The Johnsons Change Their Life Insurance Coverage Harry and Belinda Johnson spend $20 per month on life insurance in the form of a premium on a $10,000, paid-at-65 cash-value policy on Harry that his parents bought for him years ago. Belinda has a group term i-surance policy from her employer with a face amount of $200,000. By choosing a group life insurance plan from his menu of employee benefits, Harry now has $100,000 of group term life insurance. Harry and Belinda have decided that, because they have no children, they could reduce their life insurance needs by protecting one another's income for only four years, assuming the survivor would be able to fend for himself or herself after that time. They also realize that their savings fund is so low that it would have no bearing on their life insurance needs. Harry and Belinda are basing their calculations on a projected 4 percent rate of return after taxes and inflation. They also estimate the following expenses: $15,000 for final expenses, $20,000 for readjustment expenses,

and $5,000 for repayment of short-term debts.

Requirement:

(a) Should the $3,000 interest earnings from Harry's trust fund be included in his annual income for the purposes of calculating the likely dollar loss if he were to die? (See the discussions about the Johnsons in Chapter 1 beginning on page 34.) Explain your response.

(b) Based on your response to the previous question, how much more life insurance does Harry need? Use the Run the Numbers worksheet on page 366 to arrive at your answer.

(c) Repeat the calculations to arrive at the additional life insurance needed on Belinda's life.

(d) How might the Johnsons most economically meet any additional life insurance needs you have determined they may have?

e) In addition to their life insurance planning, how might the Johnsons begin to prepare for their retirement years?

2. A Markov chain with state space {1, 2, 3} has transition probability matrix 00 0.3 0.1 a: 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (b) What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (c) Consider a general three-state Markov chain with transition matrix 3011 3012 1013 P = P21 P22 P23 1031 P32 P33 Give an example of a specic set of probabilities jag-'3; for which the Markov chain is not irreducible (there is no single right answer to this1 of course l]. Consider a standard chessboard with an 8 x 8 grid of possible locations. We define a Markov chain by randomly moving a single chess piece on this board. The initial location Xo is sampled uniformly among the 82 = 64 squares. At time t, the piece then chooses Xt+1 by sampling uniformly from the set of legal moves given its current location Xt. For a description of legal chess moves, see: http://en. wikipedia. org/wiki/Rules_of_chess#Basic_moves. a) Suppose the chess piece is a king, which can move to any of the 8 adjacent squares. Is the Markov chain irreducible? Is the Markov chain aperiodic? b) Suppose the chess piece is a bishop. Is the Markov chain irreducible? Is the Markov chain aperiodic? c) Suppose the chess piece is a knight. Is the Markov chain irreducible? Is the Markov chain aperiodic?SECTION 5.2. The Weak Law of Large Numbers Problem 4. In order to estimate f, the true fraction of smokers in a large population. Alvin selects n people at random. His estimator Mn is obtained by dividing Sn, the number of smokers in his sample, by n, i.e., Mn = Sn. Alvin chooses the sample size n to be the smallest possible number for which the Chebyshev inequality yields a guarantee that P(IMn - 1 Z E) 56, where e and o are some prespecified tolerances. Determine how the value of n recom- mended by the Chebyshev inequality changes in the following cases. (a) The value of e is reduced to half its original value. (b) The probability o is reduced to half its original value