Question

Consider a lottery with three possible outcomes:

• \(125 will be received with probability .2

• \)100 will be received with probability .3

• $50 will be received with probability .5

a. What is the expected value of the lottery?

b. What is the variance of the outcomes?

c. What would a risk-neutral person pay to play the lottery?

Short answer

a. The expected value of the lottery will be $80

b. The variance of the outcomes will be $975.

c. A risk-neutral person will only pay the expected value of the lottery which is $80.

Step-by-step solution

Expected value of the lottery

The expected value of the lottery will be the same as the weight of returns of their probabilities.The probabilities at each price have been given:

At the price of $125, the probability is 0.2

At the price of $100, the probability is 0.3

At the price of $50, the probability is 0.5

The expected value of the lottery will be:

$$\begin{gathered} \mathrm{EV}=\mathrm{P}\left(\mathrm{x}\right)\times \mathrm{X} \\ \mathrm{EV}=\left(0.2\right)\left(125\right)+\left(0.3\right)\left(100\right)+\left(0.5\right)\left(50\right) \\ =25+30+25 \\ =80 \end{gathered}$$

The expected value of the lottery is $80.

Variance of the outcome

The outcome variance is the squared deviation from the mean, which is weighted by the probabilities.For finding the variance, each value should be squared and multiplied by the probability. Then all value should be summed up, and the square of the expected value should be deducted from it.

Given: Expected value = $80. The probability at a price:

125 is 0.2

100 is 0.3, and

50 is 0.5

The variance of the outcome will be:

$$\begin{gathered} Variance=\sum _{}^{}{x}^{2}p-{\mu }^2 \\ {\sigma }^2=\left(0.2\right){\left(125-80\right)}^2+\left(0.3\right){\left(100-80\right)}^2+\left(0.5\right){\left(50-80\right)}^2 \\ =405+120+405 \\ =975 \end{gathered}$$

The variance of the outcome will be $975.

The amount that a risk-neutral person pays

A risk-neutral individual has a mindset of one who is indifferent to risk while making an investment.They won’t simply focus on the risk of the investments which they have made.

The amount that a risk-neutral person pays should be $80. A risk-neutral person always tries to avoid the risk. Their utility on expected wealth from the utility will be the expected value of the lottery that is $80.