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Chapter 6 Homework Pg. 268 ST-2; ST-4 Pg. 270 6-6; 6-10 Pg. 271 6-12 Pg. 274 6-25 Pg. 268 ST-2 In the introduction to this chapter we asked whether you would prefer to invest $5,500 today and receive either $7,020 in five years or $8,126 in eight years. You should now be able to determine which investment alternative is better. a. Based only on the return you would earn from each investment, which is better? b. Can you think of any factors other than the expected return that might be important to consider when choosing between the two investment alternatives? Pg. 268 ST-2a 0 1 2 3 4 5 6 7 8 Option A -5,500 7, 020 Option B -5,500 8,126 FVn = PV(1+i)n Option A Option B 7,020 = 5,500(1+i)5 8,126 = 5,500(1+i)8 7,020/5,500 = (1+i)5 8,126/5,500 = (1+i)8 1.2764 = (1+i)5 1.4775 = FVIFi,8 1.27641/5 = (1+i) Table Lookup 1.0500 = (1+i) 1.4775 = FVIF5%,8 i = 5% Pg. 268 ST-2b Other Factors to Consider: – Risk of 8-year investment vs. 5-year – Timing – Rate Expectations Pg. 268 ST-4 Assume that it is now January 1, 2000, and you will need $1,000 on January 1, 2004. Your bank compounds interest at an eight percent annual rate. ‘00 8% ‘01 ‘02 ‘03 ‘04 1,000 ‘00 ‘01 Pg. 268 ST-4 ‘03 ‘02 ‘04 8% PV=? 1,000 a. How much must you deposit on January 1, 2001, to have a balance of $1,000 on January 1, 2004? PV=FV[1/(1+i)n] PV = 1,000[1/(1+0.08)3] PV = 1,000[0.7938] PV = 793.83 ‘00 ‘01 Pg. 268 ST-4 ‘03 ‘02 ‘04 8% PMT PMT PMT PMT 1,000 b. If you want to make equal payments to each January 1 from 2001 through 2004 to accumulate the $1,000, how large must each of the four payments be? FVAn = PMT(FVIFAi,n) 1,000 = PMT(4.5061) 1,000/4.5061 = 221.92 ‘00 ‘01 Pg. 268 ST-4 ‘03 ‘02 ‘04 8% 750 FV c. If your father were to offer either to make the payments calculated in part (b) ($221.92) or to give you a lump sum of $750 on January 1, 2001, which would you choose? Input: 3 8 -750 0 N I/Y PV PMT FV Output: 944.78 You should take the payments of $221.92 (FV = $1,000) instead of the $750 on 1/1/01 which will only be worth $944.78. ‘00 ‘01 Pg. 268 ST-4 ‘03 ‘02 ‘04 ?% 750 1,000 d. If you have on $750 on January 1, 2001, what interest rate, compounded annually, would you have to earn to have the necessary $1,000 on January 1, 2004? FV = PV(1+i)n 1,000 = 750(1+i)3 1,000/750 = (1+i)3 1.33331/3 = (1+i) 1.1006 = (1+i) i =10.06% ‘00 ‘01 Pg. 268 ST-4 ‘03 ‘02 ‘04 ?% -186.29 -186.29 -186.29 -186.29 1,000.00 e. Suppose you can deposit $186.29 on each January 1 from 2001 through 2004, but you still need $1,000 on January 1, 2004. What interest rate, with annual compounding, must you seek out to achieve your goal? Input: 1,000 4 0 -186.29 N I/Y PV PMT FV Output: 19.99 ‘00 ‘01 Pg. 268 ST-4 ‘03 ‘02 ‘04 8% -400 PMT PMT PMT PMT PMT PMT 1,000 f. To help you reach your $1,000 goal, your father offers to give you $400 on January 1, 2001. You will get a part-time job and make six additional payment of equal amounts each six months thereafter. If all of this money is deposited in a bank that pays eight percent, compounded semiannually, how large must each of the six payments be? Part 1 Input: 6 4 400 0 N I/Y PV PMT FV Output: 506.13 Part 2 Input: 6 4 0 493.87 N I/Y PV PMT FV Output: 74.46 ‘00 ‘01 Pg. 268 ST-4 ‘03 ‘02 ‘04 8% g. What is the effective annual rate being paid by the bank in part f? Effective Annual Rate = EAR = [1+(isimple/m)]m-1 Where: m = # of compounding periods per year EAR = [1+(.08/2)]2-1 EAR = 8.16% Pg. 270 6-6 Find the present values of the following cash flow streams under the following conditions: Year Cash Stream A Cash Stream B 1 $100 $300 2 400 400 3 400 400 4 400 400 5 500 100 Pg. 270 6-6 a. i = 8% Stream A: Stream B: 8% Rate CF Year FV PVIFi,n PV CFo= 0.00 1 $100 0.9259 $92.59 C01= 300.00 2 $400 0.8573 $342.94 F01= 1.00 3 $400 0.7938 $317.53 C02= 400.00 4 $400 0.7350 $294.01 F02= 3.00 5 $300 0.6806 $204.17 C03= 