**FLOW VALUATION METHODS**

**The Two Financial Flows That Matter**

Paramount to understanding the basics of valuation is recognising the financial flows within the firm. Any business is affected by two financial flows:

**The first** and in my opinion the most important is its cash flow, which is the cash generation and expenditure of the firm that determine the short term success and the long term survival of the business.

The **second flow** is where revenues are recognized as chargeable to the period in question and accounting costs are matched to the process of earning that revenue and then allocated to different categories of accounts.

When valuing the equity of the firm, the most obvious flow is the cash flow received by the investors. In terms of what is paid to the investors by the firm (and ignoring any share repurchase scheme) all they receive is a flow of variable dividend payments. Therefore the investor can be regarded as holding a variable interest rate bond of indefinite term. However, many firms do not pay dividends and for these some alternative flow measure is necessary.

**One approach** is to use projections of the **annual free cash flow** to the **equity investors** which is retained within the firm. Another method is to use some measure of residual value from the company’s accounts again projected over the lifetime of the firm. The **challenge** with these methods is that they rely upon (a) a calculation of an appropriate discount rate and (b) the forecasting and projection of the future flow measures.

One of the annual free cash flow proxy, is the dividend paid to equity investors. A common valuation technique is the dividend valuation model.

**The Dividend Valuation Model **

A return to equity investors is the total monetary gain or loss to the investor, both from capital gain and dividends, over a specified holding period. So over the next holding period (however long that may be) the return is the difference between the current price and end of period price plus any dividend paid.

or by rearrangement:

Because this model refers to a future time period it is described as an *ex-ante *model and the return (r_{1}), the price at the end of the period (p_{1}) and the dividend receivable (d_{1}) are all expected values

Given that the return formula (i) is purely definitional the question then arises as to what is the simplest set of assumptions necessary to create an **equity valuation model** which is not dependent upon some future estimate of the value of the equity concerned.

**The three assumptions that dramatically simplify the model**:

**(i)** The firm is a going concern which implies there is no foreseeable prospect of its failure,

**(ii)** The rate of return in future periods is expected to be constant which implies that there is a flat term structure on equity returns and,

**(iii)** There is an expectation of a constant rate of change in the dividends paid over the indefinite life of the firm.

**Invoking these assumptions** we can demonstrate that the current price of the firm’s equity is simply the present value of the future dividend stream (which ‘grows’ at a constant **rate ‘g’**) accruing to the investor. In fact the rate of growth can be assumed to be a zero or indeed a negative value without doing violence to the underlying integrity of the model.

The resulting model derived is as follows:

Which resolves to the dividend growth model first derived by ** John Burr Williams in 1938**:

‘A stock is worth the present value of its future dividends, with future dividends dependent on future earnings. Value thus depends on the distribution rate for earnings, which rate is itself determined by the reinvestment needs of the business’, *Williams, J,B. 1938 The Theory of Investment Value*

This insight was further developed by ** Myron Gordon in 1962** into what is now known as Gordon’s Growth Model:

Thus the current share price is a function of just three variables: the dividend paid during the last 12 months (D_{0}), the rate of return required by equity investors when discounting the dividend flow receivable by them (r_{e}) and the expected growth rate attaching to those dividends over the life time of the firm (g).

The **three assumptions** upon which the dividend growth model is based (i) to (iii) are an example of the rigorous application of **Ockham’s Razor** to the problem of how expectations are built in the equity market. Each assumption is designed to reduce the model to a progressively simpler form whilst retaining the key features of the valuation process.

**One common objection** to the dividend growth model is that it violates **Modigliani and Miller’s** dividend irrelevance hypothesis. This is a misconception partly brought about by focusing on the ‘dividend’ in the model’s name and forgetting that the model values both dividends and dividend growth and partly because of a misclassification error. Dividend growth is generated by, amongst other things, the firm’s ability to retain earnings for future investment so in reality the model is valuing dividends and reinvestment capacity which are both created by earnings. That is exactly Modigliani and Miller’s position with respect to valuation.

The dividend growth model can be classified, depending on how you view growth as either an accounting flow model where earnings are either distributed or retained for growth or as a cash flow model where the shareholder’s value the cash flows they receive as dividends and as capital gain. Given the importance of the model and its analogues we will review the assumptions upon which it is based in some detail.

__The Perpetuity Assumption__

The dividend growth model represents the limiting value of a firm’s equity in the hands of its shareholders. In principle, the value of the shares is assumed to be the present value of an infinite stream of constantly growing dividend payments discounted at the investors’ required rate of return. Any finite stream of discounted dividend payments assuming the same rates of growth and return will necessarily have a present value less than the value generated by this model.

Normally, a limited company is assumed to have an indefinite life. This is embedded in the accountant’s ‘going concern’ concept. However this is a fairly restricted concept in that it requires that management should prepare their accounts on a ‘going concern basis’ unless they ‘intend to liquidate the entity, cease trading, or have no realistic alternative to do so’ (IAS1). In a market based setting we mean something more than this in that shareholders do not have any expectation that the company will cease trading and will therefore continue indefinitely.

Where there is a significant difference between the expected rate of growth of the firm and the rate of return required by the equity investors then the cumulative present value of the dividends paid to investors rapidly approaches the present value of a stream of dividends extending into perpetuity. However, where the difference is narrow the model does not resolve towards its limiting value quickly and even assuming a 60 year life there may be a considerable difference between the value generated by the model and an equivalent dividend pattern but assuming a finite life of 60 years (i.e., p_{61} = 0).

