Lesson 4 of 4 11–14% of the exam

By Daniil Tarakanov Updated Oct 21, 20262027 curriculum

Matrix pricing

Find actively traded bonds with similar credit quality, coupons and maturities, compute their yields, average them by maturity, interpolate to the target bond's maturity, then discount the target's cash flows at that yield. Underwriters use the same idea to estimate the spread a new bond must pay.

8 min cram, 34 min full read. Based on the 2027 CFA® Program Level I curriculum, Fixed Income, Learning Module 6. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Pricing a bond from its yield to maturity

On the exam

Describe matrix pricing

Describe matrix pricing. Expect the four steps and, often, a short calculation: interpolate a yield between two maturities, then price a bond at it. Expect one question on what makes a good comparable bond.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

How do you price a bond that has not traded recently?

Most bonds trade rarely, and a new bond has no trading history at all. Matrix pricing estimates the price from actively traded comparable bonds : bonds with similar credit quality, maturity and coupon rate.

The four steps of matrix pricing

Estimate the price of a bond that rarely trades

1. Pick comparables Memorize frequently traded; similar credit quality, maturity and coupon
2. Compute yields Memorize YTM of each, then average by maturity
3. Interpolate Memorize a straight line to the target maturity
4. Price the target Memorize discount its coupons and par at that yield

InteractiveMatrix pricing: a yield for a bond that has not traded

Try this 0 of 3

Drag the target to 5 years
Cut the target's coupon until it prices below par
Show the new issue spread and push it above 60 bps

Four comparable bonds trade; the target does not. Drag the target maturity on the chart, change the comparables' prices or the target's coupon.

Show

Target maturity4 years

Target coupon (annual)5.00%

Bond A price (3y, 4% coupon)99.10

Bond B price (3y, 6% coupon)104.60

Bond C price (6y, 3.5% coupon)93.80

Bond D price (6y, 5.5% coupon)103.40

Target 4 years yield4.477%

Target price (5.00% coupon)101.88

Trades atPremium

3-year comparables
6-year comparables
Target
Straight line between averages

Averages: 4.329% at 3 years, 4.772% at 6 years.
Bond	Coupon	Price	YTM
A (3y)	4.00%	99.10	4.326%
B (3y)	6.00%	104.60	4.332%
C (6y)	3.50%	93.80	4.710%
D (6y)	5.50%	103.40	4.834%

4 years is 1 year into the 3-year gap between the 3-year and 6-year averages (weight 0.3333), so the target's yield is 4.477% and the 5.00% bond is worth 101.88 (a premium).

Show working

1. Comparables of the same credit quality: A and B at 3 years, C and D at 6 years.

2. Their yields to maturity: A 4.326%, B 4.332%, C 4.710%, D 4.834%. 3-year average 4.329%; 6-year average 4.772%.

3. Interpolate: 4.329% + (4 − 3) ÷ (6 − 3) × (4.772% − 4.329%) = 4.477%.

4. Price the target: N = 4, I/Y = 4.4767, PMT = 5, FV = 100 → 101.88.

Interpolate yields, not prices: the comparables have different coupons, so their prices are not comparable but their yields are.

Interpolate and price on your calculator

4.771955−4.329134=×1÷3+4.329134=4.477, the target yield
2NDCLR TVMP/Y = 1
4N4.4767I/Y5PMT100FV
CPTPV−101.88

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

4.771955ENTER4.329134−1×3÷4.329134+4.477, the target yield
fCLEAR FIN
4n4.4767i5PMT100FV
PV−101.88

ENTER enter (on the HP, puts the number on the stack) · f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

No trade? Borrow the yields of bonds like it and draw a straight line.

What a good comparable shares with the target

Comparable bond

Credit quality Memorize same or very similar issuer risk
Time to maturity Memorize close to the target’s
Coupon rate Memorize similar, since the coupon shapes the yield

How do you interpolate the yield?

Comparable bonds rarely have exactly the target’s maturity, so draw a straight line between the yields on either side and read off the target’s maturity. This is linear interpolation .

Linear interpolation of a yield Use it

y = y_1 + \frac{t - t_1}{t_2 - t_1} \times (y_2 - y_1)

Worked example Pricing an illiquid 4-year, 5% annual bond

The target is a 4-year bond with a 5% annual coupon that rarely trades. Four comparable bonds of the same credit quality trade actively, shown with their yield to maturity (YTM).

Bond	Maturity	Coupon (annual)	Price	YTM
A	3 years	4.0%	99.10	4.326%
B	3 years	6.0%	104.60	4.332%
C	6 years	3.5%	93.80	4.710%
D	6 years	5.5%	103.40	4.834%

Average by maturity: 3-year (4.326% + 4.332%) ÷ 2 = 4.329%; 6-year (4.710% + 4.834%) ÷ 2 = 4.772%.
Interpolate to 4 years, one-third of the way from 3 to 6: 4.329% + (1 ÷ 3) × (4.772% − 4.329%) = 4.477%.
Price the target at 4.477%, with the coupon as PMT (payment) and par as FV (future value): N = 4, I/Y = 4.477, PMT = 5, FV = 100 gives 101.88.

