The CAPM, or Capital Asset Pricing Model, is one of the models that describe the relationship between market risk and expected returns for an asset. We can use CAPM to determine how much of the expected return can be explained by the associated market risks. It is an important tool to determine the cost of equity for a firm. The cost of equity is used to determine the hurdle rate in the investment decision.

The formula of CAPM is:

*Expected Return = Riskfree Rate + Beta x (Expected Return on Market Portfolio – Risk-free Rate)*

The difference between expected return on market portfolio and the risk-free rate is called the market risk premium. It is the premium that investors demand for an investment riskier than the risk-free rate.

Thus, to use the model we need three inputs:

- The current risk-free rate
- The expected market risk premium
- The beta of the analyzed asset

## Risk-free rate

A risk-free asset is an asset where the future return is both certain and known. That means the actual return is equal to the expected return. Of course strictly speaking there are no risk-free assets but for practical purposes we assume that for some very safe investments the risk is so low that it can be ignored.

The conventional practice to estimate risk-free rates is to use the government bond rate, with government being the one that controls issuing the currency. For example, for US dollar we use the rate on a ten-year US treasury bond.

Note that not all government securities are risk-free. Some governments face risk of default so the rates on the bonds they issue will not be risk-free.

In case of a government with default risk, you can include the local currency default spread in your calculation. The risk-free rate will then be the government bond rate adjusted for (subtracted by) the default spread for the local currency as determined by the currency rating.

## Market Risk Premium

The second input required by the CAPM is the (market) Equity Risk Premium. This is the premium of an investment relative to a risk-free investment. Thus, (1) the premium is greater than 0, (2) the premium increases with risk aversion of investors and (3) the premium increases with the riskiness of the investment.

There are two ways of estimating the Equity Risk Premium. Either (1) use historical data or (2) use future expectations as input.

### Historical Equity Risk Premium

Historical data has as main advantage that factual information is available. If you choose to use historical data, make sure to use a long enough time window, as well as ensure it’s consistent with the risk-free rate and use a compounded average. However, always be aware that historical data is backward looking, noisy and subject to selection bias.

### Implied Equity Risk Premium

A relatively new approach to estimating the Equity Risk Premium is the forward looking “Implied ERP” by Mr. Aswath Damodaran.

The implied ERP model employs a basic discounted cash flow model. It equates the current value of the market with the expected future cash flow growth and solves the equation for the expected return rate.

The inputs needed to calculate the implied ERP are:

- Current value of the market
- Expected cash flow growth of the market
- Terminal growth rate of the market

To practically calculate the implied ERP, you need a proxy for the market. One option is to use the S&P 500 index as proxy for the market. The S&P 500 is a stock market index based on the market cap of 500 large companies having common stock listed on the NYSE, NASDAQ, or the Cboe BZX Exchange.

To calculate the expected cash flow growth, Mr. Damodaran relies on analysts to forecast future retained earnings as well as future returns to shareholder in the form of dividends and buybacks. The time window is set to the next 5 years

The terminal growth rate is set equal to the risk-free rate.

The implied expected return on the stocks is then subtracted by the risk-free rate to come to the implied equity risk premium. In the example of using the S&P 500 as market proxy, we have the implied equity risk premium for the US.

### Country Risk

In the example above, we assume the S&P 500 is a good proxy for the US market in general. But what about other countries?

If we follow the theory of the implied equity risk premium, we should calculate the implied ERP for any country using a proxy similar to the S&P 500. However, this is not very practical. Mr. Damodaran proposes a composite way of estimating ERP for countries. The composite way adds the risk of a specific country relative to the risk of a mature market.

First, you estimate an equity risk premium for a mature market either by using the backward looking historical ERP or a forward looking implied ERP.

Then, define what you consider to be a mature market. One option would be to go by country’s local currency rating and associated default spread. In that case, any AAA rated country is considered mature.

Finally, estimate the additional risk premium for non-mature markets. There are two options:

- Default spread for the country, estimated based on the sovereign ratings or the CDS market
- Scaled up default spread, where you adjust the default spread upwards for the additional risk in equity markets.

## Beta

The beta of a firm or stock (asset) measures its exposure to the market risk. It indicates both the volatility of the asset as well as how the volatility correlates to the general market. The beta is normalized around 1, meaning that the weighted average of all betas of firms in the market is 1.

**Beta > 1**: either returns that are more volatile than the market, or returns that are not very correlated with the market- Example: cutting-edge technology companies are typically more volatile than the market because they have a faster innovation pace

**Beta = 1**: the returns are as volatile as the market, or correlated strongly with the market- Example: large and mature companies tend to follow the market behavior

**Beta < 1:**the returns are less volatile than the market, or are not very correlated with the market- Example: firms that operate in stable industries which produce common-use products will typically have a low beta

**Beta <= 0**: the asset is market risk inducing- Example: the price of gold typically correlates inversely with the market and thus firms operating in this industry will have a negative beta.

### What Impacts Beta?

There are three main factors that will impact the beta of an asset.

