Why CAPM Sensitivity Matters
A CAPM estimate is only as informative as its inputs. A small change in the risk-free rate, beta, or market risk premium can alter the estimated required return, but the size of that effect depends on both the input change and the model’s structure. Sensitivity analysis makes this dependence explicit: it shows how much the estimate changes when one input changes and the others are held fixed.
For a meaningful comparison, use rates with the same currency, time horizon, and nominal-or-real basis. In the standard one-period CAPM, the required return is an annual expected return; the risk-free rate and market risk premium must be annual and defined consistently with it. The calculations below are model-based comparisons, not claims that any input or return is certain.
Marginal Sensitivity to Each Input
When the three inputs are \(r_f\), \(\beta\), and \(\mathrm{MRP}\), the CAPM equation is linear in each input when the other two are fixed. The marginal sensitivities are the coefficients on those inputs:
These derivatives describe the effect of a small change, and in this linear form they also give the exact effect of changing just one input. For example, if the risk-free rate rises by 0.25 percentage points while beta and the market risk premium stay fixed, the required return rises by 0.25 percentage points. A beta increase of 0.10 changes the required return by \(0.10 \times \mathrm{MRP}\); a market-risk-premium increase of 0.50 percentage points changes it by \(\beta \times 0.50\) percentage points.
The distinction follows directly from the definition of the market risk premium. If the expected market return is fixed, then \(\mathrm{MRP}=E[R_m]-r_f\). Substituting that expression into CAPM gives a different-looking, but equivalent, form:
Taking the change in \(r_f\) while holding \(E[R_m]\) and \(\beta\) fixed therefore gives a coefficient of \(1-\beta\). This is the reason that the same risk-free-rate change can raise or lower a CAPM estimate, depending on the assumption about the market input and on beta.
Deriving the One-Input Effects
The sensitivities can also be seen without calculus. Start with the CAPM estimate \(r=r_f+\beta\mathrm{MRP}\). If only beta changes by \(\Delta\beta\), the new estimate is \(r_f+(\beta+\Delta\beta)\mathrm{MRP}\). Subtracting the original estimate leaves \(\mathrm{MRP}\Delta\beta\). The same subtraction for a change in the market risk premium leaves \(\beta\Delta\mathrm{MRP}\). When only \(r_f\) changes while MRP is fixed, the risk-free-rate term itself changes by \(\Delta r_f\), and the premium term does not change.
Thus, with the three CAPM inputs treated independently, the exact one-input effects are:
Worked Example: Comparing Three One-Input Changes
A hypothetical asset has a risk-free rate of 3.20%, beta of 1.25, and market risk premium of 5.60%. Estimate its required return, then compare the effects of (a) a 0.20 percentage-point increase in the risk-free rate, holding MRP fixed; (b) a 0.10 increase in beta, holding the rates fixed; and (c) a 0.40 percentage-point increase in MRP, holding the other inputs fixed. Identify the largest effect.
Method. Use \(r=r_f+\beta\mathrm{MRP}\), then multiply each input change by its corresponding sensitivity. Express all rates as percentages and report changes in percentage points.
Initial estimate. The beta-weighted premium is \(1.25 \times 5.60\%=7.00\%\). Therefore \(r=3.20\%+7.00\%=10.20\%\).
(a) Risk-free rate. Holding MRP fixed, the sensitivity is 1. The change is \(1 \times 0.20=0.20\) percentage points. The new estimate is \(10.20\%+0.20\%=10.40\%\).
(b) Beta. The sensitivity is MRP, or 5.60 percentage points per one-unit beta change. The change is \(5.60\% \times 0.10=0.56\) percentage points. The new estimate is \(10.20\%+0.56\%=10.76\%\).
(c) Market risk premium. The sensitivity is beta. The change is \(1.25 \times 0.40=0.50\) percentage points. The new estimate is \(10.20\%+0.50\%=10.70\%\).
The beta change has the largest effect: 0.56 percentage points, compared with 0.50 and 0.20 percentage points. Check: each revised estimate equals the original 10.20% plus its calculated change. The common mistake avoided here is comparing the raw input changes without multiplying by the relevant sensitivity.
Risk-Free Rate: Specify What Is Fixed
The risk-free rate deserves particular care because MRP is defined using that same rate. The phrase “hold the other inputs fixed” must be interpreted precisely. If the inputs are \(r_f\), beta, and MRP, holding MRP fixed means the expected market return moves along with \(r_f\). If the expected market return is held fixed instead, MRP must change in the opposite direction by the same amount.
Worked Example: Changing the Risk-Free Rate While Holding the Market Return Fixed
A hypothetical asset has beta 1.40. Initially, the risk-free rate is 2.50% and the expected market return is 7.50%. The risk-free rate then rises to 3.00%, while the expected market return remains 7.50%. Calculate the initial and revised CAPM estimates, and explain the direction of change.
Method. Calculate MRP as expected market return minus the risk-free rate at each point. Since the market return is fixed, do not hold MRP fixed.
Initially, \(\mathrm{MRP}=7.50\%-2.50\%=5.00\%\). The initial required return is \(2.50\%+1.40 \times 5.00\%=2.50\%+7.00\%=9.50\%\).
After the risk-free rate rises, \(\mathrm{MRP}=7.50\%-3.00\%=4.50\%\). The revised required return is \(3.00\%+1.40 \times 4.50\%=3.00\%+6.30\%=9.30\%\). The estimate falls by \(9.30\%-9.50\%=-0.20\) percentage points.
The same result follows from the fixed-market-return sensitivity: \((1-\beta)\Delta r_f=(1-1.40)\times0.50=-0.20\) percentage points. Because beta exceeds 1, the lower risk premium more than offsets the rise in the risk-free rate. Check: both calculations give 9.30%. The common mistake avoided is increasing \(r_f\) while incorrectly leaving MRP at 5.00%, despite the stated fixed expected market return.
