Realized Semivariance: Measuring Upside and Downside Volatility

For intraday log returns (r_i) observed over one fixed interval, realized variance is the sum of squared returns, (RV=\sum_i r_i^2). Squaring keeps both gains and losses positive and gives larger moves more influence. The statistic measures variation accumulated within the interval; it does not tell us whether that variation came from upward or downward moves.

Realized semivariance retains the sign before squaring. The positive component collects squared returns above zero, and the negative component collects squared returns below zero. Their sum equals realized variance when they use the same observations and convention. A zero return contributes zero to both. The equality is an accounting identity, not evidence that either side is a separate causal source of risk.

This sign split is useful when the research question concerns directional variation—for example, whether past negative-return variation contains different information about future volatility than past positive-return variation. The article by Barndorff-Nielsen, Kinnebrock, and Shephard introduced realized semivariance as a high-frequency measure of downside variation. Their working paper develops its relation to quadratic variation and negative jumps.

For returns (r_1,\ldots,r_n), define

\[ RS^+ = \sum_{i=1}^{n} r_i^2\mathbf{1}(r_i>0), \qquad RS^- = \sum_{i=1}^{n} r_i^2\mathbf{1}(r_i<0), \]

where \(\mathbf{1}(\cdot)\) is one when its condition is true and zero otherwise. Then

\[ RV = \sum_{i=1}^{n}r_i^2 = RS^+ + RS^-. \]

Some papers use different labels or superscripts, so a report should state which term means positive returns and which means negative returns. The formula above uses uncentered returns relative to zero: it does not subtract the sample mean, a target return, or a risk-free rate. A sample semivariance calculated around a target or mean is a different statistic.

The units are squared return units. If returns are in basis points, each component is in basis-points squared. If decimal log returns are used, the result is in decimal-return squared. Changing from percent returns to basis-point returns multiplies the numerical squared measure by 10,000, so the unit must travel with the number.

Suppose a hypothetical session has six returns in basis points:

\[ (2,-1,3,-4,0,-2)\;\text{bp}. \]

The positive returns contribute \(2^2+3^2=13\,\text{bp}^2\). The negative returns contribute \((-1)^2+(-4)^2+(-2)^2=21\,\text{bp}^2\). The zero return contributes nothing. The total realized variance is

\[ RV=13+21=34\,\text{bp}^2. \]

The negative-side share in this tiny example is \(21/34\), about 61.8%, and the positive-side share is about 38.2%. These are shares of this invented sum, not probabilities of a future loss or estimates of a stable long-run asymmetry. Six returns are far too few for inference. The arithmetic is only a check of the definitions.

The realized-volatility guide covers the unsplit sum of squared returns. The bipower-variation guide addresses a different question: estimating continuous variation when infrequent jumps may be present. Directional semivariance and jump-robust variation can be studied together, but one does not substitute for the other.

If the latent log-price follows a continuous semimartingale and sampling becomes increasingly fine under the relevant regularity conditions, positive- and negative-return semivariances each converge to half of the continuous quadratic variation. In that setting, sign-specific finite-sample totals can differ even though the underlying continuous variation has no persistent directional split; sampling randomness matters.

When jumps occur, the squared size of a jump is assigned to the positive or negative component according to its sign. This preserves information that total realized variance discards, but it does not mean every negative return is a jump or that every positive return is benign. A large continuous move and a jump can both produce a large squared return. The return sign classifies the observed increment, not its economic cause.

The conceptual figure below separates positive and negative increments into two visual groups. It is not a chart of data, a test result, or a claim that the underlying process has a persistent directional bias.

<!-- learn:illustration --> <!-- Text-free concept: positive and negative return increments feed separate collections of squared-magnitude tiles, while their combined area echoes total realized variation. No measured data or market direction is implied. -->

Realized semivariances can serve as separate predictors in a model for future volatility. Patton and Sheppard’s study of the S&P 500 and individual stocks found that past negative-return variation was more strongly related to future volatility than positive-return variation in their samples and specifications. That is a reported empirical result for their design, not a universal market law or a guarantee that a new strategy will forecast well.

To make a forecasting claim, specify the forecast target and horizon, compare against a suitable benchmark, and evaluate out of sample. A difference between \(RS^-\) and \(RS^+\) on one day is descriptive; it does not establish predictive power. The regression may also need to account for volatility persistence, overlapping horizons, parameter uncertainty, and data selection. Forecast errors should be assessed with a loss function that matches the decision.

For a broader multivariate extension, Bollerslev, Li, Patton, and Quaedvlieg develop realized semicovariances that decompose realized covariance according to the signs of paired high-frequency returns. Their Econometrica paper shows why same-sign and mixed-sign components can contain different information. A univariate positive/negative split is simpler and should not be presented as a full covariance or portfolio-risk decomposition.

The phrase “downside semivariance” is used in more than one way. The realized measure here squares high-frequency returns that are below zero. A target-based downside statistic instead squares returns below a chosen threshold, often a target or minimum acceptable return. A sample calculation may center observations around a mean. Those definitions answer different questions and can produce different values even from the same return series.

Neither realized semivariance is a Value-at-Risk or Expected Shortfall measure. VaR describes a quantile of a specified loss distribution and horizon; Expected Shortfall summarizes losses beyond a quantile under a stated convention. A sum of negative intraday squared returns has neither of those interpretations. It also is not a complete measure of drawdown, which depends on the order and path of returns.

If a chart or model labels \(RS^-\) “downside risk,” define the term directly beneath it. State whether returns are measured from zero or a target, which sampling frequency is used, and whether the object is an interval statistic, forecast input, or portfolio measure. The GARCH volatility guide discusses conditional variance dynamics, while realized semivariance supplies a directional high-frequency measurement that might be considered as an input.

The sign split depends on the observed return sequence. Sampling more often can add bid–ask bounce, price discreteness, stale quotes, and timestamp effects. A small recorded price change may reverse sign when the price proxy or sampling grid changes. Microstructure noise can therefore alter both squared magnitude and which side receives a return. The equality \(RV=RS^++RS^-\) still holds for that chosen series, but it does not validate the sampling design.

For a robust comparison, document the price field, sampling interval, session boundary, treatment of zero returns, outlier rules, and whether returns are synchronous across assets. Check sensitivity across defensible sampling choices. If jump identification or noise robustness is the central question, use an estimator or test designed for that target rather than interpreting the sign split as a correction. The two-scale realized-volatility guide explains one alternative noise correction.

Statistical inference also needs a sampling distribution or a suitable resampling design. The positive and negative components are formed from the same path and may be dependent. Overlapping windows can add further dependence. A confidence interval or forecast comparison must match those features; comparing two point estimates alone does not test that their underlying predictive roles differ.

A clear report names the asset or portfolio return series, the observation window, the return definition, the sampling grid, and the units. It states the formulas for \(RS^+\) and \(RS^-\), confirms that their sum reproduces realized variance, and explains how zeros and missing observations were handled. When the measure enters a forecasting model, report the forecast horizon, benchmark, evaluation period, and loss function.

Avoid converting a realized semivariance directly into an annualized volatility without stating the aggregation assumptions. Daily realized semivariance can be summed across days as a descriptive annual total under a chosen convention, but its square root is not automatically a downside standard deviation with a known distribution. Annualization does not repair biased price inputs, sparse sampling, or selection effects.

The main use is disciplined measurement: keep the direction of each observed return while retaining the quadratic weighting used in realized variation. The result can reveal a pattern worth testing. It cannot identify the cause by itself, measure tail probability, or turn a historical asymmetry into a trading signal.