Jamin Chen's  Instablog

The Basics

When you are about to select a stock for long term holding, I mean you intend to hold it for several years, you face a difficult task of choosing between dividend yield and the growth of the value of the stock itself.

Here is a simple formula to help you on the selection.

Let me go through with you how I arrive at the final formula.

Suppose you start with A1 number of shares of XYZ Company for which you pay P1 per share. It offers D1 of dividend per share the first year. So, at the end of the first year, you own A1 shares and you have received dividends equal to (A1 * D1).

If the price rises R1% above P1 at the end of the first year and you reinvest all the dividends you have received, the total number of share you own after the dividend reinvestment is A2

(1) A2 = A1 + (A1 * D1) / [P1 * (1 + R1)] = A1 [1 + (D1/P1) / (1 + R1)]

If you repeat the same process for another year, at the end of the second year you own

(2) A3 = A2 [1 + (D2/P2) / (1 + R2)] = A1 [1 + (D1/P1) / (1 + R1)] * [1 + (D2/P2) / (1 + R2)]

Now, if you repeat this n number of years, at the end of the n-th year, which is also the start of the (n+1)-th year, you own

(3) An+1 = A1 [1 + (D1/P1) / (1 + R1)] * [1 + (D2/P2) / (1 + R2)] … [1 + (Dn/Pn) / (1 + Rn)]

And, the total value of your investment at the start of the (n+1)th year, which is also the end of the nth year, is

(4) Vn+1 = An+1 * Pn+1 = An+1 * P1 * (1 + R1) * (1 + R2) * … (1 + Rn)

Here,

(5) Pn+1 = P1 * (1 + R1) * (1 + R2) * … (1 + Rn)

Please note that Vn+1, An+1, and Pn+1 are all values at the start of (n+1)th year.

When you extend these formulas to the future performance of your investment, you need to make some assumptions and simplifications.

First, for the number of years you are going to hold this investment, you assume its share price will rises at the same rate every year, or

(6) R = constant = R1 = R2 = … = Rn

Next, for the same time period, you assume the dividend will increase in proportion to the share price. This implies the share price increases in proportion to the profitability the company and the company will also raise its dividend in proportion to its profit. It thus assumes a constant dividend yield, Y, expressed in fraction, over the chosen time period,

(7) Y = constant = D1/P1 = D2/P2 = … = Dn/Pn

And, the above formulas become

(8) An+1 = A1 * [1 + Y/ (1 + R)]^n

(9) Pn+1 = P1 * (1 + R)^n

(10) Vn+1 = An+1 * Pn+1 = A1 * [1 + Y/(1 + R)]^n * P1 * (1 + R)^n = A1 * P1 * [1 + R + Y]^n

Since the initial investment V1 = A1*P1, we have

(11) Vn+1 / V1 = [1 + R + Y]^n

The last formula above makes the selection of stocks fairly simple. It says if both the rate of growth, R, and the dividend rate, Y, remain the same over the time period of your consideration, they have equal weights on the future value of your investment. For example, the following stocks will produce the same result:

• A stock paying a yearly dividend yield of 5% and its price growing at 3% a year
• A stock paying a yearly dividend yield of 4% and its price growing at 4% a year
• A stock paying a yearly dividend yield of 3% and its price growing at 5% a year
• A stock paying no dividend at all yet its price growing at 8% a year

Of course, you will get higher return from a stock with both a high rate of growth, R, and a high dividends rate, Y.

These are purely mathematical calculations. Still, they can help you in choosing the stock for you investment. However, you must realize that there are many practical implications to each of these cases.

In applying the results above to actual stocks, there are some complications. The main complication arises from the fact the stock price, P, is changing literally every second, or even microsecond. To take care of this complication, you will have to assume some kind of "average" price of the stock based on its average P/E ratio and treat the difference between the instantaneous price or P/E ratio and the average price or P/E ratio as a deviation from the average. In other words, people expect a certain P/E ratio for a given stock and you assume that the stock price will somehow "revert to average" once in a while and fluctuate around the expected P/E ratio.

