Affine Term Structure Models

A full yield curve has dozens of maturities on it, each with its own quoted rate. Storing and forecasting dozens of numbers separately is unwieldy, and it throws away the fact that nearby maturities move together. An affine term structure model compresses the whole curve into a handful of underlying factors — often just two or three, like "overall level" and "steepness" — and reconstructs every maturity's yield as a straight-line combination of those factors. "Affine" simply means "linear plus a constant": each yield is $(\text{constant}) + (\text{loading}) \times (\text{factor})$, nothing more exotic.

Where the factors and loadings come from

The factors ($X$) are usually not directly observable rates — they are statistical constructs like "level" and "slope" extracted from the curve, or in models like Vasicek and CIR, the instantaneous short rate itself, which follows a mean-reverting process (see The Ornstein-Uhlenbeck Process). The loadings ($a_\tau$, $b_\tau$) come from either a no-arbitrage condition — solved once per model, valid for all maturities and all time — or from a purely descriptive curve-fitting exercise, as in Nelson-Siegel. Both produce the same affine shape; only true no-arbitrage models like Vasicek and CIR guarantee the resulting prices cannot be arbitraged against each other.

Worked example: two factors, one curve

Take a two-factor model with a level factor $L$ and a slope factor $S$, and a maturity loading on the slope that fades with maturity: $b_\tau = (1-e^{-\tau/2})/(\tau/2)$. Suppose today $L = 4.00\%$ and $S = -1.50\%$.

Five-year yield. The loading at $\tau=5$ is $b_5 = (1-e^{-2.5})/2.5 = (1-0.0821)/2.5 = 0.367$.

$$y_5 = 4.00 + (-1.50)(0.367) = 3.45\%$$

One-year yield. The loading at $\tau=1$ is $b_1 = (1-e^{-0.5})/0.5 = (1-0.6065)/0.5 = 0.787$.

$$y_1 = 4.00 + (-1.50)(0.787) = 2.82\%$$

Two numbers, $L$ and $S$, just generated an entire upward-sloping curve — 2.82% at one year, 3.45% at five years, and every other maturity follows the same formula with its own loading. Move $S$ from $-1.50\%$ to $-0.50\%$ tomorrow and every maturity's yield updates instantly and consistently, because they are all tied to the same two knobs.

loading b₁ = 0.787 loading b₅ = 0.367

Vasicek: the classic no-arbitrage example

The Vasicek model treats the short rate itself as the single factor, mean-reverting: $dr = a(b-r)\,dt + \sigma\, dW$, pulled toward a long-run level $b$ at speed $a$ with volatility $\sigma$. Solving the no-arbitrage pricing equation for a zero-coupon bond produces exactly the affine form $y_\tau = A_\tau - B_\tau \, r/\tau$, where $B_\tau = (1-e^{-a\tau})/a$ is a closed-form loading on today's short rate and $A_\tau$ is a closed-form constant that accounts for the average level to which the rate is expected to revert and a convexity adjustment for its volatility. The payoff of the affine structure is exactly this: bond prices and yields come out in closed form, with no numerical simulation needed, for any maturity at once.

Where it shows up

Affine models are the workhorse for anything that needs the whole curve to move consistently: pricing interest-rate options and swaptions, forecasting the curve a central bank meeting at a time, and decomposing a yield into an expected-rates component and a term-premium component (the ACM model is a multi-factor affine model built for exactly this). Their appeal is compression — three or four factors instead of forty maturities — at the cost of forcing the curve into whatever shapes those factors can produce, which occasionally misses a genuine kink or twist the real market is pricing.

Extending a single-factor Vasicek model to two or three correlated factors (still affine, still closed-form) is the standard way practitioners fix this: one factor drives the general level, a second drives the slope, and the resulting model can fit an upward-sloping, flat, or inverted curve depending on where the factors sit that day, all from the same fixed set of loadings.

Key terms

• Affine — a quantity expressed as constant plus linear combination of factors: $a + b\cdot X$.
• Factor — an underlying driver (short rate, level, slope) shared across all maturities.
• Loading — the maturity-specific weight applied to a factor to produce that maturity's yield.
• No-arbitrage restriction — the condition, present in Vasicek/CIR but not in curve-fitting models like Nelson-Siegel, that guarantees loadings are internally consistent across maturities.