A stock just paid a dividend of $$ \$ 5.05 $$ and is expected to maintain a constant dividend growth rate of 4.2 percent indefinitely. If the current stock price is $$ \$ 69 $$, what is the required return on the stock?

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UpStudy AI · Accepted Answer

To find the required return on the stock, we can use the Gordon Growth Model (also known as the Dividend Discount Model for a stock with constant growth). The formula is given by:

$$
r = \frac{D_1}{P_0} + g
$$

Where:
- $$ r $$ = required return
- $$ D_1 $$ = dividend expected next year
- $$ P_0 $$ = current stock price
- $$ g $$ = growth rate of the dividend

### Step 1: Calculate $$ D_1 $$

The dividend expected next year ($$ D_1 $$) can be calculated using the formula:

$$
D_1 = D_0 \times (1 + g)
$$

Where:
- $$ D_0 = 5.05 $$ (the most recent dividend)
- $$ g = 0.042 $$ (the growth rate)

### Step 2: Substitute the values

Now, we can calculate $$ D_1 $$:

$$
D_1 = 5.05 \times (1 + 0.042)
$$

### Step 3: Calculate $$ r $$

After calculating $$ D_1 $$, we can substitute $$ D_1 $$, $$ P_0 $$, and $$ g $$ into the required return formula to find $$ r $$.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$5.05\left(1+0.042\right)$$
- step1: Add the numbers:
  $$5.05\times 1.042$$
- step2: Multiply the numbers:
  $$5.2621$$
Calculate or simplify the expression $$ (D_1 / 69) + 0.042 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{D_{1}}{69}+0.042$$
- step1: Convert the expressions:
  $$\frac{D_{1}}{69}+\frac{21}{500}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{D_{1}\times 500}{69\times 500}+\frac{21\times 69}{500\times 69}$$
- step3: Multiply the numbers:
  $$\frac{D_{1}\times 500}{34500}+\frac{21\times 69}{500\times 69}$$
- step4: Multiply the numbers:
  $$\frac{D_{1}\times 500}{34500}+\frac{21\times 69}{34500}$$
- step5: Transform the expression:
  $$\frac{D_{1}\times 500+21\times 69}{34500}$$
- step6: Reorder the terms:
  $$\frac{500D_{1}+21\times 69}{34500}$$
- step7: Multiply the numbers:
  $$\frac{500D_{1}+1449}{34500}$$
Calculate or simplify the expression $$ (5.2621 / 69) + 0.042 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{5.2621}{69}\right)+0.042$$
- step1: Divide the terms:
  $$\frac{52621}{690000}+0.042$$
- step2: Convert the expressions:
  $$\frac{52621}{690000}+\frac{21}{500}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{52621}{690000}+\frac{21\times 1380}{500\times 1380}$$
- step4: Multiply the numbers:
  $$\frac{52621}{690000}+\frac{21\times 1380}{690000}$$
- step5: Transform the expression:
  $$\frac{52621+21\times 1380}{690000}$$
- step6: Multiply the numbers:
  $$\frac{52621+28980}{690000}$$
- step7: Add the numbers:
  $$\frac{81601}{690000}$$
### Step 1: Calculate $$ D_1 $$

We calculated $$ D_1 $$ as follows:

$$
D_1 = 5.05 \times (1 + 0.042) = 5.2621
$$

### Step 2: Calculate the Required Return $$ r $$

Now, we substitute $$ D_1 $$, $$ P_0 $$, and $$ g $$ into the required return formula:

$$
r = \frac{D_1}{P_0} + g = \frac{5.2621}{69} + 0.042
$$

Calculating this gives:

$$
r \approx 0.118262
$$

### Final Result

Thus, the required return on the stock is approximately:

$$
r \approx 11.83\%
$$

This means that investors would require a return of about 11.83% on this stock given the current price and expected growth in dividends.
The required return on the stock is approximately 11.83%.

A stock just paid a dividend of \( \$ 5.05 \) and is expected to maintain a constant dividend growth rate of 4.2 percent indefinitely. If the current stock price is \( \$ 69 \), what is the required return on the stock?

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