# Calculation of Dyson’s cost of equity capital with CAPM formula

## Use of CAPM formula

The cost of capital and the expected rate of return, which are all used as a discount rate in the DCF model, have the same meaning. Firms as capital consumers may use the cost of capital as the discount rate for future cash flows. Investors as money suppliers can use the expected rate of return for that.

#### Discount rate (for calculating equity value) = cost of equity capital = shareholders’ expected rate of return

The cost of equity capital for companies is paying dividends and granting voting rights to shareholders. The expected rate of return for investors is receiving dividend income and exercising voting rights. However, investors could obtain capital gain by selling stocks in the secondary market. From the company’s standpoint, only its shareholders have changed, and the cost of capital remains the same as before. When organised them, it is as follows.

Comparison of companies’ cost of equity capital and investors’ expected rate of return

The cost of equity capital and expected return are theoretically the same, but the practical things are different, as mentioned above.

##### That is because the primary (issuance) market and the secondary (trading) market are different.

Companies as issuers do not usually trade stocks in the secondary market after issuing them in the primary market but continuously pay dividends and grant voting rights.

On the other hand, investors receive dividend incomes and exercise voting rights, and they could also earn capital gains when selling the stocks in the market. Therefore, strictly speaking, the calculation method of the discount rate between companies and investors should be different. However, since no theory has been differently established that represents the two positions, both should choose the same method.

The most widely used method for calculating the cost of equity capital is CAPM (Capital Asset Pricing Model). Therefore, we will focus on the CAPM formula from now.

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## CAPM formula

CAPM formula is the one to calculate the cost of equity capital introduced by American economist Jack Treynor and three others. The formula expresses the relationship between market risk premium and an individual stock or portfolio’s sensitivity to it. The CAPM formula is as follows.

#### Ke = Rf + βi (Rm-Rf)

Ke: cost of equity capital, Rf: risk-free interest rate, Rm: return on market portfolio m,
Rm-Rf: market risk premium, βi: sensitivity of stock i to market risk premium (systemic risk or market risk)

In the CAPM formula, the cost of equity capital consists of a risk-free interest rate, a market risk premium, and a stock’s sensitivity to the market risk premium. Each component is summarised as follows.

• Rf is the rate of return provided an asset without investment risk, ie, a risk-free asset. In practice, it refers to the 10-year government bond yield of the country in which the stock is traded.
• Rm is the expected return on the market portfolio. A market portfolio refers to a completely diversified portfolio in which unsystematic (firm-specific) risk disappears, and only systematic risk exists. In practice, we use the historical rate of return of the stock market index on which the stock is traded.
• ‘Rm-Rf’ is a market risk premium. It refers to the stock market’s excess return rate, which exceeds the interest rate of the risk-free asset. In other words, it means the extra return you can earn by investing in stocks instead of the safest government bond.
• βi is the beta of individual stock i, which is the part that exceeds or is less than the market risk premium by investing in stock i. βi refers to the degree to which stock i fluctuates when the stock market fluctuates. Arithmetically, if β is greater than 1, stock i fluctuates more than the stock market volatility, and if it is less than 1, it fluctuates less. You can estimate the β of a private company by referring to the β of the publicly traded similar companies.

Let’s interpret the CAPM formula again by referring to the above. You can obtain the return (Rf) provided by the bond without any risk by investing in a government bond. If you invest in the stock market instead of that riskless bond, you are exposed to the market risk so that you would expect an excess return (Rm-Rf). By investing in an individual stock (or individual portfolio) in the stock market, you are exposed to the stock’s sensitivity to changes in the stock market, which is reflected in βi.

## Calculation of Dyson’s cost of equity capital with CAPM formula

Let’s understand it more closely by applying it to a real company and finding the cost of capital. Dyson Ltd is a British private technology company founded in 1991 by James Dyson. It designs and manufactures home appliances such as vacuum cleaners, air purifiers, hairdryers, bladeless fans, heaters and lighting. It is a global company that has expanded not only to the UK but also to Asia, Europe, and North America.

If you apply the CAPM FORMULA to estimate Dyson’s cost of equity capital, you can predict the following procedure.

1. Step 1: When you invest in a UK 10-year government bond, you can get a risk-free rate of return.
2. Step 2: If you invest in the UK stock market instead of the risk-free asset, you should require an excess return equal to the stock market risk.
3. Step 3: In the case that you choose Dyson out of the UK stocks, you are exposed to the Dyson’s individual sensitivity to the UK stock market risk.

The following is Dyson’s cost of equity capital calculated by sequentially applying the above procedure.

Dyson’s cost of equity capital calculation (Source: Thompson Reuters)

Rf above is the UK’s 10-year government bond yield. Since Dyson is a British company, you should choose a UK government bond with a risk-free rate. It is 0.26% as of December 2020 when referring to Thomson Reuters. It means you can earn a return rate of 0.26% without any risk of loss.

But if you risk investing in the UK stock market, you should expect a corresponding excess return. The UK representative index is FT100. Historically, FT100 has provided a market risk premium of 6.40% over the risk-free asset.

Dyson’s comparable companies’ beta (Source: Thomson Reuters)

You have selected Dyson out of the many UK stocks, so you ​should utilise β ​to reflect the Dyson’s own sensitivity to FT100. However, since Dyson is a private company, it does not have its own β because it has no market price. To find Dyson’s β, you may refer to the β of the listed companies similar to Dyson. When used the equity screener of Yahoo Finance, ten British companies were found in the sectors of home appliance, electronic components, and science and technology equipment.

The above diagram shows the list of them and the β of each company. The average β is 1, which means that the Dyson’s stock price changes by 1% when the FT100 index changes 1%. Dyson’s cost of equity capital calculated according to the CAPM formula is 6.66% (= 0.26% + 1.00 X 6.40%).

The 6.66% is used as a discount rate for Dyson’s valuation. You can also interpret it in another way. Dyson raised equity capital at the cost of 6.66%, so the company should make a profit margin higher than 6.66%. In other words, the return on equity capital should be higher than the cost of equity capital. Then, theoretically, the Dyson’s equity market value gets higher than the book value.

## Limitations of CAPM formula

In practice, many analysts use the CAPM formula to estimate the cost of equity capital, but it has obvious limitations. The model is based on unrealistic assumptions; all assets in the world are traded in the investment market, there are no transaction costs, and all investors have access to the same information (the most significant error).

In addition, only β measures the stock’s individual risk. In other words, it does not take into account a company’s specific circumstances or risks such as company size, its unique business risks, and whether it is listed or not.

To overcome these limitations, you may add the company’s individual risks with your own qualitative judgment to the costs earned from CAPM. Then, the CAPM formula could change as follows:

#### Ke = Rf + βi (Rm-Rf) + λi

Ke: cost of equity capital, Rf: risk-free interest rate, Rm: return on market portfolio m,
Rm-Rf: market risk premium, βi: sensitivity of stock i to systematic risk, λi: individual risk of stock

Let’s consider the case of Dyson. Dyson is a private company. Therefore, the β of its peer group does not reflect the Dyson’s stock liquidity risk because β was calculated with only listed companies. If you consider the premium for the stocks’ illiquidity risk as 20%, Dyson’s cost of equity capital could reach 7.99%.

7.99%= 0.26% + 1.00 X 6.40% + 6.66% X 20%