1 Chapter 4 The Time Value of Money (Part 2) LEARNING OBJECTIVES 1. Compute the future value of multiple cash flows. 2. Determine the future value of an annuity. 3. Determine the present value of an annuity. (Slide 4-2) 4. Adjust the annuity formula for present value and future value for an annuity due and understand the concept of a perpetuity. 5. Distinguish between the different types of loan repayments: discount loans, interestonly loans, and amortized loans. 6. Build and analyze amortization schedules. 7. Calculate waiting time and interest rates for an annuity. 8. Apply the time value of money concepts to evaluate the lottery cash flow choice. 9. Summarize the ten essential points about the time value of money. IN A NUTSHELL In Part two of this 2-part unit on the time value of money topic, the author discusses and illustrates how the time value of money equation can be modified and used for calculations involving the compounding and discounting of interest in cash flow streams that are more complex that mere lump sums. Real life situations seldom involve single outflow/inflow types of cash flow streams. More often, we are faced with periodic outflows such as loan, rent, or lease payments and/or periodic inflows such as retirement annuities. In this chapter, we learn how to calculate the present and future values of more complex cash flow streams such as those involving unequal cash flows, ordinary annuities and annuities due. In addition, the different methods by which loans can be paid off; and the method of setting up and analyzing amortization schedules associated with mortgages and other installment loans are also covered. LECTURE OUTLINE 4.1 Future Value of Multiple Payment Streams (Slides 4-3 to 4-7) In the case of investments involving unequal periodic cash flows, we can calculate the future value of the cash flows by treating each of the cash flows as a lump sum and calculate its future value over the relevant number of periods. The individual future values are then summed up to get the future value of the multiple payment streams. 76
2 Chapter 4 n The Time Value of Money (Part 2) 77 It is best to use a time line, as shown in Figure 4-1 in the text, which clearly shows each cash flow, the respective number of periods over which interest is to be compounded, and the interest rate that will apply. There is a shorter alternative way to solve the future value of a stream of unequal periodic cash flows, which involves using the Net Present Value (NPV) function of a financial calculator or a spreadsheet. We can first compute the net present value (at t = 0) of the stream of uneven cash flows at the given rate of interest, and then using the NPV as the present value, we find the future value of a lump sum at the end point of the cash flow stream. This method is shown in Example 1 below. Example 1: Future Value of an Uneven Cash Flow Stream Jim deposits $3,000 today into an account that pays 10% per year, and follows it up with 3 more deposits at the end of each of the next three years. Each subsequent deposit is $2,000 higher than the previous one. How much money will Jim have accumulated in his account by the end of three years? T 0 T 1 T 2 T 3 $3,000 $5,000 $7,000 $9,000 $3,000 (1.10) 3 $5,000 (1.10) 2 $7,000 (1.10) 1 $7, $6, $3, We use the Future Value of a Single Sum formula and compound each cash $26, flow for the relevant number of years over which interest will be earned. Then we sum up the compounded values to get the accumulated value of Jim s deposits at the end of three years as shown below: FV = PV (1 + r) n FV of Cash Flow at T 0 = $3,000 (1.10) 3 = $3, = $3, FV of Cash Flow at T 1 = $5,000 (1.10) 2 = $5, = $6, FV of Cash Flow at T 2 = $7,000 (1.10) 1 = $7, = $7, FV of Cash Flow at T 3 = $9,000 (1.10) 0 = $9, = $9, Total = $26, Note: Students should be reminded that cash flows can only be added up if they occur at the same point in time as at the end of Year 3.
