FN-202 Chapter 4

Part 2 Important Financial Concepts Chapters in this Part Chapter 4 Time Value of Money Chapter 5 Risk and Return Chapter 6 Interest Rates and Bond Valuation Chapter 7 Stock Valuation Integrative Case 2: Encore International Chapter 4 Time Value of Money ̈ Instructor’s Resources Overview This chapter introduces an important financial concept: the time value of money. The PV and FV of a sum, as well as the present and future values of an annuity, are explained. Special applications of the concepts include intra-year compounding, mixed cash flow streams, mixed cash flows with an embedded annuity, perpetuities, deposits to accumulate a future sum, and loan amortization. Numerous business and personal financial applications are used as examples. The chapter drives home the need to understand time value of money at the professional level because funding for new assets and programs must be justified using these techniques. Decisions in a student’s personal life should also be acceptable on the basis of applying timevalue-of-money techniques to anticipated cash flows. Study Guide The following Study Guide examples are suggested for classroom presentation: Example Topic 5 6 10 More on annuities Loan amortization Effective rate ̈ Suggested Answer to Chapter Opening Critical Thinking Question From your knowledge of the cost of name-brand and generic drugs, how do you believe the market share of generics compares with the market share of name-brand prescription drugs in dollar terms? Although generics accounted for 53% of prescriptions written in late 2006, they accounted for only 10% of the total market in dollar terms. This clearly demonstrates the price differential between generic and name-brand drugs. Chapter 4 ̈ Time Value of Money 73 Answers to Review Questions 1. Future value (FV), the value of a present amount at a future date, is calculated by applying compound interest over a specific time period. Present value (PV), represents the dollar value today of a future amount, or the amount you would invest today at a given interest rate for a specified time period to equal the future amount. Financial managers prefer present value to future value because they typically make decisions at time zero, before the start of a project. 2. A single amount cash flow refers to an individual, stand alone, value occurring at one point in time. An annuity consists of an unbroken series of cash flows of equal dollar amount occurring over more than one period. A mixed stream is a pattern of cash flows over more than one time period and the amount of cash associated with each period will vary. 3. Compounding of interest occurs when an amount is deposited into a savings account and the interest paid after the specified time period remains in the account, thereby becoming part of the principal for the following period. The general equation for future value in year n (FVn) can be expressed using the specified notation as follows: FVn = PV × (1 + i)n 4. A decrease in the interest rate lowers the future amount of a deposit for a given holding period, since the deposit earns less at the lower rate. An increase in the holding period for a given interest rate would increase the future value. The increased holding period increases the FV since the deposit earns interest over a longer period of time. 5. The present value of a future amount indicates how much money today would be equivalent to the future amount if one could invest that amount at a specified rate of interest. Using the given notation, the present value of a future amount (FVn) can be defined as follows: ⎛ 1 ⎞ PV = FV ⎜ n ⎟ ⎝ (1 + i ) ⎠ 6. An increasing required rate of return would reduce the present value of a future amount, since future dollars would be worth less today. Looking at the formula for present value in Question 5, it should be clear that by increasing the i value, which is the required return, the present value interest factor would decrease, thereby reducing the present value of the future sum. 7. Present value calculations are the exact inverse of compound interest calculations. Using compound interest, one attempts to find the future value of a present amount; using present value, one attempts to find the present value of an amount to be received in the future. 8. An ordinary annuity is one for which payments occur at the end of each period. An annuity due is one for which payments occur at the beginning of each period. The ordinary annuity is the more common. For otherwise identical annuities and interest rates, the annuity due results in a higher FV because cash flows occur earlier and have more time to compound. 74 Gitman • Principles of Managerial Finance, Twelfth Edition 9. The PV of an ordinary annuity, PVAn, can be determined using the formula: PVAn = PMT × (PVIFAi%,n) where: PMT = the end of period cash inflows PVIFAi%,n = the PV interest factor of an annuity for interest rate i and n periods. The PVIFA is related to the PVIF in that the annuity factor is the sum of the PVIFs over the number of periods for the annuity. For example, the PVIFA for 5% and 3 periods is 2.723, and the sum of the 5% PVIF for periods one through three is 2.723 (0.952 + 0.907 + 0.864). 10. The FVIFA factors for an ordinary annuity can be converted for use in calculating an annuity due by multiplying the FVIFAi%,n by 1 + i. 11. The PVIFA factors for an ordinary annuity can be converted for use in calculating an annuity due by multiplying the PVIFAi%,n by 1 + i. 12. A perpetuity is an infinite-lived annuity. The factor for finding the present value of a perpetuity can be found by dividing the discount rate into 1.0. The resulting quotient represents the factor for finding the present value of an infinite-lived stream of equal annual cash flows. 13. The future value of a mixed stream of cash flows is calculated by multiplying each year’s cash flow by the appropriate future value interest factor. To find the present value of a mixed stream of cash flows multiply each year’s cash flow by the appropriate present value interest factor. There will be at least as many calculations as the number of cash flows. 14. As interest is compounded more frequently than once a year, both (a) the future value for a given holding period and (b) the effective annual rate of interest will increase. This is due to the fact that the more frequently interest is compounded, the greater the quantity of money accumulated and reinvested as the principal value. In situations of intra-year compounding, the actual rate of interest is greater than the stated rate of interest. 15. Continuous compounding assumes interest will be compounded an infinite number of times per year, at intervals of microseconds. Continuous compounding of a given deposit at a given rate of interest results in the largest value when compared to any other compounding period. 