**It can be observed that higher financial leverage increases equity beta. However, the relationship between the unleveraged asset or enterprise beta (the beta of the underlying operating business), and leveraged equity beta that is commonly applied in practice, is incomplete.**

**We explain the relevance of asset betas in equity valuation and why it is important to analyse the beta of debt finance and the value, and riskiness, of the debt interest tax shield when delevering and relevering equity beta.**

Financial leverage increases both the risk and the expected return for equity investors. Enterprise (business or operating) risk and return are shared between the different claim holders – primarily debt and equity investors. With debtholders taking a prior claim, their exposure to business risk is mitigated but, consequently, they are willing to accept a lower return on their investment. A larger amount of debt magnifies the returns for equity investors, but also concentrates business risk. Changing capital structure does not directly alter the total business risk of the enterprise, but it does affect how that risk affects each claim.

The effect of leverage on equity beta can be seen in practice by comparing the observed equity betas of stocks with different financial leverage and in the changes in equity beta of a single stock over time. We highlighted the latter effect in our article ‘Calculating and analysing the drivers of equity beta’, which included an interactive model that calculates the beta for any investment for which price data is available through the ‘rich data’ functionality in Microsoft Excel (which is most global listed stocks). We used cruise company Carnival Corp. to illustrate the effect of higher financial leverage on the stock equity beta.

**Carnival Corp. – Current and historical trend for 2-year beta**

To reproduce this chart, use the stock ticker XNYS:CCL in the downloadable model and select the Global MSCI Index for the US time zone. We used daily data and a 2-year interval for the charts.

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Prior to the Covid-19 pandemic Carnival had an equity beta close to 1.0, which was relatively stable from 2013 to 2019. The business disruption caused by Covid-19 resulted in significant additional borrowings, a fall in stock price, and an increase in financial leverage. Higher leverage increased the stock volatility and the equity beta, which has been significantly over 2.0 for the last 3 years. Based on a 2-year view, and using daily price data, the historical Carnival equity beta is currently 2.28.

Although book value leverage (balance sheet debt compared with shareholders’ equity) is commonly used in equity analysis, it is market leverage, based on the values of debt and equity, that affects equity beta. Market values of financing claims, not their book values, matter when analysing risk and beta.

In the table below, we show the historical market leverage and equity beta of Carnival in 2023 compared with 2019. While we think some of the increase in equity beta is due to a higher level of underlying business risk (we estimate the implied underlying asset beta has risen from about 0.8 to over 1.0 – see below), most of the increase in equity risk is due to higher financial leverage.

**Carnival Corp. – Equity beta and market leverage**

The Footnotes Analyst estimates.

Equity beta is based on 2-years of daily price data. Market leverage is balance sheet net debt divided by market capitalisation. The amounts represent our approximation of average leverage over the 2-year period covered by the price data used for the equity beta calculation.

Understanding that beta and risk are affected by financial leverage is important but not sufficient – investors should also be able to quantify the effect. This is particularly important when using the Capital Asset Pricing Model to determine cost of equity and investment discount rates.

**Quantifying the leverage effect on beta**

One of the key features of betas is that they can be combined through a simple weighted average calculation. For example, the beta for a portfolio is equal to the weighted average of the betas of the constituent investments, with the weightings being the market values of those investments. The same effect also applies to claims on a business, the beta of the enterprise is the weighted average of the betas of the claims on that enterprise.^{1}The beta of the enterprise is often referred to as an asset beta. We use the terms interchangeably.

Ignoring taxation for the moment (see below) this gives:

## {~~ \sf Beta_{enterprise}~ = ~ Beta_{equity} ~x~ \dfrac{E}{D + E} + ~ Beta_{debt} ~x~ \dfrac{D}{D + E} ~~~~...~1}

E | Fair value of equity |

D | Fair value of debt |

This formula is often quoted in a simplified form where the beta of debt is assumed to be zero, and therefore the last term can be ignored.