We demonstrate this effect in exhibit **102.3.1** where we show the value of a firm of different life times (from 1 to 250 years) assuming a 10p initial dividend growing at the rates as shown and with a required rate of return on equity of 8 per cent. Note how a high growth rate brings the model into agreement with the perpetuity (i.e., the point where the curve become horizontal) much more quickly than when lower growth rates are applied.

One way of dealing with this problem is to use a time restricted version of the growth model:

This suggests that the dividend growth model is only likely to be a fair representation of corporate value where the growth return spread is greater than 4 per cent. In periods of high nominal returns this may be reasonable, in periods where both real and nominal returns on equity are very low this severely limits the validity of the model.

__The constant return assumption__

The rate of return which we assume investors use to discount the dividend flow from a company should be the rate that they would use to discount an earnings stream of that risk in the market. Given that dividends (and the growth on dividends) are an equity flow we should assume that the rate to use is the minimum required rate of return for an investment of that risk.

The capital asset pricing model is usually used as a predictor of the equity investors’ required rate of return. In principle the capital asset pricing model is a one period model in that it measures the expected return over a single holding period. The extent that it is legitimate to extend the rate of return predicted by the capital asset pricing model over a long series of holding periods is open to question. ** Bansal, Dittmar and Kiku (2005)** were concerned with this issue. Using a process called

**these researchers investigated the relationship between asset beta and consumption beta over long time horizons. The**

*stochastic co-integration***demonstrated is that in the short run, variations in returns are principally explained by transitory price shocks but over the long run it is dividend shocks that are the major source of return volatility. However, although dividends appear to be the strongest predictor of consumption betas over the longer time horizon, and the consumption betas are closely related to the asset betas, there is little to suggest the consistency in the term structure that the valuation model requires. This throws a question mark over the use of the model if long run returns are not valued by investors in the same way as short run returns. One answer to this problem is that over the very long run errors in the discount rate will have a receding impact upon current valuations. However, near term errors are likely to be much more significant.**

*Bansal et al study*__The Constant Dividend Growth Assumption__

What this implies is that future dividends grow at a constant, predictable rate. At first sight this appears to be very unlikely but the important question when building a valuation model is not what we think might happen but what the market as a whole expects to happen. Given our earlier discussion about the pricing process in markets variability and inconsistency in growth assumptions are likely to be cancelled out and a single dominant expectation of future growth emerge. Our simplest assumption is that future growth is a constant percentage (predicting changing rates adds a layer of complexity which we can incorporate at a later stage if the circumstances suggest that is appropriate). The model also requires what we will term the long run equilibrium growth rate, that is a growth rate which assumes that the firm’s opportunities for earning a rate of return in excess of the cost of its capital have been exhausted (which we assume will be the case over the long run).

**How value varies with growth**

The dividend discount model has some important **implications**.

**Firs**t it places a limit on the long run rate of growth (g) which cannot exceed the investors’ required rate of return when discounting dividends. This is often taken as a fatal flaw with the model but a moments thought reveals that it is not the model that is at fault. If a firm’s limit on growth is constrained by its ability to create new investment through retention then in the very long run the rate of return on new investment must converge on the investors required rate of return (to assume otherwise implies that the firm can keep finding positive net present value projects indefinitely). If the firm reinvests all of its surpluses then the maximum rate of return it will earn is the rate of return required by its equity investors and that will be the rate of growth of its capital account. If it only reinvests 50 per cent of its earnings (say) then the maximum growth rate will be 50 per cent of the shareholders’ required rate, and so on. What it cannot do is consistently reinvest more than 100 per cent of what it earns.

The **second** implication concerns the volatility of value and its relationship to growth. Where growth is low relative to the shareholders required rate of return then the volatility of the share price will also be low, where growth is high volatility will be high. If growth relative to return is very high then even minor changes in investor sentiment towards future growth rates can have sudden and dramatic effects upon market values. Much has been talked about bubbles and crashes in the equity markets with the most recent and spectacular occurring in the years 2007 – 2008 Global Financial Crises. However, what the model suggests is that these are not due to ‘irrational exuberance’ in the pricing of shares **(Robert Schiller, 2000)** nor necessarily to the domination of the market by speculators.

The model suggests that prices will always rise steeply when beliefs about future growth rates converge on the required rates of return on equity. The Gordon Growth Model does, therefore, give us some very interesting insights into the way real markets may work. However, concerns about the veracity of earnings numbers, or indeed, when faced with a company that does not pay dividends, may lead us to an alternative flow valuation method.

**The Free Cash Flow to Equity Model**

This modelling procedure is based upon the concept behind the dividend valuation model except that instead of dividends we employ the Free Cash Flow to Equity (FCFE) after reinvestment in their place. The intuition behind the model is slightly different in that instead of working from the definition of return we work from the idea that the value of the firm to its investors is the discounted value of the future net cash flow available either for payment as dividend or for reinvestment. Working on the basis of the net free cash flow to equity per share then the price of the share is given by:

Following the same intuition as before the growth rate will be the rate of cash reinvestment (b_{c}) times the rate of return on reinvestment.

This model is a ‘pure cash flow’ model and as such there is no ambiguity about the reinvestment rate – it should be the firm’s current internal rate of return on future investment. However, given that we are again trying to impute a long term rate of growth we can make the case again that the simplest assumption is, that over the longer run, the firm’s internal rate of return will be driven down to its cost of capital and so the formula above becomes: