1、FINANCIAL MANAGEMENT Chapter 9 Time Value of Money Instructor: Dr. Renee, Ren Gu The School of Economics and Commerce SCUT, Guangzhou 2013/11/21 Todays Contents SECTION ONE BASIC CONCEPTS 1.1 WHAT IS TIME VALUE? 1.2 WHAT IS PRESENT VALUE? 1.3 WHAT IS FUTURE VALUE? 1.4 CASH FLOW TIME LINE 1.1 What Is
2、 Time Value? Money has a time value associated with it and therefore a dollar received today is worth more than a dollar received in the future. The investor demands that financial rent, the return on an investment, be paid on his or her funds. Understanding the effective rate on a business loan, et
3、c., is dependent on using the time value of money. The investor/lender The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time 1.2 What is present value (PV)? How much the amount that you will receive in the future are worth now? (It is
4、 the value on a given date of a future payment.) DISCOUNTING It is widely to provide a means to compare cash flows at different times on a meaningful “like to like“ basis. 1.3 What is future value (FV)? How much what you receive now grows to when compounded at a given rate? Future value is the value
5、 of an asset at a specific date. The future value and present value of a dollar is based on the number of periods involved and the going interest rate. CAPITALIZATION 1.4 Analytic tool cash flow time line Definition Line depicting the operating activities and cash flows for a firm over a particular
6、period. 0 1 2 3 4 -100 5% 500 Cash Flow Time line of FV 0 1 2 3 4 -1000 10% 1100 （$1,0001.10） 1,210 （1.10） 1,331 （1.10） 1,464 （1.10） ? Cash Flow Time line of PV 1,464 1.1 time 0 1 2 3 4 ? 10% 1.1 1.1 1.1 1,000 SECTION TWO SINGLE AMOUNT VS ANNUITY 1. Future Value of A Single Amount 2. Present Value o
7、f A Single Amount 3. Future Value of An Annuity 4. Present Value of An Annuity 5. Determining the Annuity Value 6. Determining the Yield on an Investment To measure the value of an amount that is allowed to grow at a give interest rate over a period of time. 1.Future Value- Single Amount Assume an i
8、nvestor has $1000 and wishes to know its worth after four years if it grows at 10% per year. 1st year $1,0001.10=$1,100 2nd year $1,1001.10=$1,210 3rd year $1,2101.10=$1,331 4th year $1,3311.10=$1,464 Cash Flow Time line time 0 1 2 3 4 -1000 10% 1100 1,210 1,331 1,464 （$1,0001.10 ） （1.10 ） （1.10 ） （
9、1.10 ） ? ) i + PV(1 = FV n FV = Future Value PV = Present Value i = the interest rate per period n= the number of compounding periods Simple Formula: In this case, PV=$1,000, i=10%,n=4,so FV=? FV=$1,000 (1.1) 4 =$1,464 Quicker process ： Appendix A The formula may be restated as: FVx PV = FV IFThe re
10、lationship may be expressed by the following formula: the interest factor The FV IFterm is found in Table 9-1 Quicker Process an interest rate table periods 1% 2% 4% 6% 8% 10% 1 1.010 1.020 1.040 1.060 1.080 1.100 2 1.020 1.040 1.082 1.124 1.166 1.210 3 1.030 1.061 1.125 1.191 1.260 1.331 4 1.041 1.
11、082 1.170 1.262 1.360 1.464 5 1.051 1.104 1.217 1.338 1.469 1.611 10 1.105 1.219 1.480 1.791 2.159 2.594 20 1.220 1.486 2.191 3.207 4.661 6.727 Future Value of $1 (Table 9-1) 2.Present Value - Single Amount The present value of a future sum is the amount invested today, at a given interest rate, tha
12、t will equal the future sum at a specified point in time. discount rate Cash Flow Time Line 1,464 1.1 time 0 1 2 3 4 ? 10% 1.1 1.1 1.1 1,000 Relationship of PV and FV The relationship may be expressed in the following formula: ) i + (1 1FV = PV n The formula may be restated as: PVx FV = PV IF The PV
13、 IFterm is found in Table 9-2 Quicker process ： Appendix B Discount Factor Present Value Discount Factor = DF = PV of $1 Discount Factors can be used to compute the present value of any cash flow. DF r t = + 1 1 ( )Present Value of $1 (Table 9-2) periods 1% 2% 4% 6% 8% 10% 1 0.990 0.980 0.962 0.943
14、0.926 0.909 2 0.980 0.961 0.925 0.890 0.857 0.826 3 0.971 0.942 0.889 0.840 0.794 0.751 4 0.961 0.924 0.855 0.792 0.735 0.683 5 0.951 0.906 0.822 0.747 0.681 0.621 10 0.905 0.820 0.676 0.558 0.463 0.386 20 0.820 0.673 0.456 0.312 0.215 0.149 Valuing an Office Building Step 1: Forecast cash flows Cos
15、t of building = C 0= 350 Sale price in Year 1 = C 1= 400 Step 2: Estimate opportunity cost of capital If equally risky investments in the capital market offer a return of 7%, then Cost of capital = r = 7% Valuing an Office Building Step 3: Discount future cash flows Step 4: Go ahead if PV of payoff
16、exceeds investment 374 ) 07 . 1 ( 400 ) 1 ( 1 = = = + +r C PV 24 374 350 = + = NPVNet Present Value r C + + 1 C = NPV investment required - PV = NPV 1 03.Future Value - Annuity 3.1 Definition An annuity represents consecutive payments or receipts of equal amount. Note: The annuity value is normally
