How the spot, par and forward curves relate
When the spot curve slopes up, forward rates sit above it and (with positive rates) par rates just below it. When it is flat, all three are equal. When it slopes down (inverted), forwards sit below the spot curve and par rates just above. Forwards move most because they are the marginal rates.
Before this: Spot rates and pricing a bond with the spot curve, Par rates from spot rates, Implied forward rates: from spot to forward and back
On the exam
- Compare the spot curve, par curve, and forward curve
Compare the spot, par and forward curves. Expect a chart question asking which line is which, or a shape-to-relationship item, and a short calculation of a forward rate from two spot rates.
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 are the three curves linked?
All three curves describe default-free rates, and the last two are calculated from the first. The spot curve holds yields on zero-coupon bonds that start today. The par curve holds the coupon rates that price par bonds off the spot curve. The forward curve holds rates for periods that start in the future. Because the par and forward curves come from the spot curve, its shape decides theirs.
Try this 0 of 3
- Turn on the quiz and name all three lines
- Find a curve where every forward sits below the spot curve
- On the humped curve, read a year where the spot curve is falling
Pick a shape, then twist the spot curve. Tap the chart to read all three rates at a year.
- Spot
- Par
- Forward (one-year)
Spot rates from 3.20% to 5.13% over 10 years. One-year forwards range from 3.20% to 5.76%; par rates from 3.20% to 5.03%. At year 4: spot 4.400%, par 4.362%, forward 5.305%.
At year 4 the spot curve is rising, so the 3y1y forward (5.305%) is above the spot rate (4.400%); the par rate (4.362%) is just below it.
Why: a longer spot rate is an average of the shorter spot and the newest forward, so the forward must sit on the far side of a rising (or falling) average. The par rate blends all the spot rates up to maturity, weighted toward the last one, so it stays close.
Show working
Discount factors: dₜ = 1 ÷ (1 + zₜ)ᵗ. d₄ = 1 ÷ (1 + 4.40%)⁴ = 0.84178.
Par rate: (1 − d₄) ÷ (d₁ + … + d₄) = (1 − 0.84178) ÷ 3.62712 = 4.362%.
Forward 3y1y: (1 + z₄)⁴ ÷ (1 + z₃)³ − 1 = 1.187960 ÷ 1.128112 − 1 = 5.305%.
| Year | Spot | Par | Forward |
|---|---|---|---|
| 1 | 3.200% | 3.200% | 3.200% |
| 2 | 3.700% | 3.691% | 4.202% |
| 3 | 4.100% | 4.077% | 4.905% |
| 4 | 4.400% | 4.362% | 5.305% |
| 5 | 4.610% | 4.559% | 5.454% |
| 6 | 4.770% | 4.707% | 5.574% |
| 7 | 4.890% | 4.817% | 5.613% |
| 8 | 4.980% | 4.899% | 5.612% |
| 9 | 5.060% | 4.970% | 5.702% |
| 10 | 5.130% | 5.032% | 5.762% |
- 1.041y^x3=STO1(1 + z₃)³
- 1.044y^x4=÷RCL1=−1=0.05305, the 3y1y forward
y^x power · STO store · RCL recall a stored number
- 1.044ENTER4y^x(1 + z₄)⁴
- 1.041ENTER3y^x÷1−0.05305, the 3y1y forward
ENTER enter (on the HP, puts the number on the stack) · y^x power
Display: 0.05305 = 5.305%
Where do the par and forward curves sit?
- Upward sloping Memorize the normal shape
- Par curve slightly below spot
- Forward curve above spot
-
- Flat Memorize
- Par curve equal to spot
- Forward curve equal to spot
-
- Downward sloping (inverted) Memorize
- Par curve slightly above spot
- Forward curve below spot
-
| Spot curve | Par curve | Forward curve |
|---|---|---|
| upward-sloping curve | slightly below spot | above spot |
| flat curve | equal to spot | equal to spot |
| inverted curve (downward sloping) | slightly above spot | below spot |
Two short arguments explain the table.
Forwards. A longer spot rate is a geometric average of the shorter spot rate and the forward rate for the extra period. If that average rises with maturity, the newest slice must be above it. If it falls, the newest slice is below it. A flat curve has no change to explain, so every forward equals the spot rate.
Par rates. A par rate blends all the spot rates up to its maturity, with the most weight on the final one. On an upward curve the earlier, lower spot rates pull the blend a little under the final spot rate, so par sits just below spot. On an inverted curve the earlier rates are higher and par sits just above. (This par rule assumes positive rates; see the note on negative rates below.)
What does an upward curve look like in numbers?
Take the spot rates from the first three lessons.
| Maturity | Spot | Par | 1-year forward ending then |
|---|---|---|---|
| 1 year | 3.200% | 3.200% | 3.200% |
| 2 years | 3.700% | 3.691% | 4.202% |
| 3 years | 4.100% | 4.077% | 4.905% |
| 4 years | 4.400% | 4.362% | 5.305% |
Par is a few hundredths of a percent below spot. The forward is far above spot, and it rises faster. At 4 years the spot rate is 4.40% but the 3y1y forward is 5.305%, nearly a full point higher.
What does an inverted curve look like in numbers?
Exam tip Spot curve up or down is the first thing to read. The forward curve then goes further in the same direction, and par stays close to spot on the “earlier” side.
Go deeper Inverted curves, expectations and the cycle
An inverted spot curve implies forward rates below today’s short rates, so it is often read as a sign that the market expects rates to fall, for example as policy tightens and then growth slows. How far that reading holds is a matter for the Economics topic and for the Level II theories of the term structure.
Go deeper Humped curves
If the spot curve rises and then falls, the forward curve crosses it at the peak: above it on the way up and below it on the way down. Par rates stay close to the spot curve throughout.
