Lesson 3 of 4 11–14% of the exam

By Daniil Tarakanov Updated Oct 20, 20262027 curriculum

Implied forward rates: from spot to forward and back

An implied forward rate is the rate for a loan starting in the future that makes investing short and rolling over pay the same as investing long today. Get it from two spot rates, chain forwards back into spot rates, and price bonds with either set: the answer is the same.

12 min cram, 48 min full read. Based on the 2027 CFA® Program Level I curriculum, Fixed Income, Learning Module 9. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Spot rates and pricing a bond with the spot curve, Par rates from spot rates

On the exam

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

Define forward rates and calculate forward rates from spot rates, spot rates from forward rates, and bond prices from forward rates. Expect 2y1y and 1y2y style items, a chain of three one-year forwards, a bond priced off forwards, and a breakeven decision.

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.

What is an implied forward rate?

Memorize

Suppose you have two ways to invest for four years: buy a 4-year zero-coupon bond, or buy a 3-year zero and reinvest the proceeds for one more year. The first locks in today’s 4-year spot rate. The second depends on the one-year rate in three years’ time, which is unknown today. The rate that would make the two routes pay the same is the implied forward rate , also called the forward yield . It is the extra return the market is offering for going one more year out.

That makes it a breakeven reinvestment rate . If you expect the future rate to be higher, rolling the shorter bond is better. If you expect lower, buy the long bond now.

Buy long or buy short and roll?

Compare your expected future rate with the forward ratethe forward is the breakeven reinvestment rate

Expected rate above the forward Memorize buy the shorter bond and reinvest
Expected rate below the forward Memorize buy the longer bond now
Expected rate equal to the forward indifferent: both routes end at the same value

Market practice names forwards in a short code. The 3y1y notation reads “three-year into one-year”, or “3’s, 1’s”: a one-year rate starting three years from now. The first number is when the rate starts and the second is its length.

Reading the AyBy notation

AyBy forward ratefirst number: when it starts; second: how long it lasts

3y1y Memorize one-year rate starting in 3 years; needs the 3-year and 4-year spots
1y2y Memorize two-year rate starting in 1 year; needs the 1-year and 3-year spots
0y1y starts today: just the 1-year spot rate

InteractiveImplied forward rates: two routes, one answer

Try this 0 of 3

Expect a rate above the forward and see which route wins
Find a forward that sits below the spot rate for the same end date
Price the 4-year 5% bond with its forward rates

Spot rates (annual):

z₁%z₂%z₃%z₄%

Forward starts at A (years from now):

and ends at B:

Implied forward 1y2y4.553%

3-year spot4.10%

1 grows to (long)1.12811

Both routes turn 1 into 1.12811. The implied forward 1y2y (4.553%) is the one future rate that makes waiting cost nothing. It is above the 3-year spot rate because spot rates rise from year 1 to year 3.

Show working

No arbitrage: (1 + z₁) × (1 + 1y2y)² = (1 + z₃)³.

1y2y = [(1 + 4.10%)³ ÷ (1 + 3.20%)]^(1/2) − 1 = (1.128112 ÷ 1.032000)^(1/2) − 1 = 4.553%.

Check: 1.032000 × (1 + 4.553%)² = 1.12811, the same as 1.12811 on the long route.

1y2y on your calculator

1.041y^x3=÷1.032=(1 + z₃)³ ÷ (1 + z₁)
y^x21/x=square root: per year
−1=0.04553

y^x power · 1/x reciprocal

1.041ENTER3y^x1.032÷(1 + z₃)³ ÷ (1 + z₁)
21/xy^xsquare root: per year
1−0.04553

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Display: 0.04553 = 4.553%

A forward rate is the future rate that makes waiting cost nothing.

How do you calculate a forward rate from spot rates?

Growing money at the long spot rate must equal growing it at the shorter spot rate and then at the forward rate. Write that equality and solve for the forward.

Implied forward rate Memorize

(1 + z_A)^{A}\,(1 + f_{A,B})^{B} = (1 + z_{A+B})^{A+B} \;\Rightarrow\; f_{A,B} = \left[\frac{(1 + z_{A+B})^{A+B}}{(1 + z_A)^{A}}\right]^{1/B} - 1

Here $A$ is when the forward starts, $B$ is its length in years, and $z_{A+B}$ is the spot rate for the end date.