## {~~ \sf Beta_{enterprise}~ = ~ Beta_{equity} ~x~ \dfrac{E}{D + E} ~~~~...~2}

E | Fair value of equity |

D | Fair value of debt |

Rearranging the above gives perhaps the most commonly quoted (and most intuitive) version of this calculation, whereby the equity beta is expressed as a function of asset (enterprise) beta and market leverage:

## {~~ \sf Beta_{equity}~ = ~ Beta_{enterprise} ~x~ \Big(1+\dfrac{D}{E}\Big) ~~~~...~3}

E | Fair value of equity |

D | Fair value of debt |

While the assumption of the debt beta being zero may sometimes not materially affect the answer, this is often not the case. Even for companies with investment grade credit ratings, debt holders are subject to a share of underlying business risk. In our view, debt betas and the sharing of business risk with debt holders are greater than many investors appreciate.

It is possible to calculate a debt beta from price data in the same manner as for an equity beta. It is also possible to observe average debt betas using the price changes for bond funds. The beta of a bond fund is itself the weighted average of the betas for the underlying bond constituents. The chart below shows the results for 15 US bond ETFs.

**Bond ETF beta compared with average rating of constituent investments**

Data derived for a selection of US bond ETFs using the Footnotes Analyst Beta Calculator Model.

The two ETFs with betas close to or equal to zero are a US government bond ETF and a fund investing in high quality (94% AAA) CDOs. The middle group, with a beta of 0.2 to 0.3, are portfolios of investment grade corporate bonds where the credit ratings of constituent bonds are mostly in the range of BBB to AA. The average bond rating for the portfolio is derived from the disclosed asset allocation for each ETF. The final group represents portfolios of high yield / sub-investment grade bonds.

The chart shows there is a clear relationship between bond beta and bond rating and that the betas for even investment grade bonds are not insignificant. There is an approximate 0.09 increase in bond beta for each major step in bond rating.

Obviously, the impact of debt on an equity beta varies, but be particularly careful when leverage is significant and where debt ratings are lower. In our view, it is best to apply the more comprehensive calculation (formula 1 above), that includes explicit allowance for the bond beta, and not the shortened version that is often applied in practice.

**The impact of taxation**

The tax advantage of debt arises because debt interest is tax deductible when calculating corporate tax liabilities, but returns to equity investors are not. The present value of these debt tax savings is called the debt interest tax shield. Higher leverage increases the value of this tax shield, which in turn increases the value of the equity shareholder interest.

The asset beta of an enterprise is generally expressed as the beta of the business excluding the tax shield. The debt tax shield is regarded as a function of how the business is financed. The enterprise asset beta is therefore the weighted average of the betas of equity, debt and the value of the tax shield (which has a negative weighting).

## { ~\sf Beta_{enterprise}~ = ~ Beta_{equity} ~x~ \dfrac{E}{E+D-TS} + ~ Beta_{debt} ~x~ \dfrac{D}{E+D-TS} - ~ Beta_{TS} ~x~ \dfrac{TS}{E+D-TS}~~...~4}

E | Fair value of equity |

D | Fair value of debt |

TS | Value of the debt interest tax shield |

How the debt tax shield affects the impact of leverage on beta depends on the level of risk assumed for the tax shield itself.

If the beta of the tax shield is equal to the beta of the enterprise, the above actually simplifies to the initial equation above (formula 1) and taxation can be ignored. However, in our article ‘Valuing the debt interest tax shield’ we argue that the tax shield for current debt (and any changes in debt arising from a change in funding policy) should be valued using the cost of debt and, therefore, has a level of risk equal to the debt beta. Only incremental tax shields arising from additional debt due to business growth have a level of risk equal to that of the enterprise itself. Reflecting this in the above produces the following.

## { ~~\sf Beta_{enterprise}~ = ~ Beta_{equity} ~x~ \dfrac{E}{E+D-TSp} + ~ Beta_{debt} ~x~ \dfrac{D-TSp}{E+D-TSp} ~~~~...~5}

E | Fair value of equity |

D | Fair value of debt |

TSp | Value of the debt interest tax shield arising from current debt and forecast changes in debt due to a change in debt policy |

Where TS_{p} is the value of the tax shield arising from debt policy – the debt in place plus any incremental changes in debt not related to business growth.

