Key Concepts and SkillsBe able to compute the future value of multiple cash flowsBe able to compute the present value of multiple cash flowsBe able to compute loan paymentsBe able to find the interest rate on a loanUnderstand how loans are amortized or paid offUnderstand how interest rates are quoted

Chapter OutlineFuture and Present Values of Multiple Cash FlowsValuing Level Cash Flows: Annuities and PerpetuitiesComparing Rates: The Effect of Compounding PeriodsLoan Types and Loan Amortization

Basic DefinitionsPresent Value earlier money on a time lineFuture Value later money on a time lineInterest rate exchange rate between earlier money and later moneyDiscount rateCost of capitalOpportunity cost of capitalRequired return

Future ValuesSuppose you invest $1000 for one year at 5% per year. What is the future value in one year?Interest = 1000(.05) = 50Value in one year = principal + interest = 1000 + 50 = 1050Future Value (FV) = 1000(1 + .05) = 1050Suppose you leave the money in for another year. How much will you have two years from now?FV = 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50

Future Values: General FormulaFV = PV(1 + r)tFV = future valuePV = present valuer = period interest rate, expressed as a decimalT = number of periodsFuture value interest factor = (1 + r)t

Effects of CompoundingSimple interest Compound interestConsider the previous exampleFV with simple interest = 1000 + 50 + 50 = 1100FV with compound interest = 1102.50The extra 2.50 comes from the interest of .05(50) = 2.50 earned on the first interest payment

Future Values Example 2Suppose you invest the $1000 from the previous example for 5 years. How much would you have?FV = 1000(1.05)5 = 1276.28The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1250, for a difference of $26.28.)

Future Values Example 3Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?FV = 10(1.055)200 = 447,189.84What is the effect of compounding?Simple interest = 10 + 200(10)(.055) = 210.55Compounding added $446,979.29 to the value of the investment

Future Value as a General Growth FormulaSuppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?FV = 3,000,000(1.15)5 = 6,034,072

Present ValuesHow much do I have to invest today to have some amount in the future?FV = PV(1 + r)tRearrange to solve for PV = FV / (1 + r)tWhen we talk about discounting, we mean finding the present value of some future amount.When we talk about the value of something, we are talking about the present value unless we specifically indicate that we want the future value.

Present Value One Period ExampleSuppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?PV = 10,000 / (1.07)1 = 9345.79Calculator1 N7 I/Y10,000 FVCPT PV = -9345.79

Present Values Example 2You want to begin saving for you daughters college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?PV = 150,000 / (1.08)17 = 40,540.34

Present Values Example 3Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest?PV = 19,671.51 / (1.07)10 = 10,000

Present Value Important Relationship IFor a given interest rate the longer the time period, the lower the present valueWhat is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%5 years: PV = 500 / (1.1)5 = 310.4610 years: PV = 500 / (1.1)10 = 192.77

Present Value Important Relationship IIFor a given time period the higher the interest rate, the smaller the present valueWhat is the present value of $500 received in 5 years if the interest rate is 10%? 15%?Rate = 10%: PV = 500 / (1.1)5 = 310.46Rate = 15%; PV = 500 / (1.15)5 = 248.58

The Basic PV Equation - RefresherPV = FV / (1 + r)tThere are four parts to this equationPV, FV, r and tIf we know any three, we can solve for the fourthIf you are using a financial calculator, be sure and remember the sign convention or you will receive an error when solving for r or t

Discount RateOften we will want to know what the implied interest rate is in an investmentRearrange the basic PV equation and solve for rFV = PV(1 + r)tr = (FV / PV)1/t 1If you are using formulas, you will want to make use of both the yx and the 1/x keys

Discount Rate Example 1You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest?r = (1200 / 1000)1/5 1 = .03714 = 3.714%Calculator the sign convention matters!!!N = 5PV = -1000 (you pay 1000 today)FV = 1200 (you receive 1200 in 5 years)CPT I/Y = 3.714%

Discount Rate Example 2Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?r = (20,000 / 10,000)1/6 1 = .122462 = 12.25%

Discount Rate Example 3Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5000 to invest. What interest rate must you earn to have the $75,000 when you need it?r = (75,000 / 5,000)1/17 1 = .172688 = 17.27%

Finding the Number of PeriodsStart with basic equation and solve for t (remember your logs)FV = PV(1 + r)tt = ln(FV / PV) / ln(1 + r)You can use the financial keys on the calculator as well, just remember the sign convention.

Number of Periods Example 1You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years

Number of Periods Example 2Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. If the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?

