Financial Education

I showed how to estimate the expected annual return implied by the current stock price for the Gordon growth model and the H-model. For more complicated models the return cannot be derived from a single formula, but a spreadsheet model can help an investor quickly figure it out through trial and error.

Back in August I used a three-stage dividend discount model to value Rockwell Automation (ROK).  In the model:

The 14% consensus 5-year growth rate was assumed to be a market assumption for the five-year first stage

The growth rate was expected to decline at a linear rate (the H-Model) for 10 years, with a terminal growth rate of 7% (the long-term S&P 500 average growth rate.)

The investor’s required return is 10% and the initial dividend is $1.16. The resulting valuation of $63.25 was fairly close to the $69.00 market price, indicating that those assumptions may be quite close to the expectations of the average investor. We can estimate how close by reversing the model to calculate any input – in this case, we want to see the implied return an investor will earn assuming the growth assumptions are correct.

By plugging in the dividend stream and the formula used to estimate r from the H-model into a spreadsheet, we can first check our math by inputting a 10% return estimate. It comes out to the $63.25 we got the first time, so the math checks.

Since the estimated value is lower than the $69.00 market price, buying at $69 would mean we would earn less than 10%. If we plug in 9%, though, we way overshoot. It gives an present value of $96.24. Obviously the market expectation is much closer to 10% than to 9%.

Let’s try 9.8%.

Getting close, but back on the low side. 9.7%?

Almost there. How about 9.75%?

Virtualy on the nose. If our assumptions that growth will be 14% for five years, then decline linearly for 10 years to a final stable rate of 7% are correct, we can expect to earn about 9.75% per year if we pay $69.00 for ROK.

It may seem counter-intuitive to value a stock that doesn’t pay dividends using a dividend discount model. But if the investor believes the company will pay dividends in the future the model can still be used. Consider an investor with a required return of 10% looking at a stock that is not expected to pay dividends for the next five years, but that will pay a $1.00 dividend in year six and the dividend is expected to grow 7% annually thereafter. All that need be done is compute the Gordon growth value at year 5, then discount it back for five years.

The Gordon growth model says Value = D1/(r-g), or in this case $1.00/(0.10 – 0.07) = $33.33. But that is the value five years from now. Discounting that to the present at 10% annually gives a present value of $20.70, which is what this investor should be willing to pay for the stock today.

Under IAS 16, as under US GAAP, property plant and equipment are initially recorded on the balance sheet at cost, and systematically charged to expense as depreciation. Unlike GAAP, IAS permits upward revaluation (to the fair value as of the revaluation date) of property, plant and equipment.

Typically, when assets are revalued downward the charge flows through the income statement. In the case of upward revaluation, however, the change typically bypasses the income statement and is made directly to equity. The exception to these general rules is when the revaluation reverses a previous revaluation. In such cases, the reporting mimics that of the original revaluation. So downward reversals go straight to equity, while upward reversals are reported in the P&L.

Investors commonly use the Price/Earnings or P/E multiple as a gauge of how expensive a stock is trading. The Gordon growth model can be recast to indicate a justified P/E multiple based on the fundamentals (assuming the model inputs are correctly specified.)

The justified P/E based on trailing earnings is P/E = (D(1+g)/E)/(r-g) or [(payout ratio)(1+g)]/(r-g)

So, for a company with $1.00 in earnings per share and a $0.60 dividend (60% payout ratio) expected to grow at 5% and requiring an 8% return, the P/E should be (0.4 * 1.05)/(0.08 – 0.05) = 0.63/0.03 =21x and the stock should trade at $21.00.

Based on forward earnings, the formula is simply (payout ratio)/(r-g).

Many investors seek to exploit temporary market inefficiencies by buying a security they believe is underpriced and shorting a similar security they believe is overpriced. This is designed to limit fundamental risk, or the chance that bad news will hurt the investment. Since many types of news will affect all companies in the industry, shorting a competitor can cushion the risk. However, no two securities are a perfect match. For example, the trader who buys Citibank and shorts Bank of America runs the risk that a piece of negative news will hit Citigroup exclusively.

