A firm issues three-month commercial paper with $200,000 face value and receives $192,000. What is the EAR the firm is paying for these funds?
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To calculate the effective annual rate (EAR), we need to know the face value of the commercial paper, the amount received, and the time period. In this case, we are given:
Face value of commercial paper (PV) = $200,000
Amount received (P) = $192,000
Time period (t) = 3 months
In order to calculate the EAR, we can use the formula:
EAR = [(1 + r/n)^n - 1]
where:
r = nominal interest rate per compounding period
n = number of compounding periods per year
Step 1: Calculate the interest paid (I) on the commercial paper:
I = PV - P
I = $200,000 - $192,000
I = $8,000
Step 2: Calculate the nominal interest rate per compounding period (r):
r = I / P
r = $8,000 / $192,000
r = 0.0417 or 4.17%
Step 3: Calculate the number of compounding periods per year (n):
Since the time period is given as 3 months, we need to convert it to a yearly time frame. There are 12 months in a year, so:
n = 12 / 3
n = 4
Step 4: Calculate the EAR:
EAR = [(1 + r/n)^n - 1]
EAR = [(1 + 0.0417/4)^4 - 1]
EAR = [(1 + 0.0104)^4 - 1]
EAR = (1.0104)^4 - 1
EAR = 1.0412 - 1
EAR = 0.0412 or 4.12%
Therefore, the firm is paying an effective annual rate (EAR) of 4.12% for these funds.
To find the effective annual rate (EAR) that the firm is paying for the funds, we need to use the formula for EAR:
EAR = (1 + r/m)^m - 1
where:
r = nominal interest rate
m = number of compounding periods per year
In this case, since the commercial paper is for three months, the compounding period is quarterly (four times a year). Therefore, m = 4.
To calculate the nominal interest rate (r), we need to find the interest earned on the commercial paper. The interest earned can be determined by subtracting the amount received ($192,000) from the face value ($200,000):
Interest earned = Face value - Amount received
Interest earned = $200,000 - $192,000
Interest earned = $8,000
Now, we can calculate the nominal interest rate (r) using the formula:
r = Interest earned / Amount received
r = $8,000 / $192,000
r = 0.0417 (rounded to four decimal places)
Finally, we can plug in the values for r and m into the EAR formula to find the EAR:
EAR = (1 + r/m)^m - 1
EAR = (1 + 0.0417/4)^4 - 1
EAR = (1 + 0.0104)^4 - 1
EAR = (1.0104)^4 - 1
EAR = 0.0417 or 4.17% (rounded to two decimal places)
Therefore, the effective annual rate (EAR) that the firm is paying for these funds is 4.17%.