The relationship between the PVA and the interest rate is PVA falls as r

# The relationship between the pva and the interest

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The relationship between the PVA and the interest rate is: PVA falls as r increases, and PVA rises as r decreases FVA rises as r increases, and FVA falls as r decreases The present values of \$7,800 per year for 20 years at the various interest rates given are: [email protected]% = \$7,800{[1 – (1/1.10) 20 ] / .10} = \$66,405.80 [email protected]% = \$7,800{[1 – (1/1.05) 20 ] / .05} = \$97,205.24 [email protected]% = \$7,800{[1 – (1/1.15) 20 ] / .15} = \$48,822.79 Calculator Solution: Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation. Enter 20 10% \$7,800 66,405.80 ± 0.1% 97,205.24 ± 0.1% 48,822.79 ± 0.1%
N I/Y PV PMT FV Solve for \$66,405.80 Enter 20 5% \$7,800 N I/Y PV PMT FV Solve for \$97,205.24 Enter 20 15% \$7,800 N I/Y PV PMT FV Solve for \$48,822.79 Problem 4-41 EAR versus APR You have just purchased a new warehouse. To finance the purchase, you’ve arranged for a 35-year mortgage for 85 percent of the \$4,100,000 purchase price. The monthly payment on this loan will be \$18,200. What is the APR on this loan? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) Annual percentage rate % What is the EAR on this loan? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) Effective annual rate % Explanation: Here, we are finding interest rate for an annuity cash flow. We are given the PVA, number of periods, and the amount of the annuity. We need to solve for the number of payments. We should also note that the PV of the annuity is not the amount borrowed since we are making a down payment on the warehouse. The amount borrowed is: -2-2 5.27 ± 1% 5.40 ± 1%
Amount borrowed = .85(\$4,100,000) = \$3,485,000 The time line is: 0 1 –\$3,485,000 \$18,200 \$18,200 \$18,200 \$18,200 \$18,200 \$18,200 \$18,200 \$18,200 Using the PVA equation: PVA = \$3,485,000 = \$18,200[{1 – [1 / (1 + r )] 420 } / r ] Unfortunately, this equation cannot be solved to find the interest rate using algebra. To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. If you use trial and error, remember that increasing the interest rate decreases the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find: r = .439% The APR is the monthly interest rate times the number of months in the year, so: APR = 12(.439%) = 5.27% And the EAR is: EAR = (1 + .00439) 12 – 1 = .0540 or 5.40% Calculator Solution: Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation. Enter 420 .85(\$4,100,000) ±\$18,200 N I/Y PV PMT FV Solve for .439% APR = .439% × 12 = 5.27% Enter 5.27% 12 NOM EFF C/Y Solve for 5.40%
Problem 4-42 Present Value and Break-Even Interest Consider a firm with a contract to sell an asset for \$155,000 four years from now. The asset costs \$91,000 to produce today. Given a relevant discount rate on this asset of 14 percent per year, calculate the profit the firm will make on this asset. (Negative amount should be indicated by a minus sign. Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) Firm's profit(loss) \$ At what rate does the firm just break even? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) Break-even interest % Explanation: The time line is: 0 PV \$1 The profit the firm earns is just the PV of the sales price minus the cost to produce the asset. We find the PV of the

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• Spring '16

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