Lesson 3 of 4 6–9% of the exam

By Daniil Tarakanov Updated Oct 20, 20262027 curriculum

Implied forward rates from spot rates

Investing for B years at the B-year spot rate must give the same result as investing for A years and then locking in a rate for the remaining B − A years. That locked-in rate is the implied forward rate: (1 + z_A)^A (1 + IFR)^(B−A) = (1 + z_B)^B.

8 min cram, 32 min full read. Based on the 2027 CFA® Program Level I curriculum, Derivatives, Learning Module 5. 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: Term Structure: Spot, Par and Forward Curves, Time Value of Money in Finance

On the exam

Explain how forward rates are determined for interest rate forward contracts and describe the uses of these forward rates

Explain how forward rates are determined from spot rates. Expect a 1y1y or 2y1y forward computed from two spot rates, a discount factor, the meaning of a label such as 1y4y, and a sub-year version with periodic 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.

What are spot rates and discount factors?

Interest rates have a term structure: different maturities carry different rates. The spot rate for maturity i, z_i, is the yield on a single cash flow paid at time i. Because it prices a bond with no coupons, it is also called the zero rate . For a one-period instrument the yield to maturity and the zero rate are the same.

The price equivalent of a zero rate is the discount factor : the present value of one currency unit paid at time i.

Discount factor Memorize

DF_i = \frac{1}{(1 + z_i)^i}

Our curve for this lesson: z₁ = 2.7%, z₂ = 3.3%, z₃ = 3.8%. That gives DF₁ = 0.9737, DF₂ = 0.9371 and DF₃ = 0.8941.

Rearrange itDiscount factor and zero rate: solve for either

The question asks for

The 2-year spot rate is 3.30%. The 2-year discount factor is closest to:

Spot (zero) rate, z_i%Maturity, i (years)

D F_{i} = \frac{1}{( 1 + z _{i} ) ^{i}}

1 ÷ (1 + 0.033)^2 = 0.9371

Discount factor, DF_i0.9371

On your calculator

1.033y^x2=1/x0.937129

y^x power · 1/x reciprocal

1.033ENTER2y^x1/x0.937129

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

Where do spot rates come from?

Know the idea

Most traded bonds pay coupons, so spot rates are extracted from them one maturity at a time. This is called bootstrapping . The one-year bond has a single cash flow, so its yield is z₁. Substitute z₁ into the two-year bond’s price equation and solve for z₂.

What is an implied forward rate?

The implied forward rate is the breakeven reinvestment rate that links a shorter and a longer spot rate. Take two ways to invest for B years:

invest today to the longer maturity B at the spot rate z_B, or
invest to the shorter maturity A at z_A and then reinvest for the remaining B − A years at a rate locked in today.

The no-arbitrage condition for rates says both routes must end with the same money. If they did not, a trader could borrow on one route and lend on the other. The locked-in rate that makes them equal is the forward rate.

Implied forward rate Memorize

(1 + z_A)^A\,(1 + IFR_{A,B-A})^{B-A} = (1 + z_B)^B

The label tells you the period: the first number is when it starts, the second is how long it runs.

Reading a forward-rate label such as 2y3y

F(start, tenor)first number: when the rate starts; second number: how long it runs

1y1y Memorize one-year rate starting in one year; ends at year 2
2y1y Memorize one-year rate starting in two years; ends at year 3
1y2y Memorize two-year rate starting in one year; ends at year 3
2y5y Memorize five-year rate starting in two years; ends at year 7

Rearrange itImplied forward rate: solve for any input

The question asks for

The 2-year spot rate is 3.30% and the 3-year spot rate is 3.80%, both annually compounded. The implied forward rate for the 1-year period starting in 2 years is closest to:

Shorter spot rate, z_A%Shorter maturity, A (years)Longer spot rate, z_B%Longer maturity, B (years)

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

(1.038^3 ÷ 1.033^2)^(1 ÷ 1) − 1 = 4.81%

Implied forward rate, IFR (A to B)4.81%

On your calculator

(1.038y^x3)÷(1.033y^x2)=1.048073
−1=0.048073

y^x power

1.038ENTER3y^x1.033ENTER2y^x÷1.048073
1−0.048073

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

InteractiveImplied forward rates from the spot curve

Try this 0 of 3

Flatten the curve until the forward steps sit on the spot line
Make 1y1y lower than the 1-year spot rate
Pick 2y1y and raise z₃ by 0.30 (to 4.10% in the worked example)

Drag the spot points (or type them), then pick which forward to read.

