Fundamentals of corporate finance brealey chapter 04 the time value of money

Solutions to Chapter The Time Value of Money Note: Unless otherwise stated, assume that cash flows occur at the end of each year a b c d 100/(1.08)10 100/(1.08)20 100/(1.04)10 100/(1.04)20 a b c d 100  (1.08)10 100  (1.08)20 100  (1.04)10 100  (1.04)20 With simple interest, you earn 4% of $1000, or $40 each year There is no interest on interest After 10 years, you earn total interest of $400, and your account accumulates to $1400 With compound interest, your account grows to 1000  (1.04)10 = $1480 Therefore $80 is interest on interest FV = 700 = = = = $46.32 $21.45 $67.56 $45.64 = = = = $215.89 $466.10 $148.02 $219.11 PV = 700/(1.05)5 = $548.47 Present Value Years Future Value Interest Rate* a $400 11 $684 5% = ()1/11 – b $183 $249 8% = ()1/4 – c $300 $300 0% = ()1/7– To find the interest rate, we rearrange the equation FV = PV  (1 + r)n to conclude that r = ()1/n - To use a financial calculator for (a) enter PV= (-)400, FV = 684, PMT = 0, n = 11 and compute the interest rate 4­1 Copyright © 20096 McGraw-Hill Ryerson Limited You should compare the present values of the two annuities a b Discount Rate 5% 20% Present Value of 10-year, $1000 annuity 7721.73 4192.47 Present Value of 15-year, $800 annuity 8303.73 3740.38 When the interest rate is low, as in part (a), the longer (i.e., 15-year) but smaller annuity is more valuable because the impact of discounting on the present value of future payments is less severe When the interest rate is high, as in part (b), the shorter but higher annuity is more valuable In this case, with the 20 percent interest rate, the present value of more distant payments is substantially reduced, making it better to take the shorter but higher annuity PV = 200/1.05 + 400/1.052 + 300/1.053 = 190.48 + 362.81 + 259.15 = $812.44 In these problems, you can either solve the equation provided directly, or you can use your financial calculator setting PV = ()400, FV = 1000, PMT = 0, i as specified by the problem Then compute n on the calculator a 400  (1 + 04)t = 1,000 t = 23.36 periods b 400  (1 + 08)t = 1,000 t = 11.91 periods c 400  (1 + 16)t = 1,000 t = 6.17 periods Note: To solve directly, use the natural log function, ln For example, for (a), ln[ (1.04)t ] = ln[1000/400] t × ln[1.04] = 0.91629 t = 0.91629/.03922 = 23.36 period Using the calculator: PV = (-)400, FV = 1000, i = 4, compute n to get n = 23.36 a b c d PV = 100 × PVIFA(.08,10) = 100 × 6.7101 = 671.01 PV = 100 × PVIFA(.08,20) = 100 × 9.8181 = 981.81 PV = 100 × PVIFA(.04,10) = 100 × 8.1109 = 811.09 PV = 100 × PVIFA(.04,20) = 100 × 13.5903 = 1,359.03 10 a b c d FV = 100 × FVIFA(.08,10) = 100 × 14.4866 = 1,448.66 FV = 100 × FVIFA(.08,20) = 100 × 45.7620 = 4,576.20 FV = 100 × FVIFA(.04,10) = 100 × 12.0061 = 1,200.61 FV = 100 × FVIFA(.04,20) = 100 × 29.7781 = 2,977.81 4­2 Copyright © 20096 McGraw-Hill Ryerson Limited 11 APR Compounding Period Per Period Rate, APR/m Effective annual rate 12 a 12% month (m = 12/yr) 12/12 =.01 1.01  = 1268 = 12.68% b 8% months (m = 4/yr) 08/4 = 02 1.02  = 0824 = 8.24% c 10% months (m = 2/yr) 10/2 = 05 1.05  = 1025 = 10.25% 12 Effective Annual Rate, EAR Compounding Period Number of Periods per year, m a 10.0% month 12 1.1 – = 008 b 6.09% months 1.0609  = 03 c 8.24% months 1.0824  = 02 Per period rate, (1+EAR)1/m -1 APR, m × per period rate 1/12 12×.008 = 096 = 9.6% 1/2 1/4 2×.03 = 06 = 6% 4×.02 = 08 = 8% 13 We need to find the value of n for which 1.08n = You can solve to find that n = 9.01 years On a financial calculator you would enter PV = ()1, FV = 2, PMT = 0, i = and then compute n 14 Semiannual compounding means that the 8.5 percent loan really carries interest of 4.25 percent per half year Similarly, the 8.4 percent loan has a monthly rate of percent APR Period m Effective annual rate = (1 + per period rate) m – 8.5% 8.4% months month 12 (1.0425)2  = 0868 = 8.68% (1.007)12  = 0873 = 8.73% 4­3 Copyright © 20096 McGraw-Hill Ryerson Limited Choose the 8.5 percent loan for its slightly lower effective rate 15 APR = 1%  52 = 52% 52 EAR = (1 + 01)  = 6777 = 67.77% 16 Our answer assumes that the investment was made at the beginning of 1900 and now it is the end of 2008 Thus the investment was for 106 years (2008 – 1900 + 1) a b 17 1000  (1.05)109 = $204,001.61 PV  (1.05)109 = 1,000,000 implies that PV = $4,901.92 $1000  1.05 = $1050.00 $1050  1.05 = $1102.50 $52.50 First-year interest = $50 Second-year interest = $1102.50  $1050 = After years, your account has grown to 1000  (1.05)9 = $1551.33 After 10 years, your account has grown to 1000  (1.05)10 = $1628.89 Interest earned in tenth year = $1628.89  $1551.33 = $77.56 18 Method 1: If you earned simple interest (without compounding) then the total growth in your account after 25 years would be 4% per year  25 years = 100%, and your money would double With compound interest, your money would grow faster, and therefore would require less than 25 years to double Method 2: Another quick way to answer the question is with the Rule of 72 Dividing 72 by gives 18 years, which is less than 25 The exact answer is 17.673 years, found by solving 2000 = 1000  (1.04)n [On your calculator, input PV = (-) 1000, FV = 2000, i = 4, PMT = 0, and compute the number of periods.] 