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Work out the expected return on a portfolio of two assets

CFA® Level I · Portfolio Construction · 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 simple average of the component expected returnsWhy not: A plain average ignores how much money sits in each asset. It is correct only when every weight is equal.
  3. 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.

A portfolio is just a basket of investments, and the expected return is the average return you would hope to earn from it. It is not a plain average, because bigger holdings count for more. Formula: Portfolio expected return = (weight A x return A) + (weight B x return B) + ..., where a weight is the share of your money in that asset. Example: 70% in an asset expected to return 5% and 30% in one expected to return 9% gives 0.70 x 5% + 0.30 x 9% = 3.5% + 2.7% = 6.2%. The result always sits between the lowest and highest returns.

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.

We need the average return, giving more importance to the asset that holds more money. Formula: Expected return = (weight A x return A) + (weight B x return B). Working: (0.60 x 10%) + (0.40 x 5%) = 6.0% + 2.0% = 8.0%. The 7.5% option is the simple average of 10% and 5%, which wrongly treats both assets as if they held the same amount. Since more money sits in the 10% asset, the answer must be above 7.5%.

The hardest tier

Two assets each have a standard deviation of 20% and are held equally. If their correlation is +1, portfolio standard deviation is:

  1. 20% - correct
  2. 14.1%Why not: The zero-correlation answer, 20% divided by the square root of two. At perfect correlation there is no diversification to capture.
  3. 0%Why not: Zero volatility requires a correlation of −1, where the two movements cancel exactly.

Standard deviation is a measure of how much returns swing. A correlation of +1 means the two assets move perfectly together, so nothing cancels out. Formula at correlation +1: Portfolio standard deviation = (weight A x SD A) + (weight B x SD B). Working: (0.5 x 20%) + (0.5 x 20%) = 10% + 10% = 20%. This is the one case where risk simply averages, just like return does. The 14.1% answer is for a correlation of 0, and 0% would need a correlation of -1.

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