CHAPTER 2 How to Calculate Present Values

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1 CHAPTER How to Calculate Present Values Answers to Problem Sets. If the discount factor is.507, then.507 x. 6 = $. Est time: DF x 39 = 5. Therefore, DF =5/39 =.899. Est time: PV = 374/(.09) 9 = 7.. Est time: PV = 43/ /(.5 ) + 797/(.5 3 ) = = $,003. Est time: FV = 00 x.5 8 = $ Est time: NPV =, /.09 = 4.67 (cost today plus the present value of the perpetuity). Est time: PV = 4/(.4.04) = $40. Est time: a. PV = /.0 = $0. b. Since the perpetuity will be worth $0 in year 7, and since that is roughly double the present value, the approximate PV equals $5. You must take the present value of years 7 and subtract from the total present value of the perpetuity: PV = (/.0)/(.0) 7 = 0/= $5 (approximately). - 4 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

2 c. A perpetuity paying $ starting now would be worth $0, whereas a perpetuity starting in year 8 would be worth roughly $5. The difference between these cash flows is therefore approximately $5. PV = $0 $5= $5 (approximately). d. PV = C/(r g) = 0,000/(.0-.05) = $0,000. Est time: a. PV = 0,000/(.05 5 ) = $7,835.6 (assuming the cost of the car does not appreciate over those five years). b. The six-year annuity factor [(/0.08) /(0.08 x (+.08) 6 )] = You need to set aside (,000 six-year annuity factor) =, = $55,475. c. At the end of six years you would have.08 6 (60,476-55,475) = $7,935. Est time: a. FV =,000e. x 5 =,000e.6 = $,8.. b. PV = 5e. x 8 = 5e -.96 = $.94 million. c. PV = C (/r /re rt ) =,000(/. /.e. x5 ) = $3,9. Est time: a. FV = 0,000,000 x (.06) 4 =,64,770. b. FV = 0,000,000 x ( +.06/) (4 x ) =,704,89. c. FV = 0,000,000 x e (4 x.06) =,7,49. Est time: a. PV = $00/.0 0 = $ b. PV = $00/.3 0 = $9.46. c. PV = $00/.5 5 = $ by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

3 d. PV = $00/. + $00/. + $00/. 3 = $40.8. Est time: a. DF r r = = 0.50%. b. DF ( r ) (.05) c. AF = DF + DF = =.74. d. PV of an annuity = C [annuity factor at r% for t years]. Here: $4.65 = $0 [AF3] AF3 =.465 e. AF3 = DF + DF + DF3 = AF + DF3.465 =.74 + DF3 Est time: 06-0 DF3 = The present value of the 0-year stream of cash inflows is: Thus: PV $70,000 $886, (.4) NPV = $800,000 + $886, = +$86, At the end of five years, the factory s value will be the present value of the five remaining $70,000 cash flows: PV $70,000 $583, (.4) Est time: by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

4 5. NPV 0 t 0 C t (.) t $50,000 $57,000 $75,000 $80,000 $85,000 $380, Est time: 0-05 $9,000 $9,000 $80,000 $68,000 $50, $3, a. Let St = salary in year t. PV 30 t ,000 (.05) t (.08) ( ) t 30 (.05) ( ) (.08) 4 0, 30 $760,66.53 b. PV(salary) x 0.05 = $38,033.3 Future value = $38,033.3 x (.08) 30 = $38,74.30 c. PV C r r ( r) t $ 38,74.30 C (.08) C $ 38, (.08) $38, Est time: Period Present Value 0 400, by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

5 +00,000/. = +89, ,000/. = +59, ,000/. 3 = +3, Total = NPV = $6,58.56 Est time: We can break this down into several different cash flows, such that the sum of these separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project. (All dollar figures are in millions.) Cost of the ship is $8 million PV = $8 million Revenue is $5 million per year, and operating expenses are $4 million. Thus, operating cash flow is $ million per year for 5 years. PV $ million $8.559 million (.08) Major refits cost $ million each and will occur at times t = 5 and t = 0. PV = ($ million)/ ($ million)/.08 0 = $.88 million. Sale for scrap brings in revenue of $.5 million at t = 5. PV = $.5 million/.08 5 = $0.473 million. Adding these present values gives the present value of the entire project: NPV = $8 million + $8.559 million $.88 million + $0.473 million NPV = $.56 million Est time: a. PV = $00,000. b. PV = $80,000/. 5 = $0, c. PV = $,400/0. = $95,000. d. PV $9,000 $07, (.) e. PV = $6,500/( ) = $9, Prize (d) is the most valuable because it has the highest present value. Est time: by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

