Forward Rates and Term Structures¶

Forward rates describe future borrowing/lending rates implied by today's yield curve. They are central to term-structure modeling, pricing FRAs and swaps, and understanding no-arbitrage dynamics of interest rates.

Simple Forward Rates from Discount Factors¶

Definition¶

Given discount factors $P(0,T_1)$ and $P(0,T_2)$ with $0 < T_1 < T_2$, the simple forward rate over the period $[T_1, T_2]$ is defined by the no-arbitrage relation:

\[ P(0,T_1) = P(0,T_2) \cdot \left[1 + F(0; T_1, T_2) \cdot (T_2 - T_1)\right] \]

Solving for the forward rate:

\[ F(0; T_1, T_2) = \frac{1}{T_2 - T_1} \left(\frac{P(0,T_1)}{P(0,T_2)} - 1\right) \]

Economic Interpretation¶

The forward rate $F(0; T_1, T_2)$ is: - The rate that can be locked in today for borrowing/lending over $[T_1, T_2]$ - The rate that makes a forward-starting loan have zero initial value - The break-even rate for the period, given the current term structure

Time-t Forward Rates¶

More generally, the forward rate observed at time $t$ for the period $[T_1, T_2]$ is:

\[ F(t; T_1, T_2) = \frac{1}{T_2 - T_1} \left(\frac{P(t,T_1)}{P(t,T_2)} - 1\right) \]

This forward rate evolves stochastically as market conditions change.

Continuously Compounded Forward Rates¶

Using continuous compounding, the forward rate $f_c(0; T_1, T_2)$ satisfies:

\[ e^{-z(0,T_1) T_1} = e^{-z(0,T_2) T_2} \cdot e^{f_c(0; T_1, T_2)(T_2 - T_1)} \]

Solving:

\[ f_c(0; T_1, T_2) = \frac{z(0,T_2) T_2 - z(0,T_1) T_1}{T_2 - T_1} \]

This can be rewritten as:

\[ f_c(0; T_1, T_2) = \frac{-\log P(0,T_2) + \log P(0,T_1)}{T_2 - T_1} \]

Instantaneous Forward Rate¶

Definition¶

The instantaneous forward rate $f(t,T)$ is obtained by taking the limit as the accrual period shrinks to zero:

\[ f(t,T) := \lim_{\Delta \to 0} f_c(t; T, T+\Delta) = -\frac{\partial}{\partial T} \log P(t,T) \]

This fundamental relationship can be inverted:

\[ P(t,T) = \exp\left(-\int_t^T f(t,u) \, du\right) \]

Relationship to Zero Rates¶

The zero rate is the average of instantaneous forward rates:

\[ z(t,T) = \frac{1}{T-t} \int_t^T f(t,u) \, du \]

Equivalently, differentiating:

\[ f(t,T) = z(t,T) + (T-t) \frac{\partial z(t,T)}{\partial T} \]

This shows that: - If the yield curve is flat ($z$ constant), then $f(t,T) = z$ - If the yield curve is upward sloping ($\partial z/\partial T > 0$), then $f(t,T) > z(t,T)$ - If the yield curve is downward sloping, then $f(t,T) < z(t,T)$

The Short Rate¶

The instantaneous short rate is the limit:

\[ r_t = f(t,t) = \lim_{T \to t^+} f(t,T) \]

This is the rate for infinitesimally short borrowing at time $t$.

Forward Rate Agreement (FRA)¶

A Forward Rate Agreement is a contract to exchange: - A fixed rate $K$ payment - A floating rate (typically LIBOR) payment

at a future settlement date, based on a notional principal.

FRA Payoff¶

At settlement time $T_1$, the FRA payoff (to the receiver of floating) is:

\[ \text{Payoff} = N \cdot (L(T_1; T_1, T_2) - K) \cdot (T_2 - T_1) \]

discounted back to $T_1$, where $L(T_1; T_1, T_2)$ is the realized LIBOR rate.

