Spot, Forward & Term-Structure Models

The spot curve tells you the rate to lock in today for money repaid at each horizon. But it secretly encodes something richer: the rates the market has already priced for periods that haven’t started yet — the rate for the year from year 1 to year 2, say. Those are forward rates, and they fall straight out of the spot curve by no-arbitrage. Once you can read them, you can ask the deeper question this lesson builds toward: how do rates themselves move over time? That’s the job of term-structure models — and the simplest, Vasicek and CIR, introduce the mean-reverting dynamics that bridge fixed income into the world of stochastic processes.

Forward rates: the bridge between two spots

Analogy. You can get money from today to year 2 two ways: (a) invest at the 2-year spot rate directly, or (b) invest at the 1-year spot, then roll into a one-year deposit at year 1. For these to give the same result — and they must, or arbitrage appears — the rate on that second-year roll is pinned down. That pinned-down rate is the forward rate $f_{1,2}$, the bridge spanning the gap between the 1-year and 2-year spots.

The no-arbitrage relation. Compounding through the forward must equal compounding through the longer spot:

More generally, the forward rate covering the interval from $t_1$ to $t_2$ satisfies

Worked example. From the bootstrap last lesson, $z_1 = 4.00\%$ and $z_2 = 5.02\%$. The implied one-year forward starting in year 1:

The forward (6.06%) sits above both spots (4.00%, 5.02%). That’s the rule on an upward curve: because the 2-year spot is an average of the first-year rate and the second-year forward, and the second-year forward must exceed the average to pull it above the first-year rate. Forwards always overshoot the spot curve’s slope — they’re the marginal rate, not the average.

What forwards mean: expectations vs. term premium

The expectations hypothesis. In its pure form, this says the forward rate equals the market’s expected future spot rate: $f_{1,2} = \mathbb{E}[z_1^{\text{(next year)}}]$. If true, an upward-sloping curve would mean the market expects rates to rise, and an inverted curve that it expects them to fall. It’s a clean story — and only partly true.

The term premium. Investors generally demand extra yield to lock money up for longer (more rate risk, less liquidity), so the forward rate usually exceeds the expected future spot by a term (or liquidity) premium:

So a normally upward-sloping curve doesn’t necessarily mean rates are expected to rise — part of the slope is just the premium for holding long bonds. Untangling the two is one of the central, unsolved debates of fixed income.

Modelling how rates move: the short rate

Why a single random number won’t do. A stock price is often modelled as a random walk that wanders off in any direction. But interest rates can’t behave like that — they don’t drift to infinity or below some floor for long; central banks and economic forces tug them back toward a “normal” level. A realistic rate model needs mean reversion: a pull back toward a long-run average whenever the rate strays. The central object these models track is the short rate $r_t$ — the instantaneous, very-short-term interest rate — from which the whole curve can be derived.

The Ornstein–Uhlenbeck idea. A mean-reverting rate obeys dynamics of the form

where $\theta$ is the long-run mean level, $\kappa$ is the speed of reversion (how hard the rate is yanked back), $\sigma$ is volatility, and $dW_t$ is the random shock. Whenever $r_t$ sits above $\theta$, the drift term $\kappa(\theta - r_t)$ is negative and pulls it down; below $\theta$, it pushes up. The rate is on a leash.

Vasicek and CIR at a glance

Vasicek (1977). The first tractable short-rate model uses constant volatility $\sigma$ in the equation above:

Its great virtue is that it’s analytically solvable — bond prices and the whole yield curve come out in closed form, and the short rate is normally distributed. Its great flaw: because the shock is constant-size and additive, the model lets rates go negative with positive probability. For decades that was considered a fatal embarrassment; since 2014’s negative-rate experiments in Europe and Japan, it’s looked oddly prescient.

Cox–Ingersoll–Ross / CIR (1985). CIR patches the negative-rate hole by scaling volatility with $\sqrt{r_t}$:

As the rate approaches zero, the $\sqrt{r_t}$ term shrinks the shock to nothing, so the rate gets gently floored at zero (under the Feller condition, $2\kappa\theta \ge \sigma^2$, it never hits zero at all). CIR keeps much of Vasicek’s tractability while guaranteeing non-negative rates — at the cost of a messier (non-central chi-squared) distribution.

When to use it

Use forward rates whenever you need the market’s locked-in rate for a future period — pricing forward-starting swaps, valuing a floating leg, or reading the curve’s recession signal. Use short-rate models when you need to simulate how rates and the curve might evolve — pricing interest-rate options, running scenario analysis on a rate book, or computing a model-based VaR. Pick Vasicek for analytic convenience and when negative rates are acceptable or even desired; pick CIR when a strict zero floor matters. For serious desk work, you’ll graduate to multi-factor models — but the mean-reverting intuition starts here.

Putting it together

The spot curve hides the forward rates the market has priced for future periods, extractable by no-arbitrage: the forward is the bridge between two spots, overshooting the curve’s slope (above the spots on an upward curve). Forwards equal expected future spots plus a term premium, which is why an inverted curve signals expected cuts and recession. To model how rates themselves move, we track the short rate with mean-reverting dynamics — Vasicek (constant vol, can go negative) and CIR (√r vol, floored at zero) — one-factor models that teach the mean-reversion intuition you carry straight into stochastic processes.

Next — credit spreads and default risk — we leave the default-free world and price the possibility that the borrower doesn’t pay: decomposing a risky yield into the risk-free rate, the expected loss, and the risk premium that compensates you for bearing uncertainty.