CFA Level II Exam · The Term Structure and Interest Rate Dynamics

Spot Rates and Forward Rates for CFA Level II

Updated 7 October 2026 · Fact-checked

A spot rate is today's yield on a zero-coupon bond for a given maturity. A forward rate is a rate agreed today for a loan starting later. To solve questions, link the two with no-arbitrage: (1+z_B)^B = (1+z_A)^A × (1+f)^(B−A). Rearrange for the unknown.

Understand Spot Rates and Forward Rates

A spot rate is the yield today on a zero-coupon bond that matures at a given date. A set of spot rates across maturities is the spot curve. Spot rates are used to discount a single cash flow at the right maturity, so they are the building blocks for valuing any bond.

A forward rate is an interest rate fixed today for a loan that starts at a future date and runs for a stated period. The notation f(A,B) means the rate for a loan of B−A years starting A years from now. For example, f(2,3) is the one-year rate two years ahead, often called the 2y1y forward rate.

The two are linked by no-arbitrage. Investing for B years at the spot rate must give the same result as investing for A years at the spot rate and then reinvesting at the forward rate for the remaining period. If the results differed, you could lock in a risk-free profit. That equality gives the forward rate formula.

The forward rate model compares the forward rate with the spot rate that actually occurs later. If the future spot rate equals the forward rate, the bond earns the current spot-implied return. If the realised future spot rate is below the forward rate, the bond's return exceeds the implied return. If it is above the forward rate, the return falls short of the implied return.

Bootstrapping goes the other way. Starting from par yields or from a set of one-year forward rates, you build spot rates one maturity at a time. Each new spot rate is the geometric average of the one-year forward rates up to that maturity.

Key formulas to remember

Forward rate from spot rates: (1 + z_B)^B = (1 + z_A)^A × (1 + f(A,B−A))^(B−A); z_A and z_B are annual spot rates. Solve for f by dividing and taking the (B−A) root.
Spot rate from one-year forwards: (1 + z_N)^N = (1 + z_1)(1 + f(1,1))(1 + f(2,1))…(1 + f(N−1,1)); The spot rate is the geometric mean of the one-year forward rates. Subtract 1 after taking the Nth root.
Forward price of a zero-coupon bond: F(A,B−A) = P(B) ÷ P(A); P(T) is the price of a $1 face zero maturing at T. This is the price today, for delivery at time A, of a zero that matures at time B. Use it when prices are given instead of rates.
Forward rate model (value of a bond): If future spot rate = forward rate, return over the period = current spot rate for that period; If the future spot rate is lower than the forward rate, the bond's return beats the implied return. If higher, it falls short.
Bond value from spot rates: PV = CF_1 ÷ (1+z_1) + CF_2 ÷ (1+z_2)² + … + CF_N ÷ (1+z_N)^N; Use the spot rate matching each cash flow's date.
Bond value from forward rates: PV = CF_1 ÷ (1+f(0,1)) + CF_2 ÷ [(1+f(0,1))(1+f(1,1))] + CF_3 ÷ [(1+f(0,1))(1+f(1,1))(1+f(2,1))] + …; f(0,1) is the 1-year spot rate and f(1,1) is the one-year rate one year ahead. Discount each cash flow through successive one-year forward rates. It gives the same value as spot discounting.

How to solve Spot Rates and Forward Rates questions

Use this method for any spot or forward rate question in an item set.

1Read the vignette and the exhibit. Note whether the given rates are spot rates, par yields, forward rates or zero-coupon prices, and whether they are annual.
2Identify what is asked: a forward rate, a spot rate, a bond price, or a return comparison.
3Draw a timeline marking the start date A, end date B and the length of the forward period.
4Write the no-arbitrage identity: (1+z_B)^B = (1+z_A)^A × (1+f)^(B−A).
5Substitute the given values and compute the left and right terms with the correct powers. Keep several decimals until the final step.
6Take the (B−A) root of the ratio, then subtract 1 to get the forward rate.
7For return or valuation questions, compare the expected future spot rate with the forward rate and apply the forward rate model.
8Check the answer: with an upward sloping curve, forward rates should be above spot rates.

Quickest way: Ratio-and-root shortcut for one-year forwards

When to use it: Use when you need the one-year forward rate between years A and A+1 and the question gives two spot rates.

Compute (1+z_B)^B and (1+z_A)^A on your calculator.
Divide the first by the second. The result is 1 + f for a one-year forward.
Subtract 1. This is the forward rate.
For a longer forward period, take the root of the ratio equal to the number of years in the forward period.
Sanity check: forward rate must lie above the longer spot rate when the curve slopes upward.

