Portfolio variance, covariance and correlation

These formulas are the portfolio mathematics you met in the quantitative methods topic, now put to work. Here the inputs come from historical returns, and the question is why a mix of assets is less risky than the average of its parts.

How do you get the inputs from historical returns?

Use it

Four statistics describe a pair of assets: each one’s mean and standard deviation, then how they move together. For a sample of T periods the mean is the simple average, and the variance, covariance and standard deviation divide by T − 1.

$$\bar{R} = \frac{\sum_{t=1}^{T} R_t}{T}$$

$$s^{2} = \frac{\sum_{t=1}^{T} (R_t - \bar{R})^{2}}{T - 1}$$

Sample variance measures spread around the mean in squared units; its square root is the standard deviation. Covariance measures whether two returns tend to be on the same side of their own means at the same time.

$$\mathrm{Cov}_{1,2} = \frac{\sum_{t=1}^{T} (R_{1,t} - \bar{R}_1)(R_{2,t} - \bar{R}_2)}{T - 1}$$

Covariance has awkward units (percent squared) and no natural scale. Dividing by both standard deviations gives the correlation coefficient , which is always between −1 and +1.

$$\rho_{1,2} = \frac{\mathrm{Cov}_{1,2}}{\sigma_1\,\sigma_2}$$

$$\mathrm{Cov}_{1,2} = \rho_{1,2}\,\sigma_1\,\sigma_2$$

Four years of returns for two funds:

Year	Fund X	Fund Y
1	12%	5%
2	−4%	3%
3	18%	1%
4	6%	7%

Go deeper Population versus sample formulas

If the returns are the whole population, divide by T instead of T − 1. Historical returns are a sample of what could have happened, so the exam uses T − 1 unless a question says otherwise. The portfolio formulas themselves do not change: they take σ and covariance as given.

Go deeper Using the statistics worksheets

Both exam calculators can report the standard deviations and the correlation from the data directly. On the BA II Plus, enter the pairs in the data worksheet, switch the statistics worksheet to the two-variable type and step down to read the sample standard deviations and r. On the HP 12C, enter each pair as y, ENTER, x, then Σ+, and read the standard deviations and r from the statistics keys. The covariance is then r × s_x × s_y. Check the display against the hand calculation above the first time you do it.

Fund X and Fund Y on your calculator

TI BA II Plus HP 12C

1. 2NDDATA2NDCLR WORK open and clear the data worksheet
2. 12ENTER↓5ENTER↓4+|-ENTER↓3ENTER↓18ENTER↓1ENTER↓6ENTER↓7ENTER X then Y for each year
3. 2NDSTAT2NDSET until LIN shows
4. ↓ keep pressing [↓] to read Sx 9.3808, Sy 2.5820 and r −0.2752

2ND second function (the shifted label above a key) · DATA data worksheet · CLR WORK clear the worksheet · ENTER enter (on the HP, puts the number on the stack) · +|- change sign · STAT statistics worksheet · SET change the setting

1. fCLEAR Σ
2. 5ENTER12Σ+3ENTER4CHSΣ+1ENTER18Σ+7ENTER6Σ+ y first, then x
3. gs sx = 9.3808, then [x≷y] for sy = 2.5820
4. 0gŷ,rx≷y r = −0.2752

f gold shift key · CLEAR Σ clear the statistics registers · ENTER enter (on the HP, puts the number on the stack) · Σ+ add a data point · CHS change sign · g blue shift key · s standard deviation · ŷ,r estimated y and correlation · x≷y swap the last two numbers

How risky is a two-asset portfolio?

Memorize

The portfolio’s expected return is the weighted average of its assets’ expected returns. Correlation plays no part in it.

$$E(R_p) = w_1\,E(R_1) + w_2\,E(R_2)$$

Risk is different. Portfolio variance has three terms: the variance of each asset scaled by its squared weight, plus twice the weights times the covariance. Written with correlation, the covariance is ρσ₁σ₂. The portfolio standard deviation is the square root.

$$\sigma_p^{2} = w_1^{2}\sigma_1^{2} + w_2^{2}\sigma_2^{2} + 2\,w_1 w_2\,\rho_{1,2}\,\sigma_1\sigma_2$$

Change only the correlation and everything else follows. The weights fix the return at 8.6%; the risk changes.

