Chap005

nullnullMcGraw-Hill/IrwinCopyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.Key Concepts and SkillsKey Concepts and SkillsBe able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute loan payments Be able to find the interest rate on a loan Understand how loans are amortized or paid off Understand how interest rates are quotedChapter OutlineChapter Outline6.1 Future and Present Values of Multiple Cash Flows 6.2 Valuing Level Cash Flows: Annuities and Perpetuities 6.3 Comparing Rates: The Effect of Compounding Periods 6.4 Loan Types and Loan AmortizationAsk and AnswerAsk and Answer Read book page 145-152 and answer following questions: Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount rate is 7% What is the value of the CFs at year 5? What is the value of the CFs today?Ask and Answer SolutionsAsk and Answer SolutionsFuture Value: Multiple Cash Flows Example 1Future Value: Multiple Cash Flows Example 1If you deposit $100 in one year, $200 in two years and $300 in three years. How much will you have in three years at 7 percent interest? How much in five years if you don’t add additional amounts?Future Value: Multiple Cash Flows Time LineFuture Value: Multiple Cash Flows Time LineFuture Value: Multiple Cash Flows CalculatorFuture Value: Multiple Cash Flows CalculatorYear 1 CF: 2 ,; 100 S.; 7 -; %0 = 114.49 Year 2 CF: 1 ,; 200 S.; 7 -; %0 = 214.00 Year 3 CF: 0 ,; 300 S.; 7 -; %0 = 300.00 Total FV3 = 628.49 Total FV5 = 628.49 * (1.07)2 = 719.56Future Value: Multiple Cash Flows Example 2Future Value: Multiple Cash Flows Example 2Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%? Future Value: Multiple Cash Flows Time LineFuture Value: Multiple Cash Flows Time Line$100012345$300$136.05 $349.92 $485.97X (1.08)4 =X (1.08)2 =Future Value: Multiple Cash Flows Formula and CalculatorFuture Value: Multiple Cash Flows Formula and CalculatorFV = $100(1.08)4 + $300(1.08)2 = $136.05 + $349.92 = $485.97Future Value: Multiple Cash Flows Example 3Future Value: Multiple Cash Flows Example 3Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? FV = $ 500 x (1.09)2 = $ 594.05 + $ 600 x (1.09) = $ 654.00 = $1,248.05Future Value: Multiple Cash Flows Example continuedFuture Value: Multiple Cash Flows Example continuedHow much will you have in 5 years if you make no further deposits? First way: FV = $500(1.09)5 + $600(1.09)4 = $1,616.26 Second way – use value at year 2: FV = $1,248.05(1.09)3 = $1,616.26Future Value: Multiple Cash Flows CalculatorFuture Value: Multiple Cash Flows CalculatorFuture Value: Multiple Cash Flows Example 4Future Value: Multiple Cash Flows Example 4You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in 3 years? How much in 4 years?Future Value: Multiple Cash Flows FormulaFuture Value: Multiple Cash Flows FormulaFind the value at year 3 of each cash flow and add them together. Year 0: FV = $7,000(1.08)3 = $ 8,817.98 Year 1: FV = $4,000(1.08)2 = $ 4,665.60 Year 2: FV = $4,000(1.08)1 = $ 4,320.00 Year 3: value = $ 4,000.00 Total value in 3 years = $21,803.58 Value at year 4 = $21,803.58(1.08)= $23,547.87Future Value: Multiple Cash Flows CalculatorFuture Value: Multiple Cash Flows CalculatorPresent Value: Multiple Cash Flows Example 1Present Value: Multiple Cash Flows Example 1You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?Present Value: Multiple Cash Flows Formula and CalculatorPresent Value: Multiple Cash Flows Formula and CalculatorPV = $1,000 / (1.1)1 = $ 909.09 PV = $2,000 / (1.1)2 = $1,652.89 PV = $3,000 / (1.1)3 = $2,253.94 PV = $4,815.92Present Value: Multiple Cash Flows Example 2Present Value: Multiple Cash Flows Example 2You are offered an investment that will pay $200 in year 1, $400 the next year, $600 the following year, and $800 at the end of the 4th year. You can earn 12% on similar investments. What is the most you should pay for this one?Present Value: Multiple Cash Flows Time LinePresent Value: Multiple Cash Flows Time Line01234200400600800178.57318.88427.07508.411,432.93= 