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# Security Market Line

If you were told that you could invest in an asset and earn a rate of return, you would weigh it against the risk, the asset price, and the opportunity cost. Imagine all investors on the market and their choices about the assets they want to invest in. To draw inferences at the macro level, we need to aggregate all the market participants' preferences. This would allow determining how investing in a risky asset should be compensated for each possible level of risk. Well, luckily, the Security Market Line does just that! Eager to learn how the appropriate rate of return for any asset on the market can be determined, irrespective of the corresponding risk level? Keep scrolling to find out!

## Security Market Line Definition

Before diving into the definition of the security market line, we need to understand the investors' compensation.Every investor needs to be compensated when investing in an asset. But what does this compensation consist of? The first part is the opportunity cost of parting with their money or the time value of money. The investor could have invested their money elsewhere and earned at least a risk-free interest rate. This risk-free rate is considered the minimum compensation any investor needs to receive when investing in a risky asset. Why so? Because investing in risk-free securities such as U.S. government bonds would provide the same return without the added risk that the risky investment brings.This brings us to the next part that investors need to be compensated for: non-diversifiable risk. Different assets carry different risk levels; thus, the compensation, or the risk premium, that the investor should receive varies.Bringing the two parts that the investors need to be compensated for together yields the following equation:$$E(R)=r_f+RP$$Where:$$E(R)$$ - the average expected rate of return$$r_f$$ - compensation for the time value of money or the risk-free rate$$RP$$ - risk premium

A risk premium - is the compensation an investor receives for non-diversifiable risk.

Now, what does the risk of investment depend on? How can we determine which investment is riskier than the other? The answer is - beta. A beta of an investment is the degree to which an asset co-moves with the rest of the market. The takeaway is that the assets with large betas carry more non-diversifiable risk than those with small betas. This means that investors will need to be compensated more for assets with significant betas. In other words, such investments would require more substantial risk premia.

A beta of an investment is the degree to which an asset co-moves with the rest of the market.

## Security Market Line Formula

Understanding the basic principle of returns and learning about the definition of beta brings us to the security market line formula.The $$SML$$ equation or formula is:$$E(R_i)=r_f+RP=r_f+\beta_i\times(RP_M)=r_f+\beta_i\times[E(R_M)-r_f]$$Where:$$E(R_i)$$ - expected return on an asset$$r_f$$ - the risk-free rate$$RP$$ - total risk premium$$\beta_i$$ - asset's beta$$RP_M$$ - market risk premium$$E(R_M)$$ - expected return on a market portfolioAs we've seen above, the logic of investor compensation holds here. When securities are priced according to the rule of investor compensation, they will be located on the $$SML$$. However, if the securities are over or undervalued, they will be located above or below the $$SML$$.

The security market line plots the average expected rates of return on assets against their risk levels. It has a positive slope and an intercept at the risk-free rate.

## Derivation of Security Market Line

Let's go through a graphical derivation of the security market line - $$SML$$. As the average expected rate of return on an investment depends on the asset risk level, we can plot all the average returns against their risk.We can choose the average expected rate of return to be on the vertical axis and the assets' betas on the horizontal axis. The risk-free rate is the bare minimum for which the investors need to be compensated. This means that the average expected rate of return for any asset will not be less than the risk-free rate. Let's mentally mark it at some point on the vertical axis. We assume that the risk-free rate is positive.

Now, let's think about if there are any investments that we can already determine here. Turns out - we know two already!

One is a risk-free asset such as a U.S. short-term government bond. As it is assumed to be practically risk-free, its beta is $$0$$.

The second one is the market portfolio.

A market portfolio is a hypothetical portfolio comprised of all the assets in the market. It follows the rule that asset weights in such a portfolio should be proportional to the relative quantity of each asset in the market.

As the market portfolio is well-diversified, its beta will be equal to $$1$$. We now have the two points; the only thing left to determine is the slope! As we know that investors need to be compensated more for higher risk levels, the slope of the $$SML$$ will need to be positive. Now we can plot the $$SML$$. Take a look at Figure 1 below.

Figure 1. Security market line, StudySmarter Originals

Figure 1 above shows the $$SML$$. On the horizontal axis, there is the risk level as measured by the assets' beta. On the vertical axis, there is the average expected rate of return. The $$SML$$ passes through the risk-free asset and the market portfolios. The risk-free portfolio marked by $$R_f$$ carries zero risk, while the market portfolio denoted by $$M$$ carries a risk measured when beta equals one.

We have now derived the $$SML$$ graphically.

The security market line underpins the idea that any asset on the market should compensate the investors for the time value of money and the risk that those assets carry.

## The Slope and the Intercept of the Security Market Line

The slope of the security market line, $$SML$$, is solely determined by investors' expectations about the risk and compensation that they need to receive for this risk.

The $$SML$$ will be steeper the more risk-averse investors are. For any increase in the risk level, they will require greater compensation. However, if investors are less risk-averse, the $$SML$$ will be less steep. For any increase in the risk level, they will not require as much compensation as they would if they were more risk-averse.The intercept of the $$SML$$ is the risk-free rate. The Federal Reserve's policy affects it, and investors' preferences play no role in determining the intercept of the $$SML$$. However, this is precisely why investors carefully observe all the moves of the Federal Reserve. Any changes in the risk-free rate affect what investors perceive the $$SML$$ to be at any given time. This, in turn, significantly affects the rates of return and the prices of securities.

