>> solve the following please

Suppose that you will receive annual payments of $11,000 for a period of 10 years. The first payment will be made 5 years from now. If the interest rate is 5%, what is the present value of this stream of payments? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Present value You believe you will need to have saved $400,000 by the time you retire in 40 years In order to live comfortably. If the Interest rate is 5% per year, how much must you save each year to meet your retirement goal? (Do not round intermediate calculations, Round your answer to 2 decimal places.) Annual savings What is the value of a perpetuity that pays $100 every 6 months forever? The Interest rate quoted on an APR basis Is 6.8% (Do not round intermediate calculations. Round your answer to 2 decimal places.) Value of a perpetuity Good news: You will almost certainly be a millionaire by the time you retire in 40 years. Bad news: The Inflation rate over your lifetime will average about 2.9%. a. What will be the real value of $1 million by the time you retire in terms of today's dollars? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Real value b. What real annuity (In today's dollars) will $1 million support if the real Interest rate at retirement is 2.1% and the annuity must last for 10 years? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Real annuity5] You have 515,050 in savings, and in five years would like to buy a used carthat would cost $15,565. What is the minimum interest rate you would need to earn on your 516,565 to succeed in your goal? 5] You buy a house, and take out a mortgage for 51?5,DI]'D at a rate of 3.525% with equal annual payments over the next 35 years. What will your payments be? T] You want to retire 40 years from now, and have 53 million in your retirement accourrt at that time. You will make equal payments each year to the account. If you can earn a 5% annual return on your account, how much should you set aside each year? 31A tenyear bond has annual payments that start at 51000 one year from now and increase by 3.5% each year. If the discount rate is 4.25%, what is the present value of the bond? 9] After graduating from ILI, you have a student loan that must he paid off. Your lender gives you two choices: {a} pay a xed amount of $2565 each year [starting a year from now] for 1!] years, or {b} pay escalating amounts that start 515613 {a year from now] and increase by 5% each year for 12 years. Assuming a discount rate of 8%, which has a lower present value? 10] An investment costs 5265!] up front and 5255!] five years from now. It yields returns of SAILJD every fourth year [in years 4, B, 12, etc} forthe next 2!] years. In addition, in year 20, it will pay off an add'rtional amount of 52505. If the discount rate is 1.5%, is this a worthwhile investment? What is the standard deviation of a portfolio which is comprised of $3.400 invested in stock 5 and $3,600 in stock T? Returns if State Probability of State of Recession 1.36 points a. If you borrow $2,300 and agree to repay the loan in six equal annual payments at an interest rate of 11%, what will your payment be? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Amount of payment b. What will your payment be if you make the first payment on the loan immediately instead of at the end of the first year? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Amount of paymentLet X1, X2 and X3 be three independent and identically distributed random variables with P(X1 = -1) = 1 - p and P(X1 = 1) = p, where p E (0, 1) is a fixed parameter. We form a new sequence of random variables Y1 = X1, Y2 = X1X2 and Y3 = X1X2X3- (a) Obtain the marginal PMFs of Yz and Y's. (b) Obtain the conditional PMFs of Yz and Y3 given Yi = y1 for y1 = -1, 1. (c) Obtain the joint PMFs of each of (Y1, Y2), (Yi, Ya) and (Y2, Ya). (d) Compute E[Y,Y2], E[YlY;] and E[Y2Ys]. (e) Are Y, and Y2 independent? Are Y, and Y3 independent? Are Y2 and Y3 independent?2. (5 points) Let Xo, X1, X2, X4, Xs, Xe be i.i.d random variables, each with E[X;] = 2, and Var(X,) = 1. Define a new random process Y1, Y2, Y3, YA, Y's such that: Y1 = X2 - X1 Y2 = X3 - X2 Y3 = X1 - X3 YA = Xs - XA Y's = X6 - X's (a) (1 points) Find E[Y, + Y2 + Y3 + Y4 + Ys] (b) (2 points) Find Cov(Y1, Y2) (c) (2 points) Find Var(Yi + Y2 + Y3)MARKOV CHAINS 1. Let Y1, Y3, Ys,..., be a sequence of independent and identically distributed random variables with P(Y2k+1 = 1) = P(Y2k+1 = -1) = = k = 0,1,2 ... and define Yzk Yak+1 for k = 1, 2, 3,... Y2k-1 (1) Show that (Yk : k =1, 2,...) is a sequence of independent and identically distributed random variables. Hint: You may use the fact that, if X, Y are two variables that take only two values and E(XY) = E(X)E(Y), then X, Y are independent. (ii) Explain whether or not (Yk : k = 1, 2,...} constitutes a Markov chain. (iii) (a) State the transition probabilities pij(n) = P(Ym+n = j | Ym = i) of the sequence {Yk : k = 1, 2.... } 72 (b) Hence show that these probabilities do not depend on the current state and that they satisfy the Chapman-Kolmogorov equations.\f3. [10 pts] Consider the simple linear regression model yi = 60+81(xi-x) +ci(i= 1,2,..,n), where x= ni=1 xi. From the least-squares criterion S(80,81), find the least-squares estimators of 80 and 1 for this model. Hint: Do not expand the term (xi-x). That is, do not expand nti= 1 yi*(xi-x) as nZi=1 yixi- nZi=1 yi x for easier computations. Also remember what ni=1(xi-x) is. 3. [10 pts] Consider the simple linear regression model yi = Po + Bi(x; - I) + &; (i=1,2,...,n), where I = _, I;. From the least-squares criterion S(Bo, 81), find the least-squares estimators of Bo and B, for this model. n n Hint: Do not expand the term (r; - I). That is, do not expand ) yi(x, - F) as ur - 1= 1 1=1 n it for easier computations. Also remember what E(x - 1) is. i-1 i-11. Some (More) Math Review a) Let N = 3. Expand out all the terms in this expression: Cov Xi b) Now write out all the terms using the formula from last class for variance of a sum: Var( X:) = _Var(X) + > > Cov(X, X;) i-1 1=1 i-lj=1ifi Verify that (a) and (b) are giving you the same thing. Hint: Cov(X, X) = Var(X). c) Suppose that D is a roulette wheel that takes on the values {1, 2, 3} all with probability 1/3. What is the expected value of this random variable? What about the variance? d) Now suppose that M is another roulette wheel that takes on the values {1, 2,3} all with probability 1/3. Solve for the expected value of 1/M. e) Finally, suppose that D and M are independent. Solve for: E Hint: You do not need to do any new calculations here. Just remember that for independent RVs, E(XY) = E(X)E(Y). f) Does E(D/M) = E(D)/ E(M)