3 78 Brooks n Financial Management: Core Concepts, 2e ALTERNATIVE METHOD: Using the Cash Flow (CF) key of the calculator, enter the respective cash flows. That is, CF0 = -$3000; CF1 = -$5000; CF2 = -$7000; CF3 = -$9000; Next calculate the NPV using I = 10%; NPV = $20,092.41; Finally, using PV = -$20,092.41; n = 3; i = 10%; PMT = 0; CPT FV = $26, Future Value of an Annuity Stream (Slides 4-8 to 4-13) Often, we are faced with financial situations which involve equal, periodic outflows/inflows. Such payment streams are known as annuities. Examples of an annuity stream include rent, lease, mortgage, car loan, and retirement annuity payments. An annuity stream can begin at the start of each period as is true of rent and insurance payments or at the end of each period, as in the case of mortgage and loan payments. The former type is called an annuity due while the latter is known as an ordinary annuity stream. This section covers ordinary annuities. Annuities due will be covered in a later section. Although the future value of an ordinary annuity stream can be calculated by using the same process that was explained in section 4.1 above, there is a simplified formula which makes the process much easier. The formula for calculating the future value of an annuity stream is as follows: ( r) + = 1 n FV PMT 1 r where PMT is the term used for the equal periodic cash flow, r is the rate of interest, and n is the number of periods involved. The item that PMT is multiplied by is known as the Future Value Interest Factor of an Annuity (FVIFA). It can be either calculated using the equation or got from a table provided in Appendix A-3. Of course, table values are only available for discrete interest rates and time periods. Note: The length of the period can be a day, week, quarter, month, or any other equal unit of time, not just a year as is often misunderstood by students. The rate of interest, however, is often given on an annual basis and must be accordingly adjusted and used in the problem. Example 2: Future Value of an Ordinary Annuity Stream Jill has been faithfully depositing $2,000 at the end of each year since the past 10 years into an account that pays 8% per year. How much money will she have accumulated in the account? This problem could be solved by first calculating the FV of each of the year end deposits for the respective number of years involved, and then summing them all up at the end of year 10 as shown below: Future Value of Payment One = $2, = $3, Future Value of Payment Two = $2, = $3,701.86
4 Chapter 4 n The Time Value of Money (Part 2) 79 Future Value of Payment Three = $2, = $3, Future Value of Payment Four = $2, = $3, Future Value of Payment Five = $2, = $2, Future Value of Payment Six = $2, = $2, Future Value of Payment Seven = $2, = $2, Future Value of Payment Eight = $2, = $2, Future Value of Payment Nine = $2, = $2, Future Value of Payment Ten = $2, = $2, Total Value of Account at the end of 10 years $28, FORMULA METHOD It is much quicker to solve this problem using the following formula: ( r) + = 1 n FV PMT 1 r where, PMT = $2,000; r = 8%; and n=10 i.e. the number of deposits involved. The FVIFA would equal [((1.08) 10-1)/.08] = , and the FV = $ = $28, USING A FINANCIAL CALCULATOR N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28, USING AN EXCEL SPREADSHEET Enter =FV(8%, 10, -2000, 0, 0); Output = $28, Rate, Nper, Pmt, PV,Type Type is 0 for ordinary annuities and 1 for annuities due USING FVIFA TABLE A- 3 Find the FVIFA in the 8% column and the 10 period row; FVIFA = FV = = $ Present Value of an Annuity (Slides 4-14 to 4-19) If we are interested in finding out the value of a series of equal periodic cash flows at the current point in time, we can either sum up the discounted values of each periodic cash flow (PMT) for the related number of periods or use the following simplified formula: 1 PV = PMT 1 ( 1+ r) r n
5 80 Brooks n Financial Management: Core Concepts, 2e The last portion of the equation, 1 1 ( 1+ r) n, is the Present Value Interest Factor of r an Annuity (PVIFA). The values of various PVIFAs are displayed in Appendix A-4 for different combinations of discrete interest or discount rates (r) and the number of payments (n). A practical application of such a calculation would be in calculating how much to have saved up in an account prior to a child attending college or prior to retirement so as to be able to withdraw equal annual amounts each year over a required number of years. Example 3: Present Value of an Annuity John wants to make sure that he has saved up enough money prior to the year before which his daughter begins college. Based on current estimates, he figures that college expenses will amount to $40,000 per year for 4 years (ignoring any inflation or tuition increases during the 4 years of college). How much money will John need to have accumulated in an account that earns 7% per year, just prior to the year that his daughter starts college? Here the known variables are: n=4; PMT=$40,000; and r = 7%. What we have to solve for is the PV of the annuity series, which can be done by any one of 4 methods, i.e. formula, financial calculator, spreadsheet, or PVIFA factor using Table A-4. FORMULA METHOD: Using the following equation: 1 PV = PMT 1 ( 1+ r) r n 1. Calculate the PVIFA value for n=4 and r=7%. 1 1 ( ) = [ 1 ( ) ] 0.07 = Then, multiply the annuity payment by this factor to get the PV, PV = $40, = $135, FINANCIAL CALCULATOR METHOD: It is important to remind students that the calculator must be in END mode so that the payments are treated as an ordinary annuity. Set the calculator for an ordinary annuity (END mode) and then enter: N= 4; PMT = 40,000; I = 7; FV=0; CPT PV = 135, SPREADSHEET METHOD: Enter =PV(7%, 4, 40,000, 0, 0); Output = $135,488.45
