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

Fixed Income, LOS weight share 0.8 percent of the 365 Level I learning outcomes.

Fixed IncomeThe Term Structure of Interest Rates: Spot, Par, and Forward Curves

Candidates who can compute a forward rate correctly still average two spot rates when the exam asks what a flat curve implies, because the reflex to reach for the calculator is stronger than the reflex to check whether a calculation was even required.

Before you watch

Answer these three first. Getting them wrong now is normal, and it helps the lesson stick. Reveal the answers when you're done, then read on.

1. The 1-year spot rate is 3.00% and the 2-year spot rate is 4.00%. The 1-year forward rate one year from now, f(1,1), is closest to:

Answer: B. (1+S2)^2 = (1+S1)(1+f(1,1)): (1.04)^2 / 1.03 = 1.0501, so f(1,1) = 5.01%. Forward rates are compounded geometrically from spot rates, never averaged; averaging (3.50%) is the exam's most common wrong-answer trap.

2. A bond's arbitrage-free value, computed by discounting each cash flow at the spot rate matching its own maturity, is $1,050. The bond currently trades at $1,020. This means:

Answer: C. Arbitrage compares market price to the arbitrage-free value, not to par. At $1,020 versus a fair value of $1,050, the bond is cheap: buy the bond, sell the replicating strip portfolio at $1,050, and capture $30 risk-free, an action that pushes the bond's price back toward $1,050.

3. The spot curve is upward sloping. Relative to the spot rate of matching tenor, the implied forward rate for a period further out on the curve is most likely:

Answer: A. On an upward-sloping spot curve, the marginal (forward) rate for the incremental period must exceed the blended spot rate that already averages in the lower earlier-period rates; the forward rate is always more extreme than the spot rate in the direction the curve slopes.

The lesson

Runtime 8 minutes 27 seconds, measured from the published video.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to explain the relationship among spot, par and forward rates, describe how bootstrapping builds a spot curve from par rates, explain what a forward rate actually represents, describe the ordering of the three curves under an upward, flat or downward sloping curve, and compare the four theories of the term structure's shape.

A spot rate is the yield on a single cash flow received at one future date. It is exactly the yield to maturity of an equivalent zero-coupon bond of that same maturity. The spot curve is simply that rate plotted across every maturity. A coupon bond's true, arbitrage-free price discounts each of its cash flows at the spot rate matching that specific cash flow's own maturity, never at one blended rate. A coupon paid in year one and the principal repaid in year ten are genuinely different assets with genuinely different timing risk. The par curve is a related but distinct idea. The par rate for a given maturity is the coupon rate that prices a bond of that maturity at exactly 100 once its cash flows are discounted along the spot curve. On an upward-sloping curve, a par bond's earlier coupons get discounted at lower, earlier spot rates. That pulls the par rate for any given maturity slightly below the spot rate of that same maturity.

A forward rate is not a forecast of where rates will actually go. It is the break-even reinvestment rate implied by two points already sitting on today's spot curve. It is the rate that leaves an investor exactly indifferent between locking in the long spot rate today or investing short and rolling over into the forward rate later. The notation f(j,k) reads left to right as starting in j years, lasting k years. Solving for it requires compounding the two spot rates against each other, (1+S(j+k))^(j+k) = (1+Sj)^j x (1+f(j,k))^k, never averaging them; averaging two spot rates is the exam's standard wrong-answer distractor for exactly this reason.

Bootstrapping builds the spot curve one maturity at a time, using only what has already been solved. The one-year spot rate always equals the one-year par rate outright, since a single cash flow has no intermediate coupon to discount separately. From there, the two-year par bond's price equation, with its year-one coupon now discounted at the already-known one-year spot rate, is solved for the unknown two-year spot rate, and every later maturity repeats that same pattern, building forward off everything solved before it.

The relative position of the par, spot and forward curves is fully determined by the curve's own slope, and this ordering can be read off directly without recalculating anything. On an upward-sloping curve, the forward curve sits above the spot curve, which sits above the par curve, at every maturity beyond the shortest. On a flat curve, all three curves collapse into exactly one line, since there is no slope left for compounding or blending to distort. On a downward-sloping curve, the ordering fully reverses: par sits above spot, which sits above forward.

