Lower Partial Moments

Two funds both have an annualized volatility of 15%. Fund A's return distribution is symmetric, occasional big gains balanced by occasional big losses. Fund B's distribution is lopsided: mostly small, steady gains, with rare but severe crashes. Ordinary variance treats these two funds as equally risky, because variance counts a surprise on the upside exactly the same as a surprise on the downside. No investor actually feels that way; a big unexpected gain is not a risk event. Lower partial moments fix this by only counting the outcomes that fall short of some target.

The analogy before any symbols

A parent tracking a child's exam scores against a passing threshold of 60 does not lose any sleep over a score of 95 versus a score of 75, both are comfortably above the bar and the difference does not matter for the thing being worried about. A score of 40 versus a score of 55, both below the bar, is a completely different story, and the parent cares intensely about how far below 60 the score fell. Ordinary variance is like tracking distance from the average score in both directions equally; a lower partial moment is like tracking distance below the passing threshold only, ignoring how far above it a good score happened to land.

The mechanics

Choose a target return $\tau$ (often 0, or the risk-free rate, or a required minimum return). The lower partial moment of order $n$ is:

$LPM_n(\tau) = E\Big[\big(\max(\tau - R, 0)\big)^n\Big].$

In words: for every observed return $R$, compute how far below the target it fell, treat any return above the target as contributing zero (there's no shortfall to count), raise that shortfall to the power $n$, and average across all observations. The order $n$ controls how much extra weight severe shortfalls get relative to mild ones: $n=0$ just counts the frequency of falling short of the target, ignoring size entirely; $n=1$ averages the size of shortfalls, treating a shortfall twice as large as exactly twice as bad; $n=2$, called the semivariance when $\tau$ equals the mean, squares the shortfall before averaging, so a shortfall twice as large counts four times as much, mirroring how ordinary variance treats deviations but only on the downside.

The Sortino ratio, a popular alternative to the Sharpe ratio, uses the square root of $LPM_2$ (called downside deviation) in place of ordinary standard deviation in the denominator: $\text{Sortino} = (E[R]-\tau)/\sqrt{LPM_2(\tau)}$. A strategy with lopsided upside, like Fund B in the opening example but flipped, steady small losses with rare large gains, gets penalized less by the Sortino ratio than by the Sharpe ratio, because its (desirable) upside surprises no longer count against it.

Ordinary variance is built from the squared distance from the mean on both sides of this curve equally. A lower partial moment only uses the shaded region below the chosen target; everything to the right of the target contributes nothing, no matter how far right it goes.

Worked example: two funds, same variance, different LPM

Five years of annual returns. Fund A: +25%, −10%, +18%, −8%, +5% (mean 6%, symmetric-ish spread). Fund B: +9%, +8%, +7%, +6%, −30% (mean 0%, one severe crash). Both have standard deviation near 14 to 15%. Using target $\tau = 0$: Fund A's shortfalls are only the two negative years, $-10\%$ and $-8\%$, giving $LPM_1(0) = \tfrac15\big(0+0.10+0+0.08+0\big) = 0.036$, or 3.6%. Fund B's only shortfall is the one crash year, $-30\%$, giving $LPM_1(0) = \tfrac15(0+0+0+0+0.30) = 0.06$, or 6.0%. Despite similar ordinary volatility, Fund B's downside risk by this measure is nearly double Fund A's, because ordinary variance was crediting Fund B's four steady positive years as "risk" right alongside its one crash, while $LPM_1$ correctly ignores those four good years and isolates the one that actually hurt.

Worked example: order $n$ changes the ranking

Same two funds, target $\tau=0$. Using $n=2$ (semivariance): Fund A's $LPM_2(0) = \tfrac15(0.10^2+0.08^2) = \tfrac15(0.01+0.0064) = 0.00328$, downside deviation $\sqrt{0.00328} \approx 5.7\%$. Fund B's $LPM_2(0) = \tfrac15(0.30^2) = \tfrac15(0.09) = 0.018$, downside deviation $\sqrt{0.018} \approx 13.4\%$. The gap between the two funds widens sharply going from $n=1$ (6.0% vs 3.6%, a 1.7x ratio) to $n=2$ (13.4% vs 5.7%, a 2.4x ratio), because squaring gives extra weight specifically to Fund B's one severe crash relative to Fund A's two moderate losses. Choosing $n$ is choosing how much to punish severity versus frequency of shortfall.

What this means in practice

Lower partial moments and the Sortino ratio are common in strategies with deliberately asymmetric payoffs, trend-following, options-selling, or anything designed to have a "small steady gains, rare large loss" or "small steady losses, rare large gain" shape, where ordinary variance either overstates or understates the risk an investor actually cares about. They require choosing both a target $\tau$ and an order $n$, two extra judgment calls that plain variance does not need, which is the tradeoff for a measure that better matches how losses are actually felt.