19 We solve 422.21  (1 + r)10 = 1000 This implies that r = 9% [On your calculator, input PV = (-)422.21, FV = 1000, n = 10, PMT = 0, and compute the interest rate.] 4­4 Copyright © 20096 McGraw-Hill Ryerson Limited 20 The number of payment periods: n = 12 × = 48 If the payment is denoted PMT, then PMT  annuity factor( %, 48 periods) = 8,000 PMT = $202.90 The monthly interest rate is 10/12 = 8333 percent Therefore, the effective annual interest rate on the loan is (1.008333) 12  = 1047 = 10.47 percent 21 a PV = 100  annuity factor(6%, periods) = 100  = $267.30 b 22 a If the payment stream is deferred by an extra year, each payment will be discounted by an additional factor of 1.06 Therefore, the present value is reduced by a factor of 1.06 to 267.30/1.06 = $252.17 This is an annuity problem with PV = (-)80,000, PMT = 600, FV = 0, n = 20  12 = 240 months Use a financial calculator to solve for i, the monthly rate on this annuity: i = 5479% 12 EAR = (1 +.005479)  = 06776 = 6.776% APR = 12 × monthly interest rate = 12 × 5479% = 6.5748%, compounded monthly 23 b Again use a financial calculator and enter n = 240, i = 5%, FV = 0, PV = ()80,000 and compute PMT = $573.14 a With PV = 9,000 and FV = 10,000, the annual interest rate is defined by 9,000  (1 + r) = 10,000, which implies that r = 11.11% b Your present value is 10,000 (1  d), and the future value you pay back is 10,000 Therefore, the annual interest rate is determined by: PV  (1 + r) = FV 4­5 Copyright © 20096 McGraw-Hill Ryerson Limited [10,000 (1 – d)]  (1 + r) = 10,000 1+r=  r= 1=>d Since < d < 1, then – d < and d/(1 – d) > d So r must be greater than d c With a discount interest loan, the discount is calculated as a fraction of the future value of the loan In fact, the proper way to compute the interest rate is as a fraction of the funds borrowed Since PV is less than FV, the interest payment is a smaller fraction of the future value of the loan than it is of the present value Thus, the true interest rate exceeds the stated discount factor of the loan 24 If we assume cash flows come at the end of each period (ordinary annuity) when in fact they actually come at the beginning (annuity due), we discount each cash flow by one period too many Therefore we can obtain the PV of an annuity due by multiplying the PV of an ordinary annuity by (1 + r) Similarly, the FV of an annuity due also equals the FV of an ordinary annuity times (1 + r) Because each cash flow comes at the beginning of the period, it has an extra period to earn interest compared to an ordinary annuity 25 a Solve for i in the following equation: 10,000 = 275 × PVIFA(i, 48) Using the calculator, set PV = -10,000, PMT = 275, FV = 0, n = 48 and solve for i i= 1.19544% per month APR = 12 × 1.19544% = 14.3453% EAR = (1 + 0119544)12 – = 153271, or 15.3271% b Annual payment = 12 × 275 = 3,300 Repeat the steps in (a) to find the EAR of this car loan to see which loan is charging the lower interest rate: Solve for i in the following equation: 10,000 = 3,330 × PVIFA(i, 4) Using the calculator, set PV = -10,000, PMT = 3,300, FV = 0, n = and solve for i i= 12.11% per month Little Bank's loan interest rate of 12.11% is less than the EAR of 15.53% on Big Bank's loan With a lower interest rate, Little Bank's loan is better c Find the annual loan payment, P, such that 10,000 = X × PVIFA(15.3271%, 4) Using the calculator, set PV = -10,000, FV = 0, n = 4, i = 15.3271 and solve for PMT = $3,525.86 By comparison, 12 times $275 per month is $3,300 The annual payment on a 4-year loan equivalent to $275 per month for 48 months is greater than 12 times the monthly payment of $275 because of the benefit of delaying payment to the end of each year The borrower gets to delay payment and therefore is better off If Little Bank doesn't charge at least $3,525.86 annually, it earns less on its loan than Big Bank earns on its loan 4­6 Copyright © 20096 McGraw-Hill Ryerson Limited 26 27 a Compare the present value of the lease to cost of buying the truck PV lease = 8,000 × PVIFA(7%, 6) = -$38,132.32 It is cheaper to lease than buy because by leasing the truck will cost only $38,132.32, rather than $40,000 Of course, the crucial assumption here is that the truck is worthless after years If you buy the truck, you can still operate it after years If you lease it, you must return the truck and replace it b If the lease payments are payable at the start of each year, then the present value of the lease payments are: PV annuity due lease = 8,000 + 8,000 × PVIFA(7%, 5) = 8,000 + 32,801.58 = $40,801.58 Note too that PV of an annuity due = PV of ordinary annuity  (1 + r) Therefore, with immediate payment, the value of the lease payments increases from its value in the previous problem to $38,132  1.07 = $40,801 which is greater than $40,000 (the cost of buying a truck) Therefore, if the first payment on the lease is due immediately, it is cheaper to buy the truck than to lease it a Compare the PV of the payments Assume the product sells for $100 Installment plan: Down payment = 25 × 100 = 25 Three installments of 25 × 100 = 25 PV = 25 + 25  annuity factor(6%, years) = $91.83 Pay in full: Payment net of discount = $90 Choose this payment plan for its lower present value of payments Note: The pay-in-full payment plan will have the lowest present value of payments, regardless of the chosen product price b 28 Installment plan: PV = 25  annuity factor(6%, years) = $86.63 Now the installment plan offers the lower present value of