FMP 18: Mortgages and Mortgage-Backed Securities
A mortgage is a loan secured on property, residential or commercial, and a mortgage-backed security is an investment manufactured out of the cash flows that a portfolio of such loans throws off. The U.S. market is the one worth studying in detail, given its size and its importance to fixed-income investors.
Two lengths dominate the American residential market: 15 years and 30 years. Within either, the borrower chooses between a rate locked for the term and a rate that moves. The floating version, an adjustable-rate mortgage or ARM, holds its rate steady for several years before resetting off a published index. The one-year Treasury rate, known here as the constant maturity Treasury rate, is one common choice; a cost-of-funds index, the average interest expense carried by lenders in a region, is another.
Who carries the interest rate risk
An adjustable-rate mortgage moves interest rate risk off the lender and onto the household, so it is the safer product for the lender and the riskier one for the borrower. Borrowers do not accept that trade for nothing: lenders pay for it up front by setting the opening rate on an ARM below the rate on a comparable fixed-rate loan.
The prepayment option
The rest of this lesson concentrates on fixed-rate mortgages, and the feature that makes them awkward to value is an option the borrower holds. At any moment the balance can be handed back and the contract closed. That right is American-style, exercisable at any date rather than one fixed date, and it is called the prepayment option.
Two situations trigger it most often. The property changes hands, so the loan attached to it is cleared. Or market rates fall far enough that the household borrows again more cheaply and settles the old loan with the proceeds. There is usually no penalty for either, which makes the option genuinely valuable.
Whatever the borrower gains, the investor in a mortgage-backed security loses. Cash returned early has to be put back to work, and large volumes come back exactly when reinvestment rates are poor, since falling rates are what set the wave off. No analyst can value a mortgage portfolio sensibly while ignoring that option.
The quoted rate has to be expressed on the same compounding basis as the payment schedule, and the two do not always agree. Take a Canadian mortgage quoted at 4% with semi-annual compounding. Restated with monthly compounding it becomes 0.03967, or 3.967%. American mortgage rates already carry monthly compounding, so the quote is divided by twelve directly.
The payment is set by one condition: the equal monthly payments must have a present value, discounted at the mortgage rate, equal to the sum advanced. Since every payment is identical, the discounting collapses into a geometric series, and the schedule reduces to one equation linking the amount borrowed A, the annual rate R compounded monthly, the payment X, and a life of T years.
Full amortization is what separates a mortgage from a bond. A Treasury bond repays its face value in one lump at maturity, whereas a mortgage returns capital gradually, leaving nothing outstanding after the final payment, since each payment settles that month’s interest and retires a slice of principal.
A household borrows USD 250,000 on a 30-year U.S. mortgage. The fixed rate is 6% with monthly compounding.
An amortization table sets out, month by month, how the constant payment divides between interest and repayment of capital when nothing is prepaid. Interest is charged on the balance outstanding, and that balance only falls, so the interest slice shrinks every month while the principal slice grows to fill the gap. Early payments are almost entirely interest, late payments almost entirely principal.
Stay with the USD 250,000 loan at 6% per year, which is 0.5% per month, and a payment of USD 1,498.88.
| Month | Balance at month end | Principal repaid | Interest paid | Interest share of payment |
|---|---|---|---|---|
| 0 | 250,000.00 | |||
| 1 | 249,751.12 | 248.88 | 1,250.00 | 83.4% |
| 2 | 249,501.00 | 250.12 | 1,248.76 | 83.3% |
| 3 | 249,249.63 | 251.37 | 1,247.51 | 83.2% |
| 4 | 248,997.00 | 252.63 | 1,246.25 | 83.1% |
| 356 | 5,921.30 | 1,461.96 | 36.92 | 2.5% |
| 357 | 4,452.03 | 1,469.27 | 29.61 | 2.0% |
| 358 | 2,975.42 | 1,476.62 | 22.26 | 1.5% |
| 359 | 1,491.42 | 1,484.00 | 14.88 | 1.0% |
| 360 | 0.00 | 1,491.42 | 7.46 | 0.5% |
Source: GARP, Financial Markets and Products, Chapter 18. The final column is derived, not printed in the source.
