XYZ Corporation invests $[tex]6,000[/tex] into 91-day treasury bills with an interest rate of [tex]2.6[/tex]%. If the broker charges a $[tex]25[/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 × 360 days invested .
Calculate the denominator of the yield formula: denominator = amount invested × 360 days invested + commission .
Calculate the yield: yield = denominator interest .
Convert the yield to a percentage and round to the nearest hundredth: 2.56% .
Explanation
Identify Given Information and Formula First, let's identify the given information:
Amount invested: $6 , 000
Interest rate: 2.6% = 0.026
Days invested: 91 days
Commission: $25
The formula for the yield is given as:
yield = amount invested × 360 days invested + commission amount invested × interest rate × 360 days invested
We will use this formula to calculate the yield.
Substitute Values and Calculate Now, let's substitute the given values into the formula:
yield = 6000 × 360 91 + 25 6000 × 0.026 × 360 91
First, calculate the numerator:
6000 × 0.026 × 360 91 = 6000 × 0.026 × 0.252777... ≈ 39.4233
Next, calculate the denominator:
6000 × 360 91 + 25 = 6000 × 0.252777... + 25 = 1516.666... + 25 = 1541.666...
Now, divide the numerator by the denominator:
yield = 1541.666... 39.4233 ≈ 0.02557
To express the yield as a percentage, multiply by 100:
yield ≈ 0.02557 × 100 = 2.557%
Finally, round the yield to the nearest hundredth:
yield ≈ 2.56%
Final Answer Therefore, the yield is approximately 2.56% .
Examples
Understanding yield calculations is crucial in finance. For instance, if you're comparing different investment options like treasury bills or bonds, knowing the yield helps you determine which investment provides a better return relative to the risk and costs involved. In this case, calculating the yield on a treasury bill helps an investor understand the actual percentage return they'll receive after accounting for the broker's commission, allowing for a more informed investment decision.
