# CH05 - Chapter 5 Interest Rates Note All problems in this...

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Chapter 5 Interest Rates Note: All problems in this chapter are available in MyFinanceLab. An asterisk (*) indicates problems with a higher level of difficulty. 1. a. Since six months is 6 1 24 4 = of two years, using our rule += 1/4 (1 0.2) 1.0466. So the equivalent six-month rate is 4.66%. b. Since one year is half of two years = 1/2 (1.2) 1.0954. So the equivalent one-year rate is 9.54%. c. Since one month is 1/24 of two years, using our rule 1/24 (1 0.2) 1.00763. So the equivalent one-month rate is 0.763%. 2. If you deposit \$1 into a bank account that pays 5% per year for three years you will have 3 (1.05) 1.15763 = after three years. a. If the account pays 21 /2% per six months then you will have 6 (1.025) 1.15969 = after three years, so you prefer 21 /2% every six months. b. If the account pays 71 /2% per 18 months then you will have 2 (1.075) 1.15563 = after three years, so you prefer 5% per year. c. If the account pays 1/2% per month then you will have 36 (1.005) 1.19668 = after three years, so you prefer 1/2% every month. 3. Plan: Draw a timeline to fully understand the timing of the cash flows. Determine the present value of the bonus payments. Execute: 0 7 14 42 70,000 70,000 70,000 Because 7 (1.06) 1.50363, = the equivalent discount rate for a seven-year period is 50.363%. Using the annuity formula 6 70,000 1 PV 1 \$126,964 0.50363 (1.50363) ⎛⎞ =− = ⎜⎟ ⎝⎠ Evaluate: The present value of the bonus payments is \$126,964.

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42 Berk/DeMarzo/Harford • Fundamentals of Corporate Finance 4. Plan: Determine the EAR for each investment option. Execute: For \$1 invested in an account with 10% APR with monthly compounding you will have 12 0.1 1 \$1.10471 12 ⎛⎞ += ⎜⎟ ⎝⎠ So the EAR is 10.471%. For \$1 invested in an account with 10% APR with annual compounding you will have (1 0.1) \$1.10. So the EAR is 10%. For \$1 invested in an account with 9% APR with daily compounding you will have 365 0.09 1 1.09416 365 So the EAR is 9.416%. Evaluate: One dollar invested at 10% APR compounded monthly will grow to \$1.10471 in one year. This is greater than the values for the other two investments and therefore is superior. 5. Using the formula for converting from an EAR to an APR quote APR 11 . 0 5 k k Solving for the APR 1/ APR ((1.05) 1) k k =− With annual payments k = 1, so APR = 5%. With semiannual payments k = 2, so APR = 4.939%. With monthly payments k = 12, so APR = 4.889%. 6. Plan: Determine the present value of the annuity. Execute: Using the PV of an annuity formula with N = 10 payments and C = \$100 with r = 4.067% per six-month interval, since there is an 8% APR with monthly compounding: 8%/12 = 0.6667% per month, or (1.006667)^6 1 = 4.067% per six months. = 10 PV 100 1 \$808.39 0.04067 1.04067 Evaluate: The PV of the annuity is \$808.39.
Chapter 5 Interest Rates 43 7. Plan: Draw a timeline to demonstrate when the tuition payments will be needed. Then calculate the PV of the tuition payments. Execute: 0 1 2 1 4 0 1 2 8 10,000 10,000 10,000 4% APR (semiannual) implies a semiannual discount rate of = 4% 2 2%. So, 8 10,000 1 PV 1 0.02 (1.02) \$73,254.81 ⎛⎞ =− ⎜⎟ ⎝⎠ = Evaluate: You will have to deposit \$73,254.81 in the bank today in order to be able to make the tuition payments over the next four years.

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CH05 - Chapter 5 Interest Rates Note All problems in this...

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