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Notes: Compound Interest and Annual Yield I Notes: Compound Interest and Annual Yield I. Compound Interest A. The Formula • P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years. nt r F P1 n • Think of P as the present value, and F as the future value of the deposit. Number of Times Compounded Period Interest Times Credited Credited per year Annual year 1 Semiannual 6 months 2 Quarterly quarter 4 Monthly month 12 Ex 1: Suppose you invest $5000 in an IRA account that compounds quarterly at 5.5%. How much money will be in an account after 1 year? 41 0.055 F 50001 1 4 nt r F P1 $5280 .72n After 10 years? 410 0.055 F10 50001 nt 4 r F P1 n $8633.85 Decrease in value: When something is decreasing in value (such as a new car) we can use the compound interest formula. However, we will need to use subtraction instead of addition. Ex 2: Your parents bought a car two years ago for $32,000. They are going to give it to you when you graduate high school next year. If the value of the car decreases 15% each year, how much will it be worth by the time they give it to you? 1(3) .15 F 320001 = $19652 1 If you need to solve for time, use the log or ln function. Use either 4t one, just be consistent. .052 50000 320001 4 Ex 3: Lets say we want to know how long it will take $32,000 to 1.5625 1.0134t grow to $50,000 invested in an 4t account that has 5.2% annual ln1.5625 ln1.013 interest compounded quarterly. ln1.5625 (4)(t)ln1.013 nt r ln1.5625 (4)(t)ln1.013 F P1 n ln1.5625 (t) 4ln1.013 8.64 (t) B. Continuous Compounding A Pe rt • P = principal amount invested •r = the interest rate • t = the number of years interest is being compounded • A = the compound amount, the balance after t years Ex 1: Ten thousand dollars is invested at 6.5% interest compounded continuously. When will the investment be worth $41,787? A Pe rt Since the interest rate is 6.5%, r = 0.065. Since ten thousand dollars is being invested, P = 10,000. And 41,787 10,000e0.065t since the investment is to grow to become $41,787, A = 41,787. We will make the appropriate substitutions and then solve for t. 4.1787 e0.065t Divide by 10,000. ln 4.1787 0.065t Rewrite the equation in logarithmic form. 22 t Divide by 0.065 and solve for t. Therefore, the $10,000 investment will grow to $41,787, via 6.5% interest compounded continuously, in 22 years. II. Yield • One may compare investments with different interest rates and different frequencies of compounding by looking at the values of P dollars at the end of one year, and then computing the annual rates that would produce these amounts without compounding. • Such a rate is called the effective annual yield, annual percentage yield, or simply yield. This is not to be confused with annual percentage rate (which we will discuss later). A. Calculating Effective Annual Yield y = (1 + r/n)n – 1 Ex 1: Find the annual yield for an investment that has an annual interest rate of 8.4% compounded monthly. • ANSWER: y = (1 + .084/12)12 – 1 • y = (1.007)12 – 1 = 0.087310661 = 8.73% • The yield will usually be greater than the interest rate. • Note the interest rate is sometimes called the nominal interest rate. B. Effective Annual Yield for Continuous Compounding • The effective annual yield, y, for compounding continuously at an annual interest rate of r is: P er P y P OR y er 1 Example: • In our previous example with compound interest, when we compounded quarterly, after one year we had: 4(1) 0.055 F1 5000 1 $5280.72 4 • To find the effective annual yield, y, notice that we gained $280.72 on interest after a year compounded quarterly. That interest represents a gain of 5.61% on $5000: 280.72 y 0.0561 5000 Beware: APY vs APR APR is the annual rate of interest without taking into account the compounding of interest within that year. Alternatively, APY does take into account the effects of intra- year compounding. This seemingly subtle difference can have important implications for investors and borrowers. Here is a look at the formulas for each method: For example, a credit card company might charge 1% interest each month; therefore, the APR would equal 12% (1% x 12 months = 12%). This differs from APY, which takes into account compound interest. The APY for a 1% rate of interest compounded monthly would be 12.68% [(1 + 0.01)^12 – 1= 12.68%] a year. If you only carry a balance on your credit card for one month's period you will be charged the equivalent yearly rate of 12%. However, if you carry that balance for the year, your effective interest rate becomes 12.68% as a result of compounding each month. -Investopedia.com.
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