1. $1.67 million.
2. $5 million.
3. $3.33 million.
4. $1.1 million.
Choose the correct option.
The correct answer is option 2. The present value of a share equals real earnings per share divided by the appropriate real interest rate at which these earnings should be discounted. The appropriate interest rate is the real interest rate inclusive of an allowance for risk. The real interest rate (both contracted and realized) on bonds is .08 - .03 = .05. Add .01 to this to cover the additional risk of owning shares in the firm rather than government bonds and we arrive at a real capitalization rate of .06. The value of the firm is thus $5 million ($3/.06 per share multiplied by 100,000 shares).
If you chose option 1 you must have assumed that only dividends matter to the owner of the firm. This is not the case because earnings of the firm that are not paid out as dividends are by definition invested in expansion of the firm's capital stock, leading to equivalent increases in future earnings and in the value of the firm. This will appear as future capital gains.
If you chose option 3 you must have thought that the appropriate interest rate at which the firm's assets should be discounted in calculating the present value is the nominal interest rate (the nominal rate on bonds plus 1 percent risk premium). The correct interest rate is the real interest rate because the question states that the real earnings of the firm are not expected to change---the firm's nominal earnings (and hence the market value of its shares) will thus tend to move up and down with the general price level. When the future path of real earnings is known, we discount by the real interest rate to get the present value. When all we have is the future path of nominal earnings, we obtain the present value by discounting future nominal earnings by the nominal interest rate, which includes a premium for expected inflation.
Choice 4 combines the error from choosing option 1 with the error from choosing option 3 .