The dividend discount model and other discounted cash flow approached define the value of a stock as a function of the current cash flow, growth and a required rate of return. Normally these models are used to derive a valuation, which is then compared to the current stock price to determine whether the stock is “overvalued” or “undervalued.” The determination will be affected by the assumptions made regarding required return and growth.

An alternative is to reverse the model and use the current stock price to determine the average assumptions being implicitly made by investors. For example, consider a stock with a $20.00 current share price and $1.00 in expected annual dividends. The consensus long-term growth estimate is 6%, and the investor believes this growth rate reflects the typical belief of market participants.

Since the Gordon growth model defines Value = D1/(r-g) we can substitute what is known to solve for the required return r. $20.00 = $1.00/(r – 0.06) and r = 0.11 or 11%. This market-implied r can further be compared to a required return calculated using a formula such as the Capital Asset Pricing Model. If the market-implied return is higher than the CAPM required return it may indicate that the stock is undervalued (will earn a higher return than is “efficient.”)

The Gordon growth model is a type of dividend discount model used to value companies expected to grow at a constant rate forever. Most valuation models forecast growth for a certain time period before reverting to a Gordon growth model to estimate the ending value.

The Gordon growth model formula is V(0) = [D(0)(1+g)]/(r-g)

where V(0) is the value today, D(0) is the current annual dividend, g is the annual growth rate and r is the required return on the stock. This can also be expressed as V(0) = D(1)/(r-g) where D(1) is next year’s dividend. If the dividend grows in line with the expected growth rate the two formulae are equivalent.

Rapidly growing companies typically have high Price/Earnings (P?E) ratios. The growth duration model relates the P/E ratio to its rate of growth and the expected duration of that growth. It is a function of expected earnings per share growth, the required rate of return on the stock, and the dividend payout ratio. Since most growth companies pay little or no dividend, if one assumes a constant risk between any two securities the entire difference in P/E can be attributed to the difference in expected growth. Namely, the current prices should be in direct proportion to the expected future earnings ratio that will prevail when the growth rates are equal (time T).

For growth company GC and the non-growth company (or overall market) MI, the relationship is:

{[P(GC)/E(GC)]/[P(MI)/E(MI)]} = {[1+G(GC)+D(GC)]/[1+G(MI)+D(MI)]}^T

where D = the dividend yield and G = the current growth rate.

To solve for T, you take the log of both sides:

ln{[P(GC)/E(GC)]/[P(MI)/E(MI)]} = Tln{[1+G(GC)+D(GC)]/[1+G(MI)+D(MI)]}

In a principal trade, a fund manager trades through a dealer who guarantees the trade will be executed at a certain discount or premium to the current price in the market. The dealer acts as a principal, taking the risk related to the opposite side of the trade, and the discount or premium compensates the dealer for taking this risk. In return, the manager is able to minimize the opportunity cost of placing a large trade.

Although opportunity costs are minimized, total transaction costs related to a principal trade can be large due to both direct costs and the market impact (in the form of the discount or premium.)

The value of any asset must equal the present value of its future cash flows, discounted at a rate that reflects its inherent risk. Since neither the future cash flows nor the appropriate discount rate can be known with certainty, valuation is inherently an estimation.

One type of discount rate is the Weighted Average Cost of Capital (WACC). This is the rate that reflects the risk inherent to the firm, and would be an appropriate rate to use when discounting free cash flow to the firm (FCFF). The WACC is calculated as:

(Percentage of debt financing X Cost of debt financing) + (Percentage of equity financing X Cost of equity financing).

The cost of debt financing is typically the average yield to maturity on the company’s bonds.  The cost of equity financing must be estimated using one of several methods such as the Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT), the Gordon growth model or another method.