CH-8-9 Sources of Short-Term Financing – Foundations of Financial Management : Book Guide

This Document Contains Chapters 8 to 9 Chapter 8 Discussion Questions 8-1. It is advisable to borrow in order to take a cash discount when the cost of borrowing is less than the cost of forgoing the discount. If it cost us 36 percent to miss a discount, we would be much better off finding an alternate source of funds for 8 to 10 percent. 8-2. Larger firms tend to be in a net creditor position because they have the financial resources to be suppliers to credit. The smaller firm must look to the larger manufacturer or wholesaler to help carry the firm's financing requirements. 8-3. The prime rate is the rate that a bank charges its most creditworthy customers. The average customer can expect to pay one or two percent (or more) above prime. In competitive markets banks may actually charge preferred customers less than prime. 8-4. The use of a compensating balance or minimum required account balance allows the banker to generate a higher return on a loan because not all funds are actually made available to the borrower. A $125,000 loan with a $25,000 compensating balance requirement means only $100,000 is being provided on a net basis. This benefit to the lender need not be a disadvantage to the borrower. The borrower may, in turn, receive a lower quoted interest rate and certain gratuitous services because of the compensating balance requirement. Bankers have tended towards eliminating both compensating balances and gratuitous services. 8-5. The stated interest rate is the percentage rate unadjusted for time or method of repayment. The annual interest rate is the true rate and considers all these variables. A 5 percent stated rate for 90 days provides a 20 percent annual rate. The financial manager should recognize the annual rate as the true cost of borrowing. An effective rate would include any compounding effects over the relevant period. 8-6. Commercial paper can be either purchased or issued by a corporation. To the extent one corporation purchases another corporation's commercial paper as a short-term investment, it is a current asset. Conversely, if a corporation issues its own commercial paper, it is a current liability. 8-7. In comparison to bank borrowing, commercial paper can generally be issued at below the prime rate. Furthermore, there are no compensating balance requirements, though the firm is required to maintain approved credit lines at a bank. Finally, there is a certain degree of prestige associated with the issuance of commercial paper. 8-8. A bankers’ acceptance offers the guarantee of payment from a chartered bank. 8-9. Major types of collateralized short-term loans include: a. Pledging accounts receivable: borrowing with receivables as collateral. b. Factoring account receivables: selling accounts at a discount to a finance company. c. Borrowing with inventory as collateral through (1) Blanket inventory lien-general claim against inventory or collateral. No specific items are marked or designated. (2) Trust receipt-borrower holds the inventory in trust for the lender. Each item is marked and has a serial number. When the inventory is sold, the trust receipt is canceled and the funds go into the lender's account. (3) Warehousing-the inventory is physically identified, segregated, and stored under the direction of an independent warehouse company that controls the movement of the goods. If done on the premises of the warehousing firm, it is termed public warehousing. An alternate arrangement is field warehousing whereby the same procedures are conducted on the borrower's property. 8-10. A public offering backed by an asset (accounts receivable) as collateral. Essentially a firm sells its receivables into the securities markets. 8-11. Hedging means to engage in a transaction that partially or fully reduces a prior risk exposure. In selling a financial futures contract, if interest rates go up, one is able to buy back the contract at a profit. This will help to offset the higher interest charges to a corporation or other business entity. Hedging involves the matching of maturities of assets and liabilities to reduce risk. 8-12. Pledging receivables entails providing a lender with the ability to take over control of the receivables, if appropriate, in exchange for the lender providing a loan. In effect the receivables back up the loan. Factoring involves selling receivables to a lender or factor in exchange for immediate funds. When pledged the receivables stay on a firm’s balance sheet along with a newly created loan and cash, whereas in factoring the receivables are replaced on the firm’s balance sheet as cash, without the loan entry. Internet Resources and Questions 1. http://www.m-x.ca/produits_taux_int_bax_en.php 2. http://www.globeinvestor.com/v5/data/rates/ http://www.financialpost.com/personal-finance/rates/loans-personal.html Problems 8-1. Cost of forgoing the cash discount 365 d% × K DIS = 100% − d% f (date) - d (date )) a. K DIS = 2% 365 × = .2483 = 24.83% 100% − 2% f (40 ) − d (10 ) b. K DIS = 2% 365 × = .4966 = 49.66% 100% − 2% f (30) − d (15) c. K DIS = 2% 365 × = .2128 = 21.28% 100% − 2% f (45) − d (10 ) d. K DIS = 3% 365 × = .0664 = 6.64% 100% − 3% f (180 ) − d (10 ) 8-2. Startus Ltd. a. Accounts payable forgoing discount: Annual purchases/365 × Final due date = $7,360,000/ 365 × 60 = $1,209,863 Accounts payable taking discount: Annual purchases/365 × Discount period = $7,360,000/ 365 × 10 = Additional financing available = 201,644 $1,008,219 b. Cost of forgoing the cash discount 2% 365 K DIS = × = .1490 = 14.90% 100% − 2% f (60 ) − d (10 ) 8-3. Hobbit Shoes a. COGS = Average inventory × turnover rate = $510,000 × 5 = $2,550,000 Average accounts payable = COGS/ 365 × average payment period = $2,550,000/ 365 × 40 = $279,452 b. Annual sales 8-4. = Average A/R × 365/Average collection period = $315,000 × 365/35 = $3,285,000 Hugh Warner K DIS = 3% 365 × = .1881 = 18.81% 100% − 3% f (70 ) − d (10 ) In this problem, Mr. Warner has the use of funds for only 60 extra days (70 – 10), instead of 70 extra days (80 – 10). Mr. Warner suppliers are offering terms of 3/10, net 80. Mr. Warner is actually accepting terms of 3/10, net 70. 8-5. Little Kimi Clothiers K DIS = 2% 365 × = .0993 = 9.93% 100% − 2% f (90 ) − d (15) Little Kimi should accept the bank’s terms and borrow at 6% to take the cash discount. The cost of forgoing the discount is 9.93%. 