FRA Pricing¶

The fair fixed rate $K$ that makes the FRA have zero initial value is precisely the forward rate:

\[ K^* = F(0; T_1, T_2) \]

The value of an existing FRA with rate $K$ is:

\[ V_{\text{FRA}}(0) = N \cdot (F(0; T_1, T_2) - K) \cdot (T_2 - T_1) \cdot P(0, T_2) \]

Term Structure Representations¶

A term structure can be equivalently represented by any of these curves:

Representation	Notation	Contains
Discount curve	$T \mapsto P(0,T)$	Prices of zero-coupon bonds
Zero curve	$T \mapsto z(0,T)$	Spot rates for each maturity
Forward curve	$T \mapsto f(0,T)$	Instantaneous forward rates

All three contain equivalent information (given sufficient smoothness) but serve different purposes:

Discount curve: Direct use in present value calculations
Zero curve: Comparison across maturities, yield analysis
Forward curve: Rate expectations, model inputs (HJM)

Conversion Formulas Summary¶

From	To	Formula
$P(0,T)$	$z(0,T)$	$z = -\frac{1}{T}\log P$
$P(0,T)$	$f(0,T)$	$f = -\frac{\partial}{\partial T}\log P$
$z(0,T)$	$P(0,T)$	$P = e^{-zT}$
$z(0,T)$	$f(0,T)$	$f = z + T\frac{\partial z}{\partial T}$
$f(0,T)$	$P(0,T)$	$P = e^{-\int_0^T f(u)du}$
$f(0,T)$	$z(0,T)$	$z = \frac{1}{T}\int_0^T f(u)du$

Forward Rates and Expectations¶

The Expectations Hypothesis¶

Under the pure expectations hypothesis, forward rates equal expected future spot rates:

\[ F(0; T_1, T_2) = \mathbb{E}[z(T_1, T_2)] \]

However, this hypothesis is empirically rejected. The observed relationship is:

\[ F(0; T_1, T_2) = \mathbb{E}^{\mathbb{P}}[z(T_1, T_2)] + \text{term premium} \]

Risk-Neutral Expectations¶

Under the risk-neutral measure $\mathbb{Q}$, forward rates do equal expected future rates:

\[ F(0; T_1, T_2) = \mathbb{E}^{\mathbb{Q}}[L(T_1; T_1, T_2)] \]

This is the foundation of forward measure pricing (Section 10.5).

Practical Notes on Forward Rate Curves¶

Sensitivity to Interpolation¶

Forward rates are derivatives of the discount curve, making them sensitive to interpolation choices:

Linear interpolation on zero rates produces discontinuous forwards
Cubic spline interpolation can produce oscillating forwards
Monotone convex methods balance smoothness and stability

Forward Rate Volatility¶

Instantaneous forward rates inherit volatility from the term structure: - Short-end forwards are more volatile - Long-end forwards are more stable - This affects HJM model specification

Negative Forward Rates¶

Forward rates can be negative when: - The yield curve is sufficiently inverted - Market rates are near or below zero

This does not indicate arbitrage but requires care in lognormal models.

Key Takeaways¶

Forward rates are implied by ratios of discount factors via no-arbitrage
The instantaneous forward rate satisfies $f(t,T) = -\partial_T \log P(t,T)$
Zero rates are averages of forward rates; forwards are marginal rates
FRA pricing directly uses forward rates
Term structure representations (discount, zero, forward) are mathematically equivalent
Forward rates are sensitive to curve construction choices

Exercises¶

Exercise 1. Given zero rates $R(0,1) = 3.0\%$ and $R(0,2) = 3.5\%$ (continuously compounded), compute the forward rate $f(0; 1, 2)$. Verify by showing that investing at $R(0,1)$ for one year then at $f(0;1,2)$ for another year yields the same result as investing at $R(0,2)$ for two years.

