What's the relationship between spot rates and forward rates, and how do you derive one from the other?

Spot rates and forward rates are two sides of the same coin — they're just different ways of expressing the term structure of interest rates.

Definitions:

• Spot rate (z_n): The yield on a zero-coupon bond maturing in n periods. It's the rate for lending money from today to time n.
• Forward rate (f(j,k)): The implied rate for lending money from time j to time j+k, locked in today. It's derived from spot rates.

The No-Arbitrage Relationship:

Investing for 2 years at the 2-year spot rate must produce the same result as investing for 1 year at the 1-year spot rate, then rolling over at the 1-year forward rate one year from now:

(1 + z_2)^2 = (1 + z_1)^1 x (1 + f(1,1))^1

Solving for the forward rate:

f(1,1) = [(1 + z_2)^2 / (1 + z_1)] - 1

Worked Example:

Crestview Bank quotes these annual spot rates:

• 1-year: z_1 = 3.00%
• 2-year: z_2 = 3.75%
• 3-year: z_3 = 4.20%

Derive the 1-year forward rate, 1 year from now — f(1,1):

f(1,1) = [(1.0375)^2 / (1.03)] - 1 = [1.07641 / 1.03] - 1 = 1.04507 - 1 = 4.507%

Derive the 1-year forward rate, 2 years from now — f(2,1):

f(2,1) = [(1.042)^3 / (1.0375)^2] - 1 = [1.13124 / 1.07641] - 1 = 1.05092 - 1 = 5.092%

Intuition: Notice that forward rates are increasing (4.507% then 5.092%) even though spot rates rise more gradually. When the spot curve is upward-sloping, forward rates lie above spot rates and rise faster. This is because the forward rate must "compensate" for the lower rate earned in the earlier period to make the compounded result match the longer spot rate.

Key Relationship: If the spot curve is upward-sloping, forward rates > spot rates. If the spot curve is flat, forward rates = spot rates. If the spot curve is downward-sloping (inverted), forward rates < spot rates.

Explore more term structure concepts in our CFA Level I Fixed Income course.