The market values should be added probabilistically without any hesitation:

(1/3)*(96*9M+100*8M+101*7M)/100 = $7.903M

However, the buyer won't pay $7.903M at delivery - he must pay a certain unit price for the actually delivered balance. How to compute this unit price? Let us first find the average balance as (1/3)*(9M+8M+7M)/100 = $8M, then relate $7.903M to that amount: (7.903/8.0)*100=98.79.

Why have we used the average balance as the denominator? Compare two settlement contracts:

Contract 1: pay random fair value at settlement. We checked above that it is worth $7.903M.
Contract 2: pay 98.79% for every dollar actually delivered. This is worth

(1/3)*(98.79*9+98.79*8+98.79*7)/100 = $7.903M too!

Hence, values of Contract 1 and Contract 2 are identical to one another, which ends the proof.

In particular, we learned that a non-TBA forward contract is worth 21 basis points (6-7 ticks) less than a TBA forward contract, which constitutes the value of the balance delivery option, in our example. The option would be almost worthless for the next-month delivery because there exists little (if any) correlation between future rates and the next-month balance (recall the prepay lag). The option matters for remote settlements (2 and more months), especially for CMOs. Indeed, many CMO classes may amortize quickly before settlement. If, for the same price profile as in the above example, the balances had been $9M, $7M, and $5M (instead of 9, 8, and 7), the value loss would have been nearly half-point.

The ultimate forward settlement formula

Suppose the PV% denotes the expected present value of an MBS' cashflow; it is measured in percentage points off the trade-date balance. Note that an MBS investor is not entitled to receive any proceeds until the first month after the settlement. Similarly FP% is the forward price >>>