13.2 Mortgages and Mortgage-Backed Securities

Mortgages and Mortgage-Backed Securities

FMP–18 connects household mortgage credit to capital-market instruments. The FRM focus is cash-flow mechanics of fixed-rate mortgages, how pools become pass-through MBS, how CMOs and strips reallocate risk, and how dealers and investors value MBS with prepayment models and option-adjusted spreads.

Residential Mortgage Products

Common U.S. residential products:

Borrowers hold a prepayment option: they may refinance when rates fall, sell the house, or default (credit-driven prepayment). That embedded call is why MBS behave unlike option-free Treasuries.

Fixed-Rate Payment and Amortization

For a fixed-rate fully amortizing loan with principal B0, periodic rate i, and n payments, the level payment A satisfies:

A = B0 × [i(1+i)^n] / [(1+i)^n − 1]

Interest in a period = i × beginning balance. Principal = A − interest. Balance declines until the final payment.

Worked amortization example

Loan $200,000, 6% annual, monthly payments, 30 years. Monthly rate i = 0.06/12 = 0.005; n = 360.

(1+i)^n = (1.005)^360 ≈ 6.0226.

A = 200,000 × [0.005 × 6.0226] / [6.0226 − 1] = 200,000 × 0.030113 / 5.0226 ≈ 200,000 × 0.0059955 ≈ $1,199.10.

Month 1 interest = 200,000 × 0.005 = $1,000. Principal = 1,199.10 − 1,000 = $199.10. Ending balance ≈ $199,800.90.

Month 2 interest = 199,800.90 × 0.005 ≈ $999.00. Principal ≈ $200.10. Early payments are mostly interest; later payments are mostly principal.

Securitization, Pools, and TBAs

Securitization pools mortgages and issues securities backed by pool cash flows. Agency pass-throughs (Fannie Mae, Freddie Mac, Ginnie Mae) carry agency credit guarantees (Ginnie is backed by the U.S. government; Fannie/Freddie have agency guarantee structures). Investors still bear prepayment and interest-rate risk, not (in the agency case) mortgage credit risk in the same way as non-agency RMBS.

TBA (To-Be-Announced) trading is the forward market for agency pass-throughs. Counterparties agree on agency, coupon, maturity, face amount, and settlement month; the actual pools are announced just before settlement within delivery standards. TBAs create liquidity comparable to Treasury futures for the MBS market.

WAC, WAM, SMM, and CPR

Pool summary statistics:

Relationship (standard):

CPR = 1 − (1 − SMM)^12

SMM = 1 − (1 − CPR)^(1/12)

Worked SMM/CPR conversion

If SMM = 0.5% = 0.005, CPR = 1 − (1 − 0.005)^12 = 1 − 0.995^12. 0.995^12 ≈ 0.9416, so CPR ≈ 1 − 0.9416 = 5.84%.

If CPR = 20%, SMM = 1 − (1 − 0.20)^(1/12) = 1 − 0.80^(1/12). 0.80^(1/12) ≈ 0.9816, so SMM ≈ 1.84%.

PSA (Public Securities Association) benchmark: CPR ramps from 0.2% in month 1 to 6% in month 30, then flat at 6% (100% PSA). 200% PSA doubles those CPRs.

Agency Pass-Through Trading

Pass-through investors receive a pro-rata share of interest (net of servicing and guarantee fees) and principal (scheduled + prepaid). The pass-through coupon is below WAC by the servicing/guarantee strip. Trading conventions:

• Price quotes in percent of par (often 32nds historically; decimalization varies by platform)
• Settle via TBA or specified pool (“spec”) trades
• Higher-coupon pools trade with more prepayment sensitivity when deep refinancable

Specified pools can command pay-ups versus TBAs when collateral characteristics (loan size, geography, LTV) imply more valuable or more predictable prepayment behavior.

CMOs, IOs, and POs

Collateralized mortgage obligations (CMOs) tranche pass-through cash flows into classes with different average lives and prepayment exposures.

IO and PO strips split a pass-through: IO price often rises when rates rise (prepayments slow); PO price often rises when rates fall (prepayments speed)—opposite sensitivities that appear in hedging vignettes.

Worked IO/PO intuition

A PO bought at 80 (discount) receives $100 principal eventually. Faster prepayments increase value (earlier cash). An IO on the same notional receives interest on remaining balance; faster prepayments destroy the balance and crush IO value.

Dollar Rolls

A dollar roll is a pair of TBA trades: sell a TBA for near settlement and buy a TBA for later settlement (or the reverse). Economically it finances an MBS position: you give up the payments over the roll period and the drop (forward price difference) embeds financing, expected paydowns, and delivery optionality. When rolls are “special,” implied financing can be cheaper than GC repo—dealers and money managers monitor roll specialness as a relative-value signal.

Prepayment Option and Modeling Components

Prepayment models typically include:

1. Refinancing incentive — rate vs WAC; S-curve of refinancing propensity
2. Seasoning — new loans prepay slowly; ramp toward a steady state
3. Burnout — after rates fall, remaining borrowers are less responsive
4. Seasonality — turnover higher in summer
5. Turnover / relocation — baseline housing turnover even when rates are high
6. Credit / curtailment — defaults and partial prepayments

Media effect and yield-curve shape can also matter. Model risk is large: two dealers can produce different OASs on the same bond.

MBS Monte Carlo Valuation Steps

Standard Monte Carlo MBS valuation (rate-tree or rate-path framework):

1. Simulate many interest-rate paths consistent with the benchmark curve and a volatility assumption (arbitrage-free or calibrated dynamics).
2. Along each path, project mortgage rates and apply the prepayment model to generate monthly principal and interest cash flows for the MBS (net of fees).
3. Discount path cash flows along the path using short rates plus a trial spread (or using pathwise discount factors).
4. Average present values across paths to get a model price for a given spread.
5. Solve for the OAS that makes model price equal market price (or price at a quoted OAS).

Because cash flows are path-dependent (prepayments depend on the history of rates, burnout, etc.), a single binomial tree with path-independent cash flows is insufficient—hence Monte Carlo.

Worked conceptual OAS solve

Market price = 101.00. At OAS = 0 over the Treasury curve, model price = 102.50 (too rich → need positive OAS to lower PV). Increase OAS until model price = 101.00; suppose that OAS = 45 bp. That 45 bp is the option-adjusted spread after “paying for” the prepayment option through the simulation.

OAS: Uses and Challenges

Uses

• Compare MBS and mortgages with different coupons and structures on an option-adjusted basis
• Relative-value ranking across pools, CMOs, and hedged portfolios
• Risk: OAS duration / convexity from bumping curves inside the same simulation engine

Challenges

• Results inherit prepayment-model error and volatility calibration error
• OAS is not a promised yield; it is a model spread
• Different systems (different vol surfaces, different prepay models) produce different OASs
• Negative OAS can appear if the market price is rich to the model or if the prepay model understates option cost

Exam Synthesis

Compute level payments and first-month splits cleanly. Convert SMM ↔ CPR without mixing monthly and annual. For structure questions, ask who absorbs prepayment risk (PAC vs support; IO vs PO). For valuation, recite the Monte Carlo steps and interpret OAS as a model-dependent residual spread—not a pure credit spread on agency MBS.