payments a PMT  annuity factor(12%, years) = 1000 PMT  3.6048 = 1000 PMT = $277.41 b If the first payment is made immediately instead of in a year, the annuity factor will be greater by a factor of 1.12 Therefore PMT  (3.6048  1.12) = 1000 PMT = $247.69 4­7 Copyright © 20096 McGraw-Hill Ryerson Limited 29 This problem can be approached in two steps First, find the PV of the $10,000, 10-year annuity as of year 3, when the first payment is exactly one year away (and is therefore an ordinary annuity) Then discount the value back to today Using a financial calculator, 1) PMT = 10,000; FV = 0; n = 10; i = 6% Compute PV3 = $73,600.87 2) PV0 = = = $61,796.71 A second way to solve the problem is the take the difference between a 13-year annuity and a 3-year annuity, valued as of the end of year 0: PV of delayed annuity = 10,000 × PVIFA(6%,13) – 10,000 × PVIFA(6%,3) = 10,000 × (8.852683 – 2.673012) = 10,000 × 6.179671 = $61,796.71 30 Note: Assume that this is a Canadian mortgage The monthly payment is based on a $175,000 loan with a 300-month (12 × 25 years) amortization The posted interest rate of percent has a 6-month compounding period Its EAR is (1 + 06/2) – = 0609, or 6.09% The monthly interest rate equivalent to 6.09% annual is (1.0609) 1/12 – = 004939, or 0.4939% PMT  annuity factor(.4939%, 300) = 175,000 PMT = $1,119.71 When the mortgage expires in years, there will be 20 years remaining in the amortization period, or 240 months The loan balance in five years will be the present value of the 240 payments: Loan balance in years = $1,119.71  Annuity factor (.4939%, 240 periods) = $157,208 31 The EAR of the posted 7% rate is (1 + 07/2)2 – = 071225 The monthly interest rate equivalent is (1.071225)1/12 – = 00575, or 0.575% The payment on the mortgage is computed as follows: PMT  annuity factor (.575%, 300 periods) = 350,000 PMT = $2,451.44 per month If you pay the monthly mortgage payment in two equal installments, you will pay $2,451.44/2, or $1,225.72 every two weeks Thus each year you make 26 payments The bi-weekly equivalent of the 7% posted interest rate is (1.071225) 1/26 – = 002649, or 2649% every two weeks Now calculate the number of periods it will take to pay off the mortgage: 4­8 Copyright © 20096 McGraw-Hill Ryerson Limited $1,225.72  Annuity factor (.2649%, n periods) = $350,000 Using the calculator: PMT = 1,225.49, PV = (-)350,000, i = 2649 and compute n = 533.84 This is the number of bi-weekly periods Divide by 26 to get the number of years: 533.84/26 = 20.5 years If you pay bi-weekly, the mortgage is paid off 5.5 years sooner than if you pay monthly 32 a Input PV = (-)1,000, FV = 0, i = 8%, n = 4, compute PMT which equals $301.92 b Time c 33 Loan Balance $1,000.00 $778.08 $538.41 $279.56 Year End Interest Due on Balance $ 80.00 $62.25 $43.07 $22.37 Year End Payment $301.92 $301.92 $301.92 $301.92 — Amortization of Loan $221.92 $239.67 $258.85 $279.56 — 301.92  annuity factor (8%, years) = 301.92 × 2.5771 = $778.08, which equals the loan balance after one year The loan repayment is an annuity with present value $4248.68 Payments are made monthly, and the monthly interest rate is 1% We need to equate this expression to the amount borrowed, $4248.68, and solve for the number of months, n [On your calculator, input PV = () 4248.68, FV = 0, i = 1%, PMT = 200, and compute n.] The solution is n = 24 months, or years The effective annual rate on the loan is (1.01) 12  = 1268 = 12.68% 34 The present value of the $2 million, 20-year annuity, discounted at 8%, is $19,636,295 If the payment comes one year earlier, the PV increases by a factor of 1.08 to $21,207,198 35 The real rate is zero With a zero real rate, we simply divide her savings by the years of retirement: $450,000/30 = $15,000 per year 36 Per month interest = 6%/12 = 5% per month FV in year (12 months) = 1000  (1.005)12 = $1,061.68 4­9 Copyright © 20096 McGraw-Hill Ryerson Limited FV in 1.5 years (18 months) = 1000  (1.005)18 = $1,093.93 37 You are repaying the loan with an annuity of payments The PV of those payments must equal $100,000 Therefore, 804.62  annuity factor(r, 360 months) = 100,000 which implies that the interest rate is 750% per month [On your calculator, input PV = ()100,000, FV = 0, n = 360, PMT = 804.62, and compute the interest rate.] The effective annual rate is (1.00750)12  = 0938 = 9.38% If the lender is a Canadian financial institution, the quoted rate will be the APR for a 6-month compounding period: (1 + )2 – = 0938 = (1.0938)1/2 -1 = 04585 APR = × [(1.0938)1/2 -1] = 0917 or 9.17%, which is lower than the effective annual rate Note: A simpler APR calculation is 750%  12 = 9% However, this is not how Canadian mortgage lenders calculate their APRs 38 39 EAR = e.04 -1 = 1.0408 -1 = 0408 = 4.08% The PV of the payments under option (a) is 11,000, assuming the $1,000 rebate is paid immediately The PV of the payments under option (b) is $250  annuity factor(1%, 48 months) = $9,493.49 Option (b) is the better deal 40 100  e.10×6 = $182.21 100  e.06×10 = $182.21 41 Your savings goal is 30,000 = FV You currently have in the bank 20,000 = PV The PMT = (-) 100 and r = 5% Solve for n to find n = 44.74 months 4­10 Copyright © 20096 McGraw-Hill Ryerson Limited $100,000  annuity factor(4.85%, years) = $434,749 b 69 If cash flow is level in nominal terms, use the 8% nominal interest rate to discount The annuity factor is now 