Number of Periods Example 2 ContinuedHow much do you need to have in the future?Down payment = .1(150,000) = 15,000Closing costs = .05(150,000 15,000) = 6,750Total needed = 15,000 + 6,750 = 21,750Compute the number of periodsPV = -15,000FV = 21,750I/Y = 7.5CPT N = 5.14 yearsUsing the formulat = ln(21,750 / 15,000) / ln(1.075) = 5.14 years

Multiple Cash Flows Future Value Example 6.1Find the value at year 3 of each cash flow and add them together.Today (year 0): FV = 7000(1.08)3 = 8,817.98Year 1: FV = 4,000(1.08)2 = 4,665.60Year 2: FV = 4,000(1.08) = 4,320Year 3: value = 4,000Total value in 3 years = 8817.98 + 4665.60 + 4320 + 4000 = 21,803.58Value at year 4 = 21,803.58(1.08) = 23,547.87

Multiple Cash Flows FV Example 2Suppose 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(1.09)2 + 600(1.09) = 1248.05

Multiple Cash Flows Example 2 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 = 1616.26Second way use value at year 2:FV = 1248.05(1.09)3 = 1616.26

Multiple Cash Flows FV Example 3Suppose 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%?FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97

Multiple Cash Flows Present Value Example 6.3Find the PV of each cash flows and add themYear 1 CF: 200 / (1.12)1 = 178.57Year 2 CF: 400 / (1.12)2 = 318.88Year 3 CF: 600 / (1.12)3 = 427.07Year 4 CF: 800 / (1.12)4 = 508.41Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93

Example 6.3 Timeline

Multiple Cash Flows PV Another ExampleYou are considering an investment that will pay you $1000 in one year, $2000 in two years and $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?PV = 1000 / (1.1)1 = 909.09PV = 2000 / (1.1)2 = 1652.89PV = 3000 / (1.1)3 = 2253.94PV = 909.09 + 1652.89 + 2253.94 = 4815.93

Multiple Uneven Cash Flows Usingthe CalculatorAnother way to use the financial calculator for uneven cash flows is to use the cash flow keysTexas Instruments BA-II PlusPress CF and enter the cash flows beginning with year 0.You have to press the Enter key for each cash flowUse the down arrow key to move to the next cash flowThe F is the number of times a given cash flow occurs in consecutive yearsUse the NPV key to compute the present value by entering the interest rate for I, pressing the down arrow and then computeClear the cash flow keys by pressing CF and then CLR Work

Decisions, DecisionsYour broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment?Use the CF keys to compute the value of the investmentCF; CF0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1NPV; I = 15; CPT NPV = 91.49No the broker is charging more than you would be willing to pay.

Saving For RetirementYou 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%?Use cash flow keys:CF; CF0 = 0; C01 = 0; F01 = 39; C02 = 25000; F02 = 5; NPV; I = 12; CPT NPV = 1084.71

Saving For Retirement Timeline0 1 2 39 40 41 42 43 44 0 0 0 0 25K 25K 25K 25K 25KNotice that the year 0 cash flow = 0 (CF0 = 0)The cash flows years 1 39 are 0 (C01 = 0; F01 = 39)The cash flows years 40 44 are 25,000 (C02 = 25,000; F02 = 5)

Annuities and Perpetuities DefinedAnnuity finite series of equal payments that occur at regular intervalsIf the first payment occurs at the end of the period, it is called an ordinary annuityIf the first payment occurs at the beginning of the period, it is called an annuity duePerpetuity infinite series of equal payments

Annuities and Perpetuities Basic FormulasPerpetuity: PV = C / rAnnuities:

Annuities and the CalculatorYou can use the PMT key on the calculator for the equal paymentThe sign convention still holdsOrdinary annuity versus annuity dueYou can switch your calculator between the two types by using the 2nd BGN 2nd Set on the TI BA-II Plus If you see BGN or Begin in the display of your calculator, you have it set for an annuity dueMost problems are ordinary annuities

Annuity Example 6.5You borrow money TODAY so you need to compute the present value.48 N; 1 I/Y; -632 PMT; CPT PV = 23,999.54 ($24,000)Formula:

Annuity Sweepstakes ExampleSuppose 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?PV = 333,333.33[1 1/1.0530] / .05 = 5,124,150.29

Buying a HouseYou 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 and 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 - ContinuedBank loanMonthly income = 36,000 / 12 = 3,000Maximum payment = .28(3,000) = 840PV = 840[1 1/1.005360] / .005 = 140,105Total PriceClosing costs = .04(140,105) = 5,604Down payment = 20,000 5604 = 14,396Total Price = 140,105 + 14,396 = 154,501

Finding the PaymentSuppose 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?20,000 = C[1 1 / 1.006666748] / .0066667C = 488.26

Finding the Number of Payments Example 6.6Start with the equation and remember your logs.1000 = 20(1 1/1.015t) / .015.75 = 1 1 / 1.015t1 / 1.015t = .251 / .25 = 1.015tt = ln(1/.25) / ln(1.015) = 93.111 months = 7.75 yearsAnd this is only if you dont charge anything more on the card!

