Confusions about calculations in Utility Theory Paper
The notation is a bit confusing but what the author is computing is the expected utility of the investment, not the utility of the expected outcome.

Since the investment results in a wealth of \$120 with probability 0.5 and a wealth of \$90 with probability 0.5, the expected utility equals
\begin{align*}
\mathbb{E}U &amp; = 0.5 \times (-\dfrac{10^6}{90^3}) + 0.5 \times (-\dfrac{10^6}{120^3}) \\
&amp; \sim -0.98 
\end{align*}

Notice that this value is lower than the utility of the expected outcome, which is approximately equal to $-0.86$ as you wrote. This is because the agent is risk-averse and must therefore be compensated for the risk.