16. The nominal annual rate is the contractual rate that is quoted to the borrower by the lender. The effective annual rate, sometimes called the true rate, is the actual rate that is paid by the borrower to the lender. The difference between the two rates is due to the compounding of interest at a frequency greater than once per year. APR is the annual percentage rate and is required by “truth in lending laws” to be disclosed to consumers. This rate is calculated by multiplying the periodic rate by the number of periods in one year. The periodic rate is the nominal rate over the shortest time period in which interest is compounded. The APY, or annual percentage yield, is the effective rate of interest that must be disclosed to consumers by banks on their savings products as a result of the “truth in savings laws.” These laws result in both favorable and unfavorable information to consumers. The good news is that rate quotes on both loans and savings are standardized among financial institutions. The negative is that the APR, or lending rate, is a nominal rate, while the APY, or saving rate, is an effective rate. These rates are the same when compounding occurs only once per year. Chapter 4 Time Value of Money 75 17. The size of the equal annual end-of-year deposits needed to accumulate a given amount over a certain time period at a specified rate can be found by dividing the interest factor for the future value of an annuity for the given interest rate and the number of years (FVIFAi%,n) into the desired future amount. The resulting quotient would be the amount of the equal annual end-of-year deposits required. The future value interest factor for an annuity is used in this calculation: PMT = FVn FVIFA i %,n 18. Amortizing a loan into equal annual payments involves finding the future payments whose PV at the loan interest rate just equals the amount of the initial principal borrowed. The formula is: PMT = 19. a. PVn PVIFA i %,n Either the present value interest factor or the future value interest factor can be used to find the growth rate associated with a stream of cash flows. The growth rate associated with a stream of cash flows may be found by using the following equation, where the growth rate, g, is substituted for k. PV = FVn (1 + g ) To find the rate at which growth has occurred, the amount received in the earliest year is divided by the amount received in the latest year. This quotient is the PVIFi%;n. The growth rate associated with this factor may be found in the PVIF table. b. To find the interest rate associated with an equal payment loan, the present value interest factors for a one-dollar annuity table would be used. To determine the interest rate associated with an equal payment loan, the following equation may be used: PVn = PMT × (PVIFAi%,n) Solving the equation for PVIFAi%,n we get: PVIFAi %,n = PVn PMT Then substitute the values for PVn and PMT into the formula, using the PVIFA table to find the interest rate most closely associated with the resulting PVIFA, which is the interest rate on the loan. 76 Gitman • Principles of Managerial Finance, Twelfth Edition 20. To find the number of periods it would take to compound a known present amount into a known future amount you can solve either the present value or future value equation for the interest factor as shown below using the present value: PV = FV × (PVIFi%,n) Solving the equation for PVIFi%,n we get: PVIFi %, n = PV FV Then substitute the values for PV and FV into the formula, using the PVIF table for the known interest rate find the number of periods most closely associated with the resulting PVIF. The same approach would be used for finding the number of periods for an annuity except that the annuity factor and the PVIFA (or FVIFA) table would be used. This process is shown below. PVn = PMT × (PVIFAi%,n) Solving the equation for PVIFAi%,n we get: PVIFAi %,n = ̈ PVn PMT Suggested Answer to Critical Thinking Question for Focus on Practice Box As a reaction to problems in the subprime area, lenders are already tightening lending standards. What effect will this have on the housing market? The tightening of lending standards following the subprime fiasco will likely further depress home prices, which in 2007 were already undergoing their steepest, widest decline in history. If the housing-market slump persists, companies that are heavily involved with the manufacture and sale of durable household goods also will feel the impact of the slow housing market. ̈ Suggested Answer to Critical Thinking Question for Focus on Ethics Box What effect can a bank’s order of process (cashing checks presented on the same day from smallest to largest or from largest to smallest) have on NSF fees? If a customer experiences one nonsufficient funds (NSF) fee, he or she may encounter additional fees. The order of process can have a significant effect on the number of NSF fees paid. For example, suppose that you had $500 in your checking account and wrote seven checks totaling $630. The seven checks are for $400, $13, $50, $70, $25, $40, and $32. If they all arrive at the bank at the same time, the bank could clear the last six checks and bounce only the $400 check. In that case, you would pay one NSF. However, if the bank cleared the biggest ones first, the $400 check would clear and so would the $70 check. The $50 check, next in size, would bounce and so would the other smaller checks. In that sequence of checks, you would pay five NSF fees. In response to negative publicity on this issue, most major banks have adopted a “low to high” posting policy, but you should inquire as to your bank’s specific policy. Chapter 4 ̈ Time Value of Money 77 Answers to Warm-Up Exercises E4-1. Future value of a lump sum investment Answer: FV = $2,500 × (1 + 0.007) = $2,517.50 E4-2. Finding the future value Answer: Since the interest is compounded monthly, the number of periods is 4 × 12 = 48 and the monthly interest rate is 1/12th of the annual rate. FV48 = PV × (1 + I)48 where I is the monthly interest rate I = 0.02 ÷ 12 = 0.00166667 FV48 = ($1,260 + $975) × (1 + 0.00166667)48 FV48 = ($2,235) × 1.083215 = $2,420.99 If using a financial calculator, set the calculator to 12 compounding periods per year and input the following: PV = $2,235 I/year = 2 N = 48 (months) Solve for FV × FV = $2,420.99 Note: Not all financial calculators work in the same manner. Some require the user to use the CPT (Compute) button. Others require the user to calculate the monthly interest rate and input that amount rather than the annual rate. The steps shown in the solution manual will be the inputs needed to use the Hewlett Packard 10B or 10BII models. They are similar to the steps followed when using the Texas Instruments BAII calculators. If using a spreadsheet, the solution is: Column A Column B Cell 1 Cell 2 Cell 3 Future value of a single amount Present value Interest rate, pct per year compounded monthly Cell 4 Number of months = 4 × 12 Cell 5 Future value = FV(B3,B4,0,–B2,0) $2,235 = 2/12 Cell B5 = $2,420.99 E4-3. Comparing a lump sum with an annuity Answer: This problem can be solved in either of two ways. Both alternatives can be compared as lump sums in net present value terms or both alternatives can be compared as a 25-year annuity. In each case, one of the alternatives needs to be converted. Method 1: Perform a lump sum comparison. Compare $1.3 million now with the present value of the twenty-five payments of $100,000 per year. In this comparison, the present value of the $100,000 annuity must be found and compared with the $1.3 million. (Be sure to set the calculator to 1 compounding period per year.) 78 Gitman • Principles of Managerial Finance, Twelfth Edition PMT = –$100,000 N = 25 I = 5% Solve for PV PV = $1,409,394 (greater than $1.3 million) Choose the $100,000 annuity over the lump sum. Method 2: Compare two annuities. Since the $100,000 per year is already an annuity, all that remains is to convert