Finding the Number of Payments Another ExampleSuppose you borrow $2000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan?2000 = 734.42(1 1/1.05t) / .05.136161869 = 1 1/1.05t1/1.05t = .8638381311.157624287 = 1.05tt = ln(1.157624287) / ln(1.05) = 3 years

Finding the RateSuppose 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?Sign convention matters!!!60 N10,000 PV-207.58 PMTCPT I/Y = .75%

Annuity Finding the Rate Without aFinancial CalculatorTrial and Error ProcessChoose an interest rate and compute the PV of the payments based on this rateCompare the computed PV with the actual loan amountIf the computed PV > loan amount, then the interest rate is too lowIf the computed PV < loan amount, then the interest rate is too highAdjust the rate and repeat the process until the computed PV and the loan amount are equal

Future Values for AnnuitiesSuppose you begin saving for your retirement by depositing $2000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years?FV = 2000(1.07540 1)/.075 = 454,513.04

Annuity 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?FV = 10,000[(1.083 1) / .08](1.08) = 35,061.12

Annuity Due Timeline35,016.12

Perpetuity Example 6.7Perpetuity formula: PV = C / rCurrent required return:40 = 1 / rr = .025 or 2.5% per quarterDividend for new preferred:100 = C / .025C = 2.50 per quarter

Effective Annual Rate (EAR)This is the actual rate paid (or received) after accounting for compounding that occurs during the yearIf you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.

Annual Percentage RateThis is the annual rate that is quoted by lawBy definition APR = period rate times the number of periods per yearConsequently, to get the period rate we rearrange the APR equation:Period rate = APR / number of periods per yearYou should NEVER divide the effective rate by the number of periods per year it will NOT give you the period rate

Computing APRsWhat is the APR if the monthly rate is .5%?.5(12) = 6%What is the APR if the semiannual rate is .5%?.5(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 RememberYou ALWAYS need to make sure that the interest rate and the time period match.If you are looking at annual periods, you need an annual rate.If you are looking at monthly periods, you need a 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 appropriately if you have payments other than monthly

Computing EARs - ExampleSuppose you can earn 1% per month on $1 invested today.What is the APR? 1(12) = 12%How much are you effectively earning?FV = 1(1.01)12 = 1.1268Rate = (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(4) = 12%How much are you effectively earning?FV = 1(1.03)4 = 1.1255Rate = (1.1255 1) / 1 = .1255 = 12.55%

EAR - FormulaRemember that the APR is the quoted rate

Decisions, Decisions IIYou 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 II ContinuedLets 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 = .00014383562FV = 100(1.00014383562)365 = 105.39Second Account:Semiannual rate = .0539 / 2 = .0265FV = 100(1.0265)2 = 105.37You have more money in the first account.

Computing APRs from EARs If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:

APR - 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 APRsSuppose you want to buy a new computer system and the store is willing to sell it to allow you to make monthly payments. The entire computer system costs $3500. The loan period is for 2 years and the interest rate is 16.9% with monthly compounding. What is your monthly payment?Monthly rate = .169 / 12 = .01408333333Number of months = 2(12) = 243500 = C[1 1 / 1.01408333333)24] / .01408333333C = 172.88

Future 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?Monthly rate = .09 / 12 = .0075Number of months = 35(12) = 420FV = 50[1.0075420 1] / .0075 = 147,089.22

Present 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?Daily rate = .055 / 365 = .00015068493Number of days = 3(365) = 1095FV = 15,000 / (1.00015068493)1095 = 12,718.56

Continuous CompoundingSometimes investments or loans are figured based on continuous compoundingEAR = eq 1The e is a special function on the calculator normally denoted by exExample: What is the effective annual rate of 7% compounded continuously?EAR = e.07 1 = .0725 or 7.25%

Pure Discount Loans Example 6.12Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments.If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?PV = 10,000 / 1.07 = 9345.79

Interest Only Loan - ExampleConsider a 5-year, interest only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually.What would the stream of cash flows be?Years 1 4: Interest payments of .07(10,000) = 700Year 5: Interest + principal = 10,700This cash flow stream is similar to the cash flows on corporate bonds and we will talk about them in greater detail later.

