Upside Potential Ratio
An asymmetric performance measure that evaluates the opportunity for gains relative to the risk of losses, using Upper Partial Moments to capture upside potential and Lower Partial Moments to quantify downside risk.
Overview
The Upside Potential Ratio (UPR) was introduced by Sortino, Van der Meer, and Plantinga as an extension of the downside risk framework. It addresses a subtle but important gap in the Sortino Ratio: while the Sortino Ratio uses the full average excess return in the numerator (which includes compensation for both upside and downside deviations from the mean), the UPR isolates only the upside component.
The UPR directly measures the ratio of upside opportunity to downside risk by pairing the first-order Upper Partial Moment (expected gain above a threshold) with the square root of the second-order Lower Partial Moment (downside deviation). This construction is grounded in the theory of partial moments developed by Fishburn (1977) and aligns with behavioral insights from Kahneman and Tversky's (1979) Prospect Theory, which demonstrates that investors evaluate outcomes relative to a reference point and are loss-averse.
The ratio is particularly useful for strategies that aim to capture upside while limiting downside exposure, such as covered call writing, protective put strategies, or any investment with an asymmetric return profile. A UPR greater than 1.0 indicates that the expected upside (first moment) exceeds the downside risk (second moment root), suggesting an attractive risk-reward trade-off.
Mathematical Formulation
Core Formula
The Upside Potential Ratio at threshold is defined as:
where is the first-order Upper Partial Moment (measuring average upside) and is the second-order Lower Partial Moment (measuring downside variance). The threshold is typically set to the risk-free rate, zero, or the investor's minimum acceptable return.
Upper Partial Moment (UPM)
The first-order Upper Partial Moment captures the expected gain above the threshold:
This is the average of all positive deviations from the threshold. Returns at or below the threshold contribute zero. The UPM quantifies the "opportunity" embedded in the return distribution -- how much upside the strategy delivers on average.
Lower Partial Moment (LPM)
The second-order Lower Partial Moment captures the downside variance:
This is the average of the squared negative deviations from the threshold. The squaring amplifies the impact of larger losses, similar to how variance emphasizes large deviations. The square root of is the downside deviation, the same denominator used in the Sortino Ratio.
General Partial Moments Framework
The UPR belongs to the broader family of partial moment ratios. The general forms of partial moments of order are:
The order parameter controls the sensitivity to the magnitude of deviations. First-order moments () weight all deviations linearly; second-order moments () emphasize larger deviations quadratically. The UPR uses a first-order upper moment (linear upside) paired with a second-order lower moment (quadratic downside), reflecting a risk-averse stance that weights downside more heavily.
Comparison with Sortino Ratio
| Feature | Sortino Ratio | Upside Potential Ratio |
|---|---|---|
| Numerator | Mean excess return () | First-order Upper Partial Moment |
| Denominator | Downside deviation () | Downside deviation () |
| Numerator includes | Both upside and downside return contributions to the mean | Only upside contributions above the threshold |
| Symmetry treatment | Asymmetric denominator; symmetric numerator | Fully asymmetric (both numerator and denominator) |
| Best for | General downside risk evaluation | Strategies designed to capture upside with limited downside |
The Sortino Ratio and UPR share the same denominator but differ in the numerator. The Sortino Ratio uses the full mean excess return, which can be diluted by losses. The UPR uses only the upside component, providing a purer measure of the gain-to-loss opportunity. For symmetric distributions, both metrics convey similar information; for asymmetric distributions, the UPR can reveal opportunities that the Sortino Ratio misses (or vice versa).
Advantages & Limitations
Advantages
- Fully asymmetric: Both the numerator and denominator treat upside and downside separately, providing a complete asymmetric analysis.
- Prospect Theory alignment:Consistent with Kahneman and Tversky's framework: investors evaluate gains and losses relative to a reference point with different sensitivities.
- Isolates upside opportunity: Directly measures the upside potential rather than the aggregate mean return, useful for strategies with embedded optionality.
- Flexible threshold:The threshold can be tailored to the investor's specific minimum acceptable return.
Limitations
- Less intuitive: More complex than the Sharpe or Sortino Ratios; the partial moment framework is less familiar to most investors.
- Threshold sensitivity: Like the Sortino Ratio, results depend on the choice of threshold, complicating cross-study comparisons.
- Sample size requirements: Requires sufficient data points both above and below the threshold for reliable estimates.
- Limited adoption: Not as widely reported or standardized as the Sharpe and Sortino Ratios, reducing the availability of benchmark comparisons.
- Moment order choice: The specific choice of first-order upper and second-order lower moments is a modeling decision that may not suit all investor risk profiles.
References
- Sortino, F. A., & Price, L. N. (1994). "Performance Measurement in a Downside Risk Framework." Journal of Investing, 3(3), 59-64.
- Fishburn, P. C. (1977). "Mean-Risk Analysis with Risk Associated with Below-Target Returns." American Economic Review, 67(2), 116-126.
- Kahneman, D., & Tversky, A. (1979). "Prospect Theory: An Analysis of Decision under Risk." Econometrica, 47(2), 263-291.