3.99271 and the cash flow stream is worth only $399,271 a $1 million will have a real value of $1 million/(1.03) 45 = $264,439 b At a real rate of 2%, this can support a real annuity of $228,107/annuity factor(2%, 20 years) = $264,439/16.3514 = $16,172 To solve this on a calculator, input n = 20, i = 2, PV = 264,439, FV = 0, and compute PMT 70 According to the Rule of 72, at an interest rate of 8%, it will take 72/8 = years for your money to double For it to quadruple, your money must double, and then double again This will take approximately 18 years 71 (1.23)12 – = 10.99 Prices increased by 1,099 percent per year 72 Using the perpetuity formula, the 4% consol will sell for £4/.06 = £66.67 The 1/2% consol will sell for £2.50/.06 = £41.67 73 The savings calculator can be reached directly from the following link: http://stockgroup.financialpost.com/basicsavingscalculator.asp?sh=bscalc Total value after 30 years without any savings = $1,000 x (1.06 30) = $5,743.49 In the savings calculator enter Years to Save = 30, Total Cost of Expense = $0, Current Savings = $1,000, Deposit Amount = $0, Compounded Annual Rate of Return = 6%, and click Calculate, you will get the same result Total value with after 30 years with $200 savings per month: (1 + monthly rate) 12 –1 = 06  monthly rate = 004867551 Number of periods = 30 x 12 = 360 months Assuming the payments are made at the start of each period: Total Value = $5,743.49 + $200 x FVIFA(360, 4867551%) x (1.004867551) = $5,743.49 + $195,851.31 = $201,594.80 In the calculator enter $200 as Deposit Amount and choose Monthly Deposit Frequency, click calculate and you will get the same result with some rounding errors 4­21 Copyright © 20096 McGraw-Hill Ryerson Limited 74 75 Expected Result: Using "To Buy or To Lease" calculator from www.smartmoney.com, enter $20,000 as the Price of Car, $350 as Down Payment, $350 as Monthly payment of lease, 36 months as Lease term, 10% Rate of return, and $10,000 as the value of the car at the end of the lease The calculator will calculate the value of alternative investments at the end of lease term of $11,510 which is more than the value of the car at the end of the lease ($10,000) Thus, lease is a better option a $30,000  annuity factor(10%, 15 years) = $228,182 b Fin the annual payment, PMT, such that PMT × future value annuity factor(10%, 30 years) = 228,182 Using the calculator, PV = 0, n=30, i= 10%, FV= (-) 228,182 Compute PMT = 1,387 You must save $1,387 annually c 1.00  (1.04)30 = $3.24 d We repeat part (a) using the real rate of 1.10/1.04 – = 0577 or 5.77% The retirement goal in real terms is $30,000  annuity factor(5.77%, 15 years) = $295,797 e The future value of your 30-year saving stream must equal this value So we solve for payment (PMT) in the following equation PMT  future value annuity factor(5.77%, 30 years) = $295,797 PMT  75.930 = $295,797 PMT = 3,896 You must save $3896 per year in real terms This value is much higher than the answer to (b) because the rate at which purchasing power grows is less than the nominal interest rate, 10% f 76 If the real amount saved is $3,896 and prices rise at percent per year, then the amount saved at the end of one year in nominal terms will be $3,896  1.04 = $4,052 The thirtieth year will require nominal savings of 3,896  (1.04)30 = $12,636 In the 113 years since the capture of Ned Kelley, from 1880 to 1993, one dollar invested in the bank would have grown to be $35.14 (= × (1.032) 113 ) By contrast, that same dollar invested in the Australian stock market would have grown to $28, 431.22 (= $1 × (1.095)113) My clients are reasonable people but believe that $1 is a ridiculously low amount Given the 3% annual inflation, the 4­22 Copyright © 20096 McGraw-Hill Ryerson Limited real value in 1993 of $1 paid in 1880 is only $0.035 (= $1/(1.03) 113) Surely the efforts of the trackers is worth more than ½ cents! We think a reasonable payment is $14,233 each, half way between the value of $1 invested in the bank and the value of a $1 invested in the stock market 77 The interest rate per three months is 12%/4 = 3% So the value of the perpetuity is $100/.03 = $3,333 78 FV = PV  (1 + r1)  (1 + r2) =  (1.08)  (1.10) = $1.188 PV = = = $0.8418 79 You earned compound interest of 8% for years and 6% for 13 years Your $1000 has grown to 1000  (1.08)8  (1.06)13 = $3,947.90 80 The answer to this question can be found in various ways The key is to pick a common point in time to measure all cash flows Here we pick today as the common reference point Monthly interest rate = 1.061/12 – = 004868 = 4868% Present value today of the funds need for boat: 150,000/(1.06) = 118,814 Funds need for monthly expenses (this is an annuity due) = (2200 + 2200×annuity factor(.4868%, 23 months))/ (1.06) = 37,333.29 Funds need for emergencies = 45,000/(1.06)5 = 33,626.62 Total funds needed = 118,814 + 37,333.29 + 33,626.62 = 189,774 The present value of the savings stream must equal the present value of the expenditures: PMT × annuity factor (.4868%,60 months) = 189,774 The monthly savings must be $3,654.9 (Expect slight variations due to rounding) 81 Interest rate on parents’ car loan = 024/12 = 002 = 2% Monthly car repayment: PMT × annuity factor (.2%, 48) = 4,000 Using the calculator to find PMT = $87.48 Monthly opportunity cost of funds = (1.06) 1/12 – = 004868 = 4868% Summary of Car Costs: 4­23 Copyright © 20096 