### Time Value of Money and Present Value

Date: 14/11/2012 52. Annuities: You are saving for the college education of your two children. They are two years apart in age; one will begin college 15 years from today and the other will begin 17 years from today. You estimate your children’s college expenses to be $23,000 per year per child, payable at the beginning of each school year. The annual interest rate is 5. 5 percent. How much money must you deposit in account each year to fund your children’s education? Your deposits begin one year from today. You will make your last deposit when your oldest child enters college. Assume four years of college

Solution: Cost of 1 year at university = 23,000 N=4 I=5. 5% PMT=23,000 CPT PV = 80,618. 45 For the first child the PV = 80,618. 45/ (1. 055) ^14 = $38,097. 81 For the second child the PV = 80,618. 45/ (1. 055) ^16 = $34,229. 07 Therefore the total cost today of your children’s college expense will be the addition of the 2 = $72,326. 88 This is the present value of my annual savings, which are an annuity, so to get the amount I am supposed to save each year would be: PV=72,326. 88 N=15 I=5. 5 CPT PMT = 7,205. 6 57. Calculating Annuity Values: Bilbo Baggins wants to save money to meet three objectives.

First, he would like to be able to retire 30 years from now with retirement income of $25,000 per month for 20 years, with the first payment received 30 years and 1 month from now. Second, he would like to purchase a cabin in Rivendell in 10 years at an estimated cost of $350,000. Third, after he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $750,000 to his nephew Frodo. He can afford to save $2,100 per month for the next 10 years. If he can earn an 11 percent EAR before he retires and an 8 percent EAR after he retires, how much will he have to save each month in years 11 through 30? Solution:

First we get the FV of the 2,100 savings over 10 years Bilbo Baggins can afford to save $2,100 dollars per month for the next 10 years therefore at 10 years he would have saved: PMT = 2,100 I = 10. 48 / 12 = 0. 873 N = 10 x 12 = 120 CPT FV = $442,201. 15 So after 10 years he would be able to purchase his yacht at the price of $350,000, and he would be left with a balance of $92,201. 15 This $92,201. 15 will be our current PV at year 10. At year 30, the year when Bilbo retires, the $92,201. 15 would become 92,201. 15*(1. 11) ^20 = $620,283. 23 Second we have to find out how much the inheritance of 750,000 would be at year 30: 750,000/1. 8^20= $160,911. 16 Third In order for him to be able to withdraw a sum of 25,000 per month for the next 20 years after his retirement, we should now calculate this annuity’s present value: N= 20 x 12 = 240 I= 7. 72 / 12 = 0. 643 PMT= 25,000 CPT PV = $3,052,135. 26 Adding up the PV’s of the $750,000 and the annuity, we will get $3,213,046. 32 We will subtract the future value at year 30 of the $92,201. 15 ($620,283. 23) which we saved at year 10 from $3,213,046. 32 to get $2,592,763. 09 We are now left with an annuity that pays $2,592,763. 09 at year 30, and a time period of 20 years (yr11-30) To calculate the yearly PMT, we have

FV= $2,592,763. 09 I= 10. 48 / 12 = 0. 873 N= 20 x 12 = 240 CPT PMT = 3,207. 33 Therefore the monthly PMT Bilbo would have to save each month through years 11-30 would be = $3,207. 33 34. Valuing bonds: Mallory Corporation has two different bonds, currently outstanding. Bond M has a face value of $20,000 and matures in twenty years. The bond makes no payments for the first six years, then pays $1,200 every 6 months over the subsequent eight years, and finally pays $1,500 every 6 months over the last years. Bond N also has a face value of $20,000 and a maturity of 20 years; it makes n coupon payments over the life of the bond.

If the required return on both these bonds is 10% compounded semiannually, what is the current price of bond M? Of bond N? Solution: The price of a bond is equal to PV of expected future cash flows Bond M: Face value 20,000 Present value of 20,000 = 20,000/ (1. 05) ^40 = $2,840. 91 First we need to get the present value of the annuity for the 1,500 semiannual PMTs at year 14 Present Value of Annuity = $13,295 $13,295 becomes $3,391 at year 0 We then get the annuity of the 1,200 semiannual PMTs at year 6, and then at Present Value $13,005 at year 6 with a PV of $7,242 at year 0 The sum of the 3 PV’s gives us the value of the bond ,841 + 3,391 + 7,242 = $13,474 Bond N Face value 20,000 Present value of 20,000 = 20,000/ (1. 05) ^40 = $2,840. 91 38. Non-constant growth: Storico Co. just paid a dividend of aud 3. 5 per share. The company will increase its dividend by 20% next year, and will then reduce its dividend growth rate by 5% per year, until it reaches the industry average of 5% industry average growth, after which the company will keep a constant growth rate forever. If the required return on Storico stock is 13%, what will a share of stock sell for today? Solution : D0 = $3. 5 D1= 3. 5*1. 2= $4. 2 D2= 4. 2*1. 15= $4. 3 D3=4. 83*1. 1= $5. 31 D4=5. 31*1. 05= $5. 58 Since the first 4 periods are different we get the PV of each one alone, then as of the 4th year we get the perpetuity of the rest, and sum them up to get the final NPV We now get the PV of each Dividend PV D1 = 4. 2/ (1. 13) = $3. 72 PV D2 = 4. 83/ (1. 13) ^2 = $3. 78 PV D3 = 5. 31/ (1. 13) ^3 = $3. 68 So the PVs of D1+D2+D3 = $11. 18 NPV of perpetuity at constant growth = 5. 58(0. 08) / (1. 13) ^3 = 69. 75 / (1. 13) ^3 = $48. 34 NPV perpetuity + NPV dividends = NPV price of stock today 48. 34 + 11. 18 = $59. 52