Curve shape

Forward

Spot rates z (%)

z₁z₂z₃

Implied forward 1y1y3.90%

Shortcut (no compounding)3.900%

USD 100 for 2 years grows to106.71

Spot rate
One-year forward
1y1y: 1 year, starting in 1 year

USD 100 two ways (interest earned)

Interest at the 2-year spot rate
Interest in year 1
Interest at the forward 1y1y

DF = 1 ÷ (1 + z)^t, the price today of 1 paid at t
Maturity	Spot z	Discount factor
1 year	2.70%	0.9737
2 years	3.30%	0.9371
3 years	3.80%	0.8941

Both routes end at USD 106.71, so neither beats the other. 1y1y is above the 2-year spot rate because spot rates rise over this stretch.

Show working

No arbitrage: (1 + z₁) × (1 + 1y1y) = (1 + z₂)².

1y1y = [(1 + 3.30%)² ÷ (1 + 2.70%)] − 1 = (1.067089 ÷ 1.027000) − 1 = 3.904%, about 3.90%.

Same thing with discount factors: (DF₁ ÷ DF₂) − 1 = (0.973710 ÷ 0.937129) − 1 = 3.904%.

Shortcut: (2 × 3.30% − 1 × 2.70%) ÷ 1 = 3.900%. Close here, but it ignores compounding and drifts further off as rates and gaps grow, so use the formula.

1y1y on your calculator

1.033x²÷1.027=(1 + z₂)² ÷ (1 + z₁)
−1=0.039035

x² square

1.033ENTER×1.027÷(1 + z₂)² ÷ (1 + z₁)
1−0.039035

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

Display: 0.039035 = 3.90%

A forward rate is the rate that makes waiting and locking in equally good.

Each forward is higher than the spot rates on either side because the spot curve slopes upward. Flatten the curve and the forwards collapse onto the spot rates.

The building blocks of interest rate derivatives pricing

Spot (zero) rate, z_ithe yield on one cash flow at time i

Discount factor, DF_i Memorize the price equivalent: what 1 paid at time i is worth today
Implied forward rate Memorize the breakeven reinvestment rate linking a shorter and a longer zero rate
- Forward rate agreement (FRA) a contract that fixes the rate for one forward period at the implied forward rate
- Swap rate built from a strip of forward rates (swap pricing module)

Exam tip Do not average. 2 × 3.3% − 2.7% = 3.9% looks right here, but it breaks down with longer gaps and more decimals. Use the compounding equation.

How do I handle periods shorter than a year?

Market reference rates are quoted for months, not years. Convert every rate to the same periodicity first, then apply the same equation to the periodic rates, and annualize at the end.

Converting between compounding frequencies Use it

\left(1 + \frac{APR_m}{m}\right)^m = \left(1 + \frac{APR_n}{n}\right)^n

What are forward rates used for?

The implied forward rate is the no-arbitrage fixed rate for a forward rate agreement, the subject of the next lesson. It also prices each period of a swap. The same method works for any underlying with a term structure, such as credit spreads or implied volatility, but the exam concentrates on interest rates.

Go deeper Why forward rates are not forecasts

An implied forward rate is a breakeven rate, not a prediction of where spot rates will be. It is what you can lock in today. If investors demand extra yield for holding longer bonds, forward rates tend, on average, to sit above the spot rates that actually turn up later.

Memorize this

MemorizeForward-rate labels and the equation behind each

Label	Starts in	Runs for	Equation
1y1y	1 year	1 year	$(1+z_1)(1+f)=(1+z_2)^2$
2y1y	2 years	1 year	$(1+z_2)^2(1+f)=(1+z_3)^3$
1y2y	1 year	2 years	$(1+z_1)(1+f)^2=(1+z_3)^3$

Memory hook Left number = when it starts; right number = how long it lasts. Start plus length is the maturity of the longer spot rate: both routes must end with the same money.

Know it cold $DF_i = 1/(1+z_i)^i$ . $(1+z_A)^A(1+IFR)^{B-A} = (1+z_B)^B$ . Upward-sloping spot curve means forwards above spot rates. Never average the spot rates.

