1 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons MT 480 Unit 2 CHAPTER 5 The Time Value of Money

2 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  How does a manager determine the value of a series of future cash flows, whether paying for an asset or evaluating a project?  We refer to this value as the time value of money (TVM). The Time Value of Money  What is the value of a stream of future cash flows today?

3 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  TVM is based on the belief that people prefer to consume goods today rather than wait to consume similar goods tomorrow.  People have a positive time preference. The Time Value of Money Consuming Today or Tomorrow  Money has a time value because a dollar today is worth more than a dollar tomorrow.

4 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  Today’s dollar can be invested to earn interest or spent.  Value of a dollar invested (positive interest rate) grows over time.  Rate of interest determines trade-off between spending today versus saving. The Time Value of Money Consuming Today or Tomorrow

5 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons The Time Value of Money Future Value versus Present Value  Future value measures what one or more cash flows are worth at the end of a specified period.  Present value measures what one or more cash flows that are to be received in the future will be worth today (at t=0).  Financial decisions are evaluated either on a future value basis or present value basis.

6 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons The Time Value of Money  Discounting is the process of converting future cash flows to their present values.  Compounding is the process of earning interest over time. Future Value versus Present Value

7 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Future Value and Compounding Single Period Investment  We can determine the value of an investment at the end of one period if we know the interest rate to be earned by the investment.  If you invest for one period at an interest rate of i, your investment, or principle, will grow by (1 + i) per dollar invested.  The term (1+ i) is the future value interest factor, often called simply the future value factor.

8 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Future Value and Compounding Two-Period Investing  After the first period, interest accrues on original investment (principle) and interest earned in preceding periods.  A two-period investment is simply two single- period investments back-to-back.

9 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  The principal is the amount of money on which interest is paid.  Simple interest is the amount of interest paid on the original principal amount only.  Compounding interest consists of both simple interest and interest-on-interest. Future Value and Compounding Two-Period Investing

10 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Future Value and Compounding  General equation to find the future value after any number of periods. The Future Value Equation  We can use financial calculators or future value tables to find the future value factor at different interest rates and maturity periods.  The term (1 + i) n is the future value factor.

11 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons where: FV n = future value of investment at the end of period n PV = original principle (P 0 ) or present value i = the rate of interest per period, which is often a year n = the number of periods The general equation to find the future value is: Future Value and Compounding

12 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Compounding More Frequently Than Once a Year The more frequently the interest payments are compounded, the larger the future value of $1 for a given time period. where: m = number of compounding periods in a year Future Value and Compounding

13 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons When interest is compounded on a continuous basis, we can use the equation below. where: e = exponential function which is about Future Value and Compounding

14 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Continuous compounding example Future Value and Compounding Your grandmother wants to put $10,000 in a savings account at a bank. How much money would she have at the end of five years if the bank paid 5 percent annual interest compounded continuously?

15 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Present Value and Discounting Present value calculations state the current value of a dollar in the future.  This process is called discounting, and the interest rate i is known as the discount rate.  The present value (PV) is often called the discounted value of future cash payments.  The present value factor is more commonly called the discount factor.

16 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons The equation below gives us the general equation to find the present value after any number of periods. Present Value and Discounting

17 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Present Value and Discounting  The further in the future a dollar will be received, the less it is worth today.  The higher the discount rate, the lower the present value of a dollar.

18 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Finding the Interest Rate A number of situations will require you to determine the interest rate (or discount rate) for a given stream of future cash flows.  to determine the interest rate on a loan.  to determine a growth rate.  to determine the return on an investment. For an individual investor or a firm, it may be necessary.

19 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Compound Growth Rates Compound growth occurs when the initial value of a number increases or decreases each period by the factor (1 + growth rate).  Examples include population growth, earnings growth.

20 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons CHAPTER 6 Discounted Cash Flows and Valuation 20

21 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Multiple Cash Flows  Many business situations call for computing present value of a series of expected future cash flows.  Determining market value of security.  Deciding whether to make capital investment.  Process similar to determining future value of multiple cash flows. Present Value of Multiple Cash Flows 21

22 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  Next, calculate present value of each cash flow using equation 5.4 from the previous chapter. Present Value of Multiple Cash Flows  Finally, add up all present values.  Sum of present values of stream of future cash flows is their current market price, or value.  First, prepare timeline to identify magnitude and timing of cash flows. 22 Multiple Cash Flows

23 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Annuities and Perpetuities  Individual investors may make constant payments on home or car loans, or invest fixed amount year after year saving for retirement.  Many situations exist where businesses and individuals would face either receiving or paying constant amount for a length of period. Level Cash Flows 23

24 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  Annuity: any financial contract calling for equally spaced level cash flows over finite number of periods. Annuities and Perpetuities  Perpetuity: contract calling for level cash flow payments to continue forever.  Ordinary annuities: constant cash flows occurring at end of each period. 24 Level Cash Flows

25 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Present Value of an Annuity  Can calculate present value of annuity same way present value of multiple cash flows is calculated.  Becomes tedious with large no. of payments.  Instead, simplify equation 5.4 in chapter 5 to obtain annuity factor.  Results in equation 6.1 that can be used to calculate the annuity’s present value. 25 Level Cash Flows

26 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons 26 Level Cash Flows

27 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Finding Monthly or Yearly Payments Example 27 Level Cash Flows You have just purchased a $450,000 condominium. You were able to put $50,000 down and obtain a 30- year fixed rate mortgage at percent for the balance. What are your monthly payments?

