Because investors are, on average, risk averse, we should expect that there is a positive relation between expected returns and *expected* volatility—the greater the expected volatility, the greater the rate of return required. Conversely, we should expect to see a negative relationship between returns and *unexpected* volatility as investors increase the discount rate they use to value future expected earnings on risky assets.

We should also expect, during periods of heightened uncertainty, that investors, in a flight to safety, would be willing to accept lower required returns on safe assets.

The Sharpe ratio was developed to create a measure of risk-adjusted returns. It measures returns per unit of risk, with “risk” being defined as volatility. Importantly, the Sharpe ratio assumes returns are normally distributed, which is not always the case.

**Skewness**

While volatility is certainly a measure of risk, and a good one, it’s not the only one. There are other measures of risk that investors care about, including skewness and kurtosis.

Skewness measures the asymmetry of a distribution. In terms of the market, the historical pattern of returns doesn’t resemble a normal distribution, and so demonstrates skewness. Negative skewness occurs when the values to the left of (less than) the mean are fewer but farther from it than values to the right of (greater than) the mean.

For example, the return series of -30%, 5%, 10% and 15% has a mean of 0%. There is only one return less than zero, and three that are higher. The single negative return is much farther from zero than the positive ones, so the return series has negative skewness. Positive skewness, on the other hand, occurs when values to the right of (greater than) the mean are fewer but farther from it than values to the left of (less than) the mean.

Investors have a preference for assets with positive skewness. This is evidenced by an investor’s willingness to accept low, or even negative, expected returns when an asset exhibits positive skewness.

The classic example of positive skewness is lottery tickets—the expected return is -50% (the government only pays out about 50% of the sales proceeds), and the vast majority end up worthless, but investors hope to hit the big jackpot.

Some examples of assets that exhibit both positive skewness and poor returns are IPOs, “penny stocks,” stocks in bankruptcy and small-cap growth stocks with high investment and low profitability (such as dot-com stocks).

On the other hand, investors generally dislike assets with negative skewness. High-risk asset classes (such as stocks) typically exhibit negative skewness. Because they dislike them, such assets must provide a large risk premium. And historically, such assets have provided high returns as compensation for their risk.

**Kurtosis**

Kurtosis measures the degree to which exceptional values—those much larger or much smaller than the average—occur more frequently (high kurtosis) or less frequently (low kurtosis) than in a normal (bell-shaped) distribution. High kurtosis results in exceptional values that are called “fat tails.”

Fat tails indicate a higher percentage of very low and very high returns than would be expected in a normal distribution. Low kurtosis results in “thin tails” and a wide middle to the curve. In other words, more values are closer to the average than would be found in a normal distribution, and tails are thinner.

Summarizing, if an investment exhibits both negative skewness and high kurtosis, it has the potential for large losses. That investment should have a large risk premium in order to compensate investors for accepting risk, which they prefer to otherwise avoid.

Another type of risk that is unrelated to the distribution of returns is liquidity. Investments with high liquidity (low trading costs) and no restrictions (such as mutual funds, which have daily liquidity and very low, if any, trading costs) should have lower expected returns than investments with lower levels of liquidity. For example, the greater trading costs of less liquid small stocks helps explain their higher historical returns and provide a risk-based explanation for their higher expected returns. Investments with “gates,” which restrict liquidity, such as partnerships that invest in private equity, hedge funds and interval funds, should carry risk premiums as compensation—if they do not, you should not invest.

We’ll now look at three investments that exhibit negative skewness and high kurtosis.

**Reinsurance Risk**

The returns to reinsurance investments are not normally distributed—far from it. As an example, a reinsurance fund might have an expected return of 7%. In a very good year, with few to no losses incurred, the return might be 12%. However, in a bad year, such as 2017, the losses could easily reach double digits. And in an exceptionally bad year, the losses could be even larger.

Because the number of good years has far exceeded the number of really bad years, the Sharpe ratio has been exceptionally high. However, as noted earlier, the Sharpe ratio is not a good measure of risk when returns are not normally distributed.

Reinsurance returns are negatively skewed with a long, but thin left (bad) tail. In addition, because reinsurance requires investing in contracts that typically are at least one year in length, reinsurance has liquidity risk that should be compensated for. Once these additional issues are considered, the risk-adjusted returns to reinsurance would look more like the risk-adjusted returns to other risky assets (such as stocks and bonds). And that is exactly what we should expect from an efficient market.

Two examples of reinsurance funds investors could consider are Stone Ridge’s Reinsurance Risk Premium Interval Fund (SRRIX) and Pioneer’s ILS Interval Fund (XILSX). These are both interval funds with limited quarterly liquidity (minimum of 5% per quarter).