This arithmetic describes how principal is retired once a rate has been quoted, and nothing more. What the cash flows are worth on a given day is settled by the term structure prevailing then together with the expected cost of the prepayment option.
Individual mortgages are not investable in any practical sense, so they are gathered into portfolios, usually called mortgage pools, whose loans resemble one another in type, rate and origination date. Two statistics describe such a pool. The weighted-average coupon, or WAC, averages the interest rates; the weighted-average maturity, or WAM, averages the months each loan still has to run. Both weight by the share of outstanding principal a loan contributes.
A two-loan pool makes the mechanics visible. Put a USD 200,000 mortgage paying 4% alongside a USD 400,000 mortgage paying 5%. The larger loan carries twice the weight, so the WAC is 4.667%, nearer 5% than 4%. Give those loans 340 and 280 months left to run and the WAM is 300 months.
A pool holds three mortgages: USD 100,000 with 330 months to maturity at 4%, USD 200,000 with 310 months at 5%, and USD 300,000 with 290 months at 6%.
Other pool statistics
Four more descriptors travel with a pool. The average loan balance is total outstanding principal divided by the number of mortgages. The pool factor, used above, expresses current principal as a percentage of the original amount, and it can only fall. The weighted-average FICO score summarises borrower creditworthiness on a scale from 300 to 850, with many lenders treating a score above 650 as acceptable. The weighted-average loan-to-value ratio, or LTV, compares each loan against the assessed value of its property. Both of those weight by original principal. The geographical distribution records where the financed properties sit.
Prepayments are tracked pool by pool. In any month some borrowers clear their loans outright and others make partial repayments, known as curtailments. The single monthly mortality rate, or SMM, is the percentage of outstanding principal repaid ahead of schedule that month. Scheduled principal is excluded entirely.
A monthly rate is awkward to compare across pools, so it is annualized. The conditional prepayment rate, or CPR, also called the constant prepayment rate, is what an SMM would deliver over a year of repetition. Survival is the way to see it: a fraction 1 minus SMM of the principal remains after one month, that fraction squared after two, and raised to the twelfth power after a year.
A pool starts a month with USD 500,000,000 of principal outstanding. Scheduled principal for the month is USD 2,000,000. The servicer actually collects USD 6,980,000 of principal.
The PSA benchmark
A single CPR ignores loan age, and age matters, since households rarely prepay in the first months of a mortgage. The Public Securities Association, or PSA, built a benchmark around that fact: rates rise for the first 30 months after origination and stay flat afterwards. Under the standard model, called 100% PSA, month one carries an annualized rate of 0.2%, each following month adds another 0.2%, and month 30 reaches 6%. Multiples scale the ramp, so the 150% PSA model opens at 0.3% and plateaus at 9%.
Three institutions in the United States buy mortgages from banks and assemble them into pools: the Government National Mortgage Association, known as Ginnie Mae or GNMA, the Federal National Mortgage Association, known as Fannie Mae or FNMA, and the Federal Home Loan Mortgage Corporation, known as Freddie Mac or FHLMC. Only the first is an arm of the government. The other two are private companies, described as government-sponsored enterprises, whose obligations carry no explicit federal guarantee, although the market assumes an implicit one. Both ran into serious difficulty after the 2007-2008 financial crisis, and the U.S. Treasury injected capital.
A bank that sells its mortgages avoids carrying 30-year loans on its own balance sheet, and the cash released funds the next set of home buyers. That recycling is the public policy point, since it keeps money available for households seeking mortgages. Loans bought by the agencies must first clear standards on size and credit quality.
Pass-through securities
The simplest security written against a pool is a pass-through, in which every investor holds a proportional claim and earns an identical return. Collections reach investors after the agency deducts its fees for guaranteeing and servicing the loans. Pools are assembled so that the return comes in 50-basis point steps, such as 3%, 3.5% and 4%. Calling that return a coupon invites a false comparison with a bond, since the money arrives monthly and each remittance mixes interest with returned principal.
The agency guarantee covers default and stops there. If borrowers stop paying, the agency makes investors whole; if they repay early, investors absorb that themselves, reinvesting the cash at the low rates that set the refinancing wave off.