28 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Preparing a Loan Amortization Schedule  Amortization: the way the borrowed amount (principal) is paid down over life of loan.  Monthly loan payment is structured so each month portion of principal is paid off; at time loan matures, it is entirely paid off. 28 Level Cash Flows

29 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  Amortized loan: each loan payment contains some payment of principal and an interest payment. Preparing a Loan Amortization Schedule  Loan amortization schedule is a table showing:  loan balance at beginning and end of each period.  payment made during that period.  how much of payment represents interest.  how much represents repayment of principal. 29 Level Cash Flows

30 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  With amortized loan, larger proportion of each month’s payment goes towards interest in early periods.  As loan is paid down, greater proportion of each payment is used to pay down principal. Preparing a Loan Amortization Schedule  Amortization schedules are best done on a spreadsheet. 30 Level Cash Flows

31 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Finding the Interest Rate  The annuity equation can also be used to find interest rate or discount rate for an annuity.  To determine rate of return for the annuity, we need to solve equation for the unknown value i.  Other than using trial and error approach, easier to solve using financial calculator. 31 Level Cash Flows

32 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Future Value of an Annuity  Future value annuity calculations usually involve finding what a savings or investment activity is worth at some future point.  E.g. saving periodically for vacation, car, house, or retirement.  We can derive the future value annuity equation from the present value annuity equation (equation 6.1). This results in equation Level Cash Flows

33 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Future Value of an Annuity Equation 33 Level Cash Flows

34 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Perpetuities  A perpetuity is constant stream of cash flows that goes on for infinite period.  In stock markets, preferred stock issues are considered to be perpetuities, with issuer paying a constant dividend to holders.  Equation for present value of a perpetuity can be derived from present value of an annuity equation with n tending to infinity. 34 Level Cash Flows

35 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Important relationship between present value of annuity and a perpetuity. Perpetuities  Just as perpetuity equation was derived from present value annuity equation, one can also derive present value of an annuity from the equation for a perpetuity. 35 Level Cash Flows

36 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  Annuity is called an annuity due when there is an annuity with payments being incurred at beginning of each period rather than at end. Annuity Due  Rent or lease payments typically made at beginning of each period rather than at end. 36 Level Cash Flows

37 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Annuity Due  Annuity transformation method shows relationship between ordinary annuity and annuity due.  Each period’s cash flow thus earns extra period of interest compared to ordinary annuity.  Present or future value of annuity due is always higher than that of ordinary annuity. 37 Level Cash Flows Annuity due = Ordinary annuity value  (1+i) (6.4)

38 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Annuity Due Example The value of the annuity due shown in Exhibit 6.7B is: 38 Level Cash Flows Annuity due = $3,312  (1.08) = $3,577

39 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  In addition to constant cash flow streams, one may have to deal with cash flows that grow at a constant rate over time.  These cash-flow streams called growing annuities or growing perpetuities. Cash Flows That Grow at a Constant Rate 39

40 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Growing Annuity  Business may need to compute value of multiyear product or service contracts with cash flows that increase each year at constant rate.  These are called growing annuities.  Example of growing annuity: valuation of growing business whose cash flows increase every year at constant rate. 40 Cash Flows That Grow at a Constant Rate

41 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Growing Annuity  Use this equation to value the present value of growing annuity (equation 6.5) when the growth rate is less than discount rate. 41 Cash Flows That Grow at a Constant Rate

42 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Growing Perpetuity  When cash flow stream features constant growing annuity forever.  Can be derived from equation 6.5 when n tends to infinity and results in the following equation: 42 Cash Flows That Grow at a Constant Rate

43 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons  Interest rates can be quoted in financial markets in variety of ways.  Most common quote, especially for a loan, is annual percentage rate (APR).  APR represents simple interest accrued on loan or investment in a single period; annualized over a year by multiplying it by appropriate number of periods in a year. Effective Annual Interest Rate 43

44 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Calculating the Effective Annual Rate (EAR)  Correct way to compute annualized rate is to reflect compounding that occurs; involves calculating effective annual rate (EAR).  Effective annual interest rate (EAR) is defined as annual growth rate that takes compounding into account. 44 Effective Annual Interest Rate

45 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Calculating the Effective Annual Rate (EAR) EAR = (1 + Quoted rate/m) m – 1 (6.7) m is the # of compounding periods during a year.  EAR conversion formula accounts for number of compounding periods, thus effectively adjusts annualized interest rate for time value of money.  EAR is the true cost of borrowing and lending. 45 Effective Annual Interest Rate

46 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Consumer Protection Acts and Interest Rate Disclosures  Truth-in-Lending (1968) ensures that true cost of credit was disclosed to consumers, so they could make sound financial decisions.  Truth-in-Savings Act provides consumers accurate estimate of return they would earn on investment. 46 Effective Annual Interest Rate

47 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Consumer Protection Acts and Interest Rate Disclosures  Require that APR be disclosed on all consumer loans and savings plans, and prominently displayed on advertising and contractual documents.  Note that EAR, not APR, is the appropriate rate to use in present and future value calculations. 47 Effective Annual Interest Rate

48 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Problem 5.18

49 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Problem 5.20

50 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Problem 5.28

51 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Problem 6.4

52 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Problem 6.6

53 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Problem 6.18

54 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Problem 6.26

55 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons Questions