Its important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an interest rate only in terms of loans and savings accounts.Point out that we are just using algebra when deriving the FV formula. We have 1000(1) + 1000(.05) = 1000(1+.05)It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay 1276.28 in 5 years. Show the students that if they enter the 1000 as negative, the FV will compute as a positive number.

Also, you may want to point out the change sign key on the calculator. There seem to be a few students each semester that have never had to use it before.

Calculator: N = 5; I/Y = 5; PV = 1000; CPT FV = -1276.28Calculator: N = 200; I/Y = 5.5; PV = 10; CPT FV = -447,198.84Calculator: N = 5; I/Y = 15; PV = 3,000,000 CPT FV = -6,034,072Point out that the PV interest factor = 1 / (1 + r)tThe remaining examples will just use the calculator keys.Key strokes: 1.08 yx 17 +/- = x 150,000 =

Calculator: N = 17; I/Y = 8; FV = 150,000; CPT PV = -40,540.34The actual number computes to 9999.998. This is a good place to remind the students to pay attention to what the question asked and be reasonable in their answers. A little common sense should tell them that the original amount was 10,000 and that the calculation doesnt come out exactly because the future value is rounded to cents.

Calculator: N = 10; I/Y = 7; FV = 19,671.51; CPT PV = -10,000Calculator: 5 years: N = 5; I/Y = 10; FV = 500; CPT PV = -310.46N = 10; I/Y = 10; FV = 500; CPT PV = -192.77Calculator: 10%: N = 5; I/Y = 10; FV = 500; CPT PV = 310.4615%: N = 5; I/Y = 15; FV = 500; CPT PV = 248.58It is very important at this point to make sure that the students have more than 2 decimal places visible on their calculator.

Efficient key strokes for formula: 1200 / 1000 = yx 5 1/x = - 1 = .03714

If they receive an error when they try to use the financial keys, they probably forgot to enter one of the numbers as a negative.Calculator: N = 6; FV = 20,000; PV = 10,000; CPT I/Y = 12.25%Calculator: N = 17; FV = 75,000; PV = 5,000; CPT I/Y = 17.27%

This is a great problem to illustrate how TVM can help you set realistic financial goals and maybe adjust your expectations based on what you can currently afford to save.Remind the students that ln is the natural logarithm and can be found on the calculator.

The rule of 72 is a quick way to estimate how long it will take to double your money. # years to double = 72 / r where r is a percent.Calculator: I/Y = 10; FV = 20,000; PV = 15,000; CPT N = 3.02 yearsPoint out that the closing costs are only paid on the loan amount, not on the total amount paid for the house.The book discusses that there are two ways to work this problem. The first method, computing the FV one year at a time and adding the cash flows as you go along, is illustrated in Example 6.1 in the book. The slides illustrate the other method, finding the future value at the end for each cash flow and then adding.

Point out that you can find the value of a set of cash flows at any point in time, all you have to do is get the value of each cash flow at that point in time and then add them together.

The students can read the example in the book. It is also provided here.

You 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 three years? In four years?

Point out that there are several ways that this can be worked. The book works this example by rolling the value forward each year. The presentation will show the second way to work the problem.

Calculator:Today (year 0 CF): 3 N; 8 I/Y; -7000 PV; CPT FV = 8817.98Year 1 CF: 2 N; 8 I/Y; -4000 PV; CPT FV = 4665.60Year 2 CF: 1 N; 8 I/Y; -4000 PV; CPT FV = 4320Year 3 CF: value = 4,000Total value in 3 years = 8817.98 + 4665.60 + 4320 + 4000 = 21,803.58Value at year 4: 1 N; 8 I/Y; -21803.58 PV; CPT FV = 23,547.87

I entered the PV as negative for two reasons. (1) It is a cash outflow since it is an investment. (2) The FV is computed as positive and the students can then just store each calculation and then add from the memory registers, instead of writing down all of the numbers and taking the risk of keying something back into the calculator incorrectly.Calculator:Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV = 594.05Year 1 CF: 1 N; -600 PV; 9 I/Y; CPT FV = 654.00Total FV = 594.05 + 654.00 = 1248.05

Calculator:First way:Year 0 CF: 5 N; -500 PV; 9 I/Y; CPT FV = 769.31Year 1 CF: 4 N; -600 PV; 9 I/Y; CPT FV = 846.95Total FV = 769.31 + 846.95 = 1616.26Second way use value at year 2:3 N; -1248.05 PV; 9 I/Y; CPT FV = 1616.26

FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97

Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV = 136.05Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV = 349.92Total FV = 136.05 + 349.92 = 485.97The students can read the example in the book.