Go deeper Negative spot rates
The forward rule is unchanged when rates are negative. If the 1-year spot is −0.30% and the 2-year spot is −0.10%, the 1y1y forward is 0.999² ÷ 0.997 − 1 = 0.100%. The curve slopes up, so the forward is above the spot rates, and it is positive even though both spot rates are negative.
The par rule is the one that flips. With the same two spot rates, the 2-year par rate is (1 − d₂) ÷ (d₁ + d₂) = −0.0999%, a touch above the −0.10% spot rate. With negative spot rates, an upward curve puts par slightly above spot, and an inverted curve puts it slightly below. Questions on negative rates test exactly this.
Memorize this
MemorizeSpot, par and forward by curve shape
| Spot curve | Par curve | Forward curve |
|---|---|---|
| Upward sloping | Slightly below spot | Above spot |
| Flat | Equal to spot | Equal to spot |
| Downward sloping (inverted) | Slightly above spot | Below spot |
Memory hook Forwards exaggerate the slope, par rates hug the spot curve. Up: par under, forward over. Down: the reverse.
Know it cold Forwards are marginal rates and move most. Par rates are a blend of spot rates, so they stay close to the spot curve. The par column assumes positive rates: with negative spot rates the par side flips (an upward curve then has par slightly above spot).
MemorizeWhy the forward curve leads
Memory hook A longer spot rate is an average of the shorter spot rate and the forward. If the average is rising, the newest slice must be above it.
Know it cold Rising spot curve: forward above spot. Falling spot curve: forward below spot.
- Upward sloping Memorize the normal shape
- Par curve slightly below spot
- Forward curve above spot
-
- Flat Memorize
- Par curve equal to spot
- Forward curve equal to spot
-
- Downward sloping (inverted) Memorize
- Par curve slightly above spot
- Forward curve below spot
-
Quick example
The 1-year spot is 6.0% and the 2-year spot is 5.2%. The 1y1y forward is 1.052² ÷ 1.06 − 1 = 4.41%, below both spot rates: an inverted curve.
- 1.052y^x2=÷1.06−1= 0.0441
y^x power
- 1.052ENTER2y^x1.06÷1− 0.0441
ENTER enter (on the HP, puts the number on the stack) · y^x power
How it is asked
-
Ordering of the three curves
The spot curve slopes upward. Which ordering is most likely at 5 years?
par < spot < forward. Par above spot or forward below spot are the traps.
-
Flat curve, forward rate
Spot curve flat at 3.0%. The 4y1y forward?
3.0%. Equal spot rates at every maturity leave nothing for forwards to add.
-
Which line is which on a chart
Two lines: the spot curve and one other that moves more and crosses above it where the spot curve rises.
The line that exaggerates the slope is the forward curve.
-
Forward from two spots on an inverted curve
1-year spot 5.9%, 2-year spot 5.4%. 1y1y?
1.054² ÷ 1.059 − 1 = 4.90%. The 2-year spot (5.40%) and the average (5.65%) are the traps.
Exam traps
- Putting the forward curve below the spot curve when the spot curve slopes upward.
- Putting the par curve above the spot curve when the spot curve slopes upward.
- Thinking a flat spot curve implies different forward rates. Flat spot means forwards equal spot.
- Assuming negative spot rates mean negative forwards. A rising curve can give positive forwards from negative spot rates.
- Applying the par rule to negative rates unchanged. With negative spot rates on an upward curve, par sits slightly above spot, not below.
Checkpoint
5 exam-style questions. About 90 seconds each.
1Spot rates are positive and the spot curve slopes upward. Which ordering is most likely correct at a 5-year maturity?
2The spot curve is flat at 3.0% for all maturities. The 4y1y forward rate is:
3Spot rates are 5.0% (1 year) and 4.6% (2 years). The 1y1y forward rate is closest to:
4A government's spot curve is inverted, with all spot rates positive. Compared with the spot curve, the forward curve and the par curve are most likely:
5A government's spot rates are all negative, from −0.60% at 1 year rising to −0.20% at 5 years. Compared with the spot curve, its par curve most likely lies:
Key takeaways
- Upward spot curve: forwards above, par slightly below.
- Flat spot curve: spot, par and forward are all equal.
- Inverted spot curve: forwards below, par slightly above.
- Negative spot rates flip only the par side: on an upward curve, par then sits slightly above spot.
- Forwards are the marginal rates, so they swing more than the spot curve and par curve.
Quick example
The 1-year spot is 6.0% and the 2-year spot is 5.2%. The 1y1y forward is 1.052² ÷ 1.06 − 1 = 4.41%, below both spot rates: an inverted curve.
- 1.052y^x2=÷1.06−1= 0.0441
y^x power
- 1.052ENTER2y^x1.06÷1− 0.0441
ENTER enter (on the HP, puts the number on the stack) · y^x power
All Fixed Income formulas, every lesson
Practice exam-style questions on Term Structure: Spot, Par and Forward Curves
Questions about this lesson?
Where this comes back
- Interest Rate Risk and Return Fixed Income
- Effective Duration, Key Rate Duration and Empirical Duration Fixed Income
- Pricing and Valuing Forward Contracts and FRAs Derivatives
- Monetary Policy Economics
Questions candidates ask
Why is the forward curve above the spot curve?
When spot rates rise with maturity, each additional year must earn more than the average so far to lift the average. So forward rates exceed spot rates on an upward curve.
What does an inverted yield curve imply for forward rates?
Forward rates are below spot rates. The curve implies lower short-term rates in the future than the rates of today.
Are par rates above or below spot rates?
With positive rates: just below when the curve slopes upward, just above when it slopes downward, and equal when it is flat. The gap is small. With negative spot rates the par side flips.
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