8-6. K DIS Chris Angle 1% 365 = × = .0737 = 7.37% 100% − 1% f (60 ) − d(10 ) Chris should not accept the bank’s terms and borrow at 8%. He should take the cash discount. The cost of forgoing the discount is 7.37%. 8-7. Treasury Bills 100 − P 365 100 − 98.671 365 × =r= × = 0.0502 = 5.02% P d 98.671 98 8-8. Treasury Bills 100 − P 365 100 − 97.171 365 × =r= × = 0.0584 = 5.84% P d 97.171 182 8-9. Cy Burr Spacers Company a. Accounts receivable = Average daily credit sales × average collection period = $7,000 × 28 = $196,000 Accounts payable = Average daily credit purchases × average payment period = $6,000 × 20 = $120,000 Net credit position = Accounts receivable – Account payable Net credit position = $196,000 – $120,000 = $76,000 b. Accounts receivable will remain at Accounts payable = $6,000 × 35 Net credit position $196,000 210,000 ($14,000) The firm has improved its cash position and cash flow. Instead of extending $76,000 more in credit (funds) than it is receiving, it has reversed the position and is the net recipient of $14,000 in credit. 8-10. Sampson Orange Juice Company a. Accounts receivable = Average daily credit sales × average collection period = $9,000 × 34 = $306,000 Accounts payable = Average daily credit purchases × average payment period = $7,500 × 30 = $225,000 Net credit position = Accounts receivable – Account payable Net credit position = $306,000 – $225,000 = $81,000 b. Accounts receivable will remain at Accounts payable = $7,500 × 45 Net credit position $306,000 337,500 ($31,500) The firm has improved its cash position and cash flow. Instead of extending $81,000 more in credit (funds) than it is receiving, it has reversed the position and is the net recipient of $31,500 in credit. 8-11. Your Bank R ANNUAL = $25 365 I 365 × = × = 0.1014 = 10.14% P d $2,000 45 $25   rEFF = 1 +  $ 2 , 000   365 45 8-12. − 1 = 0.1060 = 10.60% Your (Other) Bank R ANNUAL = I 365 $45 365 × = × = 0.1095 = 10.95% P d $3,000 50 365 rEFF $45  50  = 1 +  − 1 = 0.1148 = 11.48% $ 3 , 000   8-13. Dr. Painkiller One year’s interest = $3,000 ×0.08 = $240 R DIS = 8-14. I 365 $240 365 × = × = 0.0870 = 8.70% P-I d $3,000 − $240 365 Marty Not R DIS = I 365 $215 365 × = × = 0.0840 = 8.40% P-I d $8,000 − $215 120 8-15. Mo and Chris’s Sporting Goods Company $ cost of loan = Amount borrowed × interest rate × = $14,500 × 0.12 × days loan outstanding 365 20 = $95.34 365 8-16. Dr. Budget At prime: At LIBOR + commitment fee: $25,000 × .04 = $1,000 $25,000 .03 = $750 + $125 = $875 LIBOR interest is: LIBOR loan is cheaper. 8-17. Birthdaybook Amount to be borrowed = a. = Amount needed (1 − c) $125,000 $125,000 = = $138,889 (1 − .10) .90 b. Interest to be paid: $138,888.89 × .07 = $9722.22 I 365 $9,722.22 365 × = × P−B d $125,000 365 = 0.0778 = 7.78% R COMP = 8-18. Beasley Furniture Company Annual rate of interest with 20% compensating balance: RCOMP = 365 $27,000 365 I × = × = 0.1125 = 11.25% P−B d $300,000 − $60,000 365 RCOMP OR: I 9% = = = 0.1125 = 11.25% (1 − c) (1 − .20 ) Installment loan with compensating balance: RINSTALL = 2 × Annual number of payments × I (Total number of payments + 1) × P 2 × 12 × $27,000 $648,000 = (12 + 1) × ($300,000 − $60,000 ) (13) × $240,000 $648,000 = $3,120,000 = 0.2077 = 20.77% RINSTALL = 8-19. Randall Corporation Annual rate of interest: I 365 $16,000 365 × = × P−B d $200,000 − $30,000 * 365 = 0.0941 = 9.41% RCOMP = *Compensating balance = 20% × $200,000 = Normal funds = Restricted compensating balance = 8-20. $40,000 10,000 $30,000 Brandon Blue Sox Annual rate of interest: 365 $986 * 365 I × = × RCOMP = $20,000 − $1,500 * * 180 P−B d = 0.1081 = 10.81% * (10% × $20,000) × 180/365 = **Compensating balance = 15% × $20,000 = Normal funds = Restricted compensating balance = 8-21. $986 $3,000 1,500 $1,500 Telco Casino Corp. Annual rate of interest with 2% administration fee (paid up front). Interest = $300,000 × 0.035 = $10,500. = $10,500/ ($300,000 – $6,000) = $10,500/ $294,000 = 0.0357 = 3.57% 8-22. Your Company a. simple interest with a 10% compensating balance: I 365 $400,000 * 365 × = × P−B d $5,000,000 − $500,000 * * 365 = 0.0889 = 8.89% RCOMP = * $5,000,000 × 0.08 = ** $5,000,000 × 0.10 = $400,000 $500,000 b. discounted interest: 365 $400,000 365 I × = × $5,000,000 − $400,000 365 P-I d = 0.0870 = 8.70% R DIS = c. an installment loan with 12 payments: 2 × 12 × $400,000 $9,600,000 = (12 + 1) × $5,000,000 (13) × $5,000,000 $9,600,000 = $65,000,000 = 0.1477 = 14.77% RINSTALL = d. discounted interest with a 1% administrative fee: $400,000/ ($5,000,000 – $400,000 – $50,000) = $400,000/ $4,550,000 = 0.0879 = 8.79% 8-23. Your borrowing a. $500/ 4,000 = 12.5% Use formula 8–6 for b, c, and d. b. R INSTALL = 2 × 2 × $500 $2,000 = = 0.1667 = 16.67% (3) × $4,000 $12,000 c. RINSTALL = 2 × 4 × $500 $4,000 = = 0.2000 = 20.00% (5) × $4,000 $20,000 d. R INSTALL = 2 × 12 × $500 $12,000 = = 0.2308 = 23.08% (13) × $4,000 $52,000 8-24. Circuit Cycle Company RINSTALL = R INSTALL = 2 × Annual number of payments × I (Total number of payments + 1) × P 2 × 1 2 × $12,000 $288,000 = = 0.1730 = 17.30% (36 + 1) × $45,000 $1,665,000 8-25. Millbanke Waters Annualized yield on discounted paper: 100 − P 365 100 − 97.912 365 × =r= × = 0.0938 = 9.38% P d 97.912 83 Effective annual yield = 9.72%  100 − 97.912  rEFF = 1 +  97.912   8-26. 365 83 − 1 = 0.0972 = 9.72% Bankers’ Acceptance 100 − P 365 100,000 − 97,915 365 × =r= × = 0.0864 = 8.64% P d 97,915 90 Effective annual yield = 8.92% FV = 100,000 PV = 97,915 (neg.) CPT % I/Y =? 8-27. PMT = 0 N = 90/ 365 K. Kompany a. Cost of forgoing the cash discount: 365 d% × K DIS = 100% − d% f (date) - d (date )) K DIS = 2.5% 365 × = .2080 = 20.80% 100% − 2.5% f (60 ) − d (15) Effective annual yield = 22.80% Choose the bank at 18%. b. No security requirements More convenient and more readily available No life story required 8-28. Reynolds Corporation Cost of forgoing the cash discount: 365 d% × K DIS = 100% − d% f (date) - d (date )) K DIS = 2% 365 × = .1655 = 16.55% 100% − 2% f (55) − d (10 ) We use 55 days instead of 40 days as the final due date because Reynolds’ suppliers have effectively made this the due date even though the stated due date is 40 days. Annual rate of interest; 20% compensating balance requirement: RCOMP = I 14% = = 0.1750 = 17.50% (1 − c) (1 − 0.20 ) The annual cost of the loan, 17.5%, is more than the cost of passing up the discount, 16.55%. Reynolds Corporation should continue to pay in 55 days and pass up the discount. 