# Discounted cash flow approach

## Introduction

There are two methods to the discounted cash flow approach:

• Present value of dividends.

• Free cash flow.

## Present value of dividends

### Introduction

We know from the bond market that the plain vanilla bond has a finite life and pays a fixed rate of interest. We also know that * yield to maturity *is an average rate of return over the period of the bond. The price of the bond (= value = present value) is discounted value of the fixed coupons and the principal repayable at maturity, discounted at the yield to maturity (ytm). The formula is:

where

cr = coupon rate (cr / 2 if six-monthly)

ytm = yield to maturity (ytm / 2 if six-monthly)

n = number of periods (years x 2 if six-monthly).

This is nothing else than the classical PV-FV formula: it discounts future cash flows (coupons and principal) at the ytm to present value.

We also know that in the case of a perpetual bond, and a perpetual preference share, the formulae are:

Price (perpetual bond) = fixed coupon rate / required rate (ytm)

Price (perpetual preference share) = fixed dividend rate / required rate.

This formula (they are the same) is nothing else than the classical PV-FV formula: it discounts future cash flows (coupons / dividends) to present value.

Ordinary shares are nothing else than perpetual bonds / preference shares, but without a fixed dividend. Thus, with equities there is no finite period of investment, and therefore no face value to be repaid. Equities are permanent capital and only pay dividends.

At this stage a reminder of the perpetual bond / preference share formula is required (cf = annual cash flow = interest or dividends; rrr = required rate of return):

PV = [cf / (1 + rrr)1] + [cf / (1 + rrr)2] + ...[cf/ (1 + rrr)3] + ... o°.

This simplifies to:

PV = cf / rrr

### Dividend discount model

In the case of ordinary shares, the pricing formula may be written as (D = dividend):

As in the case of perpetual preference shares, this simplifies to:

PV = D / rrr.

This model is called the **dividend discount model **(DDM), and it determines that the present value of a share is equal to the discounted value of future dividend flows (which are here assumed to be constant), at the rrr.

### Constant growth dividend discount model

This model is not applicable to ordinary shares because it ignores the fact that * dividends grow over time. *In the case of growing dividends, the formula may be written as (D = dividends in year 1, year 2, year 3...etc. to infinity):

However, there is a * big problem *here: it is not possible to make forecasts of dividend flows deep into the future. Thus, this formula becomes a

*rather than a useful tool, which leads us the so-called*

**principle**

**Gordon constant-growth DDM.**The formula above is made practical by assuming that the immediate past dividend (which of course was observed) will grow in the future at a * constant rate of growth. *The formula now becomes:

where

D0 = past dividend

D = assumed growth rate in dividends.

This simplifies to:

If the * past *dividend of share XYZ was LCC6.0, the dividend growth rate is 8% (based on past growth rates), and the rrr = 14%, then D1 = D0 x 1.08 = LCC6.0 x 1.08 = LCC6.48, and the present value of this share is:

PV = LCC6.48 / (0.14 - 0.08) = LCC6.48 / 0.06

= LCC108.00.

It will be apparent that in terms of this **constant-growth DDM **(or CGDDM), the PV of the share will be higher under the following conditions:

• As the rrr falls the PV rises. Example: if the rrr = 12%, D0 = LCC6.00, Dg = 8%, then the PV of the share is LCC162.00 [LCC6.48 / (0.12 - 0.08)].

• If the growth rate in dividends is higher, the PV rises. Example: if rrr = 14%, D0 = LCC6.00, Dg = 9.5%, the PV of the share is LCC146.00 [LCC6.57 / (0.14 - 0.095)].

### Multi-stage growth model

Constancy in the growth in dividends is applicable to mature companies (and the valuation model can be called the * infinite period *CGDDM), but not to young companies whose dividend growth is higher in the early stages of operations and constant later. The valuation model in this case is called a multi-stage growth model. We assume the growth rates in dividends as indicated in Table 3 [we also assume the current dividend is LCC2 (D0) and the rrr = 14%]38.

The PV of the company's share is LCC94.36. It will be evident that estimating the dividend growth rate and how long each phase will last is fraught with problems.

**Table 3: Assumed dividend growth rates and computation of PV of share**

### Required rate of return

It is now necessary to talk about the rrr. The rrr can be any number desired, but it must be above the risk-free rate, because this is the lowest rate that can be earned without assuming any risk. Thus, the rrr is made up of two parts:

rrr = rfr + rp (risk premium).

The Capital Asset Pricing Model (CAPM) provides us with a neat explanation of risk. According to the CAPM, the risk premium is made up of two parts:

• The additional return that investing in shares offers above the rfr.

• The volatility of the particular share relative to the market as a whole, i.e. the beta ((3). If a share has a beta of 2, this means that the share has a tendency to rise twice as much as the market over the chosen period of time, i.e. when the chosen index rises by z percent over a period, the share has a tendency to rise by 2 x z percent.

The additional return is the extent to which the return on the market (mr) exceeds the rfr (mr - rfr). Thus the rrr is:

rrr = rfr + (mr - rfr)(3.

The CAPM thus states that the rrr depends on the risk-free rate, the risk premium associated with investing in shares, and the risk associated with the specific share.

How the model is used should be apparent. An example will be useful. If the rfr = 8.0%, the market is expected to rise 14%, and the security has a beta of 1.7, the rrr is equal to:

rrr = rfr + (mr - rfr)(3

= 8.0 + (14.0 - 8.0)1.7 = 8.0 + (6.0)1.7 = 8.0 + 10.2

= 18.2%.

Assuming the past dividend of share XYZ = LCC6.0, the Dg = 8%, its value is:

PV = (D0 x (1 + D )) / (rrr - D )

0 g g

= (LCC6 x (1.08)) / (0.182 - 0.08) = LCC6.48 / 0.102 = LCC63.53.