues to year 5 when the money has grown from \$100 to \$161.05 in Year 5.

Compound Interest Homework Problem Consider you found an investment that pays 15% annual interest, which sounds fantastic to you! This makes you invest \$1000 in to it, how much will you have in 3 years? How much will you have in 7 years? And at the end of 7 years, how much interest will you have earned, and how much of it will be compounding interest?
Note: We recommend you try this problem without looking at the solution below first, however if you get stuck, feel free to take a peak!

 a) Future Value = \$1 x (1 + r)t = =\$1 x (1 + 0.15)3 = \$1.153 = 1.520875 x \$1000 = \$1,520.875 b) Future Value = \$1 x (1 + r)t = =\$1 x (1 + 0.15)7 = \$1.157 = 2.66002 x \$1000 = \$2,660.02

You invested \$1000 x 15% = \$150 simple interest each year, thus total simple interest in 7 years = \$150 x 7 = \$1,050

 Compound Interest = Total Interest – Simple Interest Compound Interest = \$1,660.02 -\$1050 Compound Interest = \$610.02