###### Updated January 23, 2024

# CAPM Formula Explained: Navigating the Relationship Between Risk and Return

# CAPM Formula Explained: Navigating the Relationship Between Risk and Return

# CAPM Formula Explained: Navigating the Relationship Between Risk and Return

Mike Zaccardi, CFA, CMT

Investing Master Class

Being a successful investor is a tough business. There are so many variables to consider regarding the market, individual companies, and even knowing your own biases. When it comes to analyzing the value and expected return of an asset, one of the most widely known approaches is the Capital Asset Pricing Model (CAPM).

CAPM is a pricing model that aims to simplify factors to allow an investor to better understand the relationship between systematic, or market-wide, risk and return. While offering key insights as to the payoff potential of a stock, there are assumptions and limitations with CAPM that some finance practitioners view as unrealistic.

In the end, CAPM is just one of many ways to go about analyzing companies to include in your portfolio. While controversial, CAPM remains widely used in finance classrooms, mutual fund companies, and Wall Street banks as a useful tool for estimating returns on stocks.

## Understanding CAPM

Let’s jump right into Portfolio Management 101. The CAPM formula is:

ERi = Rf +βi (ERm −Rf )

where:

ERi = expected return of a stock

Rf = risk-free rate (usually a short-term Treasury yield)

βi = beta of a stock

ERm = expected return of the market

(ERm −Rf ) = market risk premium

For non-finance people, that might be an intimidating formula. Once you know the inputs, however, performing the calculation is rather straightforward. Before we dig into the numbers, let’s set the stage by reviewing the context and importance of CAPM to better understand its power. The point of CAPM is to figure out the expected return of an asset, usually on an individual stock, using existing market variables. CAPM is based on the notion that the higher the risk-free Treasury rate, volatility of a stock relative to the market, and market risk premium, then the more you should expect to earn on the investment.

At a high level, CAPM describes the relationship between systematic risk and the expected performance of single stocks. The goal is to identify which companies have the potential to deliver the most value over time. What is controversial, though, is that a key assumption with CAPM is that the higher a stock’s beta (its risk relative to the market), the higher the return. Some studies have shown that it is actually low volatility stocks that perform better than ones with high betas. Still, CAPM is all about risk and return and is an important part of determining a company’s cost of equity capital, along with it being a stock valuation tool.

CAPM is rooted in several assumptions about markets and investors. First, all investors are seen as risk-averse and have identical time horizons. Access to unlimited amounts of capital is also assumed along with no consideration for taxes, inflation, or transaction costs. Furthermore, CAPM is based on the notion that all investments can be accessed by all investors. Of course, these assumptions are arguably unrealistic, something CAPM critics are quick to point out.

## The CAPM Inputs

As for the components of CAPM, the expected return on the market is normally taken from historical analysis of long-term performance. For large-cap US stocks, the S&P 500’s compounded annual growth rate has been about 10% while US small caps have returned slightly more than that, so inputting about 12% could make sense for small-sized firms. The risk-free rate is often the yield on a short-term Treasury bill, though some analysts may choose the rate on the 10-year Treasury note since that is less influenced by the Federal Reserve. The risk-free rate is then subtracted from the expected return on the market to give us the market risk premium. The only variable left is beta.

## What is Beta?

Beta is a gauge of how volatile a stock has been compared to the overall market. For instance, if the S&P 500 rises 1% and a stock jumps 2%, the stock’s beta for that single session would be 2.0. Of course, using a length dataset (often years) and assessing various intervals (daily, weekly, and monthly terms) helps to determine a true beta. In short, a “risky” stock will have a beta above one while a less-volatile stock will have a beta under one. While rare, some stocks or exchange-traded funds (ETFs) have a negative beta – perhaps shares in precious metal companies and funds that aim to return the inverse of the market.

A security’s beta is then multiplied by the market risk premium and the risk-free rate is added to that product. The resulting percentage is deemed the required return by the investor. It’s also the discount rate that can be applied to find the intrinsic value of a stock.

## CAPM Example

So, let’s run through an example now that we are CAPM aficionados.