Firstly, the **industry **a firm is operating in will impact its beta. Since the beta is a measurement of how exposed a firm is to market risk, firms that produce goods or services that are heavily dependent on the season will have a higher beta. For example: a firm that makes ice-cream cones will have greater volatility in earnings than a company that makes toilet paper. Ice cream is in high demand during the summer, but low demand during the winter, whereas toilet paper is used during all seasons. That higher volatility in earnings will result in a higher beta.

Secondly, the **operating leverage** will impact a firm’s beta too. Companies with a higher operating leverage, or fixed operating costs, will see their earnings vary more. For example, a ‘bad summer’ will result in lower ice cream sales. A business that has invested in a ice cream stall with inside seating and air conditioning will have higher fixed costs than someone who’s just selling at the side of the street. When revenues are lower, the higher fixed costs means lower earnings.

Thirdly, the **financial leverage** will impact a firm’s beta as well. A firm with debt has to repay interest fees which are deducted from the firm’s earnings. The interest payments are fixed, regardless of the revenue. Therefore a “bad sales season” will have a larger impact on the firm’s earnings.

Lastly, it’s important to note that the beta of a firm is the **market-value weighted average** of the businesses of the firm. A large corporate which operates in many industries will have a different beta than the industry-focused firms it competes with in those markets. This principle also applies to your personal investment portfolio; the beta of your total investment will be the weighted average of the betas of your assets.

### Measuring Beta: Top-Down or Bottom-Up

When measuring beta, you can either choose a backward looking (top-down) or a forward looking (bottom-up ) approach.

**Bottom-Up Beta**

The bottom-up beta can be estimated using the following steps:

- Find the businesses or industries the firm operates in and determine their impact on sales or operating income
- Find the unlevered betas of other firms in these businesses
- Determine the weighted average of the unlevered betas
- Lever up using the firm’s debt/equity ratio

The main advantage of a bottom-up approach is that this approach reflects the current or even future mix of the businesses that the firm is in.

**Top-Down Beta**

The top-down beta can be estimated by regressing the stock returns against market returns. The formula is:

Rj = a + b x Rm

Where,

- Rj is the stock returns
- Rm is the market returns.
- b, or slope, corresponds to the beta of a stock and measures its riskiness
- a, or intercept, is a performance measurement of the stock.
- R², or R-squared, is an estimate of the proportion of the variance attributed to market risk

The key challenge of using regression to estimate the beta is the data selection.

Choosing a larger period to evaluate a stock will yield a larger data set but may not accurately reflect the change a firm is going through during that period. Similarly, choosing shorter interval periods between returns will yield a larger data set, but may be affected by noise caused by lack of trading

**Unlevered and Levered Beta**

The term (un)levered beta usually refers to the debt of a firm. All top-down beta estimates are levered because the estimate is based on stock prices. Stock prices are set by the market and take into account the mix of equity and debt of a firm.

To calculate the unlevered beta (UB) you can use the following formula:

UB = Levered Beta / (1 + (1 – tax rate) x (D/E))

Where the tax rate is the marginal tax rate for the firm, and D/E is the average debt equity ratio during the regression period

## Calculating Cost of Equity using CAPM

Finally, we return to the formula of CAPM:

*Expected Return = Riskfree Rate + Beta x (Market Equity Risk Premium)*

Where,

- Risk-free rate is the long-term government bond
- Beta is the weighted average of the risk exposure of all the firm’s businesses to the general market
- Market Equity Risk Premium is the weighted average of market risk premium of all the markets the firm operates in

Practically, we can choose to calculate the risk of operating in a mature market like the US and add risk associated with operating in a different country. In that case, we can say:

*Expected Return = Country Gov’t Bond Rate + (Levered Beta*) * x (Mature Market Risk Premium + Country Risk Premium)*

Where,

- Country Government Bond Rate is the government in control of issuing the currency
- If there’s a default risk associated with the government bond rate, adjust the borrowing rate for the default risk

### What About Private Companies?

The methods to estimating the cost of equity discussed above are very useful when evaluating public companies. However, for private companies things are more complicated. The lack of stock prices and historical returns make it difficult to calculate how exposed the company is to market risk (beta).

There are two main ways to estimate a beta for non-traded assets: use comparable firms (bottom-up) or use accounting earnings (top-down).

When evaluating the levered betas of comparable firms, it’s important to still deleverage and releverage using the private firm’s debt equity ratio. Again, it’s difficult to estimate the market value of the firm’s equity or debt, so it’s reasonable to use book value instead.

**R-squared**

Last but not least, remember that beta is a measure of risk added on to a diversified portfolio. However, the owners of most private companies are not diversified. Therefore using beta only to come to the cost of equity will underestimate the cost of equity of the private firm.

To adjust for the added risk of an undiversified portfolio, we can use R-squared of the regression as it measures the proportion of risk that is market risk.

*Beta = Market Beta / Correlation of sector with market*

The correlation factor of industries like technology are relatively low compared to more mature industries. If you dig into it, you will quickly find that a private technology company is a risky business!