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1 Chapter 6 The Time Value of Money: Annuities and Other Topics Chapter 6 Contents Learning Objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate present and future values of each. 2. Calculate the present value of a level perpetuity and a growing perpetuity. 3. Calculate the present and future value of complex cash flow streams. Principles Used in Chapter 6 Principle 1: Money Has a Time Value. This chapter applies the time value of money concepts to annuities, perpetuities and complex cash flows. Principle 3: Cash Flows Are the Source of Value. This chapter introduces the idea that principle 1 and principle 3 will be combined to value stocks, bonds, and investment proposals. 6-2

2 Ordinary Annuities An annuity is a series of equal dollar payments that are made at the end of equidistant points in time such as monthly, quarterly, or annually over a finite period of time. If payments are made at the end of each period, the annuity is referred to as ordinary annuity. Example 6.1 How much money will you accumulate by the end of year 10 if you deposit $3,000 each for the next ten years in a savings account that earns 5% per year? Could solve by using the equation for computing the future value of an ordinary annuity. 6-3 The Future Value of an Ordinary Annuity FV n = FV of annuity at the end of nth period. PMT = annuity payment deposited or received at the end of each period i = interest rate per period n = number of periods for which annuity will last Easy to make errors when using the Equation. Very Easy to handle using Financial Calculators 6-4

4 Solving for PMT in an Ordinary Annuity Instead of figuring out how much money will be accumulated (i.e. FV), determine how much needs to be saved/accumulated each period (i.e. PMT) in order to accumulate a certain amount at the end of n years. In this case, know the values of n, i, and FV n in equation 6-1c and determine the value of PMT. 6-7 Solve for PMT in an Ordinary Annuity Example 6.2: Suppose you would like to have $25,000 saved 6 years from now to pay towards your down payment on a new house. If you are going to make equal annual end-ofyear payments to an investment account that pays 7 per cent, how big do these annual payments need to be? Using a Financial Calculator. N=6; I/Y = 7; PV = 0; FV = 25,000 PMT = -3,

5 Checkpoint 6.1 Class Problem Solve for an Ordinary Annuity Payment How much must you deposit in a savings account earning 8% annual interest in order to accumulate $5,000 at the end of 10 years? 6-9 Checkpoint 6.1: Class problem If you can earn 12 percent on your investments, and you would like to accumulate $100,000 for your child s education at the end of 18 years, how much must you invest annually to reach your goal? 6-10

6 Solving for Interest Rate in Ordinary Annuity Solve for interest rate that must be earned on an investment that will allow savings to grow to a certain amount of money by a future date. In this case, we know the values of n, PMT, and FV n when using the financial calculator and we need to determine the value of I/Y Solve for Interest Rate in an Ordinary Annuity Example Example 6.3: In 20 years, you are hoping to have saved $100,000 towards your child s college education. If you are able to save $2,500 at the end of each year for the next 20 years, what rate of return must you earn on your investments in order to achieve your goal? Using a Financial Calculator N = 20; PMT = -$2,500; FV = $100,000 PV = $0 I/Y = 6.77% 6-12

7 Solving for the Number of Periods in an Ordinary Annuity You may want to calculate the number of periods it will take for an annuity to reach a certain future value, given interest rate. Example 6.4: Suppose you are investing $6,000 at the end of each year in an account that pays 5%. How long will it take before the account is worth $50,000? Using a Financial Calculator I/Y = 5.0; PV = 0; PMT = -6,000; FV = 50,000 N = 7.14 years 6-13 The Present Value of an Ordinary Annuity The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future. PV annuity reflects how much you would (should) pay (today) for a constant set of cash flows that would be received each period for a fixed number of periods and given a constant interest rate (required rate of return). For example, compute the PV of ordinary annuity to answer the question: How much should be paid today (lump sum equivalent) for receiving $3,000 every year for the next 30 years if the interest rate is 5%? 6-14

8 The Present Value of an Ordinary Annuity Example with Logic Figure 6-2 shows the lump sum equivalent ($2,106.18) of receiving $500 per year for the next five years at an interest rate of 6% The Present Value of an Ordinary Annuity Example with Financial Calculator Compute the present value of an ordinary annuity in which $500 per year is received for the next five years at an interest rate of 6%. PMT = 500; N = 5; I/Y = 6; FV = 0 Solve PV = -2,

9 Checkpoint 6.2 Class Problem The Present Value of an Ordinary Annuity Your grandmother has offered to give you $1,000 per year for the next 10 years. What is the present value of this 10-year, $1,000 annuity discounted back to the present at 5 percent? 6-17 Amortized Loans An amortized loan is a loan paid off in equal payments consequently, the loan payments are an annuity. Examples: Home mortgage loans, Auto loans In an amortized loan, the present value can be thought of as the amount borrowed, n is the number of periods the loan lasts for, i is the interest rate per period, future value takes on zero because the loan will be paid of after n periods, and payment is the loan payment that is made. 6-18