## Security Market Line Example

Let's look at a security market line example when the asset is either over or undervalued.Take a look at Figure 2 below.

Figure 2. Security market line and arbitrage, StudySmarter Originals

Asset $$A$$ lies above the $$SML$$ because it is undervalued or underpriced. Think about it: for a given level of risk, the return it provides is way too high. Investors would aim to buy such an asset quickly, pushing its price up. The price is inversely related to the rate of return; thus, the rate of return should go down.Asset $$B$$ lies below the $$SML$$ because it is overvalued or overpriced. The return it provides is too low for a given level of risk. Investors would aim to sell such an asset quickly, pushing its price down. The price is inversely related to the rate of return; thus, the rate of return should go up.Arbitrage implies that over time both security $$A$$ and security $$B$$ would both lie on the $$SML$$.When an asset is located precisely on the $$SML$$, it is considered to be priced correctly. Its average expected rate of return accurately compensates investors for the risk they are taking and the time value of money.

A security is underpriced if the return it provides for a given level of risk is too high.

A security is overpriced if the return it provides for a given level of risk is too low.

Arbitrage in the $$SML$$ context means that overtime securities will be priced to appropriately compensate investors for the risk levels and the time value of money.

## Security Market Line vs. Capital Market Line

The security market line $$SML$$ is different from the capital market line - $$CML$$. The $$CML$$ plots the returns over and above the risk-free rate of efficient portfolios against portfolio standard deviations.The $$SML$$, however, plots the returns over and above the risk-free rate for individual assets against their risk levels.The horizontal axis for the $$CML$$ is the portfolio standard deviation as it best represents the risk incurred by an efficient portfolio of assets. The horizontal axis for the $$SML$$ is the beta of individual assets as it best represents the risk that a particular investment could potentially introduce into a portfolio.

An efficient portfolio is a portfolio that is comprised of the market and the risk-free asset.

Now that you've understood how the $$SML$$ is different from the $$CML$$ let's see why these are often confused. Well, for one, the names sound very similar, but that's not the end of the story. In fact, $$SML$$ can be constructed from the $$CML$$ by plotting all the stocks within the efficient frontier onto one line! How exactly to plot the $$SML$$ and the $$CML$$ is outside the scope of this article.

## Security Market Line - Key takeaways

• A risk premium - is the compensation an investor receives for non-diversifiable risk.
• A beta of an investment is the degree to which an asset co-moves with the rest of the market.
• The security market line plots the average expected rates of return on assets against their risk levels. It has a positive slope and an intercept at the risk-free rate.
• A market portfolio is a hypothetical portfolio comprised of all the assets in the market. It follows the rule that asset weights in such a portfolio should be proportional to the relative quantity of each asset in the market.
• An efficient portfolio is a portfolio that is comprised of the market and the risk-free asset.

The security market line plots the average expected rates of return on assets against their risk levels. It has a positive slope and an intercept at the risk-free rate.

The security market line is calculated by using the following equation:

E(R)=Rf+RP

The security market line equation is:
E(R)=Rf+RP

It allows for finding the compensation investors would like to receive for the risk they are exposed to when purchasing any asset.

The security market line has a positive slope and an intercept at the risk-free rate.

## Final Security Market Line Quiz

Question

What is the main idea behind the security market line?

The security market line underpins the idea that any asset on the market should compensate the investors for the time value of money and the risk that those assets carry.

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Question

What does the security market line plot?

The security market line plots the average expected rates of return on assets against their risk levels. It has a positive slope and an intercept at the risk-free rate.

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Question

A risk premium - is the compensation an investor receives for non-diversifiable risk.

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Question

Explain the idea behind an investment's beta?

A beta of an investment is the degree to which an asset co-moves with the rest of the market.

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Question

What is a market portfolio?

A market portfolio is a hypothetical portfolio comprised of all the assets in the market. It follows the rule that asset weights in such a portfolio should be proportional to the relative quantity of each asset in the market.

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Question

What is an efficient portfolio?

An efficient portfolio is a portfolio that is comprised of the market and the risk-free asset.

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Question

What is the security market line equation?

The security market line is calculated by using the following equation:
$$E(R_i)=r_f+RP=r_f+\beta_i\times(RP_M)=r_f+\beta_i\times[E(R_M)-r_f]$$

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Question

What are the advantages of the security market line?

It allows for finding the compensation investors would like to receive for the risk they are exposed to when purchasing any asset.

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Question

What are the characteristics of the security market line?

The security market line has a positive slope and an intercept at the risk-free rate.

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Question

When is a security considered underpriced?

A security is underpriced if the return it provides for a given level of risk is too high.

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Question

When is a security considered overpriced?

A security is overpriced if the return it provides for a given level of risk is too low.

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Question

What is arbitrage in the context of SML?

Arbitrage in the SML context means that overtime securities will be priced to appropriately compensate investors for the risk levels and the time value of money.

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Question

How is the security market line (SML) different from the capital market line (CML)?

The CML plots the risk premia of efficient portfolios against portfolio standard deviations.
The SML, however, plots the risk premia for individual assets against their risk levels.

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