An investor buys 100 shares of a $40 stock that pays an annual cash dividend of $2 a share (a 5 percent dividend yield) and signs up for the dividend reinvestment plan.
A. If neither the dividend nor the price changes, how many shares will the investor have at the end of ten years? How much will the position in the stock be worth?
B. If the price of the stock rises by 6 percent annually but the dividend remains at $2 a share, how many shares are purchased each year for the next ten years? How much is the total position worth at the end of ten years?
C. If the price of the stock rises by 6 percent annually but the dividend rises by only 3 percent annually, how many shares are purchased each year for the next ten years? Since dividend plans credit fractional shares, use three decimal places in parts (b) and (c).
A. If neither the dividend nor the price changes, to determine the number of shares the investor will have at the end of ten years, you need to consider the dividend reinvestment plan. The investor will use the cash dividend received each year to purchase additional shares at the prevailing market price.
The initial investment is 100 shares. Since the dividend yield is 5%, the investor will receive a $2 dividend per share each year. Therefore, the investor will receive $2 x 100 = $200 in the first year.
Assuming the price of the stock remains constant at $40, the investor can purchase $200 / $40 = 5 additional shares at the end of the first year.
At the beginning of the second year, the investor will have 100 initial shares + 5 additional shares = 105 shares. Using the same calculation, the investor will receive $2 x 105 = $210 as a dividend in the second year. This amount can then be used to purchase additional shares at the prevailing market price.
By repeating this process for ten years, the investor will have accumulated additional shares based on the annual dividend and the market price. At the end of ten years, you sum up the initial shares and the accumulated additional shares to find the total number of shares the investor will have.
B. If the price of the stock rises by 6% annually, this will affect the calculations for the number of shares purchased each year and the total position worth at the end of ten years.
Following the same approach as in part A, the investor will initially have 100 shares. However, each year, the market price will increase by 6%, while the dividend remains at $2 per share.
Let's calculate the number of shares purchased each year for ten years:
- In the first year, the investor receives a $200 dividend ($2 x 100 shares) and can purchase additional shares. Since the market price has risen by 6%, the price per share is now $40 + ($40 x 6%) = $42.40. Therefore, the investor can purchase $200 / $42.40 = 4.716 shares.
- In the second year, the investor will have 100 initial shares + 4.716 additional shares = 104.716 shares. The dividend received will be $2 x 104.716 = $209.432. The market price for the second year will be $42.40 + ($42.40 x 6%) = $45.02, allowing the investor to purchase $209.432 / $45.02 = 4.653 shares.
- This process is repeated for ten years to calculate the total number of shares purchased.
To determine the total position worth at the end of ten years, you sum up the value of the initial shares and the value of the accumulated additional shares. Multiply the number of shares owned by the market price at the end of ten years to calculate the total position worth.
C. If the price of the stock rises by 6% annually, but the dividend only rises by 3% annually, the calculation for the number of shares purchased each year will change.
Following the same method as in part B, you need to adjust the dividend growth rate to 3%. Let's calculate the number of shares purchased each year for ten years:
- In the first year, the investor receives a $200 dividend ($2 x 100 shares) and can purchase additional shares. The market price per share is $40 + ($40 x 6%) = $42.40. Using the adjusted dividend growth rate of 3%, the dividend for the second year will be $2 + ($2 x 3%) = $2.06 per share. Therefore, the investor can purchase $200 / $42.40 = 4.716 shares.
- Repeat this process for ten years to calculate the total number of shares purchased.
To determine the total position worth at the end of ten years, you sum up the value of the initial shares and the value of the accumulated additional shares. Multiply the number of shares owned by the market price at the end of ten years to calculate the total position worth.