2Time is Money$100 in your hand today is worth more than $100 in one yearMoney earns interestThe higher the interest, the faster your money growsQ: How much would $1,000 promised in one year be worth today if the bank paid 5% interest?A: $ If we deposited $ after one year we would have earned $47.62 ($ × .05) in interest. Thus, our future value would be $ $47.62 = $1,000.Example
3Time is Money Present Value The amount that must be deposited today to have a future sum at a certain interest rateThe discounted value of a sum is its present value
4Outline of Approach Deal with four different types of problems Amount Present valueFuture valueAnnuity
5Outline of Approach Mathematics Time lines For each type of problem an equation will be presentedTime linesGraphic portrayal of a time value problem12Helps with complicated problems
6Amount Problems—Future Value The future value (FV) of an amountHow much a sum of money placed at interest (k) will grow into in some period of timeIf the time period is one yearFV1 = PV + kPV or FV1 = PV(1+k)If the time period is two yearsFV2 = FV1 + kFV1 or FV2 = PV(1+k)2If the time period is generalized to n yearsFVn = PV(1+k) n
7Amount Problems—Future Value The (1 + k)n depends onSize of k and nCan develop a table depicting different values of n and k and the proper value of (1 + k)nCan then use a more convenient formulaFVn = PV [FVFk,n]These values can be looked up in an interest factor table.Q: If we deposited $438 at 6% interest for five years, how much would we have?A: FV5 = $438(1.06)5 = $438(1.3382) = $586.13Example
8Table 5-1: The Future Value Factor for k and n FVFk,n = (1+k)n 6%51.3382
9Other Issues Problem-Solving Techniques The Opportunity Cost Rate Three of four variables are givenWe solve for the fourthThe Opportunity Cost RateThe opportunity cost of a resource is the benefit that would have been available from its next best use
10Financial Calculators Work directly with equationsHow to use a typical financial calculator in time valueFive time value keysUse either four or five keysSome calculators distinguish between inflows and outflowsIf a PV is entered as positive the computed FV is negative
11Financial Calculators I/YPMTFVPVNumber of time periodsInterest rate (%)Future ValuePresent ValuePaymentBasic Calculator Keys
12Financial Calculators I/YPMTPVFV1650004,716.98AnswerQ: What is the present value of $5,000 received in one year if interest rates are 6%?A: Input the following values on the calculator and compute the PV:Example
13The Expression for the Present Value of an Amount The future and present values factors are reciprocalsEither equation can be used to solve any amount problemsSolving for k or n involves searching a table.
14The Expression for the Present Value of an Amount—Example PVPMTFV3-850983.965.0I/YAnswerOr you could look up the interest factor of [850 ] in the proper table with n=3.ExampleQ: What interest rate will grow $850 into $ in three years?A: Using the FV equation of FV = PV (1 + i)n, the answer is:
15The Expression for the Present Value of an Amount—Example Q: How long does it take money invested at 14% to double?A: The future value must be twice the present value, so make up some values for present and future values that meet that criterion, such as $1 doubling to $2.NPVPMTFV5.29-1214I/YAnswerOr you could look up the interest factor of 2.00 [2 1] in the proper table with k=14%.Example
16Annuity Problems Annuity A finite series of equal payments separated by equal time intervalsOrdinary annuitiesPayments occur at the end of the time periodsAnnuity duePayments occur at the beginning of the time periods
21The Future Value of an Annuity—Developing a Formula Thus, for a 3-year annuity, the formula isFVFAk,n
22The Future Value of an Annuity—Solving Problems There are four variables in the future value of an annuity equationThe future value of the annuity itselfThe paymentThe interest rateThe number of periodsHelps to draw a time line
23The Future Value of an Annuity—Solving Problems—Example Q: The Brock Corporation owns the patent to an industrial process and receives license fees of $100,000 a year on a 10-year contract for its use. Management plans to invest each payment until the end of the contract to provide funds for development of a new process at that time. If the invested money is expected to earn 7%, how much will Brock have after the last payment is received?A: Using the FVAn = PMT[FVFAk,n] equation and looking up the interest factor of the annuity at an n of 10 and a k of 7, we obtain an interest factor of This gives up a FVA10 of $100,000[ ] = $1,381,640.FVNPMTI/Y1,381,645101000007PVAnswerExample
24The Sinking Fund Problem Companies borrow money by issuing bonds for lengthy time periodsNo repayment of principal is made during the bonds’ livesPrincipal is repaid at maturity in a lump sumA sinking fund provides cash to pay off a bond’s principal at maturityProblem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem
25The Sinking Fund Problem –Example Q: The Greenville Company issued bonds totaling $15 million for 30 years. The bond agreement specifies that a sinking fund must be maintained after 10 years, which will retire the bonds at maturity. Although no one can accurately predict interest rates, Greenville’s bank has estimated that a yield of 6% on deposited funds is realistic for long-term planning. How much should Greenville plan to deposit each year to be able to retire the bonds with the money put aside?A: The time period of the annuity is the last 20 years of the bond issue’s life. Input the following keystrokes into your calculator.PMTNFVI/Y407,768.352015,000,0006PVAnswerExample
26Compound Interest and Non-Annual Compounding Earning interest on interestCompounding periodsInterest is usually compounded annually, semiannually, quarterly or monthlyInterest rates are quoted by stating the nominal rate followed by the compounding period
28The Effective Annual Rate Effective annual rate (EAR)The annually compounded rate that pays the same interest as a lower rate compounded more frequently
29The Effective Annual Rate—Example Q: If 12% is compounded monthly, what annually compounded interest rate will get a depositor the same interest?A: If your initial deposit were $100, you would have $ after one year of 12% interest compounded monthly. Thus, an annually compounded rate of 12.68% [($ $100) – 1] would have to be earned.