11 Loan Amortization Schedule Logic We can observe the following from the table: Size of each payment remains the same. However, Interest payment declines each year as the amount owed declines and more of the principal is repaid. Amortized Loans with Monthly Payments Many loans such as auto and home loans require monthly payments. Requires converting n to number of months and computing the monthly interest rate Amortized Loans with Monthly Payments -Example Example 6.6 You have just found the perfect home. However, in order to buy it, you will need to take out a $300,000, 30-year mortgage at an annual rate of 6 percent. What will your monthly mortgage payments be? Using a Financial Calculator N=360; I/Y = 0.5; PV = 300,000; FV = 0 Solve PMT = -1, per month 6-22

12 Checkpoint 6.3 Additional Complexity and Concept Integration Determining the Outstanding Balance of a Loan that will be Refinanced Let s say that exactly ten years ago you took out a $200,000, 30-year mortgage with an annual interest rate of 9 percent and monthly payments of $1, But since you took out that loan, interest rates have dropped. You now have the opportunity to refinance your loan at an annual rate of 7 percent over 20 years. You need to know what the outstanding balance on your current loan is so you can take out a lower-interest-rate loan and pay it off. If you just made the 120th payment and have 240 payments remaining, what s your current loan balance? 6-23 Checkpoint 6.3 Logical Solution Flow 6-24

14 Annuities Due Annuity due is an annuity in which all the cash flows occur at the beginning of the period. For example, rent payments on apartments are typically annuity due as rent is paid at the beginning of the month. (pay in advance of using resource or asset) The examples and logic will illustrate that both the future value and present value of an annuity due are larger than that of an ordinary annuity. In each case, all payments are received or paid earlier Difference between an ordinary annuity and an annuity due Ordinary Annuity I% Annuity Due PMT PMT PMT I% PMT PMT PMT 6-28

15 Solving for FV: 3-year Ordinary Annuity of $100 at 10% $100 payments occur at the end of each period but there is no PV. INPUTS N I/YR PV PMT FV OUTPUT Solving for FV: 3-Year Annuity Due of $100 at 10% Now, $100 payments occur at the beginning of each period. FVA due = FVA ord (1 + I) = $331(1.10) = $ Alternatively, set calculator to BEGIN mode and solve for the FV of the annuity: BEGIN INPUTS N I/YR PV PMT FV OUTPUT

16 Annuities Due: Present Value Since with annuity due, each cash flow is received one year earlier, its present value will be discounted back for one less period. Solution approach on Calculator is to compute present value of n-1 period ordinary annuity and then add 1 payment at face value Perpetuities A perpetuity is an annuity that continues forever or has no maturity. For example, a dividend stream on a share of preferred stock. There are two basic types of perpetuities: Growing perpetuity in which cash flows grow at a constant rate, g, from period to period. Level perpetuity in which the payments are constant rate from period to period. Used to value infinite life stuff like financial assets (common and preferred stock) and corporations. 6-32

18 Checkpoint 6.4 Class Problem Compute Present Value of a Level Perpetuity How much would you pay, today, for a perpetuity of $500 paid annually that can be transferred to all future generations of your family? The appropriate discount rate is 8 percent? 6-35 Present Value of a Growing Perpetuity In growing perpetuities, the periodic cash flows grow at a constant rate each period. The present value of a growing perpetuity can be calculated using a simple mathematical equation. 6-36

19 PV of a Growing Perpetuity (formula) PV = Present value of a growing perpetuity PMT period 1 = Payment made at the end of 1 st period i = rate of interest used to discount the growing perpetuity s cash flows g = the rate of growth in the payment of cash flows from period to period 6-37 Checkpoint Example Compute PV of a Growing Perpetuity What is the present value of a perpetuity stream of cash flows that pays $500 at the end of year one but grows at a rate of 4% per year indefinitely? The discount rate is 8%. 6-38

20 Checkpoint 6.5: Class problem What is the present value of a stream of payments where the year 1 payment is $90,000 and the future payments grow at a rate of 5% per year? The interest rate used to discount the payments is 9% Complex Cash Flow Streams The cash flows streams in the business world may not always involve one type of cash flows. The cash flows may have a mixed pattern. For example, different cash flow amounts mixed in with annuities. Business analysis solutions involve organizing and valuing these mixed cash flows. For example, figure 6-4 summarizes the cash flows for Marriott. 6-40

21 Complex Cash Flow Streams Important to understand this figure 6-41 Checkpoint 6.6 Class Problem Compute Present Value of a Complex Cash Flow Stream What is the present value of cash flows of $500 at the end of years through 3, a cash flow of a negative $800 at the end of year 4, and cash flows of $800 at the end of years 5 through 10 if the appropriate discount rate is 5%? 6-42

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