Mortgages and Mortgage-Backed Securities: formula sheet

Key formulas

Monthly mortgage payment

PMT = L × r ÷ [1 − (1 + r)^(−n)]

L = loan amount, r = periodic (monthly) rate = annual rate ÷ 12, n = number of payments. Level-payment, fully amortizing loan, payments at end of month.

Interest portion of a payment

Interest(t) = Balance(t−1) × r

Uses the balance at the start of the period, before the payment.

Principal portion of a payment

Principal(t) = PMT − Interest(t)

New balance = old balance − principal portion.

Loan-to-value ratio

LTV = Loan amount ÷ Property value

Lenders typically use the lower of purchase price and appraised value.

Remaining balance

B(k) = PMT × [1 − (1 + r)^(−(n − k))] ÷ r

Present value of the remaining n − k payments, after k payments are made.

Interest-only payment

Payment = L × r

Balance stays at L during the interest-only period.

CPR to SMM

SMM = 1 − (1 − CPR)^(1/12)

CPR and SMM as decimals. Compounding survival, not dividing by 12.

SMM to CPR

CPR = 1 − (1 − SMM)^12

Inverse of the above.

SMM definition

SMM = Prepayment in month t ÷ (Beginning balance − Scheduled principal payment)

The denominator is the balance after scheduled principal, not the beginning balance.

100% PSA CPR

CPR = 6% × (t ÷ 30) for t ≤ 30; CPR = 6% for t > 30

t is the age of the loan in months. Equals 0.2% per month for the first 30 months.

PSA at speed x%

CPR = (x ÷ 100) × 100% PSA CPR

For example 150% PSA at month 20 gives 1.5 × 4% = 6%.

Prepayment amount

Prepayment = SMM × (Beginning balance − Scheduled principal)

Total principal received = scheduled principal + prepayment.

Pass-through rate (net coupon)

Net coupon = WAC − servicing fee − guarantee fee

Investors earn interest at the net coupon, not the WAC. Fees are annual rates.

Pool factor and current balance

Factor = current balance ÷ original balance; current balance = original face × factor

The factor is published monthly. It falls with scheduled principal and prepayments.

Monthly interest to investors

Interest = (net coupon ÷ 12) × beginning-of-month balance

Use the balance at the start of the month. Pass-throughs use a 30/360 convention.

Level monthly mortgage payment

Payment = B × r ÷ (1 − (1 + r)^−n), with r = WAC ÷ 12 and n = WAM in months

Scheduled principal = payment − interest at the WAC. This uses the pool's WAC, not the net coupon.

SMM from CPR

SMM = 1 − (1 − CPR)^(1/12)

CPR is annual. SMM is the single monthly mortality rate. Do not divide CPR by 12.

Prepayment amount

Prepayment = SMM × (beginning balance − scheduled principal)

Prepayment applies to the balance left after the scheduled principal.

Total investor cash flow and ending balance

Cash flow = net interest + scheduled principal + prepayment; ending balance = beginning balance − scheduled principal − prepayment

Interest is not part of the balance reduction.

100% PSA benchmark

CPR = 0.2% × month (months 1 to 30), then 6% from month 30 on

Other PSA speeds scale this: 200% PSA doubles CPR at each month.

Principal priority in sequential-pay

Principal to Tranche A until paid off, then B, then C

Interest is paid on each tranche's outstanding balance every period. Cash flows of tranches sum to the collateral cash flows.

Conservation of cash flow

Σ tranche cash flows = collateral cash flows

Tranching redistributes risk. It does not create or destroy it.

Tranche interest

Interest = Coupon rate × Outstanding balance × (1 ÷ 12) for monthly pay

Use beginning-of-period balance. Ignore servicing fees unless given.

PAC protection

PAC schedule holds if prepayment speed stays between the lower and upper PSA bounds

Outside the collar, or after the support tranche is exhausted, the PAC schedule can fail.

Prepayment effect on strips

Faster prepayment: PO value ↑, IO value ↓. Slower prepayment: PO value ↓, IO value ↑

IO effective duration is often negative when prepayment effects dominate. PO effective duration is large and positive.

Effective duration

D_eff = (P₋ − P₊) ÷ (2 × P₀ × Δy)

P₋ is price when the curve shifts down by Δy, P₊ when it shifts up. Prepayment cash flows are re-estimated in each case. Δy as a decimal (0.001 = 10 bp).

Effective convexity

C_eff = (P₋ + P₊ − 2 × P₀) ÷ (P₀ × Δy²)

Negative for typical MBS where prepayment option is in play.

Price change approximation

ΔP ÷ P ≈ −D_eff × Δy + ½ × C_eff × (Δy)²

Convexity term is negative for negatively convex MBS, so it reduces the gain from falling rates.

Option cost

Option cost = Z-spread − OAS

Measures the spread given up for the borrowers' prepayment option.

