Certainly! Let's break down the steps to determine the effective rate of an 8% 13-week Treasury bill with a [tex]$10,000 face value in a clear and detailed manner.
### Step 1: Understanding the Given Information
- Face Value: $[/tex]10,000
- Annual Interest Rate: 8% (or 0.08 as a decimal)
- Treasury Bill Duration: 13 weeks
- Weeks in a Year: 52 weeks
### Step 2: Calculate the Discount Amount
The discount amount for the Treasury bill can be calculated using the formula:
[tex]\[ \text{Discount Amount} = \text{Face Value} \times \left( \text{Annual Interest Rate} \times \frac{\text{Treasury Bill Duration}}{\text{Weeks in a Year}} \right) \][/tex]
Plugging in the values:
[tex]\[ \text{Discount Amount} = 10000 \times \left( 0.08 \times \frac{13}{52} \right) \][/tex]
Performing the multiplication inside the parentheses first:
[tex]\[ 0.08 \times \frac{13}{52} = 0.08 \times 0.25 = 0.02 \][/tex]
Now, calculating the discount amount:
[tex]\[ \text{Discount Amount} = 10000 \times 0.02 = 200 \][/tex]
So, the discount amount is \[tex]$200.
### Step 3: Calculate the Purchase Price
The purchase price of the Treasury bill is the face value minus the discount amount:
\[ \text{Purchase Price} = \text{Face Value} - \text{Discount Amount} \]
Substituting the known values:
\[ \text{Purchase Price} = 10000 - 200 = 9800 \]
So, the purchase price is \$[/tex]9,800.
### Step 4: Calculate the Effective Rate
The effective rate can be calculated using the formula:
[tex]\[ \text{Effective Rate} = \frac{\text{Discount Amount}}{\text{Purchase Price}} \times \frac{\text{Weeks in a Year}}{\text{Treasury Bill Duration}} \][/tex]
Substituting the known values:
[tex]\[ \text{Effective Rate} = \frac{200}{9800} \times \frac{52}{13} \][/tex]
First, simplifying the fraction:
[tex]\[ \frac{200}{9800} = \frac{1}{49} \approx 0.0204 \][/tex]
Then, calculating the other fraction:
[tex]\[ \frac{52}{13} = 4 \][/tex]
Finally, computing the effective rate:
[tex]\[ \text{Effective Rate} = 0.0204 \times 4 = 0.08163265306122448 \][/tex]
So, the effective rate is approximately [tex]\(0.0816\)[/tex] or 8.16%.
### Summary
- Discount Amount: \[tex]$200
- Purchase Price: \$[/tex]9,800
- Effective Rate: Approximately 8.16%
Therefore, the effective rate of an 8% 13-week Treasury bill with a \$10,000 face value is approximately 8.16%.
