XYZ Corporation invests $[tex]$13,000$[/tex] into 91-day treasury bills with an interest rate of [tex]$1.8 \%$[/tex]. If the broker charges a $[tex]$20$[/tex] commission, what is the yield?
[tex]
\begin{array}{c}
\text { yield }=[?] \%
\\
\text { yield }=\frac{\text { amount invested (interest rate) }\left(\frac{\text { days invested }}{360 \text { days }}\right)}{\text { amount invested }\left(\frac{\text { days invested }}{360 \text { days }}\right)+\text { commission }}
\end{array}
[/tex]
Answer (1)
Calculate the interest earned using the formula: Interest = 13000 × 0.018 × 360 91 = 58.96666...
Calculate the denominator using the formula: Denominator = 13000 × 360 91 + 20 = 3306.111...
Calculate the yield: Yield = 3306.111... 58.96666... = 0.017835...
Convert the yield to a percentage and round to two decimal places: Yield percentage = 0.017835... × 100 ≈ 1.79%
Explanation
Understanding the Problem We are given the formula for the yield: yield = amount invested ( 360 days days invested ) + commission amount invested (interest rate) ( 360 days days invested ) We are also given the following information:
Amount invested = $13 , 000
Interest rate = 1.8% = 0.018
Days invested = 91 days
Commission = $20
Our goal is to calculate the yield percentage.
Calculating the Interest First, we calculate the interest earned. This is the numerator of the yield formula: Interest = Amount invested × Interest rate × 360 Days invested Interest = 13000 × 0.018 × 360 91 Interest = 13000 × 0.018 × 0.252777... Interest = 58.96666...
Calculating the Denominator Next, we calculate the denominator of the yield formula: Denominator = Amount invested × 360 Days invested + Commission Denominator = 13000 × 360 91 + 20 Denominator = 13000 × 0.252777... + 20 Denominator = 3286.111... + 20 Denominator = 3306.111...
Calculating the Yield Now, we calculate the yield by dividing the interest by the denominator: Yield = Denominator Interest Yield = 3306.111... 58.96666... Yield = 0.017835...
Converting to Percentage Finally, we convert the yield to a percentage by multiplying by 100: Yield percentage = Yield × 100 Yield percentage = 0.017835... × 100 Yield percentage = 1.7835% Rounding to two decimal places, the yield is 1.78% .
Final Answer The yield percentage is approximately 1.79% .
Examples
Understanding yield is crucial in finance. For instance, if you're comparing different investment options like bonds or savings accounts, knowing the yield helps you determine which investment provides the best return relative to the amount invested. Suppose you're deciding between two bonds: Bond A with a slightly higher interest rate but a higher commission, and Bond B with a lower interest rate but a lower commission. Calculating the yield for each bond helps you make an informed decision by considering both the interest earned and the costs involved. This ensures you choose the investment that maximizes your return after accounting for all expenses.
