# Why does a risk-return relationship that has historically been positive confirm risk averse investors?

As I am reading through a corporate finance textbook I came across the following figure that plots the relationship between risk and return for different asset classes:

The textbook states:

Figure 10.6 is consistent with our view that investors are risk averse. Riskier investments must offer investors higher average returns to compensate them for the extra risk they are taking on.

I can't wrap my head around this. Why can we conclude that investors have to be risk averse, based on a positive relationship between risk and return? If we assume investors are risk seeking, wouldn't the relationship be the same? Risk seeking investors love risk and would buy those portfolios with the highest risk (portfolio of small stocks for example).

Textbook and source: (Berk, Demarzo): Corporate Finance 2019

When more investors buy something, its return rate goes down (because its price goes up but the return does not change).

If investors didn't care about risk, this curve would be flat. If one of the investments had a higher return, investors would buy it until it didn't. If one of them had a lower return, investors would stop buying it until it didn't.

If investors loved risk, the highest-risk investments would be the most expensive and have the lowest returns. Even though the high-risk investment returned 5% on average and the low-risk investment returned 20% on average, investors would be saying "Ooh boy! I want some of that risk even though the average return is lower!". That's what it means to love risk.

• @xxgaryxx yes, I mean that Sep 10, 2021 at 11:59

Risk aversion measures the degree to which someone prefers a sure thing to a gamble.

If people are risk-averse that means they would, all else equal, prefer sure return to risky return, even if the expected return is the same. For example, a risk-averse person would rather invest into an asset that yields \$50 with certainty than in an asset that would have 50% chance of yielding \$0 and 50% chance of yielding \$100. So if the return on risky and risk-free assets would be equal nobody would buy the risky asset. But at the same time, there is always some higher return that would be sufficiently large to convince the risk-averse person to take on risk. What the large return is will depend on the degree of risk aversion. For example, a risk-averse person would always prefer \$50 with certainty to a gamble of 50/50 chance of winning \$100 and \$0, but if you increase the expected value of the gamble to maybe 50/50 chance of winning \$500 vs 0\$, the risk-averse person may be willing to take the gamble instead of just taking the certain \\$50.

This is why risky assets have to offer a premium, which you can think of as compensation for the loss of welfare or pain from taking on the risk. Also, if the risk increases naturally risk-averse people will demand more and more as compensation for the risk.

You can think about it from different viewpoints. All answers are true. I'll just try to explain it from another angle.

We are on the same page that everybody loves higher returns. The question is whether you also like higher risk or not. Loosely speaking, if you dislike risk, you are risk-averse.

Now suppose in the equilibrium (basically what we observe in the market), we have stock A and B, and we know that the expected return of stock A is higher. Suppose the risk of A is higher than B (so we are looking at the graph you posted). Is it possible investors are risk-lover? Then why should anyone buy stock B when stock A has both higher return and risk and people like return and risk? The only way to rationalise this positive relationship is to assume people do tradeoff risk and return, implying they are risk-averse.

Ps: in this simple analysis, we are treating investors as a homogenous group; in a more realistic analysis, if we allow for heterogeneous investors, we actually can have a minority of investors who are risk-lover (and they all end up buying at the upper right corner of the plot).