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Chapter 5 The Time Value of Money Pr. Zoubida SAMLAL 1 The Interest Rate Which would you prefer -- $10,000 today or $10,000 in 5 years? Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY!! 2 Why TIME? Why is TIME such an important element in your decision? TIME allows you the opportunity to postpone consumption and earn INTEREST. 3 Types of Interest Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent). • Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). 4 Simple Interest Formula Formula SI = P0(i)(n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods 5 Simple Interest (FV) • What is the Future Value (FV) of the deposit? FV = P0 + SI = $1,000 + $140 = $1,140 • Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. 6 Simple Interest (PV) • What is the Present Value (PV) of the previous problem? The Present Value is simply the $1,000 you originally deposited. That is the value today! • Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. 7 Types of TVM Calculations • There are many types of TVM calculations • The basic types will be covered in this review module and include: – – – – Present value of a lump sum Future value of a lump sum Present and future value of cash flow streams Present and future value of annuities • Keep in mind that these forms can, should, and will be used in combination to solve more complex TVM problems 8 Future Value Single Deposit FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070 Compound Interest You earned $70 interest on your $1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest. 9 Future Value Single Deposit FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070 FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) = $1,000(1.07)(1.07) P0 (1+i)2 = $1,000(1.07)2 = $1,144.90 = You earned an EXTRA $4.90 in Year 2 with compound over simple interest. 10 General Future Value Formula FV1 = P0(1+i)1 FV2 = P0(1+i)2 etc. General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVIFi,n) 11 Single-Sum Future Value Number of Periods 2% 1 2 3 4 5 1,02000 1,04040 1,06121 1,08243 1,10408 Table A-1 Discount Rate 4% 6% 8% 1,04000 1,08160 1,12486 1,16986 1,21665 1,06000 1,12360 1,19102 1,26248 1,33823 1,08000 1,16640 1,25971 1,36049 1,46933 10% 1,10000 1,21000 1,33100 1,46410 1,61051 What factor do we use? 12 Single-Sum Future Value Table A-1- FV of 1$ Number of Periods 2% 1 2 3 4 5 1.02000 1.04040 1.06121 1.08243 1.10408 1.04000 1.08160 1.12486 1.16986 1.21665 1.06000 1.12360 1.19102 1.26248 1.33823 1.08000 1.16640 1.25971 1.36049 1.46933 $10,000 x 1.25971 = $12,597 Present Value Discount Rate 8% 6% 4% Factor 10% 1.10000 1.21000 1.33100 1.46410 1.61051 Future Value 13 Example of FV of a Lump Sum • 1. How much money will you have in 5 years if you invest $100 today at a 10% rate of return? Draw a timeline i = 10% $100 0 1 2 3 ? 4 5 2. Write out the formula using symbols: FVt = CF0 * (1+r)t 14 Example of FV of a Lump Sum 3. Substitute the numbers into the formula: FV = $100 * (1+.1)5 4. Solve for the future value: FV = $161.05 5. Check answer using a financial calculator: i = 10% n=5 PV = $100 PMT = $0 FV = ? 15 General Present Value Formula PV0 = FV1 (1+i)-1 PV0 = FV2(1+i)-2 etc. General Future Value Formula: PV0 = FVn (1+i)-n or PV0 = FVn (PVIFi,n) 16 Single-Sum Present Value Table A-2- PV of 1$ Number of Periods 4% 2 .92456 .89000 .85734 .82645 .79719 4 .85480 .79209 .73503 .68301 .63552 6 .79031 .70496 .63017 .56447 .50663 8 .73069 .62741 .54027 .46651 .40388 Discount Rate 6% 8% 10% 12% What factor do we use? 17 Single-Sum Problems Table Table A-2-A-2 PV of 1$ Number of Periods 4% 2 .92456 .89000 .85734 .82645 .79719 4 .85480 .79209 .73503 .68301 .63552 6 .79031 .70496 .63017 .56447 .50663 8 .73069 .62741 .54027 .46651 .40388 $20,000 x .63552 Future Value Discount Rate 6% 8% 10% Factor = 12% $12,710 Present Value 18 Example of PV of a Lump Sum • 1. How much would $100 received five years from now be worth today if the current interest rate is 10%? Draw a timeline i = 10% $100 ? 