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

Riding the Yield Curve and Forward Contract Returns

Updated 7 October 2026 · Fact-checked

The forward pricing model says a forward rate is the rate that makes two investment routes equal: (1 + z_A)^A × (1 + f)^B = (1 + z_(A+B))^(A+B). Riding the yield curve means buying a longer bond and selling it as it rolls down an upward-sloping curve. If forward rates are realized, the bond earns the current spot rate for the holding period.

Understand Return of Forward Contracts and Riding the Yield Curve

A spot rate is the yield on a zero-coupon bond for a given maturity. A forward rate is a rate agreed today for a loan that starts later. The forward pricing model links them by no-arbitrage. Investing for A+B years at the long spot rate must give the same wealth as investing for A years at the short spot rate and then rolling over at the forward rate for B years. Whatever is left over after the short spot growth is the forward rate.

The same logic prices a forward contract on a zero-coupon bond. The forward price is the bond's price at the later maturity divided by the growth of money to the delivery date. In discount-factor form, F = P(A+B) ÷ P(A). Per 100 of face value, it equals 100 ÷ (1 + f)^B, where f is the forward rate for the bond's remaining life at delivery.

A forward contract has zero value at the start. Later, its value depends on how the forward price has changed. The holder of the contract gains if the new forward price for the same delivery is higher than the contract price. The gain is paid at settlement, so you discount the difference back to today with the spot rate for the time remaining to settlement.

Riding the yield curve, or rolldown, is a strategy for an upward-sloping curve. You buy a bond whose maturity is longer than your horizon. As time passes, the bond gets shorter and its yield falls along the curve, so its price rises. Rolldown return is the return you get if the spot curve stays unchanged. It combines coupon income, pull to the lower yield, and any accretion.

The key link for the exam: if the spot curve at the horizon equals the curve implied by today's forward rates, every bond earns the current one-period spot rate over that period. Realized return above that rate means rates ended up below the forwards. Return below it means rates ended up above the forwards. The forward rate is the break-even.

Key formulas to remember

Forward pricing model: (1 + z_(A+B))^(A+B) = (1 + z_A)^A × (1 + f(A,B))^B; f(A,B) is the B-year rate starting A years from now. Solve for f by dividing, then take the B-th root and subtract 1.
Forward rate from spot rates: f(A,B) = [(1 + z_(A+B))^(A+B) ÷ (1 + z_A)^A]^(1/B) − 1; Use annual compounding unless the vignette states otherwise.
Forward price of a zero-coupon bond: F(A,B) = P(A+B) ÷ P(A) = 100 ÷ (1 + f(A,B))^B per 100 face; P is today's zero-coupon price (discount factor). A is the time to delivery and B is the bond's life at delivery.
Value of a forward contract at time t: V_t = [F_t − F_0] ÷ (1 + z_(A−t))^(A−t); F_0 is the contract price. F_t is the current forward price for the same delivery. The long gains if F_t > F_0. Discount at the spot rate for the time left to delivery.
Rolldown (horizon) total return: Return = (Price at horizon + coupons received − Price now) ÷ Price now; For rolldown, price the bond at the horizon using the unchanged spot curve for its remaining maturity.
Forward-realized return rule: If the future spot curve = today's forward curve, return over the period = current spot rate for that period; Holds for any bond, whatever its maturity or coupon. It is the break-even test for the view that rates will beat or miss the forwards.

How to solve Return of Forward Contracts and Riding the Yield Curve questions

Use this order for any question on forward rates, forward prices or rolldown returns. Read the vignette first to find the spot curve, the coupon, the horizon and the compounding.

1Identify what is asked: a forward rate, a forward price, a contract value, or a horizon return. Note the compounding convention.
2Pull the spot rates from the exhibit. Check whether the exhibit gives spot rates, par rates or forward rates. Do not mix them.
3For a forward rate, apply the pricing model. Divide the long growth factor by the short growth factor, take the root for B years, and subtract 1.
4For a forward price, find the forward rate for the bond's remaining life, then price per 100 as 100 ÷ (1 + f)^B. Or divide discount factors.
5For a contract value, compute the new forward price for the same delivery. Take new minus contract price, then discount at the spot rate for the remaining time to delivery. Flip the sign for the short.
6For rolldown, price the bond today, then price it at the horizon on the unchanged curve using the shorter maturity's spot rate for each remaining cash flow. Add coupons received and divide by the starting price.
7Compare with the forward benchmark. If the curve ends where forwards imply, the return equals the spot rate for the period. Use it to judge whether a view beats the market.
8Check reasonableness: on an upward-sloping curve, forwards lie above spots and rolldown return exceeds the shorter spot rate.

Quickest way: Growth-factor shortcut

When to use it: Use it for forward rate and forward price items where the exhibit gives annual spot rates and you need an answer in under two minutes.

Write the growth factors (1 + z)^t for the two maturities. Do not convert to percentages yet.
Divide the longer factor by the shorter factor. This is the forward growth factor.
For a one-year forward rate, subtract 1 and you are done. For longer, take the root.
For a forward price per 100, divide 100 by the forward growth factor.
For a forward-realized return item, skip the full price calculation. The answer is the current spot rate for the holding period.
Eliminate answer options that move the wrong way. On an upward curve, the forward rate must exceed the long spot rate.

