As I write this, Apple stock (AAPL) is selling for $105.76 per share. Should you buy Apple at this price? If you already own Apple, should you sell it? Buying and selling stock is a gamble. These are bets. How can we decide if Apple is priced fairly? What is the “expected value” of Apple stock?
There are many ways in which one can value a stock. If everyone used the same method, then the market would behave in a very different manner. Currently, when someone buys a share of Apple stock, there is someone else ready to sell a share of Apple stock. We can safely assume that they do not have the same view or valuation of Apple.
One standard approach to valuation is the dividend discount model or DDM. The idea is that an analyst predicts all future dividends to be paid by a company, and adds them up, with an appropriate discount the further off the predicted dividend payment. The sum of those discounted dividends is the value of the stock. If that value is greater than the current stock price, then the stock is undervalued. For example, if my dividend discount model for Apple results in a value of $120, then the current price is too low, and I should buy Apple.
Rather than add up an infinite series, we can used a closed form of the DDM:
Stock Price = (next year’s projected dividend) / (cost of capital for company  dividend growth rate)
We can work through an example. Apple currently pays an annual dividend of $2.08 per share per year. Apple’s dividend growth rate has been 8.5%. So, next year’s dividend should be $2.25, which is $2.08 * 1.085. Apple’s cost of capital has been 10.3%. (Apple’s WACC or weighted average cost of capital) We get the following.
Stock price = 2.25 / (.103  .085) = 2.25 / .018 = $125
Thus, Apple stock is undervalued by roughly $20 per share, if you believe this model.
The dividend discount model is based on two ideas: (1) projecting a series of cash flows, and (2) calculating the time value of money. First, we can look at a single cash flow. Suppose that I owe you $100. Would you prefer that I give it to you today or a year from now? For the purpose of this exercise, we can assume that I am completely trustworthy. I pose no credit risk. You are 100% certain that if I promise to pay you in a year, you will get your money.
Still, you really would rather have the money today, right? One reason is that if you had the money today, you could invest it and presumably have more than $100 after a year. In the old days (say 10 years ago), a person could invest her money in a bond or certificate of deposit and expect to earn 5% or so per year. In that world, your $100 would grow to $105 after a year. If you let me wait a year before paying you back, you are losing money. In other words, $100 today should be valued at $105 a year from now.
Now, ask the question in reverse. Suppose that I really do not have to pay you $100 until a year from now. If you prefer to get paid today, it would be fair for me to pay an amount $X today that would be equivalent to $100 in a year. For example, if I invest $X at 5% with the goal of getting $100 in a year, what is X? This is simple algebra.
X * 1.05 = 100
Therefore, X = 100 / 1.05 = $95.24
X is the present value of $100 in a year with a discount rate of 5%.
OK. Suppose instead of paying you in one year, the deal is that I pay you in two years. If I have $100 today and invest it at 5% a year for two years, the interest on the second year includes interest on the interest I earned the first year. That is, the interest compounds. The algebra is pretty much the same as before.
X * 1.05 * 1.05 = 100
Therefore, X = 100 / (1.05 * 1.05) = $90.79
In fact, if I don’t have to pay you the $100 for N years, the formula is:
X = 100 / 1.05^{N}
Instead of a single cash flow, imagine that I offer to pay you $100 every year for the next 30 years. What is a fair price for that deal? It is the sum of the discounted yearly cash flows.
X = 100 / 1.05 + 100 / (1.05 * 1.05) + … + 100 / 1.05^{30}
Another name for this calculation is net present value or NPV. Many business decisions can be formulated in terms of net present value. Consider the problem stated earlier.
As an executive at Ford, I need to build a new auto plant. The options are a $40 million plant in Ohio that can produce 1000 cars a month with union labor, with an expected lifetime of 12 years and a $20 million dollar plant in South Carolina that can produce 500 cars a month with nonunion labor and tax incentives and an expected lifetime of 10 years. Which is best?
Each option can be modeled using net present value. If an option has a negative NPV, then it should always be rejected. If the NPV is 0, you should be indifferent between choosing that option or not. If two options both have a positive NPV, you should chose the one with the higher NPV.
To solve the problem above, we need to know more information. We can assume in each case that it takes two years to build the factory (which is included in the lifetime of the plant), and the cost is split evenly over those two years. Each car produced generates $10,000 in profit, excluding labor costs. Annual labor costs are $6500 per car for union workers in $4000 per car for non union workers. The discount rate is 5%.
Year 
Item 
Ohio: (Cost) Profit 
SC: (Cost) Profit 
1 
Construction 
($20,000,000) 
($10,000,000) 
2 
Construction 
($20,000,000) 
($10,000,000) 
3 
12,000 cars / 6,000 cars 
$42,000,000 
$36,000,000 
4 
12,000 cars / 6,000 cars 
$42,000,000 
$36,000,000 
5 
12,000 cars / 6,000 cars 
$42,000,000 
$36,000,000 
6 
12,000 cars / 6,000 cars 
$42,000,000 
$36,000,000 
7 
12,000 cars / 6,000 cars 
$42,000,000 
$36,000,000 
8 
12,000 cars / 6,000 cars 
$42,000,000 
$36,000,000 
9 
12,000 cars / 6,000 cars 
$42,000,000 
$36,000,000 
10 
12,000 cars / 6,000 cars 
$42,000,000 
$36,000,000 
11 
12,000 cars 
$42,000,000 

12 
12,000 cars 
$42,000,000 

NPV 

($1,375,494) 
$874,668 
The Ohio factory, which produces more cars each month and lasts two more years, has a negative NPV, while the South Carolina project has a positive NPV. The choice is obvious.
However, there are number of unstated assumptions in this model. Try to name as many as you can. What process can you invoke to identify all the assumptions?
All assumptions are not equal. What if the factory comes in over budget? or under budget? What if interest rates go higher or lower? What if auto sales go up or down? What if the factory lasts longer or shorter than assumed? What if labor costs increase or decrease?
It is not hard to model these explicit variations in assumptions to determine which if any of these would change your decision. This process is known as sensitivity analysis. It can be quite complicated by a variety of factors including correlated input variables (changing on input implicitly changes another), and nonlinearity (we will discuss this later in the course),
As a practical matter, the NPV calculation is standard in most spreadsheets. (That’s how I calculated it.) Spreadsheets also are great at performing sensitivity analysis.
This practice of deciding among competing long term investments in a firm is called Capital Budgeting. If a firm has a budget of $100 million and has a dozen possible independent long term projects, none of which cost more that $100 million, the process is to calculate the NPV of each and to rank the projects accordingly. The firm should fund as many of the projects with the highest NPVs as possible.
References
Sensitivity Analysis, Wikipedia
https://en.wikipedia.org/wiki/Sensitivity_analysis
hw2: NPV, Sensitivity Analysis, and Explanation
Slade, Automated Decision Systems 
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