How does the Nelson-Siegel model describe the yield curve, and what do its parameters represent?

The Nelson-Siegel model describes the yield curve as a function of maturity using three components, each with an intuitive economic interpretation.

The Formula:

y(T) = β0 + β1 x [(1 - e^(-T/λ)) / (T/λ)] + β2 x [(1 - e^(-T/λ)) / (T/λ) - e^(-T/λ)]

Where:

• y(T) = yield at maturity T
• β0, β1, β2 = parameters
• λ = decay factor controlling where the hump occurs

What Each Parameter Controls:

Parameter	Component	Shape Contribution	Intuition
β0	Level	Constant at all maturities	Long-term yield level
β1	Slope	Decays from 1 to 0 as T increases	Short vs. long rate spread
β2	Curvature	Starts at 0, humps, returns to 0	Medium-term hump/dip
λ	Decay	Controls speed of exponential decay	Location of the hump

Intuitive Explanation:

• β0 (Level): This is the long-run yield the curve approaches as maturity goes to infinity. If β0 = 4.5%, all maturities converge toward 4.5%.

• β1 (Slope): This determines whether the curve slopes up or down.

  • β1 < 0 → upward sloping (normal curve)
  • β1 > 0 → downward sloping (inverted curve)
  • At T=0: yield = β0 + β1, so the short rate equals β0 + β1

• β2 (Curvature): This creates the hump or trough in the middle of the curve. It adds a bulge around medium maturities without affecting the short end or long end.

Example — Fitting a Normal Curve (Ridgeway Capital, fictional):

Parameter	Value	Effect
β0	4.50%	Long-term anchor
β1	-2.00%	Upward slope (short rate = 4.5% - 2.0% = 2.5%)
β2	1.50%	Medium-term hump
λ	2.0	Hump peaks around 2-3 years

Resulting yields:

• 0.25 year: ~2.8%
• 2 year: ~3.9%
• 5 year: ~4.2%
• 10 year: ~4.4%
• 30 year: ~4.5%

Advantages:

• Parsimonious (only 4 parameters describe the entire curve)
• Parameters map to economically meaningful factors
• Smooth and well-behaved at all maturities
• Used by central banks and fixed income desks globally

Exam Tip: CFA Level II tests whether you understand which parameter controls level, slope, and curvature. Remember: β0 = level (long end), β1 = slope (short vs. long), β2 = curvature (hump in the middle).

Practice yield curve modeling in our CFA Level II question bank.