4 The Time Value Of Money

Introduction This chapter introduces the concepts and skills necessary to understand the time value of money and its applications.

Notation I denotes simple interest i denotes the interest rate per period n denotes the number of periods PMT denotes cash payment (annuities only) PV denotes the present value dollar amount T denotes the tax rate t denotes time PV0 = principal amount at time 0 FVn = future value n time periods from time 0

Simple Interest Simple Interest Interest paid on the principal sum only I = PV0  i  n FVn = PV0 + I = PV0 + PV0  i  n

Compound Interest Compound Interest Interest paid on the principal and on prior interest that has not been paid or withdrawn FV1 = PV0(1+i)1 FV2 = FV1(1+i)1 = PV0(1+i)2 FV3 = FV2(1+i)1 = PV0(1+i)2(1+i)1 = PV0(1+i)3

Future Value of a Cash Flow At the end of year n for a sum compounded at interest rate i is FVn = PV0(1 + i)n Formula See Figure 4.1. In Table I in the text, (FVIFi,n) shows the future value of $1 invested for n years at interest rate i: FVIFi,n = (1 + i)n Table I When using the table, FVn = PV0(FVIFi,n) See Figure 4.2.

Future Value of a Cash FVn = PV0(1 + i)n

Tables Have Three Variables Interest factors (IF) Time periods (n) Interest rates per period (i) If you know any two, you can solve algebraically for the third variable.

Present Value of a Cash Flow PV0 = FVn[1/(1+i)n] Formula PVIFi, n = [1/(1+i)n] Table II PV0 = FVn(PVIFi, n) Table II See Figure 4.3.

Present Value of a Cash Flow PV0 = FVn[1/(1+i)n]

Example Using Formula What is the PV of $100 one year from now with 12 percent (annual) interest compounded monthly? PV0 = $100  1/(1 + .12/12)(12 1) = $100  1/(1.126825) = $100  (.88744923) = $ 88.74

Example Using Table II PV0 = FVn(PVIFi, n) = $100(.887) From Table II = $ 88.70

Annuity A series of equal dollar CFs for a specified number of periods Ordinary annuity is where the CFs occur at the end of each period. Annuity due is where the CFs occur at the beginning of each period.

Future Value of an Ordinary Annuity Formula for IF FVANn = PMT(FVIFAi, n) Table III

Future Value of an Ordinary Annuity Suppose Ms. Jefferson receives a three-year ordinary annuity of $1,000 per year and deposits the money in a savings account at the end of each year. The account earns interest at a rate of 6% compounded annually. How much will her account be worth at the end of the three-year period?

Future Value of an Ordinary Annuity See Figure 4.6. FVAN3 = PMT(FVIFA0.06, 3) = $1,000(3.184) = $3,184

Present Value of an Ordinary Annuity Formula PVAN0 = PMT(PVIFAi, n) Table IV

Present Value of an Ordinary Annuity What is the present value of an ordinary $1,000 annuity received at the end of each year for five years discounted at a 6% rate? See Figure 4.8. PVAN0 = PMT(PVIFA0.06, 5) = $1,000(4.212) = $4,212

Annuity Due Future Value of an Annuity Due FVANDn = PMT(FVIFAi, n)(1 + i) Table III Present Value of an Annuity Due PVAND0 = PMT(PVIFAi, n)(1 + i) Table IV

Annuity Due Consider the case of Jefferson cited earlier. If she deposits $1,000 in a savings account at the beginning of each year for the next three years and the account earns 6% interest, compounded annually, how much will be in the account at the end of three years? See Figure 4.7. FVAND3 = PMT(FVIFA0.06, 3)(1 + 0.06) = $1,000(3.375) = $3,375

Annuity Due Consider the case of a five-year annuity of $1,000 each year, discounted at 6% interest rate. What is the present value of this annuity if each payment is received at the beginning of each year? See Figure 4.9. PVAND0 = PMT(PVIFA0.06, 5)(1 + 0.06) = $1,000(4.465) = $4,465

Other Important Formulas Sinking Fund PMT = FVANn  (FVIFAi, n) Table III Payments on a Loan PMT = PVAN0  (PVIFAi, n) Table IV Present Value of a Perpetuity PVPER0 = PMT  i Need better title for this slide

Example: Sinking Fund Problem Suppose the Omega Graphics Company wishes to set aside an equal, annual, end-of-year amount in a “sinking fund account” earning 10% per annum over the next five years. The firm wants to have $5 million in the account at the end of five years to retire (pay off) $5 million in outstanding bonds. How much must be deposited in the account at the end of each year?