6 Chapter 4 n The Time Value of Money (Part 2) 81 Rate, Nper, Pmt, FV,Type PVIFA TABLE (APPENDIX A- 4) METHOD Since the rate of interest (7%) and the number of withdrawals (4) are both discrete values, we could easily solve this problem by obtaining the PVIFA value from the table in Appendix A-4 i.e and multiplying it by the PMT ($40,000) to get a PV of $ Notice the slight rounding error! 4.4 Annuity Due and Perpetuity (Slides 4-20 to 4-25) Annuity due: Certain types of financial transactions such as rent, lease, and insurance payments involve equal periodic cash flows that begin right away or at the beginning of each time interval. This type of annuity is known as an annuity due. Figure 4.5 in the text shows both types of annuities on a time line. An annuity due stream is scaled back for one period as shown by the arrows. Note that when calculating the PV of an annuity due stream one less period of interest would be required for each payment, since the first cash flow begins right away. Likewise, when calculating the FV of the cash flow stream at the end of 4 periods, an additional period of interest would apply to each periodic cash flow, since the 4 th payment occurs at the beginning of the 4 th year. Figure 4.5 Ordinary Annuity versus Annuity Due T 0 T 1 T 2 T 3 T 4 #1 $100 $100 $100 $100 Ordinary Annuity For problems involving an annuity due, the equations used to calculate the PV and FV of an ordinary annuity can simply be adjusted by multiplying them by the term (1+r). That is, 1 1 n ( + ) 1 r PV = PMT + r r Or ( 1 ) PV annuity due = PV ordinary annuity (1+r) AND #2 $100 $100 $100 $100 Annuity Due
7 82 Brooks n Financial Management: Core Concepts, 2e Or ( r) n 1+ 1 FV = PMT + r r ( 1 ) FV annuity due = FV ordinary annuity (1+r) When using a financial calculator we must set the mode to BGN for an annuity due or END for an ordinary annuity and proceed just as we would any other PV or FV problem. In the case of a spreadsheet, the Type of the cash flow, within the =PV or =FV functions is set to 0 or omitted for an ordinary annuity and 1 for an annuity due. Example 4: Annuity Due versus Ordinary Annuity Let s say that you are saving up for retirement and decide to deposit $3,000 each year for the next 20 years into an account which pays a rate of interest of 8% per year. By how much will your accumulated nest egg vary if you make each of the 20 deposits at the beginning of the year, starting right away, rather than at the end of each of the next twenty years? Given information: PMT = -$3,000; n=20; i= 8%; PV=0; Future value of an ordinary annuity = Future value of an annuity due = FV = PMT ( r) 1+ n 1 r = $3,000 [((1.08) 20-1)/.08] = $3, = $137, ( r) n 1+ 1 FV = PMT + r r = $137, (1.08) = $148, Note: If we set the mode in the calculator to END; and enter: ( 1 ) PMT = -$3,000; n=20; i= 8%; PV=0; and CPT FV; we get FV =$137, Likewise, if we set the mode to BGN; we get FV = $148, Similarly, using a spreadsheet and setting the Type to 0 we get FV= $137, i.e. the PV of an ordinary annuity and if we set the Type to 1; we get FV = 148, i.e. FV of an annuity due 4.6
8 Chapter 4 n The Time Value of Money (Part 2) 83 Perpetuity A Perpetuity is an equal periodic cash flow stream that will never cease. One example of such a stream is a British consol, which is a bond issued by the British government which promised to pay a specified rate of interest forever, without ever repaying the principal. The PV of a perpetuity is calculated by using the following equation: PMT PV = r Example 5: PV of a Perpetuity If you are considering the purchase of a consol that pays $60 per year forever, and the rate of interest you want to earn is 10% per year, how much money should you pay for the consol? Here, r=10%, PMT = $60; and PV = ($60/.1) = $600 is the most you should pay for the consol. Remind students that r should be in decimals. 4.5 Three Loan Payment Methods (Slides 4-26 to 4-30) Depending on the terms agreed upon at the time of issue, borrowers can typically pay off a loan in one of 3 ways: 1. They can pay off the principal (the original loan amount that you borrowed) and all the interest (the amount the lender charges you for borrowing the money) at one time at the maturity date of the loan. This is called a discount loan. 2. They can make periodic interest payments and then pay the principal and final interest payment at the maturity date. This is called an interest-only loan. 3. They can pay both principal and interest as they go by making equal payments each period. This is called an amortized loan. Example 6: Discount versus Interest- only versus Amortized loans Roseanne wants to borrow $40,000 for a period of 5 years. The lenders offers her a choice of three payment structures: 1. Pay all of the interest (10% per year) and principal in one lump sum at the end of 5 years; 2. Pay interest at the rate of 10% per year for 4 years and then a final payment of interest and principal at the end of the 5 th year; 3. Pay 5 equal payments at the end of each year inclusive of interest and part of the principal. Under which of the three options will Roseanne pay the least interest and why? Calculate the total amount of the payments and the amount of interest paid under each alternative.