Four theories explain why the curve takes the shape it does. Pure expectations says the curve reflects only the market's forecast of future short-term rates, with forward rates as unbiased predictors of those future spot rates. Liquidity preference keeps that same foundation but adds a liquidity premium longer-maturity bonds must offer, which biases forward rates upward without guaranteeing an upward-sloping curve outright, since a strong enough expectation of falling rates can still outweigh the premium. Market segmentation says each maturity range is its own separate market, with investors who never migrate between segments regardless of the yield differential on offer elsewhere. Preferred habitat is a variation on segmentation: investors have a preferred maturity zone but will leave it if offered enough extra yield to compensate for the discomfort of moving.

Normal, flat and inverted yield curves yield maturity normal flat inverted
The same axes, three shapes. Long rates usually sit above short rates (normal); sometimes they sit level (flat); sometimes short rates sit above long rates (inverted), which the market reads as a recession warning.

The trap

A forward rate computed from two spot rates is never the arithmetic average of the two; that average is the exam's standard wrong-answer choice, and the correct forward rate always comes from the geometric no-arbitrage compounding equation instead.

Learning outcomes covered by this module

Verbatim from the 2026 CFA Level I topic outline. Every practice question and key rule below is tagged to one of these where the stem and explanation make the match clear.

  1. define spot rates and the spot curve, and calculate the price of a bond using spot rates
  2. define par and forward rates, and calculate par rates, forward rates from spot rates, spot rates from forward rates, and the price of a bond using forward rates
  3. compare the spot curve, par curve, and forward curve

Key rules

Written from this module's own lesson and the 2026 CFA Level I topic outline, in teaching order, each tagged to the learning outcome it belongs to where that is clear.

LOS 01

The spot curve prices each individual future cash flow at its own zero-coupon rate

A spot rate is the yield on a single cash flow received at one future date, the yield to maturity of an equivalent zero-coupon bond of that maturity, and the spot curve is the plot of these rates across maturities. A coupon bond's arbitrage-free price is the sum of each of its cash flows discounted at the spot rate matching that cash flow's own maturity, not at one blended rate, because each cash flow is economically equivalent to its own separate zero-coupon bond.

LOS 02

The par curve is the coupon rate that prices a bond of each maturity exactly at par, and it is derived from, not identical to, the spot curve

The par rate for a given maturity is the coupon rate that makes a bond of that maturity price at exactly 100 when its cash flows are discounted along the spot curve; because the par bond's coupons before maturity are typically discounted at lower earlier spot rates on an upward-sloping curve, the par rate for a given maturity sits slightly below the spot rate of that same maturity for every maturity beyond the first year, where the two are always identical since there is only one cash flow.

LOS 02

A forward rate is the break-even reinvestment rate implied by two points on today's spot curve, not a forecast of the future

The forward rate f(j,k), a k-year rate starting j years from now, is the rate that makes an investor exactly indifferent between investing at the spot rate for j+k years versus investing at the spot rate for j years and then rolling over into the forward-implied rate for the remaining k years: (1+S(j+k))^(j+k) = (1+Sj)^j x (1+f(j,k))^k. Solving this equation for the forward rate requires geometric compounding, never simple averaging of the two spot rates, and solving it for k>1 requires taking the k-th root of the ratio, a step frequently skipped under exam time pressure.

LOS 01

Bootstrapping builds the spot curve one maturity at a time from observed par rates, using only what was already solved

Because the one-year spot rate always equals the one-year par rate (a single cash flow has no intermediate coupons to discount separately), bootstrapping starts there and proceeds sequentially: the two-year par bond's price equation, with its year-one coupon now discounted at the already-known one-year spot rate, is solved for the unknown two-year spot rate, and each subsequent maturity repeats this process using every spot rate solved in the prior steps. No later step ever needs data beyond what previous steps already produced.

LOS 03

The relative position of the par, spot, and forward curves is fully determined by whether the curve is upward sloping, flat, or downward sloping

On an upward-sloping curve: forward curve sits above spot curve, which sits above par curve, for every maturity beyond the shortest. On a flat curve, all three curves coincide exactly, because with no slope there is nothing for compounding or blending to distort. On a downward-sloping curve, the ordering fully reverses: par curve sits above spot curve, which sits above forward curve. Recognizing which of these three regimes applies answers most curve-comparison questions without any calculation.

