Illustration of Classic Covariance.

The red and green “clover” pattern illustrates how traditional risk can be modeled. The red “leaves” are triggered when both the portfolio and the “other asset” move together in concert. The green leaves are triggered when the portfolio and asset move in opposite directions.

Each event represents a moment in time, say the closing price for each asset (the portfolio or the new asset). A common time period is 3-years of total-return data [37 months of price and dividend data reduced to 36 monthly returns.]

### Plain English

When a portfolio manager considers adding a new asset to an existing portfolio, she may wish to see how that asset’s returns would have interacted with the rest of the portfolio. Would this new asset have made the portfolio more or less volatile? Risk can be measured by looking at the time-series return data. Each time the asset and the portfolio are in the red, risk is added. Each time they are in the green, risk is subtracted. When all the reds and greens are summed up there is a “mathy” term for this sum: *covariance*. “Variance” as in change, and “co” as in together. Covariance means the degree to which two items move together.

If there are mostly red events, the two assets move together most of the time. Another way of saying this is that the assets are highly *correlated*. Again, that is “co” as in together and “related” as in relationship between their movements. If, however, the portfolio and asset move in opposite directions most of the time, the green areas, then the covariance is lower, and can even be negative.

### Covariance Details

It is not only the whether the two assets move together or apart; it is also the degree to which they move. Larger movements in the red region result in larger covariance than smaller movements. Similarly, larger movements in the green region reduce covariance. In fact it is the product of movements that affects how much the sum of covariance is moved up and down. Notice how the clover-leaf leaves move to the center, (0,0) if either the asset or the portfolio doesn’t move at all. This is because the product of zero times anything must be zero.

**Getting Technical**: The clover-leaf pattern relates to the angle between each pair of asset movements. It does not show the affect of the magnitude of their positions.

If the incremental covariance of the asset to the portfolio is less than the variance of the portfolio, a portfolio that adds the asset would have had lower overall variance (historically). Since there is a tenancy (but no guarantee!) for asset’s correlations to remain somewhat similar over time, the portfolio manager might use the covariance analysis to decide whether or not to add the new asset to the portfolio.

### Semi-Variance: Another Way to Measure Risk

Semi-variance Visualization

After staring at the covariance visualization, something may strike you as odd — The fact that when the portfolio and the asset move UP together this increases the variance. Since variance is used as a measure of risk, that’s like saying *the risk of positive returns*.

Most ordinary investors would not consider the two assets going up together to be a bad thing. In general they would consider this to be a good thing.

So why do many (most?) risk measures use a risk model that resembles the red and green cloverleaf? Two reasons: 1) It makes the math easier, 2) history and inertia. Many (most?) textbooks today still define risk in terms of variance, or its related cousin standard deviation.

There is an alternative risk measure: semi-variance. The multi-colored cloverleaf, which I will call the yellow-grey cloverleaf, is a visualization of how semi-variance is computed. The grey leaf indicates that events that occur in that quadrant are ignored (multiplied by zero). So far this is where most academics agree on how to measure semi-variance.

### Variants on the Semi-Variance Theme

However differences exist on how to weight the other three clover leaves. It is well-known that for measuring covariance each leaf is weighted equally, with a weight of 1. When it comes to quantifying semi-covariance, methods and opinions differ. Some favor a (0, 0.5, 0.5, 1) weighting scheme where the order is weights for quadrants 1, 2, 3, and 4 respectively. [As a decoder ring Q1 = grey leaf, Q2 = green leaf, Q3 = red leaf, Q4 = yellow leaf].

Personally, I favor weights (0, 3, 2, -1) for the asset versus portfolio semi-covariance calculation. For asset vs asset semi-covariance matrices, I favor a (0, 1, 2, 1) weighting. Notice that in both cases my weighting scheme results in an average weight per quadrant of 1.0, just like for regular covariance calculations.

### Financial Industry Moving toward Semi-Variance (Gradually)

Semi-variance more closely resembles how ordinary investors view risk. Moreover it also mirrors a concept economists call “utility.” In general, losing $10,000 is more painful than gaining $10,000 is pleasurable. Additionally, losing $10,000 is more likely to adversely affect a person’s lifestyle than gaining $10,000 is to help improve it. This is the concept of utility in a nutshell: losses and gains have an asymmetrical impact on investors. Losses have a bigger impact than gains of the same size.

Semi-variance optimization software is generally much more expensive than variance-based (MVO mean-variance optimization) software. This creates an environment where larger investment companies are better equipped to afford and use semi-variance optimization for their investment portfolios. This too is gradually changing as more competition enters the semi-variance optimization space. My guestimate is that currently about 20% of professionally-managed U.S. portfolios (as measured by total assets under management, AUM) are using some form of semi-variance in their risk management process. I predict that that percentage will exceed 50% by 2018.