Cumulants and the Cumulant Generating Function

Moments — mean, then $E[X^2]$, then $E[X^3]$ — describe a distribution's shape, but they behave badly when you combine independent random variables: the third moment of a sum is not the sum of the third moments, because cross-terms creep in through the algebra of expanding $(X+Y)^3$. Cumulants are a different, cleverer bookkeeping of the exact same information, engineered specifically so that combining independent variables becomes simple addition, with no cross-terms ever appearing.

The analogy: repacking the same boxes so they stack cleanly

Imagine moments are boxes of oddly mixed sizes — easy to describe individually, but when you stack two people's moving boxes together, some boxes have to be repacked because their shapes clash and overlap awkwardly. Cumulants are the same total "stuff" (they carry exactly the same information as moments, and you can convert between the two), but repacked into uniform crates that are guaranteed to just stack in a simple pile when you combine two independent shipments — no repacking, no overlap corrections, just addition, crate by crate.

Writing it down

The cumulant generating function is the natural log of the moment generating function:

$$K_X(t) = \ln E\big[e^{tX}\big] .$$

In words: take the usual moment-generating machinery, $E[e^{tX}]$, and log it. The cumulants $\kappa_n$ are the Taylor coefficients of $K_X(t)$ around $t=0$ (analogous to how ordinary moments are the Taylor coefficients of $E[e^{tX}]$ itself, before the log). The first few translate to familiar quantities: $\kappa_1 = E[X]$ (the mean), $\kappa_2 = \operatorname{Var}(X)$ (the variance), $\kappa_3 = E[(X-\mu)^3]$ (feeds skewness), and $\kappa_4$ feeds excess kurtosis. The property that makes all of this worth the trouble:

$$X, Y \text{ independent} \implies K_{X+Y}(t) = K_X(t) + K_Y(t) .$$

In words: cumulant generating functions of independent variables simply add, term by term at every order — because $K$ is a log, and logs turn the products that MGFs naturally combine under ($M_{X+Y}=M_XM_Y$ for independent variables) into sums. Every individual cumulant $\kappa_n$ inherits this: $\kappa_n(X+Y) = \kappa_n(X) + \kappa_n(Y)$, at every order $n$, with zero cross-terms — a property raw moments never have beyond the first.

Worked example 1: variance of a sum, moments versus cumulants

Let $X, Y$ be independent with means $0$ for simplicity, variances $\operatorname{Var}(X)=4$, $\operatorname{Var}(Y)=9$. Via cumulants: $\kappa_2(X+Y) = \kappa_2(X)+\kappa_2(Y) = 4+9=13$ — done in one step, since $\kappa_2$ is variance. Via raw second moments directly: $E[(X+Y)^2] = E[X^2] + 2E[X]E[Y] + E[Y^2]$ (using independence to split $E[XY]=E[X]E[Y]$) $= 4 + 2(0)(0) + 9 = 13$. Same answer here only because the means are zero — with nonzero means the raw-moment version needs the cross-term $2E[X]E[Y]$ explicitly tracked, while the cumulant version needs no adjustment at all, by construction.

Worked example 2: skewness of a sum of two skewed variables

Let $X$ have $\kappa_3(X) = 2$ (positively skewed) and $Y$ have $\kappa_3(Y) = -0.5$ (mildly negatively skewed), independent, with $\operatorname{Var}(X)=1$, $\operatorname{Var}(Y)=1$. Third cumulant of the sum: $\kappa_3(X+Y) = 2 + (-0.5) = 1.5$, by pure addition. Convert to standardised skewness using $\operatorname{Var}(X+Y)=1+1=2$: skewness $= \kappa_3/\kappa_2^{3/2} = 1.5 / 2^{1.5} = 1.5/2.828 = 0.530$. Contrast: $X$ alone has skewness $2/1^{1.5}=2.0$ and $Y$ alone has $-0.5/1^{1.5}=-0.5$ — simply averaging those two numbers, $(2 + (-0.5))/2 = 0.75$, would be a tempting but wrong shortcut; the correct combination goes through the cumulants and the variance scaling, giving $0.530$, not $0.75$.

What this means in practice

Portfolio return skewness and kurtosis — how much tail risk a combination of positions carries — are computed by adding cumulants of the individual positions (when independent, or with correction terms when not), never by naively averaging skewness numbers. Edgeworth expansions, which approximate a non-normal distribution using its low-order cumulants as corrections to a normal, are the standard tool for pricing under fat tails and rely entirely on this additive structure.