Describe the method of producing scenarios used in the historical simulation over the first three decisions (two years).

a) How many yield curves need to be assessed at the beginning of '64 for the years '64, '65, and '66?

b) Draw a decision tree depicting the probability that each of the yield curves indicated in (a) occurs. What use is made of the uncertainty in the estimates of the change in the twenty-year rate (thirty-year rate) as a fraction of the change in the one-year rate? How could this be modified?

c) Show how to compute the parameters of the functional form for each of the yield curves forecast for the start of '65.

d) Explain how you might assess the required information in actual practice, using a model similar to that described in Exercise 1. Draw a distinction between short-term forecasting (3 months) and long-term forecasting (2 years).

Exercise 1

A bank considering using the BONDS model to aid in portfolio planning divides its portfolio into two pools of funds-U.S. Governments and all grades of municipals. The bank managers forecast a yield curve for their group of municipals and are willing to treat them as one category. The investment portfolio decisions are revised monthly, but no securities with less than three months to maturity are purchased.

a) Assuming that the bank employs a two-year planning horizon with four planning periods of 3 months, 3 months, 6 months, and one year, how many constraints of each type will their model have, using 5-point approximations to the uncertainties in the first two periods and 3-point approximations in the last two periods? (Assume no initial holdings, but include the remaining constraints of Table 14.1.)

b) The bank feels that it can aggregate the purchase decisions on individual securities into the following maturities: 3 months, 6 months, 1, 2, 3, 5, 10, and 20 years for U.S. Governments and 1, 2, 3, 4, 5, 10, 20, and 30 years for the municipal group. How many decision variables will the model sketched in (a) have? (Again, assume no initial holdings.)

c) In fact, the bank does have initial holdings of both Governments and municipals. The bank is willing to aggregate these holdings in the same maturities as used for new purchases. How many additional constraints and variables need to be added to account for the initial holdings?

d) How is a sub problem defined for the model described in (a), (b), and (c)? How many such sub problems are there? Why is it impossible to combine the sub problem from initial holdings with those of subsequent purchases?

e) How many constraints does the restricted master have? How would you find an initial basic feasible solution to the restricted master?