Implied forward rate

The spot curve also hides a second object inside it: the forward rate, the rate the curve implies for a future period, starting at some point m₁ years from now and running to m₂ years from now. It falls straight out of a no-arbitrage argument. Consider two ways to turn ₹100 (or $100) into money at time m₂: (a) invest it directly at the m₂-year spot rate for the whole period, or (b) invest it at the m₁-year spot rate for m₁ years, then reinvest the proceeds for the remaining (m₂ − m₁) years at whatever rate is needed to match strategy (a) exactly. If those two strategies did not produce the same terminal value, an arbitrageur could borrow at the cheaper one and lend at the dearer one for a riskless profit, so in an arbitrage-free market, the rate that makes strategy (b) match strategy (a) is fully determined by today's spot curve, and that determined rate is the forward rate f(m₁, m₂):

(1 + s₂/2)^(2m₂) = (1 + s₁/2)^(2m₁) × (1 + f/2)^(2(m₂−m₁)) ⟺ f = 2 × [ (DF(m₁)/DF(m₂))^(1/(2(m₂−m₁))) − 1 ]

Is an implied forward rate a forecast of future rates?

Two things to hold about what a forward rate is and is not. First, it is a mathematical, arbitrage-free fact about today's curve, nothing about deriving it required anyone to forecast anything; two already-observed spot rates fully determine it. Second, and this is the trap the term-structure theories exist to defuse, a forward rate being arbitrage-free does not automatically mean it equals the market's best guess of what the future spot rate will actually turn out to be. Whether it does, and by how much it might not, is exactly the question the three term-structure theories answer.

{
  "skill": "Drafting or debugging a curve-bootstrapping script without silently shipping a wrong curve",
  "use": "Have an AI assistant draft the first version of a bootstrap/forward-rate script, or explain why a specific line of your own script disagrees with a hand-computed number — then treat the reprice-and-roll-forward proof, not the AI's explanation, as the actual verdict on whether the script is correct.",
  "tools": ["Chat/code assistants — Claude, ChatGPT, Copilot", "A notebook environment (Jupyter/Colab) to run and inspect every intermediate discount factor"],
  "prompt": "Write a Python function that bootstraps a semi-annual spot curve from a sparse dict of {maturity: par_yield}, using linear interpolation to fill the grid and sequential discount-factor solving. Then write a second function that reprices each ORIGINAL input bond off the resulting curve and prints the price and the difference from 100. Do not tell me the curve is correct — show me the reprice differences for every input bond.",
  "verify": "An AI-written bootstrap can look completely correct — clean code, sensible variable names, a smooth output table — while silently using annual compounding where the instrument is semi-annual, or including today's own maturity in the 'already known' sum one step early. The only real check is Step 3 of this module's algorithm: discount every ORIGINAL input bond's cash flows through the finished curve and confirm an exact reprice to par, then independently verify the forward-rate roll-forward identity numerically. Recompute both by hand for at least one maturity before trusting the script on the rest (Guardrail #3, Recompute).",
  "diy": "You must be able to state the bootstrap formula and the forward-rate identity from memory and hand-verify one step of each — this Core section is exactly that. AI accelerates writing the loop; it cannot certify that the loop is correct.",
  "market": "IN"
}