Compound Interest Section 5.2

Introduction Re-investing your interest income from an investment makes your money grow faster over time! This is what compound interest does. Compound interest uses the same information as simple interest, but what is new is the frequency of compounding n. n=1 annual, n=2 semi-annual, n=4 quarterly, n=12 monthly, n=52 weekly, n=365 daily.

Compound Interest Formula If P represents the present value, r the annual interest rate, t the time in years, and n the frequency of compounding, then the future value is given by the formula: F = P( 1 + r/n) nt

Example Suppose you invest $32,000 into a certificate of deposit that has an annual interest rate of 5.2% compounded annually for 3 years. ANSWER: Use the compound interest formula. F = 32000(1+.052/4) (4)(3) = 32000(1.013) 12 = $37,364.86

Annual Yield To compare different savings plans, you need to have a common basis for making the comparisons. The annual yield of a compound interest investment is the simple interest rate that has the same future value the compound rate would have in one year.

Derivation of yield Future Value Compound = Future value simple P(1 + r/n) nt = P(1 + yt) Since this computation is done for 1 year, we set t = 1. P(1+ r/n) n = P(1 + y) Since P appears on both side, we divide by P and P disappears. (1 + r/n) n = 1 + y, now solve for y by subtracting 1 from both sides. The formula for yield is y = (1 + r/n) n – 1.

Example yield calculation Find the annual yield for an investment that has an annual interest rate of 8.4% compounded monthly. ANSWER: y = ( /12) 12 – 1 y = (1.007) 12 – 1 = = 8.73% The yield will usually be greater than the interest rate. Note the interest rate is sometimes called the nominal interest rate.

Continuous Compounded Interest What would happen if we let the frequency of compounding get very large. That is we would compound not just every hour, or every minute or every second but for every millisecond! What happens is that the expression (1 +r/n) nt goes to e rt. This e is the famous Euler number. It’s value is the irrational number … The future value formula is F = Pe rt. The annual yield for continuously compounded interest is y = e r – 1.

Example of Continuous Compound interest. Consider the $32,000 from the earlier example. Now we will invest the money in an account that has 5.2% annual interest compounded continuously for 3 years. What is the future value? ANSWER: F = 32000e (.052)(3) = $37, Note this investment option is only greater by $ What is the yield for this investment? ANSWER: y = e.052 – 1 = = 5.34%

For those who know logs Sometimes we would like to know how long an investment will take to grow to a certain value. This type of question involves solving an exponential equation. The technique for solving these types of equations is taking the natural logarithm of both sides of the equation.

Example using the natural log The symbol for natural log of x is ln(x). Lets say we want to know how long it will take $32,000 to grow to $50,000 invested in an account that has 5.2% annual interest compounded quarterly. We use the formula F = P(1 + r/n) nt = 32000( /4) 4t Note the unknown is in the exponent. Divide both sides of the equation by 32000, and also simplify the inside of the parentheses. This will give = (1.013) 4t. Now take the natural log of both sides. ln(1.5625) = ln((1.013) 4t ) By rule, we can take the 4t and move it to the front of the ln(1.013). Thus the equation is now ln(1.5625) = (4t)ln(1.013). Now divide both sides of the equation by 4ln(1.013) so that you have t isolated. Thus t = ln(1.5626)/(4ln(1.013) = 8.64 years or 8 years and 8 months.