McGraw-Hill Ryerson Limited Car payment Operating cost Total costs 87.48 87.48 200 200 200 200 287.48 287.48 Month 87.48 … … 47 87.48 48 87.48 200 … 200 287.48 … 287.48 87.48 Present value of car costs = 200 + 287.48 × annuity factor (.4868%, 47 months) + 87.48/(1.004868) 1/48 = $12,320.5 You have to decide whether to charge your friends at the beginning or the end of each month In this calculation, we assume that your friends will pay you at the start of each month The total amount of money needed each month to cover the car costs, given a 6% EAR is: PMT × annuity due factor (.4868%, 48) = 12,320.5 Switch your calculator to the annuity due setting, and solve for PMT =$287 [n=48, i=.4868, FV=0, PV= (-)12,320.5] If your three friends are to cover the cost of the car, they each must pay $287/3 = $95.67 a month If you share in the cost, dividing it four ways, you each pay $287/4 = $71.75 Does it make financial sense to buy the car? Given that the cost of a monthly bus pass is $80, it does not make sense to charge your friends much more than $80 per month, unless the bus trip to school is extremely long relative to the time taken in the car Likewise, you too will not want to pay much more than $80 a month either Of course, you will have access to the car at times when your friends not It depends on the value of the convenience of having access to the car If you charge yourself $80 a month, then you should ask your friends for $69 a month Perhaps that might be viewed as a fair trade-off We have not considered the impact of inflation on the costs of operating the car The car payments won’t change with inflation but the operating costs will You can redo the analysis assuming a 2% annual rate of inflation and see how much higher must be the monthly charge One final note: You may want to consider the benefits for the air quality of taking public transit to school We have not factored into our analysis the economic costs to society of air pollution from the car 82 a Assume there are 26 events per year (52 weekends/2) In years, you attend 26×5 = 130 events The bi-weekly interest rate is (1.09) 1/26 – = 4­24 Copyright © 20096 McGraw-Hill Ryerson Limited 00332 = 332% (assuming that 9% rate is an effective annual rate) Cost of Renting Cost of renting a van per weekend = $100 Mileage charge per weekend = 200 km × $.5 = $100 Fuel costs per weekend = 200 km × $.75 = $150 Total cost = $350 per weekend PV of cost of renting = 350 × annuity factor(.332%, 130 events) = $36,904.6 Cost of Owning Weekend operating costs = 200 km × ($.25 + $.75) = $200 PV of weekend operating costs = 200 × annuity factor(.332%, 130 events) = $21,088.3 Cost today of buying van = $20,000 Expected selling price in years = (1 - 1) × 20,000 = 11,809.8 PV of selling price in years = 11,809.8/(1.09)5 = 7,675.6 PV of insurance (assume insurance is paid in advance) = 1200 × annuity due factor(9%, years) = 5,087.7 Total cost of owning = 21,088.3 + 20,000 - 7,675.6 + 5,087.7 = $38,500.4 Extra cost of purchasing the van rather than renting = $38,500.4 - $36,904.6 = $1,595.8 Although the total cost is higher, the van is available to drive at other times If costs of another vehicle can be saved, then it makes sense to buy Otherwise, it is cheaper to rent, if inflation is not considered b We use the principle of discounting real cash flows at the real discount rate Assume all cash flows are in current year dollars, including the expected resale value of the car The real discount rate is (1.09/1.03) – = 05825 effective annual rate The bi-weekly equivalent rate is (1.05825) 1/26 – = 00218 = 218% Recalculate the present value of the costs at the real discount rate: Cost of Renting PV of cost of renting = 350 × annuity factor(.218%, 130 events) = $39,583.6 Cost of Owning PV of weekend operating costs = 200 × annuity factor(.218%, 130 events) = $22,619.2 Cost today of buying van = $20,000 PV of selling price in years = 11,809.8/(1.05825)5 = $8,898.2 PV of insurance (assume insurance is paid in advance) = 1200 × annuity due factor(5.825%, years) = $5,374.8 Total cost of owning = 22,619.2 + 20,000 - 8,898.2 + 5,374.8 = $39,095.8 4­25 Copyright © 20096 McGraw-Hill Ryerson Limited Now the cost of owning is less than the cost of renting Why? Because the cost of buying of the van does not inflate but the selling price does 83 a You can either discount real cash flows at the real discount rate or nominal cash flows at the nominal discount rate We use real cash flows and discount rate Real discount rate = - = 01923, or 1.923% PV college costs = × 10,000 × annuity due factor(1.923%, years) = $32,138.7 The family makes 10 payments, starting today Using the annuity due formula, the annual payment is PMT × annuity due factor(1.923%, 10 years) = $32,138.7 PMT = $2,889.89 b If the family waits one year, they have years to accumulate the required funds The annual payment is PMT × annuity due factor(1.923%, years) = $32,138.7 PMT = $3,242.65 They must save $3,242.65 - $2,889.89 = $352.76 more each year if they delay the start of their savings program for one year 84 The first cash flow, C1, is $35,000 Assume it will be received at the end of the first year Using the formula for the present value of a perpetually growing stream of cash flows, P0 = g = r - = 08 - = 08 - 0603 = 0197 = 1.97% 85 a The maximum price is equal to the present value of all of the cash flows that can