OAS condition

Market price = average over paths of PV of cash flows discounted at (path rates + OAS)

OAS is solved by trial and error until the simulated average equals the market price.

Pool loss

Pool loss = Collateral balance × Default rate × Loss severity

Loss severity = 1 − recovery rate. Use the loss amount, not the default amount, in the waterfall.

Loss allocation (waterfall)

Loss to a tranche = min(tranche balance, remaining loss after junior tranches)

Losses hit the equity or junior tranche first, then move up. Senior tranches lose only after all junior balances are gone.

Subordination (credit enhancement) level

Subordination to a tranche = Balance of all tranches junior to it ÷ Total pool balance

A larger percentage means more loss absorption below that tranche.

Overcollateralization

OC = Collateral balance − Total bond balance; OC % = OC ÷ Collateral balance

OC is a cushion that absorbs losses before any bond loses principal.

Excess spread

Excess spread = Weighted average coupon on loans − (bond coupons + fees)

Expressed as an annual percentage of the pool. It is the first line of defence in many deals.

Agency vs non-agency

Agency: prepayment risk. Non-agency: prepayment risk + credit risk

Agency guarantee covers credit risk. Non-agency relies on credit enhancement.

Quick revision

• A level-payment mortgage has a fixed payment; early payments are mostly interest and later ones mostly principal.
• Payment is found from the loan amount, periodic rate and number of payments, using a financial calculator or the annuity formula.
• Monthly interest = outstanding balance × annual rate ÷ 12; principal = payment − interest.
• CPR is an annualized prepayment rate; SMM is the monthly rate. SMM = 1 − (1 − CPR)^(1/12).
• PSA 100% is a benchmark prepayment ramp; 200% PSA means twice the benchmark speed.
• Pass-through investors receive interest, scheduled principal and prepayments, net of servicing and guarantee fees.
• Lower rates usually raise prepayments, shortening life (contraction risk); higher rates slow prepayments, lengthening life (extension risk).
• CMO tranching redistributes prepayment risk; it does not remove it from the pool.
• MBS typically show negative convexity because of the borrower's prepayment option.
• OAS is the spread over the benchmark curve after removing the value of the embedded prepayment option.
• Effective duration is estimated by shifting rates and repricing with prepayment assumptions that change with rates.
• The subprime crisis involved weak underwriting, rating reliance, misaligned incentives and correlated defaults.

Common mistakes

• Using the annual rate in the payment formula for a monthly mortgage. Fix: Always divide the annual rate by 12 and multiply years by 12 before computing.
• Calculating interest on the original loan every month. Fix: Interest each month = the previous month-end balance × r. Only month 1 uses the original loan.
• Computing SMM as CPR ÷ 12. Fix: Use SMM = 1 − (1 − CPR)^(1/12). CPR ÷ 12 is only an approximation and is slightly too low.
• Applying SMM to the beginning balance. Fix: Subtract scheduled principal first, then multiply by SMM.
• Using the WAC to compute interest paid to investors. Fix: Investor interest always uses the net coupon. Use the WAC only for the borrowers' payment and scheduled principal.
• Treating Fannie Mae and Freddie Mac guarantees as backed by the full faith and credit of the US government, like Ginnie Mae. Fix: Ginnie Mae: explicit US government backing. Fannie and Freddie: GSE guarantees, with no explicit government backing, though they have been in conservatorship since 2008.
• Saying CMOs eliminate prepayment risk Fix: Remember that total risk is conserved. CMOs reallocate it, so the support tranche carries what the PAC avoids.
• Thinking the PAC is protected at any prepayment speed Fix: PAC protection holds only within the PSA band and while the support tranche remains outstanding.
• Using modified duration for an MBS. Fix: Use effective duration with repriced cash flows under each shifted curve whenever prepayments depend on rates.
• Leaving out the factor 2 in effective duration. Fix: The denominator is 2 × P₀ × Δy. Check that the result is near a sensible number such as 4 to 6.

Exam tips

• Practice the calculator payment keystrokes until they are automatic. Most numerical questions reduce to this.
• Remember that interest uses the opening balance each month. Questions often test month 2 or later.
• Expect conceptual questions on prime vs subprime, ARM caps and negative amortization. Link each to who bears which risk.
• Watch the compounding convention: US mortgages quote a nominal annual rate compounded monthly, so use rate ÷ 12.
• Higher LTV and weaker credit mean higher default risk. Use this to eliminate wrong options quickly.
• Memorise the 100% PSA path: 0.2% per month to 6% at month 30, then flat. Most numeric questions start here.
• Always do the CPR to SMM conversion exactly; options often differ by the CPR ÷ 12 trap.
• For conceptual questions, name the direction: falling rates lead to faster prepayment, contraction risk and negative convexity.