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}\right)}{\text { amount invested }\left(\frac{\text { days invested }}{360}\right)+\text { commission }}
\end{array}
[/tex]
Give your answer as a percent rounded to the nearest hundredth.
Answer (1)
Calculate the interest earned: I n t eres t = 6000 × 0.026 × 360 91 = 40.08 .
Calculate the denominator: De n o mina t or = 40.08 + 25 = 65.08 .
Calculate the yield: Yi e l d = 65.08 40.08 = 0.61586 .
Express as a percentage and round: Yi e l d % = 0.61586 × 100 ≈ 61.59% . The final answer is 61.59 .
Explanation
Understanding the Problem Let's break down this investment problem step by step so you can understand how to calculate the yield. We're given the following information:
Amount invested: $6,000
Interest rate: 2.6% (or 0.026 as a decimal)
Days invested: 91 days
Commission: $25
We need to find the yield, which is the return on the investment, taking into account the commission paid to the broker.
Calculating the Interest Earned First, let's calculate the interest earned from the treasury bills. The formula for simple interest is:
I n t eres t = P r in c i p a l × R a t e × T im e
In our case:
P r in c i p a l = $6 , 000 R a t e = 0.026 T im e = 360 91 (since the investment is for 91 days out of a 360-day year)
So, the interest earned is:
I n t eres t = 6000 × 0.026 × 360 91
Determining the Interest Value Now, let's calculate the value of the interest earned:
I n t eres t = 6000 × 0.026 × 360 91 = 40.083333...
So, the interest earned is approximately $40.08.
Calculating the Denominator Next, we need to calculate the denominator of the yield formula, which includes the interest earned and the commission:
De n o mina t or = I n t eres t + C o mmi ss i o n
De n o mina t or = 40.083333 + 25 = 65.083333...
So, the denominator is approximately $65.08.
Calculating the Yield Now we can calculate the yield using the formula:
Yi e l d = De n o mina t or I n t eres t = 65.083333 40.083333
Yi e l d = 0.61586...
Converting to Percentage To express the yield as a percentage, we multiply by 100:
Yi e l d % = 0.61586 × 100 = 61.586...%
Rounding to Nearest Hundredth Finally, we round the percentage to the nearest hundredth:
Yi e l d % ≈ 61.59%
Final Answer Therefore, the yield of the investment, after considering the broker's commission, is approximately 61.59%.
Examples
Understanding investment yields is crucial in finance. For instance, suppose you're comparing two different treasury bills. One offers a slightly higher interest rate but comes with a broker commission, while the other has a lower rate but no commission. By calculating the yield as we did, you can determine which bill gives you a better return on your investment after accounting for all costs. This kind of analysis helps in making informed financial decisions, whether you're investing in treasury bills, stocks, or other financial instruments. Let's say Treasury Bill A has an interest rate of 3% and a commission of $30, while Treasury Bill B has an interest rate of 2.7% and a commission of $10. Calculating the yield for each will show which is the better investment.
