What relationship indicates higher risk with payoffs/variability in investments?

The option indicating that a wider range of payoffs suggests more risk is correct because it aligns with fundamental concepts in finance regarding the risk-return trade-off. In investment contexts, risk is typically characterized by the potential variability in returns. A wider range of possible payoffs signifies that an investment's returns could vary greatly from low to high, which inherently introduces uncertainty and potential fluctuations.

When an investment has a diverse set of outcomes, the investor faces a greater likelihood of experiencing extreme values, either significantly underperforming or outperforming expectations. This variability can cause anxiety and requires a higher risk tolerance from the investor. Consequently, a wider range of payoffs is associated with increased risk because it denotes greater uncertainty about future returns, which is a critical factor for investors in making decisions about where to allocate their funds.

In contrast, a narrower range of payoffs would indicate that returns are more predictable and stable, implying lower risk associated with the investment. Understanding this relationship helps in evaluating investments and guiding strategic financial decisions.