• Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers
 2.1.2 Risk 2.1.4 Fair Division Procedure

## 2.1.3 Risk Aversion

Meaning of risk aversion

It is often said that investors are risk averse. What does this mean?

We have alluded to the idea that investors tend not to prefer risk, all things being equal. Consider a couple investment possibilities, each presenting you with a choice between certainty and risk:

• Flip a coin. You can earn a certain \$4, or you can flip a coin and receive \$10 if it's heads.
• Roll a die. You can receive a certain \$30,000, or you can roll a six-sided die and receive \$10,000 times whatever number comes up.

In each case, what are the expected returns? Which would you choose? Your answers reflect your risk aversion. Consider the expected return in the coin-flip case --

 COIN FLIP Possible returns Probability Expected return Play it safe Certain \$4.00 100% \$4.00 Flip a coin Heads \$10.00 50% \$5.00 Tails \$0 50% \$0 Total \$5.00

Consider the expected return in the dice-roll case --

 Roll Possible return Probability Expected return 1 \$10,000 16.7% \$1,667 2 \$20,000 16.7% \$ 3,333 3 \$30,000 16.7% \$ 5,000 4 \$40,000 16.7% \$ 6,667 5 \$50,000 16.7% \$ 8,333 6 \$60,000 16.7% \$10,000 Total 100% \$35,000

In both cases, your expected return is higher by taking the risk -- that is, flipping the coin or rolling the die. If you already have lunch money and you enjoy thrills, you might forego the certain \$4.00 and flip the coin in the hope of \$10.00. But if you were worried about paying your school loans (which now total \$30,000), you might not want to roll the die, even though on average you could expect to receive \$35,000 and perhaps even \$60,000. But the risk of rolling a 1 or 2 makes you shy away.

Components of risk aversion

Notice that your risk aversion reflects two components --

• your financial circumstances -- in the coin flip, you already had lunch covered and you were risking discretionary moneys
• the dispersion of the outcomes -- in the die roll, there was a wide range of outcomes (from \$10,000 to \$60,000) that made you sensitive to variation
"Do the Wealthy Risk More Money? An Experimental Comparison"

BY: ANTONI BOSCH-DOMèNECH
Universitat Pompeu Fabra
JOAQUIM SILVESTRE
University of California, Davis
Department of Economics
Date: April 20, 2003

Are poor people more or less likely to take money risks than wealthy folks? We find that risk attraction is more prevalent among the wealthy when the amounts of money at risk are small (not surprising, since ten dollars is a smaller amount for a wealthy person than for a poor one), but, interestingly,
for the larger amounts of money at risk the fraction of the nonwealthy displaying risk attraction exceeds that of the wealthy. We also replicate our previous finding that many people display risk attraction for small money amounts, but risk aversion for large ones. We argue that preferences yielding 'risk attraction for small money amounts, together with risk aversion for larger amounts, at all levels of wealth,' while contradicting the expected utility hypothesis, may be well-defined, independently of reference points, on the choice space.

Risk preferring

A British man who sold all his possessions, including his clothes, stood in a rented tuxedo on Sunday surrounded by family and friends and bet everything on a single spin of the roulette wheel. (More>>)

Certainty equivalence

Financial analysts often equate risk and variability. The greater the variability, the greater the risk -- and vice versa. But why does greater variability translate into greater risk -- and thus a lower valuation? To illustrate this relationship, consider two investment options:

 Investment A Returns Probability Value \$900 .10 \$90 \$1,000 .80 \$800 \$1,100 .10 \$110 Expected return \$1,000 Investment B Returns Probability Value \$0 .30 \$0 \$1000 .40 \$400 \$2000 .30 \$600 Expected return \$1,100

Although the expected return is the same for both, Investment B is riskier. The range of outcomes (variability or volatility) is greater for Investment B.

How much would you pay for Investment A compared to Investment B? Disregarding for now the time value of money, it should be obvious you would pay at least \$900 for Investment A -- you are certain to receive at least this. In fact, you would probably pay close to \$1,000, given the high likelihood (80%) of earning \$1,000 and the small dispersion from the mean. Let's say, you view Investment A and \$980 as equivalent. That is, Investment A's "certainty equivalent" is \$980.

But Investment B is more volatile and has a different certainty equivalent. You would probably be reluctant to pay even \$900 for Investment B, given the significant possibility (30%) of a zero return. Although Investment B's possible returns of \$1,000 and \$2,000 might be attractive, they are not certain. Perhaps you would pay only \$800 for Investment B -- its certainty equivalent.

This illustrates how an asset's riskiness affects its value. The greater the uncertainty of cash flows, the lower their value. Risk can be factored into the valuation process by either increasing the discount rate or decreasing the value of risky returns.

Example

For example, if you own a beach condo that you expect to sell in three years for \$120,000, its value may depend on the likelihood of this return.

 Certainty Risk If a federal government housing authority has contractually promised to buy the condo for \$120,000 in 3 years, you could consider the condo investment to have the attributes of a 3-year T-bill -- that is, there is no risk of less-than-full payment. If the prevailing risk-free rate on a three-year T-bill is 5.2% (the current time value of money), you can determine the present value of the condo. If you expect the condo's value will be \$120,000 in 3 years, but could fluctuate between \$80,000 and \$200,000 depending on market conditions, you might decide the condo investment has the attributes of high-risk common stock. If the prevailing expected return for high-risk common stock is 18% (required return), you can determine the condo's value. You might also conclude that \$95,000 represents the "certainty equivalence" for your three-year investment. You would then calcultate its present value using a risk-free rate.

Often the appropriate discount rate will be the most significant contention in a valuation. As the discount rate rises, the expected value falls. For example, an 8% discount rate results in a valuation that is twice as large as a 16% discount rate.

 2.1.2 Risk 2.1.4 Fair Division Procedure