CFA Level I Exam · The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Yield to Maturity, Yield Spreads and Yield Curve Movements

Updated 7 October 2026 · Fact-checked

Yield to maturity is the single rate that discounts a bond's cash flows to its price. Spot rates discount each cash flow separately. Spreads measure extra yield over a benchmark: G-spread over government yields, I-spread over swap rates, Z-spread over the whole spot curve. Compare the curve shape and shift to judge returns.

Understand Yield to Maturity, Yield Spreads and Curve Movements

A bond's yield to maturity (YTM) is one discount rate applied to every cash flow. It makes the present value of the coupons and principal equal to the market price. It assumes you hold to maturity, receive all payments and reinvest coupons at the YTM. A spot rate is different. It is the yield on a zero-coupon bond for one specific maturity. Discounting each cash flow at its own spot rate gives the arbitrage-free price, so spot rates are the more precise tool.

A yield spread is the extra yield a bond pays over a benchmark. The G-spread is the bond's YTM minus the YTM of a government bond of the same maturity. If no government bond matches, you interpolate between two government bonds. The I-spread is the bond's YTM minus the swap rate of the same maturity. The Z-spread (zero-volatility spread) is a constant spread added to every spot rate on the benchmark curve so that the discounted cash flows equal the bond price. G-spread and I-spread compare one yield with one yield. The Z-spread uses the whole curve, so it is better when the curve is steep. For a bond with an embedded option, the OAS is the Z-spread minus the option's value in spread terms. For a callable bond, OAS is lower than the Z-spread.

The yield curve plots yield against maturity. The common shapes are upward sloping (normal), flat, inverted (short rates above long rates) and humped (rises, peaks, then falls). Curves move in a few ways. A parallel shift moves all maturities by the same amount. Steepening means long rates rise relative to short rates. Flattening means the gap narrows. A butterfly (curvature) change moves the middle of the curve relative to the short and long ends.

Riding the yield curve (rolling down the curve) is a strategy for an upward-sloping curve. You buy a bond with a maturity longer than your horizon and sell it at the horizon. If the curve stays unchanged, the bond ages into a shorter maturity with a lower yield. Its price rises, so you earn more than the short-term rate. The gain disappears if the curve shifts up enough, because the price falls. Forward rates are the break-even: the bond's return equals the shorter-term alternative if future spot rates equal today's forward rates.

Key formulas to remember

Bond price using YTM: PV = Σ CF_t ÷ (1 + y)^t; One rate y for all cash flows. Solve y with the calculator.
Bond price using spot rates: PV = Σ CF_t ÷ (1 + z_t)^t; Each cash flow uses the spot rate z_t for its own maturity.
G-spread: G-spread = bond YTM − government bond YTM (same maturity); Interpolate between two government bonds if no exact match exists.
I-spread: I-spread = bond YTM − swap rate (same maturity); Benchmark is the interest rate swap curve, not government bonds.
Z-spread: PV = Σ CF_t ÷ (1 + z_t + Z)^t; Z is constant across all maturities. Found by trial and error or a solver.
OAS for an option bond: OAS = Z-spread − option value (in spread terms); Callable: OAS < Z-spread. Putable: OAS > Z-spread.
Forward rate from spot rates: (1 + z_B)^B = (1 + z_A)^A × (1 + f(A, B−A))^(B−A); f(A, B−A) is the rate for a loan of B−A years starting in A years.
Rolling-down return (unchanged curve): Return = P(horizon) ÷ P(today) − 1; P(horizon) uses the spot rate for the bond's remaining maturity from today's curve.

How to solve Yield to Maturity, Yield Spreads and Curve Movements questions

Use this method for any question on YTM, spreads or curve movements.

1Identify what is asked: a YTM, a spread type, a price from spot rates, a forward rate, or a return from a curve change.
2Name the benchmark. Government yield means G-spread. Swap rate means I-spread. Whole spot curve means Z-spread.
3If the question gives spot rates, discount each cash flow at its own rate. Do not use one YTM.
4For forward rates, use the spot-rate identity and solve for the forward term. Keep the years straight.
5For riding the yield curve, price the bond today and at the horizon using the unchanged curve, then compute P(horizon) ÷ P(today) − 1, adding coupons if any are received.
6If the curve shifts, reprice at the new rates. Compare with the unchanged-curve result.
7Check the sign and size: spreads are normally positive for a risky bond, and a rolling-down gain should exceed the short rate on an upward curve.
8Eliminate options that use the wrong benchmark or the wrong maturity.

Quickest way: Shortcut for spread and roll-down questions

When to use it: Use when you have about 90 seconds and the options are far apart.

For G-spread or I-spread, subtract the two yields. Check the benchmark named in the stem.
For Z-spread, test the middle option first. If the computed price is too high, the spread is too small, so go up. If too low, go down.
For roll-down on a zero-coupon bond bought with n years left and sold after h years, state: return = (1 + z_n)^n ÷ (1 + z_(n−h))^(n−h) − 1.
Use the calculator: to compute (1.04)^3, enter 1.04, press y^x, enter 3, press =, giving 1.124864.
Discard any option below the short-term rate if the curve is upward sloping and unchanged.

