A15.1 MONTH‐END BALANCES

This section shows that, under the principal amortization schedule described in the text, the balance outstanding at any time equals the present value of the remaining payments at the original mortgage rate. Let be the term of the mortgage, in months; let be the mortgage rate; let be the monthly payment, and let be the balance outstanding at the end of month , . By definition, as discussed in the text,

(A15.1)

If the balance outstanding at the end of month does equal the present value of the remaining payments at the rate , then,

(A15.2)

(A15.3)

According to the logic of the amortization table, the interest component of the payment for month is , and the principal component is . Because is, by definition, the present value of the remaining payments at the start of the mortgage, this section needs to prove that, for any ,

(A15.4)

To prove this, rearrange terms and then substitute for and from Equations (A15.2) and (A15.3), respectively,

(A15.5)

(A15.6)

(A15.7)

(A15.8)

which is clearly true for any parameters of the problem.

A15.2 PRICING MBS WITH TERM STRUCTURE MODELS

This section very briefly discusses three problems that arise in the context of pricing MBS in term structure models: path dependence, proxies for the mortgage rate, and factors apart from interest rates.

As for path dependence, the burnout and media effects imply that the value of an MBS depends not only on the current term structure of interest rates, but also on the history of the term structure. Pricing by backward induction, however, described in earlier chapters of the book, does not naturally accommodate path dependence: at any node of a binomial tree, for example, there is no memory of how the interest rate process moved from its current state to that particular node of the tree.

A popular solution for pricing path‐dependent claims is Monte Carlo simulation. To price a security in this framework, in a one‐factor setting, proceed as follows. First, generate a large number of paths of interest rates at the frequency and to the horizon desired. For this purpose, paths are generated using a particular risk‐neutral process for the short‐term rate. Second, calculate the cash flows of the security along each path. In the mortgage context, this would include the security's scheduled payments along with its prepayments. Burnout and the media effect can be implemented in this framework, because each path is available in its entirety as cash flows are calculated. Third, starting at the end of each path, calculate the discounted value of the security's cash flows along each path using the interest rates along that path. Fourth, compute the value of the security as the average of the discounted values across paths.

To connect Monte Carlo simulation with the pricing approach used elsewhere in the book, recall Equation (A11.5), reproduced here for convenience,

(A15.9)

where is the short‐term rate in period , is the value of a claim in periods, and is the price of the claim today. In light of the discussion in this section, the term inside the hard brackets is analogous to the discounted value of a security along one path. The expectation is analogous to the average of those discounted values across paths.

In the Monte Carlo framework, measures of interest rate sensitivity can be computed by shifting the initial term structure in some manner, repeating the valuation process, and calculating the difference between the initial and shifted prices. As a numerical matter, paths should not be regenerated between the initial and shifted valuations, as that would introduce noise into the sensitivity calculations.

A general drawback of Monte Carlo simulation is that is does not naturally accommodate the valuation of American‐ or Bermudan‐style options, because the value of holding an option at any state typically requires pricing by backward induction. While methodological advances are now used to overcome this problem,¹ the issue does not really arise when valuing the prepayment option. As it is generally accepted that homeowner behavior is not well explained as the optimal exercise of a fixed income option, a path‐dependent prepayment function, which is well‐suited to Monte Carlo simulation, is the preferred approach.

The second problem addressed in this section relates to the mortgage rate. Valuing an MBS along a path requires both the benchmark of discounting rates as well as the mortgage rate. Discounting can be done at benchmark rates plus a spread, but the incentive of a prepayment model depends on the current mortgage rate. The difficulty is that calculating the fair mortgage rate at a single date on a single path of a Monte Carlo simulation is a problem of the same order of magnitude as the original problem of pricing a particular MBS as of the current state! Common practice, therefore, is to build a simple model of the mortgage rate as a function of benchmark rates and, perhaps, volatility, for example, an estimated regression model of the mortgage rate as a function of one or more benchmark rates and rate volatility. It may not be trivial to compute long‐term benchmark rates along a path of short‐term rates, but this difficulty can usually be overcome with a closed‐form solution for long‐term rates or with a numerical approximation consistent with the short‐term rate process.

The third problem is that defaults and turnover depend on variables other than interest rates, like housing prices, which are not traditionally included in the pricing of fixed income securities. Incorporating these variables, including their correlations with interest rates, is yet another challenge of pricing MBS.