BD_SM_c05 - Chapter 5 Interest Rates 5-1. a. Since 6 months...

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Chapter 5 Interest Rates 5-1. a. Since 6 months is 61 24 4 = of 2 years, using our rule () 1 4 1 0.2 1.0466 += So the equivalent 6 month rate is 4.66% b. Since one year is half of 2 years 1 2 1.2 1.0954 = So the equivalent 1 year rate is 9.54% c. Since one month is 1 24 of 2 years, using our rule 1 24 1 0.2 1.00763 So the equivalent 1 month rate is 0.763% 5-2. If you deposit $1 into a bank account that pays 5% per year for 3 years you will have .. = 3 1 05 1 15763 after 3 years a. If the account pays 1 2 2 % per 6 months then you will have 6 1.025 1.15969 = after 3 years, so you prefer 1 2 2 % every 6 months b. If the account pays 1 2 7 % per 18 months then you will have 2 1.075 1.15563 = after 3 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 3 years, so you prefer 1 2 % every month 5-3. Timeline: 0 7 14 42 0 1 2 6 7 1.06 1.50363 = So the equivalent discount rate is 50.363%. Using the annuity formula 6 70,000 1 PV 1 $126,964 0.50363 1.50363 =− = ⎛⎞ ⎜⎟ ⎝⎠
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36 Berk/DeMarzo Corporate Finance 5-4. For a $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 a $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 a $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% 5-5. Using the formula for converting from an EAR to an APR quote k APR 11 . 0 5 k Solving for the APR 1 k APR 1.05 1 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% 5-6. Using the PV of an annuity formula with N = 10 payments and C = $100 with r = 4.067% per 6 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 6 months. 10 PV 100 1 $808.39 .04067 1.04067 =
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Chapter 5 Interest Rates 37 5-7. Timeline: 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 =− = ⎛⎞ ⎜⎟ ⎝⎠ 5-8. Using the formula for computing the discount rate from an APR quote: 5 Discount Rate 0.41667% 12 == 5-9. Timeline: 0 1 2 3 4 60 –8,000 C C C C C 5.99 APR monthly implies a discount rate of 5.99 0.499167% 12 = Using the formula for computing a loan payment 60 8,000 C $154.63 11 1 0.00499167 1.00499167
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38 Berk/DeMarzo Corporate Finance 5-10. Timeline: 0 1 2 3 4 360 –150,000 C C C C C () 1 12 1 0.05375 1.0043725 += So 3 8 5 % EAR implies a discount rate of 0.43725% Using the formula for computing a loan payment 360 150,000 C $828.02 11 1 0.0043725 1.0043725 == ⎛⎞ ⎜⎟ ⎝⎠ 5-11. Timeline: 56 57 58 360 0 1 2 304 2,356 2,356 2,356 To find out what is owed compute the PV of the remaining payments using the loan interest rate to compute the discount rate: 6.375 Discount Rate 0.53125% 12 304 2,356 1 PV 1 $354,900 0.0053125 1.0053125 =− = 5-12. First we need to compute the original loan payment Timeline #1: 0 1 2 3 360 –800,000 C C C C 1 4 5 % APR (monthly) implies a discount rate of 5.25 0.4375% 12 =
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Chapter 5 Interest Rates 39 Using the formula for a loan payment () 360 800,000 0.004375 C $4, 417.63 1 1 1.004375 × == ⎛⎞ ⎜⎟ ⎝⎠ Now we can compute the PV of continuing to make these payments The timeline is Timeline #2: 222 223 224 225 360 0 1 2 3 138 4,417.63 4,417.63 4,417.63 4,417.63 Using the formula for the PV of an annuity 138 4, 417.63 1 PV 1 $456,931.41 0.004375 1.004375 =− = So, you would keep $1,000,000 - $456,931 = $543,069.
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BD_SM_c05 - Chapter 5 Interest Rates 5-1. a. Since 6 months...

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