Fixed Income Analytics No-Arbitrage Bootstrapping

Spot vs. Forward Rate Curve & Bootstrapping Lab

Bootstrap zero-coupon spot yield curves from coupon-bearing par bonds and extract no-arbitrage implied forward rates ($f_{t_1, t_2}$). Analyze benchmark 1y1y, 2y3y, and 5y5y forward metrics, examine the Pure Expectations Hypothesis, and isolate term premium differentials.

Macroeconomic Regimes:
Yield Curve & Tenor Inputs
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Y
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1y1y Forward

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1Y rate in 1 year
2y3y Forward

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3Y rate in 2 years
5y5y Forward

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Central bank terminal metric
Custom Forward

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Term Structure Curves Visualizer

Comparison of the Zero-Coupon Spot Curve $s(t)$ and 1-Year Implied Forward Curve $f(t-1, t)$. Forward rates sit above spot rates during positive slope regimes.

Forward Premium / Spread ($f_{t-1, t} - s_t$ in basis points). Positive spread indicates upward momentum; negative values signal curve inversion.

Decomposition of Forward Rates into Pure Market Expected Future Rate ($E[r_t]$) vs. Liquidity Term Premium buffer.

Bootstrapped Spot Rates & 1-Year Forward Schedule Table
Tenor ($t$) Par Yield ($y_t$) Zero Spot Rate ($s_t$) Discount Factor ($P_t$) 1-Year Forward ($f_{t-1, t}$) Forward Spread ($f - s$) Exp. Rate ($E[r]$)
Sensitivity Matrix 1: 1Y Spot vs. 2Y Spot on 1y1y Implied Forward (%)
Rows: 1-Year Spot Rate (%) • Columns: 2-Year Spot Rate (%). Evaluates how steepness between 1Y and 2Y magnifies 1y1y forward rate moves.
Sensitivity Matrix 2: 5Y Spot vs. 10Y Spot on 5y5y Forward Rate (%)
Rows: 5-Year Spot Rate (%) • Columns: 10-Year Spot Rate (%). Shows how institutional 5y5y expectations respond to curve steepening and flattening.

Term Structure & Forward Rate Self-Assessment

1. If the zero-coupon spot yield curve is upward sloping ($s_2 > s_1$), what must be true about the 1-year forward rate beginning in 1 year ($f_{1, 2}$)?

2. Why does the recursive bootstrapping method require knowledge of shorter-term spot rates before calculating longer-term spot rates?

3. According to the Liquidity Preference Theory, why do implied forward rates typically exceed expected future spot rates?

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Frequently Asked Questions: Spot vs. Forward Rate Analytics

An implied forward rate is the interest rate agreed upon today for a loan or investment that begins on a future date and matures at a later date. Under the no-arbitrage principle, investing for t2 years must yield the same return as investing for t1 years and rolling into a forward rate: (1 + s_t2)^t2 = (1 + s_t1)^t1 × (1 + f(t1, t2))^(t2 - t1). Solving for f gives: f(t1, t2) = [ (1 + s_t2)^t2 / (1 + s_t1)^t1 ]^(1 / (t2 - t1)) - 1.

Because the spot rate s_t2 represents a geometric average of the prior spot rate s_t1 and the forward rate f(t1, t2). For a longer spot rate to exceed a shorter spot rate (s_t2 > s_t1), the incremental forward rate f(t1, t2) must be strictly higher than both to pull the cumulative average upward. Conversely, in an inverted curve, forward rates lie below spot rates.

The 5y5y ('5-year, 5-year forward') rate is the 5-year interest rate expected to prevail 5 years in the future. Central banks like the Federal Reserve and ECB monitor 5y5y forward rates because they filter out near-term cyclical policy noise, providing a clean measure of long-run structural market expectations, equilibrium neutral rates (r*), and anchored inflation expectations.

Bootstrapping is a recursive mathematical method. Because a 1-year par bond has only one cash flow, its spot rate equals its par yield (s_1 = y_1). For a 2-year par bond, the year-1 coupon is discounted using the known 1-year spot rate, allowing the unknown 2-year zero-coupon spot rate s_2 to be solved analytically from the par equation: 100 = C / (1 + s_1) + (100 + C) / (1 + s_2)². This is repeated inductively for all maturities.

According to the Pure Expectations Hypothesis, yes. However, empirical market data shows that forward rates are systematically biased upward by an unobserved 'term premium'—an extra return demanded by investors to compensate for the duration and inflation risk of committing capital over longer horizons (Liquidity Preference Theory).