# II: Share Valuation Theories

## How to Value a Share

### Introduction

The key to understanding the basic measures of stock market performance (price, yield, P/E ratio and cover) used by investors to analyse trading decisions requires a theoretical appreciation of the relationship between a share's price and its return (dividend or earnings) using various models based on discounted revenue theory.

To set the scene, we shall keep the analysis simple by outlining the theoretical determinants of share price with particular reference to the capitalisation of a perpetual annuity using both a dividend yield, and earnings yield. Detailed consideration of the controversy as to whether dividends or earnings are a prime determinant of share price will be covered in Chapter Three.

### The Capitalisation Concept

Discounted revenue theory defines an investment's present value (PV) as the sum of its relevant periodic cash flows (Ct) discounted at an appropriate opportunity cost of capital, or rate of return (r) on alternative investments of equivalent risk over time (n). Expressed algebraically:

The equation has a convenient property. If the investment's annual return (r) and cash receipts (Ct) are constant and tend to infinity, (Ct = C = C2 = C3 = C°° ) their PV simplifies to the formula for the capitalisation of a constant perpetual annuity:

The return term (r) is called the capitalisation rate because the transformation of a cash flow series into a capital value (PV) is termed "capitalisation". With data on PV°° and r, or PV°° and Ct, we can also determine Ct and r respectively. Rearranging Equation (2) with one unknown:

Activity 1

The previous PV equations are vital to your understanding of the various share valuation models that follow. They also underpin the remainder of this study. If you are unsure of their theory and application, then I recommend that you download Strategic Financial Management (SFM) from the author's bookboon series and read Chapters Two and Five before you continue.

Having completed this reading, you will appreciate that shares may be traded either cum-div or ex-div, which means they either include (cumulate) or exclude the latest dividend. For example, if you sell a share cum-div today for P0 the investor also receives the current dividend D0. Excluding any transaction costs, the investor therefore pays a total price of (D0 + P0). Sold ex -div you would retain the dividend. So, the trade is based on current price (P0) only.

This distinction between cum-div and ex-div is important throughout the remainder of our study because unless specified otherwise, we shall adopt the time-honored academic convention of defining the current price of a share using an ex-div valuation.

### The Capitalisation of Dividends and Earnings

Irrespective of whether shares are traded cum-div or ex-div, their present values can be modeled in a variety of ways using discounted revenue theory. Each depends on a definition of future periodic income (either a dividend or earnings stream) and an appropriate discount rate (either a dividend or earnings yield) also termed the equity capitalisation rate.

For example, given a forecast of periodic future dividends (Dt) and a shareholder's desired rate of return (Ke) based on current dividend yields for similar companies of equivalent risk:

The present ex-div value (P0) of a share held for a given number of years (n) should equal the discounted sum of future dividends (Dt) plus its eventual ex-div sale price (Pn) using the current dividend yield (Ke) as a capitalisation rate.

Expressed algebraically:

Rewritten and simplified this defines the finite-period dividend valuation model:

Likewise, given a forecast for periodic future earnings (Et) and a desired return (Ke) based on current earnings yields of equivalent risk:

The present ex-div value (P0) of a share held for a given number of years (n) equals the sum of future earnings (Et) plus its eventual ex-div sale price (Pn) all discounted at the current earnings yield (Ke).

Algebraically, this defines the finite-period earnings valuation model:

Activity 2

A logical approach to financial analysis is to make simplifying assumptions that rationalise its complexity. A classic example is the derivation of a series of dividend and earnings valuations, other than the finite model. Some are more sophisticated than others, but their common purpose is to enable investors to assess a share's performance under a variety of conditions.

To illustrate the point, briefly summarize the theoretical assumptions and definitions for the following models based on your reading of SFM (Chapter Five) or any other source material.

The single-period dividend valuation The general dividend valuation The constant dividend valuation

Then give some thought to which of these models underpins the data contained in stock exchange listings published by the financial press worldwide.

We know that the finite-period dividend valuation model assumes that a share is held for a given number of years (n). So, today's ex div value equals a series of expected year-end dividends (Dt) plus the expected ex-div price at the end of the entire period (Pn), all discounted at an appropriate dividend yield (Ke) for shares in that risk class. Adapting this formulation we can therefore define:

- The single-period dividend valuation model

Assume you hold a share for one period (say a year) at the end of which a dividend is paid. Its current ex div value is given by the expected year-end dividend (Dt) plus an ex-div price (Pt) discounted at an appropriate dividend yield (Ke).

- The general dividend valuation model

If a share is held indefinitely, its current ex div value is given by the summation of an infinite series of year-end dividends (Dt) discounted at an appropriate dividend yield (Ke). Because the share is never sold, there is no final ex-div term in the equation.

- The constant dividend valuation model

If the annual dividend (Dt) not only tends to infinity but also remains constant, and the current yield (Ke) doesn't change, then the general dividend model further simplifies to the capitalisation of a perpetual annuity.