8-29. Reynolds Corporation (Continued) Annual rate of interest; 10% compensating balance requirement: I 14% RCOMP = = = 0.1556 = 15.56% (1 − c) (1 − 0.10 ) The answer now changes. The annual cost of the loan, 15.56%, is less than the cost of passing up the discount. Reynolds Corporation should borrow the funds and take the discount. 8-30. Burt’s Department Store a. Annual rate on bank loan: $8,100 365 I 365 R ANNUAL = × = × = 0.1643 = 16.43% P d $300,000 60 b. Cost of forgoing cash discount 3% 365 K DIS = × = .1881 = 18.81% 100% − 3% f (70) − d (10) c. Yes, because the cost of borrowing is less than the cost of losing the discount. Amount to be borrowed = d. = e. Amount needed (1 − c) $300,000 $300,000 = = $375,000 (1 − .20) .80 I 365 $10,125 365 × = × P−B d $375,000 − $75,000 60 = 0.2053 = 20.53% RCOMP = No, do not borrow with a compensating balance of 20 percent since the annual rate is greater than the cost of forgoing the cash discount. 8-31. Neveready Flashlights, Inc. a. Annual rate on bank loan: I 365 $5,500 365 R ANNUAL = × = × = 0.1115 = 11.15% P d $300,000 60 b. Cost of forgoing cash discount 2% 365 K DIS = × = .1241 = 12.41% 100% − 2% f (70 ) − d(10 ) c. Yes, because the cost of borrowing is less than the cost of losing the discount. Amount to be borrowed = d. = e. Amount needed (1 − c) $300,000 $300,000 = = $352,941.18 (1 − .15) .85 I 365 $6,850 365 × = × P−B d $300,000 60 = 0.1389 = 13.89% R COMP = No, do not borrow with a compensating balance of 15 percent since the annual rate is greater than the cost of forgoing the cash discount. 8-32. Rockette Dancers a. Annual commitment fee Interest expense = 0.0025 × $1,000,000 = $2,500 = $1,000,000 × 14% × 60/365 = $23,013.70 Annual rate of interest: I 365 $2,500 + $23,013.70 * 365 × = × $1,000,000 60 P d = 0.1552 = 15.52% R ANNUAL = *Note: The interest cost also includes the commitment fee. b. Cost of forgoing the cash discount: K DIS = 3% 365 × = .1881 = 18.81% 100% − 3% f (70 ) − d (10 ) c. Discounted commercial paper 100 − P 365 2.65 365 × =r= × = 0.1656 = 16.56% P d 100 − 2.65 60 P = Price = 100 – 2.65 = 97.35 Choose the bank loan. It is the cheapest alternative. 8-33. Macco Bakers a. Annual commitment fee Interest expense = $5,000 = $500,000 × 8% × 90/365 = $9,863.01 Annual rate of interest: I 365 $5,000 + $9,863.01 * 365 × = × P d $500,000 90 = 0.1206 = 12.06% R ANNUAL = *Note: The interest cost also includes the commitment fee. b. Cost of forgoing the cash discount: K DIS = 2% 365 × = .0827 = 8.27% 100% − 2% f (100 ) − d (10 ) c. Discounted commercial paper 100 − P 365 2.05 365 × =r= × = 0.0849 = 8.49% P d 100 − 2.05 90 Choose forgoing the cash discount. 8-34. Ajax Box Company a. Midland Bank Annual interest rate 2 × 4 × $8,000 (4 + 1) × ($100,000 − $20,000 − $8,000 ) = 0.1778 = 17.78% RINSTALL = Central Bank Annual interest rate 2 × 12 × $8,000 (12 + 1) × ($100,000 − $10,000 ) = 0.1641 = 16.41% RINSTALL = Choose Central Bank since it has the lowest annual interest rate. b. The numerators stay the same as in part (a) but the denominator increases to reflect the use of more money because compensating balances are already maintained at both banks. Midland Bank Annual interest rate 2 × 4 × $8,000 R INSTALL = (4 + 1) × ($100,000 − $8,000 ) = 0.1391 = 13.91% Central Bank Annual interest rate 2 × 12 × $8,000 RINSTALL = (12 + 1) × $100,000 = 0.1477 = 14.77% c. The compensating balance assumption changed interest rates as follows: Interest rate Midland With compensating balance 17.78% Without compensating balance 13.91 Difference in cost 3.87% Central 16.41% 14.77 1.64% Yes. If compensating balances are maintained at both banks in the normal course of business, then Midland should be chosen over Central Bank. The annual cost of its loan will be less. 8-35. a. Marla Maple Sugar Company 0 - 30 days A C G K Total loan % loan Amount $ 60,000 70,000 30,000 210,000 370,000 90% $333,000 31 - 40 days F I L Total loan % loan Amount $ 220,000 40,000 60,000 320,000 80% $256,000 Total loan % loan Amount $ 120,000 50,000 170,000 70% $119,000 41-45 days B E Maximum Loan = $333,000 + $256,000 + $119,000 = $708,000 b. Loan balances Interest, 15% annual One month's interest 8-36. $708,000 0.0125 per month $ 8,850 Savage Coasters Bank cost: Interest: $630,000 × 14% × 1/12 = Processing charge = $900,000 × .5% = Factor cost: Interest: $630,000 × 15% × 1/12 = Processing fee = $900,000 × 2.5% = Less credit department savings = Choose the factor. 8-37. Bolton Iron Works $ 7,350 4,500 $ 11,850 $ 7,875 22,500 (21,000) $ 9,375 a. Sales price: December Government bond contract (Sale takes place in July) Purchase price, December Government bond contract: (10% price decline) .9 × $72,000 = Gain per contract Number of contracts Profit on futures contracts $72,000 64,800 7,200 5 $ 36,000 b. Profit occurred because the bond value fell due to increasing rates. This meant the subsequent purchase price was less than the initial sales price. c. Increased interest cost $40,500 Profit from hedging 36,000 Net cost $ 4,500 $4,500 Net cost = = 0.1111 = 11.11% $40,500 The net cost is 11.11%. This means 88.89% of the increased interest cost was hedged away. d. If interest rates went down, there would be a loss on the futures contracts. The lower interest rates would lead to higher bond prices and a purchase price that exceeded the original sales price. MINI CASE Fresh and Fruity Foods, Inc. (Short-term Financing) Purpose: The student must focus on accounts receivable as an investment (use of funds) and the financial advantages of reducing the commitment to this asset. At the same time the firm is also considering reductions to its accounts payable balance in order to take cash discounts. This alternative will call for additional bank financing and comparative costs must be carefully assessed. The case utilizes many calculations that are covered in the text, but places them in a more complex, decision oriented framework. Suggested Questions: a. Using the data in the income statement and the balance sheet that follow, compute the company’s average collection period (ACP) in days. Use a 365- b. c. d. e. f. g. day year when calculating sales per day. Compute the cost, as a percent, that the company is paying for not taking the suppliers’ discounts. (The suppliers’ terms are 2/10, net 60; but note from the bottom of the balance sheet that Fresh & Fruity has been taking 67 days to pay its suppliers). Assume Alice Plummer’s first initiative to offer a 10 percent discount was implemented, and the company’s average collection period dropped to 32 days. If net sales per day remained the same, as Alice expects, what would be the new accounts receivable