610 CHAPTER 10 Consumer Mathematics Example 3 highlights how the annual percentage yield can be used to help an investor compare different investments. In the United States and in many countries, banks are legally required to state the APY when they are quoting rates on fixed investments. Present Value You may wonder about what amount of money you must deposit in an account today to have a certain amount of money in the future. For example, how much must you deposit in an account today at a given rate of interest so that it will accumulate to $25,000 to pay some of your child's college expenses in 4 years? The principal, p, that would have to be invested now is called the present value. If we solve the compound interest formula for p, we have the following formula for determining the present value. Example 3 Comparing Bank Certificates of Deposit Elias is planning to invest his income tax refund in a 1-year certificate of deposit (CD). He is comparing the rates at two banks. Aktia Bank is offering a CD with 4.5% interest compounded daily. Nordea Bank is offering a CD with 4.6% interest compounded semiannually. Calculate the annual percentage yield (APY), to the nearest hundredth of a percent, for the CD at a) Aktia Bank. b) Nordea Bank. c) Assuming all other factors are equal, which CD would be the better investment? Solution a) To calculate the APY at Aktia Bank, we use the compound interest formula with p r 1, 0.045, = = n 360, = and t 1. = A p r n 1 1 1 0.045 360 1.0460 n t ( ) (360 1) = ⎛ + ⎝ ⎞ ⎠ = ⎛ + ⎝⎜ ⎞ ⎠⎟ ≈ ⋅ ⋅ From this result we subtract 1 to obtain 1.0460 1 0.0460, − = or 4.60%. Thus, the APY at Aktia Bank is 4.60%. b) To calculate the APY at Nordea Bank, we use the compound interest formula with p r 1, 0.046, = = n 2, = and t 1. = A p r n 1 1 1 0.046 2 1.0465 n t ( ) (21) = ⎛ + ⎝ ⎞ ⎠ = ⎛ + ⎝⎜ ⎞ ⎠⎟ ≈ ⋅ ⋅ From this result we subtract 1 to obtain 1.0465 1 0.0465, − = or 4.65%. Thus, the APY at Nordea Bank is 4.65%. c) Since the CD at Nordea Bank has the higher APY, it would be the better investment. 7 Now try Exercise 31 Learning Catalytics Keyword: Angel-SOM-10.3 (See Preface for additional details.) Present Value Formula p A r n 1 n t ( ) = ⎛ + ⎝ ⎞ ⎠ ⋅ where p is the present value, or the principal to invest now, A is the amount to be accumulated in the account, r is the annual interest rate as a decimal number, n is the number of compounding periods per year, and t is the time in years.
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