From CAPM to the Security Market Line
The CAPM relates an asset’s required return to its systematic risk, measured by beta. The security market line (SML) is the graph of that relationship: required return on the vertical axis and beta on the horizontal axis. Constructing it means finding the line’s vertical intercept and slope, then using the resulting equation to calculate returns at chosen beta coordinates.
The model is applied for a single, consistent investment horizon and in a common currency. Assume the CAPM equilibrium setting: investors can invest in a risk-free asset, share common expectations about returns and risk, and the market portfolio is efficient. Within this model, beta measures an asset’s sensitivity to the market portfolio. The SML describes CAPM-implied required returns; it is not a schedule of returns that will actually be realized.
Here, \(R_f\) is the risk-free rate for the period, \(E[R_M]\) is the expected return on the market portfolio over that period, and \(E[R_j]\) is the required return for asset \(j\) according to CAPM. Beta, \(\beta_j\), has no units. The difference \(E[R_M]-R_f\) is the market risk premium, the additional expected return on the market portfolio above the risk-free rate.
Intercept, Slope, and Key Coordinates
Write the CAPM equation in the form of a line, \(y=b+mx\). The vertical coordinate \(y\) is the required return, the horizontal coordinate \(x\) is beta, the intercept \(b\) is \(R_f\), and the slope \(m\) is the market risk premium. Thus, the line starts at \((0,R_f)\) and passes through the market portfolio coordinate \((1,E[R_M])\). These points provide useful checks when drawing or calculating from an SML.
The intercept follows by setting beta to zero: the beta-related component becomes zero, so the CAPM return is \(R_f\). Setting beta to one gives \(R_f+(E[R_M]-R_f)=E[R_M]\). The slope is the change in required return for a one-unit increase in beta. For example, a slope of 6 percentage points per beta unit means that increasing beta by 0.5 increases the CAPM return by 3 percentage points, holding the line’s inputs fixed.
These properties follow directly from the CAPM equation, rather than from an independent assumption about how a graph should look. The equation is linear in beta, so each additional beta unit contributes the same market-risk-premium amount to the required return. A negative beta places an asset to the left of the zero-beta intercept; the same equation applies there. The result may be below the risk-free rate when the market risk premium is positive, but beta is not a return and should not be expressed as a percentage.
Constructing the Line from Market Inputs
Use the risk-free rate and expected market return for the same investment horizon, expressed in the same rate units.
Subtract the risk-free rate from the expected market return. This difference, not the full market return, is the SML slope.
Use the risk-free rate as the intercept, then check the coordinates \((0,R_f)\) and \((1,E[R_M])\).
Multiply beta by the market risk premium and add the risk-free rate. Keep rates in percentage points or decimals consistently.
Worked Example: Plotting Returns at Four Beta Coordinates
A hypothetical investment analysis uses an annual risk-free rate of 3% and an annual expected market return of 9%. Construct the SML equation and calculate the required returns for assets with betas \(-0.5\), \(0\), \(0.75\), and \(1.5\). Identify the intercept, slope, and market coordinate.
Method. First calculate the market risk premium, then substitute each beta into the line. This avoids treating the 9% market return as the slope.
Step 1: Find the slope and write the equation. The market risk premium is \(9\%-3\%=6\%\). Therefore, the intercept is 3%, the slope is 6 percentage points per unit of beta, and the SML is \(E[R_j]=3\%+\beta_j(6\%)\).
Step 2: Calculate each coordinate. For \(\beta=-0.5\), \(E[R_j]=3\%+(-0.5)(6\%)=3\%-3\%=0\%\). For \(\beta=0\), \(E[R_j]=3\%+0(6\%)=3\%\). For \(\beta=0.75\), \(E[R_j]=3\%+0.75(6\%)=3\%+4.5\%=7.5\%\). For \(\beta=1.5\), \(E[R_j]=3\%+1.5(6\%)=3\%+9\%=12\%\).
Step 3: Check reference coordinates. The intercept coordinate is \((0,3\%)\), which agrees with the beta-zero calculation. At market beta one, the line gives \(3\%+1(6\%)=9\%\), so the market coordinate is \((1,9\%)\), as required.
The four asset coordinates are \((-0.5,0\%)\), \((0,3\%)\), \((0.75,7.5\%)\), and \((1.5,12\%)\). These values are exact with no intermediate rounding. The common mistake this method avoids is using 9% as the slope, which would incorrectly produce \(3\%+\beta(9\%)\).
Reading an SML from Two Coordinates
Sometimes an SML is specified by two beta-return coordinates instead of by \(R_f\) and \(E[R_M]\). A straight line’s slope is the change in vertical coordinate divided by the change in horizontal coordinate. Once the slope is known, substitute one coordinate into the line equation to solve for the intercept. This procedure is valid when the two coordinates have different betas and are on the same CAPM line.
For a CAPM SML, \(m\) is the market risk premium and \(b\) is the risk-free rate. If the market portfolio’s coordinate is also known or can be inferred, its return must equal \(b+m\), because its beta is one. This is a useful consistency check on coordinates or calculations.
Worked Example: Recovering the SML from Two Points
Two coordinates on a hypothetical CAPM line are \((0.25,4.5\%)\) and \((1.25,10.5\%)\), where each first value is beta and each second value is the required return. Find the SML intercept and slope, infer the risk-free rate and expected market return, and calculate the required return at beta \(-0.5\).
Method. Calculate rise over run to obtain the slope, then use a known coordinate to obtain the intercept. The main trap is subtracting the two returns but dividing by a return rather than by the change in beta.