Solution to Exercise 1

The continuously compounded forward rate over $[T_1, T_2]$ is:

\[ f_c(0; T_1, T_2) = \frac{z(0, T_2) \cdot T_2 - z(0, T_1) \cdot T_1}{T_2 - T_1} \]

With $z(0, 1) = 0.03$, $z(0, 2) = 0.035$, $T_1 = 1$, $T_2 = 2$:

\[ f_c(0; 1, 2) = \frac{0.035 \times 2 - 0.03 \times 1}{2 - 1} = \frac{0.070 - 0.030}{1} = 0.040 = 4.0\% \]

Verification: Investing at $z(0, 1) = 3\%$ for one year and then at $f_c(0; 1, 2) = 4\%$ for another year yields:

\[ e^{0.03 \times 1} \cdot e^{0.04 \times 1} = e^{0.03 + 0.04} = e^{0.07} \]

Investing at $z(0, 2) = 3.5\%$ for two years yields:

\[ e^{0.035 \times 2} = e^{0.07} \]

Both strategies produce $e^{0.07} \approx 1.07251$, confirming the no-arbitrage consistency.

Intuitively, the forward rate of 4.0% is higher than either zero rate because the upward-sloping zero curve requires the marginal (forward) rate over the second year to exceed the average rate over two years.

Exercise 2. Derive the simply compounded forward rate $F(t; T_1, T_2) = \frac{1}{T_2 - T_1}\left(\frac{P(t,T_1)}{P(t,T_2)} - 1\right)$ from the no-arbitrage condition that prevents riskless profit from lending over $[t, T_2]$ versus lending over $[t, T_1]$ and rolling into a forward contract.

Solution to Exercise 2

Consider an investor at time $t$ with two strategies to deploy capital over $[t, T_2]$:

Strategy A (Direct lending): Lend $1 from $t$ to $T_2$ at the spot rate. The amount received at $T_2$ is:

\[ \frac{1}{P(t, T_2)} \]

since $P(t, T_2)$ is the price at $t$ of $1 received at $T_2$.

Strategy B (Roll-over with forward): Lend $1 from $t$ to $T_1$ at the spot rate, and simultaneously lock in a forward contract to lend the proceeds from $T_1$ to $T_2$ at rate $F(t; T_1, T_2)$. The amount received at $T_2$ is:

\[ \frac{1}{P(t, T_1)} \cdot \bigl[1 + F(t; T_1, T_2)(T_2 - T_1)\bigr] \]

Both strategies are riskless (the forward rate is locked in at time $t$). By no-arbitrage, they must produce the same terminal amount:

\[ \frac{1}{P(t, T_2)} = \frac{1}{P(t, T_1)} \cdot \bigl[1 + F(t; T_1, T_2)(T_2 - T_1)\bigr] \]

Rearranging:

\[ \frac{P(t, T_1)}{P(t, T_2)} = 1 + F(t; T_1, T_2)(T_2 - T_1) \]

Solving for the forward rate:

\[ F(t; T_1, T_2) = \frac{1}{T_2 - T_1}\left(\frac{P(t, T_1)}{P(t, T_2)} - 1\right) \]

If this condition were violated — say, $F$ were higher — one could borrow over $[t, T_2]$, lend over $[t, T_1]$, and enter the forward to lock in a riskless profit (arbitrage). The direction reverses if $F$ were lower.

Exercise 3. If $P(0, T) = e^{-0.03T - 0.002T^2}$, compute the instantaneous forward rate $f(0, T) = -\frac{\partial}{\partial T}\ln P(0, T)$ and the zero rate $R(0,T)$. Show that the forward curve lies above the zero curve when the zero curve is upward-sloping.

Solution to Exercise 3

Given $P(0, T) = e^{-0.03T - 0.002T^2}$, we have $\ln P(0, T) = -0.03T - 0.002T^2$.