9 84 Brooks n Financial Management: Core Concepts, 2e Method 1: Discount Loan Since all the interest and the principal is paid at the end of 5 years we can use the FV of a lump sum equation to calculate the payment required, i.e. FV = PV (1 + r) n FV 5 = $40,000 (1+0.10) 5 = $40, = $64, Interest paid = Total payment - Loan amount Interest paid = $64, $40,000 = $24, Method 2: Interest- Only Loan Method 3: Amortized Loan Annual Interest Payment (Years 1-4) = $40, = $4,000 Year 5 payment = Annual interest payment + Principal payment = $4,000 + $40,000 = $44,000 Total payment = $16,000 + $44,000 = $60,000 Interest paid = $20,000 To calculate the annual payment of principal an interest we can use the PV of an ordinary annuity equation and solve for the PMT value using n = 5; I = 10%; PV = $40,000, and FV = 0. That is: or PV PMT = 1 1 r ( 1+ r) $40,000 PMT = n ( ) 5 $40,000 PMT = = $10, Total payments = 5 $10,551.8 = $52, Interest paid = Total Payments - Loan Amount = $52, $40,000 Interest paid = $12, Comparison of Total payments and interest paid under each method Loan Type Total Payment Interest Paid Discount Loan $64, $24, Interest-only Loan $60, $20, Amortized Loan $52, $12, So, the amortized loan is the one with the lowest interest expense, since it requires a higher annual payment, part of which reduces the unpaid balance on the loan and thus results in less interest being charged over the 5-year term. 4.9
10 Chapter 4 n The Time Value of Money (Part 2) Amortization Schedules (Slides 4-31 to 4-33) An amortization schedule is a tabular listing of the allocation of each loan payment towards interest and principal reduction, which can help borrowers and lenders figure out the payoff balance on an outstanding loan. To prepare an amortization schedule, we must first compute the amount of each equal periodic payment (PMT) using the PVIFA equation or the appropriate financial calculator keys. Next, we calculate the amount of interest that would be charged on the unpaid balance at the end of each period, minus it from the PMT, reduce the loan balance by the remaining amount, and continue the process for each payment period, until we get a zero loan balance. Example 7: Loan amortization schedule Prepare a loan amortization schedule for the amortized loan option given in Example 6 above. What is the loan payoff amount at the end of 2 years? PV = $40,000; n=5; i=10%; FV=0; CPT PMT = $10, Year Beg. Bal. Payment Interest Prin. Red. End Bal. 1 40, , , , , , , , , , , , , , , , , , , , , , , The loan payoff amount at the end of 2 years is $26, Waiting Time and Interest Rates for Annuities (Slides 4-34 to 4-36) Problems involving annuities typically have four variables, i.e. the present (PV)or future value (FV)of the annuity stream, the amount of the annuity (PMT), the interest rate (r), and the number of annuities (n). If any three of the four variables are given, we can easily solve for the fourth one. This section deals with the procedure of solving problems where either n or r is not given. Examples of such problems include: finding out how many deposits it would take to reach a retirement or investment goal; figuring out the rate of return required to reach a retirement goal given fixed monthly deposits, etc. The problems can be solved easily by using a financial calculator or a spreadsheet. Use of an equation to solve the problem can be tedious. Example 8: Solving for the number of annuities involved Martha wants to save up $100,000 as soon as possible so that she can use it as a down payment on her dream house. She figures that she can easily set aside $8,000 per year and earn 8% annually on her deposits. How many years will Martha have to wait before she can buy that dream house? Method 1: Using a financial calculator INPUT? TVM KEYS N I/Y PV PMT FV