The trick

Authored only where a key rule has an arbitrary number, list, or formula shape worth a memory device; a module with none of those has no tricks here, on purpose.

Forward rates are compounded from spot rates, never averaged

(3%+4%)/2 = 3.5% is always a wrong-answer trap when the exam asks for a forward rate from two spot rates; the correct method is (1+S_long)^n / (1+S_short)^m, minus one, with an (n-m)-th root taken whenever the forward period itself spans more than one year.

f(j,k) reads left to right as 'starting in j years, lasting k years'

f(1,2) is a 2-year rate starting 1 year from now; f(2,1) is a 1-year rate starting 2 years from now. These use different exponents in the compounding equation and produce different numbers; drawing a small timeline before setting up the equation prevents the swap.

Curve ordering on a sloped curve: upward means forward > spot > par; downward means the exact reverse

Memorize the upward case (forward highest, par lowest) and simply flip it for the downward case; on a flat curve all three curves collapse into one line and there is nothing to order.

The method

Authored, ordered steps for answering this module's question types; a calculation module's calculator-dependent step ends with a bracketed BA II Plus keystroke sequence.

  1. For a bond-valuation question, discount every individual cash flow at the spot rate matching its own maturity, then sum, rather than applying one blended yield to maturity.
  2. For a forward-rate question, identify j (years until the forward period starts) and k (length of the forward period) from the f(j,k) notation, set up (1+S(j+k))^(j+k) = (1+Sj)^j (1+f(j,k))^k, and solve, taking the k-th root whenever k exceeds one.
  3. For a bootstrapping question, solve maturities in order starting from one year, using only spot rates already solved in prior steps to discount intermediate coupons of the current par bond.
  4. For a curve-comparison question, first determine whether the curve is upward sloping, flat, or downward sloping, then apply the corresponding fixed ordering of par, spot, and forward curves rather than recalculating.
  5. For a mispricing question, compare the market price to the arbitrage-free value, not to par, and buy the cheaper of the bond and the replicating zero-coupon portfolio while selling the more expensive one.

One card

Condensed from the key rules and tricks above, nothing new. What you'd want on one index card the night before.

Practice questions

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. 10 question(s) available for this unit.

Question 1Exam level

The 1-year spot rate is 3.00% and the 2-year spot rate is 4.00%. The 1-year forward rate one year from now, f(1,1) is closest to:

How sure are you?

Correct: A. The correct answer is 5.01%. Using (1+S2)^2 = (1+S1) x (1+1f1): (1.04)^2 = (1.03) x (1+1f1). (1.0816)/(1.03) = 1.0501. Forward rate = 5.01%. The exam tests whether candidates correctly apply the no-arbitrage compounding relationship rather than simply averaging (3%+4%)/2=3.5%, which is wrong..
B. Simple average of 3% and 4%. Intuitive shortcut that ignores compounding. Forward rates are compounded, not averaged. Averaging is only valid if rates are continuously compounded and the difference is infinitesimal.
C. You might double S2 and subtracts S1: 2x4% - 3% = 5%, then rounds down to 4.5% as 'reasonable'. The arithmetic subtraction approach ignores the geometric nature of compounding. Must use (1+S2)^n / (1+S1)^m - 1.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 2Exam level

The 1-year spot rate is 2.50%, the 2-year spot rate is 3.00%, and the 3-year spot rate is 3.50%. The 1-year forward rate two years from now, f(2,1) is closest to:

How sure are you?

Correct: A. The correct answer is 4.51%. Formula: (1+S3)^3 = (1+S2)^2 x (1+2f1). (1.035)^3 = (1.03)^2 x (1+2f1). 1.108718 / 1.0609 = 1.04510. Forward rate = 4.51%. The first subscript is when the period starts, the second is the length..
B. You might read 3-year spot rate and assumes that is also the forward rate for the third period. Spot rates and forward rates are different. The 3-year spot rate is a blended rate covering all three periods; the forward rate is the marginal rate for just the third period.
C. You might linearly extrapolates: if spot rates rise by 0.5% per year, forward rate = 3.5% + 0.5% = 4%. Forward rates are derived geometrically from spot rates. Linear extrapolation ignores compounding and will always give the wrong answer.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 3Exam level

An analyst observes the following spot rates: S1 = 2%, S2 = 3%, S3 = 4%. She calculates that a 3-year bond's arbitrage-free price is $980. The bond currently trades at $960. What arbitrage trade should she execute, most likely?

