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

Yield Spreads and the Term Structure for CFA Level I

Updated 7 October 2026 · Fact-checked

The yield curve plots yields against maturity. Spreads measure a bond's extra yield over a benchmark: G-spread uses government yields, I-spread uses swap rates, Z-spread adds one constant spread to every spot rate, and OAS removes embedded option value. Forward rates come from comparing spot rates of different maturities.

Understand Yield Spreads and the Term Structure

The term structure of interest rates shows how yields differ by maturity. A spot rate is the yield on a zero-coupon bond for a given maturity. The yield curve is the plot of yields against maturity. Common shapes are upward-sloping (normal), inverted (short rates above long rates), flat, and humped. Slope reflects expectations about future short rates, plus compensation for risk.

A forward rate is an interest rate agreed today for a loan that starts in the future. It is implied by spot rates through no-arbitrage. Investing for two years at the 2-year spot rate must give the same result as investing for one year at the 1-year spot rate and then rolling over at the one-year rate that starts in one year. That rolled-over rate is the forward rate. Notation such as "2y3y" means a 3-year rate starting 2 years from now.

A yield spread is a bond's yield minus a benchmark yield. It pays you for credit risk, liquidity risk and option features. The G-spread is the bond's yield to maturity minus the government bond yield at the same maturity (interpolated if needed). The I-spread is the bond's yield to maturity minus the swap rate at the same maturity. Both are single-point spreads. They ignore the shape of the curve.

The Z-spread (zero-volatility spread) is the constant spread added to every spot rate on the benchmark curve so that the present value of the bond's cash flows equals its market price. It uses the whole curve, so it is more precise than G- or I-spread when the curve is steep.

The option-adjusted spread (OAS) is the Z-spread after removing the value of the embedded option. The Z-spread includes the effect of the option: it is higher than OAS for a callable bond and lower than OAS for a putable bond. Use one convention: OAS = Z-spread − option value, with the option value in bp. The option value is positive for a call held by the issuer and negative for a put held by the investor. For a callable bond, the issuer holds the option, so investors are paid extra yield for it. The option value is positive, so the Z-spread is larger than OAS. For a putable bond, the investor holds the option and gives up yield for it, so the option value is negative and OAS is greater than the Z-spread. In that case OAS = Z-spread + |put value|. OAS is the clean credit and liquidity compensation, which makes bonds with and without options comparable.

Key formulas to remember

Forward rate from spot rates: (1 + z_B)^B = (1 + z_A)^A × (1 + f_(A,B−A))^(B−A); z is the spot rate for the maturity in years. f is the forward rate for a loan of B−A years starting at A. Solve for f. Use annual compounding unless told otherwise.
One-year forward rate starting in one year: f(1,1) = (1 + z2)² ÷ (1 + z1) − 1; Special case of the formula above, with A = 1 and B = 2.
Bond price using spot rates and Z-spread: PV = CF1 ÷ (1 + z1 + Z)¹ + CF2 ÷ (1 + z2 + Z)² + … + CFn ÷ (1 + zn + Z)ⁿ; Z is the constant Z-spread. It is found by trial and error until PV equals the market price.
G-spread: G-spread = bond YTM − government yield at same maturity; Interpolate between two government bonds if no exact match exists.
I-spread: I-spread = bond YTM − swap rate at same maturity; Benchmark is the interest rate swap curve, not government bonds.
OAS: OAS = Z-spread − option value (in bp); The Z-spread includes the effect of the option: higher for callable, lower for putable. Option value is positive for a call held by the issuer and negative for a put held by the investor. Callable: OAS < Z-spread. Putable: OAS = Z-spread + |put value|, so OAS > Z-spread.

How to solve Yield Spreads and the Term Structure questions

Use this method for any question on spreads, the yield curve or forward rates.

1Identify what is asked: a curve shape, a forward rate, or a spread measure. Note the benchmark named (government, swap or spot curve).
2For a spread question, decide which measure applies: single-point against government (G), single-point against swaps (I), whole spot curve (Z), or option-adjusted (OAS).
3For G-spread and I-spread, subtract the benchmark yield at the same maturity from the bond's YTM. Convert to basis points (1% = 100 bp).
4For OAS, check who holds the option. Callable: OAS = Z-spread − option value. Putable: OAS is higher than Z-spread.
5For a forward rate, write the no-arbitrage equation with the longer spot rate on the left and the shorter spot rate times the forward on the right. Use decimals.
6Solve by dividing, then take the root of the forward period (for one-year forwards, no root is needed). Subtract 1.
7Sense-check: an upward-sloping curve means forward rates sit above spot rates; an inverted curve means forwards sit below.
8Eliminate options that fail the check, then pick the one that matches your calculation.

Quickest way: Forward rate shortcut and spread logic check

When to use it: Use when you have about 90 seconds and the question gives spot rates or lists several spread values.