Rf = risk-free rate: 4%

βi = beta of the stock: 1.2

ERm = expected return of the market: 10%

Now, we can use the CAPM formula to calculate the expected return (ERi) for a particular stock:

ERi = Rf +βi (ERm − Rf )

Plug in the values:

ERi = 0.04 + 1.2 x (0.10-0.04)

Calculate the result:

ERi = 0.04 + 1.2 x 0.06

ERi = 0.04 + 0.072

ERi = 0.0112, or 11.2%

So, in this example, the expected return for the stock, calculated using the CAPM, is 11.2%. This is the return that an investor would anticipate for holding this particular stock, considering its risk profile in relation to the overall market.

## Systematic and Non-Systematic Risk

Memorizing a formula is one thing, but recognizing the importance of its logic is another ballgame. It is crucial to realize that CAPM focuses on systematic, or market-wide risk factors. That's essentially what beta measures. When managing a portfolio of stocks, bonds, and other assets, though, other issues can come into play, thus limiting the usefulness of CAPM.

For instance, if you own shares of a company that misses badly on its earnings estimates for a given quarter, the stock’s return may fall well short of what CAPM implies. Another firm-specific issue could be a scandal among its top executives (see: Enron and Worldcom). Non-systematic risk cannot be completely wiped away through diversification – a reality investors must always remember.

## Diversification, the Capital Market Line, and the Efficient Frontier

Diversification is an important piece of the CAPM puzzle. While calculating the expected return on a stock is helpful, CAPM is commonly used from a broader portfolio perspective. Computer programs can determine the expected return on thousands of stocks along with their expected risk levels. Those data are then plotted on a graph along with the Capital Market Line (CML), also called the Capital Allocation Line (CAL). The CML is a line starting on the y-axis at the risk-free rate and then plots upward based on how much risk is assumed. Modern Portfolio Theory (MPT) asserts that the more risk an investor accepts, the greater the expected return.

## The Capital Market Line

Source: CFA Institute

Any portfolio that plots under and to the right of the CML is considered inefficient due to a poor tradeoff between return and risk. The goal, using CAPM, is to build a portfolio of assets that plots on the CML, thereby optimizing for expected returns and risk. Portfolio managers build many different allocation strategies, plotting each relative to the CML, forming what’s known as the efficient frontier - a portfolio on the efficient frontier that rests on the CML offers the maximum return for the least risk. The efficient frontier is based on the same assumptions as CAPM.

## Capital Allocation Line and Optimal Risky Portfolio on the Efficient Frontier of Risky Assets

Source: CFA Institute

## CAPM’s Disadvantages

Critics of CAPM say that there are too many unrealistic assumptions tied to the formula and its practicality is limited when it comes to portfolio management. For one thing, academics have poked holes in beta’s usefulness. Professors Eugene Fama and Ken French found that differences in betas across stocks and through time offered little in the way of prediction power for future returns. Moreover, the linear relationship between beta and returns fails to hold over short timeframes, making it nearly worthless for short-term traders.

Another problem is that CAPM assumes a normal distribution of returns. Empirical data show that stock returns are far from normal in the statistical sense. Finally, the risk-free rate is constantly changing. In the early 1980s, for instance, you could have used a rate north of 10% based on the short-term Treasury yields at the time. Then during the Fed’s zero-interest-rate policy last decade, the risk-free rate was below 1%. Minor adjustments to the risk-free rate can result in wildly different expected returns with CAPM.

## CAPM Alternatives

Given sharp criticism and the apparent flaws in CAPM, other pricing models have come about over the decades. The Arbitrage Pricing Theory (APT) incorporates many other factors into its formulation, including macroeconomic and firm-specific variables. Another formula taught in college finance classrooms around the world is the Fama-French 3-Factor Model, which layers additional variables on top of CAPM’s beta. Even what’s known as the Behavioral Asset Pricing Model has come about, which puts a high emphasis on human biases and our preference, at times, to be loss averse, not risk averse. The list of CAPM alternatives will surely grow as further research is conducted.

