4. The Uris & Warren Ratings Agency rates bonds on a simplied scale with just three categories: A, B,

and C.T he Professors Pension Fund has all its money invested in two bonds, X and Y , both of which are

currently rated B. Over the course of the next year the ratings of the bonds may change; the end-of-year

value (in millions) of each bond depends on its end-of-year rating as in the following table:

Rating X Value Y Value

A

100

100

B

75

75

C

50

50

The joint distribution of the end-of-year ratings of the two bonds is given by the following table:

Y

A

B

C

A

0.2

0.05

0

X

B

0

0.4

0.1

C

0

0.05

0.2

(a) Find the probability that bond X will be rated C at the end of one year.

(b) Find the expected value of the year-end value of bond X.

(c) The Pension Fund buys an insurance contract that will pay 20 million if either bond is downgraded

to C.If both bonds are downgraded the contract still pays 20; if neither is downgraded the contract pays

nothing.F ind the expected value of the payo of the insurance contract.