With this in mind, the rate of the rise in a stock's price per year, R, will now be the same as the rate of the rise in a stock's earnings.

Now, if E is the earning per year of a stock, and (P/E) is the price/earnings ratio expected of the stock, and since Y = D/P, we have,

(12) Y = Y *(P/E) / (P/E) = (D/P) * (P/E) / (P/E) = (D/E) /(P/E)

The formula above implies the ratio of dividends and earnings (D/E) is a constant.

The formula (11) above becomes

(13) Vn+1 / V1 = [1 + R + (D/E)/(P/E)]^n

The last formula indicates you will get a better return from stocks having

• A higher earnings growth rate, R,
• A higher dividends to earnings ratio, D/E, and
• A lower price to earnings ratio, P/E.

We are now ready to apply the last formula above in evaluating stocks for investment. However, as illustrated in the following cases, the application is not straight forward. Still, when applied properly, this formula can give you a good indication of how good a particular stock may perform, if its R, D/E, and P/E all hold up in the coming years. As always, when these and other parameters of a stock start to deviate from their established "norms" it is time for you to reevaluate your investment in that stock.

Applications

Here, we are going to apply the formulas developed above on four stocks, T, MCD, GOOG, and AAPL.

Stock 1: AT&T, Inc. (T)

T is a widely held dividend stock.

In the table below, the earnings per share, E/S, are obtained from CNBC website and the dividends per share, D/S, from Yahoo website. The prices per share, P, are also obtained from Yahoo website. Here the prices chosen are that of the first trading days of the corresponding years.

The growth rates, R, are calculated from E/S of two consecutive years. For example, for the year 2006,

R = 1.94/2.34 - 1 = -0.170

The average given in column 8 is a simple average of the 5 data available. Note that year 2012 is not yet over and, therefore, there is no actual E/S available. Though estimates are available for the year 2012, for simplicity sake, they are not used.

The dividends/earnings ratios, D/E, are calculated by dividing D/S with E/S. For example, for the year 2006,

D/E = 1.332/2.34 = 0.5692

The average in column 8 is also a simple average.

The P/E ratios are calculated by dividing P with E/S. For example, for the year 2006,

P/E = 24.71/2.34 = 10.559

The average in column 8 is also a simple average. Please note that the P/E ratio calculated here may not be the same as that normally used. Here, for a given year, it is calculated from the earnings of that year and the price of the stock at the beginning of the same year. The selection of the price is quite arbitrary here. However, for simplicity sake, the price used here is the same as the price used to calculate the number of shares that can be purchased at the start of the year from the dividends received during the preceding year (see the paragraph immediately below).

The number of shares starts with 1 and for year 2007 it is calculated according to formula (1) above,

A2 = A1 + (A1 * D1) / P2 = A1 * (1 + D1/P2) = 1 * (1 + 1.332/34.95) = 1.0381

Please note that it is assumed that all dividends are received at the end of the year and it is reinvested on the first of the year immediately after.

From the averages in the table above, the following are calculated:

(D/E)/(P/E) = 0.7213/14.5578 = 0.0495

1 + R + (D/E)/(P/E) = 1 - 0.0069 + 0.0495 = 1.0426

And, from formula (13) above, each dollar you invested in this stock on the first of 2006 should grow to 1.2846 by the first of 2012:

Vn+1 / V1 = [1 + R + (D/E)/(P/E)]^n = 1.04266 = 1.2846

The actual number is

1.3369 (No. of shares you own at the beginning of 2012) x 30.38 (Price of the stock at the beginning of 2012) / 24.71 (Price of the stock at the beginning of 2006) = 1.6437

In this case, the simplified formula (13) indicates a return (1.2846) that is lower than the actual return (1.6437). This is caused by the high P/E ratios of years 2007 and 2008.

Stock 2: McDonald's Corp (MCD)

MCD is a good growth company and it also pays very good dividends.