## The Bottom Line

The CAPM is among the most popular methods by which to value stocks and forecast returns. While not without its flaws, the formula, based on Modern Portfolio Theory, helps investors understand the quantitative relationship between expected risk and return. Managing portfolios is no easy task, and CAPM helps practitioners make better decisions in financial markets.

Being a successful investor is a tough business. There are so many variables to consider regarding the market, individual companies, and even knowing your own biases. When it comes to analyzing the value and expected return of an asset, one of the most widely known approaches is the Capital Asset Pricing Model (CAPM).

CAPM is a pricing model that aims to simplify factors to allow an investor to better understand the relationship between systematic, or market-wide, risk and return. While offering key insights as to the payoff potential of a stock, there are assumptions and limitations with CAPM that some finance practitioners view as unrealistic.

In the end, CAPM is just one of many ways to go about analyzing companies to include in your portfolio. While controversial, CAPM remains widely used in finance classrooms, mutual fund companies, and Wall Street banks as a useful tool for estimating returns on stocks.

## Understanding CAPM

Let’s jump right into Portfolio Management 101. The CAPM formula is:

ERi = Rf +βi (ERm −Rf )

where:

ERi = expected return of a stock

Rf = risk-free rate (usually a short-term Treasury yield)

βi = beta of a stock

ERm = expected return of the market

(ERm −Rf ) = market risk premium

For non-finance people, that might be an intimidating formula. Once you know the inputs, however, performing the calculation is rather straightforward. Before we dig into the numbers, let’s set the stage by reviewing the context and importance of CAPM to better understand its power. The point of CAPM is to figure out the expected return of an asset, usually on an individual stock, using existing market variables. CAPM is based on the notion that the higher the risk-free Treasury rate, volatility of a stock relative to the market, and market risk premium, then the more you should expect to earn on the investment.

At a high level, CAPM describes the relationship between systematic risk and the expected performance of single stocks. The goal is to identify which companies have the potential to deliver the most value over time. What is controversial, though, is that a key assumption with CAPM is that the higher a stock’s beta (its risk relative to the market), the higher the return. Some studies have shown that it is actually low volatility stocks that perform better than ones with high betas. Still, CAPM is all about risk and return and is an important part of determining a company’s cost of equity capital, along with it being a stock valuation tool.

CAPM is rooted in several assumptions about markets and investors. First, all investors are seen as risk-averse and have identical time horizons. Access to unlimited amounts of capital is also assumed along with no consideration for taxes, inflation, or transaction costs. Furthermore, CAPM is based on the notion that all investments can be accessed by all investors. Of course, these assumptions are arguably unrealistic, something CAPM critics are quick to point out.

## The CAPM Inputs

As for the components of CAPM, the expected return on the market is normally taken from historical analysis of long-term performance. For large-cap US stocks, the S&P 500’s compounded annual growth rate has been about 10% while US small caps have returned slightly more than that, so inputting about 12% could make sense for small-sized firms. The risk-free rate is often the yield on a short-term Treasury bill, though some analysts may choose the rate on the 10-year Treasury note since that is less influenced by the Federal Reserve. The risk-free rate is then subtracted from the expected return on the market to give us the market risk premium. The only variable left is beta.

## What is Beta?

Beta is a gauge of how volatile a stock has been compared to the overall market. For instance, if the S&P 500 rises 1% and a stock jumps 2%, the stock’s beta for that single session would be 2.0. Of course, using a length dataset (often years) and assessing various intervals (daily, weekly, and monthly terms) helps to determine a true beta. In short, a “risky” stock will have a beta above one while a less-volatile stock will have a beta under one. While rare, some stocks or exchange-traded funds (ETFs) have a negative beta – perhaps shares in precious metal companies and funds that aim to return the inverse of the market.

A security’s beta is then multiplied by the market risk premium and the risk-free rate is added to that product. The resulting percentage is deemed the required return by the investor. It’s also the discount rate that can be applied to find the intrinsic value of a stock.

## CAPM Example

So, let’s run through an example now that we are CAPM aficionados.