the $1.3 million into a 25-year annuity. PV = –$1.3 million N = 25 years I = 5% Solve for PMT PMT = $92,238.19 (less than $100,000) Choose the $100,000 annuity over the lump sum. You may use the table method or a spreadsheet to do the same analysis. E4-4. Comparing the present value of two alternatives Answer: To solve this problem you must first find the present value of the expected savings over the 5-year life of the software. Year 1 2 3 4 5 Savings Estimate $35,000 50,000 45,000 25,000 15,000 Present Value of Savings $32,110 42,084 34,748 17,710 9,749 $136,401 Since the $136,401 present value of the savings exceeds the $130,000 cost of the software, the firm should invest in the new software. You may use a financial calculator, the table method or a spreadsheet to find the PV of the savings. Chapter 4 Time Value of Money 79 E4-5. Compounding more frequently than annually Answer: Partners’ Savings Bank: i ⎞ ⎛ FV1 = PV × ⎜ 1 + ⎟ ⎝ m⎠ FV1 = $12,000 × (1 + 0.03/2)2 m×n FV1 = $12,000 × (1 + 0.03/2)2 = $12,000 × 1.030225 = $12,362.70 Selwyn’s: FV1 = PV × (ei × n ) = $12,000 × (2.71830.0275×1 ) = $12,0000 × 1.027882 = $12,334.58 Joseph should choose the 3% rate with semiannual compounding. E4-6. Determining deposits needed to accumulate a future sum Answer: The financial calculator input is as follows: FV = –$150,000 N = 18 I = 6% Solve for PMT. ̈ P4-1. PMT = $4,853.48 Solutions to Problems LG 1: Using a time line Basic a, b, and c d. Financial managers rely more on present value than future value because they typically make decisions before the start of a project, at time zero, as does the present value calculation. 80 Gitman • Principles of Managerial Finance, Twelfth Edition P4-2. LG 2: Future value calculation: FVn = PV × (1 + I)n Basic Case A B C D P4-3. FVIF12%,2 periods = (1 + 0.12)2 = 1.254 FVIF6%,3 periods = (1 + 0.06)3 = 1.191 FVIF9%,2 periods = (1 + 0.09)2 = 1.188 FVIF3%,4 periods = (1 + 0.03)4 = 1.126 LG 2: Future value tables: FVn = PV × (1 + I)n Basic Case A a. 2 = 1 × (1 + 0.07)n 2/1 = (1.07)n 2 = FVIF7%,n 10 years < n < 11 years Nearest to 10 years Case B a. 2 = 1 × (1 + 0.40)n 2 = FVIF40%,n 2 years < n < 3 years Nearest to 2 years Case C a. 2 = 1 × (1 + 0.20)n 2 = FVIF20%,n 3 years < n < 4 years Nearest to 4 years Case D a. 2 = 1 × (1 + 0.10)n 2 = FVIF10%,n 7 years < n < 8 years Nearest to 7 years P4-4. b. 4 = 1 × (1 + 0.07)n 4/1 = (1.07)n 4 = FVIF7%,n 20 years < n < 21 years Nearest to 20 years b. 4 = (1 + 0.40)n 4 = FVIF40%,n 4 years < n < 5 years Nearest to 4 years b. 4 = (1 + 0.20)n 4 = FVIF20%,n 7 years < n < 8 years Nearest to 8 years b. 4 = (1 + 0.10)n 4 = FVIF40%,n 14 years < n <15 years Nearest to 15 years LG 2: Future values: FVn = PV × (1 + I)n or FVn = PV × (FVIFi%,n) Intermediate Case A FV20 = PV × FVIF5%,20 yrs. FV20 = $200 × (2.653) FV20 = $530.60 Calculator solution: $530.66 Case B FV7 = PV × FVIF8%,7 yrs. FV7 = $4,500 × (1.714) FV7 = $7,713 Calculator solution: $7,712.21 Chapter 4 P4-5. 81 C FV10 = PV × FVIF9%,10 yrs. FV10 = $10,000 × (2.367) FV10 = $23,670 Calculator solution: $23,673.64 D FV12 = PV × FVIF10%,12 yrs. FV12 = $25,000 × (3.138) FV12 = $78,450 Calculator solution: $78,460.71 E FV5 = PV × FVIF11%,5 yrs. FV5 = $37,000 × (1.685) FV5 = $62,345 Calculator solution: $62,347.15 F FV9 = PV × FVIF12%,9 yrs. FV9 = $40,000 × (2.773) FV9 = $110,920 Calculator solution: $110,923.15 LG 2: Personal finance: Time value: FVn = PV × (1 + I)n or FVn = PV × (FVIFi%,n) Intermediate a. c. P4-6. Time Value of Money (1) FV3 = PV × (FVIF7%,3) FV3 = $1,500 × (1.225) FV3 = $1,837.50 Calculator solution: $1,837.56 b. (1) Interest earned = FV3 – PV Interest earned = $1,837.50 –$1,500.00 $337.50 (2) FV6 = PV × (FVIF7%,6) FV6 = $1,500 × (1.501) FV6 = $2,251.50 Calculator solution: $2,251.10 (2) Interest earned = FV6 – FV3 Interest earned = $2,251.50 –$1,837.50 $414.00 (3) FV9 = PV × (FVIF7%,9) FV9 = $1,500 × (1.838) FV9 = $2,757.00 Calculator solution: $2,757.69 (3) Interest earned = FV9 – FV6 Interest earned = $2,757.00 –$2,251.50 $505.50 The fact that the longer the investment period is, the larger the total amount of interest collected will be, is not unexpected and is due to the greater length of time that the principal sum of $1,500 is invested. The most significant point is that the incremental interest earned per 3-year period increases with each subsequent 3 year period. The total interest for the first 3 years is $337.50; however, for the second 3 years (from year 3 to 6) the additional interest earned is $414.00. For the third 3-year period, the incremental interest is $505.50. This increasing change in interest earned is due to compounding, the earning of interest on previous interest earned. The greater the previous interest earned, the greater the impact of compounding. LG 2: Personal finance: Time value Challenge a. (1) FV5 = PV × (FVIF2%,5) FV5 = $14,000 × (1.104) FV5 = $15,456.00 Calculator solution: $15,457.13 (2) FV5 = PV × (FVIF4%,5) FV5 = $14,000 × (1.217) FV5 = $17,038.00 Calculator solution: $17,033.14 b. The car will cost $1,582 more with a 4% inflation rate than an inflation rate of 2%. This increase is 10.2% more ($1,582 ÷ $15,456) than would be paid with only a 2% rate of inflation. 82 Gitman • Principles of Managerial Finance, Twelfth Edition P4-7. LG 2: Personal finance: Time value Challenge Deposit Now: Deposit in 10 Years: FV40 = PV × FVIF9%,40 40 FV40 = $10,000 × (1.09) FV40 = $10,000 × (31.409) FV40 = $314,090.00 Calculator solution: $314,094.20 FV30 = PV10 × (FVIF9%,30) 30 FV30 = PV10 × (1.09) FV30 = $10,000 × (13.268) FV30 = $132,680.00 Calculator solution: $132,676.78 You would be better off by $181,410 ($314,090 – $132,680) by investing the $10,000 now instead of waiting for 10 years to make the investment. P4-8. P4-9. LG 2: Personal finance: Time value: FVn = PV × FVIFi%,n Challenge a. $15,000 = $10,200 × FVIFi%,5 FVIFi%,5 = $15,000 ÷ $10,200 = 1.471 8% < i < 9% Calculator solution: 8.02% c. $15,000 = $7,150 × FVIFi%,5 FVIFi%,5 = $15,000 ÷ $7,150 = 2.098 15% < i < 16% Calculator solution: 15.97% b. $15,000 = $8,150 × FVIFi%,5 FVIFi%,5 = $15,000 ÷ $8,150 = 1.840 12% < i < 13% Calculator solution: 12.98% LG 2: Personal finance: Single-payment loan repayment: FVn = PV × FVIFi%,n Intermediate a. FV1 = PV × (FVIF14%,1) FV1 = $200 × (1.14) FV1 = $228 Calculator solution: $228 c. FV8 = PV × (FVIF14%,8) FV8 = $200 × (2.853) FV8 = $570.60 Calculator solution: $570.52 P4-10. LG 2: Present value calculation: PVIF = Basic Case A B C D PVIF = 1 ÷ (1 + 0.02)4 = 0.9238 2 PVIF = 1 ÷ (1 + 0.10) = 0.8264 3 PVIF = 1 ÷ (1 + 0.05) = 0.8638 2 PVIF = 1 ÷ (1 + 0.13) = 0.7831 b. FV4 = PV × (FVIF14%,4) FV4 = $200 × (1.689) FV4 = $337.80 Calculator solution: $337.79 1 (1 + i)n Chapter 4 Time Value of Money 83 P4-11. LG 2: Present values: PV = FVn × (PVIFi%,n) Basic Case A B C D E Calculator Solution PV12%,4yrs PV8%, 20yrs PV14%,12yrs PV11%,6yrs PV20%,8yrs = $7,000 = $28,000 = $10,000 = $150,000 = $45,000 $4,448.63 $6,007.35 $2,075.59 $80,196.13 $10,465.56 × 0.636 = $4,452 × 0.215 = $6,020 × 0.208 = $2,080 × 0.535 = $80,250 × 0.233 = $10,485 P4-12. LG 2: Present value concept: PVn = FVn × (PVIFi%,n) Intermediate a. PV = FV6 × (PVIF12%,6) PV = $6,000 × (.507) PV = $3,042.00 Calculator solution: $3,039.79 c. PV = FV6 × (PVIF12%,6) PV = $6,000 × (0.507) PV = $3,042.00 Calculator solution: $3,039.79 b. PV = FV6 × (PVIF12%,6) PV = $6,000 × (0.507) PV = $3,042.00 Calculator