Expected Value: What is Risk Aversion?

In its simplest sense, risk aversion is the concept that when a person is faced with two or more comparable decisions that he or she will choose the decision with the least risk associated with it. The topic of risk aversion is generally grouped in with cost-benefit analysis discussions because the goal behind the analysis is to find a way to maximize the overall utility received from goods and services. Thus, when these two concepts are combined, an individual is aiming to minimize risk while maximizing utility.

To even properly compare the risks associated with any decision to be made, one has to take into account the expected values of each decision. Expected value is an average expectation of the outcome of an event if that event was repeated in an identical sense numerous times. More formally, the expected value is the sum of the probability of all possible outcomes multiplied with its corresponding outcome value. An easy application of this concept can be applied to the roll of a six-sided die. The probability of each outcome is 1/6 and the corresponding outcome value is just each of the numbers. So the equation for this problem would look like:

E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6)
= (1 +2 + 3 + 4 + 5 + 6)/6
= 3.5

Obviously, 3.5 is not an option when rolling a six-sided die, but the purpose of using the expected value equation is to find the average of the outcomes. A common application of expected value is gambling and determining the odds of winning through the equation.

Now that both expected value and risk aversion have been explained, it is time to tie them together and see how the two topics overlap. As mentioned earlier, risk-averse individuals tend to choose the option of least risk when posed with items that yield similar outcomes. This means that if a decision needs to be made between Option A and Option B and Option A involves much less risk than Option B yet it yields similar results as Option B then Option A will be the more desirable choice. An example could be taken from the health economics arena where a health administrator is given the task of choosing among different managed care contracts. The hospital the administrator is representing has a diminishing marginal utility from income and has narrowed its search of managed care contracts down to two. One of the contracts has the potential ability to greatly stabilize future net incomes up to 23% for the hospital but has a higher risk of failure than the other contract which can only assure that future net incomes will be stabilized by 18%. Given the net income stabilization rates only differ by five points, the health administrator could be more tempted to take on the contract that assures the lower rate because of the lower risk that accompanies it.

Problems:

1. A highly risk-averse investor is considering the addition of an asset to a 10-stock portfolio. The two securities under consideration both have an expected return equal to 15 percent. However, the distribution of possible returns associated with Asset A has a standard deviation of 12 percent, while Asset B's standard deviation is 8 percent. Which asset should the risk-averse investor add to his/her portfolio?
a.. Asset A.
b. Asset B.
c. Both A and B.
d. Neither A nor B.
e. Cannot tell without more information.

The answer to this question is (b) because if the two securities yield the same return but Asset B varies from that return by only 8 percent and Asset A varies by 12 percent then the more secure investment would be Asset B.

2. An American roulette wheel has 38 equally likely outcomes just like a die has 6 equally likely outcomes. Suppose a bet was placed on a number and it pays 70-to-1. This means that you are paid 70 times your bet in addition to the $1 you put in the bet, so you end up getting 71 times your bet. If all 38 possible outcomes on the roulette wheel are considered, the expected value of the profit resulting from a $1 bet on a single number is what? What does this value tell you?

The answer is E(X) is equal to : - $1(37/38) + $70(1/38) $0.8684. This means that the person betting can expect to win $0.87 per every dollar bet on the roulette game.



Sources:

Expected Value. Wikipedia. 17 Feb 2007. < http://en.wikipedia.org/wiki/Expected_value>

Neun, Stephen P., and Rexford E. Santerre. Health Economics: Theories, Insights, and Industry Studies. 4th Ed. Mason: Thomson South-Western, 2007.

The Theory of Risk Aversion. Economics New School. 17 Feb 2007. <http://cepa.newschool.edu/het/essays/uncert/aversion.htm>