balance? How much cash was freed up by the reduction in accounts receivable? What is the new accounts payable balance if the money is used to pay off suppliers? As a result of Alice’s first initiative described in part c, Fresh & Fruity is able to take advantage of the 2 percent discount on one-third of its purchases (see the income statement). What will be the cash discount figure on the income statement? What effect does this have on net income (after taxes)? The simplest way to get this figure is to multiply the cash discount figure by (1 – Tax rate) and add this figure to the net income after tax figure on the income statement. Also what is the effect on the return-on-sales ratio shown toward the bottom of the balance sheet? Consider the effect on the return-onequity ratio as well. Alice’s second initiative calls for Fresh & Fruity to obtain a bank loan of a sufficient size to enable the company to take all suppliers’ discounts. What is the minimum size of this loan? Hint: To take all suppliers’ discounts, the average payment period must be 10 days, and net purchases will be Purchases – (Purchases from Figure 1 × .02). Assume all this happens, and solve the following formula for the new accounts payable balance, using: Accounts payable = Average payment period × Purchase per day* Now compare the accounts payable you just solved with the new accounts payable balance you found in part c. The difference is the size of the loan that is required. Assume Fresh & Fruity obtains an 8 percent loan for one year in the amount you solved in part e, and it reduces its accounts payable balance accordingly. Now the company is taking 2 percent discounts on all purchases and paying 8 percent a year on the loan balance. What is the net gain from taking the discounts and paying the interest on a before-tax basis? (on an aftertax basis?) (Optional) Suppose the 8 percent loan that Fresh & Fruity obtained was a discount loan, and the bank further required a 20 percent compensating balance of the full loan amount. What is the annual rate of interest to Fresh & Fruity? How does this compare to your answer in question b for the cost of not taking a cash discount? Answers a. Average collection period = Accounts receivable/ Average daily credit sales Accounts receivable = $209,686 Average daily credit sales = $1,179,000/ 365 = $3,230 Average collection period = $209,686/ $3,230 = 64.92 days b. Cost of forgoing the cash discount: 2% 365 K DIS = × = .1307 = 13.07% 100% − 2% f (67 ) − d (10) The formula tells us that Fresh and Fruity is effectively paying 13.07% interest to delay paying the discounted amount for 57 days (the 67 days on which they pay less the 10 day discount period). c. Average collection period × Average daily credit sales = New accounts receivable 32 × $3,230 = $103,360 Freed-up cash = Old accounts receivable − New accounts receivable Old accounts payable − Funds from accounts receivable New accounts payable d. Purchases (Figure 1) 1/3 exposed to purchase discount 2% purchase discount savings $209,686 103,360 $106,326 $180,633 106,326 $ 74,307 $969,000 $323,000 $ 6,460 With the firm in a 33 percent tax bracket, a savings of $6,460 will produce $4,328 in aftertax income. The answer is equal to the cost savings × (1 - T). $6,460 (1 − .33) = $6,460 (.67) = $4,328 This means total income will now be: Old income New aftertax income Total aftertax income Return on sales will be: Net income (aftertax)/ Sales $50,623 4,328 $54,951 = $54,951/ $1,179,000 = 0.0466 = 4.66% This, of course, represents an improvement over the old figure of 4.29%. Return on equity will be: Net income (aftertax)/ Equity = $54,951/ $123,600 = 44.46% This, also, represents an improvement over the old ratio of 40.96%. (Note: This firm has a particularly high return on equity because of rapid asset turnover and high current liabilities). If the added profit is included in equity, the return is 42.95% ($54,951/ $127,928). e. Accounts payable = Average payment period × Purchases per day Average payment period = 10 days Purchases per day = [969,000 − (.02 × 969,000)]/365 = [969,000 − $19,380]/ 365 = $2,602 Accounts payable = 10 × $2,602 = $26,020 Accounts payable from question c $74,307 Accounts payable from question e 26,020 Size of loan required $48,287 This is the size of the loan required to take all cash discounts in 10 days. f. The cost is the 8 percent interest on the bank loan of $48,287 or $3,863. The gain is the cash discounts taken of $19,380. The net gain before tax is $15,517 ($19,380 − $3,863). On an aftertax basis this translates to a gain of $10,396 ($15,517 × 0.67). g. First determine the amount of funds on which interest must be paid. $48,287 − (.08 × $48,287) − (.20 × $48,287) = $48,287 − $3,863 − $9,657 = $34,767 Then divide the interest payment by this value. Interest/ Useable funds $3,863/ $34,767 = 0.1111 or 11.11% The cost goes up from 8% to 11.11%. However, this value is still less than the cost of forgoing the cash discount of 13.07%, computed in part (b). Thus, it is advantageous to borrow and take the cash discount. Note: Alert students may point out that Fresh & Fruity still needs $48,287 in cash no matter what kind of loan it is. Therefore if the interest is to be charged on a discounted basis, and a compensating balance is required, Fresh & Fruity must borrow a larger amount to make up for it. Solve for the larger amount using algebra where L is the larger amount. L − (.08 × L) − (.20 × L) L − .08L − .20L L − .28L .72L L L = $48,287 = $48,287 = $48,287 = $48,287 = $48,287/ .72 = $67,065 Chapter 9 Discussion Questions 9-1. The future value represents the expected worth of a single amount, whereas the present value represents the current worth. Future value FV = PV (1 + i) n Present value PV = FV 1 (1 + i ) n 9-2. The present value of a single amount is the discounted value for one future payment, whereas the present value of an annuity represents the discounted value of a series of consecutive payments of equal amount. 9-3. Money has a time value because funds received today can be reinvested to reach a greater value in the future. A person would rather receive $1 today than $1 in ten years, because a dollar received today, invested at 6 percent, is worth $1.791 after ten years. 9-4. Inflation makes a dollar today worth more than a dollar in the future. Because inflation tends to erode the purchasing power of money, funds received today will be worth more than the same amount received in the future. The objective should be to place current funds in real assets (such as real estate or capital assets) that will go up in value with the rate of inflation. 