Diversification & Portfolio Risk

In the last post we worked out the return on any portfolio: weight each asset’s return by its share of equity and add them up. We might expect risk to work the same way, taking the weighted average of the individual standard deviations to get the portfolio’s.

That guess is wrong. A portfolio’s standard deviation is almost always less than the weighted average of its parts. Two assets that move in opposite directions can cancel some of each other’s volatility, and what is left is smoother than either one alone. The key ingredient is correlation.

This post works out where that cancellation comes from, how far we can push it, and what it cannot remove.

Covariance and Correlation

To get at portfolio risk we need to measure how two assets move relative to each other, and covariance is that measure. It is the average product of the two assets’ deviations from their own means, taken across all $N$ observations in the sample:

Cov (r_{A}, r_{B}) = \frac{1}{N} t = 1 \sum N (r_{A, t} - \overset{r}{ˉ}_{A}) (r_{B, t} - \overset{r}{ˉ}_{B})

where $r_{A, t}$ is the return on asset A in period $t$ and $\overset{r}{ˉ}_{A}$ is its sample mean. Setting B equal to A recovers the variance formula from the last post.

When both assets are above their means in the same periods, the products are positive and the covariance is positive. When one is above its mean in the periods when the other is below, the products are negative and the covariance is negative. When no consistent pattern exists, the positives and negatives cancel and the covariance is near zero.

Covariance is hard to interpret because its units are “percent squared.” A covariance of 48 between two stocks means little on its own. Correlation fixes that by dividing out both standard deviations:

ρ_{A B} = \frac{Cov ( r _{A} , r _{B} )}{σ _{A} \times σ _{B}}

where $σ_{A}$ and $σ_{B}$ are the standard deviations of the two assets.

Correlation $ρ$ ranges from -1 to +1. At +1, the two assets move in perfect lockstep: every time one rises 2%, the other rises by a proportional amount. At -1, they move in perfect opposition. At 0, no linear relationship exists between them.

A few examples make the sign concrete. Two airline stocks have high positive correlation (both get hurt when fuel prices spike, both benefit from strong travel demand). An airline and an oil producer have negative correlation (expensive fuel hurts airlines but helps drillers). An airline and a pharmaceutical company have low correlation (their fortunes are driven by unrelated forces). The sign and magnitude of $ρ$ tell us how much diversification benefit to expect from holding both.

Two-Asset Portfolio Variance

For two risky assets A and B held at weights $w_{A}$ and $w_{B}$ , the portfolio variance is:

σ_{p}^{2} = (w_{A} σ_{A})^{2} + (w_{B} σ_{B})^{2} + 2 w_{A} w_{B} ρ_{A B} σ_{A} σ_{B}

where $σ_{p}$ is the standard deviation of the portfolio.

The first two terms are the “own-risk” contributions: each asset’s variance, scaled by the square of its weight. Those two are what we would get by adding up the weighted variances independently. The third term is the cross-term, and it is our new ingredient, because it is the only place $ρ$ appears.

How much the cross-term helps depends entirely on the sign and size of $ρ$ , so take three cases.

When $ρ = 1$ , the cross-term is at its maximum and $σ_{p}$ simplifies to the weighted average of the individual standard deviations. No diversification benefit at all, and the portfolio is as risky as the naive weighted average predicts.

When $ρ < 1$ , the cross-term shrinks. The portfolio standard deviation falls below the weighted average. The lower the correlation, the bigger the gap.

When $ρ < 0$ , the cross-term actively subtracts from portfolio variance. The assets are partially offsetting each other’s moves.

Take Asset X with $σ = 20%$ and Asset Y with $σ = 30%$ , held at 60% in X and 40% in Y. We run the formula at three correlations.

With $ρ = 1$ (perfect correlation):

σ_{p}^{2} = (0.60 \times 20)^{2} + (0.40 \times 30)^{2} + 2 (0.60) (0.40) (1) (20) (30) = 144 + 144 + 288 = 576

σ_{p} = 24.0%

That is the weighted average we would have guessed, $0.60 \times 20 + 0.40 \times 30 = 24.0%$ . No diversification at all.

With $ρ = 0.2$ :

σ_{p}^{2} = 144 + 144 + 2 (0.60) (0.40) (0.2) (20) (30) = 144 + 144 + 57.6 = 345.6

σ_{p} = 18.6%

That result is below either asset’s individual $σ$ . Our 60/40 mix of a 20% and a 30% volatility asset has only 18.6% volatility, and correlation alone produced that.

With $ρ = - 0.5$ :

σ_{p}^{2} = 144 + 144 + 2 (0.60) (0.40) (- 0.5) (20) (30) = 144 + 144 - 144 = 144

σ_{p} = 12.0%

The cross-term now subtracts from variance instead of adding to it, cancelling one of the two own-risk terms outright. A 12.0% portfolio out of a 20% asset and a 30% asset.

Drag the $ρ$ slider below to redraw portfolio $σ$ across every weighting of the two assets.