0 1 2 3 4 5 The arrow represents the flow of money and the numbers under the timeline represent the time period. Note that time period zero is today. 19 Example of PV of a Lump Sum 2. Write out the formula using symbols: PV = CFt / (1+r)t 3. Insert the appropriate numbers: PV = 100 / (1 + .1)5 4. Solve the formula: PV = $62.09 5. Check using a financial calculator: FV = $100 n=5 PMT = 0 i = 10% PV = ? 20 Annuities Annuity requires the following: (1) Periodic payments or receipts (called rents) of the same amount, (2) The same-length interval between such rents, and (3) Compounding of interest once each interval. Two Types Ordinary annuity - rents occur at the end of each period. Annuity Due - rents occur at the beginning of each period. 21 Annuities Future Value of an Ordinary Annuity Rents occur at the end of each period. No interest during 1st period. Future Value Present Value $20,000 20,000 0 1 2 20,000 3 20,000 4 20,000 5 20,000 6 20,000 7 20,000 8 22 Future Value of an Ordinary Annuity Future Value Present Value $20,000 20,000 0 1 2 20,000 3 20,000 4 20,000 5 20,000 6 20,000 7 20,000 8 Bayou Inc. will deposit $20,000 in a 12% fund at the end of each year for 8 years beginning December 31, Year 1. What amount will be in the fund immediately after the last deposit? What table do we use? 23 Future Value of an Ordinary Annuity Table A-3- FV of an annuity payment of 1$ per year Number of Periods 4% 6% 2 4 6 8 10 2.04000 4.24646 6.63298 9.21423 12.00611 2.06000 4.37462 6.97532 9.89747 13.18079 Discount Rate 8% 10% 12% 2.08000 4.50611 7.33592 10.63663 14.48656 2.10000 4.64100 7.71561 11.43589 15.93743 2.12000 4.77933 8.11519 12.29969 17.54874 What factor do we use? 24 Future Value of an Ordinary Annuity Table A-3- FV ofTable an annuity A-3 payment of 1$ per year Number of Periods 4% 6% 2 4 6 8 10 2.04000 4.24646 6.63298 9.21423 12.00611 2.06000 4.37462 6.97532 9.89747 13.18079 $20,000 Deposit x Discount Rate 8% 10% 12% 2.08000 4.50611 7.33592 10.63663 14.48656 2.10000 4.64100 7.71561 11.43589 15.93743 2.12000 4.77933 8.11519 12.29969 17.54874 12.29969 Factor = $245,994 Future Value 25 Example of FV of an Annuity 2. Write out the formula using symbols: FVAt = PMT * {[(1+r)t –1]/r} 3. Substitute the appropriate numbers: FVA20 = $100 * {[(1+.15)20 –1]/.15 4. Solve for the FV: FVA20 = $100 * 102.4436 FVA20 = $10,244.36 26 Example of FV of an Annuity 5. Check using calculator: Make sure that the calculator is set to one period per year PMT = $100 n = 20 i = 15% FV = ? 27 Present Value of an Ordinary Annuity Present Value $100,000 100,000 100,000 100,000 100,000 100,000 ..... 0 1 2 3 4 19 20 Jaime Yuen wins $2,000,000 in the state lottery. She will be paid $100,000 at the end of each year for the next 20 years. How much has she actually won? Assume an appropriate interest rate of 8%. What table do we use? 28 Present Value of an Ordinary Annuity A-4 payment of 1$ per year Table A-4- PV ofTable an annuity Number of Periods 4% 6% 1 5 10 15 20 0.96154 4.45183 8.11090 11.11839 13.59033 0.94340 4.21236 7.36009 9.71225 11.46992 Discount Rate 8% 0.92593 3.99271 6.71008 8.55948 9.81815 10% 0.90900 3.79079 6.14457 7.60608 8.51356 12% 0.89286 3.60478 5.65022 6.81086 7.46944 What factor do we use? 29 Present Value of an Ordinary Annuity 6-4 Table A-4- PV ofTable an annuity payment of 1$ per year Number of Periods 4% 6% 1 5 10 15 20 0.96154 4.45183 8.11090 11.11839 13.59033 0.94340 4.21236 7.36009 9.71225 11.46992 $100,000 Receipt x Discount Rate 8% 0.92593 3.99271 6.71008 8.55948 9.81815 9.81815 Factor = 10% 0.90900 3.79079 6.14457 7.60608 8.51356 12% 0.89286 3.60478 5.65022 6.81086 7.46944 $981,815 Present Value 30 Formulas of Annuities Present value of an annuity: PVA = PMT * {[1-(1+r)-t]/r} Future value of an annuity: FVA t = PMT * {[(1+r)t –1]/r} 31 How about a stream of payments that are NOT equal?? 