For a 1y1y forward: compute (1 + z2)² ÷ (1 + z1) − 1. On the TI BA II Plus, key 1.025 [y^x] 2 [=] [÷] 1.02 [=] [−] 1 [=], giving 0.030025, which is 0.0300 when rounded to four decimal places.
Rough check: the forward is about 2 × z2 − z1 when rates are small. For 2.5% and 2.0%, that gives 3.0%.
If the curve slopes up, discard any option where the forward is below the long spot rate.
For spread questions, start from the Z-spread. Callable: the option value is positive, so subtract it to get OAS. Putable: the option value is negative, so add the size of the put value.
Check units. Spreads are in bp. If the answer looks like 1.2 instead of 120, you left it in percent.

Common mistakes in Yield Spreads and the Term Structure

Averaging spot rates to get a forward rate.
Candidates think a 2-year rate is the simple average of two one-year rates.
Fix: Use the compounding equation. Divide the 2-year growth factor by the 1-year growth factor, then subtract 1.
Treating Z-spread and OAS as the same thing for a bond with an embedded option.
Both are described as spreads over the spot curve.
Fix: Z-spread includes the effect of the option: higher for a callable bond, lower for a putable bond. OAS removes it. Callable: OAS = Z-spread − option value, so OAS is smaller. Putable: OAS is larger.
Adding the call option value to Z-spread to get OAS.
Candidates forget that the issuer owns the call, so the investor's extra yield must be removed.
Fix: The investor is paid extra yield for the call. Take that extra away to get OAS. Putable bonds work the other way.
Using the wrong benchmark for I-spread.
G-spread and I-spread look alike in formulas.
Fix: G is for government. I is for interest rate swap. Use the benchmark the question names.
Forgetting to subtract 1 from the growth factor.
The calculator shows a growth factor such as 1.0300, which looks like the answer.
Fix: Always subtract 1 at the end. 1.0300 means a 3.00% rate.
Calling a downward-sloping curve always a recession signal and ignoring its definition.
Candidates mix up the definition with market commentary.
Fix: An inverted curve simply has short-maturity yields above long-maturity yields. Learn the definition first. Theories explain why it occurs.

Worked examples

Example 1

The 1-year spot rate is 2.0% and the 2-year spot rate is 2.5%, both annually compounded. The implied one-year forward rate starting one year from now is closest to: A) 2.50%, B) 3.00%, C) 3.50%.

Show the solution

Write the no-arbitrage equation: (1.025)² = (1.02) × (1 + f).
Compute (1.025)² = 1.050625.
Divide by 1.02: 1.050625 ÷ 1.02 ≈ 1.030025.
Subtract 1: f ≈ 0.030025, or about 3.0025%, which is closest to 3.00%.
Check: the curve slopes up, so the forward is above the 2-year spot of 2.5%. Option A is wrong.

Answer: B) 3.00%

Example 2

A callable corporate bond has a YTM of 5.40%. The interpolated government bond yield at the same maturity is 4.20%, and the swap rate at that maturity is 4.60%. The bond's Z-spread is 128 bp and the embedded call option is worth 35 bp. What is the bond's OAS? A) 93 bp, B) 128 bp, C) 163 bp.

Show the solution

Note that the bond is callable. The issuer holds the option, so the Z-spread includes the effect of the call and is higher than OAS.
Use OAS = Z-spread − option value.
OAS = 128 bp − 35 bp = 93 bp.
For reference, G-spread = 5.40% − 4.20% = 120 bp and I-spread = 5.40% − 4.60% = 80 bp. These are not asked for.
Check: OAS is smaller than Z-spread, as expected for a callable bond. Option C (163 bp) adds the option value and fails this check.

Answer: A) 93 bp

Exam tips

Questions often give a callable or putable bond and ask whether OAS is above or below Z-spread. Decide who holds the option first, then answer.
For forward rate questions, write the equation before touching the calculator. It prevents setup errors. Many wrong options come from averaging rates or forgetting to subtract 1.
Use the option order. Numerical options run from smallest to largest, so check whether your answer fits one end. An upward-sloping curve should give a forward rate above the spot rate.
Memorise the one-line definitions: G = government, I = swap, Z = whole spot curve, OAS = Z minus option value. Many items test only these.
There is no penalty for wrong answers. If you are stuck, eliminate options that break the curve-shape logic and then choose.

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

Yield Spreads and the Term Structure in other exams

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

Yield Spreads and the Term Structure: frequently asked questions

What is the difference between G-spread, I-spread and Z-spread?

G-spread is the bond's YTM minus a government bond yield at the same maturity. I-spread is the bond's YTM minus the swap rate at the same maturity. Z-spread is one constant spread added to every spot rate on the benchmark curve so the bond's discounted cash flows equal its price.

How do I calculate a forward rate from spot rates?

Set the growth of the longer spot rate equal to the growth of the shorter spot rate times the forward rate. For a 1y1y forward, f = (1 + z2)² ÷ (1 + z1) − 1. For longer periods, take the appropriate root of the forward period.

What is option-adjusted spread in simple terms?

OAS is the spread a bond pays over the benchmark curve after taking out the value of its embedded option. It shows the extra yield for credit and liquidity risk only. It lets you compare bonds with and without options.

What shapes can the yield curve take?

The main shapes are upward-sloping (normal), inverted, flat and humped. An upward-sloping curve has longer-maturity yields above shorter ones. An inverted curve has the reverse.