Rf = risk-free rate: 4%

βi = beta of the stock: 1.2

ERm = expected return of the market: 10%

Now, we can use the CAPM formula to calculate the expected return (ERi) for a particular stock:

ERi = Rf +βi (ERm − Rf )

Plug in the values:

ERi = 0.04 + 1.2 x (0.10-0.04)

Calculate the result:

ERi = 0.04 + 1.2 x 0.06

ERi = 0.04 + 0.072

ERi = 0.0112, or 11.2%

So, in this example, the expected return for the stock, calculated using the CAPM, is 11.2%. This is the return that an investor would anticipate for holding this particular stock, considering its risk profile in relation to the overall market.

## Systematic and Non-Systematic Risk

Memorizing a formula is one thing, but recognizing the importance of its logic is another ballgame. It is crucial to realize that CAPM focuses on systematic, or market-wide risk factors. That's essentially what beta measures. When managing a portfolio of stocks, bonds, and other assets, though, other issues can come into play, thus limiting the usefulness of CAPM.

For instance, if you own shares of a company that misses badly on its earnings estimates for a given quarter, the stock’s return may fall well short of what CAPM implies. Another firm-specific issue could be a scandal among its top executives (see: Enron and Worldcom). Non-systematic risk cannot be completely wiped away through diversification – a reality investors must always remember.

## Diversification, the Capital Market Line, and the Efficient Frontier

Diversification is an important piece of the CAPM puzzle. While calculating the expected return on a stock is helpful, CAPM is commonly used from a broader portfolio perspective. Computer programs can determine the expected return on thousands of stocks along with their expected risk levels. Those data are then plotted on a graph along with the Capital Market Line (CML), also called the Capital Allocation Line (CAL). The CML is a line starting on the y-axis at the risk-free rate and then plots upward based on how much risk is assumed. Modern Portfolio Theory (MPT) asserts that the more risk an investor accepts, the greater the expected return.

## The Capital Market Line

Source: CFA Institute

Any portfolio that plots under and to the right of the CML is considered inefficient due to a poor tradeoff between return and risk. The goal, using CAPM, is to build a portfolio of assets that plots on the CML, thereby optimizing for expected returns and risk. Portfolio managers build many different allocation strategies, plotting each relative to the CML, forming what’s known as the efficient frontier - a portfolio on the efficient frontier that rests on the CML offers the maximum return for the least risk. The efficient frontier is based on the same assumptions as CAPM.

## Capital Allocation Line and Optimal Risky Portfolio on the Efficient Frontier of Risky Assets

Source: CFA Institute

## CAPM’s Disadvantages

Critics of CAPM say that there are too many unrealistic assumptions tied to the formula and its practicality is limited when it comes to portfolio management. For one thing, academics have poked holes in beta’s usefulness. Professors Eugene Fama and Ken French found that differences in betas across stocks and through time offered little in the way of prediction power for future returns. Moreover, the linear relationship between beta and returns fails to hold over short timeframes, making it nearly worthless for short-term traders.

Another problem is that CAPM assumes a normal distribution of returns. Empirical data show that stock returns are far from normal in the statistical sense. Finally, the risk-free rate is constantly changing. In the early 1980s, for instance, you could have used a rate north of 10% based on the short-term Treasury yields at the time. Then during the Fed’s zero-interest-rate policy last decade, the risk-free rate was below 1%. Minor adjustments to the risk-free rate can result in wildly different expected returns with CAPM.

## CAPM Alternatives

Given sharp criticism and the apparent flaws in CAPM, other pricing models have come about over the decades. The Arbitrage Pricing Theory (APT) incorporates many other factors into its formulation, including macroeconomic and firm-specific variables. Another formula taught in college finance classrooms around the world is the Fama-French 3-Factor Model, which layers additional variables on top of CAPM’s beta. Even what’s known as the Behavioral Asset Pricing Model has come about, which puts a high emphasis on human biases and our preference, at times, to be loss averse, not risk averse. The list of CAPM alternatives will surely grow as further research is conducted.

## The Bottom Line

The CAPM is among the most popular methods by which to value stocks and forecast returns. While not without its flaws, the formula, based on Modern Portfolio Theory, helps investors understand the quantitative relationship between expected risk and return. Managing portfolios is no easy task, and CAPM helps practitioners make better decisions in financial markets.

