Solutions-Chapter5

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You have found three investment choices for a one-year deposit: 10% APR compounded monthly, 10% APR compounded annually, and 9% APR compounded daily. Compute the EAR for each investment choice. (Assume that there are 365 days in the year.)
For a $1 invested in an account with 10% APR with monthly compounding you will have:
 0.1 12
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:
 0.09 365
1  1.09416  365
So the EAR is 9.416%.
5-5. You are considering moving your money to new bank offering a one-year CD that pays an 8% APR with monthly compounding. Your current bank’s manager offers to match the rate you have been offered. The account at your current bank would pay interest every six months. How much interest will you need to earn every six months to match the CD?
With 8% APR, we can calculate the EAR as follows:  0.08 12
EAR= 1  =8.3%  12
1
Over six months this works out to be 1.0832 1  0.040672. Hence you need to earn 4.0672% interest
rate to match the CD.
5-7. Suppose the interest rate is 8% APR with monthly compounding. What is the present value of an annuity that pays $100 every six months for five years?
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.
11
PV 100 1  $808.39  10
.04067  1.04067 
5-9. Suppose you invest $100 in a bank account, and five years later it has grown to $134.39.
a. What APR did you receive, if the interest was compounded semiannually?
b. What APR did you receive if the interest was compounded monthly?
The EAR can be calculated as follows:
1/5
f 1/5
1  1.3439 1  6.0897% = 0.060897 
p
a) Using the formula for EAR, we can calculate the APR for semiannual compounding.
1212 APR2EAR1 121.060897 16%
b) Similarly we can calculate the APR for monthly compounding:  112   112 
APR12 EAR1 1 12 1.060897 1 5.926%
5-13. Oppenheimer Bank is offering a 30-year mortgage with an EAR of 5 3 8 %. If you plan to borrow
$150,000, what will your monthly payment be?
Timeline:
01234 360
–150,000 C C C C C
1
1  0.0537512  1.0043725
So 5 83 % EAR implies a discount rate of 0.43725% Using the formula for computing a loan payment
C 
150, 000 11
 $828.02
0.004372511.0043725360  
5-40. You firm is considering the purchase of a new office phone system. You can either pay $32,000 now, or $1000 per month for 36 months.
a. Suppose your firm currently borrows at a rate of 6% per year (APR with monthly compounding). Which payment plan is more attractive?
b. Suppose your firm currently borrows at a rate of 18% per year (APR with monthly compounding). Which payment plan would be more attractive in this case?
a. The payments are as risky as the firm’s other debt. So opportunity cost = debt rate. PV(36 month annuity of 1000 at 6%/12 per month) = $32,871. So pay cash.
b. PV(annuity at 18%/12 per mo) = $27,661. So pay over time.