9-5. FV = PV × FV IF (Appendix A) i = 12%, n = 10 3.106 (Annual) i = 6%, n = 20 3.207 (Semiannual) The more frequent compounding under the semiannual compounding assumption increases the future value. 9-6. The greater the number of compounding periods, the larger the future value. The investor should choose daily compounding over monthly or quarterly. 9-7. A deferred annuity is an annuity in which the equal payments will begin at some future point in time. 9-8. Different financial applications of the time value of money: Equipment purchase or new product decision, Present value of a contract providing future payments, Future worth of an investment, Regular payment necessary to provide a future sum, Regular payment necessary to amortize a loan, Determination of return on an investment, Determination of the value of a bond 9-9. Compounding of interest allows the accumulation of capital by earning interest on interest. If money (capital) is held outside a tax sheltered plan there will be a depletion of the ‘interest’ capital base each year as taxes are paid. Therefore less capital will be accumulated over time. Internet Resources and Questions 1. http://money.canoe.ca/calculators/ www.rbcroyalbank.com/products/mortgages/ www.bmo.com/home/personal/banking/mortgages-loans/mortgage/quick-info-andresources/mortgage-rates 2. www.bloomberg.com/personal-finance/calculators/mortgage/ Problems 9-1. Appendix B PV = FV × PV IF a. $ 8,000 × .558 b. $16,000 × .567 c. $25,000 × .315 d. $ 1,000 × .001 9-2. Appendix A FV = PV × FV IF a. $12,000 × 1.501 b. $12,000 × 5.474 c. $12,000 × 10.835 d. $12,000 × 11.467 $4,464 $9,072 $7,875 $1 Calculator (N = 10, %I/Y = 6) $ 4,467 (N = 5, %I/Y = 12) 9,079 (N = 15, %I/Y = 8) 7,881 (N = 40, %I/Y = 20) 0.68 = $ 18,012 = $ 65,688 = $130,020 = $137,604 Calculator $ 18,009 65,683 130,016 137,609 = = = = (N = 6, %I/Y = 7) (N = 15, %I/Y = 12) (N = 25, %I/Y = 10) (N = 50, %I/Y = 5) 9-3. Appendix B (a and b) PV = FV × PV IF Calculator a. $12,000 × .517 = $6,084 (N = 6, %I/Y =12) $ 6,080 b. $15,000 × .315 = $4,725 (N = 15, %I/Y =8) 4,729 Appendix D (c and d) Calculator PV A = A × PV IFA c. $5,000 × 6.710 = $33,550 (N = 10, %I/Y = 8) $ 33,550 d. $5,000 × 7.247 = $36,235 (N = 10, %I/Y = 8)(BGN) 36,234 e. $50,000 × 13.801 = $690,050 (N = 50, %I/Y = 7) 690,037 f. $50,000 × 14.767 = $738,350 (N = 50, %I/Y = 7)(BGN) 738,340 9-4. Appendix C Calculator FV A = A × FV IFA a. $8,000 × 12.578 = $100,624 (N = 10, %I/Y = 5) $ 100,623 b. $8,000 × 51.160 = $409,280 (N = 20, %I/Y = 9) 409,281 c. $8,000 × 341.590 = $2,732,720 (N = 35, %I/Y =11) 2,732,716 9-5. Appendix C (Annuity in advance) BGN Calculator FV A = A × FV IFA a. $8,000 × 13.207 = $105,656 (N = 10, %I/Y = 5) $ 105,654 b. $8,000 × 55.765 = $446,120 (N = 20, %I/Y = 9) 446,116 c. $8,000 × 379.164 = $3,033,312 (N = 35, %I/Y =11) 3,033,315 9-6. 9-7. You Invest Appendix A FV = PV × FV IF 20,000 × 1.501 = $30,020 Calculator (N = 6, %I/Y = 7) $30,015 Appendix A FV = PV × FV IF $30,020 × 2.144 = $64,363 (N = 8, %I/Y = 10) $64,339 Delia PV = FV × PV IF (Appendix B) (N= 50, %I/Y = 8) Calculator = $30,000 × .021 = $630 $640 Take the $650 today. 9-8. Mr. Sampson a. PV A = A × PV IFA (Appendix D) (N= 14, %I/Y = 8) Calculator = $6,500 × 8.244 = $53,586 $53,588 b. Mr. Sampson (in advance) (N= 14, %I/Y = 8)Calculator PV A = A × PV IFA (Appendix D) = $6,500 × (7.904 + 1) = $57,876 (BGN) $57,875 9-9. Phil Goode PV = FV × PV IF (Appendix B) (N= 50, %I/Y = 14)Calculator = $175,000 × .001 = $175 $250 9-10. Carrie Tune a. PV A = A × PV IFA (Appendix D) (N = 20, %I/Y = 10)Calculator PV A = $18,000 × 8.514 = $153,252 $153,244 Yes, the present value of the annuity is not worth $160,000. b. Carrie Tune (in advance - BGN) PV A = A × PV IFA (Appendix D) (N = 20, %I/Y = 10)Calculator PV A = $18,000 × (8.365 + 1) = $168,570 $168,569 No, the present value of the annuity is worth more than $160,000. 9-11. George Penny a. PV A = A × PV IFA (Appendix D) (N = 10, %I/Y = 6) Calculator $237,363 PV A = $32,250 × 7.360 = $237,360 Yes, the present value of the annuity is not worth $240,000. b. George Penny (BGN) PV A = A × PV IFA (Appendix D) (N = 10, %I/Y = 6) Calculator PV A = $32,250 × (6.802 + 1) = $251,615 (BGN) $251,605 No, the present value of the annuity is worth more than $240,000. 9-12. Grand Casino PV A = A × PV IFA (Appendix D) (N = 50, %I/Y = 11)Calculator PV A = $100,000 × (9.036 + 1) = $1,003,600 (BGN) $1,003,624 9-13. Joan Lucky PV A = A × PV IFA (Appendix D) (N=50, %I/Y = 12)Calculator PV A = $1,000,000 × 8.304 = $8,304,000 $8,304,498 PV = FV × PV IF (Appendix B) (N=50, %I/Y = 12%) PV = $30,000,000 × .003 = $90,000 $ 103,805 $8,304,000 + $90,000 = $8,394,000 $8,408,303 9-14. Mr. Elite FV = PV × FV IF (Appendix A) (N = 5, %I/Y = 12) Calculator = $120,000 × 1.762 = $211,440 $211,481 9-15. Dr. Sisters FV A = A × FV IFA (Appendix C) (N = 12, %I/Y = 6) Calculator = $10,500 × 16.870 = $177,135 $177,134 9-16. Doubling, Tripling If the sum is doubling, then the tabular value must equal 2. In Appendix A, looking down the 8% column, we find the factor closest to 2 (1.999) on the 9-year row. The factor closest to 3 (2.937) is on the 14-year row. Calculator: PV = 1 FV = 2 %I/Y = 8% Compute N = 9.006 PMT = 0 N =? Calculator: PV = 1 FV = 3 %I/Y = 8% Compute N = 14.275 PMT = 0 N =? 9-17. . PV 9-18. . Your Debt = FV × PV IF (Appendix B) (N = 7, %I/Y = 11) Calculator = $30,000 × .482 = $14,460 $14,450 Jack Hammer PV = FV × PV IF (Appendix B) = $2.00 × .901 = 1.80 = $2.20 × .812 = 1.79 = $2.40 × .731 = 1.75 = $33.00 × .731 = 24.12 Calculator (N= 1, %I/Y = 11) $ 1.80 (N= 2, %I/Y = 11) 1.79 (N= 3, %I/Y = 11) 1.75 (N= 3, %I/Y = 11) 24.13 $29.46 $29.47 9-19. S. Ken Flint a. PV A = A × PV IFA (Appendix D) (N = 10, %I/Y = 10)Calculator = $45,000 × 6.145 = $276,525 $276,506 b. PV = FV × PV IF (Appendix B) (N Calculator = $276,525 × .826 = $228,410 = 2, %I/Y = 10%) $228,517 Alternative Solution Deferred annuity-Appendix D PV A = $45,000(6.814 ─ 1.736) (where N=12; N=2 %I/Y=10) = $45,000 (5.078) = $228,510 (difference due to rounding in the tables) and c. PV A = A × PV IFA (Appendix D) (N = 10, %I/Y = 10)Calculator = $45,000 × (5.759 + 1) = $304,155 BGN $304,156 9-20. Cousin Berta a. FV = PV × FV IF (Appendix A) (N =40, %I/Y = 3) FV = $100,000 × 3.262 = $326,200 n Calculator $326,204 4 i   .12  Effective annual interest rate = 1 +  − 1 = 1 +  −1 b. 4 n     = 0.1255 = 12.55% 9-21. Appendix A FV = PV × FV IF