# August, 2000UT Department of Finance The Time Value of Money 4 In order to work the problems in this module, the user should have the use of a business.

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August, 2000UT Department of Finance The Time Value of Money 4 What is the Time Value of Money? 4 Compound Interest 4 Future Value 4 Present Value 4 Frequency of Compounding 4 Annuities 4 Multiple Cash Flows 4 Bond Valuation 4 What is the Time Value of Money? 4 Compound Interest 4 Future Value 4 Present Value 4 Frequency of Compounding 4 Annuities 4 Multiple Cash Flows 4 Bond Valuation

August, 2000UT Department of Finance \$1,000 today Obviously, \$1,000 today. TIME VALUE OF MONEY Money received sooner rather than later allows one to use the funds for investment or consumption purposes. This concept is referred to as the TIME VALUE OF MONEY!! The Time Value of Money \$1,000 today \$1,000 in 5 years? Which would you rather have -- \$1,000 today or \$1,000 in 5 years?

August, 2000UT Department of Finance TIME INTEREST TIME allows one the opportunity to postpone consumption and earn INTEREST. NOT having the opportunity to earn interest on money is called OPPORTUNITY COST. Why TIME?

August, 2000UT Department of Finance How can one compare amounts in different time periods? 4 One can adjust values from different time periods using an interest rate. 4 Remember, one CANNOT compare numbers in different time periods without first adjusting them using an interest rate.

August, 2000UT Department of Finance Compound Interest When interest is paid on not only the principal amount invested, but also on any previous interest earned, this is called compound interest. FV = Principal + (Principal x Interest) = 2000 + (2000 x.06) = 2000 (1 + i) = PV (1 + i) Note: PV refers to Present Value or Principal

August, 2000UT Department of Finance \$2,000 today in an account that pays 6 If you invested \$2,000 today in an account that pays 6% interest, with interest compounded annually, how much will be in the account at the end of two years if there are no withdrawals? Future Value (Graphic) 0 1 2 \$2,000 FV 6%

August, 2000UT Department of Finance FV 1 PV\$2,000 \$2,247.20 FV 1 = PV (1+i) n = \$2,000 (1.06) 2 = \$2,247.20 Future Value (Formula) FV = future value, a value at some future point in time PV = present value, a value today which is usually designated as time 0 i = rate of interest per compounding period n = number of compounding periods Calculator Keystrokes: 1.06 (2nd y x) 2 x 2000 =

August, 2000UT Department of Finance Future Value (HP 17 B II Calculator) 2 6 2000 +/- N I%Yr PV 2,247.20FV Exit until you get Fin Menu. 2 nd, Clear Data. Choose Fin, then TVM

August, 2000UT Department of Finance \$5,000 5 years John wants to know how large his \$5,000 deposit will become at an annual compound interest rate of 8% at the end of 5 years. Future Value Example 5 0 1 2 3 4 5 \$5,000 FV 5 8%

August, 2000UT Department of Finance 4 Calculator keystrokes : 1.08 2 nd y x x 5000 = Future Value Solution FV n FV 5 \$7,346.64 u Calculation based on general formula:FV n = PV (1+i) n FV 5 = \$5,000 (1+ 0.08) 5 = \$7,346.64

August, 2000UT Department of Finance Future Value (HP 17 B II Calculator) 8 5000 +/- FV N I%Yr PV 7,346.64 Exit until you get Fin Menu. 2 nd, Clear Data. Choose Fin, then TVM 5

August, 2000UT Department of Finance Double Your Money!!! Quick! How long does it take to double \$5,000 at a compound rate of 12% per year (approx.)? Rule-of-72. We will use the Rule-of-72.

August, 2000UT Department of Finance The Rule-of-72 Quick! How long does it take to double \$5,000 at a compound rate of 12% per year (approx.)? 72 Approx. Years to Double = 72 / i% 726 Years 72 / 12% = 6 Years [Actual Time is 6.12 Years]

August, 2000UT Department of Finance Present Value 4 Since FV = PV(1 + i) n. PVFV PV = FV / (1+i) n. 4 Discounting is the process of translating a future value or a set of future cash flows into a present value.

