## Fair Price of Common Stock |

**Dividend Discount Model DDM):** An equation (model) that takes the present value of future dividends and cash flows. An equation is a type of model and discount means to "take the present value of." So a dividend discount model is an equation that takes the present value of future dividends.

To determine the fair price of a common stock, we use the principle of finance known as the Valuation Principle:

The fair price of an asset is equal to the present value of the asset's future cash flows.

Since a stock has no maturity, investors can expect to receive dividends out to infinity. However, an individual investor won't hold the stock until infinity; he or she will sell the stock at some point in the future. So the cash flows for an individual investor will be (1) dividends while the stock is owned, and (2) the selling price at some point in the future. So we estimate the fair price by adding the present value of the dividends and the present value of the future selling price. This process is known as a *dividend discount model*.

This means that we need to conduct four steps: (1) determine the required rate of return using the capital asset pricing model (CAPM), (2) estimate the future dividends, (3) estimate the future selling price, and (4) take the present value of the dividends and selling price.

The required rate of return on the stock depends upon the risk level, or beta, of the stock. The capital asset pricing model (CAPM) is used to determine this required rate of return. The CAPM is an equation:

Required return = Risk-free rate of interest + beta * (Expected return on the overall market - the risk-free rate of interest)

Required return = R_{f} + beta * (Return_{market} - R_{f})

For example, if we assume a four-year holding period for the stock, we would use the 4-year Treasury note as the value of the risk-free rate of interest (let's say, 4%). Let's assume that the beta for a stock is 1.50 and that the overall stock market is expected to return 10% over the next twelve months. Inserting these values into the equation yields:

Required return = R_{f} + beta * (Return_{market} - R_{f})

Required return = 4% + 1.50 * (10% - 4%)

Required return = 13%

This is the value that we will use to determine the present value factors when we calculate the present value of the dividends and selling price.

Analysts often estimate future earnings and dividends with the use of pro forma (projected) income statements. They simply build a future income statement a line at a time, often by entering formulas into a spreadsheet. Although only one year is shown below, an analyst will often build estimated income statements for the next three or four years. (The number of years is usually based upon how far in the future the anayst thinks that he or she can see clearly, i.e., make reasonable estimates).

Here is a sample pro forma income statement, along with a brief explanation of the method or formula used to estimate the item.

Income Statement Item |
Estimate (millions) |
How do you estimate the value? |
---|---|---|

Sales | $ 2,000 | use multiple regression and surveys of buying plans of customers |

* Profit Margin |
* 40% |
find the traditional profit margin, then modifying it for current conditions |

EBDIT | 800 | |

- Depreciation |
- 250 |
use information in the Form 10-K's (annual report's) footnote section |

EBIT | 550 | |

- Interest Expense |
- 50 |
use information on debt in the Form 10-K's footnote section |

Earnings Before Taxes | 500 | |

- Taxes @40% |
- 200 |
use a tax schedule and multiply times earnings before taxes |

Earnings After Taxes | 300 | |

- Preferred Stock Dividends |
- 0 |
use information on preferred stock in the Form 10-K |

Earnings Available to Common | 300 | |

÷Shares of Common Stock Outst. |
÷ 30.0 |
look up current shares outstanding and modify for exp. repurch. or sale |

Earnings Per Share | $ 10.00 | |

* Dividend Policy Percentage |
* 20% |
using the company's dividend policy (% of sales, etc.) |

Dividends Per Share | $ 2.00 |

When the dividends per share are estimated for the next three or four years, we have completed step #1.

Sidenote: Many investment companies use this approach to estimate earnings and dividends. The spreadsheets are continually updated as soon as new information becomes available. To see the results of these estimates, obtain a copy of *Value Line Investment Survey* (available in most large libraries), which estimates earnings for the next four years for hundreds of companies. Value Line, however, only displays the estimates for two of the four years - the first year and the fourth year. It does not show the estimated earnings and dividends for years 2 and 3 in the future. These can be estimated easily however by using the geometric mean return.

We use the following formula for estimating the price of the stock in the future (let's say, four years in the future):

Price

_{4}= Earnings Per Share_{4}* Expected Price-Earnings Ratio_{4}

This formula comes from the definition for the *price-earnings ratio (P/E ratio)*:

Price/Earnings Ratio = Price divided by Earnings per share

Since this is a definition, the formula can be applied to any year's data. For example, four years from now, the P/E ratio at that time will be equal to the Price in year 4 divided by the Earnings per share for year 4. In other words:

P/E Ratio

_{4}= Price_{4}/ Earnings Per Share_{4}

This is just a definition. Notice, however, that the Price_{4} is what we are trying to estimate. Solving the above equation for Price_{4,} we arrive at this form of the equation:

Price

_{4}= Earnings Per Share_{4}* Expected Price-Earnings Ratio_{4}

Where do these numbers come from?

- The earnings per share in year 4 are estimated with the help of the pro forma income statements mentioned above.
- Follow this link to see how to estimate the future expected P/E ratio for year 4.

We now calculate the fair price by taking the present value of the dividends and selling price. Assuming a holding period of four years, the set-up would look like this:

Cash flow | x | PVF | = | PVB |
---|---|---|---|---|

1st year's dividend | x | PVF for year 1 | = | PVB for year 1 |

2nd year's dividend | x | PVF for year 2 | = | PVB for year 2 |

3rd year's dividend | x | PVF for year 3 | = | PVB for year 3 |

4th year's dividend | x | PVF for year 4 | = | PVB for year 4 (dividend) |

Price in year 4 | x | PVF for year 4 | = | PVB for year 4 (price) |

Fair price of the stock |

Assume that:

- the CAPM's value for the required rate of return is 13%.
- the dividends shown below have been estimated with the help of pro forma income statements.
- The expected P/E ratio is four years is expected to be 15 and the earnings per share in year 4 are expected to be $4.00. This means that the price four years from now is expected to be 15 * $4.00, or $60.00.
- The current price of the stock is $42.00 per share.

Would you buy the stock at the current price of $42.00? Answer by determining the fair price to pay for the stock.

Cash flow | x | PVF @13% | = | PVB |
---|---|---|---|---|

$ 0.71 | x | 0.885 | = | $ 0.63 |

$ 0.85 | x | 0.783 | = | $ 0.66 |

$ 1.01 | x | 0.693 | = | $ 0.70 |

$ 1.20 | x | 0.613 | = | $ 0.74 |

$60.00 | x | 0.613 | = | $36.78 |

$39.51 |

Would you pay $42.00 per share for the stock? No, because it is only worth $39.51 per share.