Similar calculation is also applied to MCD.

From the averages in the table above, the following values are calculated:

(D/E)/(P/E) = 0.0321

1 + R + (D/E)/(P/E) = 1.1992

Vn+1 / V1 = [1 + R + (D/E)/(P/E)]^n = 2.9743 vs. 3.4592 actual

For MCD, the simplified formula matches pretty nicely with the actual result. This also shows that MCD has a very steady past performance.

Following is a comparison of T and MCD:

Here, though the two stocks have about the same P/E ratios and while T distributed a higher percentage of earnings as dividends (D/E), the high growth rate of MCD makes it a better investment.

Case 3: Google Inc. (GOOG)

GOOG is a stock that is diametrically opposite to T. It has a high growth rate and distributes no dividends. When the same calculation is applied to it, it gives the following result:

(D/E)/(P/E) = 0

1 + R + (D/E)/(P/E) = 1.2815

Vn+1 / V1 = [1 + R + (D/E)/(P/E)]^n = 4.4290 vs. 1.5289 actual

In this case, the simplified formula indicates a much higher return than the actual. What has happened here is this: the simplified formula assumes a constant P/E ratio over the time span of interest. However, GOOG had a drastic reduction in the P/E ratio during this time span.

What is interesting here is that over the same time span, the actual return from T (1.6347) is better than GOOG (1.5289), though only slightly.

Also interesting is that GOOG earnings per share have been growing phenomenally at a rate of about 28% per year. If, a very big if, GOOG can keep it up with a reasonable P/E ratio, the return on investing in GOOG could be tremendous. This can be illustrated by applying the same calculation to a shorter time span of between 2009 and 2011 during which its P/E ratio appears to stabilize around 17. The result is as follows:

(D/E)/(P/E) = 0

1 + R + (D/E)/(P/E) = 1.2467

Vn+1 / V1 = [1 + R + (D/E)/(P/E)]^n = 1.9378 vs. 2.0709 actual

If GOOG earnings continue to grow at about 25% per year and its P/E ratio stays about 17, the return on investing GOOG would produce a return of about 25% per year.

Case 4: Apple Inc. (AAPL)

APPL has a similar earnings and P/E ratio histories like GOOG as the table below shows.

(D/E)/(P/E) = 0

1 + R + (D/E)/(P/E) = 1.6563

Vn+1 / V1 = [1 + R + (D/E)/(P/E)]^n = 20.6495 vs. 5.5014 actual

Again, the huge P/E ratios in 2006-2008 caused a great discrepancy between the return indicated by the simplified formula and the actual result. If AAPL earnings continue to grow as it has been at about 75% per year as it has been in the past two years and its P/E ratio stays around 12 as shown in the table below, it will also produce a great return.

(D/E)/(P/E) = 0

1 + R + (D/E)/(P/E) = 1.7463

Vn+1 / V1 = [1 + R + (D/E)/(P/E)]^n = 5.3253 vs. 4.5315 actual

As shown in the table below, the combination of a high earnings growth and a low P/E ratio makes AAPL a better investment than GOOG.

Table below summarizes all four cases examined above:

Here, we see AAPL has the highest potential returns which is not a surprise. However, MCD, with its high dividends payout, a good growth rate, and a lower P/E ratio, outshines GOOG. T has the lowest returns yet, as its future performance is not dependent on high growth rate, it may be less susceptible to earnings disappoints which could happen to the growth stocks when, for example, their market penetrations reach saturation.

We all know we should pick stocks having high dividends, high growth rate, and low price/earnings ratio. However, what I present here is a method of quantifying how these factors relate to each other and how they together affect the performance of a stock.

When you are trying to determine the future performance of a stock based on its past record, it is always important to remember that there is no guarantee that the future performance will be as predicted. You must also attempt to foresee coming events that may affect the performance of that stock. Still, the market has so many variables that you cannot foresee them all. However, a forecast based on actual past performance is better than none at all.

Disclosure: I am long T, DIA, GLD, MCD.