= Secant-slope existence criterion for an adjustment coefficient
{title2=$\frac{M_X(R)-1}{R}=(1+\theta)\mu$}
For positive claims with finite nonzero <expected value> $\mu$, the function $(M_X(r)-1)/r$ is continuous and strictly increasing on the positive finite-transform domain, starting at $\mu$. If $M_X$ diverges at a finite upper endpoint, or is finite for all positive arguments, the secant slope tends to infinity. In the latter case use $M_X(r)\ge\mathbb P(X\ge a)e^{ar}$ for some $a>0$ of positive tail probability. Every target $(1+\theta)\mu$ with positive <relative safety loading> therefore has exactly one positive root.
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