Why Two Lines Are Needed
The security market line (SML) and capital market line (CML) are both straight-line relationships between risk and expected return in the Capital Asset Pricing Model (CAPM). They do not, however, measure risk in the same way or apply to the same set of investments. The SML uses beta, which measures an asset’s systematic risk relative to the market. The CML uses standard deviation, which measures a portfolio’s total risk.
This difference determines which line to use. The SML describes the CAPM required expected return for individual assets and portfolios. The CML describes the expected return of efficient portfolios formed from the risk-free asset and the market portfolio. Applying a CML calculation to an individual security just because its standard deviation is known is a common error.
The Model and Its Two Relationships
Use both lines in the same idealized, one-period CAPM framework. Investors evaluate expected return and variance, share the same expectations about return distributions, and can borrow and lend at a common risk-free rate. The market portfolio is the efficient, diversified portfolio of risky assets. These assumptions are important: the CML’s efficient-portfolio result depends on the risk-free asset and the market portfolio forming the relevant efficient set. If those assumptions do not describe the situation being analyzed, the lines may not describe observed returns.
Let \(R_i\) be the one-period return on asset \(i\), \(R_M\) the return on the market portfolio, \(r_f\) the risk-free rate for the same period, and \(E[R]\) an expected return. The asset’s beta is its covariance with the market return divided by the variance of the market return. Under CAPM, the SML gives the required expected return:
The term \(E[R_M]-r_f\) is the market risk premium. The SML starts at \(r_f\) when beta is zero, and its slope is the market risk premium. Beta, rather than total standard deviation, determines the CAPM return in this equation. The formula applies to a single asset as well as to a portfolio, provided the beta is measured relative to the same market portfolio and the CAPM assumptions are being used.
The CML uses \(\sigma_p\), the standard deviation of portfolio \(p\)’s return, and \(\sigma_M\), the standard deviation of the market return. For an efficient portfolio on the CML:
The CML’s slope is the market risk premium per unit of market standard deviation. It describes the efficient risk-return tradeoff: at a given level of standard deviation, a portfolio below the CML has a lower expected return than the efficient portfolio available at that risk level. The CML is not a general return formula for every asset or portfolio. A portfolio that is not efficient does not qualify for the CML formula merely because its standard deviation is known.
| Feature | Security market line (SML) | Capital market line (CML) |
|---|---|---|
| Risk measure | Beta, \(\beta\) | Standard deviation, \(\sigma\) |
| Relationship | Expected return versus beta | Expected return versus total portfolio risk |
| Applies to | Individual assets and portfolios under CAPM | Efficient portfolios on the risk-free/market portfolio line |
| Slope | Market risk premium, \(E[R_M]-r_f\) | Market risk premium divided by \(\sigma_M\) |
Why the Relationships Match for Efficient Portfolios
A portfolio formed by investing a fraction \(x\) in the market portfolio and the remainder in the risk-free asset has a particularly useful relationship between beta and standard deviation. In this model, \(x\) can be greater than one if the investor borrows at the risk-free rate to invest more than the portfolio’s own funds in the market. For the usual CML segment with \(x\geq 0\), the portfolio’s expected return is \(r_f+x(E[R_M]-r_f)\), and its standard deviation is \(x\sigma_M\).
The beta relationship follows from covariance. The risk-free return is constant, so its covariance with the market return is zero. Thus, the covariance of the portfolio return with the market return is \(x\operatorname{Var}(R_M)\). Dividing by market variance gives the portfolio’s beta. The steps also show why this portfolio is perfectly aligned with market risk rather than carrying separate asset-specific risk.
\(R_p=xR_M+(1-x)r_f\), where \(r_f\) is constant.
\(\operatorname{Cov}(R_p,R_M)=x\operatorname{Var}(R_M)\), so \(\beta_p=\operatorname{Cov}(R_p,R_M)/\operatorname{Var}(R_M)=x\).
The risk-free component has no variance and \(x\geq 0\), so \(\sigma_p=x\sigma_M\).