Rearrange itImplied forward rate: solve for the forward or either spot rate

The question asks for

The 2-year spot rate is 3.70% and the 3-year spot rate is 4.10%. The 2y1y implied forward rate is closest to:

Years until the forward starts, ALength of the forward in years, BSpot rate for A years%Spot rate for A + B years%

f_{A, B} = [\frac{( 1 + z _{A + B} ) ^{A + B}}{( 1 + z _{A} ) ^{A}}]^{1/ B} - 1

(1.041^3 ÷ 1.037^2) − 1 = 4.905%

Implied forward rate4.90%

On your calculator

1.041y^x3=÷(1.037y^x2)=−1=0.04905, so 4.9046%

y^x power

1.041ENTER3y^x1.037ENTER2y^x÷1−0.04905, so 4.9046%

ENTER enter (on the HP, puts the number on the stack) · y^x power

How do you get spot rates from forward rates?

The same equation works backward. Multiplying the growth factors of the one-year forwards gives the growth factor of the longer spot rate, so a spot rate is the geometric mean of the forwards before it. Building spot rates from forward rates is the reverse of the calculation above.

Spot rate from one-year forward rates Memorize

(1 + z_n)^{n} = (1 + f_{0,1})(1 + f_{1,1})\cdots(1 + f_{n-1,1})

Rearrange itSpot rate from one-year forwards: solve for the spot or the last forward

The question asks for

One-year forward rates are 2.50% (0y1y), 3.10% (1y1y) and 3.60% (2y1y). The 3-year spot rate is closest to:

0y1y forward (today)%1y1y forward%2y1y forward%

z_{3} = [(1 + f_{0, 1}) (1 + f_{1, 1}) (1 + f_{2, 1})]^{1/3} - 1

(1.025 × 1.031 × 1.036)^(1/3) − 1 = 3.066%

3-year spot rate3.07%

On your calculator

1.025×1.031×1.036=y^x(1÷3)=−1=0.03066, so 3.0657%

y^x power

1.025ENTER1.031×1.036×31/xy^x1−0.03066, so 3.0657%

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

How do you price a bond with forward rates?

Each cash flow is discounted one period at a time, using the one-year forward for that period. The discount factor for a payment in year $t$ is the product of the first $t$ one-year forward growth factors. The result is the bond pricing with forward rates method, and it must equal the price from spot rates.

Bond price from forward rates Use it

PV = \frac{CF_1}{1 + f_{0,1}} + \frac{CF_2}{(1 + f_{0,1})(1 + f_{1,1})} + \frac{CF_3}{(1 + f_{0,1})(1 + f_{1,1})(1 + f_{2,1})} + \cdots

Rearrange itBond priced with forward rates: solve for price or coupon

The question asks for

One-year forward rates are 2.50% (0y1y), 3.10% (1y1y) and 3.60% (2y1y). The price per 100 of par of a 3-year bond with a 3.00% annual coupon is closest to:

0y1y forward (today)%1y1y forward%2y1y forward%Annual coupon rate%

P V = \frac{P M T}{1 + f _{0, 1}} + \frac{P M T}{( 1 + f _{0, 1} ) ( 1 + f _{1, 1} )} + \frac{P M T + F V}{( 1 + f _{0, 1} ) ( 1 + f _{1, 1} ) ( 1 + f _{2, 1} )}

2.927 + 2.839 + 94.079 = 99.845

Price per 100 of par99.85

On your calculator

103÷1.036+3=÷1.031+3=÷1.025=99.8451

103ENTER1.036÷3+1.031÷3+1.025÷99.8451

ENTER enter (on the HP, puts the number on the stack)

Exam tip A forward rate applies to one slice of time. Discounting a year-2 cash flow at the 1y1y squared treats the forward as a spot rate, the most common mistake in these questions.

Go deeper Money market forwards in months

The same equation works with periods shorter than a year. A “3m6m” forward is a six-month rate starting in three months. With simple interest on an actual/360 or actual/365 basis, the growth factors become (1 + rate × days ÷ basis), and the forward is found by dividing the long growth factor by the short one. The derivatives topic uses this for forward rate agreements.

Go deeper Forward rates are not forecasts

An implied forward rate comes from today’s prices, and it equals the market’s expected future rate only if investors demand no extra compensation for holding longer bonds. Theories of why the curve has its shape, including term premiums, are a Level II topic. For the exam, a forward is the breakeven rate, nothing more.

Go deeper Forward rate agreements

A forward rate agreement lets two parties fix today a rate for a future period. Its fair rate is the implied forward rate, which is why this lesson is the base for forward and swap pricing in the Derivatives topic.

Memorize this

MemorizeForward rate from two spot rates

Memory hook Long growth divided by short growth is what the forward period must add. Then take the root for its length.

Know it cold $(1 + z_A)^A (1 + f)^B = (1 + z_{A+B})^{A+B}$ . Divide, then raise to $1/B$ , subtract 1.

