### Project Description

Recently a professional living in Tokyo was interested in learning more about equities and **how stock prices are calculated**.

He admittedly was not intent on becoming the next Warren Buffet, nor did he have aspirations of taking up day trading or even stock picking. However, he did want to learn more about **how individual stocks are valued**, and **what determines the price**.

There are a near infinite number of variables that can effect **market prices**. Investor sentiment, macroeconomic data, geopolitics, the weather…these are just some of the day to day factors that can drive prices up, down, or sideways. In addition, there are certainly no shortage of valuation methodologies to process this data. On top of that, each investor, whether it is a pension fund investing on the behalf of millions of people, or an individual hoping to achieve reasonable returns on capital, each will have unique goals, time horizon, and risk appetite.

Accordingly, there is no one fool-proof and all encompassing formula for calculating a security’s objective and unquestionable price. A good comparison is how the theoretical physicists have yet to determine a “Grand Unifying Theory” which seeks to explain everything in the universe, reconciling the different schools of physics. However, this does not discount the importance and value of knowing the simple and fundamental Newtonian physics which governs our day to day experiences.

Take for example the basic formula F=M*A, which stands for Force = Mass * Acceleration. For those that have not taken a physics course, this formula holds that the total force of an object is simply a function of how big it is multiplied by how fast it is going. For example, a 15-car train traveling at only 5 kilometers per hour could probably smash through a brick wall, but someone swinging a baseball bat as fast as they can into the same brick wall would probably do little damage to the wall, and maybe even injure themselves. Although the train is traveling slowly, it has orders of magnitude more mass than the baseball bat, and accordingly exerts much more force on the brick wall.

Of course common sense would predict the outcome of the above experiment; but F=MA explains exactly how much force each object (the train, and the bat) are exerting on the wall, and why only one succeeds in breaking it.

We want to elaborate on one of the fundamental formulas for determining a **stock’s value**. There will be no Grand Unifying Theory for stocks here, but we would like to explain one of the F=MA caliber of formulas for stocks.

**Dividend Growth Model**

Let us first start by asking ourselves, **“what is a stock?”**. A stock is simply a partial ownership in a company. It can be a privately held company like a family business, or a large “publicly traded” company like Coca-Cola or Microsoft. The stocks you will be investing in are in the majority of cases going to be shares of **publicly traded companies**.

So the question that follows is: “what does a partial ownership in a publicly traded company mean for me?”. It means a few major points. Firstly, if the company grows in size or value, your share should also grow in value. Also, if your stock awards voting rights, occasionally you may receive a letter in the post requesting your vote on a major decision being made by the company. Lastly, and most important for today’s article, by holding a **share of stock** you would have rights to** receive a dividend** issued by the company.

One of the most fundamental methods for valuing a stock is simply looking at the dividends. Are they paying a dividend, how much is the dividend, how often do they pay, and **is the dividend increasing**? Then ask yourself, how much am I willing to pay for that future stream of cash flows? The below formula helps answer that question (for simplification we will use annual dividends)

Dividend Growth Model:

Value of stock = D / (K – G)

Whereby

D is **the dividend per share** next year

K is the investor’s **required rate of return**

G is the **expected growth rate** of the dividend

For example, a company called Grandma’s Cookies Inc sold 1 Million Cookies last year at $8 per cookie, at a cost of $7 per cookie. This would mean that the company profited $1 Million ($8 -$7 = $1 profit per cookie, for 1 Million cookies). Grandma’s Cookies currently has 1 Million shares outstanding in total **common stock**. Each year Grandma pays out all the earnings as a **dividend**, which would mean that each investor would have received a $1 **dividend per share**. Grandma is also reasonably popular, and profits (and as such, in this example dividends) have been growing by 3% each year.

How much would you be willing to pay for this stock, if you want to receive a 10% return on your capital?

Value of Stock = **Next year’s** dividend / (Your Required Rate of Return – Dividend Growth)

Value of Stock = 1.07 / (0.1 – 0.03)

Value of Stock = 1.07 / 0.07

Value of Stock = $15.29

So, in other words, given the dividend Grandma is currently paying per share, and the expected growth of the dividends; in order to achieve your goal of earning a return of 10% per year, you would be willing to pay $15.29, and not a penny more.

However, this simplified formula assumes that growth will continue at 3% per year, forever. Simply put, not all companies will fit this model of growth. For example, what if the company you are valuing is in a high growth phase, and is experiencing 30% growth in earnings each year? That certainly would be a remarkable feat for a company to maintain ad infinitum. More realistically, companies experience a short high-growth phase, and then taper off. As such, the above formula would not work, and will need to be amended to account for the short phase of high-growth.

