An investor holds a portfolio of 50 zero-coupon bonds, each one with maturity in 3 years and a face value of 100 $. Half of the bonds have an initial rating Aaa, 10% are rated B and another 40% are rated Caa. The holder of the bond receives 100$ at maturity plus and interest rate of 3%, 4% and 5% of the face value of the bond is initially rated Aaa, B and Caa respectively, provided the bond does not default. If it does default the investor recovers 70% of its face value without collecting any interest. Refer to the rating one-step transition matrix Mk , posted on D2L. a) Compute the transition probability matrix in one and three steps and write the probability that a bond rated Aaa, B or Caa defaults. b) The random variables Y Aaa, YB and Ycaa are defined as the amount received by the investor for each bond rated Aaa, B and Caa respec- tively. Compute the expected value of each variable. c) Assume default of bonds in the portfolio are independent events. Compute the expected value of the random variable Z defined as the amount received by the investor at maturity from the entire portfolio. An investor holds a portfolio of 50 zero-coupon bonds, each one with maturity in 3 years and a face value of 100 $. Half of the bonds have an initial rating Aaa, 10% are rated B and another 40% are rated Caa. The holder of the bond receives 100$ at maturity plus and interest rate of 3%, 4% and 5% of the face value of the bond is initially rated Aaa, B and Caa respectively, provided the bond does not default. If it does default the investor recovers 70% of its face value without collecting any interest. Refer to the rating one-step transition matrix Mk , posted on D2L. a) Compute the transition probability matrix in one and three steps and write the probability that a bond rated Aaa, B or Caa defaults. b) The random variables Y Aaa, YB and Ycaa are defined as the amount received by the investor for each bond rated Aaa, B and Caa respec- tively. Compute the expected value of each variable. c) Assume default of bonds in the portfolio are independent events. Compute the expected value of the random variable Z defined as the amount received by the investor at maturity from the entire portfolio