Common mistakes in Spot Rates and Forward Rates

Dividing spot rates directly instead of the compounded growth factors
The formula looks like a simple ratio of rates, so students skip the powers.
Fix: Always raise 1+z to the power of its maturity first, then divide, then take the root.
Using the wrong forward period when taking the root
Students confuse the end date B with the length B−A.
Fix: Take the root equal to B−A. For the 2y3y rate, the end is year 5 and the root is 3.
Treating par yields as spot rates
Both are quoted as yields and both appear in exhibits.
Fix: Spot rates apply to single cash flows. If given par yields, bootstrap first before computing forwards.
Reading the forward rate as a forecast that must come true
The word forward suggests prediction.
Fix: The forward rate is a no-arbitrage break-even rate. It equals the expected future spot rate only under strict assumptions.
Reversing the sign of the forward rate model conclusion
Students forget that lower rates mean higher bond prices.
Fix: If the future spot rate ends below the forward rate, the bond's price is higher than implied and its return is greater.
Rounding spot rates too early
Small rounding errors grow once raised to powers.
Fix: Keep at least five decimals in intermediate steps and round only the final answer.

Worked examples

Example 1

Vignette: An analyst uses a spot curve of 1-year 2.00%, 2-year 2.50%, 3-year 3.00% and 4-year 3.20%, all annual. Q1: What is the 1-year forward rate two years from now, f(2,1)? Q2: What is the 2-year forward rate starting in two years, f(2,2)?

Show the solution

Q1: Use (1+z_3)^3 = (1+z_2)^2 × (1+f(2,1)).
(1.03)^3 = 1.092727.
(1.025)^2 = 1.050625.
1 + f(2,1) = 1.092727 ÷ 1.050625 = 1.040071.
So f(2,1) ≈ 4.01%.
Q2: Use (1+z_4)^4 = (1+z_2)^2 × (1+f(2,2))^2.
(1.032)^4 = 1.134278 (1.032² = 1.065024; 1.065024² = 1.134277).
1.134277 ÷ 1.050625 = 1.079606.
Square root of 1.079606 = 1.03904.
So f(2,2) ≈ 3.90%.

Answer: Q1: f(2,1) ≈ 4.01%. Q2: f(2,2) ≈ 3.90%.

Example 2

Vignette: A portfolio manager holds a 2-year zero-coupon bond. The current 1-year spot rate is 3.0% and the 1-year forward rate one year ahead is 4.0%. Q1: What is the 2-year spot rate implied by these rates? Q2: One year from now, the 1-year spot rate is 3.6%. Relative to the forward rate, will a 2-year zero bought today earn more or less than the implied return over the first year? Explain.

Show the solution

Q1: (1+z_2)^2 = (1.03)(1.04) = 1.0712.
z_2 = √1.0712 − 1 = 1.03499 − 1 = 3.499%, about 3.50%.
Q2: After one year the zero has one year left and is priced at the new 1-year spot rate.
Today's price = 1 ÷ 1.0712 = 0.93353.
Price in one year if spot is 3.6% = 1 ÷ 1.036 = 0.96525.
Return = 0.96525 ÷ 0.93353 − 1 = 3.40%.
The implied return is the 1-year spot of 3.0%.
3.40% is above 3.0%.

Answer: Q1: about 3.50%. Q2: more. The realised spot rate (3.6%) is below the forward rate (4.0%), so the bond's price rises more than implied and the return is about 3.40% versus 3.0%.

Exam tips

Identify rate types first. Many wrong answers come from using a par yield where a spot rate is needed.
Set up the timeline before computing. Marking A, B and B−A prevents root errors.
Compare the future spot rate with the forward rate, not with today's spot rate, for forward rate model questions.
Sanity check direction: an upward sloping spot curve means forwards sit above spot rates.
Keep intermediate values to five or six decimals because powers amplify rounding differences.

Spot Rates and Forward Rates in other exams

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

Spot Rates and Forward Rates: frequently asked questions

What is the difference between a spot rate and a forward rate?

A spot rate is the yield today on a zero-coupon bond for a given maturity. A forward rate is the rate agreed today for a loan that begins at a later date. The forward rate is implied by the spot curve through no-arbitrage.

How do you calculate a forward rate from spot rates?

Raise each spot factor to its maturity: (1+z_B)^B and (1+z_A)^A. Divide the first by the second, then take the root equal to B−A. Subtract 1 to get the forward rate.

How does bootstrapping work?

You build spot rates one maturity at a time. Each spot rate is found so that the bond's price equals the present value of its cash flows, using the earlier spot rates for earlier cash flows. The forward rates then follow from the spot curve.

What does the forward rate model tell you?

It says that a bond earns the implied return if the future spot rate equals the forward rate. If the future spot rate is lower than the forward rate, the bond outperforms. If higher, it underperforms.