17、assumed to take place at the end of the period. (Ordinary Annuity) The future value of an annuity represents the sum of the future value of the individual flows. 3.2 ordinary annuity If we invest $1,000 at the end of each year for four years and our funds grow at 10%, what is the future value of thi
18、s annuity? The future value for the annuity is 4,641. Please try to draw the time line. Compounding process for annuity 0 1 2 3 4 $1,000 $1,0001.100=1,100 $1,0001.210=1,210 $1,0001.331=1,331 The future value for the annuity is 4,641. 1,000 1,000 1,000 10% The formula for the future value for an annu
19、ity is: FV A= A FV IFAProof process: 1 2 10 (1 ) (1 ) (1 ) (1 ) nn A F VAi Ai Ai Ai =+ + (1 ) 1 n IFA i A A FV i + = = If a wealthy relative offered to set aside $2,500 a year for you for the next 20 years, how much would you have in your account after 20 years if the funds grew at 8%? The answer is
20、 as follows: FV A= A FV IFA (n=20,I=8%) The FV IFAterm is found in Table 9-3. (P231) 3.3 Annuity Due If we invest $1,000 at the BEGINNING of each year for four years and our funds grow at 10%, what is the future value of this annuity? 0 1 2 3 4 $1,000 1,000 1,000 1,000 10% $1,0001.1 2 = 1,210 $1,000
21、1.1 3 = 1,331 $1,0001.1 4 =1464 $1,0001.1=1,100 6105 Equation: (1 ) 1 ( ) (1 ) (1 ) n IFA i FV due A i A FV i i + = += + Summary: FVx PV = FV IF PVx FV = PV IF FV A= A FV IFA1. FV of single 2. PV of single 3. ordinary annuity 4. Annuity due ( ) (1 ) IFA FV due A FV i = +4. Present Value - Annuity 4.
22、1 The present value of an annuity represents the sum of the present value of the individual flows. Time savings big money and small drawing 4.2 The formula for the present value of an annuity is: 1 1 (1 ) n A IFA i PV A PV A i + = = PV(due)=A*1+PVIFA(n-1,i) 5.Determining the Annuity Value 5.1 Annuit
23、y Equaling a Future Value The process can be reversed to find an annuity value that will grow to equal a future sum. FV A x = FV IFA A FV FV= A IFA A The FV IFAterm is found in Table 9-3 Example: Assuming we wish to accumulate $4,641 after four years at a 10 percent interest rate, how much must be s
24、et aside at the end of each of the four periods? Use Table 9-3(P231). For n=4,i=10, FV IFA=4.641, thus A =$1,000 5.2 Annuity Equality a Present Value The terms in the formula for the present value of an annuity are transformed to find A. A IFA= A PV PV PV PV= A IFA A The PV IFAterm is found in Table
25、 9-4. The annuity value equal to a present value is often associated with withdrawal of funds from an initial deposit or the repayment of a loan. Suppose your wealthy uncle presents you with $10,000 now to help you get through the next four years of college. If you are able to earn 6 percent on depo
26、sited funds, how many equal payments can you withdraw at the end of each year for four years? $10,000 $2,886 3.465 A = = EXAMPLE 1 Suppose a homeowner signs a $40,000 mortgage to be repaid over 20 years at 8 percent interest. How much must he or she pay annually to eventually liquidate the loan? EXA
27、MPLE 2 $40,000 $4,074 9.818 A = = A IFA PV A = PV ( 20, 8%) ni = =6.Determining the Yield on an Investment 6.1 Yield - Present value of a single amount a. The rate equating FV to PV must be found. IF PV= PV V F b. The first step is to determine PV IF . c. The next step is to find this value in Table
28、 9-2 so the yield may be identified. d. Interpolation may be used to find a more exacting answer. With n=3, PV IF =0.861 PV IFat 5%0.864 PV IFat 6%0.840 0.003 5% (1%) 5% 0.125% 5.125% 0.024 +=+ =6.2 Yield - Present value of an annuity a. The rate equating A to PV must be found. A PV= PV A IFA b. The
29、 first step is to determine PV IFA . c. The next step is to find this value in Table 9-4 so the yield may be identified. d. Interpolation may be used to find a more exacting answer. 7. Special considerations in Time Value Analysis A. Semi-annual,quarterly,monthly,etc compounding periods. Find the fu
30、ture value of a $1,000 investment after five years at 8 percent annual interest, compounded semiannually. We have assumed interest was compounded or discounted on an annual basis. n=5*2=10 i=8 percent / 2 =4 percent FV=PV*FV IF =$1,000*1.480=$1,480 Case 1 Case 2 Find the present value of 20 quarterl
31、y payments of $2,000 each to be received over the next five years. The stated interest rate is 8 percent per annum. PV A =A*PV IFA(n=20,i=2%) PV A =$2,000*16.351=$32,702 B. Patterns of Payment -Deferred annuity What is a deferred annuity? A deferred annuity is an annuity in which the equal payments