MemorizeAyBy notation

Memory hook "3y1y": three years until it starts, one year long. The two spot rates are at 3 and 3 + 1 = 4 years.

Know it cold First number = start, second = length. Longer spot is at start + length.

MemorizeSpot from forwards

Memory hook A spot rate is the geometric mean of the one-year forwards before it.

Know it cold $(1 + z_n)^n = (1+f_{0,1})(1+f_{1,1})\cdots(1+f_{n-1,1})$

MemorizePricing and breakeven

Memory hook Spot pricing and forward pricing give the same price. A forward is the breakeven reinvestment rate.

Know it cold Discount each cash flow by the product of one-year forwards up to its date. Expected rate above forward: buy short and roll. Below: buy long.

Reading the AyBy notation

AyBy forward ratefirst number: when it starts; second: how long it lasts

3y1y Memorize one-year rate starting in 3 years; needs the 3-year and 4-year spots
1y2y Memorize two-year rate starting in 1 year; needs the 1-year and 3-year spots
0y1y starts today: just the 1-year spot rate

Buy long or buy short and roll?

Compare your expected future rate with the forward ratethe forward is the breakeven reinvestment rate

Expected rate above the forward Memorize buy the shorter bond and reinvest
Expected rate below the forward Memorize buy the longer bond now
Expected rate equal to the forward indifferent: both routes end at the same value

Implied forward rate

(1 + z_A)^{A}\,(1 + f_{A,B})^{B} = (1 + z_{A+B})^{A+B} \;\Rightarrow\; f_{A,B} = \left[\dfrac{(1 + z_{A+B})^{A+B}}{(1 + z_A)^{A}}\right]^{1/B} - 1

Quick example

The 1-year spot rate is 2.0% and the 3-year spot rate is 3.0%. The 1y2y forward is (1.03³ ÷ 1.02)^(1/2) − 1 = 3.504%.

On your calculator

1.03y^x3=÷1.02=y^x0.5=−1= 0.0350

y^x power

1.03ENTER3y^x1.02÷0.5y^x1− 0.0350

ENTER enter (on the HP, puts the number on the stack) · y^x power

Spot rate from one-year forward rates

(1 + z_n)^{n} = (1 + f_{0,1})(1 + f_{1,1})\cdots(1 + f_{n-1,1})

Quick example

One-year forwards today, in 1 year and in 2 years: 1.8%, 2.4%, 3.0%. The 3-year spot rate is (1.018 × 1.024 × 1.030)^(1/3) − 1 = 2.399%.

On your calculator

1.018×1.024×1.03=y^x(1÷3)=−1= 0.0240

y^x power

1.018ENTER1.024×1.03×31/xy^x1− 0.0240

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Bond price from forward rates

PV = \dfrac{CF_1}{1 + f_{0,1}} + \dfrac{CF_2}{(1 + f_{0,1})(1 + f_{1,1})} + \dfrac{CF_3}{(1 + f_{0,1})(1 + f_{1,1})(1 + f_{2,1})} + \cdots

Quick example

A 3-year, 5% annual-coupon bond with one-year forwards of 1.8%, 2.4% and 3.0%: 5 ÷ 1.018 + 5 ÷ (1.018 × 1.024) + 105 ÷ (1.018 × 1.024 × 1.030) = 107.500.

On your calculator

105÷1.03+5=÷1.024+5=÷1.018= 107.5003

105ENTER1.03÷5+1.024÷5+1.018÷ 107.5003

ENTER enter (on the HP, puts the number on the stack)

How it is asked

2y1y from two spots

2-year spot 2.10%, 3-year spot 2.50%. 2y1y forward?

1.025³ ÷ 1.021² − 1 = 3.30%. Subtracting the spots (0.40%) or averaging them (2.30%) are the traps.
Spot rate from a chain of forwards

One-year forwards 1.6%, 2.2%, 2.9%. 3-year spot?

(1.016 × 1.022 × 1.029)^(1/3) − 1 = 2.23%. Forgetting the cube root gives 6.85%.
Price a bond with forward rates

1-year rate 2.6%, 1y1y forward 3.8%. 2-year, 3% annual bond?

3 ÷ 1.026 + 103 ÷ (1.026 × 1.038) = 99.64. Treating 3.8% as a 2-year spot (98.52) is the trap.
Which strategy beats the forward?

You expect the 1-year rate in 3 years to be below the 3y1y forward.

Buy the longer (4-year) bond now. Expecting a higher rate means buy the 3-year and reinvest.