1/(1.12)4 x = 1/(1.12)3 xTime (years) = 1/(1.12)2 xPresent Value: Multiple Cash Flows FormulaPresent Value: Multiple Cash Flows FormulaFind the PV of each cash flow and add them: Year 1 CF: $200 / (1.12)1 = $ 178.57 Year 2 CF: $400 / (1.12)2 = $ 318.88 Year 3 CF: $600 / (1.12)3 = $ 427.07 Year 4 CF: $800 / (1.12)4 = $ 508.41 Total PV = $1,432.93Present Value: Multiple Cash Flows Calculator 1Present Value: Multiple Cash Flows Calculator 1Present Value: Multiple Cash Flows Calculator 2Present Value: Multiple Cash Flows Calculator 2Display You Enter ‘ 0 0 0 ' 1 200 200 ' 2 400 400 ' 3 600 600 ' 4 800 800 ' 12 12 - NPV ( 1432.93Cash Flows: CF0 = 0 CF1 = 200 CF2 = 400 CF3 = 600 CF4 = 800Present Value: Multiple Cash Flows Example 3Present Value: Multiple Cash Flows Example 3You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%? Present Value: Multiple Cash Flows TimelinePresent Value: Multiple Cash Flows Timeline 0 0 0 … 25K 25K 25K 25K 25K Notice that the year 0 cash flow = 0 Cash flows years 1–38 = 0 Cash flows years 39–43 = 25,000Present Value: Multiple Cash Flows SolutionPresent Value: Multiple Cash Flows SolutionFind the PV of each cash flow and add them: Year 39 CF: $25000 / (1.12)39 = $300.91 Year 40 CF: $25000 / (1.12)40 = $268.67 Year 41 CF: $25000 / (1.12)41 = $239.88 Year 42 CF: $25000 / (1.12)42 = $214.18 Year 43 CF: $25000 / (1.12)43 = $191.23 Total PV = $1214.87Present Value: Multiple Cash Flows Example 4Present Value: Multiple Cash Flows Example 4Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in the first year and $75 in the second year. If you require a 15% return on investments of this risk, should you take the investment?Present Value: Multiple Cash Flows SolutionPresent Value: Multiple Cash Flows SolutionNo – the broker is charging more than you would be willing to pay. Display You Enter ‘ 0 0 0 ' 1 40 40 ' 2 75 75 ' 12 15 - NPV ( 91.49Annuities and PerpetuitiesAnnuities and PerpetuitiesAnnuity – finite series of equal payments that occur at regular intervals If the first payment occurs at the end of the period, it is called an ordinary annuity If the first payment occurs at the beginning of the period, it is called an annuity duePage 152Page 160Table 5.2Table 5.2Page 161Annuities and Perpetuities Basic FormulasAnnuities and Perpetuities Basic FormulasAnnuities:Annuity due: x (1+r) x (1+r) Page 153Page 159Page 161Important Points to RememberImportant Points to RememberInterest rate and time period must match! Annual periods  annual rate Monthly periods  monthly rate The Sign Convention Cash inflows are positive Cash outflows are negativeSign Convention ExampleSign Convention Example 5 , 10 - 100 S. 20 / %0 = $38.95 Implies you deposited $100 today and plan to WITHDRAW $20 a year for 5 years 5 , 10 - 100 S. 20 S/ %0 = $283.15 Implies you deposited $100 today and plan to ADD $20 a year for 5 years+CF = Cash INFLOW to YOU-CF = Cash OUTFLOW from youAnnuity Example 1Annuity Example 1You can afford $632 per month. Going rate = 1%/month for 48 months. How much can you borrow? You borrow money TODAY so you need to compute the present value. 48 , 1 - 632 S/ 0 0 %. = 23,999.54 ($24,000)Annuity – Sweepstakes Example 2Annuity – Sweepstakes Example 2Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?Annuity – Sweepstakes SolutionAnnuity – Sweepstakes SolutionPV = $333,333.33[1 – 1/1.0530] / .05 = $5,124,150.29Annuity – Buying a House Example 3Annuity – Buying a House Example 3You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000. The bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?Buying a House - ContinuedBuying a House - ContinuedBank loan Monthly income = 36,000 / 12 = 3,000 Maximum payment = .28(3,000) = 840 360 , (30*12) 0.5 - 840 S/ 0 %. = 140,805 Total Price Closing costs = .04(140,805) = 5,632 Down payment = 20,000 – 5632 = 14,368 Total Price = 140,805 + 14,368 = 155173Quick Quiz – Part 2Quick Quiz – Part 2You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value?Quick Quiz – Part 2Quick Quiz – Part 2You want to receive $5,000 per month in retirement. If you can earn .75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement?Quick Quiz – Part 2 SolutionQuick Quiz – Part 2 Solution300 ,  Months 0.75 -  Monthly rate 5000 /  Monthly Payment 0 0 %. -600,277 Finding the Number of Payments Example 1Finding the Number of Payments Example 1Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = .66667% per month). If you take a 4 year loan, what is your monthly payment?Finding the Number of Payments SolutionFinding the Number of Payments Solution4(12) = 48 , 0.66667 - 20,000 . 