solution: $3,039.79 d. The answer to all three parts are the same. In each case the same questions is being asked but in a different way. P4-13. LG 2: Personal finance: Time value: PV = FVn × (PVIFi%,n) Basic Jim should be willing to pay no more than $408.00 for this future sum given that his opportunity cost is 7%. PV = $500 × (PVIF7%,3) PV = $500 × (0.816) PV = $408.00 Calculator solution: $408.15 P4-14. LG 2: Time value: PV = FVn × (PVIFi%,n) Intermediate PV = $100 × (PVIF8%,6) PV = $100 × (0.630) PV = $63.00 Calculator solution: $63.02 84 Gitman • Principles of Managerial Finance, Twelfth Edition P4-15. LG 2: Personal finance: Time value and discount rates: PV = FVn × (PVIFi%,n) Intermediate a. (1) PV = $1,000,000 × (PVIF6%,10) PV = $1,000,000 × (0.558) PV = $558,000.00 Calculator solution: $558,394.78 (2) PV = $1,000,000 × (PVIF9%,10) PV = $1,000,000 × (0.422) PV = $422,000.00 Calculator solution: $422,410.81 (3) PV = $1,000,000 × (PVIF12%,10) PV = $1,000,000 × (0.322) PV = $322,000.00 Calculator solution: $321,973.24 b. (1) PV = $1,000,000 × (PVIF6%,15) PV = $1,000,000 × (0.417) PV = $417,000.00 Calculator solution: $417,265.06 (2) PV = $1,000,000 × (PVIF9%,15) PV = $1,000,000 × (0.275) PV = $275,000.00 Calculator solution: $274,538.04 (3) PV = $1,000,000 × (PVIF12%,15) PV = $1,000,000 × (0.183) PV = $183,000.00 Calculator solution: $182,696.26 c. As the discount rate increases, the present value becomes smaller. This decrease is due to the higher opportunity cost associated with the higher rate. Also, the longer the time until the lottery payment is collected, the less the present value due to the greater time over which the opportunity cost applies. In other words, the larger the discount rate and the longer the time until the money is received, the smaller will be the present value of a future payment. P4-16. Personal finance: LG 2: Time value comparisons of lump sums: PV = FVn × (PVIFi%,n) Intermediate a. A PV = $28,500 × (PVIF11%,3) PV = $28,500 × (0.731) PV = $20,833.50 Calculator solution: $20,838.95 C PV = $160,000 × (PVIF11%,20) PV = $160,000 × (0.124) PV = $19,840.00 Calculator solution: $19,845.43 B PV = $54,000 × (PVIF11%,9) PV = $54,000 × (0.391) PV = $21,114.00 Calculator solution: $21,109.94 b. Alternatives A and B are both worth greater than $20,000 in term of the present value. c. The best alternative is B because the present value of B is larger than either A or C and is also greater than the $20,000 offer. Chapter 4 Time Value of Money P4-17. LG 2: Personal finance: Cash flow investment decision: PV = FVn × (PVIFi%,n) Intermediate A PV = $30,000 × (PVIF10%,5) PV = $30,000 × (0.621) PV = $18,630.00 Calculator solution: $18,627.64 B PV = $3,000 × (PVIF10%,20) PV = $3,000 × (0.149) PV = $447.00 Calculator solution: $445.93 C PV = $10,000 × (PVIF10%,10) PV = $10,000 × (0.386) PV = $3,860.00 Calculator solution: $3,855.43 D PV = $15,000 × (PVIF10%,40) PV = $15,000 × (0.022) PV = $330.00 Calculator solution: $331.42 Purchase Do Not Purchase A C B D P4-18. LG 3: Future value of an annuity Intermediate a. Future value of an ordinary annuity vs. annuity due (1) Ordinary Annuity (2) Annuity Due FVAk%,n = PMT × (FVIFAk%,n) FVAdue = PMT × [(FVIFAk%,n × (1 + k)] A FVA8%,10 = $2,500 × 14.487 FVA8%,10 = $36,217.50 Calculator solution: $36,216.41 FVAdue = $2,500 × (14.487 × 1.08) FVAdue = $39,114.90 Calculator solution: $39,113.72 B FVA12%,6 = $500 × 8.115 FVA12%,6 = $4,057.50 Calculator solution: $4,057.59 FVAdue = $500 ×( 8.115 × 1.12) FVAdue = $4,544.40 Calculator solution: $4,544.51 C FVA20%,5 = $30,000 × 7.442 FVA20%,5 = $223,260 Calculator solution: $223,248 FVAdue = $30,000 × (7.442 × 1.20) FVAdue = $267,912 Calculator solution: $267,897.60 D FVA9%,8 = $11,500 × 11.028 FVA9%,8 = $126,822 Calculator solution: $126,827.45 FVAdue = $11,500 × (11.028 × 1.09) FVAdue = $138,235.98 Calculator solution: $138,241.92 E FVA14%,30 = $6,000 × 356.787 FVA14%,30 = $2,140,722 Calculator solution: $2,140,721.08 FVAdue = $6,000 × (356.787 × 1.14) FVAdue = $2,440,422.00 Calculator solution: $2,440,422.03 b. The annuity due results in a greater future value in each case. By depositing the payment at the beginning rather than at the end of the year, it has one additional year of compounding. 85 86 Gitman • Principles of Managerial Finance, Twelfth Edition P4-19. LG 3: Present value of an annuity: PVn = PMT × (PVIFAi%,n) Intermediate a. Present value of an ordinary annuity vs. annuity due (1) Ordinary Annuity (2) Annuity Due PVAdue = PMT × [(PVIFAi%,n × (1 + k)] PVAk%,n = PMT × (PVIFAi%,n) A PVA7%,3 = $12,000 × 2.624 PVA7%,3 = $31,488 Calculator solution: $31,491.79 PVAdue = $12,000 × (2.624 × 1.07) PVAdue = $33,692 Calculator solution: $33,696.22 B PVA12%15 = $55,000 × 6.811 PVA12%,15 = $374,605 Calculator solution: $374,597.55 PVAdue = $55,000 × (6.811 × 1.12) PVAdue = $419,557.60 Calculator solution: $419,549.25 C PVA20%,9 = $700 × 4.031 PVA20%,9 = $2,821.70 Calculator solution: $2,821.68 PVAdue = $700 × (4.031 × 1.20) PVAdue = $3,386.04 Calculator solution: $3,386.01 D PVA5%,7 = $140,000 × 5.786 PVA5%,7 = $810,040 Calculator solution: $810,092.28 PVAdue = $140,000 × (5.786 × 1.05) PVAdue = $850,542 Calculator solution: $850,596.89 E PVA10%,5 = $22,500 × 3.791 PVA10%,5 = $85,297.50 Calculator solution: $85,292.70 PVAdue = $22,500 × (2.791 × 1.10) PVAdue = $93,827.25 Calculator solution: $93,821.97 b. The annuity due results in a greater present value in each case. By depositing the payment at the beginning rather than at the end of the year, it has one less year to discount back. P4-20. LG 3: Personal finance: Time value–annuities Challenge a. Annuity C (Ordinary) FVAi%,n = PMT × (FVIFAi%,n) Annuity D (Due) FVAdue = PMT × [FVIFAi%,n × (1 + i)] (1) FVA10%,10 = $2,500 × 15.937 FVA10%,10 = $39,842.50 Calculator solution: $39,843.56 FVAdue = $2,200 × (15.937 × 1.10) FVAdue = $38,567.54 Calculator solution: $38,568.57 (2) FVA20%,10 = $2,500 × 25.959 FVA20%,10 = $64,897.50 Calculator solution: $64,896.71 FVAdue = $2,200 × (25.959 × 1.20) FVAdue = $68,531.76 Calculator solution: $68,530.92 Chapter 4 Time Value of Money 87 b. (1) At the end of year 10, at a rate of 10%, Annuity C has a greater value ($39,842.50 vs. $38,567.54). (2) At the end of year 10, at a rate of 20%, Annuity D has a greater value ($68,531.76 vs. $64,897.50). c. Annuity C (Ordinary) PVAi%,n = PMT × (FVIFAi%,n) Annuity D (Due) PVAdue = PMT × [FVIFAi%,n × (1 + i)] (1) PVA10%,10 = $2,500 × 6.145 PVA10%,10 = $15,362.50 Calculator solution: $15,361.42 PVAdue = $2,200 × (6.145 × 1.10) PVAdue = $14,870.90 Calculator solution: $14,869.85 (2) PVA20%,10 = $2,500 × 4.192 PVA20%,10 = $10,480 Calculator solution: $10,481.18 PVAdue = $2,200 × (4.192 × 1.20) PVAdue = $11,066.88 Calculator solution: $11,068.13 d. (1) At the beginning of the 10 years, at a rate of 10%, Annuity C has a greater value ($15,362.50 vs. $14,870.90). (2) At the beginning of the 10 years, at a rate of 20%, Annuity D has a greater value ($11,066.88 vs. $10,480.00). e. Annuity C, with an annual payment of $2,500 made at the end of the year, has a higher present value at 10% than Annuity D with an annual payment of $2,200 made at the beginning of the year. When the rate is increased to 20%, the shorter period of time to discount at the higher rate results in a larger value for Annuity D, despite the lower payment. P4-21. LG 3: Personal finance: Retirement planning Challenge a. FVA40 = $2,000 × (FVIFA10%,40) FVA40 = $2,000 × (442.593) FVA40 = $885,186 Calculator solution: $885,185.11 b. FVA30 = $2,000 × (FVIFA10%,30) FVA30 = $2,000 × (164.494) FVA30 = $328,988 Calculator solution: $328,988.05 c. By delaying the deposits by 10 years the total opportunity cost is $556,198. This difference is due to both the lost deposits of $20,000 ($2,000 × 10yrs.) and the lost compounding of interest on all of the money for 10 years. d. Annuity Due: FVA40 = $2,000 × (FVIFA10%,40) × (1 + 0.10) FVA40 = $2,000 × (486.852) FVA40 = $973,704 Calculator solution: $973,703.62 88 Gitman • Principles of Managerial Finance, Twelfth Edition FVA30 = $2,000 × (FVIFA10%,30) × (1.10) FVA30 = $2,000 × (180.943) FVA30 = $361,886 Calculator solution: $361,886.85 Both deposits increased due to the extra year of compounding from the beginning-of-year deposits instead of the end-of-year deposits. However, the incremental change in the 40 year annuity is much larger than the incremental compounding on the 30 year deposit ($88,518 versus $32,898) due to the larger sum on which the last year of compounding occurs. P4-22. LG 3: Personal finance: Value of a retirement annuity Intermediate PVA = PMT × (PVIFA9%,25) PVA = $12,000 × (9.823) PVA = $117,876.00 Calculator solution: $117,870.96 P4-23. LG 3: Personal finance: Funding your retirement Challenge a. PVA = PMT × (PVIFA11%,30) PVA = $20,000 × (8.694) PVA = $173,880.00 Calculator solution: $173,875.85 b. PV = FV × (PVIF9%,20) PV = $173,880 × (0.178) PV = $30,950.64 Calculator solution: $31,024.82 c. Both values would be lower. In other words, a smaller sum would be needed in 20 years for the annuity and a smaller amount would have to be put away today to accumulate the needed future sum. P4-24. LG 2, 3: Personal finance: Value of an annuity versus a single amount Intermediate a. PVAn = PMT × (PVIFAi%,n) PVA25 = $40,000 × (PVIFA5%,25) PVA25 = $40,000 × 14.094 PVA25 = $563,760 Calculator solution: $563,757.78 At 5%, taking the award as an annuity is better; the present value is $563,760, compared to receiving $500,000 as a lump sum. Chapter 4 Time Value of Money b. PVAn = $40,000 × (PVIFA7%,25) PVA25 = $40,000 × (11.654) PVA25 = $466,160 Calculator solution: $466,143.33 At 7%, taking the award as a lump sum is better; the present value of the annuity is only $466,160, compared to the $500,000 lump sum payment. c. Because the annuity is worth more than the lump sum at 5% and less at 7%, try 6%: PV25 = $40,000 × (PVIFA6%,25) PV25 = $40,000 × 12.783 PV25 = $511,320 The rate at which you would be indifferent is greater than 6%; about 6.25% Calculator solution: 6.24% P4-25. LG 3: Perpetuities: PVn = PMT × (PVIFAi%,∞) Basic a. b. Case PV Factor PMT × (PVIFAi%,∞) = PMT × (1 ÷ i) A 1 ÷ 0.08 = 12.50 $20,000 × 12.50 = $250,000 B 1 ÷ 0.10 = 10.00 $100,000 × 10.00 = $1,000,000 C 1 ÷ 0.06 = 16.67 $3,000 D 1 ÷ 0.05 = 20.00 $60,000 × 20.00 = $1,200,000 × 16.67 = $50,000 P4-26. LG 3: Personal finance: Creating an endowment Intermediate a. PV = PMT × (PVIFAi%,∞) PV = ($600 × 3) × (1 ÷ i) PV = $1,800 × (1 ÷ 0.06) PV = $1,800 × (16.67) PV = $30,006 Calculator solution: $30,000 b. PV = PMT × (PVIFAi%,∞) PV = ($600 × 3) × (1 ÷ i) PV = $1,800 × (1 ÷ 0.09) PV = $1,800 × (11.11) PV = $19,998 Calculator solution: $20,000 89 90 Gitman • Principles of Managerial Finance, Twelfth Edition P4-27. LG 4: Value of a mixed stream Challenge a. Cash Flow Stream A Year Number of Years to Compound 1 2 3 3 2 1 Future Value FV = CF × FVIF12%,n $ 900 × 1.405 1,000 × 1.254 1,200 × 1.120 = = = B 1 2 3 4 5 5 4 3 2 1 Calculator solution: $30,000 × 1.762 = 25,000 × 1.574 = 20,000 × 1.405 = 10,000 × 1.254 = 5,000 × 1.120 = C 1 2 3 4 4 3 2 1 Calculator solution: $ 1,200 × 1.574 = 1,200 × 1.405 = 1,000 × 1.254 = 1,900 × 1.120 = Calculator solution: $ 1,264.50 1,254.00 1,344.00 $ 3,862.50 $ 3,862.84 $ 52,860.00 39,350.00 28,100.00 12,540.00 5,600.00 $138,450.00 $138,450.79. $ 1,888.80 1,686.00 1,254.00 2,128.00 $ 6,956.80 $ 6,956.54 b. If payments are made at the beginning of each period the present value of each of the end-ofperiod cash flow streams will be multiplied by (1 + i) to get the present value of the beginningof-period cash flows. A $3,862.50 (1 + 0.12) = $4,326.00 B $138,450.00 (1 + 0.12) = $155,064.00 C $6,956.80 (1 + 0.12) = $7,791.62 P4-28. LG 4: Personal finance: Value of a single amount versus a mixed stream Intermediate Lump Sum Deposit FV5 = PV × (FVIF7%,5)) FV5 = $24,000 × (1.403) FV5 = $33,672.00 Calculator solution: $33,661.24 Chapter 4 Time Value of Money Mixed Stream of Payments Beginning of Year Number of Years to Compound FV = CF × FVIF7%,n 5 4 3 2 1 $ 2,000 × 1.403 $ 4,000 × 1.311 $ 6,000 × 1.225 $ 8,000 × 1.145 $10,000 × 1.070 1 2 3 4 5 Future Value $ 2,805.00 $ 5,243.00 $ 7,350.00 $ 9,159.00 $10,700.00 $35,257.00 $35,257.75 = = = = = Calculator solution: Gina should select the stream of payments over the front-end lump sum payment. Her future wealth will be higher by $1,588. P4-29. LG 4: Value of mixed stream Basic Cash Flow Stream Year CF × PVIF12%,n = Present Value 0.893 0.797 0.712 0.636 0.567 = = = = = – $1,786 2,391 2,848 3,816 4,536 $11,805 $11,805.51 $ 8,930 13,560 3,549 $26,039 $26,034.58 A 1 2 3 4 5 –$2000 3,000 4,000 6,000 8,000 × × × × × B 1 2–5 6 $10,000 5,000 7,000 Calculator solution: 0.893 × = a 2.712 = × 0.507 × = Calculator solution: * Sum of PV factors for years 2–5 C 1–5 6–10 $10,000 8,000 × × 3.605b c 2.045 Calculator solution: $36,050 16,360 $52,410 $52,411.34 a PVIFA for 12% over years 2 through 5 = (PVIFA 12% 5 years) – (PVIFA 12% 1 year) b PVIFA for 12% 5 years c (PVIFA for 12%,10 years) – (PVIFA for 12%,5 years) 91 92 Gitman • Principles of Managerial Finance, Twelfth Edition P4-30. LG 4: PV-mixed stream Intermediate a. Cash Flow Stream A B Year CF × PVIF15%,n = Present Value 1 $50,000 × 0.870 = $43,500 2 3 40,000 30,000 0.756 0.658 20,000 10,000 = = = = 30,240 19,740 4 5 × × × × 1 2 $10,000 20,000 3 4 5 30,000 40,000 50,000 0.572 0.497 Calculator solution: 0.870 × = 0.756 × = 0.658 × = 0.572 × = 0.497 × = Calculator solution: 11,440 4,970 $109,890 $109,856.33 $ 8,700 15,120 19,740 22,880 24,850 $ 91,290 $ 91,272.98 b. Cash flow stream A, with a present value of $109,890, is higher than cash flow stream B’s present value of $91,290 because the larger cash inflows occur in A in the early years when their present value is greater, while the smaller cash flows are received further in the future. P4-31. LG 1, 4: Value of a mixed stream Intermediate a. Chapter 4 Time Value of Money 93 b. Cash Flow Stream A Year CF × PVIF12%,n = Present Value 1 2 3–9 10 $30,000 25,000 15,000 10,000 × × × × 0.893 0.797 * 3.639 0.322 = = = = $ 26,790 19,925 54,585 3,220 $104,520 $104,508.28 Calculator solution: * c. The PVIF for this 7-year annuity is obtained by summing together the PVIFs of 12% for periods 3 through 9. This factor can also be calculated by taking the PVIFA12%,7 and multiplying by the PVIF12%,2. Alternatively, one could subtract PVIFA12%,2 from PVIFA12%,9. Harte should accept the series of payments offer. The present value of that mixed stream of payments is greater than the $100,000 immediate payment. P4-32. LG 5: Personal finance: Funding budget shortfalls Intermediate a. Year Budget Shortfall × 1 2 3 $5,000 4,000 6,000 × × × 4 5 10,000 3,000 × × PVIF8%,n = Present Value 0.926 0.857 0.794 = = = $ 4,630 3,428 4,764 0.735 0.681 = = 7,350 2,043 $22,215 $22,214.03 Calculator solution: A deposit of $22,215 would be needed to fund the shortfall for the pattern shown in the table. b. An increase in the earnings rate would reduce the amount calculated in part (a). The higher rate would lead to a larger interest being earned each year on the investment. The larger interest amounts will permit a decrease in the initial investment to obtain the same future value available for covering the shortfall. 