be generated by the property: P0 = - - + × × annuity factor(8%,50 years) = $ 1.21 million If you pay $1.21 million, you will earn 8% return on your investment If you pay less than $1.21 million, you will earn more than 8% We are assuming after 50 years no further development will take place b If the annual cash flows grow at 1.5% per annum, the value of the land increases The maximum you should be willing to pay is the present value 4­26 Copyright © 20096 McGraw-Hill Ryerson Limited of the all the future cash flows that can be generated: [ () ] P0 = - - + × × - T [ () ] =- - + × × 1- 50 = - 1.389 – 1.286 + 5.878 = $3.203 million With 1.5% growth rate of the cash inflows, the present value of all of the cash flows increases to $3.203 million This is the maximum you should be willing to pay Again, we’ve assumed that the land has no use beyond year 50 You might disagree with that assumption However, the present value of a cash flow 50 years away is quite small 86 Annual real interest rate used for discounting and investing savings: 1.06/1.03 -1 = 029126, or 2.9126% Monthly real interest rate = (1.029126)1/12 - = 002395, or 0.2395% a How much will the Smiths need to have saved by the time they retire? Calculate the present value at the end of Year 35 of all the cash flows incurred during retirement (1) Retirement income Monthly retirement income (real) = $45,000/12 = $3,750 per month Number of months of retirement = 12/year × 20 years = 240 months PV(real retirement income as of end of Yr 35) = $3,750 × PVIFA(.2395%, 240 months) = $683,918 (2) Bequest to son PV(real value of bequest as of end of Yr 35) = 500,000 × PVIF(.2395%, 240) = $281,602 (3) Value of house Real growth rate = 1.04/1.03 - = 0097087 = 97087% Expected value of house in 55 years = 250,000 × FVIF(.97087%, 55) = 425,329 PV(real value of house as of end of Yr 35) = 425,329 × PVIF(.2395%, 240) = $239,547 Total amount needed for retirement, as of end of Year 35 = 683,918 +281,602 - 239,547 = $725,973 Assume that the Smiths save an equal amount at the end of each of the 12 × 35 (420) months before their retirement They will need to save PMT such that: $725,973 = PMT × FVIFA(.2395%,420) PMT = $1,004.41 per month 4­27 Copyright © 20096 McGraw-Hill Ryerson Limited In addition to saving for their retirement, the Smiths must save for their child's education If you assume that they will save all the needed funds just as the child starts university, they will need savings of: Total required savings (real) at the end of Year = 10,000 + 10,000 × PVIFA(2.9126%, years) = 38,333.71 To reach this goal, they must save PMT per month from Year to the end of Year 8: 38,333.71 = PMT × FVIFA(.2395%, 96 months) PMT = 355.64/ month (real dollars) Monthly savings required (real dollars) Years to Real retirement savings = 1,004.41 Real university savings = 355.64 Total 1,360.05 Years to 35 Real retirement savings = 1,004.41 Note: If you assume that the Smiths will continue to save for their child's education while the child is at university, the monthly amount needed will be slightly smaller To answer the question under this assumption, for each of the $10,000 find the monthly payment needed to be saved, recognizing that each amount is due one year later For example, the monthly payment to be saved for the $10,000 needed for the first year of university, at the start of Year 8, is: 10,000 = PMT × FVIFA(.2395%, 96 months) or PMT = $92.77 For the second year of university, Year 9, the monthly payment needed is: 10,000 = PMT × FVIFA(.2395%, 108 months) or PMT = $81.24 Repeat for the third and fourth years of university (payments are $72.03 and $64.20, respectively) Then add these amounts together, keeping track of when the payments stop Years to Real retirement savings = 1,004.41 Real university savings = 310.54 (= 92.77 + 81.24 + 72.03 + 64.5) Total 1,314.95 Year Real retirement savings = 1,004.41 Real university savings = 217.77 (= 81.24 + 72.03 + 64.5) Total 1,222.18 Year 10 Real retirement savings = 1,004.41 Real university savings = 136.53 (= 72.03 + 64.5) Total 1,140.94 4­28 Copyright © 20096 McGraw-Hill Ryerson Limited Year 11 Real retirement savings = 1,004.41 Real university savings = 64.5 Total 1,068.91 Then for Years 12 - 35, same as above b The nominal mortgage interest rate = 07/12 = 0.00583, or 583% The nominal monthly mortgage payment is: 200,000 = PMT × PVIFA(.583%, 12×20) PMT = 1,550.11 The last month of the mortgage is at the end of Year 20 Real payments Real retirement savings = 1,004.41 Real mortgage payment = 858.26 (= 1,550.11/(1.03)20 ) Total payments 1,862.67 Nominal payments Nominal retirement savings = 1,814.08 Nominal mortgage payment = 1,550.11 Total payments 3,364.19 c (= 1,004.41 × (1.03) 20 ) The last month before retirement is 35 years from today The only payment is the monthly retirement savings: Real value = 1,004.41 Nominal value = 1,004.41 × (1.03)35 = $2,826.27 87 Internet: RESP Calculator Note to instructors: You may wish to give students various profiles of families with children to compare the RESP savings requirements Expected results: Students have experience thinking about the impact of inflation and interest rates on funds needed Also, the students will see a fairly well thought out online financial planning tool 88 Internet: Mortgage and Loan Calculators The purpose of this problem is to give students an opportunity to