ASSET PROPERTIES σ(A): 20.0% σ(B): 30.0% ρ: +0.50 w(A): 60% 0% 5% 10% 15% 20% 25% 30% 0% 20% 40% 60% 80% 100% Weight in Asset A Portfolio σ PORTFOLIO σ 20.8% WEIGHTED AVG σ 24.0% RISK REDUCTION 3.2 pp Portfolio σ Weighted avg (ρ=1) Drag ρ to see how correlation affects the risk curve.

At $ρ = + 1$ , the curve is a straight line connecting $σ_{A}$ on the right to $σ_{B}$ on the left, the weighted average at every mix. As $ρ$ falls, the curve bows downward, opening a region where the portfolio is less volatile than either asset alone. At $ρ = - 1$ , the curve touches zero. Two risky assets, perfectly negatively correlated, combine into a riskless portfolio.

Diversification

Adding more stocks to an equally weighted portfolio reduces $σ$ , and the same variance arithmetic, extended from two stocks to $N$ , tells us by how much. The full expression collapses to something readable if we assume every stock has the same individual $σ$ and the same pairwise correlation $ρ$ with every other stock:

σ_{p} = σ \frac{1}{N} + \frac{N - 1}{N} ρ

where $N$ is now the number of stocks held rather than the number of observations.

With $σ = 30%$ and $ρ = 0.3$ :

Stocks ( $N$ )	Portfolio $σ_{p}$
1	30.0%
10	18.2%
30	17.1%
50	16.8%

The first 10 to 20 stocks do most of the work, and after 40 or 50 the curve is nearly flat.

Where does the curve stop falling? Letting $N \to \infty$ drives $1/ N$ to zero and $(N - 1) / N$ to one, which leaves us with the correlation term alone:

σ_{p} \to σ ρ = 30% \times 0.3 = 16.4%

That is the floor. No matter how many stocks we add, portfolio risk converges to $σ ρ$ .

Drag the correlation slider below to raise and lower that floor.

Stock σ: 30.0% Avg ρ: +0.30 0% 5% 10% 15% 20% 25% 30% 1102030405060708090100 Number of Stocks Portfolio σ Idiosyncratic risk Systematic risk Floor (σ\sqrtρ = 16.4%) 1 STOCK 30.0% 10 STOCKS 18.2% 30 STOCKS 17.1% FLOOR (N\to\infty) 16.4% Higher ρ raises the systematic floor. At ρ = 0, all risk is diversifiable.

Higher $ρ$ raises the floor, leaving less risk that can be diversified away. At $ρ = 0$ , all risk is diversifiable and the curve drops toward zero. In practice, stocks are positively correlated because they share exposure to the same economy, so the floor is always above zero in real markets.

Systematic vs. Idiosyncratic Risk

The gap between that curve and its floor has a name, and so does the floor.

The risk that diversifies away is idiosyncratic risk (also called firm-specific or unsystematic risk). A CEO gets fired, a product recall hits the news, a patent lawsuit goes badly. These events affect one company but not the market as a whole, and because they are independent across firms, they average out in a large portfolio. One company’s bad luck is offset by another’s good luck, and the law of large numbers takes care of the rest.

The risk that remains is systematic risk (or market risk). Recessions, interest rate changes, pandemics, and trade wars all belong here. These forces push most stocks in the same direction simultaneously, and that shared exposure is the source of the positive correlation. No amount of diversification removes systematic risk, because every stock in the portfolio feels the same macroeconomic shock.

Look again at the diversification chart above. The shaded region between the curve and the floor is idiosyncratic risk, and it shrinks as we add stocks. The region below the floor is systematic risk, and it stays constant regardless of portfolio size. For one stock, most of the total variance may be idiosyncratic. For a well-diversified portfolio of 50 or more stocks, almost all remaining variance is systematic. Factor models and beta formalize that split later in the series.

The Free Lunch

Diversification is “the only free lunch in finance,” a line attributed to Harry Markowitz. The reason is arithmetic. Recall from the last post that our portfolio return is the weighted average of the individual returns, and the same holds for expectations:

E [r_{p}] = i \sum w_{i} \times E [r_{i}]

No correlation term appears, so expected return does not depend on how the assets co-move. Portfolio risk does depend on correlations, as we’ve seen. So we can lower risk without lowering expected return by spreading money across assets that are not perfectly correlated.

But only idiosyncratic risk can be eliminated. Systematic risk is what an investor must accept to hold equities at all. That limit implies something the CAPM post develops: the market pays no premium for bearing idiosyncratic risk, because any investor could have diversified it away for free. Only systematic risk earns a premium. Two stocks with the same systematic exposure but different amounts of idiosyncratic risk should earn the same expected return, since the extra risk can be diversified away and goes unpriced.

What’s Next

We can now say how correlation drives portfolio risk and why diversification works. If several combinations deliver the same expected return at different levels of risk, we still need a rule for choosing the best one. The next post prices the cost of volatility itself, then builds the Capital Allocation Line and the efficient frontier on top of that price. Both single out the portfolio with the highest Sharpe ratio.