First, it's important to note that the investment decision process consists of the following components:

• Estimate the dollar amount and timing of expected cash flows (this is almost always given in a question).
• Determine the required rate of return (k) in the constant dividend growth model (this is sometimes given in a question).
• Use the required return to discount the expected cash flows and add their present values to determine the estimated value of the security.
• Compare this value to what the security is actually selling for to determine if the security is over- or undervalued.

Let's start by valuing a stock with a one-year holding period. To determine the stock’s value we need to:

• Identify all expected future cash flows, which consist of all dividend(s) received and the ending value of the stock.
• Estimate the required return (k) for the investment using the capital asset pricing model: k = Rf + B(Rm – Rf)
• Discount the expected dividend and selling price at the required return.

Therefore, Value = [dividend(s) to be received] / (1 + k)^1 + [year-end price] / (1 + k)^1

Assume the dividend paid last year was \$1.25 and that it will grow next year by 6%. The risk-free-rate of return (Rf) is 5%, the return on the market (Rm) is 12%, the stock’s beta is .6571 and the stock will sell for \$15 at the end of the year. What is the value of this stock?

You can can choose to calculate in one of two ways.

### Option One: Using the formula

1. Determine the cash flows. The dividend received at the end of year one is \$1.325 (\$1.25 x 1.06). Determine the value of the stock at the end of the year of \$15 (given in the facts).
2. Determine the required rate of return. k = .05 + .6571(.12 – .05) or .096
3. Determine value. [\$1.325] / (1 + .096)^1 + \$15 / (1 + .096)^1 = \$14.90

Now, using the HP-12C time value of money keys:

1. Determine the PV of the dividend. n = 1, i = 9.6, FV = \$1.325 (\$1.25 x 1.06)
2. Solve for PV = \$1.21

– Determine the PV of the future stock price
n = 1
i = 9.6
FV = \$15 (given)
Solve for PV = \$13.69

The present value is \$1.21 for the dividend, plus \$13.69 for the stock, for a total of \$14.90.

To prove that this makes sense, \$14.90 today multiplied by a yearly return of 9.6% gives you the future value of the dividend and the stock next year of \$16.325 (\$1.325 + \$15) or \$14.09 x 1.096 = \$16.33.

I did the above example to show you that you can arrive at the answer in one of two ways. It would be great to know both computations; however, if you want to only learn one, learn the one that you feel most comfortable with.

Now let’s assume a two-year holding period on the same facts. The year two dividend is not \$1.325 but \$1.405 (\$1.325 x 1.096). Note that the \$15 for the stock and this \$1.405 dividend is not received until two years from today. Because it is two years, I raise the denominator of (1 + .096) to the second power (^2). Please notice that if it is 8 years from now, it would be raised to the eighth power (^8).

Using the formula:
– Determine the cash flows: dividend received at the end of year one is \$1.325 (\$1.25 x 1.06). The dividend received at the end of year two is \$1.405 (\$1.325 x 1.06) and the value of the stock at the end of year two is \$15 (given in the facts).
– Determine ‘k’: k = .05 + .6571(.12 – .05) or .096
– Determine value using the following formula: [\$1.325] / (1 + .096)^1 + [1.405] / (1 + .096)^2 + \$15 / (1 + .096)^2 = \$14.90
– Using the HP-12C time value of money keys:
– Determine the PV of the dividend
n = 1
i = 9.6
FV = \$1.325 (\$1.25 x 1.06)
Solve for PV = \$1.21

– Determine the PV of the year two dividend and future stock price (they are assumed paid on the same day)
n = 2
i = 9.6
FV = \$16.405 (\$15 stock price (given) + \$1.405 year two dividend)
Solve for PV = \$13.66

The present value is \$1.20 for the dividend plus \$13.66 for the stock for a total of \$14.87. The amount per the single year formula was \$14.90 for a difference of .03 due to rounding.

You use the same format for when the dividend is growing at different rates. Try this, and if you are still having difficulty, I will provide an example with how to compute the dividend when the rates are changing.