August, 2000UT Department of Finance \$4,000 years from now. Assume that you need to have exactly \$4,000 saved 10 years from now. How much must you deposit today in an account that pays 6% interest, compounded annually, so that you reach your goal of \$4,000? 5 0 5 10 \$4,000 6% PV 0 Present Value (Graphic)

August, 2000UT Department of Finance PV 0 FV\$4,000 \$2,233.58 PV 0 = FV / (1+i) 2 = \$4,000 / (1.06) 10 = \$2,233.58 Present Value (Formula) 5 0 5 10 \$4,000 6% PV 0

August, 2000UT Department of Finance Present Value (HP 17 B II Calculator) 10 6 4000 PV N I%Yr FV -2,233.57 Exit until you get Fin Menu. 2 nd, Clear Data. Choose Fin, then TVM

August, 2000UT Department of Finance \$2,500 5 years. Assume todays deposit will grow at a compound rate of Joann needs to know how large of a deposit to make today so that the money will grow to \$2,500 in 5 years. Assume todays deposit will grow at a compound rate of 4% annually. Present Value Example 5 0 1 2 3 4 5 \$2,500 PV 0 4%

August, 2000UT Department of Finance PV 0 FV n PV 0 \$2,500/(1.04) 5 4 Calculation based on general formula: PV 0 = FV n / (1+i) n PV 0 = \$2,500/(1.04) 5 = \$2,054.81 4 Calculator keystrokes: 1.04 2nd y x 5 = 2 nd 1/x X 2500 = Present Value Solution

August, 2000UT Department of Finance Present Value (HP 17 B II Calculator) 5 4 2,500 +/- N I%Yr FV 2,054.81PV Exit until you get Fin Menu. 2 nd, Clear Data. Choose Fin, then TVM

August, 2000UT Department of Finance Finding n or i when one knows PV and FV 4 If one invests \$2,000 today and has accumulated \$2,676.45 after exactly five years, what rate of annual compound interest was earned?

August, 2000UT Department of Finance (HP 17 B II Calculator) 5 2000 +/- 2,676.45 I%Yr N PV FV 6.00 Exit until you get Fin Menu. 2 nd, Clear Data. Choose Fin, then TVM

August, 2000UT Department of Finance General Formula: PV 0 FV n = PV 0 (1 + [i/m]) mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FV n,m : FV at the end of Year n PV 0 PV 0 : PV of the Cash Flow today Frequency of Compounding

August, 2000UT Department of Finance Frequency of Compounding Example 4 Suppose you deposit \$1,000 in an account that pays 12% interest, compounded quarterly. How much will be in the account after eight years if there are no withdrawals? PV = \$1,000 i = 12%/4 = 3% per quarter n = 8 x 4 = 32 quarters

August, 2000UT Department of Finance Solution based on formula: FV= PV (1 + i) n = 1,000(1.03) 32 = 2,575.10 Calculator Keystrokes: 1.03 2 nd y x 32 X 1000 =

August, 2000UT Department of Finance Future Value, Frequency of Compounding (HP 17 B II Calculator) 32 3 1000 +/- N I%Yr PV 2,575.10FV Exit until you get Fin Menu. 2 nd, Clear Data. Choose Fin, then TVM

August, 2000UT Department of Finance Annuities 4 Examples of Annuities Include: Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings u An Annuity u An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

August, 2000UT Department of Finance FVA 3 \$3,215 FVA 3 = \$1,000(1.07) 2 + \$1,000(1.07) 1 + \$1,000(1.07) 0 = \$3,215 If one saves \$1,000 a year at the end of every year for three years in an account earning 7% interest, compounded annually, how much will one have at the end of the third year? Example of an Ordinary Annuity -- FVA \$1,000 \$1,000 \$1,000 3 0 1 2 3 4 \$3,215 = FVA 3 End of Year 7% \$1,070 \$1,145

August, 2000UT Department of Finance Future Value (HP 17 B II Calculator) 1,000 +/- 3 7 FV PMT N I%Yr 3,214.90 Exit until you get Fin Menu. 2 nd, Clear Data. Choose Fin, then TVM

August, 2000UT Department of Finance PVA 3 PVA 3 = \$1,000/(1.07) 1 + \$1,000/(1.07) 2 + \$2,624.32 \$1,000/(1.07) 3 = \$2,624.32 If one agrees to repay a loan by paying \$1,000 a year at the end of every year for three years and the discount rate is 7%, how much could one borrow today? rate is 7%, how much could one borrow today? Example of anOrdinary Annuity -- PVA \$1,000 \$1,000 \$1,000 3 0 1 2 3 4 \$2,624.32 = PVA 3 End of Year 7% \$934.58 \$873.44 \$816.30

August, 2000UT Department of Finance Present Value (HP 17 B II Calculator) Exit until you get Fin Menu. 2 nd, Clear Data. Choose Fin, then TVM PMT1,000 3N 7I% Yr PV-2,624.32

August, 2000UT Department of Finance Suppose an investment promises a cash flow of \$500 in one year, \$600 at the end of two years and \$10,700 at the end of the third year. If the discount rate is 5%, what is the value of this investment today? Multiple Cash Flows Example 0 1 2 3 \$500 \$600 \$10,700 \$500 \$600 \$10,700 PV 0 5%

August, 2000UT Department of Finance Multiple Cash Flow Solution 0 1 2 3 \$500 \$600 \$10,700 \$500 \$600 \$10,700 5% \$476.19\$544.22\$9,243.06 \$10,263.47 = PV 0 of the Multiple Cash Flows