test their understanding of the time value of money concepts while exploring applications of financial tools on the internet a Checking the answer to Problem 20 with loan calculator Tips: The annual interest rate used must be the APR Banks must give interest 4­29 Copyright © 20096 McGraw-Hill Ryerson Limited rates as APRs (the per period rate × number of payments per year) but sometime also report the EAR Expected results: For the $8,000 48-month loan with a 10% APR, the personal loan calculator says the monthly payment will be $202.90 This is the same number found when we answered the problem ourselves b Using a mortgage calculator Tips: All Canadian mortgage interest rates are reported with semi-annual compounding Thus 6% annual interest means 6%, compounded semi-annually or 3% every months Expected results: Using the HSBC mortgage calculator, enter $200,000 as the mortgage amount The annual interest rate is 6% and the number of years to repay the mortgage is 25 years, the conventional amortization period for Canadian mortgages Payments type is monthly The calculator says the monthly payment for the $200,000 mortgage with the 6% stated interest is $1,279.61 This matches exactly the number found by following the procedure presented in the text: PV = (-)200,000, n = 12 × 25 = 300, i = (1 + 06/2)1/6 – = 49038622%, FV = 0, compute PMT It is important to carry many decimal places What about the other payment periods? In each case, the HSBC calculator recalculates the mortgage payment using a different number of payments per year but keeps the amortization constant In our case, the amortization stays at 25 years With semi-monthly payments, 24 payments are made each year (2 × 12 months) With bi-weekly, 26 payments are made each year (26 = 52 weeks/2) and 52 payments are made with the weekly option In part (c), we look at another way to calculate mortgage payments Weekly Mortgage Payments: Using the HSBC calculator, keep all the information the same except click on the weekly button (or pick 52 payments per year) The computed weekly mortgage payment is $294.74 Checking the HSBC weekly mortgage calculation: PV= (-) 200,000 n = 52 × 25 = 1300 weeks i = (1 + 06/2)1/26 – = 11375235% (find the weekly interest rate equivalent to 3% every months) Compute PMT =$294.74 This exactly matches the weekly payment at HSBC Bi-Weekly Mortgage Payments: The same procedure is followed to get the bi-weekly payment This time, n = 26 × 25 = 650 bi-weekly payments (there are 26 bi-weeks in a year), i = (1.03)1/13 – = 2276341% and PMT = $589.81 Again, matches HSBC’s amount 4­30 Copyright © 20096 McGraw-Hill Ryerson Limited Semi-Monthly Mortgage Payments: n = 24 × 25 = 600 payments, i = (1.03)1/12 – = 00246627 and PMT = $639.02 , matching the HSBC amount c Comparing mortgage calculators The TD website gives the same monthly mortgage amount However, for different payment periods, things get more interesting (or confusing, depending on your spirit of adventure) Two general approaches are taken in the variations on the monthly mortgage The first is to simply change the number of payments per year (e.g., weekly), keep the amortization at 25 years and calculate the new mortgage payment, as was done with the HSBC mortgage calculator The other approach is to keep the total monthly payment constant but pay it in installments (often called “rapid” or “accelerated” mortgage payments) The TD bi-weekly mortgage payment is $590.59 For HSBC, the bi-weekly payment is $589.81 TD's bi-weekly payment is calculated by multiply the monthly payment by 12 and dividing by 26: 12 × $1,279.61/26 = $590.59 The second approach gives accelerated or rapid payment mortgages The perperiod payment is the monthly payment divide by the number of payment periods in the month: for bi-weekly rapid/accelerated, for weekly (or weekly rapid) Now instead of calculating PMT, you can calculate n, the number of periods until the mortgage is paid off Since for each you pay more money each year, compared to the regular monthly payment, the mortgage is paid off more quickly Note that banks are free to call these payment options whatever they wish – you will find that some calculators call “weekly” mortgages that are really “weekly rapid” Do not be fooled by the names Bi-Weekly Rapid or Accelerated Mortgage: Take the monthly payment and divide it by to get semi-monthly payments of 1,279.61/2 = $639.81 Now instead of calculating PMT, you can calculate n, the number of periods before the mortgage is paid off The bi-weekly interest rate is 2276341%, PV = (-) 200,000, PMT = 639.81, compute n = 546.81 semi-monthly periods, or 546.8/26 = 21.03 years This works because you actually pay more each year: 26 bi-weeks × 639.81/bi-weeks = $16,635.06 per year compared with 12 months × 1,279.61/month = $15,355.32 per year d U.S Mortgages Tip: In the US, interest is compounded monthly So 6% interest means (.06/12) = 005 per month Expected results: At the www.smartmoney.com, the monthly US mortgage payment is $1,289 The difference boils down to the fact that 6%, compounded monthly is a higher interest rate than 6%, compounded semi-annually: 4­31 Copyright © 20096 McGraw-Hill Ryerson Limited US: EAR of 6%, monthly = (1.005)12 – = 0616778, or 6.168% Canada: EAR of 6%, semi-annually = (1.03)2 – = 0609, or 6.09% You can make a U.S mortgage equivalent to a Canadian mortgage by carefully ensuring