Several Inputs Change at Once
If multiple CAPM inputs change, their effects can interact. For initial inputs \(r_f,\beta,\mathrm{MRP}\), and changes \(\Delta r_f,\Delta\beta,\Delta\mathrm{MRP}\), direct expansion gives:
The last term is the interaction between the beta and MRP changes. Omitting it gives an approximation based on the starting values, not generally the exact total change. One useful exact comparison method splits that interaction evenly between the beta and MRP contributions. It calculates each contribution using the average of that input’s initial and final counterpart. The contributions then add exactly to the total change.
Worked Example: Finding the Dominant Contribution to a Joint Change
A hypothetical asset’s inputs change from \(r_f=2.80\%\), \(\beta=0.85\), and MRP \(=6.00\%\) to \(r_f=3.10\%\), \(\beta=1.05\), and MRP \(=5.60\%\). Calculate the old and new required returns, then compare the risk-free-rate, beta, and MRP contributions using an exact symmetric allocation.
Old estimate. The beta-weighted premium is \(0.85 \times 6.00\%=5.10\%\). Therefore \(r_{\mathrm{old}}=2.80\%+5.10\%=7.90\%\).
New estimate. The beta-weighted premium is \(1.05 \times 5.60\%=5.88\%\). Therefore \(r_{\mathrm{new}}=3.10\%+5.88\%=8.98\%\). The total change is \(8.98\%-7.90\%=1.08\) percentage points.
Symmetric contributions. The risk-free-rate contribution is \(3.10\%-2.80\%=+0.30\) percentage points. The beta contribution uses the average MRP: \((1.05-0.85)\times(6.00\%+5.60\%)/2=0.20\times5.80\%=+1.16\) percentage points. The MRP contribution uses the average beta: \((5.60\%-6.00\%)\times(0.85+1.05)/2=(-0.40\%)\times0.95=-0.38\) percentage points.
The contributions sum to \(0.30+1.16-0.38=1.08\) percentage points, matching the direct estimate change. The beta change is the largest contribution in magnitude. Check: the exact expansion gives \(0.30+6.00\%\times0.20+0.85\times(-0.40\%)+0.20\times(-0.40\%)=0.30+1.20-0.34-0.08=1.08\) percentage points. The symmetric allocation includes the interaction exactly, splitting its \(-0.08\) percentage-point effect equally between the beta and MRP contributions.
Common Traps and Practical Interpretation
A sensitivity is not a universal ranking of inputs. The beta sensitivity is MRP, and the MRP sensitivity is beta; the effects depend on the asset’s current inputs. To identify the dominant source of change, compare the impacts of the specified changes, not merely the sensitivity coefficients. A coefficient has to be multiplied by a relevant input change before it becomes a contribution measured in percentage points.
- Mixing fixed-input assumptions. The risk-free-rate effect is \(+1\) times its change if MRP is fixed, but \((1-\beta)\) times its change if the expected market return is fixed.
- Confusing percent with percentage points. A rise from 5.00% to 5.40% is an increase of 0.40 percentage points, not 0.40% of the starting rate.
- Ignoring units for beta. Beta is a multiplier, not a rate. A beta change of 0.10 is multiplied by MRP expressed as a rate.
- Adding separate effects when several inputs change. The beta and MRP changes include an interaction. For a precise result, recalculate the estimate or include the interaction term.
For example, with beta 1.25 and a 0.20 percentage-point increase in the risk-free rate, using \((1-\beta)\Delta r_f\) would give \((1-1.25)\times0.20=-0.05\) percentage points. That is wrong if MRP is held fixed: the correct change is \(+0.20\) percentage points. The \(-0.05\) result applies only if the expected market return, rather than MRP, is held fixed.
Check Your Understanding
For each problem, state the relevant fixed-input assumption and calculate the requested effect.
- An asset has \(r_f=4.00\%\), beta \(=0.80\), and MRP \(=6.50\%\). Calculate its required return, then find the new estimate if MRP rises by 0.30 percentage points with the other inputs fixed. (Answer: initial 9.20%; revised 9.44%, an increase of 0.24 percentage points)
- An asset has beta \(=1.30\), \(r_f=3.40\%\), and expected market return \(=8.40\%\). The risk-free rate rises by 0.20 percentage points while the expected market return stays fixed. Find the old and new required returns and the change. (Answer: old 9.90%; new 9.84%; change \(-0.06\) percentage points)
- With MRP fixed at 4.80%, an asset’s beta rises by 0.12. Separately, with beta fixed at 1.20, MRP rises by 0.35 percentage points. Compare the two required-return impacts. (Answer: beta change \(+0.576\) percentage points; MRP change \(+0.420\) percentage points; beta change has the larger impact by 0.156 percentage points)
- Inputs change from \(r_f=2.60\%\), beta \(=0.90\), and MRP \(=5.20\%\) to \(r_f=2.90\%\), beta \(=1.00\), and MRP \(=4.70\%\). Use symmetric contributions to compare the risk-free, beta, and MRP effects. (Answer: total change \(+0.32\) percentage points; contributions are \(+0.30\), \(+0.495\), and \(-0.475\) percentage points, respectively; beta is the largest contribution in magnitude)
- An asset has beta \(=0.70\), \(r_f=2.80\%\), and expected market return \(=7.80\%\). If the risk-free rate rises to 3.10% while the expected market return stays fixed, calculate both estimates and explain the sensitivity sign. (Answer: old 6.30%; new 6.39%; change \(+0.09\) percentage points, because \((1-0.70)\times0.30=+0.09\))