FRM Exam Part I · Mortgages and Mortgage-Backed Securities

MBS Duration, Convexity and OAS Explained

Q: What is negative convexity in MBS?

It means the price-yield curve bends the wrong way. As yields fall, prepayments speed up and price gains are capped. As yields rise, prepayments slow and losses are not cushioned.

Q: What is the difference between OAS and Z-spread?

The Z-spread is a constant spread over the spot curve for one fixed cash flow scenario. OAS is found across many simulated rate paths with rate-dependent prepayments. For an MBS, OAS is the Z-spread less the option cost.

Q: Why can't I use modified duration for MBS?

Modified duration assumes cash flows do not change when yields change. MBS cash flows change with prepayments, so you reprice under shifted curves and use effective duration.

Q: How is Monte Carlo used to value an MBS?

You simulate many interest rate paths and use a prepayment model to produce cash flows on each path. You discount them along that path, add a spread, and choose the spread (the OAS) so the average value equals the market price.

Updated 11 October 2026 · Fact-checked

MBS prices fall when rates rise and rise less than a normal bond when rates fall, because borrowers prepay. This is negative convexity. You measure rate risk with effective duration, found by repricing under shifted curves. OAS is the spread over the curve after removing the value of the prepayment option.

Understand Valuation and Risk of MBS: Duration, Convexity and OAS

A mortgage pass-through gives you the borrowers' scheduled payments plus prepayments. Borrowers hold a prepayment option. When rates fall, they refinance and you get principal back early, just when you must reinvest at lower rates. When rates rise, prepayments slow and the security lasts longer, just when you would prefer a short one.

This behavior creates negative convexity. For a plain bond, price rises more when yields fall than it drops when yields rise. For a current-coupon or premium MBS, price gains are capped as rates fall, because prepayments rise and the price is pulled toward par. Price losses are not cushioned when rates rise. Investors in an MBS are effectively long a bond and short a call option to the borrowers.

Because cash flows change with rates, the usual Macaulay or modified duration (which assumes fixed cash flows) is not reliable. You use effective duration, which reprices the MBS after shifting the whole curve up and down by a small amount, with a prepayment model re-estimating cash flows under each curve. Effective convexity is found the same way, and it is often negative for MBS.

To value an MBS, you generate many random interest rate paths with Monte Carlo simulation. On each path, a prepayment model gives cash flows, and you discount them along that path. The option-adjusted spread (OAS) is the constant spread added to the short rates on every path so that the average present value equals the market price.

The Z-spread is the constant spread over the spot curve that prices the cash flows of one fixed (usually projected) scenario. It ignores the option. So OAS = Z-spread − option cost. A larger gap means more embedded optionality. A higher OAS at the same risk suggests the MBS is cheaper.

Key formulas to remember

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.

How to solve Valuation and Risk of MBS: Duration, Convexity and OAS questions

Use this order for most MBS duration, convexity and OAS questions.

1Identify what is asked: effective duration, convexity, price change, OAS or Z-spread, or option cost.
2Check whether cash flows change with rates. If prepayments are modeled, use effective measures, not modified duration.
3For duration or convexity, list P₀, P₋ and P₊ and the shift Δy. Convert basis points to decimals.
4Plug into the formula and keep the sign. Negative convexity means P₋ + P₊ < 2 × P₀.
5For price change, use −D × Δy + ½ × C × Δy², then multiply by P₀.
6For spreads, remember OAS = Z-spread − option cost, and OAS is the lower number when the investor is short the option.
7Sanity check: does the answer match the rate direction and the sign of convexity?

Quickest way: Shortcut for effective duration and spread questions

When to use it: When you are given three prices and a shift, or a Z-spread and an option cost.

For duration: (P₋ − P₊) divided by 2 × P₀ × Δy. Do not forget the factor 2.
For convexity sign: compare P₋ + P₊ with 2 × P₀. If smaller, convexity is negative.
For OAS: subtract option cost from Z-spread. For option cost: subtract OAS from Z-spread.
If an option asks which rate move hurts most, pick rising rates for MBS, since extension risk has no cushion.