6 . Mr. Basset is buying a security worth $,000 now, which is its present value. The unknown is the annual payment. Using the present value of an annuity formula, we have: PV C r r ( r) t $,000 C (.08) C $,000 Est time: (.08) $, Assume the Zhangs will put aside the same amount each year. One approach to solving this problem is to find the present value of the cost of the boat and then equate that to the present value of the money saved. From this equation, we can solve for the amount to be put aside each year. PV(boat) = $,000/(.0) 5 = $,48 PV(savings) = annual savings Because PV(savings) must equal PV(boat): (.0) 5 Annual savings $, (.0) Annual savings $,48 $3, (.0) Another approach is to use the future value of an annuity formula: Annual savings (.0) 5 Annual savings = $ 3,76.0 $,000 Est time: by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

7 . The fact that Kangaroo Autos is offering free credit tells us what the cash payments are; it does not change the fact that money has time value. A 0% annual rate of interest is equivalent to a monthly rate of 0.83%: rmonthly = rannual / = 0.0/ = = 0.83% The present value of the payments to Kangaroo Autos is: $,000 $300 $8, (.0083) A car from Turtle Motors costs $9,000 cash. Therefore, Kangaroo Autos offers the better deal, i.e., the lower present value of cost. Est time: The NPVs are: $30,000 $870,000 at 5% NPV $700,000 $7, (.05) $30, ,000 at 0% NPV $700,000 $46,8.0 (.0) $30, ,000 at 5% NPV $700,000 $6,068.5 (.5) The figure below shows that the project has zero NPV at about 3.5%. As a check, NPV at 3.5% is: $30, ,000 NPV $700,000 $ (.35) -7 4 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

8 Est time: a. This is the usual perpetuity, and hence: PV C r $ $,48.57 b. This is worth the PV of stream (a) plus the immediate payment of $00: PV = $00 + $,48.57 = $,58.57 c. The continuously compounded equivalent to a 7% annually compounded rate is approximately 6.77%, because: Thus: Ln(.07) = or e =.0700 C PV r $ $, by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

9 Est time: 06-0 Note that the pattern of payments in part (b) is more valuable than the pattern of payments in part (c). It is preferable to receive cash flows at the start of every year than to spread the receipt of cash evenly over the year; with the former pattern of payment, you receive the cash more quickly. 5. a. PV = $ billion/0.08 = $.5 billion. b. PV = $ billion/( ) = $5.0 billion. c. PV $ billion $9.88 billion (.08) d. The continuously compounded equivalent to an 8% annually compounded rate is approximately 7.7%, because: Est time: 06-0 Thus: Ln(.08) = or e =.0800 PV $ billion ) e (0.077)( $0.3 billion This result is greater than the answer in Part (c) because the endowment is now earning interest during the entire year. 6. With annual compounding: FV = $00 (.5) = $, With continuous compounding: FV = $00 e (0.5 ) = $, Est time: One way to approach this problem is to solve for the present value of: () $00 per year for 0 years, and () $00 per year in perpetuity, with the first cash flow at year. If this is a fair deal, these present values must be equal, and thus we can solve for the interest rate (r). The present value of $00 per year for 0 years is: -9 4 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

10 PV $00 r (r) ( r) 0 The present value, as of year 0, of $00 per year forever, with the first payment in year, is: PV0 = $00/r. At t = 0, the present value of PV0 is: ( r) PV 0 $00 r Equating these two expressions for present value, we have: $00 $ r (r) ( r) ( r) r Using trial and error or algebraic solution, we find that r = 7.8%. Est time: Assume the amount invested is one dollar. Let A represent the investment at %, compounded annually. Let B represent the investment at.7%, compounded semiannually. Let C represent the investment at.5%, compounded continuously. After one year: FVA = $ ( + 0.) = $.0 FVB = $ ( ) = $.4 FVC = $ e (0.5 ) = $.9 After five years: FVA = $ ( + 0.) 5 = $.763 FVB = $ ( ) 0 = $.7657 FVC = $ e (0.5 5) = $.777 After twenty years: FVA = $ ( + 0.) = $ FVB = $ ( ) 40 = $9.793 FVC = $ e (0.5 ) = $9.974 The preferred investment is C. Est time: by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

11 9. Because the cash flows occur every six months, we first need to calculate the equivalent semiannual rate. Thus,.08 = ( + r/) => r = semiannually compounded APR. Therefore the rate for six months is 7.846/, or 3.93%: PV $ 00, 000 $ 00, 000 $ 846, ( ) Est time: a. Each installment is: $9,4,73/9 = $495,87. PV $495,87 $4,76, (.08) b. If ERC is willing to pay $4. million, then: Est time: 06-0 $4,0,000 $495,87 r r ( r) 9 Using Excel or a financial calculator, we find that r = 9.8%. 3. a. PV $70,000 $40, (.08) b. Year Beginningof-Year Balance ($) Year-End Interest on Balance ($) Total Year-End Payment ($) Amortization of Loan ($) End-of-Year Balance ($) 40, ,8.8 70, , , , , , , , , , , ,.87 79, ,489.7, , , , , , , , , , , , , , , , , , , ,84.8 5, , , Est time: This is an annuity problem with the present value of the annuity equal to $ million (as of your retirement date), and the interest rate equal to 8% - 4 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