You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one?

Point out that the question could also be phrased as How much is this investment worth?

Calculator:Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = -178.57Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = -318.88Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV = -427.07Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV = - 508.41Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93

Remember the sign convention. The negative numbers imply that we would have to pay 1432.93 today to receive the cash flows in the future.

Calculator:N = 1; I/Y = 10; FV = 1000; CPT PV = -909.09N = 2; I/Y = 10; FV = 2000; CPT PV = -1652.89N = 3; I/Y = 10; FV = 3000; CPT PV = -2253.94

The next example will be worked using the cash flow keys.

Note that with the BA-II Plus, the students can double check the numbers they have entered by pressing the up and down arrows. It is similar to entering the cash flows into spreadsheet cells.

Other calculators also have cash flow keys. You enter the information by putting in the cash flow and then pressing CF. You have to always start with the year 0 cash flow, even if it is zero.

Remind the students that the cash flows have to occur at even intervals, so if you skip a year, you still have to enter a 0 cash flow for that year.

You can also use this as an introduction to NPV by having the students put 100 in for CF0. When they compute the NPV, they will get 8.51. You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV.Other calculators also have a key that allows you to switch between Beg/End.The students can read the example in the book.

After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?

Note that the difference between the answer here and the one in the book is due to the rounding of the Annuity PV factor in the book.Calculator:30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29You might point out that you would probably not offer 154,501. The more likely scenario would be 154,500.

Calculator:30*12 = 360 N.5 I/Y840 PMTCPT PV = 140,105Note if you do not round the monthly rate and actually use 8/12, then the payment will be 448.30

Calculator:4(12) = 48 N; 20,000 PV; .66667 I/Y; CPT PMT = 488.26You ran a little short on your spring break vacation, so you put $1000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000.

This is an excellent opportunity to talk about credit card debt and the problems that can develop if it is not handled properly. Many students dont understand how it works and it is never discussed. This is something that students can take away from the class, even if they arent finance majors.Calculator:1.5 I/Y1000 PV-20 PMTCPT N = 93.111 MONTHS = 7.75 yearsSign convention matters!!!5 I/Y2000 PV-734.42 PMT CPT N = 3 yearsThe next slide talks about how to do this without a financial calculator.

FV = 2000(1.07540 1)/.075 = 454,513.04

Remember the sign convention!!!40 N7.5 I/Y-2000 PMTCPT FV = 454,513.04Note that the procedure for changing the calculator to an annuity due is similar on other calculators.

Calculator2nd BGN 2nd Set (you should see BGN in the display)3 N-10,000 PMT8 I/YCPT FV = 35,061.122nd BGN 2nd Set (be sure to change it back to an ordinary annuity)What if it were an ordinary annuity? FV = 32,464 (so receive an additional 2597.12 by starting to save today.)If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period.This is a good preview to the valuation issues discussed in future chapters. The price of an investment is just the present value of expected future cash flows.

Example statement:

Suppose the Fellini Co. wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell.Where m is the number of compounding periods per year

Using the calculator:

The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates.2nd I Conv NOM is the quoted rate down arrow EFF is the effective rate down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates.

Calculator:2nd I conv 5.25 NOM up arrow 365 C/Y up arrow CPT EFF = 5.39%5.3 NOM up arrow 2 C/Y up arrow CPT EFF = 5.37%It is important to point out that the daily rate is NOT .014, it is .014383562

First Account:365 N; 5.25 / 365 = .014383562 I/Y; 100 PV; CPT FV = 105.39Second Account:2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = 105.37On the calculator: 2nd I conv down arrow 12 EFF down arrow 12 C/Y down arrow CPT NOM2(12) = 24 N; 16.9 / 12 = 1.408333333 I/Y; 3500 PV; CPT PMT = -172.88

35(12) = 420 N9 / 12 = .75 I/Y50 PMTCPT FV = 147,089.22

3(365) = 1095 N5.5 / 365 = .015068493 I/Y15,000 FVCPT PV = -12,718.56Remind students that the value of an investment is the present value of expected future cash flows.

1 N; 10,000 FV; 7 I/Y; CPT PV = -9345.79