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Numerical Questions 1. You deposit \$1000 in a savings account today, and expect it will grow to \$1500 at the end of five years. Interest is compounded annually. What is the annual interest rate? A) 8.4472% B) 9.6224% C) 10% D) 10.6682% E) 50% Solution A FV = PV x (1+r) t solving for r, r = (FV/PV) 1/t – 1 = (1500/1000) 1/5 – 1 = .084472 2. Consider a perpetuity with an annual cash flow of \$1000 beginning one year from today, and an annuity due with a cash flow of \$1000 per year for t years. What is the smallest integer value of t (i.e., we want the minimum number of years) for which the annuity due has a greater present value than the perpetuity? The interest rate is 8%. Solution D The present value of the perpetuity is PV = C/r = 1000/.08 = \$12,500.00 Because the annuity due has one extra cash flow, at time zero, it will be worth more than the perpetuity for large values of t. Following the approach of equation (4.10), the PV of this annuity due is \$1000 plus the PV of an ordinary annuity of t-1 payments. Therefore if we find the length of time for an ordinary annuity to be worth exactly \$11,500, add one more year and the annuity due will be worth exactly \$12,500. I/Y = 8, PMT = 1000, PV = -11,500, FV = 0, CPT N = 32.82 This N is for t-1, so t is 33.82 and the smallest integer number of years is 34.
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