Common mistakes in Yield to Maturity, Yield Spreads and Curve Movements

Using the wrong benchmark for a spread, such as subtracting a swap rate for a G-spread.
The three spreads sound alike and all subtract a benchmark yield.
Fix: Link the letter to the benchmark: G for government, I for interest rate swap, Z for the zero-volatility (spot) curve.
Discounting at one YTM when the question gives spot rates.
YTM is the habit from earlier bond pricing.
Fix: Whenever you see a spot curve, discount each cash flow at its own spot rate.
Adding the Z-spread to the YTM instead of to each spot rate.
Confusing the Z-spread with a simple yield difference.
Fix: The Z-spread is added to every spot rate on the curve before discounting.
Saying OAS is always bigger than the Z-spread, or ignoring option type.
Forgetting who owns the option.
Fix: A callable bond has OAS below Z-spread because the issuer owns the option. A putable bond has OAS above Z-spread.
Assuming riding the yield curve always earns extra return.
Only the unchanged-curve case is remembered.
Fix: It works on an upward curve that stays unchanged. A rise in yields can wipe out or reverse the gain. On a flat or inverted curve there is no roll-down gain.
Mixing up forward rate periods, for example using f(1,1) when the question needs f(2,1).
The notation shows start and length, which are easy to swap.
Fix: Write the notation as f(start, length) and check that start + length equals the longer spot maturity.

Worked examples

Example 1

The government spot curve is 2.0% for one year and 3.0% for two years. A 2-year bond pays an annual coupon of 5% on a par of 100 and is priced at 101.93. Which Z-spread reproduces the price?
A. 0.50%
B. 1.00%
C. 1.50%

Show the solution

Write the pricing equation: 101.93 = 5 ÷ (1.02 + Z) + 105 ÷ (1.03 + Z)^2, with Z in decimals. Rates are 2.0% and 3.0%, so the first factor is (1 + 0.02 + Z) and the second is (1 + 0.03 + Z).
Test B, Z = 1.00%. First cash flow: 5 ÷ 1.03 = 4.8544.
Second cash flow: 105 ÷ (1.04)^2 = 105 ÷ 1.0816 = 97.0784.
Sum: 4.8544 + 97.0784 = 101.9328, which rounds to 101.93. This matches the price.
A smaller Z (0.50%) would give a higher price, and a larger Z (1.50%) a lower price, so neither fits.

Answer: B. The Z-spread is 1.00%.

Example 2

The spot curve for zero-coupon bonds is: 1 year 2.0%, 2 years 3.0%, 3 years 4.0%. An investor with a one-year horizon buys a 3-year zero-coupon bond and sells it after one year. The curve does not change. What is the return?
A. 2.00%
B. 6.03%
C. 8.00%

Show the solution

Price today of the 3-year zero (par 100): 100 ÷ (1.04)^3 = 100 ÷ 1.124864 = 88.8996.
After one year, the bond has two years left. Under the unchanged curve the 2-year spot rate is 3.0%.
Price at horizon: 100 ÷ (1.03)^2 = 100 ÷ 1.0609 = 94.2596.
Return = 94.2596 ÷ 88.8996 − 1 = 1.0603 − 1 = 6.03%.
This is above the 2.0% one-year spot rate, because the bond rolled down an upward-sloping curve.

Answer: B. The return is about 6.03%.

Exam tips

Read the benchmark word first: government, swap or spot curve. It decides the spread type.
Many questions test direction rather than calculation: callable OAS below Z-spread, steepening versus flattening, and which curve shape is normal.
For roll-down, remember the maturity shortens at the horizon and the lower yield applies. Check whether a coupon is received.
Numerical options go from smallest to largest. If the answer must exceed the short rate, you can often remove the smallest option immediately.
There is no penalty for a wrong answer, so always pick one after eliminating what you can.

Practice questions from The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Yield to Maturity, Yield Spreads and Curve Movements in other exams

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

Yield to Maturity, Yield Spreads and Curve Movements: 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 YTM of the same maturity. I-spread is the YTM minus the swap rate of the same maturity. Z-spread is a constant spread added to each spot rate so that the discounted cash flows equal the bond price.

How do you calculate the Z-spread in the CFA Level I exam?

Write the price as the sum of cash flows discounted at spot rate plus Z. Try the option values for Z and see which one reproduces the given price. A higher Z gives a lower price, so you can adjust in the correct direction.

What is riding the yield curve?

It is buying a bond with a longer maturity than your horizon and selling it at the horizon. On an upward-sloping curve that stays unchanged, the bond's price rises as its yield falls with a shorter remaining maturity. The strategy loses its advantage if yields rise.

What shapes can the yield curve take?

The main shapes are upward sloping (normal), flat, inverted and humped. An inverted curve has short-term yields above long-term yields. A humped curve rises to a peak at intermediate maturities and then declines.