The building blocks of interest rate derivatives pricing

Spot (zero) rate, z_ithe yield on one cash flow at time i

Discount factor, DF_i Memorize the price equivalent: what 1 paid at time i is worth today
Implied forward rate Memorize the breakeven reinvestment rate linking a shorter and a longer zero rate
- Forward rate agreement (FRA) a contract that fixes the rate for one forward period at the implied forward rate
- Swap rate built from a strip of forward rates (swap pricing module)

Reading a forward-rate label such as 2y3y

F(start, tenor)first number: when the rate starts; second number: how long it runs

1y1y Memorize one-year rate starting in one year; ends at year 2
2y1y Memorize one-year rate starting in two years; ends at year 3
1y2y Memorize two-year rate starting in one year; ends at year 3
2y5y Memorize five-year rate starting in two years; ends at year 7

Discount factor

DF_i = \dfrac{1}{(1 + z_i)^i}

Quick example

The three-year spot rate is 4.5%. DF₃ = 1 ÷ 1.045³ = 0.8763. One currency unit paid in three years is worth 0.8763 today.

On your calculator

1.045y^x3=1/x 0.8763

y^x power · 1/x reciprocal

1.045ENTER3y^x1/x 0.8763

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

Implied forward rate

(1 + z_A)^A\,(1 + IFR_{A,B-A})^{B-A} = (1 + z_B)^B

Quick example

z₂ = 2.0% and z₃ = 2.6%. The 2y1y rate satisfies 1.02² × (1 + IFR) = 1.026³, so IFR = 1.026³ ÷ 1.02² − 1 = 3.81%.

On your calculator

(1.026y^x3)÷(1.02y^x2)=−1= 0.0381

y^x power

1.026ENTER3y^x1.02ENTER2y^x÷1− 0.0381

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

Implied forward rate, solved for the forward

IFR_{A,B-A} = \left[\dfrac{(1 + z_B)^B}{(1 + z_A)^A}\right]^{1/(B-A)} - 1

Quick example

z₁ = 2.0% and z₃ = 3.3%. The 1y2y rate is (1.033³ ÷ 1.02)^½ − 1 = 3.96% a year for the two years starting in one year.

On your calculator

1.033y^x3=÷1.02= 1.0807
y^x21/x=−1= 0.0396

y^x power · 1/x reciprocal

1.033ENTER3y^x1.02÷ 1.0807
21/xy^x1− 0.0396

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

Converting between compounding frequencies

\left(1 + \dfrac{APR_m}{m}\right)^m = \left(1 + \dfrac{APR_n}{n}\right)^n

Quick example

A 2.40% rate compounded semiannually, restated with quarterly compounding: 4 × (1.012^0.5 − 1) = 2.39%.

On your calculator

1.012y^x0.5=−1=×4= 0.0239

y^x power

1.012ENTER0.5y^x1−4× 0.0239

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

How it is asked

One-year forward one year ahead

z₁ = 1.6%, z₂ = 2.9%. 1y1y?

1.029² ÷ 1.016 − 1 = 4.22%. Traps: 2.25% (average), 1.28% (second year left uncompounded).
Multi-year forward

z₁ = 2.0%, z₃ = 3.3%. 1y2y?

(1.033³ ÷ 1.02)^½ − 1 = 3.96%. Traps: 8.07% (forgot the root), 2.65% (average).
Discount factor

z₂ = 3.3%. DF₂?

1 ÷ 1.033² = 0.9371. Traps: 0.9670 (1 − z), 0.9681 (one year only).
Sub-year forward

Six-month rate 2.8%, nine-month rate 3.4% (quarterly compounding). Rate for the 3 months starting in 6?

1.0085³ ÷ 1.007² − 1 = 1.15% a quarter, × 4 = 4.60%. Trap: forgetting to annualize.

Exam traps

Averaging spot rates. The forward rate comes from the compounding equation, not from an arithmetic mean.
Forgetting the root for a multi-year forward. For a B − A year period, take the (B − A)th root.
Mixing annual and periodic rates for sub-year periods. Divide the quoted rate by the periods per year, and annualize the answer.
Using 2 × z₂ − z₁ as a formula. It is close here but fails with longer gaps and more decimals.
Reading 1y4y as a one-year rate starting in four years, or as a rate ending in year 4. The first number is the start, the second is the length.

Checkpoint

7 exam-style questions. About 90 seconds each.