Common mistakes in Return of Forward Contracts and Riding the Yield Curve

Averaging spot rates instead of using the growth-factor ratio to get a forward rate.
Candidates think forward rates are an arithmetic blend of two spot rates.
Fix: Always divide compounded growth factors. Raise each spot to its own maturity before dividing.
Using the wrong exponents, such as taking the B-th root of a ratio built with the wrong periods.
The notation f(A,B) mixes start time and length, so the two numbers get swapped.
Fix: Label A as the start and B as the length. The two spot maturities must be A and A+B.
Pricing the horizon bond with the original spot rate for its old maturity.
Candidates forget that rolldown means moving to a shorter maturity on the same curve.
Fix: At the horizon, discount each remaining cash flow at the spot rate for its new remaining time.
Forgetting the coupon received during the holding period in the total return.
The horizon price is quoted ex-coupon, so the cash already paid is easy to drop.
Fix: Always add coupons received to the horizon price before dividing by the starting price.
Valuing a forward contract without discounting the price difference.
The difference between forward prices looks like the value already.
Fix: The gain is realized at settlement. Discount it using the spot rate for the time left to delivery.
Assuming rolldown return is guaranteed.
The method assumes an unchanged curve, which sounds like a fact.
Fix: Treat it as a scenario. Actual returns depend on how the curve moves. Only if forwards are realized does the bond earn the spot rate.

Worked examples

Example 1

Vignette: An analyst uses this annual-pay zero-coupon spot curve: 1-year 2.00%, 2-year 2.50%, 3-year 3.00%. She considers a forward contract to buy a 1-year zero-coupon bond (face 100) in two years. Q1: What is the 2-year-forward 1-year rate, f(2,1)? Q2: What is the forward price of the bond per 100?

Show the solution

Q1: Growth factor for 3 years is 1.03^3 = 1.092727.
Growth factor for 2 years is 1.025^2 = 1.050625.
f(2,1) = 1.092727 ÷ 1.050625 − 1 = 1.040073 − 1 = 4.01%.
Q2: Forward price = 100 ÷ 1.040073 = 96.147.
Check with discount factors: P(3) = 1 ÷ 1.092727 and P(2) = 1 ÷ 1.050625. The ratio P(3) ÷ P(2) = 0.96147, so 96.147 per 100. This matches.

Answer: Q1: about 4.01%. Q2: about 96.147 per 100 of face value.

Example 2

Vignette: Using the same spot curve (1-year 2.00%, 2-year 2.50%, 3-year 3.00%), a manager buys a 3-year, 4% annual-coupon bond (face 100) and plans to hold it one year. Q1: What is the bond's price today? Q2: What is the total horizon return if the spot curve is unchanged after one year? Q3: What is the total return if the future spot curve equals today's implied forward curve?

Show the solution

Q1: Price = 4 ÷ 1.02 + 4 ÷ 1.050625 + 104 ÷ 1.092727.
Terms: 3.9216 + 3.8073 + 95.1747 = 102.9035.
Q2: After one year, the bond has two years left. On the unchanged curve, discount at 2.00% for year 1 and 2.50% for year 2.
Horizon price = 4 ÷ 1.02 + 104 ÷ 1.050625 = 3.9216 + 98.9887 = 102.9102.
Total return = (102.9102 + 4 − 102.9035) ÷ 102.9035 = 4.0067 ÷ 102.9035 = 3.89%.
Q3: If forwards are realized, the bond earns the 1-year spot rate. Check: the implied 1-year rate in a year is 3.0025%, and the implied 2-year spot in a year is (1.092727 ÷ 1.02)^(1/2) − 1 = 3.504%.
Horizon price = 4 ÷ 1.030025 + 104 ÷ 1.071301 = 3.8834 + 97.0782 = 100.9616.
Return = (100.9616 + 4) ÷ 102.9035 − 1 = 1.0200 − 1 = 2.00%.

Answer: Q1: 102.9035. Q2: about 3.89%. Q3: 2.00%, equal to the current 1-year spot rate.

Exam tips

Read the exhibit label carefully. Spot, par and forward rates look alike but are used differently. The forward formulas need spot rates.
If a question asks for the return when forward rates are realized, answer with the current spot rate for the holding period. It saves a long price calculation.
Rolldown items need two prices. Do the horizon price on the shorter maturity and do not drop the coupon.
On an upward-sloping curve, rolldown return beats the short spot rate when the curve is unchanged. Use that to catch wrong options.
Time is limited in the item set. Keep six decimals in growth factors, then round only the final answer.

Return of Forward Contracts and Riding the Yield Curve: frequently asked questions

What is riding the yield curve?

It is a strategy where you buy a bond with maturity longer than your horizon on an upward-sloping curve. As it ages, its yield falls along the curve and its price rises. The extra gain over the shorter spot rate is the rolldown benefit, assuming the curve stays unchanged.

How do I calculate a forward rate from spot rates?

Divide the long-maturity growth factor by the short-maturity growth factor. Then take the root for the forward period length and subtract 1. For example, f(2,1) = 1.03^3 ÷ 1.025^2 − 1.

What return does a bond earn if forward rates are realized?

It earns the current spot rate for the holding period, regardless of its maturity or coupon. That makes the forward rate the break-even level. If actual rates end below the forwards, the bond beats the spot rate. If they end above, it falls short.

How do I value a forward contract after it is created?

Find the new forward price for the same delivery date and take its difference from the original contract price. Discount the difference using the spot rate for the time left to delivery. The long gains if the new forward price is higher.