Common mistakes in Valuation and Risk of MBS: Duration, Convexity and OAS

Using modified duration for an MBS.
Modified duration is the familiar formula, but it assumes fixed cash flows.
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.
Students recall the numerator but forget the two-sided shift.
Fix: The denominator is 2 × P₀ × Δy. Check that the result is near a sensible number such as 4 to 6.
Treating OAS as larger than the Z-spread for a callable-style MBS.
Mixing up which side owns the option.
Fix: The investor is short the prepayment option, so OAS is below the Z-spread. Option cost = Z-spread − OAS.
Saying negative convexity means price falls when rates fall.
Confusing sign of convexity with direction of price.
Fix: Price still rises when rates fall. It rises by less than a positive-convexity bond, and the gain flattens.
Entering Δy as 25 instead of 0.0025.
Basis points are not converted.
Fix: Divide basis points by 10,000 before using the formula.

Worked examples

Example 1

An MBS has a price of 102.00. If the curve shifts down 50 bp, its price is 103.40. If the curve shifts up 50 bp, its price is 100.20. Compute effective duration and effective convexity.

Show the solution

Δy = 0.005. P₀ = 102.00, P₋ = 103.40, P₊ = 100.20.
Duration = (103.40 − 100.20) ÷ (2 × 102.00 × 0.005) = 3.20 ÷ 1.02 = 3.137.
Convexity numerator = 103.40 + 100.20 − 2 × 102.00 = 203.60 − 204.00 = −0.40.
Denominator = 102.00 × 0.005² = 102.00 × 0.000025 = 0.00255.
Convexity = −0.40 ÷ 0.00255 = −156.9.

Answer: Effective duration ≈ 3.14 and effective convexity ≈ −156.9, so the MBS is negatively convex.

Example 2

An MBS has a Z-spread of 118 bp and an OAS of 72 bp. (a) What is the option cost? (b) If the OAS were 95 bp at the same Z-spread, what would that imply about the embedded option?

Show the solution

Option cost = Z-spread − OAS.
(a) 118 − 72 = 46 bp.
(b) With OAS 95 bp, option cost = 118 − 95 = 23 bp.
A lower option cost means the prepayment option is worth less in the model, so the MBS is less negatively convex or less rate-volatile in the model.

Answer: (a) 46 bp. (b) The option cost would fall to 23 bp, so the embedded option is valued less.

Exam tips

Expect conceptual questions: why effective duration shrinks when rates fall and lengthens when rates rise for MBS (contraction and extension risk).
Know the OAS versus Z-spread link by heart: the gap is the option cost, and OAS is lower for MBS.
Questions on Monte Carlo valuation test that OAS is the spread making the average path value equal the market price.
Always check that the sign of your convexity matches the story of prepayments before you pick an answer.

Practice questions from Mortgages and Mortgage-Backed Securities

Valuation and Risk of MBS: Duration, Convexity and OAS in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CMA FinalDuration and Convexity

Valuation and Risk of MBS: Duration, Convexity and OAS: frequently asked questions

What is negative convexity in MBS?

It means the price-yield curve bends the wrong way. As yields fall, prepayments speed up and price gains are capped. As yields rise, prepayments slow and losses are not cushioned.

What is the difference between OAS and Z-spread?

The Z-spread is a constant spread over the spot curve for one fixed cash flow scenario. OAS is found across many simulated rate paths with rate-dependent prepayments. For an MBS, OAS is the Z-spread less the option cost.

Why can't I use modified duration for MBS?

Modified duration assumes cash flows do not change when yields change. MBS cash flows change with prepayments, so you reprice under shifted curves and use effective duration.

How is Monte Carlo used to value an MBS?

You simulate many interest rate paths and use a prepayment model to produce cash flows on each path. You discount them along that path, add a spread, and choose the spread (the OAS) so the average value equals the market price.