32 Formulas of Cash Flow stream • Future value of a cash flow stream: n FV = S [CFt * (1+r)n-t] t=0 • Present value of a cash flow stream: – n – PV = S [CFt / (1+r)n-t] t=0 33 Example of PV of a Cash Flow Stream • 1. Joe made an investment that will pay $100 the first year, $300 the second year, $500 the third year and $1000 the fourth year. If the interest rate is ten percent, what is the present value of this cash flow stream? Draw a timeline: $100 0 A-2 r=10% , n= 1 A-2 r=10% , n= 2 1 $300 2 $500 3 $1000 4 i = 10% A-2 r=10% , n= 3 A-2 r=10% , n= 4 34 Example of PV of a Cash Flow Stream Number of Periods CF1 * Factor 1 + CF2 * Factor 2 + CF3 * Factor 3 + CF4* Factor 4 Table A-2- PV of 1$ 4% Discount Rate 6% 8% 10% 12% 1 .92456 .89000 .85734 .82645 .79719 2 .85480 .79209 .73503 .68301 .63552 3 .79031 .70496 .63017 .56447 .50663 4 .73069 .62741 .54027 .46651 .40388 = Present value of a cash flow stream: n PV = S [CFt / (1+r)n-t] t=0 35 Example of PV of a Cash Flow Stream 2. Write out the formula using symbols: n PV = S [CFt / (1+r)t] t=0 OR PV = [CF1/(1+r)1]+[CF2/(1+r)2]+[CF3/(1+r)3]+[CF4/(1+r)4] 3. Substitute the appropriate numbers: PV = [100/(1+.1)1]+[$300/(1+.1)2]+[500/(1+.1)3]+[1000/(1.1)4] 36 Example of PV of a Cash Flow Stream 4. Solve for the present value: PV = $90.91 + $247.93 + $375.66 + $683.01 PV = $1397.51 5. Check using a calculator: – Make sure to use the appropriate rate of return, number of periods, and future value for each of the calculations. To illustrate, for the first cash flow, you should enter FV=100, n=1, i=10, PMT=0, PV=?. Note that you will have to do four separate calculations. 37 Example of FV of a Cash Flow Stream • 1. Joe made a decision to start saving money. He will pay $100 now year, $300 the first year, $500 the second year and $1000 the third year. If the interest rate is ten percent, what is the future value of this cash flow stream? Draw a timeline: i = 10% $100 0 $300 1 $500 2 $1000 3 4 A-1 r=10% , n= 4 + A-1 r=10% , n= 3 + A-1 r=10% , n= 2 + A-1 r=10% , n= 138 Example of FV of a Cash Flow Stream CF3* Factor 1 + CF2 * Factor 2 + CF1 * Factor 3 + CF0* Factor 4 Table A-1- FV of 1$ Number of Periods 2% 1 2 3 4 1,02000 1,04040 1,06121 1,08243 Discount Rate 4% 6% 8% 1,04000 1,08160 1,12486 1,16986 1,06000 1,12360 1,19102 1,26248 1,08000 1,16640 1,25971 1,36049 10% 1,10000 1,21000 1,33100 1,46410 = Future value of a cash flow stream: n FV = S [CFt * (1+r)n-t] 39 Rule of Thumb • • The following are simple rules that you should always use no matter what type of TVM problem you are trying to solve: 1. Stop and think: Make sure you understand what the problem is asking. You will get the wrong answer if you are answering the wrong question. 2. Draw a representative timeline and label the cash flows and time periods appropriately. 3. Write out the complete formula using symbols first and then substitute the actual numbers to solve. 4. Check your answers using a calculator. While these may seem like trivial and time consuming tasks, they will significantly increase your understanding of the material and your accuracy rate. 40 Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? We will use the “Rule-of-72”. 41 The “Rule-of-72” Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? Approx. Years to Double = 72 / i% 72 / 12% = 6 Years [Actual Time is 6.12 Years] 42 Steps to Solve Time Value of Money Problems 1. 2. 3. 4. 5. Read problem thoroughly Create a time line Put cash flows and arrows on time line Determine if it is a PV or FV problem Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) 43