a. $3,000 × 1.469 = $4,407 b. $3,000 × 1.480 = $4,440 c. $3,000 × 1.486 = $4,458 Calculator (N = 5, %I/Y = 8) $4,408 (N = 10, %I/Y = 4) $4,441 (N = 20, %I/Y = 2) $4,458 Effective annual interest rate a. (1 + 0.08)1 – 1 = 0.0800 = 8.00% b. (1 + 0.04)2 – 1 = 0.0816 = 8.16% c. (1 + 0.02)4 – 1 = 0.0824 = 8.24% 9-22. Jill Hill a. Effective annual interest rate: (1 + 0.02)4 – 1 = 0.0824 = b. A = FV A / FV IFA (Appendix C) = $100,000/ 19.249 = $5,195 (Factor interpolated between 8 & 9%) (N = 12, %I/Y = 8.24) 8.24% Calculator $5,195 c. PV = FV × FV IFA (Appendix C) Calculator = $100,000/ 20.836 = $4,799 (Factor interpolated between 8 & 9%) (N = 12, %I/Y = 8.24) BGN $4,799 9-23. Tommy Crews a. Effective annual interest rate: (1 + 0.03)2 – 1 = 0.0609 = 6.09 b. PV A = A × PV IFA (Appendix D) = $32,500 × 7.329 = $238,193 (Factor interpolated between 6 & 7%) (N = 10, %I/Y = 6.09) Calculator c. PV = FV × PV IFA (Appendix D) = $32,500 × 7.775 = $252,688 (Factor interpolated between 6 & 7%) (N = 12, %I/Y = 8.24) BGN Calculator 9-24. Your Grandfather (1st alternative) PV of $5,000 received now: $238,186 $252,692 $ 5,000 (2nd alternative) PV of annuity of $1,000 for eight years: Calculator PV A = A × PV IFA (Appendix D) = $1,000 × 5.146 = $5,146 (N = 8, %I/Y = 11) $5,146 (3rd alternative) PV of $12,000 received in eight years: PV = FV × PV IF (Appendix B) Calculator = $12,000 × .434 = $5,208 (N = 8, %I/Y = 11) $5,207 Select $12,000 to be received in eight years. Revised answers based on 12%. (1st alternative) PV of $5,000 received today: $5,000 (2nd alternative) PV of annuity of $1,000 for eight years: PV A = A × PV IFA (Appendix D) Calculator = $1,000 × 4.968 = $4,968 (N = 8, %I/Y = 12) $4,968 (3rd alternative) PV of $12,000 received in eight years: Calculator PV = FV × PV IF (Appendix B) $12,000 × .404 = $4,848 (N = 8, %I/Y = 12) $4,847 Select $5,000 now. 9-25. Your Need a. PV = FV × PV IF (Appendix B) Calculator = $23,000 × .547 = $12,581 (N = 7, %I/Y = 9) $12,582 b. A 9-26. 9-27. = FV A /FV IFA (Appendix C) = $23,000/9.200 = $ 2,500 (N = 7, %I/Y = 9) Amy Hirt FV A = A × FV IFA (Appendix C) = $300 × 18.639 = $5,592 (N = 16, %I/Y =2) FV = PV × FV IF (Appendix A) = $5,592 × 1.482 = $8,287 (14%, 3 periods) $2,500 Calculator $5,592 $8,284 Charley Dow PV (3 periods ) = $12 = 0.667 PVIF = FV $18 Return is between 14% & 15% for 3 years (Appendix B) PV IF @ 14% .675 PF IF at 14% .675 PV IF @ 15% –.658 PV IF computed –.667 .017 .008 14% + (.008/ .017) (1%) Calculator 14% + .471 (1%) = 14.47% (N = 3, PV =12, FV = 18) 14.47% 9-28. Hy Waters a. PVIF = PV (2 periods ) = $10.00 = 0.816 FV $12.25 Return is between 10% and 11% for 2 years (Appendix B) PV IF @ 10% 0.826 PV IF @ 10% PV IF @ 11% – 0.812 PV IF computed 0.826 – 0.816 0.014 = 10% + (0.010/ 0.014) (1%) = 10% + (0.714) (1%) = 10.71% 0.010 Calculator (N = 2, PV = ─ 10, FV = 12.25) 10.68% b. PVIF = PV (4 periods ) = $10.00 = 0.629 FV $15.90 Return is between 12% and 13% for 4 years (Appendix B) 0.636 PV IF @ 12% PV IF @ 12% PV IF @ 13% – 0.613 PV IF computed 0.023 = 12% + (0.007/ 0.023) (1%) = 12% + (0.304) (1%) = 12.30% 0.636 – 0.629 0.007 Calculator (N = 4, PV = ─ 10, FV = 15.90) 12.29% c. PVIF = PV (7 periods ) = $10.00 = 0.345 FV $29.00 Return is between 16% and 17% for 7 years (Appendix B) PV IF @ 16% 0.354 PV IF @ 16% – 0.333 PV IF computed PV IF @ 17% 0.021 = 16% + (0.009/ 0.021) (1%) = 16% + (0.429) (1%) = 16.43% 0.354 – 0.345 0.009 Calculator (N = 7, PV = ─ 10, FV = 29.00) 16.43% 9-29. John Foresight PV (17 periods ) = $8,370 = 0.093 PV IF = FV $90,000 Rate of return = 15% (Appendix B) 9-30. Calculator 14.99%` Chris Seals a. PVIFA = b. PV A (12 periods ) = $56,521 = 7.536 BGN (11 periods ) = 7.536 − 1 = 6.536 A $7,500 7.536 is exactly 8% for 12 periods PV IFA @ 8% 7.536 6.536 is between 9% and 10% for 11 periods (Appendix D) PV IFA @ 9% 6.805 PV IFA @ 10% – 6.495 0.310 PV IFA @ 9% PV IFA @ computed a. b. 9-31. 6.805 – 6.536 0.269 9% + (0.269/ 0.310) (1%) 9% + (0.868) (1%) = 9.87% Calculator (N = 12, PV = 56,521, PMT = 7,500) 8.00% BGN (N = 12, PV = 56,521, PMT = 7,500) 9.86% Dell Weed Calculator PV = FV × PV IF (Appendix B) = $45,000 × 0.812 = $36,540 (N = 2, %I/Y = 11) $36,523 PV = FV × PV IF (Appendix B) = $36,540 × 0.857 = $31,315 (N = 2, %I/Y =8) $31,313 PV = FV × PV IF (Appendix B) = ($31,315 + $20,000) × 0.794 = $40,744 (N = 3, %I/Y =8) $40,734 Loan amount can be $40,744 $40,734 9-32. 9-33. Dee Fine PV = FV × PV IF (Appendix B) = $30,000 × [.837 .790]/2 = $24,405 (N = 3 × 2 = 6, %I/Y = 7/ 2 = 3.5) Value of land = $24,405 + $40,000 = $64,405 $24,405 $64,405 Otto Wah PV = FV × PV IF (Appendix B) Calculator = $100,000 × [.558 (.558 .508)/ 4] = $54,550 (N = 10, %I/Y = 6.25) $54,539 A = PV A / PV IFA (Appendix D) = [$450,000 – $54,550] / [6.802 (6.802 6.515)/ 4] + 1 = $395,450/ 7.73025 = $51,156 (N = 9 + 1, %I/Y =6.25) BGN $51,170 9-34. A 9-35. Calculator Graham Bell = PV A / PV IFA (Appendix D) = $300,000/ 9.447 = $31,756 (N = 16, %I/Y = 7) Calculator $31,757 River Babylon FV = PV × FV IF (Appendix A) Calculator = $65,000 × 1.469 = $95,485 (N = 5, %I/Y = 8) $95,506 A = PV A / PV IFA (Appendix D) = $95,485/ 7.161 = $13,334 (N = 12, %I/Y = 9) $13,338 9-36. Una Day Determine the present value of an annuity during retirement: PV A = A × PV IFA (Appendix D) Calculator = $12,000 × 9.077 = $108,924 (N = 25, %I/Y = 10, PMT = 12,000) $108,924 Determine the annual deposit into an account earning 8% that is necessary to accumulate $108,924 after 20 years: A = FV A / FV IFA (Appendix C) = $108,924/ 45.762 = $2,380 (N = 20, %I/Y = 8, FV = 108,924) Calculator $2,380 9-37. Your Retirement a. PV A = A × PV IFA (Appendix D) Calculator = $60,000 × 12.042 = $722,520 (N = 22, %I/Y = 6) $722,495 A = FV A / FV IFA (Appendix C) = $722,520/ 172.317 = $4,193 (N = 35, %I/Y = 8) $4,193 b. A BGN = FV A / FV IFA (Appendix C) = $722,520/ 186.102 = $3,882 (N = 34 + 1, %I/Y = 8)BGN $3,882 9-38. Retirement Planning FV = PV × FV IF (Appendix A) = $45,000 × 2.814 = $126,630 (N = 35, %I/Y = 3) Calculator $126,624 PV A (BGN) = A × PV IFA (N =29) + A (Appendix D) = $126,630 × 13.591 + $126,630 = $1,847,658 (N = 29 + 1, %I/Y = 6)$1,847,533 A = FV A / FV IFA (Appendix C) = $1,847,658/ 172.317 = $10,722 (N = 35, %I/Y = 8) $10,722 9-39. Your Retirement (#2) FV = PV × FV IF (Appendix A) = $80,000 × 2.208 = $176,640 (N = 40, %I/Y = 2) Calculator $176,643 PV A (BGN) = A × PV IFA (N =29) + A (Appendix D) = $176,640 × 16.984 + $176,640 = $3,176,694 (N = 29 + 1, %I/Y = 4)$3,176,700 FV = PV × FV IF (Appendix A) = $350,000 × 4.292 = $1,502,200 (N = 25, %I/Y = 6) A 9-40. = FV A / FV IFA (Appendix C) = [$3,176,700 + 1,502,155]/ 154.762 = $30,233 (N = 40, %I/Y = 6) Del Monty PV = FV × PV IF (%I/Y = 9) 2,000 × 0.917 = $1,834 (N = 1) 3,500 × 0.842 = 2,947 (N = 2) (N = 3) 3,474 4,500 × 0.772 = $8,255 Calculator $1,502,155 $30,233 Calculator $1,835 2,946 3,475 $8,256 PV A = A × PV IFA (Appendix D) (N = 7, %I/Y = 9) = $5,000 × 5.033 = $25,165 (PMT = 5,000) $25,165 PV = FV × PV IF (N = 3, %I/Y = 9) = $25,165 × 0.772 = $19,427 (FV = 25,165) $19,432 Total = $8,255 + $19,427 = $27,682 ($27,687.38) $27,688 9-41. Bridget Jones PV = FV × PV IF (%I/Y = 14) 1,000 × 0.877 = $ 877 (N = 1) 2,000 × 0.769 = 1,538 (N = 2) 3,000 × 0.675 = 2,025 (N = 3) 4,000 × 0.592 = 2,368 (N = 4) 2,595 (N = 5) 5,000 × 0.519 = $9,403 Calculator $ 877 1,539 2,025 2,368 2,597 $9,406 PV A = A × PV IFA (Appendix D) (N = 10, %I/Y = 14) = $8,500 × 5.216 = $44,336 (PMT = 8,500) $44,337 PV = FV × PV IF (N = 5, %I/Y = 14) = $44,336 × 0.519 = $23,010 (FV = 44,337) $23,027 Total = $9,403 + $23,010 = $32,413 ($27,687.38) $32,433 9-42. Darla White The value of the annuity at the beginning of the year it starts (2016) is (assuming first payment at end of 2016): PV A = A × PV IFA (Appendix D) Calculator = $12,000 × 5.146 (N = 8, %I/Y = 11)$61,753(2016 Begin) = $61,752 (BGN) $68,546 (2016 End) To find the present value at the beginning of 2012 we discount from: the beginning of 2016 for 3 periods: (2012, 2013, 2014) the end of 2016 for 4 periods: (2012, 2013, 2014, 2015) PV = FV × PV IF (11%, 3 periods) Calculator = $61,752 × .693 = $45,153 $45,154 The maximum that should be paid for the annuity is $45,154. 9-43. Air N. Grabba PV = 9-44. Perpetual University PV = 9-45. A $500 = = $12,500 i − g .07 − .03 Perpetual University(2nd thought) PV = 9-47. A $2,500 = = $41,667 i .06 Air N. Grabba (2nd thought) PV = 9-46. A $500 = = $7,143 i .07 A $2,500 = = $62,500 i .06 − .02 Air N. Grabba (3rd thought)  1   1 + g   1     1 + .03    1 −  PVn = A1    = $500  1 −    = $7,678 − + − + 1 . 07 . 03 1 . 07 i g i            n 9-48. 25 Perpetual University (3rd thought)  1   1 + g   1     1 + .02    1 −  PVn = A1    = $2,500  1 −    = $42,789 − + − + 1 . 06 . 02 1 . 06 i g i           n 30 9-49. Your retirement (Growing annuity) FV = PV × FV IF (Appendix A) Calculator = $90,000 × 3.262 = $293,580 (N = 40, %I/Y = 3) $293,583 n 30  1   1 + g   1     1 + .03    1 −  PVn = A1    = $293,583  1 −    = $7,387,419 − + − + 1 . 04 . 03 1 . 04 i g i            By tables and formula: $7.387,344 FV = PV × FV IF (Appendix A) = $250,000 × 4.322 = $1,080,500 (N = 30, %I/Y = 5) Required (expressed at time 40) = $7,387,344 + $1,080,500 = $8,467,844 A = FV A / FV IFA (Appendix C) = [$8,467,844]/ 120.800 = $70,098 (N = 40, %I/Y = 5) 9-50. Calculator $1,080,486 $8,467,905 $70,099 Your Interest Rate PV IFA = PV A /A (Appendix D) (N = 5) = $9,725/ $2,500 = 3.890 Interest rate = 9 percent Calculator %i = 9% 9-51. Sarah Adia FV = PV × FV IF (Appendix A) = $15,000 × 1.260 = $18,900 (N = 3, %I/Y = 8) A = PV A /PV IFA (Appendix D) = $18,900/3.890 = $4,859 (N = 5, %I/Y = 9) 9-52. Calculator $18,896 $4,858 Your Uncle A = PV A /PV IFA (Appendix D) Calculator = $50,000/5.335 = $9,372 (N = 8, %I/Y = 10) $9,372 First payment: $50,000 × .10 = $9,372 – $5,000 = $5,000 interest $4,372 to principal $4,372 Second payment: First determine remaining principal $50,000 – $4,372 = $45,628 $45,628 $45,628 × .10 = $9,372 – $4,563 = $4,563 interest $4,809 to principal $4,563 $4,809 9-53. Jim Thomas a. A = PV A /PV IFA (Appendix D) = $70,000/8.055 = $8,690 b. $ 8,690 × 30 $ 260,700 – 70,000 $ 190,700 Calculator (N = 30, %I/Y = 12%) $8,690 annual payments years total payment repayment of principal $190,702 c. New payments at 10% A = PV A /PV IFA (Appendix D) = $70,000/ 9.427 = $7,425 (N = 30, %I/Y = 10) Difference between old and new payments $8,690 old 7,425 new $ 1,265 difference $7,426 $ 8,690 7,426 $ 1,264 P.V. of difference: Amount that could be paid to refinance. PV A = A × PV IFA (Appendix D) = $1,265 × 9.427 = $11,925 (N = 30 %I/Y = 10) $11,916 9-54. Cindy Connor a. Determine the monthly effective interest rate. PV = 1 FV = 1.0375 (7.5/ 2) PMT = 0 N=6 Compute %I/Y = 0.6154524% Determine monthly payment PV = $90,000 FV = 0 %I/Y = 0.6154524% N = 240 (20 × 12) Compute PMT= $718.74 b. Determine the weekly effective interest rate. PV = 1 FV = 1.0375 (7.5/ 2) PMT = 0 N = 26 Compute %I/Y = 0.1416925% Determine weekly payment PV = $90,000 FV = 0 %I/Y = 0.1416925% N = 1,040 (20 × 52) Compute PMT= $165.47 c. Determine the bi-weekly effective interest rate. PV = 1 FV = 1.0375 (7.5/ 2) PMT = 0 N = 13 Compute %I/Y = 0.2835858 % Determine bi-weekly payment PV = $90,000 FV = 0 %I/Y = 0.2835858 % N = 520 (20 × 26) Compute PMT= 9-55. $331.18 Mac Wire a. Determine the monthly effective interest rate. PV = 1 FV = 1.03 (6/ 2) PMT = 0 N=6 Compute %I/Y = 0.4938622 % Determine monthly payment PV = $180,000 FV = 0 %I/Y = 0.4938622 % N = 300 (25 × 12) Compute PMT= $1,151.65 b. Determine the weekly effective interest rate. PV = 1 FV = 1.03 (6/ 2) PMT = 0 N = 26 Compute %I/Y = 0.1137523 % Determine weekly payment PV = $180,000 FV = 0 N = 1,300 (25 × 52) %I/Y = 0.1137523 % Compute PMT= $265.26 c. Determine the bi-weekly effective interest rate. PV = 1 FV = 1.03 (6/ 2) PMT = 0 N = 13 Compute %I/Y = 0.2276341 % Determine bi-weekly payment PV = $180,000 FV = 0 %I/Y = 0.2276341 % N = 650 (25 × 26) Compute PMT= 9-56. $530.83 Tippi & Tyler Kanu Determine the monthly effective interest rate. PV = 1 FV = 1.0325 (6.5/ 2) PMT = 0 N=6 Compute %I/Y = 0.5344740 % Determine length of amortization payment PV = $130,000 FV = 0 %I/Y = 0.5344740 % PMT = $1,200 Compute N= Or 9-57. 