Covariance, Correlation, and Beta

Why this matters

The last lesson ended with a puzzle. A portfolio of 60% shares and 40% bonds had a standard deviation of 12.1%, where the weighted average of the two would be 14.8%. Correlation, Plainly explained the idea in words: holdings that don't have their bad years together steady each other. This lesson gives the arithmetic, because the same formula explains why 20 shares are safer than one, why 200 aren't much safer than 20, and what beta is.

Covariance

Covariance measures whether two investments tend to be above their averages at the same time. For each year, multiply one investment's distance from its average by the other's. Average the results.

Year	A's return	B's return	A's distance from 7.5%	B's distance from 5%	Product
1	20%	12%	+12.5	+7	87.5
2	−10%	−6%	−17.5	−11	192.5
3	15%	2%	+7.5	−3	−22.5
4	5%	12%	−2.5	+7	−17.5
Total					240

In years 1 and 2 the two were on the same side of their averages, which gives positive products. In years 3 and 4 they were on opposite sides, which gives negative ones. Dividing the total by one fewer than the number of years, as with a standard deviation, gives a covariance of 240 ÷ 3 = 80.

A covariance of 80 is hard to interpret, because its size depends on how volatile the two investments are. Dividing by both standard deviations fixes that.

Correlation = Covariance ÷ (Standard deviation of A × Standard deviation of B)

A's standard deviation is 13.2% and B's is 8.7%, so the correlation is 80 ÷ (13.2 × 8.7) = 0.69. Correlation always falls between −1 and +1, which makes it comparable from one pair of investments to another.

The risk of two investments together

For a portfolio of two investments, with weights that add to 1:

Portfolio variance = (Weight A)² × Variance A + (Weight B)² × Variance B + 2 × Weight A × Weight B × Covariance

The first two terms are each investment's own risk. The third is the risk they share. If the covariance is low, the third term is small, and the portfolio is steadier than its parts.

Shares: 0.60² × 0.19404²	0.013554
Bonds: 0.40² × 0.07900²	0.000998
Shared: 2 × 0.60 × 0.40 × 0.00028	0.000134
Portfolio variance	0.014687
Portfolio standard deviation	12.1%

The standard deviations and the covariance are calculated from Aswath Damodaran's yearly returns for the S&P 500 and 10-year Treasury bonds.¹ Returns are written as decimals, so 19.404% is 0.19404. The covariance of 0.00028 corresponds to a correlation of 0.02: over the whole period, shares and bonds had almost no relationship, and the shared term is close to nothing.

The correlation does all the work. Here's the same 60/40 portfolio, with the same two standard deviations, at different correlations.

Only at a correlation of exactly +1 is the portfolio's risk the weighted average of its parts. At anything less, the portfolio is steadier, and the expected return hasn't changed at all. That's the sense in which diversification is said to cost nothing: it lowers risk without lowering expected return. It's also why the correlation matters so much, and Correlation, Plainly showed that the one between shares and bonds has ranged from 0.33 to −0.65 depending on the decades measured.

The rest of this lesson is for members

Still to come in this lesson: Many investments: the floor under diversification, Beta, Three cautions, the key takeaways, and 2 quiz questions with explanations.

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