ρ	σp	Working
+1.0	16.00%	equals 0.6 × 20 + 0.4 × 10
+0.5	14.42%	√(0.0144 + 0.0016 + 0.0048) = √0.0208
0.0	12.65%	√(0.0144 + 0.0016) = √0.016
−0.5	10.58%	√(0.0144 + 0.0016 − 0.0048) = √0.0112
−1.0	8.00%	|0.6 × 20 − 0.4 × 10|

Slide the correlation from +1 down to −1. The dashed straight line is the +1 case, and the gap to its left is the diversification benefit. The weight slider moves the dot along the curve; its return follows the weights only.

Correlation between two assetsreturn is unchanged; only risk moves

• ρ = +1 Memorize σp = weighted average of the σs; no diversification
• −1 < ρ < +1 Memorize σp below the weighted average; the lower ρ, the lower σp
• ρ = 0 Use it no linear relation; still a diversification benefit
• ρ = −1 Memorize a zero-risk mix exists: w₁ = σ₂ ÷ (σ₁ + σ₂)

Three points to carry away.

• At ρ = +1 the portfolio’s σ is exactly the weighted average. No diversification is possible: two assets that rise and fall together cannot offset each other.
• Anywhere below +1 the portfolio’s σ is below that average, and the lower the correlation, the lower the risk. Diversification is this reduction in risk from combining assets that do not move in lockstep.
• At ρ = −1 a mix exists with no risk at all. Set the two weighted σs equal: the weight in asset 1 is σ₂ ÷ (σ₁ + σ₂).

$$w_1 = \frac{\sigma_2}{\sigma_1 + \sigma_2}$$

In the example this is 10 ÷ (20 + 10) = 33.3% in the equity fund: 0.333 × 20 = 6.67 and 0.667 × 10 = 6.67 cancel exactly.

The intuition fits in a small table. Fuel is one of the biggest costs of an airline and the main product of an oil producer. Next year the oil price either spikes or slumps, with equal chances.

	Oil spikes	Oil slumps	Mean	σ
Airline	3%	15%	9%	6%
Second airline	3%	15%	9%	6%
Oil producer	15%	3%	9%	6%

Half and half in the two airlines earns 3% or 15% as before: ρ = +1, no gain. Half and half in an airline and the oil producer earns exactly 9% whatever the oil price does: ρ = −1, no risk. To reduce risk you need assets that do well when your others do badly.

Exam tip Most real assets have correlations well above −1. A correlation above about 0.90 offers little diversification; a correlation below about 0.50 is attractive. Stocks and bonds have generally been the least correlated of the major pairs, and Treasury bills are close to uncorrelated with stocks.

Go deeper Why correlations rise when you need them most

Correlations estimated from history are averages. In market crises many risky assets fall together, so correlations rise and the diversification benefit shrinks just when it is most wanted. Within one country, correlations among asset classes have been fairly stable. Across countries they have been rising over recent decades with globalization. Risk (standard deviation) is more stable over time than return, which is why historical risk and correlation are used as estimates for the future more readily than historical returns are.

What happens when a portfolio has many assets?

Use it

Take an equally weighted portfolio of N assets and let σ̄² be the average variance and average covariance be the mean of all the pairwise covariances.

$$\sigma_p^{2} = \frac{\bar{\sigma}^{2}}{N} + \frac{N-1}{N}\,\overline{\mathrm{Cov}}$$

The first term shrinks as N grows: each asset’s own variance gets divided across more holdings. The second term approaches the average covariance. So for a large portfolio, covariance among the assets accounts for almost all of its risk. If all assets share the same σ and the same pairwise correlation ρ, the average covariance is ρσ² and the floor for the portfolio’s σ is √(ρσ²).

Variance of an equally weighted portfolioσ̄²/N + (N − 1)/N × average covariance

• Own variances Use it divided by N; fade away as N grows
• Average covariance Memorize stays; the floor that diversification cannot remove

The risk that remains is shared by all the assets. Adding more holdings wipes out each one’s own variance but cannot touch the covariance they have in common. The next module calls that remaining part systematic risk. If the assets were on average uncorrelated, the floor would be zero. Real assets are not.

Go deeper The covariance matrix view for three or more assets

With N assets the exact portfolio variance sums every weight pair times its covariance. The covariance of an asset with itself is its variance, so the N variances sit on the diagonal of a matrix and the N(N − 1)/2 distinct covariances sit above and below it, each counted twice. The number of covariances grows far faster than the number of assets, which is why the covariance terms dominate in large portfolios.

$$\sigma_p^{2} = \sum_{i=1}^{N}\sum_{j=1}^{N} w_i\,w_j\,\mathrm{Cov}_{i,j}$$