162.3 13.53 years Molly Reach Determine the monthly effective interest rate. PV = 1 FV = 1.03625 (7.25/ 2) PMT = 0 N=6 Compute %I/Y = 0.5952383% Determine monthly payment FV = 0 N = 300 %I/Y = 0.5952383% PMT = $750 Compute PV= 9-58. $104,761 Barbara Present value of college costs PV A = A × PV IFA (Appendix D) = $15,000 × 3.170 = $47,550 (N = 4, %I/Y =10) Calculator $47,548 Accumulation based on investing $2,000 per year for 10 years. FV A = A × FV IFA (Appendix C) = $2,000 × 15.937 = $31,874 (N = 10, %I/Y = 10) $31,875 Additional funds required 5 years from now. $47,550 PV of college costs Calculator 31,874 Accumulation based on $2,000 per year $15,676 Additional funds required $15,673 Added contribution for the next 5 years A = FV A /FV IFA (Appendix C) = $15,676/6.105 = $ 2,568 (N = 5, %I/Y = 10) 9-59. $2,567 Barbara’s Wedding Funds available after the wedding $47,550 Funding available before the wedding – 7,000 Wedding $40,550 Funds available after the wedding Less present value of vacation PV A = A × PV IFA (Appendix D) Calculator $47,548 ─ 7,000 $40,548 = $4,000 × 2.487 = $9,948 $40,550 – 9,948 $30,602 (N = 3, %I/Y = 10) $9,947 $40,548 ─ 9,947 $30,601 Remaining funds for graduate school Value of funds 3 years later for graduate school: FV = PV × FV IF (Appendix A) = $30,602 × 1.331 = $40,731 (N = 3, %I/Y =10) Calculator $40,730 Number of years of graduate education: PV A (10% ) = $40,731 = 3.170 Appendix D A $12,850 With i = 10%, N = 4 for 3.170. The answer is 4 years. Calculator: 3.9996 years PVIFA = 9-60. Middle Hockey League a. A = FV A / FV IFA (Appendix C) = $250,000/60.402 = $4,139 (N = 40, %I/Y = 2) Calculator $4,139 b. Determine how much the old payments are equal to after 16 periods: FV A = A × FV IFA (Appendix C) = $4,139 × 18.639 = $77,147 (N = 16, %I/Y = 2) $77,147 Determine how much this value will grow to after 24 periods at 3%. FV = PV × FV IF (Appendix A) = $77,147 × 2.033 = $156,840 (N = 24, %I/Y = 3) $156,824 Determine how much has to accumulate on the next 24 payments. $250,000 – 156,840 $250,000 – 156,824 $ 93,160 $ 93,176 Determine revised payment to accumulate this sum after 24 periods at 3%. A = FV A / FV IFA (Appendix C) Calculator = $93,160/ 34.426 = $2,706 (N = 24, %I/Y = 3) $2,707 Comprehensive Questions 9-61. Medical Research Corporation Offer #1: Annuity of 200,000 (years 6 thru 15): Calculator PV A = A × PV IFA (Appendix D) = $200,000 × 6.145 = $1,229,000 (N = 10, %I/Y = 10)$1,228,913 PV = FV × PV IF (Appendix B) = $1,229,000 × .621 = $763,209 (N = 5, %I/Y= 10) $ 763,059 th Possible payment in 15 year: PV = FV × PV IF (Appendix B) = 0.70 × $3,000,000 ×.239 = $501,900 (N = 15, %I/Y= 10) 502,723 Payment now: $1,000,000 1,000,000 $(763,209 + 501,900 + 1,000,000) = $2,265,109 $2,265,782 Offer #2: (% of gross profit) Year 1 2 3 4 Sales $2,000,000 2,800,000 3,920,000 5,488,000 Gross profit (60% of sales) $1,200,000 1,680,000 2,352,000 3,292,800 Payment (30% of Gross) $360,000 504,000 705,600 987,840 PV factor .909 .826 .751 .683 PV $327,240 416,304 529,906 674,695 $1,948,145 Calculator $1,948,473 Offer #3: Trust fund value after 8 years (16 periods): Calculator FV A = A × FV IFA (Appendix C) (N = 17 ─ 1, %I/Y= 5) = $200,000 × (25.840 ─ 1) = $4,968,000 (BGN) $4,968,073 Present value of trust fund: PV = FV × PV IF (Appendix B) = $4,968,000 ×.467 = $2,320,056 (N = 8, %I/Y= 10) $2,317,643 Or: (N = 16, %I/Y= 5)$2,275,931 Offer #3 is the best MINI CASE Allison Boone The case brings the time value of money into a legal settlement context, where present value concepts are frequently utilized. Many professors may also be able to draw on their own personal expertise to enhance the discussion of the case. The case deals with a high earning medical doctor and the loss to her family as a result of an accident. Select a current long-term interest rate as discount rate. This may vary. We will select 6%, close to the long-term Government of Canada bond in late 2011 plus 3%. a. @ 6% Proposal Number One: $300,000 a year for the next 20 years PV A = A × PV IFA (Appendix D) Calculator = $300,000 × 11.470 = $3,441,000 (N = 20, %I/Y = 6) $3,440,976 Plus: $500,000 a year for the remaining 20 years Step 1 PV A = A × PV IFA (Appendix D) Calculator =$500,000 × 11.470 = $5,735,000 (N = 20, %I/Y =6) $5,734,961 Step 2 PV = FV × PV IF (Appendix B) = $5,735,000 × .312 = $1,789,320 (N = 20, %I/Y= 6) Total present value: = $3,441,000 + $1,789,320 = $5,230,320 $1,788,188 $5,229,164 Proposal Number Two: $5,000,000 Proposal Number Three $50,000 a year for the next 40 years PV A = A × PV IFA (Appendix D) = $50,000 × 15.046 = $752,300 Plus: $75 million at the end of 40 years PV = FV × PV IF (Appendix B) = $75,000,000 × .097 = $7,275,000 Total present value: = $752,300 + $7,275,000 = $8,027,300 Calculator (N = 40, %I/Y = 6) $752,315 (N = 40, %I/Y = 6) $7,291,664 $8,043,979 At a discount rate of 6 percent, proposal three has the highest net present value of $8,043,979. b. @ 11% Proposal Number One: $300,000 a year for the next 20 years PV A = A × PV IFA (Appendix D) Calculator = $300,000 × 7.963 = $2,388,900 (N = 20, %I/Y =11)$2,388,998 Plus: $500,000 a year for the remaining 20 years Step 1 PV A = A × PV IFA (Appendix D) Calculator = $500,000 × 7.963 =$3,981,500 (N = 20, %I/Y =11)$3,981,664 Step 2 PV = FV × PV IF (Appendix B) = $5,735,000 × .124 = $493,706 (N = 20, %I/Y =11) $493,861 Total present value: = $2,388,900 + $493,706 = $2,882,606 $2,882,860 Proposal Number Two: $5,000,000 Proposal Number Three $50,000 a year for the next 40 years PV A = A × PV IFA (Appendix D) = $50,000 × 8.951 = $447,550 Plus: $75 million at the end of 40 years PV = FV × PV IF (Appendix B) = $75,000,000 × .015 = $1,125,000 Total present value: = $447,550 + $1,125,000 = $1,572,550 Calculator (N = 40, %I/Y =11) $447,553 (N = 40, %I/Y =11)$1,153,831 $1,601,383 At a discount rate of 11 percent, proposal two has the highest net present value of $5,000,000. c. At a relatively high discount rate of 11 percent in question 2, the later payments lose much of their value. For example, the $75 million payment as part of proposal three only has a present value of $1,125,000 at a discount rate of 11 percent as compared to $7,275,000 at six percent. For this reason, the $5 million (immediate) payment in proposal two is the most favorable at the higher discount rate. d. Punitive damages are added on to the economic damages. With the likelihood of $4 million in punitive damages, Sloan Whitaker may well want to take the case before a jury. However, we should keep in mind that offers for the out-of-court settlements have likely been influenced by the potential for punitive damages. Also, a jury verdict may be appealed and actual payment may be deferred many years into the future. Because attorneys in cases such as this often get 1/3rd of the out-of-court settlement (or the jury determined value) as their fee, Sloan Whitaker is likely to consider this matter quite seriously. Of course, the final decision will rest with the Boone family, but Samuel Boone will be strongly influenced by the attorney’s recommendation. Although this question is not a financial one, it has financial implications for the student doing the case. Solution Manual for Foundations of Financial Management Stanley B. Block, Geoffrey A. Hirt, Bartley Danielsen, Doug Short, Michael Perretta 9780071320566, 9781259268892, 9781259261015