Final answer:
To find the effective rate of interest on a 13-week Treasury bill with a 4% rate, you first convert the discount rate to an annualized rate by dividing the time period by the number of weeks in a year. As Treasury bills do not compound within the maturity period, the EAR is simply the annualized rate, which in this case is 16%.
Step-by-step explanation:
To calculate the effective rate of interest for a $10,000 Treasury bill with a 4% interest rate for 13 weeks, we use the formula for the effective annual rate (EAR). The formula considers the nominal annual interest rate and the number of compounding periods per year.
The nominal annual interest rate is given by the face value minus the price paid for the Treasury bill, divided by the price paid, and then multiplied by 100 to get the percentage. However, the question only provides the interest rate but not the purchase price, so I'm assuming this given interest rate is the discount rate. First, we need to calculate the annualized discount rate and then convert it to the effective annual rate since Treasury bills are typically discounted securities.
To calculate the annualized discount rate:
- Convert the 13 weeks to a fraction of the year: 13/52 = 0.25 years.
- Multiply the stated discount rate (4%) by the annualization factor (1/0.25): 4% * (1/0.25) = 16%.
Now, to convert the annualized discount rate to the effective annual rate (EAR):
- EAR = (1 + nominal interest rate / number of compounding periods) ^ number of compounding periods - 1.
- Since Treasury bills do not have compounding within their maturity period, the number of compounding periods is 1.
- EAR = (1 + 0.16 / 1) ^ 1 - 1 = 0.16 or 16%.
Note: If the purchase price was lower than the face value to account for the 4% interest over 13 weeks, the effective annual rate would be higher than 16%.