Investors cannot attain a portfolio below the feasible set or opportunity set because they cannot

The efficient frontier is the set of optimal portfolios that offer the highest expected return for a defined level of risk or the lowest risk for a given level of expected return. Portfolios that lie below the efficient frontier are sub-optimal because they do not provide enough return for the level of risk. Portfolios that cluster to the right of the efficient frontier are sub-optimal because they have a higher level of risk for the defined rate of return.

The efficient frontier comprises investment portfolios that offer the highest expected return for a specific level of risk.
The standard deviation of returns in a portfolio measures investment risk and consistency in investment earnings.
Lower covariance between portfolio securities results in lower portfolio standard deviation.
Successful optimization of the return versus risk paradigm should place a portfolio along the efficient frontier line.
Optimal portfolios that comprise the efficient frontier usually exhibit a higher degree of diversification.

The efficient frontier theory was introduced by Nobel Laureate Harry Markowitz in 1952 and is a cornerstone of modern portfolio theory (MPT). The efficient frontier rates portfolios (investments) on a scale of return (y-axis) versus risk (x-axis). The compound annual growth rate (CAGR) of an investment is commonly used as the return component while standard deviation (annualized) depicts the risk metric.

The efficient frontier graphically represents portfolios that maximize returns for the risk assumed. Returns are dependent on the investment combinations that make up the portfolio. A security's standard deviation is synonymous with risk. Ideally, an investor seeks to fill a portfolio with securities offering exceptional returns but with a combined standard deviation that is lower than the standard deviations of the individual securities.

The less synchronized the securities (lower covariance), the lower the standard deviation. If this mix of optimizing the return versus risk paradigm is successful, then that portfolio should line up along the efficient frontier line.

A key finding of the concept was the benefit of diversification resulting from the curvature of the efficient frontier. The curvature is integral in revealing how diversification improves the portfolio's risk/reward profile. It also reveals that there is a diminishing marginal return to risk.

Adding more risk to a portfolio does not gain an equal amount of return—optimal portfolios that comprise the efficient frontier tend to have a higher degree of diversification than the sub-optimal ones, which are typically less diversified.

The efficient frontier and modern portfolio theory have many assumptions that may not properly represent reality. For example, one of the assumptions is that asset returns follow a normal distribution.

In reality, securities may experience returns (also known as tail risk) that are more than three standard deviations away from the mean. Consequently, asset returns are said to follow a leptokurtic distribution or heavy-tailed distribution.

Additionally, Markowitz posits several assumptions in his theory, such as that investors are rational and avoid risk when possible, that there are not enough investors to influence market prices, and that investors have unlimited access to borrowing and lending money at the risk-free interest rate.

However, reality proves that the market includes irrational and risk-seeking investors, there are large market participants who could influence market prices, and there are investors who do not have unlimited access to borrowing and lending money.

One assumption in investing is that a higher degree of risk means a higher potential return. Conversely, investors who take on a low degree of risk have a low potential return. According to Markowitz's theory, there is an optimal portfolio that could be designed with a perfect balance between risk and return.

The optimal portfolio does not simply include securities with the highest potential returns or low-risk securities. The optimal portfolio aims to balance securities with the greatest potential returns with an acceptable degree of risk or securities with the lowest degree of risk for a given level of potential return. The points on the plot of risk versus expected returns where optimal portfolios lie are known as the efficient frontier.

Assume a risk-seeking investor uses the efficient frontier to select investments. The investor would select securities that lie on the right end of the efficient frontier. The right end of the efficient frontier includes securities that are expected to have a high degree of risk coupled with high potential returns, which is suitable for highly risk-tolerant investors. Conversely, securities that lie on the left end of the efficient frontier would be suitable for risk-averse investors.

The efficient frontier graphically depicts the benefit of diversification and can. The curvature of the efficient frontier shows how diversification can improve a portfolio's risk versus reward profile.

An optimal portfolio is one designed with a perfect balance of risk and return. The optimal portfolio looks to balance securities that offer the greatest possible returns with acceptable risk or the securities with the lowest risk given a certain return.

The efficient frontier rates portfolios on a coordinate plane. Plotted on the x-axis is the risk, while return is plotted on the y-axis—annualized standard deviation is typically used to measure risk, while compound annual growth rate (CAGR) is used for return.

