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Work out a required return with the capital asset pricing model

CFA® Level I · Portfolio Construction · Portfolio Risk and Return: Part II · PM.4.a

The idea

The capital asset pricing model says the return required on an asset is the risk-free rate plus beta times the market risk premium. Its central claim is that only systematic risk is rewarded: risk specific to one company can be diversified away at no cost, so no investor should be paid for bearing it. Beta measures sensitivity to the market - a beta of one moves with it, above one amplifies it, below one damps it. The model's assumptions are heroic and its empirical record is mixed, but the discipline it imposes survives the criticism: the question it forces is always whether a risk is one the market will pay you to hold.

Formulas

Required return from the capital asset pricing model

required return = rf + β × (market return − rf)

rf
the risk-free rate, usually a long government bond
β
the asset's sensitivity to the market
market return
the expected return on the market portfolio

The return an asset must offer for the systematic risk it carries. Plotted against beta this is the security market line, and it prices only risk that cannot be diversified away - an asset with a beta of zero is required to return no more than the risk-free rate however volatile it is.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the capital asset pricing model, required return equals the risk-free rate plus beta multiplied by the:

  1. inflation rateWhy not: Inflation is already embedded in the nominal risk-free rate. Beta scales the compensation for market risk, not for price changes.
  2. dividend yieldWhy not: A feature of how a company distributes cash, unrelated to the systematic risk being priced.
  3. market risk premium - correct

The capital asset pricing model gives the return an investor should demand from an asset. It starts with the risk-free rate, the return on a safe investment, and adds a reward for risk. Formula: Required return = Risk-free rate + Beta x Market risk premium. Beta says how strongly the asset moves with the market. The market risk premium is the extra return the market pays over the risk-free rate for taking market risk. Beta scales that reward up or down to match this asset. Inflation and dividend yield are not part of the formula.

A harder one

The risk-free rate is 2%, the expected market return is 9% and a stock's beta is 0.8. What is its required return?

  1. 9.00%Why not: The market return itself, which would be the answer only for a beta of exactly 1.0.
  2. 9.20%Why not: Applies beta to the whole 9% market return and then adds the risk-free rate on top. Beta scales the 7% premium.
  3. 7.60% - correct

Formula: Required return = Risk-free rate + Beta x (Market return - Risk-free rate). The bracket is the market risk premium, the extra return the market pays over a safe investment. Step 1: premium = 9% - 2% = 7%. Step 2: required return = 2% + 0.8 x 7% = 2% + 5.6% = 7.6%. Beta of 0.8 means the stock takes 80% of the market's risk, so it earns 80% of the premium. Multiplying beta by the whole 9% and adding 2% gives 9.2%, a common slip.

The hardest tier

Using the capital asset pricing model to set a required return for a project implicitly assumes the investor is:

  1. undiversifiedWhy not: The reverse. If they were undiversified, specific risk would matter to them and beta alone would understate what they need.
  2. risk neutralWhy not: A risk-neutral investor would require the risk-free rate on everything, which is what beta exists to adjust.
  3. holding a well-diversified portfolio - correct

The model says required return depends only on beta, the risk that moves with the whole market. Formula: Required return = Risk-free rate + Beta x Market risk premium. This leaves out company-specific risk, which is the risk of one business. Leaving it out only makes sense if the investor has already got rid of that risk by holding many different investments, in other words a well-diversified portfolio. An undiversified investor would still be hurt by specific risk, so beta alone would understate the return they need.

In the app: 18 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.

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