Mortgage-Backed Securities: The Products Behind the Crisis

Most people know what a mortgage is. You borrow money to buy a house, you make monthly payments, and after 20 or 30 years you own the house free and clear. But what happens to all those mortgages after the bank gives them out? They get bundled together and sold to investors. That is a mortgage-backed security. Chapter 34 of Wilmott’s book explains how these things work, why they are tricky to price, and what makes them different from every other fixed-income product.

The most interesting part of MBS modeling is not the math of interest rates. It is the math of people. Specifically, you need to model when and why homeowners decide to pay off their mortgages early.

How Individual Mortgages Work

A mortgage is a loan backed by a house. They come in two main flavors:

Fixed rate mortgages have a constant interest rate for the entire life of the loan. Your monthly payment stays the same every month, but the split between interest and principal changes. Early on, most of your payment goes to interest. By the end, most goes to principal. These are the dominant type in the US.

Floating rate mortgages have payments tied to a variable interest rate that the lender adjusts periodically. These are more common in the UK.

For a fixed rate mortgage with interest rate r_M, the monthly payment x on a loan of $1 over N years can be calculated from a simple formula. The remaining balance when there are M payments left can also be computed directly. Nothing complicated here.

As Wilmott puts it: “For more details of how mortgages work play a game of Monopoly.”

The Prepayment Problem

What makes mortgages special is prepayment. The borrower can pay off the remaining balance before the loan matures. This is like the callable feature of a bond, but with a crucial difference. With callable bonds, we assume the issuer acts optimally, calling when it is mathematically best to do so. With mortgages, homeowners do not behave optimally. They prepay for all sorts of reasons:

• They come into some money and want to be debt-free
• They move to a new house
• The house is destroyed and insurance pays off the loan
• They default and insurance covers the lender
• Interest rates fall and they refinance with a cheaper mortgage

Only that last reason approaches “rational” behavior, and even then there is enormous inertia. Many people do not bother refinancing, or do not even know they can. This is a very human problem, not a pure math problem.

What Are Mortgage-Backed Securities?

Mortgage-backed securities (MBS) are created by pooling together many individual mortgages. Investors buy a piece of the pool and receive the combined interest and principal payments from all the homeowners.

The beauty of pooling is diversification. Individual homeowners are unpredictable, but when you have thousands of them in a pool, the average prepayment behavior becomes somewhat stable and modelable. Instead of predicting what one person will do, you predict what the crowd will do on average.

Collateralized mortgage obligations (CMOs) go a step further. They take MBS pools and slice them into different pieces, or “tranches,” with different maturity profiles. A typical CMO tranche might receive interest and principal only during a certain time window.

MBS can also be stripped into components. Principal only (PO) strips receive just the principal payments. They become more valuable when prepayment increases because you get your money back faster. Interest only (IO) strips receive just the interest. These can be very risky because high prepayment means the outstanding loan balance shrinks quickly, and with it, your interest income.

Who Issues MBS?

In the US, the main issuers are the government-sponsored entities with the memorable nicknames:

• Ginnie Mae (Government National Mortgage Association, GNMA)
• Fannie Mae (Federal National Mortgage Association, FNMA)
• Freddie Mac (Federal Home Loan Mortgage Corporation, FHLMC)

Investment banks and house builders also issue private-label MBS.

Modeling Prepayment: The PSA Model

The simplest prepayment model is the Public Securities Association (PSA) model. It ignores interest rates entirely and only considers the age of the mortgage.

The assumption is simple: prepayment starts at zero for a brand-new mortgage, rises linearly at 0.2% per month for the first 30 months, then stays flat at 6% per year. These numbers are in annualized CPR (Conditional Prepayment Rate) terms.

The logic is that people are unlikely to move or refinance right after taking out a mortgage. After about two and a half years, the prepayment rate stabilizes.

Since prepayment in the PSA model is completely deterministic (depends only on time), valuing an MBS under this model is straightforward. You just adjust the cashflows for the known prepayment schedule and discount them. But there is no hedging against the risk that actual prepayment differs from the model, because the model has no random component.

Better Prepayment Models

The PSA model’s big weakness is that it ignores market rates. In reality, prepayment increases when interest rates fall because homeowners refinance. The data shows this clearly. When the spread between the existing mortgage rate (the weighted-average coupon, or WAC) and the current market rate exceeds about 1%, refinancing starts to take off.

A more realistic model separates prepayment into two effects:

prepayment(r, t) = a(t) * f(r)

The function a(t) captures the age effect, similar to the PSA model. The function f(r) captures the interest rate effect. When current rates are well below the mortgage rate, f(r) is high. When they are above, f(r) is low (but not zero, because people still move houses and come into money).

Other factors affect prepayment too. Seasonality matters (people tend to move in summer). Economic conditions like unemployment play a role. And interestingly, there is a two to three month lag between rate changes and prepayment response. People are slow to act.

Once prepayment depends on interest rates, you have a path-dependent problem on your hands. The outstanding balance at any point depends on the entire history of rates, not just the current rate.

Valuing MBS

To value an MBS with interest-rate-dependent prepayment, you need to track several variables:

• r: the spot interest rate
• P: the outstanding balance on each mortgage
• Q: the fraction of mortgages remaining in the pool
• t: time

The MBS value is a function V(r, P, Q, t). The regular payments reduce the balance deterministically. Prepayment reduces both the balance and the number of remaining mortgages randomly (through the dependence on r).

The governing equation looks like the standard bond pricing equation, but with extra terms. There are additional partial derivatives with respect to P and Q, and a cashflow term capturing both regular payments and early repayment of principal.

One nice result: when interest rates equal the mortgage rate and there is no prepayment, the solution is simply V = P. The MBS is worth exactly the outstanding balance. Makes sense.

There is also a similarity reduction that brings the problem from four variables down to three. This helps with computation, but the problem is still more complex than anything we saw in the equity options world.

The Bigger Picture

Wilmott keeps this chapter relatively brief and admits there is “much more to MBSs” than what he covers. He focuses on the conceptual framework and the key modeling challenge: prepayment.

What makes MBS fascinating from a mathematical perspective is that you are modeling human behavior, not just financial variables. The randomness is not just in interest rates. It is in millions of homeowners making individual decisions for personal reasons that have nothing to do with optimal exercise theory.

This is also what made MBS so dangerous in the 2008 financial crisis. The models assumed certain prepayment and default patterns based on historical data. When housing markets collapsed and behavior changed dramatically, the models broke down. But that story is beyond the scope of this chapter.

Key Takeaways

Mortgage-backed securities are pools of individual home loans sold to investors. The key challenge in pricing them is prepayment: homeowners can pay off their mortgages early, and they do so for reasons that are only loosely connected to interest rates. Simple models like PSA treat prepayment as deterministic. Better models make prepayment a function of current rates, introducing path dependence. The valuation requires tracking the interest rate, outstanding balance, and remaining pool fraction, making MBS one of the more computationally demanding fixed-income products. And unlike most financial models, the randomness here is partly about people, not just markets.

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