To use the efficient frontier, a risk-seeking investor selects investments that fall on the right side of the frontier. Meanwhile, a more conservative investor would pick investments that lie on the left side of the frontier.

The Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk, or the general perils of investing, and expected return for assets, particularly stocks. CAPM evolved as a way to measure this systematic risk. It is widely used throughout finance for pricing risky securities and generating expected returns for assets, given the risk of those assets and cost of capital.

The formula for calculating the expected return of an asset, given its risk, is as follows:

$\begin{aligned} &ER_i = R_f + \beta_i ( ER_m - R_f ) \\ &\textbf{where:} \\ &ER_i = \text{expected return of investment} \\ &R_f = \text{risk-free rate} \\ &\beta_i = \text{beta of the investment} \\ &(ER_m - R_f) = \text{market risk premium} \\ \end{aligned}$

Investors expect to be compensated for risk and the time value of money. The risk-free rate in the CAPM formula accounts for the time value of money. The other components of the CAPM formula account for the investor taking on additional risk.

The beta of a potential investment is a measure of how much risk the investment will add to a portfolio that looks like the market. If a stock is riskier than the market, it will have a beta greater than one. If a stock has a beta of less than one, the formula assumes it will reduce the risk of a portfolio.

A stock’s beta is then multiplied by the market risk premium, which is the return expected from the market above the risk-free rate. The risk-free rate is then added to the product of the stock’s beta and the market risk premium. The result should give an investor the required return or discount rate that they can use to find the value of an asset.

The goal of the CAPM formula is to evaluate whether a stock is fairly valued when its risk and the time value of money are compared with its expected return. In other words, by knowing the individual parts of the CAPM, it is possible to gauge whether the current price of a stock is consistent with its likely return.

For example, imagine an investor is contemplating a stock valued at $100 per share today that pays a 3% annual dividend. The stock has a beta compared with the market of 1.3, which means it is riskier than a market portfolio. Also, assume that the risk-free rate is 3% and this investor expects the market to rise in value by 8% per year.

The expected return of the stock based on the CAPM formula is 9.5%:

$\begin{aligned} &9.5\% = 3\% + 1.3 \times ( 8\% - 3\% ) \\ \end{aligned}$

The expected return of the CAPM formula is used to discount the expected dividends and capital appreciation of the stock over the expected holding period. If the discounted value of those future cash flows is equal to $100, then the CAPM formula indicates the stock is fairly valued relative to risk.

Several assumptions behind the CAPM formula have been shown not to hold up in reality. Modern financial theory rests on two assumptions:

Securities markets are very competitive and efficient (that is, relevant information about the companies is quickly and universally distributed and absorbed).
These markets are dominated by rational, risk-averse investors, who seek to maximize satisfaction from returns on their investments.

As a result, it’s not entirely clear whether CAPM works. The big sticking point is beta. When professors Eugene Fama and Kenneth French looked at share returns on the New York Stock Exchange, the American Stock Exchange, and Nasdaq, they found that differences in betas over a lengthy period did not explain the performance of different stocks. The linear relationship between beta and individual stock returns also breaks down over shorter periods of time. These findings seem to suggest that CAPM may be wrong.

Despite these issues, the CAPM formula is still widely used because it is simple and allows for easy comparisons of investment alternatives.

Including beta in the formula assumes that risk can be measured by a stock’s price volatility. However, price movements in both directions are not equally risky. The look-back period to determine a stock’s volatility is not standard because stock returns (and risk) are not normally distributed.

The CAPM also assumes that the risk-free rate will remain constant over the discounting period. Assume in the previous example that the interest rate on U.S. Treasury bonds rose to 5% or 6% during the 10-year holding period. An increase in the risk-free rate also increases the cost of the capital used in the investment and could make the stock look overvalued.

The market portfolio used to find the market risk premium is only a theoretical value and is not an asset that can be purchased or invested in as an alternative to the stock. Most of the time, investors will use a major stock index, like the S&P 500, to substitute for the market, which is an imperfect comparison.

The most serious critique of the CAPM is the assumption that future cash flows can be estimated for the discounting process. If an investor could estimate the future return of a stock with a high level of accuracy, then the CAPM would not be necessary.

