Time Value of Money Chapter 5 © 2003 South-Western/Thomson Learning

2 Time is Money $100 in your hand today is worth more than $100 in one year Money earns interest The higher the interest, the faster your money grows

3 Time is Money Present Value The amount that must be deposited today to have a future sum at a certain interest rate The discounted value of a sum is its present value

4 Outline of Approach Deal with four different types of problems Amount Present value Future value Annuity Present value Future value

5 Outline of Approach Mathematics For each type of problem an equation will be presented Time lines Graphic portrayal of a time value problem 012 Helps with complicated problems

6 Amount Problems—Future Value The future value (FV) of an amount How much a sum of money placed at interest (k) will grow into in some period of time If the time period is one year FV 1 = PV + kPV or FV 1 = PV(1+k) If the time period is two years FV 2 = FV 1 + kFV 1 or FV 2 = PV(1+k) 2 If the time period is generalized to n years FV n = PV(1+k) n

7 Amount Problems—Future Value The (1 + k) n depends on Size of k and n Can develop a table depicting different values of n and k and the proper value of (1 + k) n Can then use a more convenient formula FV n = PV [FVF k,n ] These values can be looked up in an interest factor table.

8 Other Issues Problem-Solving Techniques Three of four variables are given We solve for the fourth The Opportunity Cost Rate The opportunity cost of a resource is the benefit that would have been available from its next best use

9 Financial Calculators Work directly with equations How to use a typical financial calculator in time value Five time value keys Use either four or five keys Some calculators distinguish between inflows and outflows If a PV is entered as positive the computed FV is negative

10 The Expression for the Present Value of an Amount The future and present values factors are reciprocals Either equation can be used to solve any amount problems Solving for k or n involves searching a table.

11 Annuity Problems Annuity A finite series of equal payments separated by equal time intervals Ordinary annuities Payments occur at the end of the time periods Annuity due Payments occur at the beginning of the time periods

12 The Future Value of an Annuity— Developing a Formula Future value of an annuity The sum, at its end, of all payments and all interest if each payment is deposited when received

13 The Future Value of an Annuity— Developing a Formula Thus, for a 3-year annuity, the formula is FVFA k,n

14 The Future Value of an Annuity—Solving Problems There are four variables in the future value of an annuity equation The future value of the annuity itself The payment The interest rate The number of periods Helps to draw a time line

15 The Sinking Fund Problem Companies borrow money by issuing bonds for lengthy time periods No repayment of principal is made during the bonds’ lives Principal is repaid at maturity in a lump sum A sinking fund provides cash to pay off a bond’s principal at maturity Problem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem

16 Compound Interest and Non- Annual Compounding Compounding Earning interest on interest Compounding periods Interest is usually compounded annually, semiannually, quarterly or monthly Interest rates are quoted by stating the nominal rate followed by the compounding period

17 The Effective Annual Rate Effective annual rate (EAR) The annually compounded rate that pays the same interest as a lower rate compounded more frequently

18 The 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

19 The Effective Annual Rate The APR and EAR Annual percentage rate (APR) Is actually the nominal rate and is less than the EAR Compounding Periods and the Time Value Formulas Time periods must be compounding periods Interest rate must be the rate for a single compounding period For instance, with a quarterly compounding period the k nominal must be divided by 4 and the n must be multiplied by 4

20 The Present Value of an Annuity— Developing a Formula Present value of an annuity Sum of all of the annuity’s payments Easier to develop a formula than to do all the calculations individually PVFA k,n

21 The Present Value of an Annuity—Solving Problems There are four variables in the present value of an annuity equation The present value of the annuity itself The payment The interest rate The number of periods Problem usually presents 3 of the 4 variables

22 Spreadsheet 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.

23 Spreadsheet Solutions Complications Interest rates in entered as decimals, not percentages Of the three cash variables (FV, PMT or PV) One is always zero The other two must be of the opposite sign Reflects inflows (+) versus outflows (-)

24 Amortized Loans An amortized loan’s principal is paid off regularly over its life Generally structured so that a constant payment is made periodically Represents the present value of an annuity

25 Loan Amortization Schedules Detail the interest and principal in each loan payment Show the beginning and ending balances of unpaid principal for each period Need to know Loan amount (PVA) Payment (PMT) Periodic interest rate (k)

26 Mortgage Loans Mortgage loans (AKA: mortgages) Loans used to buy real estate Often the largest single financial transaction in an average person’s life Typically an amortized loan over 30 years During the early years of the mortgage nearly all the payment goes toward paying interest This reverses toward the end of the mortgage

27 Mortgage Loans Implications of mortgage payment pattern Early mortgage payments provide a large tax savings which reduces the effective cost of a loan Halfway through a mortgage’s life half of the loan has not been paid off Long-term loans like mortgages result in large total interest amounts over the life of the loan

28 The Annuity Due In an annuity due payments occur at the beginning of each period The future value of an annuity due Because each payment is received one period earlier It spends one period longer in the bank earning interest

29 The Annuity Due The present value of an annuity due Formula Recognizing types of annuity problems Always represent a stream of equal payments Always involve some kind of a transaction at one end of the stream of payments End of stream—future value of an annuity Beginning of stream—present value of an annuity

30 Perpetuities A perpetuity is a stream of regular payments that goes on forever An infinite annuity Future value of a perpetuity Makes no sense because there is no end point Present value of a perpetuity A diminishing series of numbers Each payment’s present value is smaller than the one before

31 Continuous Compounding Compounding periods can be shorter than a day As the time periods become infinitesimally short, interest is said to be compounded continuously To determine the future value of a continuously compounded value:

32 Multipart Problems Time value problems are often combined due to complex nature of real situations A time line portrayal can be critical to keeping things straight

33 Uneven Streams and Imbedded Annuities Many real world problems have sequences of uneven cash flows These are NOT annuities For example, if you were asked to determine the present value of the following stream of cash flows $100$200$300 Must discount each cash flow individually Not really a problem when attempting to determine either a present or future value Becomes a problem when attempting to determine an interest rate

34 Calculator Solutions for Uneven Streams Financial calculators and spreadsheets have the ability to handle uneven streams with a limited number of payments Generally programmed to find the present value of the streams or the k that will equate a present value to the stream

35 Imbedded Annuities Sometimes uneven streams of cash flows will have annuities embedded within them We can use the annuity formula to calculate the present or future value of that portion of the problem