32、will begin at some future point in time . Present value of deferred annuity A contract may call for the payment of a different amount each year over a three-year period. To determine present value, each payment is discounted to the present value and then summed.(Assume 8%) 0 1 2 3 1,000 2,000 3,000
33、926=0.926 1,714=0.857 2,382=0.794 5,022 Assume the same problem as above, but with an annuity of $1,000 that will be paid at the end of each year from the fourth through the eighth year. With a discount rate of 8%, what is the present value of the cash flows? 1. 1,000 2. 2,000 3. 3,000 4. 1,000 5. 1
34、,000 6. 1,000 7. 1,000 8. 1,000 Present Value=5,022 Five-year Annuity What about the annuity? 1.Two step solution process (1)Tabular values only discount to the beginning of the first stated period of an annuity The “present value” of the annuity = 3,933 (2) The 3,933 must finally be discounted back
35、 to the present. 2.Single step solution (1) PV IFAfor total period; (2) PV IFAfor total time period minus the deferred annuity period; (3) Subtract the value in step2 from value in step1, and multiply by A to get the PV for deferred annuity. Explored Topics About Present Value Valuing Long-Lived Ass
36、ets PV Calculation Short Cuts Compound Interest Nominal and Real Rates of Interest (inflation) Example: Present Values and Bonds, Preferred Stocks, Common Stocks (Chapter 10) Review: Present Values Discount Factor = DF = PV of $1 DF r t = + 1 1 ( ) Discount Factors can be used to compute the present
37、 value of any cash flow. Present Values Discount Factors can be used to compute the present value of any cash flow. DF r t = + 1 1 ( ) 1 1 1 1 r C C DF PV + = =Present Values Replacing “1” with “t” allows the formula to be used for cash flows that exist at any point in time t t t r C C DF PV ) 1 ( +
38、 = =Present Values Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? PV = = 3000 1 08 2 572 02 ( . ) $2, .Present Values PVs ca
39、n be added together to evaluate multiple cash flows. PV C r C r = + + + + 1 1 2 2 1 1 ( ) ( ) Present Values Given two dollars, one received a year from now and the other two years from now, the value of each is commonly called the Discount Factor. Assume r1 = 20% and r2 = 7%. 87 . 83 . 2 1 ) 07 . 1
40、 ( 00 . 1 2 ) 20 . 1 ( 00 . 1 1 = = = = + + DF DFPresent Values Example Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value. 000 , 300 000 , 100 000 , 150 2
41、Year 1 Year 0 Year + Present Values Example - continued Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value. ( ) 400 , 18 $ 900 , 261 000 , 300 873 . 2 500 ,
42、 93 000 , 100 935 . 1 000 , 150 000 , 150 0 . 1 0 Value Present Flow Cash Factor Discount Period 2 07 . 1 1 07 . 1 1 = = + + = = Total NPVShort Cuts Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods. These tools allow u
43、s to cut through the calculations quickly. Short Cuts Perpetuity - Financial concept in which a cash flow is theoretically received forever. PV C r = = lue present va flow cash ReturnShort Cuts Perpetuity - Financial concept in which a cash flow is theoretically received forever. r C PV 1 rate disco
44、unt flow cashFlow Cash of PV = =Short Cuts Annuity - An asset that pays a fixed sum each year for a specified number of years. Short Cuts Annuity - An asset that pays a fixed sum each year for a specified number of years. ( ) + = t r r r C 1 1 1 annuity of PVAnnuity Short Cut Example You agree to le
45、ase a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease? Annuity Short Cut Example - continued You agree to lease a car for 4 years at $300 per month.
46、 You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease? ( ) 10 . 774 , 12 $ 005 . 1 005 . 1 005 . 1 300 Cost Lease 48 = + = CostCompound Interest i ii iii iv v Periods Interest Value Annual
47、ly per per APR after compounded year period (i x ii) one year interest rate 1 6% 6% 1.06 6.000% 2 3 6 1.03 2= 1.0609 6.090 4 1.5 6 1.015 4 = 1.06136 6.136 12 .5 6 1.005 12= 1.06168 6.168 52 .1154 6 1.001154 52= 1.06180 6.180 365 .0164 6 1.000164 365= 1.06183 6.183 APR = Annual Percentage Rate Compou
48、nd Interest 0 2 4 6 8 10 12 14 16 18 0 3 6 9 12 15 18 21 24 27 30 Number of Years FV of $1 10% Simple 10% CompoundCompound Interest Example Suppose you are offered an automobile loan at an APR of 6% per year. What does that mean, and what is the true rate of interest, given monthly payments? Compound Interest Example - continued Suppose you are offered an automobile loan at an APR of 6% per year. What does that mean, and what is the true rate of interest, given monthly payments? Assume $10,000 loan amount. % 1678 . 6 78 . 616 , 10 ) 005 . 1 ( 000 , 10 Pmt Loan 12 = = =