Non-agency deals
Securities not issued by GNMA, FNMA or FHLMC work differently. A bank sells a mortgage portfolio to a special purpose vehicle, which issues securities against it and passes the collections through. Nobody stands behind the credit, so the dominant exposure is default risk rather than prepayment risk, and the vehicle usually issues several tranches carrying different amounts of it. This structure was central to the 2007-2008 crisis, when lending standards had loosened and realised default rates ran far above the priced level.
A pass-through is identified by three attributes: who issued it, what it pays, and how long the underlying loans were written for. Naming a GNMA 30-year 4% pool describes one completely, and that maturity label reports the original term rather than the years still left to run. What separates these securities from other high quality investments is prepayment risk, which depends on the level of interest rates and on the balances outstanding.
Trading happens in two venues. In the specified pools market, buyer and seller name an exact pool, an exact amount and a price. In the to-be-announced market, shortened to TBA, the pool is deliberately left open, and the two sides fix parameters instead: the issuer, say FNMA; the original maturity, say 30 years; the coupon, say 4.5%; the price per USD 100 of par value, say USD 104.50; the par value, say USD 100 million; and the settlement month, say August. Delivery lies ahead, so a TBA is a forward contract, and it draws far more activity than the specified pools market.
The cheapest-to-deliver option
Consider a seller who has agreed to deliver, in August, mortgages from an FNMA 30-year pool paying 4.5%, USD 100 million of par value, priced at USD 104.50 million plus accrued interest from the start of the month. Nothing in that agreement names a pool, so any qualifying August delivery satisfies it. That is a cheapest-to-deliver option, and a seller with a prepayment model works out which qualifying pool is worth least. Two business days ahead of settlement the seller declares which pool or pools are coming, and no more than three may be used.
Settlement dates inside the month are published by the Securities Industry and Financial Markets Association, or SIFMA, and normally fall on the twelfth or thirteenth. What the seller collects is the agreed price together with interest accrued since the first, every month being treated as 30 days. Rules also bound what may be delivered: inside a 30-year pool, remaining maturities must fall between 15 and 30 years.
A dollar roll pairs two TBA trades in opposite directions and in consecutive settlement months. A trader sells, for August settlement, a 30-year FNMA pool paying 4.5% with USD 100 million of par value, and buys the equivalent September position. Cash comes in now and goes out a month later, so the trade raises funds, often at an attractive rate.
A repurchase agreement does something superficially similar, but two features set a dollar roll apart. The securities coming back need not be the ones handed over, since the counterparty may return a different qualifying pool, possibly one with worse prepayment behaviour. And no interest is added to the repurchase price: the initiating party gives up one month of pool income instead, and the counterparty picks it up.
Valuing the roll
Four quantities settle the arithmetic. A is the sale price in the first month, accrued interest included. B is the repurchase price in the second month, again with accrued interest. C is the interest earned on the sale proceeds over the intervening month. D is the coupon and principal the pool would have delivered that month had it been kept.
A holding of USD 1 million par value in a pool paying 4.5% is sold at USD 102.50 during March and repurchased at USD 102.00 during April. Both payment dates are the twelfth. The sale proceeds earn 0.1% for the month. Had the pool been kept, interest and principal during the roll month would have come to 0.45% of par.
A pass-through spreads prepayment risk evenly across every holder. A collateralized mortgage obligation, or CMO, does the opposite: it carves the same pool into classes, called tranches, carrying deliberately unequal amounts of that risk.
Take three tranches funding one pool. Tranche A investors put up 30% of the principal, Tranche B investors 50%, and Tranche C investors the remaining 20%. Each earns interest on whatever principal it still has outstanding. Principal is where the structure bites: every dollar of it, scheduled repayment and prepayment alike, goes to Tranche A until Tranche A is retired. Only then does principal reach Tranche B, and Tranche C waits until Tranche B has gone.
The consequence is a lopsided distribution of risk. Tranche A is repaid first, so a wave of prepayments lands almost entirely on it, while Tranche C sits behind two cushions and is barely disturbed. Shifting the percentages redistributes that exposure.