The target’s 5% coupon is above the 4.477% yield, so it is a premium bond.

Steps 2 and 3 on your calculator

1÷3×(4.772−4.329)+4.329= 4.4767
2NDCLR TVM with P/Y set to 1
4N4.4767I/Y5PMT100FV
CPTPV −101.878

1ENTER3÷4.772ENTER4.329−×4.329+ 4.4767
fCLEAR FIN
4n4.4767i5PMT100FV
PV −101.878

Rearrange itLinear interpolation: solve for the yield or the maturity

The question asks for

Comparable bonds yield 4.329% at 3 years and 4.772% at 6 years. The yield for a 4-year maturity, by linear interpolation, is closest to:

Shorter maturity, t₁ (years)Yield at t₁%Longer maturity, t₂ (years)Yield at t₂%Target maturity, t (years)

y = y_{1} + \frac{t - t _{1}}{t _{2} - t _{1}} \times (y_{2} - y_{1})

4.329% + (1 ÷ 3) × (4.772% − 4.329%) = 4.477%

Interpolated yield4.48%

On your calculator

4−3=÷3×(4.772−4.329)+4.329=4.477

4ENTER3−6ENTER3−÷4.772ENTER4.329−×4.329+4.477

ENTER enter (on the HP, puts the number on the stack)

Interpolate yields, not prices. Prices depend on coupons and maturities in a curved way, so a straight line through them is wrong.

Go deeper Why coupons and credit quality matter for comparables

Two bonds with different coupons can have slightly different yields even at the same maturity, which is why bonds A and B (and C and D) above are averaged. A comparable with weaker credit quality yields more for reasons that have nothing to do with maturity, and averaging it in would overstate the target’s yield. Interpolation also assumes the yield curve is a straight line between the two points, which is only an approximation.

How do underwriters use the same idea for a new issue?

Before a bond is issued, the underwriter needs the yield investors will require. Take a benchmark rate , usually a government yield at the same maturity, then add the required yield spread that comparable bonds of the issuer’s credit quality pay over it.

Required yield spread Use it

s = y_{\text{bond}} - y_{\text{benchmark}}

Worked example Estimating the spread over government bonds

An issuer’s existing 6-year, 4% annual bond trades at 96.00.

Its yield, with the price as a negative PV (present value): N = 6, PV = −96.00, PMT = 4, FV = 100 gives 4.783%.
No 6-year government bond exists. The 5-year yields 4.15% and the 7-year 4.55%, so the interpolated 6-year yield is 4.35%.
Spread = 4.783% − 4.35% = 0.433%, about 43 bps.

The underwriter starts from this spread when pricing the issuer’s new bond over its own benchmark.

Step 1 and step 3

2NDCLR TVM with P/Y set to 1
6N96+|-PV4PMT100FV
CPTI/Y 4.7826
−4.35=×100= 43.26 bps, about 43

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · PMT payment · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

fCLEAR FIN
6n96CHSPV4PMT100FV
i 4.7826
4.35−100× 43.26 bps, about 43

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · PMT payment · FV future value · i interest rate per period

Rearrange itRequired yield spread: solve for the spread or the issuer's yield

The question asks for

An issuer's bond yields 4.783%. The interpolated benchmark yield at the same maturity is 4.350%. The yield spread is closest to (basis points):

Issuer's bond yield%Benchmark yield at the same maturity%

s = (y_{b o n d} - y_{b e n c hma r k}) \times 10, 000

(0.04783 − 0.0435) × 10,000 = 43.3 bps

Spread (basis points)43.3

On your calculator

4.783−4.35=×100=43.3 bps

4.783ENTER4.35−100×43.3 bps

ENTER enter (on the HP, puts the number on the stack)

Exam tip Matrix pricing is the reason many quotes on data screens are estimates and not trades. When the exam says a bond is illiquid or newly issued, think comparables, interpolation, discount.

Go deeper Limits and evaluated pricing services

Funds that must value thinly traded bonds for their daily net asset values often rely on evaluated pricing services, which use matrix pricing and similar methods. The estimate is only as good as the comparables: too few trades, different credit quality, or a curved yield curve between the end points all add error.

Memorize this

MemorizeMatrix pricing in four steps

Memory hook Look at neighbors, average, draw a line, price.

Know it cold (1) Comparable bonds. (2) Their YTMs, averaged by maturity. (3) Interpolate to the target maturity. (4) Discount the target’s cash flows at that yield.

MemorizeInterpolation

Memory hook Weight = how far along the maturity gap the target sits.

Know it cold $y = y_1 + \dfrac{t - t_1}{t_2 - t_1}(y_2 - y_1)$ . Interpolate yields, never prices.

MemorizeNew issue spread

Memory hook Issuer yield minus benchmark yield at the same maturity.

Know it cold Required yield = benchmark yield + spread. Interpolate the benchmark if no bond has the exact maturity.

