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Work out the expected return on a portfolio of two assets

CFA® Level I · Portfolio Management · Portfolio Risk and Return: Part I · PM.1.a

The idea

The expected return of a two-asset portfolio is the weighted average of the component expected returns: E(Rp) = w1 x E(R1) + w2 x E(R2), with the weights summing to one. Risk does not work this way. Portfolio variance depends on the two variances, the weights, and the covariance between the assets, so portfolio standard deviation is generally less than the weighted average of the individual standard deviations. That gap is the diversification benefit, and it is the reason risk must be assessed at portfolio level rather than asset by asset.

Formulas

Expected return of a two-asset portfolio

E(Rp) = w₁ × E(R₁) + w₂ × E(R₂)

w₁, w₂
the share of the portfolio's value in each asset, adding to one
E(R₁), E(R₂)
the expected return of each asset
E(Rp)
the expected return of the portfolio

Return is a weighted average, and only return. Risk is not, which is the whole of diversification: the return of the mix is the average of the parts while its standard deviation is less.

A practice question

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

The expected return of a portfolio is:

  1. the weighted average of the component expected returns — correct
  2. the sum of the component expected returnsWhy not: Adding unweighted returns would give a figure above the best asset. The weights, which sum to one, are what keep the result inside the range.
  3. the return of the highest-returning componentWhy not: That would be true only if everything were invested in that one asset. Any weight elsewhere pulls the average down.

Expected return combines linearly through the weights. Risk, notably, does not.

A harder one

A portfolio holds 60% in an asset expected to return 10% and 40% in one expected to return 5%. Expected portfolio return is:

  1. 7.5%Why not: The simple average of the two returns, which would be right only at equal weights. Here the 10% asset carries 60%.
  2. 8.0% — correct
  3. 6.0%Why not: Below the simple average of 7.5%, which cannot be right when the majority of the money sits in the higher-returning asset.

(0.60 × 10%) + (0.40 × 5%) = 8.0%. The 7.5% figure is the unweighted average, which ignores how much sits in each asset.

The hardest tier

Portfolio standard deviation is generally:

  1. equal to that weighted averageWhy not: That holds only in the limiting case of perfect positive correlation, where no offsetting occurs at all.
  2. greater than that weighted averageWhy not: Combining assets cannot make the result more volatile than the weighted average of the parts; the covariance term cannot exceed its perfect-correlation value.
  3. less than the weighted average of the component standard deviations — correct

Unless the assets are perfectly positively correlated, offsetting movements reduce portfolio volatility. That shortfall is the diversification benefit.

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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