CFA Level I Exam · Fixed-Income Bond Valuation: Prices and Yields

Bond Valuation Using Spot Rates and Matrix Pricing

Updated 7 October 2026 · Fact-checked

Valuing a bond with spot rates means discounting each cash flow at the spot rate for its own maturity, then adding the present values. Matrix pricing estimates the price of an illiquid bond by interpolating the yields of similar, liquid bonds, then discounting its cash flows at that yield. Spot-rate pricing uses many rates; matrix pricing uses one.

Understand Spot Rates and Matrix Pricing

A spot rate is the yield on a zero-coupon bond for a given maturity. The 3-year spot rate is the annual return you earn if you lend money today and receive one payment in 3 years. Plotting spot rates against maturity gives the spot curve (zero curve).

A coupon bond is a bundle of zero-coupon payments. The coupon due in year 1 is a 1-year payment, the coupon in year 2 is a 2-year payment, and so on. Each payment should be discounted at the spot rate for its own maturity. Adding those present values gives the bond's price. This is the no-arbitrage price: if the bond traded elsewhere, you could strip and rebuild its cash flows for a risk-free profit.

The yield to maturity (YTM) is different. It is one single rate that, applied to every cash flow, gives the same price. So YTM is a kind of average of the spot rates, weighted by the size and timing of the cash flows. Two bonds with the same maturity but different coupons usually have different YTMs, even though they sit on the same spot curve.

Matrix pricing solves a different problem. Some bonds, such as newly issued or thinly traded ones, have no reliable market price. You find liquid bonds with the same credit quality and similar coupon and maturity, then interpolate their yields to estimate a required yield for the subject bond. You then discount the subject bond's cash flows at that yield. It is an estimate, so the result is only as good as the comparable bonds.

Key formulas to remember

Bond price using spot rates: PV = PMT ÷ (1 + Z1)^1 + PMT ÷ (1 + Z2)^2 + ... + (PMT + FV) ÷ (1 + Zn)^n; Zt is the spot rate for maturity t. For semiannual bonds, use periods and the per-period spot rate (annual rate ÷ 2).
Discount factor: DFt = 1 ÷ (1 + Zt)^t; Price = Σ (cash flow t × DFt). Useful when the question gives discount factors directly.
Price using YTM: PV = Σ PMT ÷ (1 + y)^t + FV ÷ (1 + y)^N; One rate y for all cash flows. Solve with the calculator's N, I/Y, PMT, FV keys.
Linear interpolation of yield (matrix pricing): y = y1 + [(T − T1) ÷ (T2 − T1)] × (y2 − y1); T1 and T2 are the maturities of the two comparable bonds with yields y1 and y2. T is the subject bond's maturity. Comparables must have the same credit quality.

How to solve Spot Rates and Matrix Pricing questions

Decide first whether the question gives you a spot curve (use spot-rate discounting) or comparable bonds with yields (use matrix pricing). Then follow these steps.

1List every cash flow of the bond and its timing: coupons each period, plus face value with the final coupon.
2Check the periodicity. For semiannual coupons, the number of periods doubles and each rate is the annual rate ÷ 2.
3Spot-rate question: match each cash flow with the spot rate for that same maturity. Do not use one rate for all.
4Discount each cash flow: cash flow ÷ (1 + Zt)^t. Add all present values to get the price.
5Matrix pricing question: pick the two comparable bonds that bracket the subject bond's maturity (and match credit quality). Interpolate the yield with the formula.
6Price the subject bond using that interpolated yield as the single discount rate (N, I/Y, PMT, FV, CPT PV).
7Sanity check: a coupon above the discount rate gives a price above par, and a coupon below the rate gives a price below par.

Quickest way: Estimate first, then calculate only if needed

When to use it: Use under time pressure when the three options are far enough apart to eliminate two quickly.

Compare the coupon rate with the relevant discount rate. Coupon above the rate means price above par; below means below par. This often removes one option at once.
For spot-rate pricing, the answer should sit near the price at the spot rates' middle value. If spot rates rise with maturity, the price is lower than if you used the shortest spot rate for everything.
For matrix pricing, interpolate the yield first. If the subject maturity is halfway between the two comparables, the yield is the midpoint. Do this mentally.
Calculate only the largest cash flow precisely: the final payment ÷ (1 + Zn)^n. It is usually most of the price. Add rough values for the coupons and pick the closest option.
On the BA II Plus, compute each present value with N, I/Y, FV (or PMT), CPT PV, using N = 1 for each cash flow at its own rate and entering the cash flow as FV. Keep full decimals and do not round until the end.

