1. Using the cummulant generating function, show that the premium e obtained from the variance principle is given by E(S) + RVar (S), where R is a small risk aversion parameter.
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- a) Compute the (absolute) risk aversion measure dependent r(W) of utility function -e -aW Is r(W) on W?Explain the relationship between U" >0 and risk aversion.A risk-averse agent, Andy, has power utility of consumption with riskaversion coefficient γ = 0.5. While standing in line at the conveniencestore, Andy hears that the odds of winning the jackpot in a new statelottery game are 1 in 250. A lottery ticket costs $1. Assume his income isIt = $100. You can assume that there is only one jackpot prize awarded,and there is no chance it will be shared with another player. The lotterywill be drawn shortly after Andy buys the ticket, so you can ignore therole of discounting for time value. For simplicity, assume that ct+1 = 100even if Andy buys the ticket How large would the jackpot have to be in order for Andy to play thelottery? b) What is the fair (expected) value of the lottery with the jackpot youfound in (a)? What is the dollar amount of the risk premium that Andyrequires to play the lottery? Solve for the optimal number of lottery tickets that Andy would buyif the jackpot value were $10,000 (the ticket price, the odds of winning,and Andy’s…
- Can you explain how Constant Relative Risk Aversion utility function should be understood and how it works mathematicallyIn the field of financial management, it has been observed that there is a trade-off between the rate of return that one earns on investments and the amount of risk that one must bear to earn that return. a) Draw a set of indifference curves between risk and return for a person that is risk-averse (a person that does not like risk).Question 6 Which of the below indifference curve correspond to the highest risk aversion? E(r) .60 .40 .20 .09 .05 0 D and C B and A .10 .20 .30 .40 .5 D C B A
- 5. Shift-in-charge Nazar Al Rushdy: Nazar is pessimistic about the market price. What is your guidance for Nazar? The decision to employ decision trees in crucial situations has been taken by Salem Al Harthi, the plant manager. The table below presents data on demand for a duration of 6 hours along with their respective probabilities. The first row of the table provides the probability of demand for the initial three hours when a leak occurs, denoted in parentheses. Subsequently, the following three rows indicate the probabilities of high, medium, and low demand for the succeeding three hours. To illustrate, if the initial 3-hour market price was low, the probabilities of high demand, medium demand, and low demand in the next three hours are 0.2, 0.3, and 0.5, respectively. Market price High Market price Medium Initial 3-hrs (0.2) Initial 3-hrs (0.5) Market price Low Initial 3-hrs (0.3) High demand (next (0.5) (0.4) (0.2) 3 hrs) Medium demand (0.3) (0.2) (0.3) (next 3 hours) Low demand…Consider the lottery that assigns a probability r of obtaining a level of consumption CH and a probability 1-T of obtaining a low level of consumption cL an individual facing such a lottery with utility function u(c) that has the properties that more is better (that is, a strictly positive marginal utility of consumption at all levels of c) and diminishing marginal utility of consumption, u"(c) CL. Consider du(c) for the first derivative of the utility function with respect to dc d²u(c) dc2 du' (c) consumption and u"(c) which is also the derivative of the first derivative of the utility function). to be the second derivative of the utility function dcDefine the term risk aversion?
- 4) Consider investors with preferences represented by the utility function U = E(r) – Ao². (a) Draw the indifference curve representing a utility level of 10% for an in- vestor with a risk aversion parameter A = 3 in expected return-standard deviation space. (b) In the same graph, draw the indifference curve representing a utility level of 15% for an investor with a risk aversion parameter A = 3. (c) In the same graph, draw the indifference curve representing a utility level of 10% for an investor with a risk aversion parameter A = 5.The Constant Relative Risk Aversion (CRRA) utility function is a widely used specifica- tion of preferences in economics that captures risk aversion and intertemporal consump- tion smoothing. The CRRA utility function has the desirable property that the degree of risk aversion is constant and independent of the level of consumption. This means that as a household's consumption grows, its willingness to take risks remains the same. The coefficient of relative risk aversion (σ) measures the extent to which households are risk- averse and prefer a smooth consumption path over time. A higher value of σ indicates a greater degree of risk aversion and a stronger preference for consumption smoothing. Consider a two-period endowment economy with a large number of identical house- holds. Each household has the following lifetime utility function: U(j) = C+(j) 1-0 - 1 1-σ +ẞ C++1(j) 1-0 - 1 1-σ where C₁(j) and C++1(j) are consumption in periods t and t + 1 for household j, re- spectively, ẞ is…a) Explain what is meant by risk aversion, and illustrate with the help of a figure out what we mean by the term "risk premium". Suppose Donald runs hotels and casinos, which makes one very insecure income. With probability 1 the income becomes 100 and with probability 1 the 64th Donald's expected income is thus equal to 82. Further assume that the utility to Donald is a a function of income, and that it is given by U (x) = 2x 12 x is the income level. b) Calculate Donald's expected utility.