How sure are you?

Correct: A. The correct answer is Buy the bond at $960 and sell (short) the replicating portfolio of zero-coupon bonds at $980, capturing a $20 risk-free profit. The arbitrage-free price ($980) represents what it costs to replicate the bond's cash flows using individual zero-coupon bonds (STRIPS). If the bond is cheaper than the replicating portfolio, buy the cheap asset and sell the expensive one..
B. You might reverse the trade direction. Confuses which asset is cheap vs expensive. The bond at $960 is CHEAP relative to the replicating portfolio at $980. Buy cheap, sell expensive.
C. You might incorrectly focuses on the par value comparison rather than the relative pricing between bond and replicating portfolio. Arbitrage is about relative mispricing between the bond and its replicating portfolio, not about pricing relative to par.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 4Exam level

The 2-year spot rate is 4.00%. The 1-year forward rate one year from now is 5.01%. The 1-year spot rate is closest to:

How sure are you?

Correct: A. The correct answer is 3.00%. Rearrange: (1+S2)^2 = (1+S1)(1+1f1). (1.04)^2 = (1+S1)(1.0501). 1.0816 / 1.0501 = 1.03. S1 = 3.00%. This reversal of the standard formula is a common exam variant. Most candidates can solve forward-from-spot but struggle to solve spot-from-forward..
B. You might subtract: 4.00% - (5.01% - 4.00%) = 2.99%, rounds to 2%. Arithmetic subtraction ignores compounding. Must use the geometric rearrangement of the formula.
C. You might average 4.00% and 5.01% = 4.5%. Averaging never works for compounded rates in a term structure calculation.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 5Exam level

Which statement best describes the 'arbitrage-free' condition in bond valuation?

How sure are you?

Correct: A. The correct answer is A bond's market price must equal the sum of the present values of each cash flow discounted at the spot rate for that cash flow's maturity. If not, a trader can construct a riskless profit by stripping or reconstituting the bond relative to a portfolio of zero-coupon bonds..
B. YTM and risk-free rate seem related; candidates confuse the definition of no-arbitrage with the definition of fair yield. Arbitrage-free pricing is not about the level of YTM relative to risk-free rates; it is about internal consistency with the spot rate curve.
C. Par bonds trade at face value. You might confuse 'fairly priced' with 'arbitrage-free'. An arbitrage-free bond can trade at a premium or discount to par. The condition is consistency with spot rates, not equality to par.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 6Exam level

The 1-year spot rate is 3%, the 2-year spot rate is 4%, and the 3-year spot rate is 5%. The 2-year forward rate starting one year from now, f(1,2) is closest to:

How sure are you?

Correct: A. The correct answer is 6.01%. Formula: (1+S3)^3 = (1+S1)^1 x (1+1f2)^2. (1.05)^3 = (1.03) x (1+1f2)^2. 1.157625 / 1.03 = 1.12390. (1.12390)^(0.5) - 1 = 0.0601 = 6.01% per year. Key: take the nth root where n = length of the forward period (here, 2 years)..
B. You might read off the 3-year spot rate, confusing spot and forward. The 3-year spot rate and the 2-year forward starting in 1 year are entirely different quantities.
C. You might forget the square root step: computes (1.12390) - 1 = 12.39%, then halves it to 6.20% or misestimates. Forgetting the nth root for multi-period forward rates is the most common error. For f(1,2), the right side is (1+1f2)^2, requiring a square root at the end.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 7Exam level

A bond has an arbitrage-free value of $1,050 but trades in the market at $1,020. Which of the following is most likely true?

How sure are you?