1The one-year spot rate is 2.5% and the two-year spot rate is 3.5%, both annually compounded. The one-year rate one year from now (1y1y) is closest to:
2If the two-year spot rate is 3.6%, the two-year discount factor is closest to:
3When the spot curve slopes upward, implied forward rates are most likely:
4A pension fund will reinvest a bond redemption in one year and asks its dealer for the 1y4y forward rate. That rate is best described as the:
5The one-year spot rate is 2.4% and the three-year spot rate is 3.2%, both annually compounded. The annual forward rate for the two years starting in one year (1y2y) is closest to:
6The six-month rate is 4.0% and the nine-month rate is 4.4%, both annualized with quarterly compounding. The annualized implied forward rate for the three months starting in six months is closest to:
7The one-year zero rate is 3.0%. A two-year annual-coupon bond with a 4.0% coupon trades at par (100). The two-year zero rate is closest to:

Key takeaways

(1 + z_A)^A (1 + IFR)^(B−A) = (1 + z_B)^B: waiting and locking in must equal the longer spot rate.
DF_i = 1 ÷ (1 + z_i)^i is the price today of one unit paid at time i.
An upward-sloping spot curve gives implied forward rates above the spot rates.
For periods shorter than a year, work in periodic rates and annualize the result.

Discount factor

DF_i = \dfrac{1}{(1 + z_i)^i}

Quick example

The three-year spot rate is 4.5%. DF₃ = 1 ÷ 1.045³ = 0.8763. One currency unit paid in three years is worth 0.8763 today.

On your calculator

1.045y^x3=1/x 0.8763

y^x power · 1/x reciprocal

1.045ENTER3y^x1/x 0.8763

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

Implied forward rate

(1 + z_A)^A\,(1 + IFR_{A,B-A})^{B-A} = (1 + z_B)^B

Quick example

z₂ = 2.0% and z₃ = 2.6%. The 2y1y rate satisfies 1.02² × (1 + IFR) = 1.026³, so IFR = 1.026³ ÷ 1.02² − 1 = 3.81%.

On your calculator

(1.026y^x3)÷(1.02y^x2)=−1= 0.0381

y^x power

1.026ENTER3y^x1.02ENTER2y^x÷1− 0.0381

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

Implied forward rate, solved for the forward

IFR_{A,B-A} = \left[\dfrac{(1 + z_B)^B}{(1 + z_A)^A}\right]^{1/(B-A)} - 1

Quick example

z₁ = 2.0% and z₃ = 3.3%. The 1y2y rate is (1.033³ ÷ 1.02)^½ − 1 = 3.96% a year for the two years starting in one year.

On your calculator

1.033y^x3=÷1.02= 1.0807
y^x21/x=−1= 0.0396

y^x power · 1/x reciprocal

1.033ENTER3y^x1.02÷ 1.0807
21/xy^x1− 0.0396

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

Converting between compounding frequencies

\left(1 + \dfrac{APR_m}{m}\right)^m = \left(1 + \dfrac{APR_n}{n}\right)^n

Quick example

A 2.40% rate compounded semiannually, restated with quarterly compounding: 4 × (1.012^0.5 − 1) = 2.39%.

On your calculator

1.012y^x0.5=−1=×4= 0.0239

y^x power

1.012ENTER0.5y^x1−4× 0.0239

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

All Derivatives formulas, every lesson

Practice exam-style questions on Pricing and Valuing Forward Contracts and FRAs

Next lesson Forward rate agreements (FRAs): settlement and uses

Questions about this lesson?

Where this comes back

Forward rate agreements (FRAs): settlement and uses Pricing and Valuing Forward Contracts and FRAs
Pricing and Valuing Interest Rate Swaps Derivatives
Term Structure: Spot, Par and Forward Curves Fixed Income

Questions candidates ask

How do you calculate an implied forward rate?

Set the two routes equal. Investing to the longer maturity B at z_B must match investing to A at z_A and then earning the forward rate for B − A years. Solve (1 + z_A)^A (1 + IFR)^(B−A) = (1 + z_B)^B for IFR. For a one-year forward, divide the two compounded growth factors and subtract 1.

What is a 1y1y forward rate?

It is the one-year rate that starts one year from now. The first 1y is the start, the second is the length. With z₁ = 2.7% and z₂ = 3.3%, it is 1.033² ÷ 1.027 − 1 = 3.90%.

What is a discount factor?

The present value of one unit of currency paid at a future date: DF_i = 1 ÷ (1 + z_i)^i, where z_i is the spot rate for that date. It is the price equivalent of a zero rate. A 3.3% two-year rate gives a two-year discount factor of 0.9371.

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