30The Effective Annual Rate EAR can be calculated for any compounding period using the following formula:Effect of more frequent compounding is greater at higher interest rates
32The Effective Annual Rate The APR and EARAnnual percentage rate (APR)Is actually the nominal rate and is less than the EARCompounding Periods and the Time Value FormulasTime periods must be compounding periodsInterest rate must be the rate for a single compounding periodFor instance, with a quarterly compounding period the knominal must be divided by 4 and the n must be multiplied by 4
33The Effective Annual Rate – Example PMTNFVI/Y431.223015,0001PVAnswerQ: You want to buy a car costing $15,000 in 2½ years. You plan to save the money by making equal monthly deposits in your bank account, which pays 12% compounded monthly. How much must you deposit each month?A: This is a future value of an annuity problem with a 1% monthly interest rate and a 30-month time period. Input the following keystrokes into your calculator.
34The Present Value of an Annuity—Developing a Formula Sum of all of the annuity’s paymentsEasier to develop a formula than to do all the calculations individuallyPVFAk,n
35The Present Value of an Annuity—Solving Problems There are four variables in the present value of an annuity equationThe present value of the annuity itselfThe paymentThe interest rateThe number of periodsProblem usually presents 3 of the 4 variables
36The Present Value of an Annuity—Solving Problems–Example Q: The Shipson Company has just sold a large machine to Baltimore Inc. on an installment contract. The contract calls for Baltimore to make payments of $5,000 every six months (semiannually) for 10 years. Shipson would like its cash now and asks its bank to discount the contract and pay it the present (discounted) value. Baltimore is a good credit risk, so the bank is willing to discount the contract at 14% compounded semiannually. How much should Shipson receive?A: The contract represents an annuity with payments of $5,000. Adjust the interest rate and number of periods for semiannual compounding and solve for the present value of the annuity.ExamplePVNFVI/Y52,970.072075000PMTAnswerThis can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 20 and a k of 7%, resulting in PVA = $5,000[10.594] = $52,970.
37Spreadsheet Solutions Time value problems can be solved on a spreadsheet such as Microsoft Excel™ or Lotus 1-2-3™To solve for:FV use =FV(k, n, PMT, PV)PV use =PV(k, n, PMT, FV)K use =RATE(n, PMT, PV, FV)N use =NPER(k, PMT, PV, FV)PMT use =PMT(k, n, PV, FV)Select the function for the unknown variable, place the known variables in the proper order within the parentheses and input 0, for the unknown variable.
38Spreadsheet Solutions ComplicationsInterest rates in entered as decimals, not percentagesOf the three cash variables (FV, PMT or PV)One is always zeroThe other two must be of the opposite signReflects inflows (+) versus outflows (-)
39Amortized LoansAn amortized loan’s principal is paid off regularly over its lifeGenerally structured so that a constant payment is made periodicallyRepresents the present value of an annuity
40Amortized Loans—Example PMTNPVI/Y293.754810,0001.5FVAnswerThis can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 48 and a k of 1.5%, resulting in $10,000 = PMT[ ] = $Q: Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment?A: Adjust your interest rate and number of periods for monthly compounding and input the following keystrokes into your calculator.
41Amortized Loans—Example PVNFVI/Y15,053.75361500PMTAnswerExampleThis can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 36 and a k of 1%, resulting in PVA = $500[ ] = $15,Q: Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car?A: Adjust your k and n for monthly compounding and input the following calculator keystrokes.
42Loan Amortization Schedules Detail the interest and principal in each loan paymentShow the beginning and ending balances of unpaid principal for each periodNeed to knowLoan amount (PVA)Payment (PMT)Periodic interest rate (k)
43Loan Amortization Schedules—Example Q: Develop an amortization schedule for the loan demonstrated in Example 5.12.Note that the Interest portion of the payment is decreasing while the Principal portion is increasing.
44Mortgage Loans Mortgage loans (AKA: mortgages) Loans used to buy real estateOften the largest single financial transaction in an average person’s lifeTypically an amortized loan over 30 yearsDuring the early years of the mortgage nearly all the payment goes toward paying interestThis reverses toward the end of the mortgage
45Mortgage Loans Implications of mortgage payment pattern Early mortgage payments provide a large tax savings which reduces the effective cost of a loanHalfway through a mortgage’s life half of the loan has not been paid offLong-term loans like mortgages result in large total interest amounts over the life of the loan
46Mortgage Loans—Example PMTNFVI/Y1,015.65360.5979150,000PVAnswer$150,944Net interest cost$64,690Tax 30%$215,634Total Interest$150,000- Original Loan$365,634Total paymentsX number of payments$1,015.65Monthly paymentQ: Calculate the monthly payment for a 30-year 7.175% mortgage of $150,000. Also calculate the total interest paid over the life of the loan.A: Adjust the n and k for monthly compounding and input the following calculator keystrokes.