94 Gitman • Principles of Managerial Finance, Twelfth Edition P4-33. LG 4: Relationship between future value and present value-mixed stream Intermediate a. Present Value CF × PVIF5%,n = Present Value 1 $800 × 0.952 = $ 761.60 2 900 × 0.907 = 816.30 3 1,000 × 0.864 = 864.00 4 1,500 × 0.822 = 1,233.00 5 2,000 × 0.784 = 1,568.00 Year Calculator solution: $5,242.90 $5,243.17 b. The maximum you should pay is $5,242.90. c. A higher 7% discount rate will cause the present value of the cash flow stream to be lower than $5,242.90. P4-34. LG 5: Changing compounding frequency Intermediate a. Compounding frequency: FVn = PV × FVIFi%/m,n × m (1) Annual 12%, 5 years FV5 = $5,000 × (1.762) FV5 = $8,810 Calculator solution: $8,811.71 Semiannual 12% ÷ 2 = 6%, 5 × 2 = 10 periods FV5 = $5,000 × (1.791) FV5 = $8,955 Calculator solution: $8,954.24 Quarterly 12% ÷ 4 = 3%, 5 × 4 = 20 periods FV5 = $5,000 (1.806) FV5 = $9,030 Calculator solution: $9,030.56 (2) Annual 16%, 6 years FV6 = $5,000 (2.436) FV6 = $12,180 Calculator solution: $12,181.98 Semiannual 16% ÷ 2 = 8%, 6 × 2 = 12 periods FV6 = $5,000 (2.518) FV6 = $12,590 Calculator solution: $12,590.85 Chapter 4 Time Value of Money 95 Quarterly 16% ÷ 4 = 4%, 6 × 4 = 24 periods FV6 = $5,000 (2.563) FV6 = $12,815 Calculator solution: $12,816.52 (3) Annual 20%, 10 years FV10 = $5,000 × (6.192) FV10 = $30,960 Calculator solution: $30,958.68 Semiannual 20% ÷ 2 = 10%, 10 × 2 = 20 periods FV10 = $5,000 × (6.727) FV10 = $33,635 Calculator solution: $33,637.50 Quarterly 20% ÷ 4 = 5%, 10 × 4 = 40 periods FV10 = $5,000 × (7.040) FV10 = $35,200 Calculator solution: $35,199.94 b. Effective interest rate: ieff = (1 + i/m)m – 1 (1) Annual ieff = (1 + 0.12/1)1 – 1 1 ieff = (1.12) – 1 ieff = (1.12) – 1 ieff = 0.12 = 12% Semiannual ieff = (1 + 12/2)2 – 1 2 ieff = (1.06) – 1 ieff = (1.124) – 1 ieff = 0.124 = 12.4% Quarterly ieff = (1 + 12/4)4 – 1 ieff = (1.03)4 – 1 ieff = (1.126) – 1 ieff = 0.126 = 12.6% (2) Annual 1 ieff = (1 + 0.16/1) – 1 1 ieff = (1.16) – 1 ieff = (1.16) – 1 ieff = 0.16 = 16% Quarterly 4 ieff = (1 + 0.16/4) – 1 4 ieff = (1.04) – 1 ieff = (1.170) – 1 ieff = 0.170 = 17% Semiannual ieff = (1 + 0.16/2)2 – 1 ieff = (1.08)2 – 1 ieff = (1.166) – 1 ieff = 0.166 = 16.6% 96 Gitman • Principles of Managerial Finance, Twelfth Edition (3) Annual ieff = (1 + 0.20/1)1 – 1 ieff = (1.20)1 – 1 ieff = (1.20) – 1 ieff = 0.20 = 20% Semiannual ieff = (1 + 0.20/2)2 – 1 ieff = (1.10)2 – 1 ieff = (1.210) – 1 ieff = 0.210 = 21% Quarterly 4 Ieff = (1 + 0.20/4) – 1 Ieff = (1.05)4 – 1 Ieff = (1.216) – 1 Ieff = 0.216 = 21.6% P4-35. LG 5: Compounding frequency, time value, and effective annual rates Intermediate a. Compounding frequency: FVn = PV × FVIFi%,n A FV5 = $2,500 × (FVIF3%,10) FV5 = $2,500 × (1.344) FV5 = $3,360 Calculator solution: $3,359.79 C FV10 = $1,000 × (FVIF5%,10) FV10 = $1,000 × (1.629) FV10 = $16,290 Calculator solution: $1,628.89 b. Effective interest rate: ieff = (1 + i%/m)m – 1 A ieff = (1 + 0.06/2)2 – 1 ieff f = (1 + 0.03)2 – 1 ieff = (1.061) – 1 ieff = 0.061 = 06.1% C ieff = (1 + 0.05/1)1 – 1 ieff = (1 + 0.05)1 – 1 ieff = (1.05) – 1 ieff = 0.05 = 5% B FV3 = $50,000 × (FVIF2%,18) FV3 = $50,000 × (1.428) FV3 = $71,400 Calculator solution: $71,412.31 D FV6 = $20,000 × (FVIF4%,24) FV6 = $20,000 × (2.563) FV6 = $51,260 Calculator solution: $51,266.08 B ieff = (1 + 0.12/6)6 – 1 ieff = (1 + 0.02)6 – 1 ieff = (1.126) – 1 ieff = 0.126 = 12.6% D ieff = (1 + 0.16/4) – 1 ieff = (1 + 0.04)4 – 1 ieff = (1.170) – 1 ieff = 0.17 = 17% 4 c. The effective rates of interest rise relative to the stated nominal rate with increasing compounding frequency. Chapter 4 Time Value of Money 97 P4-36. LG 5: Continuous compounding: FVcont. = PV × ex (e = 2.7183) Intermediate A B C D FVcont. = $1,000 × e0.18 = $1,197.22 FVcont. = $ 600 × e1 = $1,630.97 0.56 FVcont. = $4,000 × e = $7,002.69 0.48 FVcont. = $2,500 × e = $4,040.19 Note: If calculator doesn’t have ex key, use yx key, substituting 2.7183 for y. P4-37. LG 5: Personal finance: Compounding frequency and time value Challenge a. (1) FV10 = $2,000 × (FVIF8%,10) FV10 = $2,000 × (2.159) FV10 = $4,318 Calculator solution: $4,317.85 (2) FV10 = $2,000 × (FVIF4%,20) FV10 = $2,000 × (2.191) FV10 = $4,382 Calculator solution: $4,382.25 (3) FV10 = $2,000 × (FVIF0.022%,3650) FV10 = $2,000 × (2.232) FV10 = $4,464 Calculator solution: $4,450.69 (4) FV10 = $2,000 × (e0.8) FV10 = $2,000 × (2.226) FV10 = $4,452 Calculator solution: $4,451.08 b. (1) ieff = (1 + 0.08/1)1 – 1 ieff = (1 + 0.08)1 – 1 ieff = (1.08) – 1 ieff = 0.08 = 8% (2) ieff = (1 + 0.08/2)2 – 1 ieff = (1 + 0.04)2 – 1 ieff = (1.082) – 1 ieff = 0.082 = 8.2% 365 (4) ieff = (ek– 1) ieff = (e0.08– 1) ieff = (1.0833 – 1) ieff = 0.0833 = 8.33% (3) ieff = (1 + 0.08/365) – 1 ieff = (1 + 0.00022)365 – 1 ieff = (1.0833) – 1 ieff = 0.0833 = 8.33% c. Compounding continuously will result in $134 more dollars at the end of the 10 year period than compounding annually. d. The more frequent the compounding the larger the future value. This result is shown in part a by the fact that the future value becomes larger as the compounding period moves from annually to continuously. Since the future value is larger for a given fixed amount invested, the effective return also increases directly with the frequency of compounding. In part b we see this fact as the effective rate moved from 8% to 8.33% as compounding frequency moved from annually to continuously. 98 Gitman • Principles of Managerial Finance, Twelfth Edition P4-38. LG 5: Personal finance: Comparing compounding periods Challenge a. FVn = PV × FVIFi%,n (1) Annually: FV = PV × FVIF12%,2 = $15,000 × (1.254) = $18,810 Calculator solution: $18,816 (2) Quarterly: FV = PV × FVIF3%,8 = $15,000 × (1.267) = $19,005 Calculator solution: $19,001.55 (3) Monthly: FV = PV × FVIF1%,24 = $15,000 × (1.270) = $19,050 Calculator solution: $19,046.02 (4) Continuously: FVcont. = PV × ex t 0.24 FV = PV × 2.7183 = $15,000 × 1.27125 = $19,068.77 Calculator solution: $19,068.74 b. The future value of the deposit increases from $18,810 with annual compounding to $19,068.77 with continuous compounding, demonstrating that future value increases as compounding frequency increases. c. The maximum future value for this deposit is $19,068.77, resulting from continuous compounding, which assumes compounding at every possible interval. P4-39. LG 3, 5: Personal finance: Annuities and compounding: FVAn = PMT × (FVIFAi%,n) Intermediate a. (1) Annual (2) Semiannual FVA10 = $150 × (FVIFA4%,20) FVA10 = $300 × (FVIFA8%,10) FVA10 = $300 × (14.487) FVA10 = $150 × (29.778) FVA10 = $4,346.10 FVA10 = $4,466.70 Calculator solution: $4,345.97 Calculator solution: $4,466.71 (3) Quarterly FVA10 = $75 ×.