Solution Based on Table III $5,000,000 = PMT(FVIFA0.10, 5) = PMT(6.105)  PMT = $819,001

Solution Based on the Financial Calculator 1. 5,000,000 → FV 2. 10 → %i 3. 5 → N 4. Compute 5. PMT (= -818,987.40)

Example: Payments on a Loan Suppose you borrowed $10,000 from the ICBC. The loan is for a period of four years at an interest rate of 10%. It requires that you make four equal, annual, end-of-year payments that include both principal and interest on the outstanding balance.

Solution Based on Table IV PMT = PVAN0  (PVIFAi, n) = $10,000  (PVIFA0.10, 4) = $10,000  3.170 = $3,155

Solution Based on the Financial Calculator 1. 10,000 → PV 2. 10 → %i 3. 4 → N 4. Compute 5. PMT (= -3,154.71)

Present Value of a Perpetuity Assume that Kansas City Power & Light series E preferred stock promises payments of $4.50 per year forever and that an investor requires a 10% rate of return on this type of investment. How much would the investor be willing to pay for this security? PVPER0 = PMT  i PVPER0 = $4.50  10% = $45

Present Value of an Uneven Payment Stream Algebraically, the present value of an uneven payment stream can be represented as See Figure 4.10.

Present Value of Deferred Annuities Suppose that you wish to provide for the college education of your daughter. She will begin college five years from now, and you wish to have $15,000 available for her at the beginning of each year in college. How much must be invested today at a 12% annual rate of return in order to provide the four-year, $15,000 annuity for your daughter? See Figure 4.11.

Solution Based on Tables II and III First step: calculate the present value of the four-year (ordinary) annuity PVAN4 = $15,000(PVIFA0.12, 4) = $15,000(3.037) = $45,555 Second step: calculate the present value of the (ordinary) annuity PVAN0 = PVAN4(PVIF0.12, 4) = $45,555(0.636) = $28,973

Solution Based on the Financial Calculator 1. 15,000 → PMT 2. 12 → %i 3. 4 → N 4. Compute 5. PV (= -45,560) 6. 45,560 → FV 7. 12 → %i 8. 4 → N 9. Compute 10. PV (= -28,954)

Interest Compounded More Frequently Than Once Per Year Suppose: m = # of times interest is compounded per year n = # of years Future Value Present Value

Compounding and Effective Rates Rate of interest per compounding period Effective annual rate of interest

Compounding and Effective Rates Suppose a bank offers you a loan an annual nominal interest rate of 12% compounded quarterly. What effective annual interest rate is the bank charging you?

Example Mr. Moore is 45 years old today (Dec. 31) and is beginning to plan for his retirement. He wants to set aside an equal amount at the end of each of the next 20 years so that he can retire at age 65. He expects to live to the maximum age of 85 and wants to be able to withdraw $25,000 per year from the account on his 66th through 85th birthdays.

Example The account is expected to earn 10 percent per annum for the entire period of time. Determine the size of the annual deposits that must be made by Mr. Moore.

Solution Based on Tables III and IV PVAN = PMT(PVIFA0.10,20) = $25,000(8.514) = $212,850 (needed on 65th birthday) $212,850 = PMT(FVIFA0.10,20) = PMT(57.275)  PMT = $3,716.28

Solution Based on the Financial Calculator 1. 25,000 → PMT 2. 10 → %i 3. 20 → N 4. Compute 5. PV (= -212,839) 6. 212,839 → FV 7. 10 → %i 8. 20 → N 9. Compute 10. PMT (= -3,716.08)

Example You sold 1000 shares of stock today for $80.515 per share that you paid $50 for 5 years ago. Determine the average annual rate of return on your investment, assuming the stock paid no dividends.

Solution Based on Table II PV0 = FVn(PVIFi, n); n = 5; PV0 = $50; FV5 = $80.515 $50 = $80.515(PVIFi, 5) (PVIFi, 5) = 0.621; Therefore i = 10% from Table II.

Example The Texas lottery agrees to pay the winner $500,000 at the end of each year for the next 20 years. What is the future value of this lottery if you plan to put each payment in an account earning 12 percent?

Solution Based on Table III FVAN20 = PMT(FVIFAi, n) = $500,000(FVIFA0.12, 20) = $500,000(72.052) = $36,026,000

Solution Based on the Financial Calculator 1. -500,000 → PMT 2. 12 → %i 3. 20 → N 4. Compute 5. FV (= 36,026,221.22)

Example Your firm has just leased a $32,000 BMW for you. The lease requires five beginning of the year payments that will fully amortize the cost of the car. What is the amount of the payments if the interest rate is 10 percent?

Solution Based on Table IV PVAND0 = PMT(PVIFAi, n)(1+i) $32,000 = PMT(PVIFA0.10, 5)(1+10%) $32,000 = PMT(3.791)(1.10)  PMT = $7,673.68

Solution Based on the Financial Calculator 1. 32,000 → PV 2. 10 → %i 3. 5 → N 4. Compute 5. PMT (= -8,441.52) 6. -8,441.52  1.10 = -7,674.11