Bank X has 8 subprime mortgage loans to sell to investors.

X is supposed to receive $100 as the “principal + interest” by each borrower.

However, many borrowers are low-credit ones, so the true values of many loans are much less than $100.

The true, current value of k’th loan = X(k) = 100 - 5^k, where k = 1, 2,…, 8.

Thus, the values of the 8 loans are 95, 90, …, 65, and 60. 

Each of the investors wants to buy only one mortgage loan, but no investor knows exactly which loan is worth how much.

They only know

A. 8 individual loans’ values in the pool are 95, 90, …, 65, and 60

B. the standard deviation (s.d.) of the true values of the 8 loans(here, s.d. = population s.d. -- Use the Excel function of “STDEV.P”)

The s.d. is a measure of how risky buying one of the 8 loans is, or, equivalently, the higher the s.d. is, the more risky buying a loan is.

Since investors are risk averse, they use a rule (of thumb) to value each loan in the pool in such a way that they pay “the average loan value - the s.d.” to any loan in the pool.

Equivalently, they pay “77.5 - the s.d.” to buy any loan in the pool.

Also, they calculate the s.d. values rounded off to the fifth digit after the decimal point.

(e.g.) if the real s.d.= 14.478239### -- use the s.d. = 14.47824, Or,just use the Excel STDEV.P function to compute the s.d. value of a pool or of a tranche

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Suppose X can sell the 8 loans in several tranches.

A tranche of loans must include at least 2 loans.

So, X can sell   2 tranches of 6 loans and 2 loans, 5 loans + 3 loans, or 4 + 4

                          3 tranches of 4+2+2 or 3+3+2

                          4 tranches of 2+2+2+2.

Assumptions of A. and B. for the original pool also apply to these tranches.

e.g.) tranche = X(3)+X(5)+X(7) -- investors know that the three loan values are 85, 75, and 65, and they will pay (for each loan) only “75 - the s.d. of the 3 loan values”

In order to maximize the sum of the sales prices of the 8 loans,

1) How many tranches should X create?

2) Which loans are included in each tranche? e.g.) tranche 1 = X(1)+X(5)+X(7), tranche 2 = X(2)+X(3), …

3) How much is the maximized sum of the sales prices of the 8 loans, then? (Round off your answer to an integer: e.g.) if answer = 550.78457 -- $551 )