Let rt be a log return. Suppose that r0, r1, . . . are i.i.d. N(0, 0.01^2). (a) What is the distribution of rt(8) = rt + rt−1 + rt−2 +...+ rt−7? (b) What is the covariance between r7(3) and r9(3)? (c) What is the conditional distribution r17(3) given that r16 =0.004 (d) What is the probability that the gross return over the first 10 times periods is at least 1.05

Tossa coin once, S = (H,T) with P,(H) = p and P,(T) = q. Define X(-) by X(H) = ₁ X(T) = 0. Compute and plot the probability distribution function F, (x).

Z is the present-value random variable for a whole life insurance of b payable at the moment of death of (x). You are given: (i) (ii) 8 = 0.05 (iii) The net single premium for this insurance is equal to Var(Z). Calculate b. x+t = 0.01 t≥0 (A) 1.36 (B) 1.68 (C) 2.00 (D) 2.32 (E) 2.64

The Volatility X for the S&P stock index on a given day is a normal random variable with mean = 10 and standard deviation = 2 The volatilities recorded over a 100-day period on the S&P500 are Y1, Y2, ... Y100. Assume that these Yi's are independent and identically distributed, uniform on the interval [5,15]. Let V = (Y1 + Y2 + ... + Y100)/100. What approximately is P[9.5 < V < 10.5]?

Q.#3 (a) In a certain industrial facility, accidents occur infrequently. It is known that the probability of an accident on any given day is 0.005 and accidents are independent of each other.(i) What is the probability that in any given period of 400 days there will be an accident on one day?(ii) What is the probability that there are at most three days with an accident?(iii) Find ?(2 < ? ≤ 4 /? < 3)

Q6

In finance, one example of a derivative is a financial asset whose value is determined (derived) from a bundle of various assets, such as mortgages. Suppose a randomly selected mortgage in a certain bundle has a probability of 0.04 of default. Complete parts a to d below. (a) What is the probability that a randomly selected mortgage will not default (that is, pay off)? The probability is (Type an integer or a decimal. Do not round.) (b) What is the probability that a bundle of five randomly selected mortgages will not default assuming the likelihood any one mortgage being paid off is independent of the others? Note: A derivative might be an investment that only pays when all five mortgages do not default. The probability is (Round to four decimal places as needed.) (c) What is the probability that the derivative from part (b) becomes worthless? That is, at least one of the mortgages defaults. The probability is. (Round to four decimal places as needed.) (d) In part (b), we made the…