# Capital Asset Pricing Model (CAPM)

The most popular method to calculate cost of equity is Capital Asset Pricing Model (CAPM). Why? Because it displays the relationship between risk and expected return for a company’s assets. This model is used throughout financing for calculating expected returns for assets while including risk and cost of capital.

## Cost of Equity

Also known as the required rate of return on common stock, define the cost of equity as the cost of raising funds from equity investors. It is by far the most challenging element in discount rate determination.

## Calculating Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) states that the expected return on an asset is related to its risk as measured by beta:

E(Ri) = Rf + ßi * (E(Rm) – Rf)

Or = Rf + ßi * (risk premium)

Where

E(Ri) = the expected return on asset given its beta

Rf = the risk-free rate of return

E(Rm) = the expected return on the market portfolio

ßi = the asset’s sensitivity to returns on the market portfolio

E(Rm) – Rf = market risk premium, the expected return on the market minus the risk free rate.

### Expected Return of an Asset

Therefore, the expected return on an asset given its beta is the risk-free rate plus a risk premium equal to beta times the market risk premium. Beta is always estimated based on an equity market index. Additionally, determine the beta of a company by the three following variables:

1. The type business the company is in
2. The degree of operating leverage of the company
3. The company’s financial leverage

### Risk-Free Rate of Return

Short-term government debt rate (such as a 30-day T-bill rate, or a long-term government bond yield to maturity) determines the risk-free rate of return. When cash flows come due, it is also determined. Define risk-free rate as the expected returns with certainty.

Additionally, risk premium indicates the “extra return” demanded by investors for shifting their money from riskless investment to an average risk investment. It is also a function of how risk-averse investors are and how risky they perceive investment opportunities compared with a riskless investment.

## Cost of Equity Calculation

For example, a company has a beta of 0.5, a historical risk premium of 6%, and a risk-free rate of 5.25%. Therefore, the required rate of return of this company according to the CAPM is: 5.25% + (0.5 * 6%) = 8.25%

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# Discount Rate Definition

The discount rate definition, also known as hurdle rate, is a general term for any rate used in finding the present value of a future cash flow. In a discounted cash flow (DCF) model, estimate company value by discounting projected future cash flows at an interest rate. This interest rate is the discount rate which reflects the perceived riskiness of the cash flows.

## Discount Rate Explanation

Using discount rate, explained as the risk factor for a given investment, has many benefits. The purpose is to account for the loss of economic efficiency of an investor due to risk. Investors use this rate because it provides a way to account and compensate for their risk when choosing an investment. Furthermore, this provides, with each choice, a buffer to provide for the chance of failure in an investment over time as well as many investments over a portfolio. Though risk is somewhat of a sunk cost, still include it to add a real-world element to financial calculations. It is a measure used to prevent one from becoming “calculator rich” without actually increasing personal wealth.

In DCF model, there are two methods to get discount rate: weighted average cost of capital (WACC) and adjusted present value (APV). For WACC, calculate discount rate for leveraged equity using the capital asset pricing model (CAPM). Whereas for APV, all equity firms calculate the discount rate, present value, and all else.

The Discount Rate should be consistent with the cash flow being discounted. For cash flow to equity, use the cost of equity. For cash flow to firm, use the cost of capital.

## Discount Rate Formula

A succinct Discount Rate formula does not exist; however, it is included in the discounted cash flow analysis and is the result of studying the riskiness of the given type of investment. The two following formulas provide a discount rate:

First, there is the following Weighted Average Cost of Capital formula.

Weighted Average Cost of Capital (WACC) = E/V * Ce + D/V * Cd * (1-T)

Where:
E = Value of equity
D = Value of debt
Ce = Cost of equity
Cd = Cost of debt
V = D + E
T = Tax rate

Then, there is the following Adjusted Present Value formula.

Adjusted Present Value = NPV + PV of the impact of financing

Where:
NPV = Net Present Value
PV = Present Value

### Calculation

See the following calculation of WACC and APV.

For WACC:

WACC = \$10,000/\$20,000 * \$2,000 + \$10,000/\$20,000 * \$1,000 * (1-.3) = \$1,050,000

If:
E = \$10,000
D = \$10,000
Ce = \$2,000
Cd = \$1,000
V = \$20,000
T = 30%

For APV:

APV = \$1,000,000 + \$50,000 = \$1,050,000

If:
NPV = \$1,000,000
PV of the impact of financing = \$50,000

## Discount Rate Example

For example, Donna is an analyst for an entrepreneur. Where her boss is the visionary, Donna performs the calculations necessary to find whether a new venture is a good decision or not. She does not need a discount rate calculator because she has the skills to provide value above and beyond this. Donna is the right hand woman to the entrepreneur which she aspires to be. But she first needs to prove herself in the professional world.