(FVIFA2%,40) FVA10 = $75 × (60.402) FVA10 = $4,530.15 Calculator solution: $4,530.15 b. The sooner a deposit is made the sooner the funds will be available to earn interest and contribute to compounding. Thus, the sooner the deposit and the more frequent the compounding, the larger the future sum will be. Chapter 4 P4-40. LG 6: Deposits to accumulate growing future sum: PMT = Time Value of Money FVAn FVIFA i %,n Basic Case Terms Calculation Payment A 12%, 3 yrs. $1,481.92 PMT = $5,000 ÷ 3.374 = Calculator solution: $1,481.74 B 7%, 20 yrs. $2,439.32 PMT = $100,000 ÷ 40.995 = Calculator solution: $2,439.29 C 10%, 8 yrs. $2,623.29 PMT = $30,000 ÷ 11.436 = Calculator solution: $2,623.32 D 8%, 12 yrs. $ 790.43 PMT = $15,000 ÷ 18.977 = Calculator solution: $ 790.43 P4-41. LG 6: Personal finance: Creating a retirement fund Intermediate a. PMT = FVA42 ÷ (FVIFA8%,42) PMT = $220,000 ÷ (304.244) PMT = $723.10 Calculator solution: $723.10 b. FVA42 = PMT × (FVIFA8%,42) FVA42 = $600 × (304.244) FVA42 = $182,546.40 Calculator solution: $182,546.11 P4-42. LG 6: Personal finance: Accumulating a growing future sum Intermediate FVn = PV × (FVIFi%,n) FV20 = $185,000 × (FVIF6%,20) FV20 = $185,000 × (3.207) FV20 = $593,295 = Future value of retirement home in 20 years. Calculator solution: $593,320.06 PMT = FV ÷ (FVIFAi%,n) PMT = $593,295 ÷ (FVIFA10%,20) PMT = $593,295 ÷ (57.274) PMT = $10,358.89 Calculator solution: $10,359.15 = annual payment required. 99 100 Gitman • Principles of Managerial Finance, Twelfth Edition P4-43. LG 3, 6: Personal finance: Deposits to create a perpetuity Intermediate a. Present value of a perpetuity = PMT × (1 ÷ i) = $6,000 × (1 ÷ 0.10) = $6,000 × 10 = $60,000 b. PMT = FVA ÷ (FVIFA10%,10) PMT = $60,000 ÷ (15.937) PMT = $3,764.82 Calculator solution: $3,764.72 P4-44. LG 2, 3, 6: Personal finance: Inflation, time value, and annual deposits Challenge a. FVn = PV × (FVIFi%,n) FV20 = $200,000 × (FVIF5%,25) FV20 = $200,000 × (3.386) FV20 = $677,200 = Future value of retirement home in 25 years. Calculator solution: $677,270.99 b. PMT = FV ÷ (FVIFAi%,n) PMT = $677,270.99 ÷ (FVIFA9%,25) PMT = $677,270.99 ÷ (84.699) PMT = $7,995.37 Calculator solution: $7,996.03 = annual payment required. c. Since John will have an additional year on which to earn interest at the end of the 25 years his annuity deposit will be smaller each year. To determine the annuity amount John will first discount back the $677,200 one period. PV 24 = $677,200 × 0.9174 = $621,263.28 This is the amount John must accumulate over the 25 years. John can solve for his annuity amount using the same calculation as in part b. PMT = FV ÷ (FVIFAi%,n) PMT = $621,263.28 ÷ (FVIFA9%,25) PMT = $621,263.28 ÷ (84.699) PMT = $7,334.95 Calculator solution: $7,335.81 = annual payment required. To check this value, multiply the annual payment by 1 plus the 9% discount rate. $7,335.81 (1.09) = $7996.03 Chapter 4 P4-45. LG 6: Loan payment: PMT = Time Value of Money 101 PVA PVIFAi %, n Basic Loan A C PMT = $12,000 ÷ (PVIFA8%,3) PMT = $12,000 ÷ 2.577 PMT = $4,656.58 Calculator solution: $4,656.40 B PMT = $60,000 ÷ (PVIFA12%,10) PMT = $60,000 ÷ 5.650 PMT = $10,619.47 Calculator solution: $10,619.05 PMT = $75,000 ÷ (PVIFA10%,30) PMT = $75,000 ÷ 9.427 PMT = $7,955.87 Calculator solution: $7,955.94 D PMT = $4,000 ÷ (PVIFA15%,5) PMT = $4,000 ÷ 3.352 PMT = $1,193.32 Calculator solution: $1,193.26 P4-46. LG 6: Personal finance: Loan amortization schedule Intermediate a. PMT = $15,000 ÷ (PVIFA14%,3) PMT = $15,000 ÷ 2.322 PMT = $6,459.95 Calculator solution: $6,460.97 b. End of Year Loan Payment Beginning of Year Principal 1 2 3 $6,459.95 6,459.95 6,459.95 $15,000.00 10,640.05 5,669.71 Payments Interest Principal $2,100.00 1,489.61 793.76 $4,359.95 4,970.34 5,666.19 End of Year Principal $10,640.05 5,669.71 0 (The difference in the last year’s beginning and ending principal is due to rounding.) c. Through annual end-of-the-year payments, the principal balance of the loan is declining, causing less interest to be accrued on the balance. P4-47. LG 6: Loan interest deductions Challenge a. PMT = $10,000 ÷ (PVIFA13%,3) PMT = $10,000 ÷ (2.361) PMT = $4,235.49 Calculator solution: $4,235.22 102 Gitman • Principles of Managerial Finance, Twelfth Edition b. End of Year 1 2 3 Loan Payment $4,235.49 4,235.49 4,235.49 Beginning of Year Principal $10,000.00 7,064.51 3,747.41 Payments Interest Principal End of Year Principal $1,300.00 $2,935.49 918.39 3,317.10 487.16 3,748.33 $7,064.51 3,747.41 0 (The difference in the last year’s beginning and ending principal is due to rounding.) P4-48. LG 6: Personal finance: Monthly loan payments Challenge a. PMT = $4,000 ÷ (PVIFA1%,24) PMT = $4,000 ÷ (21.243) PMT = $188.28 Calculator solution: $188.29 b. PMT = $4,000 ÷ (PVIFA0.75%,24) PMT = $4,000 ÷ (21.889) PMT = $182.74 Calculator solution: $182.74 P4-49. LG 6: Growth rates Basic a. PV = FVn × PVIFi%,n Case A PV = FV4 × PVIFk%,4yrs. $500 = $800 × PVIFk%,4yrs 0.625 = PVIFk%,4yrs 12% < k < 13% Calculator solution: 12.47% C B PV = FV9 × PVIFi%,9yrs. $1,500 = $2,280 × PVIFk%,9yrs. 0.658 = PVIFk%,9yrs. 4% < k < 5% Calculator solution: 4.76% PV = FV6 × PVIFi%,6 $2,500 = $2,900 × PVIFk%,6 yrs. 0.862 = PVIFk%,6yrs. 2% < k < 3% Calculator solution: 2.50% b. Case A Same as in a B Same as in a C Same as in a c. The growth rate and the interest rate should be equal, since they represent the same thing. Chapter 4 Time Value of Money 103 P4-50. LG 6: Personal finance: Rate of return: PVn = FVn × (PVIFi%,n) Intermediate a. PV = $2,000 × (PVIFi%,3yrs.) $1,500 = $2,000 × (PVIFi%,3 yrs.) 0.75 = PVIFi%,3 yrs. 10% < i < 11% Calculator solution: 10.06% b. Mr. Singh should accept the investment that will return $2,000 because it has a higher return for the same amount of risk. P4-51. LG 6: Personal finance: Rate of return and investment choice Intermediate a. A PV = $8,400 × (PVIFi%,6yrs.) $5,000 = $8,400 × (PVIFi%,6 yrs.) 0.595 = PVIFi%,6 yrs. 9% < i < 10% Calculator solution: 9.03% B PV = $15,900 × (PVIFi%,15yrs.) $5,000 = $15,900 × (PVIFi%,15yrs.) 0.314 = PVIFi%,15yrs. 8% < i < 9% Calculator solution: 8.02% C PV = $7,600 × (PVIFi%,4yrs.) $5,000 = $7,600 × (PVIFi%,4 yrs.) 0.658 = PVIFi%,4 yrs. 11% < i < 12% Calculator solution: 11.04% D PV = $13,000 × (PVIFi%,10 yrs.) $5,000 = $13,000 × (PVIFi%,10 yrs.) 0.385 = PVIFi%,10 yrs.. 10% < i < 11% Calculator solution: 10.03% b. Investment C provides the highest return of the four alternatives. Assuming equal risk for the alternatives, Clare should choose C. P4-52. LG 6: Rate of return-annuity: PVAn = PMT × (PVIFAi%,n) Basic $10,606 = $2,000 × (PVIFAi%,10 yrs.) 5.303 = PVIFAi%,10 yrs. 13% < i < 14% Calculator solution: 13.58% P4-53. LG 6: Personal finance: Choosing the best annuity: PVAn = PMT × (PVIFAi%,n) Intermediate a. Annuity A $30,000 = $3,100 × (PVIFAi%,20 yrs.) 9.677 = PVIFAi%,20 yrs. 8% < i < 9% Calculator solution: 8.19% Annuity B $25,000 = $3,900 × (PVIFAi%,10 yrs.) 6.410 = PVIFAi%,10 yrs. 9% < i < 10% Calculator solution: 9.03% 104 Gitman • Principles of Managerial Finance, Twelfth Edition Annuity C $40,000 = $4,200 × (PVIFAi%,15 yrs.) 9.524 = PVFAi%,15 yrs. 6% < i< 7% Calculator solution: 6.3% Annuity D $35,000 = $4,000 × (PVIFAi%,12 yrs.) 8.75 = PVIFAi%,12 yrs. 5% < i < 6% Calculator solution: 5.23% b. Annuity B gives the highest rate of return at 9% and would be the one selected based upon Raina’s criteria. P4-54. LG 6: Personal finance: Interest rate for an annuity Challenge a. Defendants interest rate assumption $2,000,000 = $156,000 × (PVIFAi%,25 yrs.) 12.821 = PVFAi%,25 yrs. 5% < i < 6% Calculator solution: 5.97% b. Prosecution interest rate assumption $2,000,000 = $255,000 × (PVIFAi%,25 yrs.) 7.843 = PVFAi%,25 yrs. i = 12% Calculator