SECTION 5.7 Financial Models 349 The effective rate of interest is the annual simple interest rate that would yield the same amount as compounding n times per year, or continuously, after one year. THEOREM Effective Rate of Interest The effective rate of interest rE of an investment earning an annual interest rate r is given by • ( ) = + − n r r n Compounding times per year: 1 1 E n • = − r e Continuous compounding: 1 E r Computing the Effective Rate of Interest—Which Is the Best Deal? Suppose you want to buy a 5-year certificate of deposit (CD). You visit three banks to determine their CD rates. American Express offers you 3.15% annual interest compounded monthly, and First Internet Bank offers you 3.20% compounded quarterly. Discover offers 3.12% compounded daily. Determine which bank is offering the best deal. Solution EXAMPLE 4 The bank that offers the best deal is the one with the highest effective interest rate. American Express First Internet Bank Discover ( ) = + − r 1 0.0315 12 1 E 12 ( ) = + − r 1 0.032 4 1 E 4 ( ) = + − r 1 0.0312 365 1 E 365 ≈ − 1.03196 1 ≈ − 1.03239 1 ≈ − 1.03169 1 = 0.03196 = 0.03239 = 0.03169 = 3.196% = 3.239% = 3.169% The effective rate of interest is highest for First Internet Bank, so First Internet Bank is offering the best deal. Now Work PROBLEM 23 3 Determine the Present Value of a Lump Sum of Money When people in finance speak of the “time value of money,” they are usually referring to the present value of money. The present value of A dollars to be received at a future date is the principal that you would need to invest now so that it will grow to A dollars in the specified time period. The present value of money to be received at a future date is always less than the amount to be received, since the amount to be received will equal the present value (money invested now) plus the interest accrued over the time period. The compound interest formula (2) is used to develop a formula for present value. If P is the present value of A dollars to be received after t years at a per annum interest rate r compounded n times per year, then, by formula (2), ( ) = ⋅ + A P r n 1 nt To solve for P, divide both sides by ( ) + r n 1 . nt The result is ( ) ( ) + = = ⋅ + − A r n P P A r n 1 or 1 nt nt Credit: Lacroix Serge/Getty Images
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