**Introduction:**

Profits, otherwise known as earnings or net income, are the lifeblood of a company and investing (as opposed to speculating) is the process of buying these profits at an attractive price. An incredible array of different pricing methods have been developed in a continuous search for ways to make informed investing decisions. Though I could not possibly make a comprehensive account of this literature here, I will discuss a few common methods as a warm up and for context.

My main goal is to introduce a method that I have been exploring that seems quite promising. This method seeks to explicitly incorporate any long term changes in both how markets price stocks and the ratio of a company's earnings that are paid out in dividends . As the stock market is currently somewhat overvalued, in the sense that we as investors are paying, on average, more money for each dollar of profit than usual, this additional flexibility seems to be especially relevant. Though the derivation is mildly involved, the resulting models (one for where the dividends are reinvested in the underlying stock and one for when the dividends are not reinvested) are straight-forward to calculate.

First, we should discuss some terminology and notation

- Price-to-earnings ratio [PE]: This is the current stock price [P] divided by the current earnings-per-share [EPS]: PE = P/EPS. PE expresses how much each dollar of profit is going to cost you. For instance, a company with a $18 share price and $1.5 EPS has a PE = 18/1.5 = 12. Hence, for this particular company, each dollar of profit will cost you $12.

- Payout ratio [PR]: The fraction of earnings that are paid out as dividends. If a company has EPS = $2 and pays a dividend of $1, then it has PR = 1/2 = 0.5.

- Discount rate [r]: This is the return I would demand in order to buy a particular type of investment with a particular return and risk profile.

As well, it seems appropriate to give some disclaimers as I can already see the comments section heating up about the folly of using EPS and PE to evaluate a stock. Yes, earnings are rather notorious for various accounting manipulations. At the conclusion of this article I briefly discuss using alternate (both GAAP and non-GAAP) accounting practices instead of earnings. However, that tomfoolery aside, they are still a crucial part of valuing a company. As pointed out by Peter Lynch in "One Up on Wall Street"

"You can see the importance of earnings on any chart that has an earnings line running alongside the stock price. On chart after chart the two lines will move in tandem, or if the stock price strays away from the earnings line, sooner or later it will come back to the earnings"

**Some Context on Valuation Using PE**

Returning to valuation, perhaps some of the most straightforward and well used metrics based on PE are price-to-earnings-to-growth [PEG] and earnings yield [EY]. PEG, popularized by Peter Lynch, divides the PE by a forecast for earnings growth [G] (that is, PEG = PE/G). Evidence suggests that companies with 0 < PEG < 1 tend to perform better than companies such that PEG < 0 or PEG > 1. Lastly, if the company issues a dividend, PEG is commonly yield [Y] adjusted by forming PE/(G+Y).

The earnings yield contains exactly the same information as PE but allows for a different interpretation. If we "flip" the PE we find EY = EPS/P = 1/PE. This can be used to conveniently compare the profit per dollar invested across many types of investments, regardless of asset class. This allows viewing as stock as a so-called "equity bond" in which the coupon is the EPS the company produces on our behalf. If an investor can invest in a risk-free treasury or money market fund at 2.5%, then this can be directly compared to the earnings yield of, say, Hershey (NYSE:HSY) (*). At close of business October 30, 2015, Hershey traded at P = $88.69 with an estimated EPS = $4.1, giving a PE = 88.69/4.1 = 21.63 and EY = 1/21.63 = 0.046. While both PE and EY are "unit-less", EY is commonly interpreted as a percent: EY = 4.6% so that is can be compared directly to, say, a bond. Compared directly to the risk-free rate of 2.5%, an investment in Hershey is demanding 210 more basis points for additional risk and the possibility of the coupon increasing.

(*) I'm going to use Hershey as a running example, not because it demonstrates my point exceptionally, but rather because it is a company I have been thinking about a lot after its recent quarterly report.

To take the "equity bond" interpretation further, many dividend growth investors use a discounted cash flow analysis to provide a quantitative measure of value. The logic goes that we can precisely compute the present value of a cash-producing asset by asking what would I pay now for an ongoing income stream? To avoid digressions, I will limit my discussion to stating a commonly used present value computation which seeks to use the "equity bond" interpretation and adjust for growth in the coupon via dividends (define D to be the forward annualized dividend issued):

present value = (dividend next year)/(r - dividend growth) = (D*(1 + dividend growth)/(r - dividend growth)

If we supply a discount rate (say the 30 year treasury bond rate + inflation + 10 = 14%) and a dividend growth estimate for Hershey (maybe 12%, which is the past 5 year CAGR), gives

present value = (0.583*4*(1.12))/(.14-.12) = $130.59.

Hence, if I plan on owning Hershey forever, and my discount rate and dividend growth assumptions remain unchanged, I would be willing to pay $130.59, 47% or so above the current market price, which indicates substantial value. This valuation of course relies an investor accurately quantifying his discount rate.

**What if the price-to-earnings or payout ratios change?**

Crucial pieces of information missing in the above, discussed measures are: what happens if either the value the market places on the earnings of a company or the fraction of earnings a company wishes to pay in dividends, changes over time? To explicitly model these type of changes, I derived from first principles the following equations for the value of an investment at time t (I_t) when starting with an initial investment of $1. Using the following notation:

- Y = D/P is the current yield

- c = (future payout ratio)/(PR) is the fraction of the long run, equilibrium payout ratio that a company is currently paying.

- k = (future price-to-earnings ratio)/ is the fraction of the long run, equilibrium price-to-earnings ratio that a company is currently being valued at by the market.