\(E[R_p]=r_f+x(E[R_M]-r_f)\). Substituting \(\beta_p=x\) into the SML gives the same return; substituting \(\sigma_p=x\sigma_M\) into the CML also gives the same return.
For such a market/risk-free combination, beta and standard deviation are related by \(\beta_p=\sigma_p/\sigma_M\). That relationship is not a general rule for every asset. For an individual risky asset, beta also depends on how strongly its returns move with the market. In particular, \(\beta_i=\rho_{iM}\sigma_i/\sigma_M\), where \(\rho_{iM}\) is the correlation between the asset and the market. If the asset has market-independent risk, its standard deviation can be high without a corresponding increase in beta.
Worked Example: Use Beta for an Individual Security
A hypothetical technology supplier has a one-year expected return of 11.0%, a beta of 1.20, and a standard deviation of 28.0%. The market portfolio has an expected one-year return of 9.0% and a standard deviation of 15.0%; the one-year risk-free rate is 3.0%. Under the CAPM assumptions, calculate the security’s required expected return. Then check whether its standard deviation can be used in the CML formula.
Method. Because the investment is an individual security, use the SML. The market risk premium is \(9.0\%-3.0\%=6.0\%\). Its required expected return is
The security’s hypothetical expected return of 11.0% is 0.8 percentage points above that CAPM required return. Its standard deviation is not a reason to replace beta with \(\sigma_i/\sigma_M\). If we incorrectly applied the CML, we would calculate \(3.0\%+(6.0\%/15.0\%)(28.0\%)=14.2\%\). That result is not justified: the CML formula is for efficient portfolios, and the problem gives an individual security, not an efficient portfolio on the CML.
Check. The SML return uses the security’s beta and market risk premium; the 10.2% result is below its stated expected return of 11.0%. The mistake avoided is substituting total security risk into a formula that requires CML eligibility.
Worked Example: Calculate Both Risk Measures for a Market Mix
A hypothetical investor places 40% of a portfolio in the market portfolio and 60% in the risk-free asset for one year. The risk-free rate is 4.0%, the market expected return is 10.0%, and the market standard deviation is 20.0%. Assume the CAPM framework, with risk-free borrowing and lending available at 4.0%. Find the portfolio’s beta, standard deviation, and expected return using both lines.
The market investment fraction is \(x=0.40\). The derivation above gives \(\beta_p=x=0.40\). Since the remaining 60% is risk-free, the portfolio standard deviation is \(0.40(20.0\%)=8.0\%\).
The market risk premium is \(10.0\%-4.0\%=6.0\%\). Using the SML:
Using the CML, its slope is \(6.0\%/20.0\%=0.30\), or 30.0% expected-return points per unit of standard deviation. Therefore:
The calculations agree because this portfolio is a market/risk-free combination on the CML and has beta \(0.40\). A direct check gives \(0.60(4.0\%)+0.40(10.0\%)=6.4\%\). The mistake avoided is treating the CML as a different required-return answer for a portfolio that lies on both lines.
Reading the Lines Without Confusing Them
The CML uses total portfolio risk. A portfolio’s standard deviation includes both market-related movements and any other return variation. The SML uses beta because CAPM compensates investors for systematic risk, not for risk that can be diversified away in a well-diversified portfolio. Consequently, a high standard deviation alone does not imply a high CAPM required return: the asset’s relationship to the market matters.
A portfolio can be evaluated against both lines, but the questions differ. Use the SML to compare its expected return with the CAPM return implied by its beta. Use the CML to ask whether its expected return is consistent with the efficient risk-return tradeoff at its standard deviation. A portfolio can have a beta that places it at a particular SML return and still not be efficient for its level of total risk.
The lines coincide for the risk-free/market combinations described above. In a CAPM equilibrium, the market portfolio itself is on both lines: its beta is 1 and its standard deviation is \(\sigma_M\). For other individual assets, do not assume the same beta-to-standard-deviation ratio. That shortcut would require the asset to move perfectly with the market, a condition not established merely by knowing its beta or standard deviation.
