# Capital Gearing and the CAPM

The CAPM defines a project's discount rate as a return equal to the risk-free rate of interest, plus the product of the market premium and the project's asset beta (a risk premium) to compensate for systematic (business) risk. However, we now know that the financial risk associated with capital gearing can also affect beta factors. So, the discount rate derived from the CAPM for investment appraisal must also be affected, but how?

Let us first consider a company funded entirely by equity that is considering a new project with the same level of risk as its existing activities. The firm's equity beta (bEU) can be used as the project's asset beta (bA) because the shareholders' return (Ke) equals the company's return (r) on a new project of equivalent risk. So, the project return that provides adequate compensation for holding shares in the company is the equity return (Ke) obtained by substituting the appropriate equity beta (bE) into the familiar CAPM formula.

The CAPM therefore offers management an important alternative to the derivation of project discount rates that use the traditional dividend and earnings valuation models explained in the * SFM *texts. In an

*(all-equity) firm, the*

**unlevered***return (Ke) defines the*

**shareholders'***cost of capital (KU) as follows:*

**company's**The question we must now ask is whether Equation (55) has any parallel if the firm is geared.

The short answer is yes. Rather than use traditional dividend, earnings and interest models to derive a WACC (explained in SFM) we can substitute an appropriately * geared *asset beta for an

*beta into the CAPM to estimate the overall return on debt and equity capital for project appraisal.*

**all-equity**7.5 Modigliani-Miller and the CAPM

Without debt in it capital structure, a company's asset beta equals its equity beta for projects of equivalent risk. However, according to MM's theory of capital structure * (op. cit.) *based on their "law of one price" and the

*process, companies that are identical in every respect apart from their gearing should also have the same asset betas. Because their business risk is the same, the factors are not influenced by methods of financing. To summarise MM's position*

**arbitrage**An ungeared company's asset beta equals its equity beta.

A geared company's asset beta is lower than its equity beta.

Irrespective of gearing, the asset beta for any company equals the equity beta of an ungeared company with the same business risk.

The asset beta (equity beta) of an unlevered company can be used to evaluate projects in the same risk class without considering their finance.

You will recall from your studies that MM's capital theory (like their dividend irrelevancy hypothesis) depends on perfect market assumptions. However, because these assumptions also underpin much else in finance (including the CAPM) for the moment we shall accept them. To illustrate the MM relationship between the beta factors of all-equity and geared companies with the same systemic business risk, let us begin with the following equation using our familiar notation in a tax less world.

If we now rearrange terms, divide through by VE and solve for bEG, the mathematical relationship between the geared and ungeared equity betas can be expressed as follows:

This equation reveals that the equity beta in a geared company equals the equity beta for an all-share company in the same class of business risk, * plus *a premium for systemic financial risk. The premium represents the difference between the all-equity beta and debt beta multiplied by the debt-equity ratio. However, the important point is that the increase in the equity beta measured by the risk premium is exactly offset by a lower debt factor as the firm gears up leaving the asset beta unaffected. In other words, irrespective of leverage, the asset betas of the two firms are still identical and equal the equity beta of the ungeared firm.

For those of you familiar with MM's capital structure hypothesis, the parallels are striking. According to MM, the expected return on equity for a geared firm (K) relative to the return (K) for an all-share firm in a taxless world equals:

This states that the return for a geared firm equals an all-equity return for the same class of business risk, * plus *a financial risk premium defined by the difference between the all-equity return and the cost of debt multiplied by the debt-equity ratio. The premium compensates shareholders for increasing exposure to financial risk as a firm gears up. As we observed in SFM, however, because the cheaper cost of debt exactly offsets rising equity yields, the overall cost of capital (WACC) is unaffected. So, irrespective of leverage, all firms with the same business risk can use the cost of equity for an all-share firm as a project discount rate before considering methods of financing.

Turning to a world of taxation, where debt is a * tax-deductible *expense with a tax rate (t), we can redefine the equity beta of a geared company from Equation (58) as follows:

And if debt is risk-free with * zero *variance, so that bD is zero, the formula simplifies to:

**Review Activity**

To illustrate the union between MM and the CAPM, consider a leveraged company in an economy where interest is tax deductible at a 20 per cent corporate rate. 20 million ordinary shares are authorized and issued at a current market value of £2.00 each * (ex-div). *The equity beta is 1.5. Debt capital comprises £10 million, irredeemable 10 per cent loan stock, currently trading at par value.

Calculate the company's asset beta and briefly explain the result.

Since the equity beta for an * ungeared *company equals the asset beta for any company in the same risk class, we can use Equation (61) to solve for bEU and hence bA as follows.

First, define the market values of equity and debt:

Next, define the geared equity beta of 1.5 assuming that debt sold at par is risk-free (bD = 0).

Finally, rearrange terms to solve for ßEU and ßA.

The result is to be expected. The asset beta should be smaller than the geared equity beta (i.e. 1.25 < 1.5) since the systemic risk associated with the asset investment is only one component of the total risk associated with the shares. The asset beta measures business risk, whereas the geared beta measures business and financial risk

# Summary and Conclusions

If management use the CAPM rather than a WACC to obtain a risk-adjusted discount rate for project appraisal, they need to resolve the following questions

Question: **Is the business risk of a project equivalent to that for the company?**

Answer: YES NO

Solution: Use the company's current

Use an equity beta for similar equity beta companies with similar projects

Question: **Is the chosen equity beta affected by capital gearing?**

Answer: YES NO

Solution: De-leverage "ungear" the Use an equity beta equivalent to an equity beta to derive an asset beta if it is not affected by gearing asset beta

Having obtained an appropriate asset beta, the project discount rate may then be calculated using the CAPM formula.

According to MM's capital structure theory, the asset betas of companies, or projects, in the same class of business risk are identical irrespective of leverage. Higher equity betas are offset by lower debt betas, just as higher equity yields offset cheaper financing, as a firm gears up

Even in a taxed world, it is possible to establish a connection between MM and the CAPM. With tax, the MM cost of equity for a geared firm is given by:

According to the CAPM, the equity costs for an ungeared and geared firm are given by:

Where:

If we assume that the company's pre-tax cost of debt (Kd) in Equation (63) equals the risk-free rate (rf) in Equations (64) and (65) we can write rf for Kd in Equation (63). If we now substitute Equations (64) and (65) into Equation (63) rearrange terms and simplify the result, we can confirm our earlier equation for a geared equity beta:

For an application of this formula and the derivation of the cost of equity using the CAPM see Exercise 7.2 in the companion text.

# Selected References

1. Modigliani, F. and Miller, M.H., "The Cost of Capital, Corporation Finance and the Theory of Investment", * American Economic Review, *Vol. XLVIII, No. 4, September 1958.

2. Miller, M.H. and Modigliani, F., "Dividend Policy, Growth and the Valuation of Shares", * Journal of Business of the University of Chicago, *Vol. 34, No. 4, October 1961.

3. Hill, R.A., __bookboon.com__

- * Strategic Financial Management, *2009.

- * Strategic Financial Management; Exercises, *2009.

- * Portfolio Theory and Financial Analyses; Exercises, *2010.