Using the CAPM to build a portfolio is supposed to help an investor manage their risk. If an investor were able to use the CAPM to perfectly optimize a portfolio’s return relative to risk, it would exist on a curve called the efficient frontier, as shown in the following graph.

The graph shows how greater expected returns (y-axis) require greater expected risk (x-axis). Modern portfolio theory (MPT) suggests that starting with the risk-free rate, the expected return of a portfolio increases as the risk increases. Any portfolio that fits on the capital market line (CML) is better than any possible portfolio to the right of that line, but at some point, a theoretical portfolio can be constructed on the CML with the best return for the amount of risk being taken.

The CML and the efficient frontier may be difficult to define, but they illustrate an important concept for investors: There is a tradeoff between increased return and increased risk. Because it isn’t possible to perfectly build a portfolio that fits on the CML, it is more common for investors to take on too much risk as they seek additional return.

In the following chart, you can see two portfolios that have been constructed to fit along the efficient frontier. Portfolio A is expected to return 8% per year and has a 10% standard deviation or risk level. Portfolio B is expected to return 10% per year but has a 16% standard deviation. The risk of Portfolio B rose faster than its expected returns.

The efficient frontier assumes the same things as the CAPM and can only be calculated in theory. If a portfolio existed on the efficient frontier, it would provide maximal return for its level of risk. However, it is impossible to know whether a portfolio exists on the efficient frontier because future returns cannot be predicted.

This tradeoff between risk and return applies to the CAPM, and the efficient frontier graph can be rearranged to illustrate the tradeoff for individual assets. In the following chart, you can see that the CML is now called the security market line (SML). Instead of expected risk on the x-axis, the stock’s beta is used. As you can see in the illustration, as beta increases from 1 to 2, the expected return is also rising.

The CAPM and the SML make a connection between a stock’s beta and its expected risk. Beta is found by statistical analysis of individual, daily share price returns compared with the market’s daily returns over precisely the same period. A higher beta means more risk, but a portfolio of high-beta stocks could exist somewhere on the CML where the tradeoff is acceptable, if not the theoretical ideal.

The value of these two models is diminished by assumptions about beta and market participants that aren’t true in the real markets. For example, beta does not account for the relative riskiness of a stock that is more volatile than the market with a high frequency of downside shocks compared with another stock with an equally high beta that does not experience the same kind of price movements to the downside.

Considering the critiques of the CAPM and the assumptions behind its use in portfolio construction, it might be difficult to see how it could be useful. However, using the CAPM as a tool to evaluate the reasonableness of future expectations or to conduct comparisons can still have some value.

Imagine an advisor who has proposed adding a stock to a portfolio with a $100 share price. The advisor uses the CAPM to justify the price with a discount rate of 13%. The advisor’s investment manager can take this information and compare it with the company’s past performance and its peers to see if a 13% return is a reasonable expectation.

Assume in this example that the peer group’s performance over the last few years was a little better than 10% while this stock had consistently underperformed, with 9% returns. The investment manager shouldn’t take the advisor’s recommendation without some justification for the increased expected return.

An investor also can use the concepts from the CAPM and the efficient frontier to evaluate their portfolio or individual stock performance vs. the rest of the market. For example, assume that an investor’s portfolio has returned 10% per year for the last three years with a standard deviation of returns (risk) of 10%. However, the market averages have returned 10% for the last three years with a risk of 8%.

The investor could use this observation to reevaluate how their portfolio is constructed and which holdings may not be on the SML. This could explain why the investor’s portfolio is to the right of the CML. If the holdings that are either dragging on returns or have increased the portfolio’s risk disproportionately can be identified, then the investor can make changes to improve returns. Not surprisingly, the CAPM contributed to the rise in the use of indexing, or assembling a portfolio of shares to mimic a particular market or asset class, by risk-averse investors. This is largely due to the CAPM message that it is only possible to earn higher returns than those of the market as a whole by taking on higher risk (beta).

The CAPM uses the principles of modern portfolio theory to determine if a security is fairly valued. It relies on assumptions about investor behaviors, risk and return distributions, and market fundamentals that don’t match reality. However, the underlying concepts of CAPM and the associated efficient frontier can help investors understand the relationship between expected risk and reward as they strive to make better decisions about adding securities to a portfolio.