12 with 5 time periods. Thus, your annual level of expenditure (C) is determined as follows: PV C t r r ( r) $,000,000 C (.08) 5 C $,000, (.08) 5 $33,659 With an inflation rate of 4% per year, we will still accumulate $ million as of our retirement date. However, because we want to spend a constant amount per year in real terms (R, constant for all t), the nominal amount (Ct) must increase each year. For each year t: R = Ct /( + inflation rate) t Therefore: PV [all Ct ] = PV [all R ( + inflation rate) t ] = $,000,000 ( 0.04) ( 0.08) ( 0.04) ( 0.08) ( 0.04)... ( 0.08) 5 R 5 R [ ] = $,000,000 R.390 = $,000,000 R = $77,95 $,000,000 ( 0.08) Alternatively, consider that the real rate is Then, redoing ( 0.04) the steps above using the real rate gives a real cash flow equal to: C $,000, (.03846) 5 $77,95 Thus C = ($77,95.04) = $85,070, C = $9,473, etc. Est time: a. PV $50,000 $430, (.055) b. The annually compounded rate is 5.5%, so the semiannual rate is: - 4 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

13 Est time: 06-0 (.055) (/) = 0.07 =.7% Since the payments now arrive six months earlier than previously: PV = $430, = $44, In three years, the balance in the mutual fund will be: FV = $,000,000 (.035) 3 = $,08,78 The monthly shortfall will be: $5,000 ($7,500 + $,500) = $6,000. Annual withdrawals from the mutual fund will be: $6,000 = $7,000. Assume the first annual withdrawal occurs three years from today, when the balance in the mutual fund will be $,08,78. Treating the withdrawals as an annuity due, we solve for t as follows: PV C ( r) t r r ( r) $,08,78 $7, t (.035) Using Excel or a financial calculator, we find that t =.38 years. Est time: a. PV = /. = $6.667 million. b. PV = $ $ million (.) c. PV = /(.-.03) = $. million.03 d. PV = $ $8. 06million. ( ) ( ) (.) Est time: a. First we must determine the -year annuity factor at a 6% interest rate. -year annuity factor = [/.06 /.06(.06) ) = by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

14 b. Once we have the annuity factor, we can determine the mortgage payment. Mortgage payment = $0,000/.4699 = $7, Year Beginning Balance ($) Year-End Interest ($) Total Year- End Payment ($) Amortization of Loan ($) End-of-Year Balance ($) 0,000.00, , , , ,563.09, , , , ,799.96, , ,08.9 8, , , , , , ,5.60 0, , , , , ,6.0 7, ,75.8 6, , , , , , , ,6.8 7, , , , , , , , ,5.73 8,5.36 7, , , , , , , , , ,6.03 7, , , ,79.6 6, , , , , , ,436.9, , , , ,436.9, , , , , , , ,4.74 3,65.4 7, , , ,609.07, , , , ,968.7,98. 7, , , , , , c. Nearly 69% of the initial loan payment goes toward interest ($,000/$7, =.688). Of the last payment, only 6% goes toward interest (987.4/7, =.06). Est time: -5 After 0 years, $7,66. has been paid off ($0,000 remaining balance of $8,338.79). This represents only 36% of the loan. The reason that less than half of the loan has paid off during half of its life is due to compound interest. 37. a. Using the Rule of 7, the time for money to double at % is 7/, or six years. More precisely, if x is the number of years for money to double, then: (.) x = -4 4 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any

15 Using logarithms, we find: x (ln.) = ln x = 6. years b. With continuous compounding for interest rate r and time period x: e rx = Taking the natural logarithm of each side: rx = ln() = Thus, if r is expressed as a percent, then x (the time for money to double) is: x = 69.3/(interest rate, in percent). Est time: Spreadsheet exercise. Est time: a. This calls for the growing perpetuity formula with a negative growth rate (g = 0.04): $ million PV 0.0 ( 0.04) $ million 0.4 $4.9 million b. The pipeline s value at year (i.e., at t = ), assuming its cash flows last forever, is: Est time: 06-0 PV C C ( g) r g r g With C = $ million, g = 0.04, and r = 0.0: PV ($ million) ( 0.04) 0.4 $0.884 million $ million Next, we convert this amount to PV today, and subtract it from the answer to Part (a): $6.34 million $4.9 million (.0) PV $3.35 million -5 4 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any