0 0 %/ = - 488.26Finding the Number of Payments Example 2Finding the Number of Payments Example 2$1,000 due on credit card Payment = $20 month minimum Rate = 1.5% per month The sign convention matters!!! Finding the Number of Payments SolutionFinding the Number of Payments Solution1.5 - 1000 . 20 S/ 0 0 %, = 93.111 months = 7.75 yearsFinding the Number of Payments Example 3Finding the Number of Payments Example 3Suppose you borrow $2,000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan? Finding the Number of Payments SolutionFinding the Number of Payments Solution5 - 2000 . 734.42 S/ 0 0 %, = 3 yearsFinding the Rate Example 1Finding the Rate Example 1Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate? Finding the Rate SolutionFinding the Rate Solution60 , 10000 . 207.58 S/ 0 0 %- =.75%Quick Quiz – Part 3Quick Quiz – Part 3You want to receive $5,000 per month for the next 5 years. How much would you need to deposit today if you can earn .75% per month? 60 , 0.75 - 5000 / 0 0 %. = -240866.87Quick Quiz – Part 3Quick Quiz – Part 3You want to receive $5,000 per month for the next 5 years. What monthly rate would you need to earn if you only have $200,000 to deposit? 60 , 200000 S. 5000 / 0 0 %- = 1.4395%Quick Quiz – Part 3Quick Quiz – Part 3Suppose you have $200,000 to deposit and can earn .75% per month. How many months could you receive the $5,000 payment? 0.75 - 200000 S. 5000 / 0 0 %, = 47.73 months ≈ 4 yearsQuick Quiz – Part 3Quick Quiz – Part 3Suppose you have $200,000 to deposit and can earn .75% per month. How much could you receive every month for 5 years? 60 , 0.75 - 200000 S. 0 0 %/ = 4151.67Future Values for AnnuitiesFuture Values for AnnuitiesSuppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years? 40 , 7.5 - 0 . 2000 S/ %0 = 454513.04Annuity DueAnnuity DueYou are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? Annuity Due TimelineAnnuity Due Timeline35,016.12PerpetuityPerpetuityPerpetuity – infinite series of equal payments. Perpetuity formula: PV = PMT / rPage 160PerpetuityPerpetuityPage 161 Example 6.7 Current required return: 40 = 1 / r r = .025 or 2.5% per quarter Dividend for new preferred: 100 = PMT / .025 PMT = 2.50 per quarterGrowing AnnuityGrowing AnnuityA growing stream of cash flows with a fixed maturityPage 162Growing Annuity ExampleGrowing Annuity ExampleA defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?Growing PerpetuityGrowing PerpetuityA growing stream of cash flows that lasts foreverPage 163Growing Perpetuity ExampleGrowing Perpetuity ExampleThe expected dividend next year is $1.30, and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream?Example: Work the Web Example: Work the Web Another online financial calculator can be found at Calculatoredge.com. Click on the Web surfer, select “Finance” calculator and “Annuity Payments” and work the following example: How much could you withdraw each year if you have $2,500,000, earn 8 % and make annual withdrawals for 35 years?Quick Quiz – Part 4Quick Quiz – Part 4You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month? Ordinary Annuity420 , 1 - 0 . 1000000 0 %/ = -155.50Quick Quiz – Part 4Quick Quiz – Part 4You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made today? Annuity Due ] 420 , 1 - 0 . 