Being a successful investor is a tough business. There are so many variables to consider regarding the market, individual companies, and even knowing your own biases. When it comes to analyzing the value and expected return of an asset, one of the most widely known approaches is the Capital Asset Pricing Model (CAPM).

CAPM is a pricing model that aims to simplify factors to allow an investor to better understand the relationship between systematic, or market-wide, risk and return. While offering key insights as to the payoff potential of a stock, there are assumptions and limitations with CAPM that some finance practitioners view as unrealistic.

In the end, CAPM is just one of many ways to go about analyzing companies to include in your portfolio. While controversial, CAPM remains widely used in finance classrooms, mutual fund companies, and Wall Street banks as a useful tool for estimating returns on stocks.

## Understanding CAPM

Let’s jump right into Portfolio Management 101. The CAPM formula is:

ERi = Rf +βi (ERm −Rf )

where:

ERi = expected return of a stock

Rf = risk-free rate (usually a short-term Treasury yield)

βi = beta of a stock

ERm = expected return of the market

(ERm −Rf ) = market risk premium

For non-finance people, that might be an intimidating formula. Once you know the inputs, however, performing the calculation is rather straightforward. Before we dig into the numbers, let’s set the stage by reviewing the context and importance of CAPM to better understand its power. The point of CAPM is to figure out the expected return of an asset, usually on an individual stock, using existing market variables. CAPM is based on the notion that the higher the risk-free Treasury rate, volatility of a stock relative to the market, and market risk premium, then the more you should expect to earn on the investment.

At a high level, CAPM describes the relationship between systematic risk and the expected performance of single stocks. The goal is to identify which companies have the potential to deliver the most value over time. What is controversial, though, is that a key assumption with CAPM is that the higher a stock’s beta (its risk relative to the market), the higher the return. Some studies have shown that it is actually low volatility stocks that perform better than ones with high betas. Still, CAPM is all about risk and return and is an important part of determining a company’s cost of equity capital, along with it being a stock valuation tool.

CAPM is rooted in several assumptions about markets and investors. First, all investors are seen as risk-averse and have identical time horizons. Access to unlimited amounts of capital is also assumed along with no consideration for taxes, inflation, or transaction costs. Furthermore, CAPM is based on the notion that all investments can be accessed by all investors. Of course, these assumptions are arguably unrealistic, something CAPM critics are quick to point out.

## The CAPM Inputs

As for the components of CAPM, the expected return on the market is normally taken from historical analysis of long-term performance. For large-cap US stocks, the S&P 500’s compounded annual growth rate has been about 10% while US small caps have returned slightly more than that, so inputting about 12% could make sense for small-sized firms. The risk-free rate is often the yield on a short-term Treasury bill, though some analysts may choose the rate on the 10-year Treasury note since that is less influenced by the Federal Reserve. The risk-free rate is then subtracted from the expected return on the market to give us the market risk premium. The only variable left is beta.

## What is Beta?

Beta is a gauge of how volatile a stock has been compared to the overall market. For instance, if the S&P 500 rises 1% and a stock jumps 2%, the stock’s beta for that single session would be 2.0. Of course, using a length dataset (often years) and assessing various intervals (daily, weekly, and monthly terms) helps to determine a true beta. In short, a “risky” stock will have a beta above one while a less-volatile stock will have a beta under one. While rare, some stocks or exchange-traded funds (ETFs) have a negative beta – perhaps shares in precious metal companies and funds that aim to return the inverse of the market.

A security’s beta is then multiplied by the market risk premium and the risk-free rate is added to that product. The resulting percentage is deemed the required return by the investor. It’s also the discount rate that can be applied to find the intrinsic value of a stock.

## CAPM Example

So, let’s run through an example now that we are CAPM aficionados.

