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# An Introduction to Stock Valuation

Discounted Dividends

The value of any asset is simply the present value of the future cash flows you expect to receive from the asset.

• In the case of stock, this means the present value of expected future dividends.

But what if the stock doesn’t pay dividends?

• All stock will at some point return cash back to stockholders (i.e., pay dividends). It is common for young firms to keep cash as retained earnings to fuel growth. Eventually though, like Apple Inc., every company returns cash to shareholders.

• Another way to think about it is, how much would you pay for a stock for which you never expected to receive cash? Certainly nothing.

## Cash Flow Structure

The first step in valuing a stock by discounted dividends is to project what the future dividends will be. It is common to divide possible future structures into 3 cases.

1. Constant dividends.
2. Constant growth rate in dividends.
3. Irregular growth in dividends.

## Constant Dividends

If we assume a stock’s dividends will be constant, and since companies are infinitely lived, the dividends are in the form of a perpetuity. Letting D be the constant dividend and k be the discount rate we’ll apply to the dividends (k comes from something like the Capital Asset Pricing Model, or CAPM, and is a function of the risk in the cash flows), we can value the stock as:

V_0 = \sum_{t=1}^\infty \frac{D}{(1+k)^t} = \frac{D}{k}

where V_0 is the value of the stock today (time 0).

• Note this uses the formula for the present value of a perpetuity.

For all the following examples say the stock pays annual dividends. Say we expect a stock to pay a constant dividend of $10 per share. The discount rate is 10%. Then the stock value is: • V_0 = \frac{D}{k} = \frac{\$10}{0.1} = \$100 ## Constant Growth Rate in Dividends A more realistic assumption is to say that dividends will grow at a constant rate g. If this is the case then the value of the stock is: V_0 = \sum_{t=1}^\infty \frac{D_1(1+g)^t}{(1+k)^t} = \frac{D_1}{k-g} assuming k>g. Note if k<g then the sum would be \infty and the last equality would not hold. Note also we cannot have k = g or you would divide by 0 which is undefined. • This formula is the present value of a growing perpetuity. ## Constant Growth Rate Example Say a firm will pay a$5 dividend next year, and dividends are expected to grow at 8% forever. The discount rate is 12%. Then today’s value of the stock is:

• V_0 = \frac{D_1}{k - g} = \frac{\$5}{0.12 - 0.08} = \$125

Sometimes this is asked in a trickier fashion. Instead, say the firm just paid a $5 dividend. Again, assume dividends are expected to grow at 8% forever, and the discount rate is 12%. In this case, what is the value of the stock today? • To use the present value of a perpetuity formula, we need next year’s dividend. We can find it easily because it will be 8% greater than last year’s. So we have D_1 = D_0(1+g) = \$5(1.08) = \$5.40. This uses the time-value-of-money formula FV = PV(1+r)^t. • Now we can find the value of the stock with: V_0=\frac{D_1}{k-g}=\frac{\$5.40}{0.12-0.08} = \$135 ## Irregular Growth in Dividends This covers all other cases. The most common scenarios are: 1. The firm pays no dividend for a number of years, after which point the firm pays a dividend which grows at some constant rate. 2. The firm’s dividends grow at rate g_1 for a number of years, then grow at g_2 thereafter. ## Example of Irregular Growth As an example of scenario 1, say a firm is not expected to pay dividends for the next 8 years. In year 9 the firm will pay a$5 dividend which is expected to grow at 3% thereafter. The discount rate k is 7%.

• We can use the formula for the value of a growing perpetuity to find the value of the stock in year 8. V_8 = \frac{D_9}{k-g} = \frac{\$5}{0.07 - 0.03} = \$125

• Now we just need to discount this value back to today: V_0 = \frac{\$125}{(1.07^8)} = \$72.75

## Discounted Dividend Stock Values App

Give stock valuation a try, and on the next slide you can check your answers with the interactive app.

• The app will use scenario 1 above for the irregular growth case.

• As you switch between the three types of growth, the inputs will automatically react.

## Credits and Collaboration

Click the following links to see the codeline-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.