Common mistakes in Spot Rates and Matrix Pricing

Discounting all cash flows at one spot rate, such as the final-maturity rate.
It feels like using YTM, which does apply one rate for all cash flows.
Fix: Spot-rate pricing needs a different rate for each cash flow date. Write Z1, Z2, ... Zn beside each payment before calculating.
Calling the YTM the same as the spot rate for that maturity.
Both are quoted as annual yields for a maturity.
Fix: A spot rate applies to one single payment at one date. YTM is one blended rate for a whole coupon bond. They are equal only when the curve is flat.
Forgetting to halve rates and double periods for semiannual bonds.
Quoted rates are annual, and students go straight to the formula.
Fix: Convert first: periods = years × 2, per-period rate = annual rate ÷ 2, coupon = annual coupon ÷ 2.
Interpolating bond prices instead of yields in matrix pricing.
Prices look like the thing being estimated.
Fix: Interpolate the yield, then compute the price from that yield. Price is not a linear function of maturity or yield.
Using comparable bonds with a different credit quality or sector.
The maturities fit, so the bonds look suitable.
Fix: Matrix pricing assumes similar credit risk. Check the rating and sector first; maturity alone is not enough.
Leaving out the face value in the last period's cash flow.
Students discount the coupons and forget the principal comes back with the final coupon.
Fix: The last cash flow is the final coupon plus face value. Write it as one figure before discounting.

Worked examples

Example 1

The annual spot rates are 2.0% for 1 year, 3.0% for 2 years and 4.0% for 3 years. A 3-year bond has a face value of USD 1,000 and pays a 5% annual coupon. The bond's price is closest to: A) USD 1,018.40, B) USD 1,029.60, C) USD 1,041.20.

Show the solution

Cash flows: USD 50 in year 1, USD 50 in year 2, USD 1,050 in year 3.
Year 1: 50 ÷ 1.02 = 49.02.
Year 2: 50 ÷ (1.03)^2 = 50 ÷ 1.0609 = 47.13.
Year 3: 1,050 ÷ (1.04)^3 = 1,050 ÷ 1.124864 = 933.45.
Sum: 49.02 + 47.13 + 933.45 = 1,029.60.
Check: the coupon of 5% exceeds all spot rates, so the price must be above par. Option B fits and the estimate is consistent.

Answer: B) USD 1,029.60

Example 2

An illiquid 5-year bond with a 4% annual coupon and USD 1,000 face value has the same credit quality as two liquid bonds. The liquid 3-year bond yields 3.00% and the liquid 7-year bond yields 4.00%. Using matrix pricing, the estimated price of the illiquid bond is closest to: A) USD 1,011.40, B) USD 1,022.58, C) USD 1,034.90.

Show the solution

The 5-year maturity is halfway between 3 and 7 years: (5 − 3) ÷ (7 − 3) = 0.5.
Interpolated yield: 3.00% + 0.5 × (4.00% − 3.00%) = 3.50%.
Price the bond at 3.50%: N = 5, I/Y = 3.5, PMT = 40, FV = 1,000, CPT PV.
Check by hand: 1.035^5 = 1.187686, so the discount factor is 0.841973.
Coupons: 40 × (1 − 0.841973) ÷ 0.035 = 40 × 4.51505 = 180.60. Principal: 1,000 × 0.841973 = 841.97.
Price = 180.60 + 841.97 = 1,022.58 (about 1,022.57–1,022.58 depending on rounding). The 4% coupon exceeds the 3.5% yield, so the price is above par, as expected.

Answer: B) USD 1,022.58

Exam tips

Read the question stem for the clue: a given spot curve means discount each cash flow separately; a given YTM or comparable yields means use one rate.
Questions often make the options close together. Eliminate by the par rule (coupon versus discount rate) first, then calculate precisely.
Interpolation questions usually place the subject maturity at a clean fraction such as one-half or one-quarter between the comparables. Do the fraction mentally and save time.
Check whether the question is semiannual. Missing this changes the answer by a clear margin and is a favourite trap.
Know the conceptual points: spot rate versus YTM, and why matrix pricing needs comparable credit quality. These can appear as pure concept questions with no calculation.

Practice questions from Fixed-Income Bond Valuation: Prices and Yields

Spot Rates and Matrix Pricing in other exams

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

Spot Rates and Matrix Pricing: frequently asked questions

What is the difference between a spot rate and yield to maturity?

A spot rate is the yield on a zero-coupon bond for one maturity, so it applies to a single cash flow. YTM is one rate that equates the price of a coupon bond to the present value of all its cash flows. The two are equal only for a zero-coupon bond or a flat curve.

What is matrix pricing for bonds?

It is a method for estimating the required yield, and so the price, of a bond that does not trade often. You use yields of liquid bonds with similar credit quality, coupon and maturity, and interpolate between them. You then discount the bond's cash flows at that yield.

How do I price a bond using the zero curve?

Take each coupon and the final principal, and discount each one at the spot rate for its own date. Add the present values together. The total is the bond's price.

Can I use my calculator to price a bond with different spot rates?

Yes, but not in one step. The TVM keys assume one rate. Compute each cash flow's present value separately, using N for its year, I/Y for its spot rate and the cash flow as FV, then add the results.