Step 1: Calculate the slope. The return difference is \(10.5\%-4.5\%=6.0\) percentage points, and the beta difference is \(1.25-0.25=1.00\). Therefore, \(m=6.0/1.00=6.0\) percentage points per beta unit.
Step 2: Calculate the intercept. Using the first point, \(b=4.5\%-6.0\%(0.25)=4.5\%-1.5\%=3.0\%\). The line is \(E[R_j]=3.0\%+6.0\%\beta_j\). Thus, the risk-free rate is 3.0% and the market risk premium is 6.0%.
Step 3: Infer the market return and calculate the requested coordinate. At beta one, the market return is \(3.0\%+6.0\%(1)=9.0\%\). At beta \(-0.5\), the required return is \(3.0\%+6.0\%(-0.5)=3.0\%-3.0\%=0\%\).
The inferred line returns 4.5% at beta 0.25 and 10.5% at beta 1.25, reproducing both given coordinates. The intercept, slope, market return, and requested return are therefore consistent with the stated line.
Care with Rates and Beta Coordinates
The SML is a relationship between returns and beta, not between returns and time periods. If the inputs are annual, the calculated required return is annual; a beta coordinate does not convert an annual rate to a monthly rate. A monthly risk-free rate cannot be combined directly with an annual expected market return. Convert the rates to the same horizon first, using the appropriate compounding convention for the rates provided.
Worked Example: Using the Market Risk Premium Correctly
A hypothetical asset has beta 1.4. The annual risk-free rate is 3.2%, and the annual expected market return is 8.2%. Calculate the asset’s annual CAPM required return. A proposed calculation uses \(1.4(8.2\%)\) as the beta-related component; determine whether it is correct.
Method. Separate the intercept from the slope before applying beta. The slope is the market return less the risk-free rate, not the market return by itself.
Step 1: Calculate the market risk premium. It is \(8.2\%-3.2\%=5.0\%\). Thus, the SML is \(E[R_j]=3.2\%+\beta_j(5.0\%)\).
Step 2: Substitute beta. \(E[R_j]=3.2\%+1.4(5.0\%)=3.2\%+7.0\%=10.2\%\). The annual CAPM required return is 10.2%.
Step 3: Correct the proposed calculation. Multiplying beta by the full market return gives \(1.4(8.2\%)=11.48\%\), and adding the risk-free rate would give an incorrect \(14.68\%\). The market return already includes the risk-free component; multiplying beta by the market risk premium and then adding \(R_f\) avoids counting that component incorrectly.
Check. The SML passes through \((1,8.2\%)\): \(3.2\%+1(5.0\%)=8.2\%\). The calculated return at beta 1.4 is 2.0 percentage points above the market return, equal to the additional \(0.4\) beta units times the 5.0% slope.
Why the Line Has This Form
The CAPM required-return equation is linear in beta. To see why the graph has the stated intercept and slope, begin with the model equation, then separate its constant and beta-dependent parts:
The risk-free rate does not depend on the asset’s beta, so it is the constant vertical value at beta zero. The market risk premium is multiplied by beta, so it is the change in required return per unit increase in beta. This is precisely the intercept-slope form of a straight line. Setting beta to one gives the market return, confirming that the line connects the risk-free coordinate to the market coordinate.
The argument depends on using the CAPM relationship and consistent period-specific rates. It does not establish that every observed or realized asset return will lie on the line. The SML gives the return required by the model for a specified beta, not a guarantee about an outcome.
Common Construction Errors
- Using the market return as the slope. The slope is \(E[R_M]-R_f\). Add \(R_f\) after multiplying beta by that difference.
- Putting beta on the vertical axis. Beta is the horizontal coordinate; required return is the vertical coordinate.
- Assuming the market coordinate has beta zero. The market portfolio is at beta one, so its coordinate is \((1,E[R_M])\). The risk-free intercept is at beta zero.
- Mixing rate horizons. Annual and monthly rates cannot be combined directly in one SML calculation.
- Treating beta as a percentage. Beta is a unit-free sensitivity measure. A beta of 1.2 is substituted as 1.2, not as 1.2%.
Check Your Understanding
For each problem, identify the relevant SML quantities and show the calculation before checking the final answer.
- A hypothetical portfolio analysis uses an annual risk-free rate of 2.5% and annual expected market return of 8.5%. State the SML intercept and slope, then calculate the required return at beta 0.5. (Answer: Intercept 2.5%; slope 6.0%; required return 5.5%.)
- A project has beta \(-0.25\). The annual risk-free rate is 4% and the annual expected market return is 11%. Calculate its CAPM required return and the market coordinate. (Answer: Required return 2.25%; market coordinate \((1,11\%)\).)
- Two coordinates on a hypothetical SML are \((0.2,5.2\%)\) and \((1.2,10.2\%)\). Find the slope, intercept, implied market return, and required return at beta 1.6. (Answer: Slope 5.0%; intercept 4.2%; market return 9.2%; required return 12.2%.)
- For an annual risk-free rate of 3.5% and an annual market risk premium of 5.5%, calculate the required return at beta 0.8 and verify the return at the market coordinate. (Answer: Required return at beta 0.8 is 7.9%; market coordinate is \((1,9.0\%)\).)
- A CAPM required return is 10.2%. The annual risk-free rate is 3.2%, and the annual expected market return is 10.2%. Find the market risk premium and the beta coordinate corresponding to that required return. (Answer: Market risk premium 7.0%; beta 1.0.)