In the case of a company that is already operating at its desired capital structure, TS_{p} equals D.T* and the above simplifies to:

## { ~~\sf Beta_{enterprise}~ = ~ Beta_{equity} ~x~ \dfrac{E}{E+D(1-T^*)} + ~ Beta_{debt} ~x~ \dfrac{D(1-T^*)}{E+D(1-T^*)} ~~~~...~6}

E | Fair value of equity |

D | Fair value of debt |

T* | The percentage tax advantage of debt |

The tax rate used above of T* is the net tax advantage of debt after allowing for the effect of the personal tax advantage of equity financing.

The value of the tax shield may be reduced if there is a lower rate of personal tax applied to returns for investors holding equity compared with debt. In many jurisdictions, equity investors pay less personal tax to allow for the corporate tax payments already charged on their earnings. This personal tax saving creates an effective equity tax shield that may partially (or even sometimes fully) offset the debt interest tax shield.

The net tax advantage of debt (T*) is given by:

## {~~ \sf T^*~ = ~ Tc - Tp'.(1 - Tc) ~~~~...~7}

T* | Effective tax advantage of debt after allowing for personal tax effects |

Tp’ | The personal tax advantage of equity |

Tc | Rate of corporate tax |

Where Tp’ is the net percentage personal tax advantage of equity (the post-tax equity return relative to the post-tax debt return for a given pre-tax return).

## {~~ \sf Tp'~ = ~\dfrac{(1 - Tpe)}{(1 - Tpd)} - 1 ~~~~...~8}

T* | Effective tax advantage of debt after allowing for personal tax effects |

Tp’ | The personal tax advantage of equity |

Tpe | Effective rate of personal tax on returns from equity investments |

Tpd | Effective rate of personal tax on returns from debt investments |

**For more about the effect of personal taxes on the net tax advantage of debt, and an explanation of the above calculations, see our article ‘Valuing the debt interest tax shield’. Also see our downloadable model ‘DCF valuation: Financial leverage and the debt tax shield’ where the above leverage adjustments to beta are applied.**

In practice it is common for the personal tax effects to be ignored and the tax advantage of debt set equal to the rate of corporate tax (i.e. T* = Tc). This is mainly because of the challenges of determining the personal tax effect where multiple types of investor and multiple tax jurisdictions are involved, rather than because the personal tax effects do not exist. This was the approach to leverage adjustments that was applied when we worked in UBS investment research. The famous (and now very old) UBS ‘EV Guide’^{2}We cannot claim any credit for the UBS publication The EV guide. It was written by two of the heads of research prior to Steve joining UBS in 1997. Like many of these investment banking reports with a long shelf life, a copy has found its way onto the web – here it is., that formed the basis of the UBS Investment Research EV based approach to valuation (at least when we were involved), included the following:

“The size of the tax wedge depends on the interaction between the corporation tax code and the income tax code, leading the analysis into the mire of imputation systems, international tax treaties, and the treatment of pension funds. We conclude that we cannot estimate the size of the tax wedge with confidence using tax code data.”

UBS 1996 EV Guide. (‘Tax wedge’ is the term used in that publication for the tax shield.)

That is not to say we think personal tax effects should always be ignored – you must make your own judgement.

**Delevering and relevering stock and sector betas**

One of the key reasons why investors should understand the link between leverage and beta is because of the need to delever and relever beta factors when calculating costs of capital. Delevering is simply calculating an asset (enterprise) beta from an observed equity beta. Levering is the opposite. Generally, asset betas cannot be observed in the market, therefore the asset betas that are relevered are those that are themselves derived by delevering from observed equity betas.

There are three main reasons for applying these calculations:

**Estimating a more reliable historical equity beta**

Equity betas that are derived from historical price data have an inherent margin of error. Like any statistic that is derived from a sample, the result is only an estimate of the true (population) statistic. For equity beta, different samples of price data produce a different answer. In part, this is due to using different time periods, but it is also due to the statistical margin of error.