I derived two models: one for when I want to directly reinvest dividends into the issuing company ("Reinvested") and one for when the dividends are instead collected and spent elsewhere ("Not-Reinvested"). See the supplementary article for a more detailed discussion of this distinction as well as examples of how I use each model in my investing decisions.

Reinvested: I_t = k * (1 + G)^t * (1 + (c * Y)/k )^t

Not-reinvested: I_t = k * (1+G)^t + c * Y * [ (1+G) + (1+G)^2 + ... + (1+G)^t ]

These expressions are comparable in difficulty to compute as finding the present value of a security. Additionally, we can turn these into a CAGR via

Reinvested [CAGR]: (I_t)^(1/t) - 1 = k^(1/t) * (1 + G) * (1 + (c * Y)/k ) - 1

Not-reinvested [CAGR]: (I_t)^(1/t) = (k(1+G)^t + c * Y * [ (1+G) + (1+G)^2 + ... + (1+G)^t ])^(1/t) - 1

**Examples:**

Let's suppose we estimate Hershey's general equilibrium PE to be 22.5 (one relatively quick and free way to get this information is by searching "hsy p/e" in WolframAlpha). I also estimate that the current year's EPS for Hershey, after correcting for certain items, is 4.1 (this can be found on Yahoo finance or by listening to the Q3 earnings call), giving a current PE = 88.69/4.1 = 21.63. Hence, I can estimate k = 22.5/21.63 = 1.04. Also, the current payout ratio is (0.535*2 + 0.583*2)/4.1 = 0.545 (**). Finding the historical payout ratio is a bit more involved. One way to get this for the last decade is from Morningstar's "Key Ratios" tab (though I don't personally use this as I don't agree with their payout ratio computations). I estimate the equilibrium payout ratio to be 0.55, which is quite close to the current PR, and hence c = .55/.545 = 1.01. Estimating the earnings growth is of course crucially important and quite difficult. However, I usually do the following: I find a variety of earnings growth rate estimates (say, from Morningstar, Zacks, Nasdaq), along with any forecasts from the company itself. I usually take a trimmed mean by taking an arithmetic average after excluding the estimates I find overly optimistic/pessimistic. Lastly, I compare this to a "back of the envelope" earnings growth rate I compute using return on equity, share buybacks, and other considerations. For Hershey I get a CAGR for EPS of 8.3%. The final ingredient is the dividend yield, which, as opposed to the payout ratio, I use the forward dividend yield of 2.63%. Lastly, I use a time frame of t = 10, which I will discuss further below.

(**) I use the total dividends issued instead of the forward dividend stream for payout ratio computations as this is the quantity companies are using for making their dividend decisions.

Hence, if I was to reinvest dividends, I would expect a CAGR for returns of 11.5%. Without dividends reinvested, I get 10.57%. Additionally, I can get a forecast of the dividend growth rate by computing c*G, which comes in a 8.38% (in this cast almost exactly the same as the EPS growth as c is almost 1). This could be compared with The 3M Company (NYSE:MMM), which I estimate to have G = 7.8%, k = 0.8, c = 0.75, and Y = 2.58% for a Reinvested CAGR of 7.97% and a Not-reinvested CAGR of 7.15%. Hence, even with a similar EPS growth the returns are much lower due to the above-average PE and PR. Note as well that I forecast the dividend growth rate to be much lower than the recent huge increases (5.85%).

The time frame parameter, t, needs to be set by the user. A larger value of t tends to decrease the impact of the c, k parameters. I either set t to be the value I think it will take for the company to return to equilibrium levels, such as by comparing earnings yield to the 10 year treasury bond. Alternatively, I will use a t that matches the forecast horizon for G (usually 3-5 years). Lastly, if I don't have a strong opinion, I will use t between 7 and 10, mostly based on the intrinsic value formula posited by Benjamin Graham in "Security Analysis".

Conclusion:

Based on the general observation that both PE and PR are quite high for many of our most venerable dividend growth companies, I wanted to develop a valuation technique that incorporated this information explicitly. While these formula do not give a "fair value" price like present value calculations, the reported total/annualized returns are still very helpful in determining the relative merits both of competing companies, but even alternate asset classes altogether.

Note that this or any valuation must be part of a more comprehensive analysis involving debt, underlying business model, etc. Also, the following changes to these models can readily be made by accounting for:

- Taxation: If the considered investment is made in a taxable account, taxes can make a tremendous difference. If you want to find the expected after tax return, or want to directly compare making investments in different companies in different types of accounts, multiplying the growth by 1-(tax rate) is the appropriate adjustment.

- Changing growth rates: Some firms, say Walmart (NYSE:WMT), are experiencing a time of negative to no growth, which will (hopefully) transition to higher growth down the line. This could be modeled by having G_1 = G_2 = G_3 = 0 followed by G_4 = G_5 = 0.08. Or, we could use one growth parameter G by finding the CAGR over that time frame (1*1*1*1.08*1.08)^(1/5) - 1 = 0.0313.

- Free cash flow: Earnings are notoriously subject to accounting manipulations, while free cash flow is relatively more robust. Hence, many investors prefer looking at free cash flow. This model can readily be adapted to reference free cash flow over earnings. I phrase the model in terms of earnings as: 1) Earnings growth estimates are easier to find 2) Information on PE and PR are readily available 3) I tend to correct earnings-related estimates for accounting manipulations and non-cash related write downs (such as a large goodwill impairment at Hershey this past 9 months). In fact, I use these formulae for evaluating both MLP and REIT by using distributable cash flow and adjusted funds from operations, respectively.

**Disclosure:** I/we have no positions in any stocks mentioned, but may initiate a long position in HSY, WMT over the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.