August, 2000UT Department of Finance Multiple Cash Flow Solution (HP 17 B II Calculator) FIN Flow(0)=? Flow(1)=? Flow(2)=? CFLO 0 500 600 Exit until you get Fin Menu. 2 nd, Clear Data. Flow(3)=? 10,700 NVP I%5 Calc Exit # Times (2) = 1 Input# Times (1) = 1 Input

August, 2000UT Department of Finance Bond Valuation Problem Find todays value of a coupon bond with a maturity value of \$1,000 and a coupon rate of 6%. The bond will mature exactly ten years from today, and interest is paid semi-annually. Assume the discount rate used to value the bond is 8.00% because that is your required rate of return on an investment such as this. Interest = \$30 every six months for 20 periods Interest rate = 8%/2 = 4% every six months

August, 2000UT Department of Finance Bond Valuation Solution (HP 17 B II Calculator) Bond Valuation Solution (HP 17 B II Calculator) Exit until you get Fin Menu. 2 nd, Clear Data FINTVM 1000 30 4 20 PV PMT FV I% YR N -864.09 0 1 2 ……….… 20 30 30 30 1000

August, 2000UT Department of Finance Welcome to the Interactive Exercises 4 Choose a problem; select a solution 4 To return to this page (slide 37), use Power Points Navigation Menu 4 Choose Go and By Title 1 1 2 2 3 3

August, 2000UT Department of Finance Problem #1 You must decide between \$25,000 in cash today or \$30,000 in cash to be received two years from now. If you can earn 8% interest on your investments, which is the better deal?

August, 2000UT Department of Finance Possible Answers - Problem 1 4 \$25,000 in cash today \$25,000 in cash today 4 \$30,000 in cash to be received two years from now \$30,000 in cash to be received two years from now 4 Either option O.K. Either option O.K. Need a Hint? Need a Hint? Need a Hint? Need a Hint?

August, 2000UT Department of Finance Solution (HP 17 B II Calculator) Problem #1 Solution (HP 17 B II Calculator) Problem #1 Exit until you get Fin Menu. 2 nd, Clear Data Choose FIN, then TVM I%YR N FV -25,720.16PV 30,000 8 2 Compare PV of \$30,000, which is \$25,720.16 to PV of \$25,000. \$30,000 to be received 2 years from now is better.

August, 2000UT Department of Finance Problem #2 4 What is the value of \$100 per year for four years, with the first cash flow one year from today, if one is earning 5% interest, compounded annually? Find the value of these cash flows four years from today.

August, 2000UT Department of Finance Possible Answers - Problem 2 4 \$400 \$400 4 \$431.01 \$431.01 4 \$452.56 \$452.56 Need a Hint?

August, 2000UT Department of Finance Solution (HP 17 B II Calculator) Problem #2 Solution (HP 17 B II Calculator) Problem #2 Exit until you get Fin Menu. 2 nd, Clear Data Choose FIN, then TVM PMT FVA=100(1.05) 3 + 100(1.05) 2 + 100(1.05) 1 + 100(1.05) 0 100 I% Y R N 431.01 4 5 FV 0123401234 100100100100

August, 2000UT Department of Finance Problem #3 4 What is todays value of a \$1,000 face value bond with a 5% coupon rate (interest is paid semi-annually) which has three years remaining to maturity. The bond is priced to yield 8%.

August, 2000UT Department of Finance Possible Solutions - Problem 3 4 \$1,000 \$1,000 4 \$921.37 \$921.37 4 \$1021.37 \$1021.37 Need a Hint? Need a Hint?

August, 2000UT Department of Finance Solution (HP 17 B II Calculator) Problem #3 Solution (HP 17 B II Calculator) Problem #3 Exit until you get Fin Menu. 2 nd, Clear Data FINTVM 1000 25 4 6 PV PMT FV I% Y R N 921.37 0 1 2 ……….… 12 25 25 25 1000

August, 2000UT Department of Finance Congratulations! 4 You obviously understand this material. Now try the next problem. 4 The Interactive Exercises are found on slide #37.

August, 2000UT Department of Finance Comparing PV to FV 4 Remember, both quantities must be present value amounts or both quantities must be future value amounts in order to be compared.

August, 2000UT Department of Finance How to solve a time value of money problem. 4 The value four years from today is a future value amount. 4 The expected cash flows of \$100 per year for four years refers to an annuity of \$100. 4 Since it is a future value problem and there is an annuity, you need to solve for a FUTURE VALUE OF AN ANNUITY.

August, 2000UT Department of Finance Valuing a Bond 4 The interest payments represent an annuity and you must find the present value of the annuity. 4 The maturity value represents a future value amount and you must find the present value of this single amount. 4 Since the interest is paid semi-annually, discount at HALF the required rate of return (4%) and TWICE the number of years to maturity (6 periods).

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