47The Annuity DueIn an annuity due payments occur at the beginning of each periodThe future value of an annuity dueBecause each payment is received one period earlierIt spends one period longer in the bank earning interest
48The Annuity Due—Example FVNPMTI/Y3,020,099 x 1.02 = 3,080,5014050,0002PVAnswerNOTE: Advanced calculators allow you to switch from END (ordinary annuity) to BEGIN (annuity due) mode.Q: The Baxter Corporation started making sinking fund deposits of $50,000 per quarter today. Baxter’s bank pays 8% compounded quarterly, and the payments will be made for 10 years. What will the fund be worth at the end of that time?A: Adjust the k and n for quarterly compounding and input the following calculator keystrokes.
49The Annuity Due The present value of an annuity due FormulaRecognizing types of annuity problemsAlways represent a stream of equal paymentsAlways involve some kind of a transaction at one end of the stream of paymentsEnd of stream—future value of an annuityBeginning of stream—present value of an annuity
50PerpetuitiesA perpetuity is a stream of regular payments that goes on foreverAn infinite annuityFuture value of a perpetuityMakes no sense because there is no end pointPresent value of a perpetuityA diminishing series of numbersEach payment’s present value is smaller than the one before
51Perpetuities—Example You may also work this by inputting a large n into your calculator (to simulate infinity), as shown below.PVNPMTI/Y25099952FVAnswerQ: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Money markets are such that investors can earn about 8% compounded quarterly on their money. How much can Longhorn sell this special security for?A: Convert the k to a quarterly k and plug the values into the equation.
52Continuous Compounding Compounding periods can be shorter than a dayAs the time periods become infinitesimally short, interest is said to be compounded continuouslyTo determine the future value of a continuously compounded value:
53Continuous Compounding—Example A: To determine the EAR of 12% compounded continuously, find the future value of $100 compounded continuously in one year, then calculate the annual return.ExampleNOTE: Some advanced calculators have a function to solve for the EAR with continuous compounding.Q: The First National Bank of Charleston is offering continuously compounded interest on savings deposits. Such an offering is generally more of a promotional feature than anything else. If you deposit $5,000 at 6½% compounded continuously and leave it in the bank for 3½ years, how much will you have? Also, what is the equivalent annual rate (EAR) of 12% compounded continuously?A: To determine the future value of $5,000, plug the appropriate values into the equation.
55Multipart ProblemsTime value problems are often combined due to complex nature of real situationsA time line portrayal can be critical to keeping things straight
56Multipart Problems—Example Q: Exeter Inc. has $75,000 invested in securities that earn a return of 16% compounded quarterly. The company is developing a new product that it plans to launch in two years at a cost of $500,000. Exeter’s cash flow is good now but may not be later, so management would like to bank money from now until the launch to be sure of having the $500,000 in hand at that time. The money currently invested in securities can be used to provide part of the launch fund. Exeter’s bank has offered an account that will pay 12% compounded monthly. How much should Exeter deposit with the bank each month to have enough reserved for the product launch?
57Multipart Problems—Example A: Two things are happening in this problemExeter is saving money every month (an annuity) andThe money invested in securities (an amount) is growing independently at interestWe have two problems that must be handled sequentially.First, we need to find the future value of $75,000Then, we subtract that future value from $500,000 to determine how much extra Exeter needs to save via the annuityThen, we’ll solve a future value of an annuity problem for the payment
58Multipart Problems—Example To find the future value of the $75,000…PVNPMTI/Y102,6438475,000FVAnswerTo find the savings annuity value14,731241397,355$500,000 - $102,645= $397,355
59Uneven Streams and Imbedded Annuities Many real world problems have sequences of uneven cash flowsThese are NOT annuitiesFor example, if you were asked to determine the present value of the following stream of cash flows$100$200$300Must discount each cash flow individuallyNot really a problem when attempting to determine either a present or future valueBecomes a problem when attempting to determine an interest rate
60Uneven Streams and Imbedded Annuities $100$200$300ExampleA: We’ll start with a guess of 12% and discount each amount separately at that rate.This value is too low, so we need to select a lower interest rate. Using 11% gives us $ The answer is between 8% and 9%.Q: Calculate the interest rate at which the present value of the stream of payments shown below is $500.
61Calculator Solutions for Uneven Streams Financial calculators and spreadsheets have the ability to handle uneven streams with a limited number of paymentsGenerally programmed to find the present value of the streams or the k that will equate a present value to the stream
62Imbedded AnnuitiesSometimes uneven streams of cash flows will have annuities embedded within themWe can use the annuity formula to calculate the present or future value of that portion of the problem