Stripped mortgage-backed securities
A second way to divide a pool separates the two kinds of cash flow outright. Interest-only securities, or IOs, take every interest payment the pool produces, and principal-only securities, or POs, take every principal payment. Both are risky, and they respond to prepayments in opposite directions. Faster prepayment pulls the principal forward, so a PO, a claim on a fixed total sum, gains as that sum arrives sooner. The same acceleration shrinks the balance on which interest is charged, so the IO loses. Slower prepayment reverses both effects.
Putting a price on the prepayment option is harder than pricing an ordinary interest rate option, because households do not exercise on rates alone. Prepayments come from four sources, and a usable model handles all four: refinancing, turnover, defaults and curtailments.
Refinancing
Refinancing means clearing one mortgage by taking out another on the same property, and a fall in market rates is the usual trigger, since it lowers the monthly payment. A borrower whose credit standing has improved can obtain a better rate unaided, and one whose property has appreciated can negotiate a larger loan, which is called cash-out refinancing.
The pull toward refinancing is captured by an incentive function, the simplest version subtracting the current mortgage rate available to borrowers, R, from the pool’s weighted-average coupon.
The second version measures the saving in present value terms rather than as a rate difference, and by scaling with loan size it reproduces an empirical regularity: larger loans prepay faster. Feed the incentive into an annualized prepayment rate of the form 1 divided by the quantity a plus b times e raised to the power minus cI, with a, b and c fitted to observed data, and an S-shaped curve results. Illustrated with the simple incentive in basis points, setting a = 4 alongside b = 0.02 and c = 25, it matches experience. Where rates have risen, so that WAC – R is negative, almost nobody refinances; as rates fall the prepayment rate climbs steeply, then flattens.
Those fitted parameters are not constants of nature; the economic environment moves them. Rising house prices push prepayments up, and burnout pushes them down. Once rates have been low for a while, the borrowers most alert to the opportunity, those with good credit, large balances or financial sophistication, have already refinanced and left. Whoever remains is less likely to act, so the pool responds weakly to a rate that would once have provoked a wave. Burnout makes the prepayment function path dependent: a seasoned pool responds to where rates have been, not only to where they are.
Turnover, defaults and curtailments
Turnover prepayments arise when the house is sold. Sales run higher in summer than in winter and are rare in the first months of a mortgage, since households seldom move straight after buying. Geography and borrower age both shift the rate. Interest rates barely enter, with one exception: a household paying a below-market rate is reluctant to give it up by moving, which is the lock-in effect.
Defaults belong in a prepayment model even though an agency pool is guaranteed. When a borrower defaults the agency settles the outstanding balance, and from the investor’s side that money arrives exactly as a prepayment would. Defaults added heavily to prepayment experience during the 2007-2008 crisis, and models predict the component from average FICO scores, LTVs and house price history. Curtailments are partial prepayments, concentrated in old loans with small balances, and when one or two years remain they can reach 5%.
Valuation starts with a prepayment model, and the model needs paths, not points, for two variables. Interest rates matter over their whole history since origination, because of burnout. House prices matter too, since sharp rises invite cash-out refinancing while sharp falls produce defaults. Pool descriptors enter as well: average loan size drives the level of prepayment, geography feeds house price and turnover assumptions, FICO scores and LTVs drive defaults, and loan age drives curtailments.
The value is then estimated by Monte Carlo simulation. Sample a month-by-month path for risk-free rates and house prices from their probability distributions, along with the spread of the mortgage rate over the risk-free rate. Apply the prepayment model month by month to obtain prepayment rates, and turn those into cash flows. Discount those back from the end of the life of the security at the rate sampled for each month, then repeat for many paths and average the present values.
Path dependence is why simulation beats a tree here. A given month’s prepayment depends on the history of rates and house prices as well as their current levels, and a tree cannot easily carry that history at each node.
A two-scenario illustration
Take a pool of new 15-year mortgages paying 4% with monthly compounding and USD 100,000 of principal, with the risk-free rate 0.5% below the mortgage rate at every maturity. If rates never move and nobody prepays, discounting at 3.5% a year values the position at USD 103,470, made up of USD 27,759 of interest and USD 75,711 of principal.