that the mortgages have the same interest rate per payment period 4­32 Copyright © 20096 McGraw-Hill Ryerson Limited Solution to Minicase for Chapter How much can Mr Road spend each year? First let's see what happens if we ignore inflation Account for Canada Pension (CPP) and Old Age Security (OAS) income of $750 per month, or $9,000 annually Account for the income from the savings account Because Mr Road does not want to run down the balance of this account, he can spend only the interest income, or 05  $12,000 = $600 annually Compute the annual consumption available from his investment account We find the 20-year annuity with present value equal to the value in the account: Present Value = annual payment  20-year annuity factor at 9% interest rate $180,000 = annual payment  9.129 Annual payment = $180,000/9.129 = $19,717 Notice that the investment account provides annual income of $19,717, which is more than the annual interest from the account (.09× 180,000 = $16,200) This is because Mr Road plans to run the account down to zero by the end of his life So Mr Road can spend $19,717 + $600 + $9,000 = $29,317 a year, comfortably above his current living expenses, which are $2,000 a month or $24,000 annually The problem of course is inflation We have mixed up real and nominal flows The CPP and OAS payments are tied to the consumer price index and therefore are level in real terms But the annuity of $19,717 a year from the investment account and the $600 interest from the savings account are fixed in nominal terms, and therefore the purchasing power of these flows will steadily decline For example, let's look out 15 years At percent inflation, prices will increase by a factor of (1.04)15 = 1.80 Income in 15 years will therefore be as follows: 4­33 Copyright © 20096 McGraw-Hill Ryerson Limited Income source CPP and OAS (indexed to CPI; fixed in real terms at $9000) Savings account Investment account (fixed nominal annuity) Total income Nominal income Real income $16,200 $ 9,000 600 333 19,717 10,954 $36,517 $20,287 Once we recognize inflation, we see that, in 15 years, income from the investment account will buy only a bit more than one-half of the goods it buys today Obviously Mr Road needs to spend less today and put more aside for the future Rather than spending a constant nominal amount out of his savings, he would probably be better off spending a constant real amount Since we are interested in level real expenditures, we must use the real interest rate to calculate the 20-year annuity that can be provided by the $180,000 The real interest rate is 4.8% (because + real interest rate = 1.09/1.04 = 1.048) We therefore calculate the real sum that can be spent out of savings as $14,200 [n = 20; i = 4.8%; PV = (–)180,000; FV = 0; compute PMT] Thus Mr Road's investment account can generate a real income of $14,200 a year The real value of CPP and OAS is fixed at $9,000 Finally, if we assume that Mr Road wishes to maintain the real value of his savings account at $12,000, then he will have to increase the balance of the account in line with inflation, that is, by 4% each year Since the nominal interest rate on the account is 5%, only the first 1% of interest earnings on the account, or $120 real dollars, are available for spending each year The other 4% of earnings must be re-invested So total real income is $14,200 + $9,000 + $120 = $23,320 To keep pace with inflation Mr Road will need to spend percent more of his savings each year After one year of inflation, he will spend 1.04  $23,320 = $24,253; after two years he will spend (1.04)2  $23,320 = $25,223, and so on The picture 15 years out looks like this: 4­34 Copyright © 20096 McGraw-Hill Ryerson Limited Income source CPP and OAS Net income from savings account (i.e., net of reinvested interest) Investment account (fixed real annuity) Total income Nominal income $16,200 Real income $ 9,000 216 120 25,560 14,200 $41,976 $23,320 Mr Road's income and expenditure will nearly double in 15 years but his real income and expenditure are unchanged at $23,320 This may be bad news for Mr Road since his living expenses are $24,000 Do you advise him to prune his living expenses? Perhaps he should put part of his nest egg in junk bonds which offer higher promised interest rates, or into the stock market, which has generated higher returns on average than investment in bonds These higher returns might support a higher real annuity — but is Mr Road prepared to bear the extra risks? Should Mr Road consume more today and risk having to sell his house if his savings are run down late in life? These issues make the planning problem even more difficult It is clear, however, that one cannot plan for retirement without considering inflation 4­35 Copyright © 20096 McGraw-Hill Ryerson Limited ... at the end of the lease The calculator will calculate the value of alternative investments at the end of lease term of $11,510 which is more than the value of the car at the end of the lease ($10,000)... half way between the value of $1 invested in the bank and the value of a $1 invested in the stock market 77 The interest rate per three months is 12%/4 = 3% So the value of the perpetuity is... Limited Another way to think about this is to recognize that the present value of the savings stream must equal the present value of the consumption stream The PV of consumption as of today is