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Work out a required return with the capital asset pricing model

CFA® Level I · Portfolio Management · 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

Beta scales the reward the market offers for bearing systematic risk to the amount this asset carries.

A harder one

The risk-free rate is 3%, the market risk premium 5% and beta 1.4. The required return is:

  1. 10.0% — correct
  2. 8.0%Why not: The answer for a beta of 1.0: 3% plus the whole premium. Beta of 1.4 scales the premium to 7%.
  3. 7.0%Why not: The risk premium alone, 1.4 times 5%, with the risk-free rate left off.

Three plus 1.4 multiplied by five gives 3 plus 7, or 10%.

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

Ignoring specific risk is only defensible for someone who has already diversified it away.

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