Now add one piece of uncertainty, resolved almost immediately. With probability 0.5 the mortgage rate rises to 6% and nothing is prepaid. With probability 0.5 it drops to 2%, and 2% of the balance remaining after each scheduled repayment is prepaid every month. The branches are valued separately and averaged.
| Scenario | Total | Change in total | Principal | Interest |
|---|---|---|---|---|
| Mortgage rate decrease to 2% | 107,716 | +4,246 | 95,370 | 12,326 |
| Mortgage rate increase to 6% | 90,528 | -12,942 | 65,269 | 25,259 |
| Average of the 6% and 2% environments | 99,122 | -4,348 | 80,320 | 18,802 |
| No interest rate change | 103,470 | 75,711 | 27,759 |
Source: GARP, Financial Markets and Products, Chapter 18, Table 18.4, reordered, with the change column derived. The interest and principal entries printed for the 2% environment sum to 107,696 rather than the 107,716 shown, and the averaged row of 18,802 is consistent with an interest present value of 12,346.
Per USD 100 of principal, uncertainty about rates and the prepayments following from it cut the price to USD 99.122 from USD 103.470. The table also prices the strips. An IO carved from this pool drops to USD 18.802 from USD 27.759, while a PO climbs to USD 80.320 from USD 75.711, so prepayment uncertainty hurts the interest strip and helps the principal strip.
Negative convexity
Look at the asymmetry in the table. A 200 basis point rise in the mortgage rate costs 12,942, while a fall of the same size gains only 4,246. An ordinary non-callable bond behaves the other way round, gaining more from a fall than it loses on an equivalent rise, which is positive convexity. Over the region where refinancing bites, the mortgage-backed security is its mirror image, and that is negative convexity.
The mechanism is the prepayment option. When rates fall, borrowers exercise it and hand back principal at par, so the security cannot appreciate as a comparable bond would, and the returned money is reinvested at the new low rates. When rates rise, prepayments slow and the principal stays outstanding longer, so the security extends just as an investor would least want it to. Rising rates lengthen it and falling rates shorten it, unhelpfully in both directions.
The option-adjusted spread, or OAS, is the amount by which the expected return on a fixed-income instrument exceeds the risk-free return once embedded options have been accounted for. For a mortgage-backed security the option in question is the prepayment option, and the spread is measured against Treasury instruments.
Calculating it reuses the simulation already described. Start with an estimate of the OAS, run the Monte Carlo valuation discounting at the Treasury rate plus that estimate, and compare the result with the market price. If the market price is above the simulated price, lower the estimate; if it is below, raise the estimate. Keep adjusting until the two agree.
Successive bisection turns that into a workable algorithm. Push the OAS down until the simulated price sits above the market price, and up until it sits below, which brackets the answer. Value the security at the midpoint and replace whichever bound it beats, halving the interval on each pass. The same random samples should be used every iteration, otherwise simulation noise is mistaken for a change in the spread.
Return to the pool valued above and suppose it trades at USD 98.00. The OAS is whatever spread, added to the discount rates, brings the present value per USD 100,000 of par value down to USD 98,000. That spread is 24.67 basis points. Discount the two scenarios at 1.7467% and 5.7467%, in place of 1.5% and 5.5%, and the value moves from USD 99.12 to USD 98.00.
What OAS is good for, and where it fails
The natural use is relative value. A pool offering an OAS of 80 basis points, meaning 0.8% above the Treasury rate, looks better than one offering 40 basis points. That comparison is only as good as the prepayment model behind it, and a model that is wrong or poorly calibrated produces a spread meaning nothing. A high OAS is a prompt to investigate rather than a signal to buy: find the institutional or technical reason the pool trades away from the market, then the assumption generating the result, and ask whether it is defensible.
Hedging carries a related difficulty. If the model does describe behaviour correctly, the interest rate exposure can be hedged approximately and an expected profit locked in. Treasuries are the obvious hedge, yet mortgage rates and Treasury rates do not move in lockstep, so residual exposure survives even a perfect prepayment model.