11 86 Brooks n Financial Management: Core Concepts, 2e Compute Method 2: Using an Excel spreadsheet Using the =NPER function we enter the following values Rate = 8%; Pmt = -8000; PV = 0; FV = ; Type = 0 or omitted; i.e. =NPER(8%,-8000,0,100000,0) The cell displays So, Martha will have to wait approximately 9 years to accumulate $100, Solving a Lottery Problem (Slides 4-37 to 4-39) In this section, the author uses the example of lottery winnings to illustrate the use of time value functions to calculate the implied interest rate given an annuity stream (annual lottery payment) versus a lump sum lottery payout. The point is that the winner can make an informed judgment regarding the 2 choices once he or she has an idea of what rate of interest is being used by the lottery authorities to calculate the annuity that is being paid over the given number of years. If the winner feels that he or she can earn a higher after-tax rate of return on the lump sum payout than the implied rate used by the authorities, the lump sum choice would be better. Of course, as pointed out by the author, other factors are being ignored here. Example 9: Calculating an implied rate of return given an annuity Let s say that you have just won the state lottery. The authorities have given you a choice of either taking a lump sum of $26,000,000 or a 30-year annuity of $1,500,000. Both payments are assumed to be after-tax. What will you do? Calculate the implied interest rate given the lump sum and the 30-year annuity Using the TVM keys of a financial calculator, enter: PV = 26,000,000; FV=0; N=30; PMT = -1,625,000; CPT I = % OR Using the =RATE(nper,pmt,pv,fv,type,guess) function of Excel enter; =RATE(30, , ,0,0); the cell will display %, which is the rate of interest which is implicitly being used by the lottery authorities to determine the 30-year annuity of $1,625,000 per year versus the $26,000,000 lump sum pay out. Choice: If you can earn an annual after-tax rate of return higher than 4.65% over the next 30 years, go with the lump sum. Otherwise, take the annuity option.
12 Chapter 4 n The Time Value of Money (Part 2) Ten Important Points about the TVM Equation (Slides 4-40 to 4-42) After covering the various topics dealing with the discounting and compounding of lump sums and annuity streams in this 2-part series, the author identifies 10 key points that students must remember going forward. It is important to advise students that a proper grasp of these 10 points is paramount to their success in dealing with more complex financial problems that lie ahead. Ten Key Points about Time Value of Money 1. Amounts of money can be added or subtracted only if they are at the same point in time. 2. The timing and the amount of the cash flow are what matters. 3. It is very helpful to lay out the timing and amount of the cash flow with a timeline. 4. Present value calculations discount all future cash flow back to current time. 5. Future value calculations value cash flows at a single point in time in the future. 6. An annuity is a series of equal cash payments at regular intervals across time. 7. The time value of money equation has four variables but only one basic equation, and so you must know three of the four variables before you can solve for the missing or unknown variable. 8. There are three basic methods to solve for an unknown time value of money variable: Method 1, using equations and calculating the answer; Method 2, using the TVM keys on a calculator; and Method 3, using financial functions from a spreadsheet. All three give the same answer because they all use the same time value of money equation. 9. There are three basic ways to repay a loan: (1) principal and interest at maturity, or discount loans, (2) interest as you go and principal at maturity, or interest-only loans, and (3) principal and interest as you go with equal and regular payments, or amortized loans. 10. Despite the seemingly accurate answers from the time value of money equation, in many situations not all the important data can be classified into the variables of present value, i.e., time, interest rate, payment, or future value. Questions 1. What is the difference between a series of payments and an annuity? What are the two specific characteristics of a series of payments that make them an annuity? An annuity is a series of payments of equal size at equal intervals. Uniform payments and equal time intervals such as months, quarters or years, are the two characteristics that make a series of payments an annuity. So, a series of payments can be an annuity but not all series of payments are annuities. If the series of payments is of different values or at different intervals, it is not an annuity.