Instantaneous forward rate:

\[ f(0, T) = -\frac{\partial}{\partial T}\ln P(0, T) = -\frac{\partial}{\partial T}(-0.03T - 0.002T^2) = 0.03 + 0.004T \]

Zero rate:

\[ R(0, T) = -\frac{\ln P(0, T)}{T} = \frac{0.03T + 0.002T^2}{T} = 0.03 + 0.002T \]

Comparison: For any $T > 0$:

\[ f(0, T) - R(0, T) = (0.03 + 0.004T) - (0.03 + 0.002T) = 0.002T > 0 \]

So $f(0, T) > R(0, T)$ for all $T > 0$, confirming that the forward curve lies above the zero curve when the zero curve is upward-sloping.

This is a consequence of the general identity $f(t, T) = z(t, T) + (T - t)\frac{\partial z}{\partial T}$. When the zero curve slopes upward ($\frac{\partial z}{\partial T} > 0$), the second term is positive, so the forward rate exceeds the zero rate. Intuitively, the zero rate is the average of forward rates up to $T$, and if forward rates are increasing, the marginal (instantaneous) rate must exceed the average.

Exercise 4. Prove the identity $R(t, T) = \frac{1}{T-t}\int_t^T f(t, u)\,du$: the zero rate is the average of the instantaneous forward rate curve over $[t, T]$.

Solution to Exercise 4

Proof. Starting from the definition $P(t, T) = e^{-z(t, T)(T - t)}$, take the logarithm:

\[ \ln P(t, T) = -z(t, T)(T - t) \]

The instantaneous forward rate is:

\[ f(t, T) = -\frac{\partial}{\partial T}\ln P(t, T) \]

Now integrate $f(t, u)$ over $u \in [t, T]$:

\[ \int_t^T f(t, u)\,du = -\int_t^T \frac{\partial}{\partial u}\ln P(t, u)\,du = -\bigl[\ln P(t, u)\bigr]_{u=t}^{u=T} \]

\[ = -\ln P(t, T) + \ln P(t, t) \]

Since $P(t, t) = 1$, we have $\ln P(t, t) = 0$, so:

\[ \int_t^T f(t, u)\,du = -\ln P(t, T) = z(t, T)(T - t) \]

Dividing both sides by $(T - t)$:

\[ \frac{1}{T - t}\int_t^T f(t, u)\,du = z(t, T) \]

This establishes that the zero rate is the arithmetic average of the instantaneous forward rate curve over $[t, T]$. $\blacksquare$

This identity has a natural interpretation: the zero rate for maturity $T$ represents the "average" cost of borrowing per unit time over $[t, T]$, while the forward rate $f(t, u)$ represents the marginal cost at each instant $u$. Averaging the marginal costs recovers the average cost.

Exercise 5. An inverted yield curve has $R(0,1) = 5\%$ and $R(0,10) = 3.5\%$. Compute the forward rate $f(0; 1, 10)$ and show it is below $3.5\%$. What does this imply about market expectations for future short-term rates under the expectations hypothesis?

Solution to Exercise 5

With $R(0, 1) = 5\%$ and $R(0, 10) = 3.5\%$ (continuously compounded), the forward rate over $[1, 10]$ is:

\[ f_c(0; 1, 10) = \frac{R(0, 10) \times 10 - R(0, 1) \times 1}{10 - 1} = \frac{0.035 \times 10 - 0.05 \times 1}{9} = \frac{0.35 - 0.05}{9} = \frac{0.30}{9} \approx 3.333\% \]

Indeed $f_c(0; 1, 10) = 3.333\% < 3.5\% = R(0, 10)$, so the forward rate is below the 10-year zero rate.

Interpretation under the expectations hypothesis: The expectations hypothesis posits that forward rates equal expected future spot rates:

\[ f_c(0; 1, 10) = \mathbb{E}[R(1, 10)] \]

The forward rate of 3.333% being below even the current 10-year rate of 3.5% implies the market expects short-term rates to decline significantly from their current level of 5%. This is consistent with the inverted yield curve: the market anticipates that the central bank will cut rates in response to an expected economic slowdown. The average expected rate over years 1 through 10 is only 3.33%, well below the current short rate.

