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# chapter5 - Chapter 5 Interest Rates 5-1 a Since 6 months is...

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Chapter 5 Interest Rates 5-1. a. Since 6 months is 6 1 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 70,000 70,000 70,000 Because ( ) 7 1.06 1.50363 = , the equivalent discount rate for a 7-year period 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: The Core 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 1 1.05 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 1 1 PV 100 1 \$808.39 .04067 1.04067 = × =
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 1 1 1 0.00499167 1.00499167 = =

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38 Berk/DeMarzo • Corporate Finance: The Core 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 1 1 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 =
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 × = =

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