100.00 F03= 1.00 PVA $1,251.25 CPT NPV I = 8.00 NPV CPT 1,300.32 Pg. 270 6-6 b. i = 0% Stream A: Stream B: 0% Rate CF Year FV PVIFi,n PV CFo= 0.00 1 $100 1.0000 $100.00 C01= 300.00 2 $400 1.0000 $400.00 F01= 1.00 3 $400 1.0000 $400.00 C02= 400.00 4 $400 1.0000 $400.00 F02= 3.00 5 $300 1.0000 $300.00 C03= 100.00 F03= 1.00 PVA $1,600.00 CPT NPV I = 0.00 NPV CPT 1,600.00 Pg. 270 6-10a Find the future values of the following ordinary annuities: a. FV of $400 each six months for five years at a simple rate of 12 percent, compounded semiannually. Input: 10 6 0 -400 N I/Y PV PMT FV Output: 5,272.32 Pg. 270 6-10b Find the future values of the following ordinary annuities: b. FV of $200 each three months for five years at a simple rate of 12 percent, compounded quarterly. Input: 20 3 0 -200 N I/Y PV PMT FV Output: 5,374.07 Pg. 270 6-10c Find the future values of the following ordinary annuities: c. The annuities described in parts (a) and (b) have the same amount of money paid into them during the five- year period and both earn interest at the same simple rate, yet the annuity in part (b) ears $101.76 more than the one in part (a) over the five years. Why does this occur? The first payment is made 3 months earlier in (b) than in (a) and the effect of more frequent compounding both contribute to the higher earnings. Pg. 271 6-12 Lorkay Seidens Inc. just borrowed $25,000. The loan is to be repaid in equal installments at the end of each of the next five years, and the interest rate is ten percent. a. Set up an amortization schedule for the loan. b. How large must each annual payment be if the loan is for $50,000? Assume that the interest rate remains at ten percent and that the loan is paid off over five years. Pg. 271 6-12a a. 10% Interest Loan Amount 25,000 Year Payment Interest Principal Balance 1 ($6,594.94) ($2,500.00) ($4,094.94) $20,905.06 2 ($6,594.94) ($2,090.51) ($4,504.43) $16,400.63 3 ($6,594.94) ($1,640.06) ($4,954.87) $11,445.76 4 ($6,594.94) ($1,144.58) ($5,450.36) $5,995.40 5 ($6,594.94) ($599.54) ($5,995.40) $0.00 ($32,974.69) ($7,974.69) ($25,000.00) In Excel: Payment = PMT(rate,Nper,PV,FV,Type) Interest = IPMT(rate,Per,Nper,PV,FV) Principle = PPMT(rate,Per,Nper,PV,FV,Type) Balance = Loan Amount -Principal Pg. 271 6-12b a. 10% Interest Loan Amount 50,000 Year Payment Interest Principal Balance 1 ($13,189.87) ($5,000.00) ($8,189.87) $41,810.13 2 ($13,189.87) ($4,181.01) ($9,008.86) $32,801.26 3 ($13,189.87) ($3,280.13) ($9,909.75) $22,891.52 4 ($13,189.87) ($2,289.15) ($10,900.72) $11,990.79 5 ($13,189.87) ($1,199.08) ($11,990.79) $0.00 ($65,949.37) ($15,949.37) ($50,000.00) In Excel: Payment = PMT(rate,Nper,PV,FV,Type) Interest = IPMT(rate,Per,Nper,PV,FV) Principle = PPMT(rate,Per,Nper,PV,FV,Type) Balance = Loan Amount -Principal Pg. 271 6-12c c. How large must each payment be if the loan is for $50,000, the interest rate is ten percent, and the loan is paid off in equal installments at the end of each of the next ten years? This loan is for the same amount as the loan in part (b), but the payments are spread out over twice as many periods. Why are these payments not half as large as the payments on the loan in part (b)? Input: 10 10 50,000 0 N I/Y PV PMT FV Output: 8,137.27 Because the payments are spread out over a longer period of time, more interest must be paid. The total interest paid on the 10-year loan is $31,373, while the total for the 5-year loan is $15,949, however the same principal ($50,000) is repaid over a longer period of time so that the total payment per year is not doubled. Pg. 274 6-25 While Steve Bouchard was a student at the University of Florida, he borrowed $12,000 in student loans at an annual interest rate of nine percent. If Steve repays $1,500 per year, how long, to the nearest year, will it take him to repay the loan? PVAn = PMT(PVIFAi,n) Input: 9 12,000 1,500 0 N I/Y PV PMT FV Output: 14.77