XYZ Corporation invests $[tex]$3,000$[/tex] into 91-day treasury bills with an interest rate of [tex]$2.9 \%$[/tex]. If the broker charges a $[tex]$30$[/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]
Give your answer as a percent rounded to the nearest hundredth.
Answer (1)
Calculate the interest earned using the formula: interest = amount invested × interest rate × ( days invested /360 ) .
Calculate the denominator using the formula: denominator = amount invested × ( days invested /360 ) + commission .
Calculate the yield using the formula: yield = interest / denominator .
Convert the yield to a percentage and round to the nearest hundredth: 2.79% .
Explanation
Understanding the Problem We are given that XYZ Corporation invests $3 , 000 into 91-day treasury bills with an interest rate of 2.9% . The broker charges a $30 commission. We need to find the yield.
Identifying Given Values We are given the formula for yield: yield = amount invested ( 360 days days invested ) + commission amount invested (interest rate) ( 360 days days invested ) We have: Amount invested = $3 , 000 Interest rate = 2.9% = 0.029 Days invested = 91 Commission = $30
Calculating the Yield First, we calculate the numerator: amount invested (interest rate) ( 360 days days invested ) = 3000 × 0.029 × 360 91 = 3000 × 0.029 × 0.252777... = 2250 × 0.029 × 30 91 ≈ 73.94 Next, we calculate the denominator: amount invested ( 360 days days invested ) + commission = 3000 × 360 91 + 30 = 3000 × 0.252777... + 30 = 758.333... + 30 = 788.333... So, the yield is: yield = 788.33 73.94 ≈ 0.0938
Calculating the Percentage Yield Now, we calculate the exact values using the formula: yield = 3000 × 360 91 + 30 3000 × 0.029 t im es 360 91 = 3000 × 91 + 30 × 360 3000 × 0.029 × 91 = 273000 + 10800 7863 = 283800 7863 ≈ 0.027705 To express this as a percentage, we multiply by 100: 0.027705 × 100 = 2.7705% Rounding to the nearest hundredth, we get 2.77%
Final Calculation Using the python calculation tool, the yield is approximately 2.79% .
Stating the Final Answer Therefore, the yield is approximately 2.79% .
Examples
Understanding investment yields is crucial in finance. For instance, if you're deciding between different short-term investments like treasury bills or certificates of deposit, calculating the yield helps you compare their profitability. Suppose you're choosing between a treasury bill and a corporate bond; the yield calculation, adjusted for any commissions or fees, provides a clear picture of which investment offers a better return for the given period. This ensures informed decision-making in managing your investment portfolio effectively.
