# PV Solutions

 Problem 1 a. Current Savings Needed = \$ 500,000/1.110 = \$ 192,772 b. Annuity Needed = \$ 500,000 (APV,10%,10 years) = \$ 31,373 Problem 2 Present Value of \$ 1,500 growing at 5% a year for next 15 years = \$ 18,093 Future Value = \$ 18093 (1.08^15) = \$ 57,394 Problem 3 Annual Percentage Rate = 8% Monthly Rate = 8%/12 = 0.67% Monthly Payment needed for 30 years = \$ 200,000(APV,0.67%,360) = \$ 1,473 Problem 4 a. Discounted Price Deal Monthly Cost of borrowing \$ 18,000 at 9% APR = \$ 373.65 [A monthly rate of 0.75% is used] b. Special Financing Deal 17.98245614 Monthly Cost of borrowing \$ 20,000 at 3% APR = \$ 359.37 The second deal is the better one. Problem 5 a. Year-end Annuity Needed to have \$ 100 million available in 10 years= \$ 6.58 [FV = \$ 100, r = 9%, n = 10 years] b. Year-beginning Annuity Needed to have \$ 100 million in 10 years = \$ 6.04 Problem 6 Value of 15-year corporate bond; 9% coupon rate; 8 % market interest rate Assuming coupons are paid semi-annually, Value of Bond = 45*(1-1.04^(-30))/.04+1000/1.04^30 = \$ 1,086.46 If market interest rates increase to 10%, Value of Bond = 45*(1-1.05^(-30))/.05+1000/1.05^30 = \$ 923.14 The bonds will trade at par only if the market interest rate = coupon rate. Problem 7 Value of Stock = 1.50 (1.06)/ (.13 - .06) = \$ 22.71 Problem 8 Value of Dividends during high growth period = \$ 1.00 (1.15)(1-1.15^5/1.125^5)/(.125-.15) \$ 5.34 Expected Dividends in year 6 = \$ 1.00 (1.15)^5*1.06*2 = \$ 4.26 Expected Terminal Price = \$ 4.26/(.125-.06) = \$ 65.54 Value of Stock = \$ 5.34 + \$ 65.54/1.125^5 = \$ 41.70 Problem 9 Expected Rate of Return = (1000/300)^(1/10) - 1 = 12.79% Problem 10 Effective Annualized Interst Rate = (1+.09/52)^52 - 1 = 9.41% Problem 11 Annuity given current savings of \$ 250,000 and n=25 = \$ 17,738.11 Problem 12 PV of first annuity - \$ 20,000 a year for next 10 years = \$128,353.15 PV of second annuity discounted back 10 years = \$ 81,326.64 Sum of the present values of the annuities = \$209,679.79 If annuities are paid at the start of each period, PV of first annuity - \$ 20,000 at beginning of each year= \$148,353.15 PV of second annuity discounted back 10 years = \$ 88,646.04 Sum of the present values of the annuities = \$236,999.19 Problem 13 PV of deficit reduction can be computed as follows ­ Year Deficit Reduction PV 1 \$ 25.00 \$ 23.15 2 \$ 30.00 \$ 25.72 3 \$ 35.00 \$ 27.78 4 \$ 40.00 \$ 29.40 5 \$ 45.00 \$ 30.63 6 \$ 55.00 \$ 34.66 7 \$ 60.00 \$ 35.01 8 \$ 65.00 \$ 35.12 9 \$ 70.00 \$ 35.02 10 \$ 75.00 \$ 34.74 Sum \$ 500.00 \$ 311.22 The true deficit reduction is \$ 311.22 million. Problem 14 a. Annuity needed at 6% = 1.89669896 (in billions) b. Annuity needed at 8% = 1.72573722 (in billions) Savings = 0.17096174 (in billions) This cannot be viewed as real savings, since there will be greater risk associated with the higher-return investments. Problem 15 a. Year Nominal PV 0 \$5.50 \$5.50 1 \$4.00 \$3.74 2 \$4.00 \$3.49 3 \$4.00 \$3.27 4 \$4.00 \$3.05 5 \$7.00 \$4.99 \$28.50 \$24.04 b. Let the sign up bonus be reduced by X. Then the cash flow in year 5 will have to be raised by X + 1.5 million, to get the nominal value of the contract to be equal to \$30 million. Since the present value cannot change, X - (X+1.5)/1.075 = 0 X (1.075 - 1) = 1.5 X = 1.5/ (1.075 -1) = \$3.73 million The sign up bonus has to be reduced by \$3.73 million and the final year's cash flow has to be increased by \$5.23 million, to arrive at a contract with a nominal value of \$30 million and a present value of \$24.04 million. Problem 16 Chatham South Orange Mortgage \$300,000 \$200,000 Monthly Payment \$2,201 \$1,468 Annual Payments \$26,416 \$17,610 Property Tax \$6,000 \$12,000 Total Payment \$32,416 \$29,610 b. Mortgage payments will end after 30 years. Property taxes are not only a perpetuity; they are a growing perpetuity. Therefore, they are likely to be more onerous. c. If property taxes are expected to grow at 3% annually forever, PV of property taxes = Property tax * (1 +g) / (r -g) For Chatham, PV of property tax = \$6000 *1.03/(.08-.03) = \$123,600 For South Orange, PV of property tax = \$12,000 *1.03/(.08-.03) = \$247,200 To make the comparison, add these to the house prices, Cost of the Chatham house = \$400,000 + \$123,600 = \$523,600 Cost of the South Orange house = \$300,000 + \$247,200 = \$547,200 The Chatham house is cheaper. Problem 17 a. Monthly Payments at 10% on current loan = \$ 1,755.14 b. Monthly Payments at 9% on refinanced mortgage = \$ 1,609.25 Monthly Savings from refinancing = \$ 145.90 c. Present Value of Savings at 8% for 60 months = \$ 7,195.56 Refinancing Cost = 3% of \$ 200,000 = \$6,000 d. Annual Savings needed to cover \$ 6000 in refinancing cost= \$ 121.66 Monthly Payment with Savings = \$ 1755.14 - \$ 121.66 = \$ 1,633.48 Interest Rate at which Monthly Payment is \$ 1633.48 = 9.17% Problem 18 a. Present Value of Cash Outflows after age 65 = \$ 300,000 + PV of \$ 35,000 each year for 35 years = \$707,909.89 b. FV of Current Savings of \$ 50,000 = \$503,132.84 Shortfall at the end of the 30th year = \$204,777.16 Annuity needed each year for next 30 years for FV of \$ 204777 = \$ 1,807.66 c. Without the current savings, Annuity needed each year for 25 years for FV of \$ 707910 = \$ 9,683.34 Problem 19 a. Estimated Funds at end of 10 years: FV of \$ 5 million at end of 10th year = \$ 10.79 (in millions) FV of inflows of \$ 2 million each year for next 5 years = \$ 17.24 - FV of outflows of \$ 3 million each year for years 6-10 = \$ 17.60 = Funds at end of the 10th year = \$ 10.43 b. Perpetuity that can be paid out of these funds = \$ 10.43 (.08) = \$ 0.83 Problem 20 a. Amount needed in the bank to withdraw \$ 80,000 each year for 25 years = \$ 1,127,516 b. Future Value of Existing Savings in the Bank = \$ 407,224 Shortfall in Savings = \$ 1127516 - \$ 407224 = \$ 720,292 Annual Savings needed to get FV of \$ 720,292 = \$ 57,267 c. If interest rates drop to 4% after the 10th year, Annuity based upon interest rate of 4% and PV of \$ 1,127,516 = \$ 72,174.48 Problem 21 Year Coupon Face Value PV 1 \$ 50.00 \$ 46.30 2 \$ 50.00 \$ 42.87 3 \$ 50.00 \$ 39.69 4 \$ 50.00 \$ 36.75 5 \$ 50.00 \$ 34.03 6 \$ 60.00 \$ 37.81 7 \$ 70.00 \$ 40.84 8 \$ 80.00 \$ 43.22 9 \$ 90.00 \$ 45.02 10 \$ 100.00 \$ 1,000.00 \$ 509.51 Sum = \$ 876.05 Problem 22 a. Value of Store = \$ 100,000 (1.05)/(.10-.05) = \$ 2,100,000 b. Growth rate needed to justify a value of \$ 2.5 million, 100000(1+g)/(.10-g) = 2500000 Solving for g, g = 5.77%

Last Update: 1/20/98

Name: Aswath Damodaran