Worked Example: A Portfolio’s SML Return Does Not Establish CML Efficiency
A hypothetical pension portfolio has a beta of 0.75 and a standard deviation of 30.0%. The market has an expected one-year return of 11.0%, a standard deviation of 15.0%, and a variance of 0.0225; the risk-free rate is 2.0%. The portfolio’s expected one-year return is 8.5%. Under CAPM, calculate its SML required return. Then calculate the CML return at 30.0% standard deviation and explain what can and cannot be concluded.
Method. First use the portfolio’s beta with the SML. The market risk premium is \(11.0\%-2.0\%=9.0\%\), so the SML return is
The stated expected return of 8.5% is 0.25 percentage points below the SML return. Next, the CML slope is \(9.0\%/15.0\%=0.60\). If the portfolio were an efficient CML portfolio at its stated standard deviation, its return would be
The given 8.5% expected return is far below the CML return of 20.0%. Under the stated CAPM framework, it is therefore not on the CML and is not the efficient portfolio available at that standard deviation. Its beta-based SML comparison and its total-risk CML comparison answer different questions. The mistake avoided is inferring that an SML return, calculated from beta, also proves that a portfolio is CML-efficient.
Common Exam Traps
- Using standard deviation in the SML. The SML uses beta. Replacing beta with an asset’s standard deviation does not produce its CAPM required return.
- Using beta as the CML risk measure. The CML uses portfolio standard deviation. Beta is not a substitute for total risk in that equation.
- Applying the CML to any risky investment. The CML describes efficient portfolios in the specified model, not every individual asset or portfolio.
- Mixing market-risk units. The CML slope is \((E[R_M]-r_f)/\sigma_M\). Use standard deviations in consistent units, such as 15% and 20%, and keep all returns on the same time horizon.
A reliable choice is to identify the object and the risk measure before calculating. If the question gives beta and asks for a CAPM required return, use the SML. If it gives a portfolio’s standard deviation and asks about the efficient risk-return tradeoff, use the CML, after confirming that the portfolio is eligible for that relationship.
Check Your Understanding
For each problem, identify the applicable line before calculating.
- A hypothetical security has beta 0.60 and standard deviation 25.0%. The risk-free rate is 3.0%, the market expected return is 9.0%, and the market standard deviation is 20.0%. Find its SML required return. What return would a mechanical CML substitution produce, and is that substitution justified for the security? (Answer: SML return 6.6%; mechanical CML result 10.5%, not justified for an individual security.)
- A portfolio invests 125% in the market portfolio and borrows the remaining 25% at the risk-free rate. The one-year risk-free rate is 4.0%, market expected return is 12.0%, and market standard deviation is 16.0%. Under the CAPM borrowing/lending assumption, find its beta, standard deviation, and expected return. (Answer: beta 1.25; standard deviation 20.0%; expected return 14.0%.)
- An asset has covariance with the market of 0.018 and the market return variance is 0.030. Its standard deviation is 24.0%. The one-year risk-free rate is 2.0% and market expected return is 11.0%. Find its beta and SML return. If the market standard deviation is 15.0%, calculate the return suggested by a CML substitution and state whether the asset’s data establish that it belongs on the CML. (Answer: beta 0.60; SML return 7.4%; mechanical CML result 16.4%, but the asset is not established as an efficient portfolio.)
- The risk-free rate is 2.0%, the market expected return is 10.0%, and market standard deviation is 20.0%. A portfolio has standard deviation 12.0% and expected return 8.0%. What would its expected return be if it were on the CML, and is the stated return consistent with being on that line? (Answer: CML return 6.8%; no, the stated 8.0% is above the CML value, so it is not on the specified CML.)
- A portfolio is formed by investing 50% in the market and 50% in the risk-free asset. The risk-free rate is 3.0%, market expected return is 9.0%, and market standard deviation is 18.0%. Find its beta, standard deviation, and expected return using the SML and CML. (Answer: beta 0.50; standard deviation 9.0%; expected return 6.0% by either line.)