More granularly, if the forward curve $f(0, T)$ is monotonically declining from 5% at $T = 0$ to some value below 3.33% for large $T$ (such that the average over $[1, 10]$ equals 3.33%), the market is pricing in a substantial and sustained rate-cutting cycle.

Exercise 6. Explain the difference between the forward rate $f(0; T, T+\Delta)$ and the expected future spot rate $\mathbb{E}[R(T, T+\Delta)]$. Under the expectations hypothesis they are equal. Under the liquidity premium hypothesis, which is larger and why?

Solution to Exercise 6

The forward rate $f(0; T, T + \Delta)$ is the rate implied by today's yield curve for lending/borrowing over the future period $[T, T + \Delta]$. It is derived purely from current discount factors:

\[ f(0; T, T + \Delta) = \frac{1}{\Delta}\left(\frac{P(0, T)}{P(0, T + \Delta)} - 1\right) \]

The expected future spot rate $\mathbb{E}^{\mathbb{P}}[R(T, T + \Delta)]$ is the physical (real-world) expectation of what the interest rate will actually be over $[T, T + \Delta]$.

Under the expectations hypothesis (EH): The EH asserts:

\[ f(0; T, T + \Delta) = \mathbb{E}^{\mathbb{P}}[R(T, T + \Delta)] \]

That is, forward rates are unbiased predictors of future spot rates. Under EH, the term structure reflects only rate expectations, with no risk premium.

Under the liquidity premium hypothesis (LPH): Borrowers prefer long-term fixed-rate funding while lenders prefer short-term lending (to maintain liquidity). This maturity mismatch creates a liquidity premium $L(T) > 0$ that increases with horizon:

\[ f(0; T, T + \Delta) = \mathbb{E}^{\mathbb{P}}[R(T, T + \Delta)] + L(T) \]

The forward rate is larger than the expected future spot rate by the liquidity premium. This premium compensates long-term lenders for bearing duration risk: if rates rise unexpectedly, long-term bond prices fall, so investors demand extra return for holding longer maturities.

The LPH explains why the yield curve is typically upward-sloping even when rates are not expected to rise — the positive liquidity premium tilts the curve upward. Empirically, the term premium (a generalization of the liquidity premium) has been estimated at 1-2% for 10-year maturities relative to the short rate, though it varies with economic conditions.

From	To	Formula
\(P(0,T)\)	\(z(0,T)\)	\(z = -\frac{1}{T}\log P\)
\(P(0,T)\)	\(f(0,T)\)	\(f = -\frac{\partial}{\partial T}\log P\)
\(z(0,T)\)	\(P(0,T)\)	\(P = e^{-zT}\)
\(z(0,T)\)	\(f(0,T)\)	\(f = z + T\frac{\partial z}{\partial T}\)
\(f(0,T)\)	\(P(0,T)\)	\(P = e^{-\int_0^T f(u)du}\)
\(f(0,T)\)	\(z(0,T)\)	\(z = \frac{1}{T}\int_0^T f(u)du\)

Representation	Notation	Contains
Discount curve	\(T \mapsto P(0,T)\)	Prices of zero-coupon bonds
Zero curve	\(T \mapsto z(0,T)\)	Spot rates for each maturity
Forward curve	\(T \mapsto f(0,T)\)	Instantaneous forward rates

Forward Rates and Term Structures¶

Simple Forward Rates from Discount Factors¶

Definition¶

Economic Interpretation¶

Time-t Forward Rates¶

Continuously Compounded Forward Rates¶

Instantaneous Forward Rate¶

Definition¶

Relationship to Zero Rates¶

The Short Rate¶

Forward Rate Agreement (FRA)¶

FRA Payoff¶

FRA Pricing¶

Term Structure Representations¶

Conversion Formulas Summary¶

Forward Rates and Expectations¶

The Expectations Hypothesis¶

Risk-Neutral Expectations¶

Practical Notes on Forward Rate Curves¶

Sensitivity to Interpolation¶

Forward Rate Volatility¶

Negative Forward Rates¶

Key Takeaways¶

Further Reading¶

Exercises¶