13 88 Brooks n Financial Management: Core Concepts, 2e 2. What effect on the future value of an annuity does increasing the interest rate have? Does a change from 4% to 6% have the same dollar impact as a change from 6% to 8%? The greater the interest rate the greater the future value of an annuity everything else held constant. Changing the interest rate from 4% to 6% will increase the annuity but with a smaller dollar increase when compared to the 6% to 8% change. For example: Annuity of $100 for five years at 4%: Future Value is $ Annuity of $100 for five years at 6%: Future Value is $ Annuity of $100 for five years at 8%: Future Value is $ Increase from 4% to 6% is $22.08 Increase from 6% to 8% is $22.95.which is higher 3. What effect on the present value of an annuity does increasing the interest rate have? Does a decrease from 7% to 5% have the same dollar impact as a decrease from 5% to 3%? Decreasing the interest rate (discount rate) increases the present value of an annuity. The impact is different as the discount rates get smaller. For example: Annuity of $100 for five years at 7%: Present Value is $ Annuity of $100 for five years at 5%: Present Value is $ Annuity of $100 for five years at 3%: Present Value is $ Decrease from 7% to 5% increases PV by $22.93 Decrease from 5% to 3% increases PV by $ What is the difference between an ordinary annuity and an annuity due? An ordinary annuity has the payments at the end of the period and an annuity due has the payment due at the start of the period. 5. What is an iterative process? An iterative process is a repetitive process where the replication of the operation produces an approximation of the desired result more and more closely. 6. What does the amortization schedule tell you about a loan repayment? An amortization schedule tells you the amount of each payment that is applied against the interest expense, the amount applied against the principal and the principal balance after the payment at each scheduled payment. 7. What does it mean that the current principal balance of a loan being repaid as an amortized loan is the present value of the future payment stream? The outstanding balance or remaining unpaid principal after the application of a scheduled payment reflects the current amount needed to pay off the loan. The remaining scheduled payment stream is another way to pay off the loan. Because both the principal and the remaining scheduled payments are sufficient to pay off the loan the current principal is therefore the present value of the remaining payments.
14 Chapter 4 n The Time Value of Money (Part 2) If you increase the number of payments on an amortized loan, does the payment increase or decrease? Why or why not? Increasing the number of payments (all else held constant) decreases the size of each payment. Reverse logic provides the rationale for the answer. If the payment was increased with the same interest payment, the loan would be paid off sooner with the higher payments. Therefore increasing the time to pay off the loan means the payments have been lowered. As the amount applied to the interest is the same but the amount applied to the principal is lower so it takes more payments to eliminate the principal. 9. If you increase the interest rate on an amortized loan, does the payment increase or decrease? Why or why not? The payment increases with a rise in interest rates all else held constant. The reason is that more of the payment is applied to the interest and so to reduce the principal at the same pace as before a higher payment is needed. 10. If you won the lottery and had the choice of the lump-sum payoff or the annuity payoff, what factors would you consider besides the implied interest rate (indifference interest rate) in selecting the payoff style? Factors such as your current wealth or debts could influence your decision. If you have a large amount of debt and want to be debt free you might elect the lump-sum option. If you are terrible at budgeting money and know you would probably squander the money you might elect a slower payment method, i.e. the annuity. If you wanted to be philanthropic you might want to take the lump sum to give more money away to important causes. These are just a few of the potential non- financial impacts that could influence your decision on the pay out style. Prepping for Exams 1. b. 2. b. 3. d. 4. a. 5. b. 6. a. 7. d. 8. c. 9. a. 10. b. Problems 1. Different Cash Flow. Given the following cash inflow at the end of each year, what is the future value of this cash flow at 6%, 9%, and 15% interest rates at the end of the seventh year?