solution: 12.0% c. $2,000,000 = PMT × (PVIFA9%,25yrs.) $2,000,000 = PMT (9.823) PMT = $203,603.79 Calculator solution: $203,612.50 P4-55. LG 6: Personal finance: Loan rates of interest: PVAn = PMT × (PVIFAi%,n) Intermediate a. Loan A $5,000 = $1,352.81 × (PVIFAi%,5 yrs.) 3.696 = PVIFAi%,5 yrs. i = 11% Loan C $5,000 = $2,010.45 × (PVIFAi%,3 yrs.) 2.487 = PVIFAk%,3 yrs. i = 10% Loan B $5,000 = $1,543.21 × (PVIFAi%,4 yrs.) 3.24 = PVIFAi%, 4 yrs. i = 9% Calculator solutions are identical. b. Mr. Fleming should choose Loan B, which has the lowest interest rate. Chapter 4 Time Value of Money 105 P4-56. LG 6: Number of years to equal future amount Intermediate A FV = PV × (FVIF7%,n yrs.) $1,000 = $300 × (FVIF7%,n yrs.) 3.333 = FVIF7%,n yrs. 17 < n < 18 Calculator solution: 17.79 years B FV = $12,000 × (FVIF5%,n yrs.) $15,000 = $12,000 × (FVIF5%,n yrs.) 1.250 = FVIF5%,n yrs. 4<n<5 Calculator solution: 4.573 years C FV = PV × (FVIF10%,n yrs.) $20,000 = $9,000 × (FVIF10%,n yrs.) 2.222 = FVIF10%,n yrs. 8<n<9 Calculator solution: 8.38 years D FV = $100 × (FVIF9%,n yrs.) $500 = $100 × (FVIF9%,n yrs.) 5.00 = FVIF9%,n yrs. 18 < n < 19 Calculator solution: 18.68 years E FV = PV × (FVIF15%,n yrs.) $30,000 = $7,500 × (FVIF15%,n yrs.) 4.000 = FVIF15%,n yrs. 9 < n < 10 Calculator solution: 9.92 years P4-57. LG 6: Personal finance: Time to accumulate a given sum Intermediate a. 20,000 = $10,000 × (FVIF10%,n yrs.) 2.000 = FVIF10%,n yrs. 7<n<8 Calculator solution: 7.27 years c. 20,000 = $10,000 × (FVIF12%,n yrs.) 2.000 = FVIF12%,n yrs. 6<n<7 Calculator solution: 6.12 years b. 20,000 = $10,000 × (FVIF7%,n yrs.) 2.000 = FVIF7%,n yrs. 10 < n < 11 Calculator solution: 10.24 years d. The higher the rate of interest the less time is required to accumulate a given future sum. P4-58. LG 6: Number of years to provide a given return Intermediate A PVA = PMT × (PVIFA11%,n yrs.) $1,000 = $250 × (PVIFA11%,n yrs.) 4.000 = PVIFA11%,n yrs. 5<n<6 Calculator solution: 5.56 years B PVA = PMT × (PVIFA15%,n yrs.) $150,000 = $30,000 × (PVIFA15%,n yrs.) 5.000 = PVIFA15%,n yrs. 9 < n < 10 Calculator solution: 9.92 years 106 Gitman • Principles of Managerial Finance, Twelfth Edition C PVA = PMT × (PVIFA10%,n yrs.) $80,000 = $10,000 × (PVIFA10%,n yrs.) 8 = PVIFA10%,n yrs. 16 < n < 17 Calculator solution: 16.89 years E PVA = PMT × (PVIFA6%,n yrs.) $17,000 = $3,500 × (PVIFA6%,n yrs.) 4.857 = PVIFA6%,n yrs. 5<n<6 Calculator solution: 5.91 years D PVA = PMT × (PVIFA9%,n yrs.) $600 = $275 × (PVIFA9%,n yrs.) 2.182 = PVIFA9%,n yrs. 2<n<3 Calculator solution: 2.54 years P4-59. LG 6: Personal finance: Time to repay installment loan Intermediate a. $14,000 = $2,450 × (PVIFA12%,n yrs.) 5.714 = PVIFA12%,n yrs. 10 < n < 11 Calculator solution: 10.21 years b. $14,000 = $2,450 × (PVIFA9%,n yrs.) 5.714 = PVIFA9%,n yrs. 8<n<9 Calculator solution: 8.38 years c. $14,000 = $2,450 × (PVIFA15%,n yrs.) 5.714 = PVIFA15%,n yrs. 13 < n < 14 Calculator solution: 13.92 years d. The higher the interest rate the greater the number of time periods needed to repay the loan fully. P4-60. Ethics problem Intermediate This is a tough issue. Even back in the Middle Ages, scholars debated the idea of a “just price.” The ethical debate hinges on (1) the basis for usury laws, (2) whether full disclosure is made of the true cost of the advance, and (3) whether customers understand the disclosures. Usury laws are premised on the notion that there is such a thing as an interest rate (price of credit) that is “too high.” A centuries-old fairness notion guides us into not taking advantage of someone in duress or facing an emergency situation. One must ask, too, why there are not market-supplied credit sources for borrowers, which would charge lower interest rates and receive an acceptable riskadjusted return. On issues #2 and #3, there is no assurance that borrowers comprehend or are given adequate disclosures. See the box for the key ethics issues on which to refocus attention (some would view the objection cited as a smokescreen to take our attention off the true ethical issues in this credit offer). Chapter 4 ̈ Time Value of Money 107 Case Finding Jill Moran’s Retirement Annuity Chapter 4’s case challenges the student to apply present value and future value techniques to a real-world situation. The first step in solving this case is to determine the total amount Sunrise Industries needs to accumulate until Ms. Moran retires, remembering to take into account the interest that will be earned during the 20-year payout period. Once that is calculated, the annual amount to be deposited can be determined. 1. 2. Total amount to accumulate by end of year 12 PVn = PMT × (PVIFAi%,n) PV20 = $42,000 × (PVIFA12%,20) PV20 = $42,000 × 7.469 PV20 = $313,698 Calculator solution: $313,716.63 3. End-of-year deposits, 9% interest: PMT = FVAn FVIFAi %, n PMT = $313,698 ÷ (FVIFA9%, 12 yrs.) PMT = $313,698 ÷ 20.141 PMT = $15,575.10 Calculator solution: $15,576.23 Sunrise Industries must make a $15,575.10 annual end-of-year deposit in years 1-12 in order to provide Ms. Moran a retirement annuity of $42,000 per year in years 13 to 32. 4. End-of-year deposits, 10% interest PMT = $313,698 ÷ (FVIFA10%,12 yrs.) PMT = $313,698 ÷ 21.384 PMT = $14,669.75 Calculator solution: $14,669.56 The corporation must make a $14,669.75 annual end-of-year deposit in years 1–12 in order to provide Ms. Moran a retirement annuity of $42,000 per year in years 13 to 32. 108 Gitman • Principles of Managerial Finance, Twelfth Edition 5. Initial deposit if annuity is a perpetuity and initial deposit earns 9%: PVperp = PMT × (1 ÷ i) PVperp = $42,000 × (1 ÷ 0.12) PVperp = $42,000 × 8.333 PVperp = $349,986 Calculator solution: $350,000 End-of-year deposit: PMT = FVAn ÷ (FVIFAi%,n) PMT = $349,986 ÷ (FVIFA9%,12 yrs.) PMT = $349,986 ÷ 20.141 PMT = $17,376.79 Calculator solution: $17,377.73 ̈ Spreadsheet Exercise The answer to Chapter 4’s Uma Corporation spreadsheet problem is located in the Instructor’s Resource Center at www.prenhall.com/irc. ̈ Group Exercises This set of deliverables concern each group’s fictitious firm. The first scenario involves the replacement of a copy machine. The first decision pertains to a choice between competing leases, while the second is choosing among purchase plans to buy the machine outright. In the first case leasing information is provided, while for the second option students are asked to get pricing information. This information is readily available on the Web, as is the needed information regarding interest rates for both the possible savings plans regarding the copy machine, and the computer upgrade scenario. For the savings plan the groups are asked to look at several deposit options while for the computer upgrade purchase an amortization schedule must be developed. Modifications or even elimination of one of these scenarios is perfectly allowable and shouldn’t affect future work. The same can be said of the final deliverable involving a simple calculation of the present value of a court-ordered settlement of a patentinfringement case. ̈ A Note on Web Exercises A series of chapter-relevant assignments requiring Internet access can be found at the book’s Companion Website at http://www.prenhall.com/gitman. In the course of completing the assignments students access information about a firm, its industry, and the macro economy, and conduct analyses consistent with those found in each respective chapter.