1000000 0 %/ = -153.96 ]Quick Quiz – Part 4Quick Quiz – Part 4You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be willing to pay? $1.50/0.03 = $50Interest RatesInterest RatesEffective Annual Rate (EAR) The interest rate expressed as if it were compounded once per year. Used to compare two alternative investments with different compounding periods Annual Percentage Rate (APR) “Quoted” The annual rate quoted by law APR = periodic rate X number of periods per year Periodic rate = APR / periods per yearReturn to Quick QuizPage 164Page 166Computing APRsComputing APRsWhat is the APR if the monthly rate is .5%? .5% X12 = 6% What is the APR if the semiannual rate is .5%? .5% X 2 = 1% What is the monthly rate if the APR is 12% with monthly compounding? 12% / 12 = 1% Can you divide the above APR by 2 to get the semiannual rate? NO. You need an APR based on semiannual compounding to find the semiannual rate.Things to RememberThings to RememberYou ALWAYS need to make sure that the interest rate and the time period match. Annual periods  annual rate. Monthly periods  monthly rate. If you have an APR based on monthly compounding, you have to use monthly periods for lump sums or adjust the interest rate accordingly. You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rateComputing EARs - ExampleComputing EARs - ExampleSuppose you can earn 1% per month on $1 invested today. What is the APR? 1%X12 = 12% How much are you effectively earning? FV = 1X(1+1%)12 = 1.1268 Rate = (1.1268 – 1) / 1 = .1268 = 12.68% Suppose if you put it in another account, you earn 3% per quarter. What is the APR? 3%X4 = 12% How much are you effectively earning? FV = 1X(1+3%)4 = 1.1255 Rate = (1.1255 – 1) / 1 = .1255 = 12.55%EAR FormulaEAR FormulaAPR = the quoted rate m = number of compounds per yearPage 165Decisions, Decisions Decisions, Decisions You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? First account: EAR = (1 + .0525/365)365 – 1 = 5.39% Second account: EAR = (1 + .053/2)2 – 1 = 5.37% Which account should you choose and why?Decisions, Decisions ContinuedDecisions, Decisions ContinuedLet’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year? First Account: Daily rate = .0525 / 365 = .00014383562 FV = 100(1.00014383562)365 = 105.39 Second Account: Semiannual rate = .053 / 2 = .0265 FV = 100(1.0265)2 = 105.37 You have more money in the first account.Computing APRs from EARs Computing APRs from EARs M = number of compounding periods per yearAPR ExampleAPR ExampleSuppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? Computing Payments with APRsComputing Payments with APRsSuppose you want to buy a new computer. The store is willing to allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years. The interest rate is 16.9% with monthly compounding. What is your monthly payment?Computing Payments with APRs SolutionComputing Payments with APRs Solution2x12 24 , 16.9 / 12 1.40833 - 3500 . 0 0 %/ = -172.88Monthly rate = .169 / 12 = .01408333 Number of months = 2 x12= 24 3,500 = C[1 – (1 / 1.01408333333)24] / .01408333 C = 172.88Future Values with Monthly CompoundingFuture Values with Monthly CompoundingSuppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years? Future Values with Monthly Compounding Solution420 , (35*12) 0.75 - (9/12) 0 . -50 / %0 = 147,089.22Future Values with Monthly Compounding SolutionMonthly rate = .09 / 12 = .0075 Number of months = 35x12 = 420 FV = 50x[1.0075420 – 1] / .0075 = 147,089.22Present Value with Daily CompoundingPresent Value with Daily CompoundingYou need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit? Present Value with Daily Compounding SolutionPresent Value with Daily Compounding Solution 1095 , (3*365) .015068493 - (5.5/365) 0 / 15,000 0 %. = -12,718.56Daily rate = .055 / 365 = .00015068493 Number of days = 3x365 = 1,095 PV = 15,000 / (1.00015068493)1095 = 12,718.56Continuous CompoundingContinuous CompoundingSometimes investments or loans are figured based on continuous compounding EAR = eq – 1 The e is a special function on the calculator normally denoted by ex Example: What is the effective annual rate of 7% compounded