15 90 Brooks n Financial Management: Core Concepts, 2e Year 1 $15,000 Year 2 $20,000 Year 3 $30,000 Years 4 through 6 $0 Year 7 $150,000 FV at 6% = $15, = $15, = $21, $20, = $20, = $26, $30, = $30, = $37, $150, = $15, = $150, FV = $21, $26, $37, $150, = $235, FV at 9% = $15, = $15, = $25, $20, = $20, = $30, $30, = $30, = $42, $150, = $15, = $150, FV = $25, $30, $42, $150, = $248, FV at 15% = $15, = $15, = $34, $20, = $20, = $40, $30, = $30, = $52, $150, = $15, = $150, FV = $34, $40, $52, $150, = $277, Future value of an ordinary annuity. Fill in the missing future values in the following table for an ordinary annuity. PMT = -$250; n= 10; PV = 0; I% = 6; CPT FV = $ PMT =-$ ;n=20; PV=0; I%=12; CPT FV = $100, PMT =-$600; n=25; PV=0; I%=4; CPT FV = $24, PMT =-$572.25; n=360; PV=0; I%=1; CPT FV = $1,999, Future value. A speculator has purchased land along the southern Oregon coast. He has taken out a ten-year loan with annual payments of $7,200. The loan rate is 6%. At the
16 Chapter 4 n The Time Value of Money (Part 2) 91 end of ten years, he believes that he can sell the land for $100,000. If he is right on the future price, did he make a wise investment? To solve this problem, we must calculate the rate of return earned on an annual investment of $7,200 over a 10-year period, with a future value of $100,000 and compare it with the interest rate paid on the loan. If the rate earned is higher, then the investment would be worth it. FV = $100,000; PMT = -$7,200; n = 10; PV = 0; CPT I% = 7.11% > 6%; Good Investment 4. Future value. Jack and Jill are saving for a rainy day and decide to put $50 away in their local bank every year for the next twenty-five years. The local Up-the-Hill Bank will pay them 7% on their account. a. If Jack and Jill put the money in the account faithfully at the end of every year, how much will they have in it at the end of twenty-five years? b. Unfortunately, Jack had an accident in which he sustained head injuries after only 10 years of savings. The medical bill has come to $700. Is there enough in the rainy day fund to cover it? Part a. FV = $50 ( )/0.07 = $ = $3, Part b. FV = $50 ( )/0.07 = $ = $ so the rainy day fund is $9.18 short of being able to cover the medical bill. 5. Future value: You are a new employee with the Metropolis Daily Planet. The Planet offers three different retirement plans for you to choose from. Plan 1 starts the first day of work and puts $1,000 away in your retirement account at the end of every year for forty years. Plan 2 starts after ten years and puts away $2,000 every year for thirty years. Plan 3 starts after twenty years and puts away $4,000 every year for the last twenty years of employment. All three plans guarantee an annual growth rate of 8%. a. Which plan should you choose if you plan to work at the Planet for forty years? b. Which plan should you choose if you only plan to work at the Planet for the next thirty years? c. Which plan should you choose if you only plan to work at the Planet for the next twenty years? d. Which plan should you choose if you only plan to work at the Planet for the next ten years? e. What do the answers in parts (a) through (d) imply about savings?
17 92 Brooks n Financial Management: Core Concepts, 2e Part a: Plan One FV = $1,000 ( )/0.08 = $1, = $259, Part a: Plan Two FV = $2,000 ( )/0.08 = $2, = $226, Part a: Plan Three FV = $4,000 ( )/0.08 = $4, = $183, Chose Plan One Part b: Plan One FV = $1,000 ( )/0.08 = $1, = $113, Part b: Plan Two FV = $2,000 ( )/0.08 = $2, = $91, Part b: Plan Three FV = $4,000 ( )/0.08 = $4, = $57, Chose Plan One Part c: Plan One FV = $1,000 ( )/0.08 = $1, = $45, Part c: Plan Two FV = $2,000 ( )/0.08 = $2, = $28, Part c: Plan Three FV = $4,000 ( )/0.08 = $4, = $0.00 Chose Plan One Part d: Plan One FV = $1,000 ( )/0.08 = $1, = $14, Part d: Plan Two FV = $2,000 ( )/0.08 = $2, = $0.00 Part d: Plan Three FV = $4,000 ( )/0.08 = $4, = $0.00 Chose Plan One Part e: The sooner you begin to save the better and that increasing the amount of savings in later years may not be sufficient to catch up to an early savings program. 6. Different cash flow. Given the following cash inflow, what is the present value of this cash flow at 5%, 10%, and 25% discount rates? Year 1: $3,000 Year 2: $5,000 Years 3 through 7: $0 Year 8: $25,000 PV at 5% = $3,000 1/ = $3, = $2, $5,000 1/ = $5, = $4, $25,000 1/ = $25, = $16, PV at 5% = $2, $4, $16, = $24, PV at 10% = $3,000 1/ = $3, = $2, $5,000 1/ = $5, = $4, $25,000 1/ = $25, = $11, PV at 10% = $2, $4, $11, = $18, PV at 25% = $3,000 1/ = $3, = $2, $5,000 1/ = $5, = $3,200.00