Rf = risk-free rate: 4%

βi = beta of the stock: 1.2

ERm = expected return of the market: 10%

Now, we can use the CAPM formula to calculate the expected return (ERi) for a particular stock:

ERi = Rf +βi (ERm − Rf )

Plug in the values:

ERi = 0.04 + 1.2 x (0.10-0.04)

Calculate the result:

ERi = 0.04 + 1.2 x 0.06

ERi = 0.04 + 0.072

ERi = 0.0112, or 11.2%

So, in this example, the expected return for the stock, calculated using the CAPM, is 11.2%. This is the return that an investor would anticipate for holding this particular stock, considering its risk profile in relation to the overall market.

## Systematic and Non-Systematic Risk

Memorizing a formula is one thing, but recognizing the importance of its logic is another ballgame. It is crucial to realize that CAPM focuses on systematic, or market-wide risk factors. That's essentially what beta measures. When managing a portfolio of stocks, bonds, and other assets, though, other issues can come into play, thus limiting the usefulness of CAPM.

For instance, if you own shares of a company that misses badly on its earnings estimates for a given quarter, the stock’s return may fall well short of what CAPM implies. Another firm-specific issue could be a scandal among its top executives (see: Enron and Worldcom). Non-systematic risk cannot be completely wiped away through diversification – a reality investors must always remember.

## Diversification, the Capital Market Line, and the Efficient Frontier

Diversification is an important piece of the CAPM puzzle. While calculating the expected return on a stock is helpful, CAPM is commonly used from a broader portfolio perspective. Computer programs can determine the expected return on thousands of stocks along with their expected risk levels. Those data are then plotted on a graph along with the Capital Market Line (CML), also called the Capital Allocation Line (CAL). The CML is a line starting on the y-axis at the risk-free rate and then plots upward based on how much risk is assumed. Modern Portfolio Theory (MPT) asserts that the more risk an investor accepts, the greater the expected return.

## The Capital Market Line

Source: CFA Institute

Any portfolio that plots under and to the right of the CML is considered inefficient due to a poor tradeoff between return and risk. The goal, using CAPM, is to build a portfolio of assets that plots on the CML, thereby optimizing for expected returns and risk. Portfolio managers build many different allocation strategies, plotting each relative to the CML, forming what’s known as the efficient frontier - a portfolio on the efficient frontier that rests on the CML offers the maximum return for the least risk. The efficient frontier is based on the same assumptions as CAPM.

## Capital Allocation Line and Optimal Risky Portfolio on the Efficient Frontier of Risky Assets

Source: CFA Institute

## CAPM’s Disadvantages

Critics of CAPM say that there are too many unrealistic assumptions tied to the formula and its practicality is limited when it comes to portfolio management. For one thing, academics have poked holes in beta’s usefulness. Professors Eugene Fama and Ken French found that differences in betas across stocks and through time offered little in the way of prediction power for future returns. Moreover, the linear relationship between beta and returns fails to hold over short timeframes, making it nearly worthless for short-term traders.

Another problem is that CAPM assumes a normal distribution of returns. Empirical data show that stock returns are far from normal in the statistical sense. Finally, the risk-free rate is constantly changing. In the early 1980s, for instance, you could have used a rate north of 10% based on the short-term Treasury yields at the time. Then during the Fed’s zero-interest-rate policy last decade, the risk-free rate was below 1%. Minor adjustments to the risk-free rate can result in wildly different expected returns with CAPM.

## CAPM Alternatives

Given sharp criticism and the apparent flaws in CAPM, other pricing models have come about over the decades. The Arbitrage Pricing Theory (APT) incorporates many other factors into its formulation, including macroeconomic and firm-specific variables. Another formula taught in college finance classrooms around the world is the Fama-French 3-Factor Model, which layers additional variables on top of CAPM’s beta. Even what’s known as the Behavioral Asset Pricing Model has come about, which puts a high emphasis on human biases and our preference, at times, to be loss averse, not risk averse. The list of CAPM alternatives will surely grow as further research is conducted.

## The Bottom Line

The CAPM is among the most popular methods by which to value stocks and forecast returns. While not without its flaws, the formula, based on Modern Portfolio Theory, helps investors understand the quantitative relationship between expected risk and return. Managing portfolios is no easy task, and CAPM helps practitioners make better decisions in financial markets.

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