Finding the apr with monthly compounding and dividing

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Unformatted text preview: ntract. The remaining amount will be the PV of the future quarterly paychecks. $39,269,529.66 – 10,000,000 = $29,269,529.66 To find the quarterly payments, first realize that the interest rate we need is the effective quarterly rate. Using the daily interest rate, we can find the quarterly interest rate using the EAR equation, with the number of days being 91.25, the number of days in a quarter (365 / 4). The effective quarterly rate is: Effective quarterly rate = [1 + (.05/365)]91.25 – 1 = .01258 or 1.258% Now, we have the interest rate, the length of the annuity, and the PV. Using the PVA equation and solving for the payment, we get: PVA = $29,269,529.66 = C{[1 – (1/1.01258)24] / .01258} C = $1,420,476.43 60. To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the interest rate quoted in the problem is only relevant to determine the total interest under the terms given. The cash flows of the loan are the $20,000 you must repay in one year, and the $17,200 you borrow today. The interest rate of the loan is: $20,000 = $17,200(1 + r) r = ($20,000 / 17,200) – 1 = .1628 or 16.28% Because of the discount, you only get the use of $17,200, and the interest you pay on that amount is 16.28%, not 14%. 61. Here, we have cash flows that would have occurred in the past and cash flows that would occur in the future. We need to bring both cash flows to today. Before we calculate the value of the cash flows today, we must adjust the interest rate, so we have the effective monthly interest rate. Finding the APR with monthly compounding and dividing by 12 will give us the effective monthly rate. The APR with monthly compounding is: APR = 12[(1.09)1/12 – 1] = 8.65% To find the value today of the back pay from two years ago, we will find the FV of the annuity (salary), and then find the FV of the lump sum value of the salary. Doing so gives us: FV = ($42,000/12) [{[ 1 + (.0865/12)]12 – 1} / (.0865/12)] (1 + .09) = $47,639.05 72 Notice we found the FV of the annuity with the effective monthly rate, and then found the FV of the lump sum with the EAR. Alternatively, we could have found the FV of the lump sum with the effective monthly rate as long as we used 12 periods. The answer would be the same either way. Now, we need to find the value today of last year’s back pay: FVA = ($45,000/12) [{[ 1 + (.0865/12)]12 – 1} / (.0865/12)] = $46,827.37 Next, we find the value today of the five year’s future salary: PVA = ($49,000/12){[{1 – {1 / [1 + (.0865/12)]12(5)}] / (.0865/12)}= $198,332.55 The value today of the jury award is the sum of salaries, plus the compensation for pain and suffering, and court costs. The award should be for the amount of: Award = $47,639.05 + 46,827.37 + 198,332.55 + 150,000 + 25,000 Award = $467,798.97 As the plaintiff, you would prefer a lower interest rate. In this problem, we are calculating both the PV and FV of annuities. A lower interest rate will decrease the FVA, but increase the PVA. So, by a lower interest rate, we are lowering the value of the back pay. But, we are also increasing the PV of the future salary. Since the future salary is larger and has a longer time, this is the more important cash flow to the plaintiff. 62. Again, to find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is in the form of a lump sum, the amount you will repay is the FV of the principal amount, which will be: Loan repayment amount = $10,000(1.09) = $10,900 The amount you will receive today is the principal amount of the loan times one minus the points. Amount received = $10,000(1 – .03) = $9,700 Now, we simply find the interest rate for this PV and FV. $10,900 = $9,700(1 + r) r = ($10,900 / $9,700) – 1 = .1237 or 12.37% With a 12 percent quoted interest rate loan and two points, the EAR is: Loan repayment amount = $10,000(1.12) = $11,200 Amount received = $10,000(1 – .02) = $9,800 $11,200 = $9,800(1 + r) r = ($11,200 / $9,800) – 1 = .1429 or 14.29% The effective rate is not affected by the loan amount, since it drops out when solving for r. 73 63. First, we will find the APR and EAR for the loan with the refundable fee. Remember, we need to use the actual cash flows of the loan to find the interest rate. With the $2,100 application fee, you will need to borrow $202,100 to have $200,000 after deducting the fee. Solving for the payment under these circumstances, we get: PVA = $202,100 = C {[1 – 1/(1.00567)360]/.00567} where .00567 = .068/12 C = $1,317.54 We can now use this amount in the PVA equation with the original amount we wished to borrow, $200,000. Solving for r, we find: PVA = $200,000 = $1,317.54[{1 – [1 / (1 + r)]360}/ r] Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives: r = 0.5752% per month APR = 12(0.5752%) = 6.90% EAR = (1 + .005752)12 – 1 = .0713 or 7.13% With the nonrefundable fee, the APR of the loan is simply the quoted APR since the fee is not considered part of the loan. So: APR = 6.80% EAR = [1 +...
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This note was uploaded on 07/10/2010 for the course FIN 6301 taught by Professor Eshmalwi during the Spring '10 term at University of Texas-Tyler.

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