Donna’s boss wants to know how much risk he has taken on his last venture. He would like, eventually, to find the discount rate business valuation to judge levels for performance and new ventures alike.

Donna’s boss gives Donna the financial information she needs for one venture. She finds the discount rate (risk) using the following equation:

WACC = \$10,000/\$20,000 * \$2,000 + \$10,000/\$20,000 * \$1,000 * (1-.3) = \$1,050,000

If:
E = \$10,000
D = \$10,000
Ce = \$2,000
Cd = \$1,000
V = \$20,000
T = 30%

Next, Donna’s boss has her find the discount rate for another venture that he is involved in. The results are below:

Adjusted Present Value = NPV + PV of the impact of financing

Where:
NPV = Net Present Value
PV = Present Value

Donna appreciates her experience with her employer. As a result, she is sure that with this experience she can find the path to mentor another just like her. 1

# Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) is an equilibrium model that measures the relationship between risk and expected return of an asset based on the asset’s sensitivity to movements in the overall stock market.

CAPM is used to price the risk of an asset or a portfolio of assets. The model is based on the idea that there are two types of risk, systematic risk and idiosyncratic risk, and that the investor should be compensated for both types of risk, as well as, the time value of money. Systematic risk refers to market risk. Idiosyncratic risk refers to the risk of an individual asset. Time value of money refers to the difference between the present value of money and the future value of money. Also, use the model to measure the required rate of return for capital budgeting projects.

The CAPM states that an asset’s expected return equals the risk-free rate plus a risk premium. The risk-free rate refers to the return on an investment without risk, such as a US Treasury Bond, and represents the time value of money. The risk premium represents the incremental return for investing in a risky asset. In the CAPM, it is defined as the market premium, or the overall stock market return less the risk-free rate, multiplied by the beta of the asset. Beta is a factor that measures an asset’s sensitivity to movements in the overall stock market. According to the CAPM, riskier assets should yield higher returns.

## The CAPM Formula

Expected Return = Risk-Free Rate + Beta (Market Return – Risk-Free Rate)

For example, if the risk free rate is 5%, the market return is 10%, and the stock’s beta is 2, then the expected return on the stock would be 15%.

15% = 5% + 2 (10% – 5%)

### Problems with Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) is based on assumptions. First, the model assumes that a riskier asset will yield a higher return. But this is not necessarily true. A risky asset could decline in value. Second, historical data determines beta. The model assumes this historical data an accurate predictor of future results. But the asset’s future volatility may not necessarily reflect its past volatility.

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# Arbitrage Pricing Theory Definition

The arbitrage pricing theory (APT) is a multifactor mathematical model used to describe the relation between the risk and expected return of securities in financial markets. It computes the expected return on a security based on the security’s sensitivity to movements in macroeconomic factors. Then use the resultant expected return to price the security.

The arbitrage pricing theory is based on three assumptions. First, that a factor model can be used to describe the relation between the risk and return of a security. Then, idiosyncratic risk can be diversified away. Finally, efficient financial markets do not allow for persisting arbitrage opportunities.

In addition, the arbitrage pricing theory calculates the expected return for a security based on the security’s sensitivity to movements in multiple macroeconomic factors. Whereas the standard capital asset pricing model (CAPM) is a single factor model, incorporating the systematic and firm specific risk related to the overall market return, the arbitrage pricing theory is a multifactor model.

Then, set up the arbitrage pricing theory to consider several risk factors, such as the business cycle, interest rates, inflation rates, and energy prices. The model distinguishes between both systematic risk and firm-specific risk. It also incorporates both types of risk into the model for each given factor.

## Arbitrage Pricing Theory Formula

The formula includes a variable for each factor, and then a factor beta for each factor, representing the security’s sensitivity to movements in that factor. Because it includes more factors, consider the arbitrage pricing theory more nuanced – if not more accurate, than the capital asset pricing model.

A two-factor version of the arbitrage pricing theory formula is as follows:

r = E(r) + B1F1 + B2F2 + e

r = return on the security
E(r) = expected return on the security
F1 = the first factor
B1 = the security’s sensitivity to movements in the first factor
F2 = the second factor
B2 = the security’s sensitivity to movements in the second factor
e = the idiosyncratic component of the security’s return Strategic CFO Lab Member Extra

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