The four steps of matrix pricing

Estimate the price of a bond that rarely trades

1. Pick comparables Memorize frequently traded; similar credit quality, maturity and coupon
2. Compute yields Memorize YTM of each, then average by maturity
3. Interpolate Memorize a straight line to the target maturity
4. Price the target Memorize discount its coupons and par at that yield

What a good comparable shares with the target

Comparable bond

Credit quality Memorize same or very similar issuer risk
Time to maturity Memorize close to the target’s
Coupon rate Memorize similar, since the coupon shapes the yield

Linear interpolation of a yield

y = y_1 + \dfrac{t - t_1}{t_2 - t_1} \times (y_2 - y_1)

Quick example

Comparables yield 5.10% at 5 years and 5.55% at 8 years. At 7 years: 5.10 + (2 ÷ 3) × 0.45 = 5.40%.

On your calculator

2÷3×0.45+5.1= 5.4

2ENTER3÷0.45×5.1+ 5.4

ENTER enter (on the HP, puts the number on the stack)

Required yield spread

s = y_{\text{bond}} - y_{\text{benchmark}}

Quick example

An issuer’s 6-year bond yields 4.78%. The interpolated 6-year government yield is 4.35%. The spread is 43 bps.

On your calculator

4.78−4.35=×100= 43 bps

4.78ENTER4.35−100× 43 bps

ENTER enter (on the HP, puts the number on the stack)

How it is asked

Interpolate, then price

Comparables yield 5.10% at 5 years and 5.55% at 8 years. Price an illiquid 7-year 5.5% annual bond.

Yield 5.10 + (2/3) × 0.45 = 5.40%; N = 7, I/Y = 5.4, PMT = 5.5 gives 100.57. The simple average (5.325%) is the trap.
What should comparables share?

Which features should comparable bonds share with the target?

Credit quality, maturity and coupon rate.
Required spread

Issuer’s 6-year bond yields 4.78%; government yields 4.15% (5-year) and 4.55% (7-year). Spread?

Benchmark 4.35%; 4.78 − 4.35 = 43 bps. One end point (23 or 63 bps) is the trap.

Exam traps

Averaging the two maturities' yields instead of interpolating at the right weight.
Interpolating prices instead of yields. Interpolate yields, then price.
Picking comparables of a different credit quality. They must have similar credit risk.
Using the nearest benchmark yield instead of interpolating when no bond has the exact maturity.

Checkpoint

3 exam-style questions. About 90 seconds each.

1Comparable bonds yield 3.80% on average at 4 years and 4.40% at 7 years. Using matrix pricing, the price of an illiquid 5-year, 4.5% annual-coupon bond of the same credit quality is closest to:
2Which features should comparable bonds share with the bond being matrix-priced? Most appropriately:
3An issuer's 6-year bond yields 3.72%. Government bonds yield 3.10% at 5 years and 3.40% at 7 years. The estimated required yield spread is closest to:

Key takeaways

Matrix pricing estimates prices for illiquid or new bonds from comparables.
Steps: comparables, yields, interpolate, price.
The same method estimates the spread a new issue needs.

Linear interpolation of a yield

y = y_1 + \dfrac{t - t_1}{t_2 - t_1} \times (y_2 - y_1)

Quick example

Comparables yield 5.10% at 5 years and 5.55% at 8 years. At 7 years: 5.10 + (2 ÷ 3) × 0.45 = 5.40%.

On your calculator

2÷3×0.45+5.1= 5.4

2ENTER3÷0.45×5.1+ 5.4

ENTER enter (on the HP, puts the number on the stack)

Required yield spread

s = y_{\text{bond}} - y_{\text{benchmark}}

Quick example

An issuer’s 6-year bond yields 4.78%. The interpolated 6-year government yield is 4.35%. The spread is 43 bps.

On your calculator

4.78−4.35=×100= 43 bps

4.78ENTER4.35−100× 43 bps

ENTER enter (on the HP, puts the number on the stack)

All Fixed Income formulas, every lesson

Practice exam-style questions on Bond Valuation: Prices and Yields

Cheat sheet · PDF · 4 A4 pages

Bond Valuation: Prices and Yields on 4 pages

What to memorize, the concept maps, every formula with a worked example and keystrokes for the TI BA II Plus and HP 12C, how each idea is asked, and the traps. Print it, or keep it on your phone.

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Next module: Yield and Yield Spread Measures for Fixed-Rate Bonds Periodicity: converting yields between compounding frequencies

Questions about this lesson?

Where this comes back

Yield and Yield Spread Measures for Fixed-Rate Bonds Fixed Income
Term Structure: Spot, Par and Forward Curves Fixed Income

Questions candidates ask

What is matrix pricing of bonds?

Estimating a bond's yield and price from the yields of actively traded bonds with similar credit quality, coupons and maturities. It is used for bonds that rarely trade and for bonds not yet issued.

Why is matrix pricing needed?

Most bonds trade rarely, so there is often no recent price to use. Dealers and pricing services estimate a price from comparable bonds instead, so many quoted prices are matrix estimates.

How do you interpolate a bond yield?

Move along a straight line between the yields at the two nearest maturities, in proportion to the target's distance from each. A target two-thirds of the way from 5 to 8 years takes two-thirds of the yield gap.

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