350 CHAPTER 5 Exponential and Logarithmic Functions To derive (6), solve = A Pert for P. Computing the Value of a Zero-Coupon Bond A zero-coupon (noninterest-bearing) bond can be redeemed in 10 years for $1000. How much should you be willing to pay for it now if you want a return of (a) 8% compounded monthly? (b) 7% compounded continuously? Solution EXAMPLE 5 (a) To find the present value of $1000, use formula (5) with = A $1000, = = n r 12, 0.08, and = t 10. ( ) ( ) = ⋅ + = + = − − ⋅ P A r n 1 $1000 1 0.08 12 $450.52 nt 12 10 For a return of 8% compounded monthly, pay $450.52 for the bond. (b) Here use formula (6) with = = A r $1000, 0.07, and = t 10. = = = − − ⋅ P Ae e $1000 $496.59 rt 0.07 10 For a return of 7% compounded continuously, pay $496.59 for the bond. Determining the Rate of Interest Required to Double an Investment What rate of interest compounded annually is needed in order to double an investment in 5 years? Solution EXAMPLE 6 If P is the principal and P is to double, then the amount A will be P2 . Use the compound interest formula with = n 1 and = t 5 to find r. ( ) ( ) ( ) = ⋅ + = ⋅ + = + + = = − ≈ − = A P r n P P r r r r 1 2 1 2 1 1 2 2 1 1.148698 1 0.148698 nt 5 5 5 5 The annual rate of interest needed to double the principal in 5 years is 14.87%. THEOREM Present Value Formulas The present value P of A dollars to be received after t years, assuming a per annum interest rate r compounded n times per year, is ( ) = ⋅ + − P A r n 1 nt (5) If the interest is compounded continuously, then = − P Ae rt (6) Now Work PROBLEM 15 4 Determine the Rate of Interest or the Time Required to Double a Lump Sum of Money = = = A P n t 2 , 1, 5 Divide both sides by P. Take the fifth root of both sides. Now Work PROBLEM 31
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