Correct: A. The correct answer is The bond is underpriced. Arbitrageurs will buy the bond at $1,020 and sell (short) the replicating portfolio of zero-coupon strips at $1,050, earning $30 per bond. This activity will push the bond price up toward $1,050, restoring equilibrium. The arbitrage-free value represents the fair value derived from the spot curve; any deviation is an arbitrage opportunity..
B. You might compare bond price to par ($1,000) rather than to the arbitrage-free value ($1,050). The benchmark for arbitrage is the arbitrage-free value ($1,050), not par. At $1,020, the bond is BELOW its fair value.
C. You might focuse on the absolute level (both above $1,000) rather than the relative mispricing. Arbitrage is about relative mispricing between the bond and replicating portfolio. If market price ≠ arbitrage-free value, an arbitrage exists regardless of the relationship to par.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 8Exam level

An investor can either invest for 2 years at the 2-year spot rate of 4% per year, or invest for 1 year at the 1-year spot rate of 3% and then reinvest at whatever rate prevails in year 2. What forward rate would most likely make the investor indifferent between the two strategies?

How sure are you?

Correct: A. The correct answer is 5.01%. This is the break-even forward rate, which is identical to f(1,1). At 5.01%, both strategies yield the same accumulated value: Strategy 1: (1.04)^2 = 1.0816. Strategy 2: (1.03)(1.0501) = 1.0816. The forward rate is the rate that makes you indifferent. It is the break-even reinvestment rate embedded in the spot curve..
B. You might assume the forward rate must equal the current long-term spot rate. The 2-year spot rate is a blended rate covering both periods. The forward rate is the marginal rate for just the second period, which must be higher than 4% to compensate for the lower first-year rate of 3%.
C. Simple arithmetic average of the two spot rates. The break-even forward rate requires the geometric calculation, not arithmetic averaging.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 9Above the exam

The 1-year spot rate is 3% and the 2-year spot rate is 5%. Combining the relationship between spot rates and forward rates, the 1-year forward rate one year from now, f(1,1), is closest to:

How sure are you?

Correct: A. The forward rate is derived from the no-arbitrage relationship between spot rates: (1 + S2)^2 = (1 + S1) x (1 + f(1,1)). (1.05)^2 = 1.1025 = 1.03 x (1 + f(1,1)); 1 + f(1,1) = 1.1025 / 1.03 = 1.0704; f(1,1) = 7.04%, closest to 7.0%. The 2-year spot rate is a geometric blend of the 1-year spot rate and the market's implied 1-year rate one year from now, so the forward rate must be solved for algebraically, not simply averaged or subtracted.
B. 4.0% comes from a simple average of the two spot rates ((3%+5%)/2), which is not how forward rates relate to spot rates; the relationship is compounding (geometric), not a simple average, which is exactly the distinction this LOS tests.
C. 5.0% is just the 2-year spot rate itself, as though the forward rate one year from now equaled the current 2-year rate; the forward rate is a DIFFERENT, derived figure implied by the gap between the 1-year and 2-year spot rates, not simply equal to either one.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

Question 10Above the exam

An investor observes an upward-sloping (positive) yield curve and, based purely on the pure expectations theory, concludes that the market expects short-term interest rates to rise steadily in the future. A more sophisticated colleague points out that the liquidity preference theory offers an alternative explanation. Combining both theories, the colleague's most likely point is that:

How sure are you?

Correct: A. The pure expectations theory attributes the ENTIRE shape of the yield curve to expected future short-term rates. The liquidity preference theory adds that investors generally demand an additional liquidity premium for holding longer-term bonds (compensation for greater price risk/less liquidity), which by itself can produce an upward-sloping curve even if the market does not actually expect short rates to rise at all; the observed slope could reflect some mix of genuine rate-rise expectations AND a liquidity premium, and the two cannot be cleanly separated just by looking at the curve's shape.
B. Liquidity preference theory does not always produce a downward-sloping curve; it adds an UPWARD bias (a positive liquidity premium that generally increases with maturity) on top of whatever the expectations component implies, which is consistent with, not contradictory to, an observed upward-sloping curve.
C. The two theories can produce very different implied conclusions about actual rate expectations from the SAME observed curve shape; that is exactly the colleague's point, that pure expectations theory overstates how much of the curve's slope reflects genuine rate expectations once a liquidity premium is also at work.

Unit: the-term-structure-of-interest-rates-spot-par-and-forward-curves

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