Answer:
Step-by-step explanation:
a. To calculate Johnathan's risk-odds for the S+/-1 deal, we need to compare the expected utility of the two outcomes (winning or losing $1). Assuming a 50/50 chance of winning or losing, the expected utility can be calculated as:
EU(S+/-1) = 0.5*u(0) + 0.5*u(2)
where u(0) is the utility of having a net gain of $0, and u(2) is the utility of having a net gain of $2.
Substituting in Johnathan's utility function, we get:
EU(S+/-1) = 0.5*(4-4^(-0/1000)) + 0.5*(4-4^(-2/1000))
= 3.992
To calculate the risk-odds, we can use the formula:
r($1) = EU(S+/-1)/(EU(S+/-1) + u(1))
where u(1) is the utility of having a net gain of $1. Substituting in Johnathan's utility function, we get:
r($1) = 3.992/(3.992 + (4-4^(-1/1000)))
= 0.5025
Therefore, Johnathan's risk-odds for the S+/-1 deal is r($1) = 0.5025.
b. To determine if Johnathan is a deltaperson, we need to compare his marginal utility of wealth (MUW) to his marginal utility of consumption (MUC) at the current level of wealth. If MUW > MUC, he is a deltaperson; if MUW < MUC, he is a deltaaverse person; if MUW = MUC, he is risk-neutral.
The marginal utility of wealth can be calculated as the derivative of Johnathan's utility function with respect to wealth:
MUW = d(u(x))/dx = 4*(ln(2))/1000 * 4^(-x/1000)
At any given level of wealth x, the marginal utility of consumption can be calculated as the derivative of Johnathan's utility function with respect to consumption:
MUC = d(u(x))/d(c) = 4*(ln(2))/1000 * 4^(-x/1000) * (1/1000)
Since both MUW and MUC are decreasing functions of wealth, we can conclude that Johnathan is a deltaperson.
c. Johnathan is not risk-neutral because his utility function is concave (diminishing marginal utility of wealth). This means that he is risk-averse and would prefer a certain outcome to an equivalent gamble, ceteris paribus.
d. To calculate Johnathan's risk-odds for the $+/-100 deal, we can repeat the same process as in part (a), but with the expected utility calculated as:
EU($+/-100) = 0.5*u(-100) + 0.5*u(100)
where u(-100) is the utility of having a net loss of $100, and u(100) is the utility of having a net gain of $100. Substituting in Johnathan's utility function, we get:
EU($+/-100) = 0.5*(4-4^(-100/1000)) + 0.5*(4-4^(-(-100)/1000))
= 2.007
Using the risk-odds formula, we get:
r($100) = EU($+/-100)/(EU($+/-100) + u(0))
= 2.007/(2.007 + (4-4^(-0/1000)))
= 0.3333
Therefore, Johnathan's risk-odds for the $+/-100 deal is r($100) = 0.3333.
e. Johnathan's risk-aversion coefficient can be calculated as the absolute value of the ratio between his marginal utility of wealth and his marginal utility of consumption, evaluated at the current level of wealth:
γ = | MUW(x)/MUC(x) |
Substituting in Johnathan's utility function and simplifying, we get:
γ = 4^(x/1000)
where x is the current level of wealth.