Where:

Ve = market value of equity;

Vd= market value of debt;

T = corporation tax rate;

ßa = the asset beta;

ße = the equity beta;

ßd = the debt beta.

ßd, the debt beta, is nearly always assumed to be zero, so the formula simplifies to:

ßa = Ve ße / Ve + Vd (1 – T)

What is meant by ßa (the asset beta), and ße (the equity beta)?

The asset beta is the beta (a measure of risk) which arises from the assets and the business the company is engaged in. No heed is paid to the gearing. An alternative name for the asset beta is the ‘ungeared beta’.

The equity beta is the beta which is relevant to the equity shareholders. It takes into account the business risk *and* the financial (gearing) risk because equity shareholders’ risk is affected by both business risk and financial (gearing) risk. An alternative name for the equity beta is the ‘geared beta’.

Note that the formula shows that if Vd = 0 (ie there is no debt), then the two betas are the same. If there is debt, the asset beta will always be less than the equity beta because the latter contains an additional component to account for gearing risk.

The formula is extremely useful as it allows us to predict the beta, and hence the cost of equity, for any level of gearing. Once you have the cost of equity, it is a straightforward process to calculate the WACC and hence discount the project.

To illustrate the use of CAPM in determining a discount rate, we will work through the following example, **Example 2**.

#### Example 2

Emway Co is a company engaged in road building. Its equity shares have a market value of $200 million and its 6% irredeemable bonds are valued at par, $50m. The company’s beta value is 1.3. Its cost of equity is 21.1%. (Note: this figure is quite high in the current economic situation and is used for illustration purposes. Currently, in a real situation, the cost of equity would be lower.)

The company is worried about the recession and is considering diversifying into supermarkets. It has investigated listed supermarkets and found one, Foodoo Co, which quite closely matches its plans. Foodoo has a beta of 0.9 and the ratio of the market value of its equity to its debt is 7:5. Emway plans that its new venture would be financed with a market value of equity to market value of debt ratio of 1:1.

The corporation tax rate is 20%. The risk free rate is 5.5%. The market return is 17.5%.

What discount rate should be used for the evaluation of the new venture?

**Solution:**

We have information supplied about a company in the right business but with the wrong gearing for our purposes. To adjust for the gearing we plan to have, we must always go through a theoretical ungeared company in the same business.

Again the beta supplied to us will be the beta measured in the market, so it will be an equity (geared) beta. Were Foodoo to be ungeared, its asset beta would be given by:

ßa = Ve ße / Ve + Vd (1 – T)

= (7 x 0.9)/(7 + 5 (1 – 0.2))

= 0.5727

This asset beta (ungeared beta) can now be adjusted to reflect the gearing ratio planned in the new venture:

0.5727 =1/(1 + 1(1 – 0.2))ße

So the planned ße will be 0.5727 x 1.8 = 1.03

Check that this makes sense. Foodoo has a gearing ratio of 7:5, equity to debt, a current beta of 0.9, and a cost of equity of 16.30 (calculated from CAPM as 5.5 + 0.9(17.5 – 5.5)). Were Foodoo ungeared, its beta would be 0.5727, and its cost of equity would be 12.37 (calculated from CAPM as 5.5 + 0.5727(17.5 - 5.5)).

Emway is planning a supermarket with a gearing ratio of 1:1. This is higher gearing, so the equity beta must be higher than Foodoo’s 0.9.

To calculate the return required by the suppliers of equity to the new project:

Required return = Risk free rate + ß (Return from market – Risk free rate)

= 5.5 + 1.03 (17.5 – 5.5) = 17.86%

17.86 is the return required by equity holders, but the new venture is being financed by a mix of debt and equity, and we need to calculate the cost of capital of this pool of finance.

Note that while *Financial Management* does not require students to undertake calculations of a project-specific WACC, they are required to understand it from a theoretical perspective.

The appropriate rate at which to evaluate the project is the WACC of the finance. Again, in the F9 and P4 exam formula sheet you will find a formula for WACC consisting of equity and irredeemable debt.

Ke = 17.86%

Kd = 6% (from the cost of the debentures already issued by Emway)

WACC = 1/(1+1) x 17.86 + 1/(1+1) x 6 (1 – 0.2) = 11.33%

In terms of the diagram used in **Example 1**, for Modigliani and Miller with tax, what we have done for Foodoo’s figures is set out below. We started with information relating to a supermarket with a gearing ratio of debt:equity of 5:7, and an implied cost of equity of 16.30%. We strip out the gearing effect to arrive at an ungeared cost of equity of 12.37, then we project this forward to whatever level of gearing we want. In this example, this is a gearing ratio of 1:1 and this implies a cost of equity of 17.86%.

Finally, we take account of the cheap debt finance that is mixed with this equity finance, by calculating the WACC of 11.33%. This is the rate which should be used to evaluate the new supermarket project, funded by debt:equity of 1:1.

**Ken Garrett is a freelance author and lecturer**