In the next formula, we take a look at Grandma’s Cookies Inc, except assuming this time Grandma somehow manages to get free PR and is showcased on Japanese Saturday morning television.

In the segment, multiple celebrities watch another celebrity hold up Grandma’s Cookie. The camera zooms in. The cookie is shaking slightly more than necessary, and it is also for some reason steaming more than you would think it should. The picture-in-picture in the bottom right of the screen reveals the other celebrities waiting in anticipation. Finally, after tense consideration, the celebrity confirms that the cookie is, in fact, “*umai*”. Grandma’s Cookies Inc then proceeds to experience not 3% growth, but instead 30% growth in year one, 20% growth in year two, 10% growth in year three, and then a modest 3% dividend growth thereafter.

Below is the formula for **computing stock value** with a short growth-phase. It has three major steps and involves **time value of money**, but it is not complicated once broken down. In short, you will need to compute the present value of future dividends during the growth phase, then the future stock value assuming steady growth as computed above, then “discount back” the future stock value to present-value.

**First Year Dividend ** = $1 * 1.30 = $1.30

**Second Year Dividend** = $1.3 * 1.20 = $1.56

**Third Year Dividend ** = $1.56 * 1.10 = $1.72

Now determine the present value of these three dividends. The **present value** is simply what you would pay today for one future cash flow, given your required rate of return. So, in the above example, for the first year dividend, how much would you pay **today **in return for $1.30 one year from now, and then $1.56 two years from now, etc., assuming you want to make a 10% return on your investment?

Present Value of 1^{st} Year Dividend = 1.30 / (1.10)^1 = $1.18

Present Value of 2^{nd} Year Dividend = 1.56 / (1.10)^2 = $1.29

Present Value of 3^{rd} Year Dividend = 1.72 / (1.10)^3 = $1.29

**Total Present Value of these three cash flows ** = $3.76

Now calculate the value of the stock once the dividend growth stabilizes at 3% growth.

Value of Stock at end of 3^{rd} Year = 4^{th} Year Dividend / (Required Rate of Return – Dividend Growth)

Value of Stock at end of 3^{rd} Year = (1.72 * 1.03) / (0.10 – 0.03)

Value of Stock at end of 3^{rd} Year = (1.77) / (0.07) = $25.29

Because this is the expected stock value at the end of the 3^{rd} year, let’s then “discount it back” to the present value. Again, this simply means, in this example, how much would I be willing to pay **today** in exchange for $25.29 **in three years**, assuming a 10% rate of return.

Present value of stock = 25.29 / (1.10)^3 = $19.00

Now for the home stretch, add up the present value of the stock, plus the present value of the three streams of cash flows calculated above.

Total Present Value = 19.00 + 1.18 + 1.29 + 1.29

Total Present Value = $22.76

So, in other words, given an expected short period of high growth then transitioning to low growth, the stock price would be valued at $22.76 per share. This is contrast to the $15.29 per share if it did not experience the short period of high growth.

Taking it one step further, remember that Grandma’s Cookies Inc. has one million total common stock (shares) outstanding. This means that the company’s total value would be $22,760,000 instead of $15,290,000. In other words, that one “*umai” *from a celebrity was worth approximately $7,470,000 to Grandma’s Cookie’s shareholders.

As mentioned, the **Gordon Dividend Growth Model** explained here is certainly not the grand unifying theorem for determining stock prices. It also relies heavily on key assumptions, which are not likely to be static, nor are they to be similar across investors. For example, people can have different required rates of return, risk appetites, and wholly different expectations for the company’s dividend growth. Similarly, in the above example, trains and baseball bats can alter their speed, they can be exposed to different kinds of friction, or hit the wall at different angles. However, the fundamentals of F=MA hold true in the majority of basic questions of physics. Likewise, the dividend growth model is a reliable foundation for understanding the intrinsic value of a particular stock.

[ Sources ]

– Dividends, Earnings, and Stock Prices – M. J. Gordon, The Review of Economics and Statistics, Vol. 41, No. 2, Part 1

– Dividend Growth Model – Paul Barnes, Wiley Encyclopedia of Management – Finance, Volume 4

– Stock Prices, Earnings, and Expected Dividends – JOHN Y. CAMPBELL, The Journal of the American Finance Association, Volume 43, Issue 3

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