Sure! Let's break down the problem step-by-step.
### Step 1: Calculate the Interest Rate
First, we need to find the interest rate provided by a similar stock.
We know:
- The price of the similar stock is [tex]$40 per share.
- The dividend of the similar stock is $[/tex]1 per quarter.
To find the quarterly interest rate for the similar stock, we use the formula:
[tex]\[ \text{Interest Rate (Quarterly)} = \frac{\text{Quarterly Dividend}}{\text{Price per Share}} \][/tex]
Plugging in the numbers:
[tex]\[ \text{Interest Rate (Quarterly)} = \frac{1 \text{ USD}}{40 \text{ USD}} = 0.025 \text{ or } 2.5\% \][/tex]
Next, convert the quarterly interest rate to an annual interest rate:
[tex]\[ \text{Annual Interest Rate} = \text{Quarterly Interest Rate} \times 4 \][/tex]
So:
[tex]\[ \text{Annual Interest Rate} = 0.025 \times 4 = 0.10 \text{ or } 10\% \][/tex]
### Step 2: Calculate the Dividend
With the annual interest rate known, we can now calculate the required annual dividend for the preferred stock of company A.
The price of company A’s preferred stock is [tex]$100 per share.
Using the formula for dividend:
\[ \text{Annual Dividend} = \text{Annual Interest Rate} \times \text{Price per Share} \]
Plugging in the values:
\[ \text{Annual Dividend} = 0.10 \times 100 \text{ USD} = 10 \text{ USD} \]
This means the preferred stock must offer an annual dividend of $[/tex]10 per share.
For completeness, we can also calculate the quarterly dividend, given that dividends are often paid quarterly:
[tex]\[ \text{Quarterly Dividend} = \frac{\text{Annual Dividend}}{4} \][/tex]
So:
[tex]\[ \text{Quarterly Dividend} = \frac{10 \text{ USD}}{4} = 2.5 \text{ USD} \][/tex]
### Summary
1) Interest Rate (Annual): 10%
2) Dividend:
- Annual Dividend: [tex]$10 per share.
- Quarterly Dividend: $[/tex]2.5 per share.
Thus, company A will have to offer an annual dividend of [tex]$10 per share (or $[/tex]2.5 per quarter) to match the interest rate of the similar issue.