18 Chapter 4 n The Time Value of Money (Part 2) 93 + $25,000 1/ = $25, = $4, PV at 25% = $2, $3, $4, = $9, Present value of an ordinary annuity. Fill in the missing present values in the following table for an ordinary annuity. Number of Payments or Years Annual Interest Rate Future Value Annuity Present Value 10 6% 0 $ % 0 $3, % 0 $ % 0 $2, Formula Answer (Rounded final answer to two decimal places) PV = $ [1-1/( ) 10 ] / 0.06 = $ = $1, PV = $3, [1-1/( ) 20 ] / 0.12 = $3, = $25, PV = $ [1-1/( ) 25 ] / 0.04 = $ = $9, PV = $2, [1-1/( ) 360 ] / 0.01 = $2, = $249, Ordinary annuity payment. Fill in the missing annuity in the following table for an ordinary annuity stream. Number of Payments or Years Annual Interest Rate Future Value Annuity Present Value 5 9% 0 $25, % $25, % 0 $200, % $96, Formula Answer (Rounded final answer to two decimal places) PMT = $25, / ([1-1/( ) 5 ] / 0.09) = $25, / = $6, PMT = $25, / ([( ) 20-1] / 0.08) = $25, / = $546.31
19 94 Brooks n Financial Management: Core Concepts, 2e PMT = $200,000 / ([1-1/( ) 30 ] / 0.07) = $200,000 / = $16, PMT = $96, / ([( ) 10-1] / 0.04) = $96, / = $8, Present value. County Ranch Insurance Company wants to offer a guaranteed annuity in units of $500, payable at the end of each year for twenty-five years. The company has a strong investment record and can consistently earn 7% on its investments after taxes. If the company wants to make 1% on this contract, what price should it set on it? Use 6% as the discount rate. Assume that it is an ordinary annuity and that the price is the same as present value. Formula Answer (Rounded final answer to two decimal places) PV = $ ([1-1/( ) 25 ] / 0.06) = $ = $6, Present value. A smooth used-car salesman who smiles considerably is offering you a great deal on a preowned car. He says, For only six annual payments of $2,500, this beautiful 1998 Honda Civic can be yours. If you can borrow money at 8%, what is the price of this car? Formula Answer (Rounded final answer to two decimal places) PV = $2, ([1-1/( ) 6 ] / 0.08) = $2, = $11, Payments. Cooley Landscaping Company needs to borrow $30,000 for a new front-end dirt loader. The bank is willing to loan the funds at 8.5% interest with annual payments at the end of the year for the next ten years. What is the annual payment on this loan for Cooley Landscaping? Payments = $30,000 / [(1 1/(1.085) 10 ) /.085] = $30,000 / = $4, Payments. Sam Hinds, a local dentist, is going to remodel the dental reception area and two new workstations. He has contacted A-Dec, and the new equipment and cabinetry will cost $18,000. A-Dec will finance the equipment purchase at 7.5% over a six-year period of time. What will Hinds have to pay in annual payments for this equipment? Payments = $18,000 / [(1 1/(1.075) 6 ) /.075] = $18,000 / = $3, Annuity due. Reginald is about to lease an apartment for the year. The landlord wants the lease payments paid at the start of the month. The twelve monthly payments are
20 Chapter 4 n The Time Value of Money (Part 2) 95 $1,300 per month. The landlord says he will allow Reginald to prepay the rent for the entire year with a discount. The one-time annual payment due at the beginning of the lease is $14,778. What is the implied monthly discount rate for the rent? If Reginald is earning 1.5% on his savings monthly, should he pay by month or take the one annual payment? Using TVM Keys from a Texas Instrument BAII Plus Calculator and rounded to two decimal places for interest percent. The P/Y and C/Y variables are set to 12. Set the MODE to BGN as this is an annuity due problem. INPUT 12-14,778 1,300 0 TVM KEYS N I/Y PV PMT FV OUTPUT This is an annual rate so with simple interest you get 12% / 12 = 1% per month. If he can get 1.5% interest per month...then his annual rate is 18% and he can generate $1, per month with the $14,778 it would take to pay off the rent. He is ahead $34.82 per month by not taking the one time payment. INPUT ,778 0 TVM KEYS N I/Y PV PMT FV OUTPUT 1, Timeline of cash flow and application of time value of money. Mauer Mining Company leases a special drilling press with annual payments of $150,000. The contract calls for rent payments at the beginning of each year for a minimum of six years. Mauer Mining can buy a similar drill for $750,000, but will need to borrow the funds at 8%. a. Show the two choices on a time line with the cash flow. b. Determine the present value of the lease payments at 8%. c. Should Mauer Mining lease or buy this drill? (Part A). Draw in a timeline with $750,000 at T 0 for the buy option and $150,000 annuity due stream for the lease option. (Part B). PV = $150,000 + $150,000 (1 1/ )/0.08 = $150,000 + $150, = $150,000 + $598, = 748,906.51
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