We include a 95% confidence interval range in our equity beta model. In the calculation of the Carnival Corp. equity beta of 2.28, we show a range of 2.04 to 2.52. Had we chosen a longer period of price data this range would be smaller, but may lead to the beta being less relevant, particularly when seeking to understand changes or where a measure that more closely reflects current risk characteristics is desired.

One way to increase confidence in beta is to make use of data for other companies in the same sector to derive a sector beta. However, it is not possible to simply use an average sector equity beta due to the differences in financial leverage. A better approach is to delever the equity betas of the group of stocks to remove their individual financial leverage. The resulting group of asset betas can be used to derive an average sector asset beta. This is then used as the asset beta of the stock in question, which can be relevered using that company’s specific financial leverage to obtain its equity beta.

**Estimating an equity beta where there is no price data**

When valuing unlisted equity instruments there is no price data to derive an equity beta directly. The only possible approach to is consider the price data for similar listed companies. The process to derive an equity beta for an unlisted stock is exactly the same as the delevering and relevering process described above.

**Estimating an unleveraged cost of equity for use in APV valuations**

A further use of an asset beta, either derived by delieering the equity beta of a single stock or, more commonly, the average of the asset betas of a group of comparable companies, is to derive a discount rate for an Adjusted Present Value (APV) valuation.

APV is an alternative to DCF valuation using WACC as a discount rate. In an APV valuation enterprise free cash flows are discounted at the unleveraged cost of equity, which itself is obtained by applying the asset beta in CAPM. The value of the debt interest tax shield is then added to this present value to obtain the target enterprise value.

**Carnival Corp. analysis**

The increase in equity beta of Carnival Corp. that we highlight above is undoubtedly primarily driven by the higher financial leverage experienced following the Covid pandemic disruption. The extent to which there has been a change in the market perception of business risk can be derived by calculating an asset beta from the equity beta before and after the change in leverage.

We use the more comprehensive calculation explained above:

## { ~~\sf Beta_{enterprise}~ = ~ Beta_{equity} ~x~ \dfrac{E}{E+D(1-T*)} + ~ Beta_{debt} ~x~ \dfrac{D-TSp}{E+D(1-T*)} ~~~~...~6}

E | Fair value of equity |

D | Fair value of debt |

TSp | Value of the debt interest tax shield arising from current debt and forecast changes in debt due to a change in debt policy |

To do so we need to identify the debt beta and tax shield value (T*).

**Debt beta:**We estimate that the average debt rating for Carnival in 2019 was ‘A’ and that this decreased to ‘BB’ for the period after 2020. Our approximation is simply for the purpose of this illustration and, although directionally correct, it may not be wholly accurate. For example, we have not attempted to allow for the different types of debt issued by Carnival or for the effects of their convertible bonds.

**Value of the tax shield:**Carnival has a very low effective tax rate due to its domicile and business model. Consequently, we have assumed a zero tax shield. The company’s international tax arrangements are complex and would require further investigation in practice.

**Carnival Corp. – Change in equity and asset beta**

The Footnotes Analyst estimates

Applying the more comprehensive delevering calculation that includes the debt beta reveals that the Carnival Corp. asset beta increased from 0.81 in 2019 to 1.01 today. The simplified approach in which the debt beta is assumed to be zero and omitted from the leverage adjustments incorrectly indicates that the asset beta has remained at 0.78 over this period. In our view the more relevant answer is that provided by the comprehensive calculation and that the asset beta and business risk of Carnival has indeed increased over this period.

Of course, this all is subject to the usual statistical error involved in all beta calculations and, additionally, the assumptions we made for our model inputs. Nevertheless, we think this example illustrates the importance of not automatically using the simplified version of the beta and leverage calculation that is often quoted in practice.

**Insights for investors**

**Equity betas increase with higher financial leverage. Observed historical equity betas reflect the average financial leverage during the period for which price changes are included.**

**The commonly quoted relationship between asset and equity betas ignores the beta of debt and the potential for the debt tax shield to be lower than the rate of corporate tax.**

**Use the more comprehensive relationship between asset and equity betas that explicitly allows for debt betas and the risk shared with debt holders.**