Exam traps

Using simple differences or averages instead of compounding. For example, 3 × 4.10% − 2 × 3.70% = 4.90% is close to the true 2y1y of 4.905% but not exact, and answer options are often a few basis points apart.
Forgetting the 1/B root for a multi-year forward, such as 1y2y.
Reading 2y1y as a two-year rate one year ahead. It is a one-year rate two years ahead.
Treating a forward rate as a spot rate when pricing: each forward applies to one period, and the discount factors multiply.

Checkpoint

5 exam-style questions. About 90 seconds each.

1The 2-year spot rate is 3.40% and the 3-year spot rate is 3.90%. The 2y1y implied forward rate is closest to:
2One-year forward rates are 2.0% (0y1y), 2.8% (1y1y) and 3.4% (2y1y). The 3-year spot rate is closest to:
3The 1-year rate today is 3.0% and the 1y1y forward rate is 4.5%. A 2-year, 4% annual-coupon bond is worth closest to:
4To calculate the 3y2y implied forward rate, an analyst needs the:
5An investor expects the one-year rate in three years to be 5.0%. The 3y1y implied forward rate is 5.3%. To earn the higher return, the investor should most likely:

Key takeaways

(1 + z_A)^A × (1 + f)^B = (1 + z_(A+B))^(A+B): solve for the forward by dividing and taking the root.
A forward rate is the breakeven reinvestment rate between buying short and buying long.
A spot rate is the geometric mean of the one-year forwards up to its maturity.
Spot-rate pricing and forward-rate pricing give the same no-arbitrage price.

Implied forward rate

(1 + z_A)^{A}\,(1 + f_{A,B})^{B} = (1 + z_{A+B})^{A+B} \;\Rightarrow\; f_{A,B} = \left[\dfrac{(1 + z_{A+B})^{A+B}}{(1 + z_A)^{A}}\right]^{1/B} - 1

Quick example

The 1-year spot rate is 2.0% and the 3-year spot rate is 3.0%. The 1y2y forward is (1.03³ ÷ 1.02)^(1/2) − 1 = 3.504%.

On your calculator

1.03y^x3=÷1.02=y^x0.5=−1= 0.0350

y^x power

1.03ENTER3y^x1.02÷0.5y^x1− 0.0350

ENTER enter (on the HP, puts the number on the stack) · y^x power

Spot rate from one-year forward rates

(1 + z_n)^{n} = (1 + f_{0,1})(1 + f_{1,1})\cdots(1 + f_{n-1,1})

Quick example

One-year forwards today, in 1 year and in 2 years: 1.8%, 2.4%, 3.0%. The 3-year spot rate is (1.018 × 1.024 × 1.030)^(1/3) − 1 = 2.399%.

On your calculator

1.018×1.024×1.03=y^x(1÷3)=−1= 0.0240

y^x power

1.018ENTER1.024×1.03×31/xy^x1− 0.0240

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Bond price from forward rates

PV = \dfrac{CF_1}{1 + f_{0,1}} + \dfrac{CF_2}{(1 + f_{0,1})(1 + f_{1,1})} + \dfrac{CF_3}{(1 + f_{0,1})(1 + f_{1,1})(1 + f_{2,1})} + \cdots

Quick example

A 3-year, 5% annual-coupon bond with one-year forwards of 1.8%, 2.4% and 3.0%: 5 ÷ 1.018 + 5 ÷ (1.018 × 1.024) + 105 ÷ (1.018 × 1.024 × 1.030) = 107.500.

On your calculator

105÷1.03+5=÷1.024+5=÷1.018= 107.5003

105ENTER1.03÷5+1.024÷5+1.018÷ 107.5003

ENTER enter (on the HP, puts the number on the stack)

All Fixed Income formulas, every lesson

Practice exam-style questions on Term Structure: Spot, Par and Forward Curves

Next lesson How the spot, par and forward curves relate

Questions about this lesson?

Where this comes back

How the spot, par and forward curves relate Term Structure: Spot, Par and Forward Curves
Interest Rate Risk and Return Fixed Income
Pricing and Valuing Forward Contracts and FRAs Derivatives
Pricing and Valuing Interest Rate Swaps Derivatives

Questions candidates ask

How do you calculate a forward rate from spot rates?

Divide the growth factor of the longer spot rate, (1 + z)^years, by that of the shorter one. Take the root for the forward period's length and subtract one.

What does 2y1y mean?

The one-year interest rate starting two years from today. The first number is when the rate starts and the second is how long it lasts.

Why is a forward rate called a breakeven reinvestment rate?

It is the future rate at which investing short and rolling over earns exactly the same as investing long today. Expect more than that and rolling wins; expect less and the long bond wins.

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