We present a novel way to understand the low uptake of index insurance using the interlinked concepts of ambiguity and compound-lottery aversion. Noting that the presence of basis risk makes index insurance a compound lottery, we derive an expression of the willingness to pay (WTP) to eliminate basis risk. Empirically, we implement this WTP measure using framed field experiments with cotton farmers in Southern Mali. In this sample, 57% of the surveyed farmers reveal themselves to be compound-risk averse to varying degrees. Using the distributions of compound-risk aversion and risk aversion in this population, we simulate the impact of basis risk on the demand for an index insurance contract. Compound-risk aversion decreases the demand for index insurance relative to what it would be if individuals had the same degree of risk aversion but were compound-risk neutral. In addition, demand declines more steeply as basis risk increases under compound-risk aversion than it does under risk neutrality. Our results highlight the importance of designing contracts with minimal basis risk if potential buyers are compound-risk averse.