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XYZ Corporation invests [tex]$11,000 into 91-day treasury bills with an interest rate of 2.3%. If the broker charges a $[/tex]20 commission, what is the yield?

Yield (%) = [tex]\(\frac{\text{Amount invested} \times \text{interest rate} \times \left(\frac{\text{days invested}}{360 \text{ days}}\right)}{\text{Amount invested} + \text{commission}}\)[/tex]

Give your answer as a percent rounded to the nearest hundredth.

Sagot :

GinnyAnswer
To solve this problem and determine the yield of the investment, we need to follow several steps systematically. Let's break it down:

### Step 1: Gather the Essential Information
- Initial investment amount ([tex]\(P\)[/tex]): \[tex]$11,000 - Annual interest rate (\(r\)): 2.3% - Number of days invested (\(t\)): 91 days - Broker's commission: \$[/tex]20
- Number of days in a year (financial convention): 360 days

### Step 2: Convert the Annual Interest Rate to a Decimal
The annual interest rate given as a percentage needs to be converted to a decimal for calculations:
[tex]\[ r = 2.3\% = \frac{2.3}{100} = 0.023 \][/tex]

### Step 3: Calculate the Interest Earned
The interest earned over the investment period can be calculated using the formula for simple interest:
[tex]\[ \text{Interest Earned} = P \times r \times \left( \frac{t}{360} \right) \][/tex]

Substituting the values we get:
[tex]\[ \text{Interest Earned} = 11,000 \times 0.023 \times \left( \frac{91}{360} \right) \][/tex]

### Step 4: Calculate the Yield
Yield is the percentage return based on the initial investment and the commission. The formula for yield, considering the commission, is:
[tex]\[ \text{Yield} = \left( \frac{\text{Interest Earned}}{P + \text{Commission}} \right) \times 100 \% \][/tex]

### Step 5: Plug in the Values and Compute
First, calculate the interest earned:
[tex]\[ \text{Interest Earned} = 11,000 \times 0.023 \times \left( \frac{91}{360} \right) = 63.952777777777776 \][/tex]

Then, calculate the yield:
[tex]\[ \text{Yield} = \left( \frac{63.952777777777776}{11,000 + 20} \right) \times 100 \][/tex]

Simplify the yield calculation:
[tex]\[ \text{Yield} = \left( \frac{63.952777777777776}{11,020} \right) \times 100 \][/tex]
[tex]\[ \text{Yield} \approx 0.5804 \][/tex]

### Step 6: Round the Yield to the Nearest Hundredth
As requested, we need the yield as a percent rounded to the nearest hundredth:
[tex]\[ \text{Yield} \approx 0.58\% \][/tex]

### Final Answer
The yield of the investment, taking into account the broker's commission, is approximately [tex]\( \boxed{0.58\%} \)[/tex].
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