Continuous-Time Time Value of Money (TVM)

Instantaneous Forward Rates

In real-world markets, interest rates are not constant over long periods. The "rate" you get for a 1-year loan might be different from the rate for a 10-year loan. To model this, we use the concept of the instantaneous forward rate.

While a standard interest rate is often quoted for a specific duration, the instantaneous forward rate, denoted as $f(t, T)$, represents the rate applicable at a specific time $t$ for a contract that expires at a future time $T$. It is the "marginal" rate for an infinitesimally small period of time.

The relationship between the instantaneous forward rate and the discount factor $P(t, T)$—which is the price at time $t$ of a zero-coupon bond maturing at time $T$—is given by:

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

This equation tells us that the forward rate is the rate of change of the logarithm of the bond price with respect to maturity. If we know the entire curve of instantaneous forward rates, we can reconstruct the entire term structure of interest rates.

Modeling the Interest Rate Term Structure

The "term structure" refers to the relationship between interest rates (or bond yields) and different maturities. In continuous time, we represent this through the Spot Rate Curve. The spot rate $r(t, T)$ is the constant continuously compounded rate that would yield the same present value as the entire forward rate curve from time $t$ to $T$.

The Relationship between Spot and Forward Rates

We can express the spot rate as the average of the instantaneous forward rates over the period. Mathematically, the spot rate $r(t, T)$ is:

$$r(t, T) = \frac{1}{T-t} \int_{t}^{T} f(t, s) \, ds$$

This integral shows that the spot rate is essentially a time-weighted average of all the instantaneous forward rates between the present and the maturity date.

Worked Example: Calculating Present Value from Forward Rates

Suppose we are given a simplified instantaneous forward rate curve where $f(0, T) = 0.02 + 0.005T$. We want to find the present value of $100 received at time $T=2$.

Step 1: Find the Spot Rate $r(0, 2)$ Using the averaging formula:

$$r(0, 2) = \frac{1}{2-0} \int_{0}^{2} (0.02 + 0.005s) \, ds$$ $$r(0, 2) = \frac{1}{2} \left[ 0.02s + \frac{0.005s^2}{2} \right]_{0}^{2}$$ $$r(0, 2) = \frac{1}{2} \left( (0.02 \cdot 2) + (0.0025 \cdot 4) \right)$$ $$r(0, 2) = \frac{1}{2} (0.04 + 0.01) = 0.025 \text{ or } 2.5\%$$

Step 2: Calculate the Present Value Now, apply the continuous discounting formula using the spot rate:

$$PV = 100 \cdot e^{-0.025 \cdot 2}$$ $$PV = 100 \cdot e^{-0.05} \approx 100 \cdot 0.9512 = \$95.12$$

Summary of Key Concepts

To master continuous-time TVM, keep this hierarchy of concepts in mind:

1. Continuous Compounding: Uses $e^{rt}$ to allow for calculus-based modeling.
2. Instantaneous Forward Rate ($f$): The rate for an infinitesimal slice of time at a specific point in the future.
3. Spot Rate ($r$): The average forward rate over a specific duration, used to discount single cash flows.
4. Term Structure: The collection of these rates across all possible maturities, forming the yield curve.

Understanding these building blocks is essential before moving into stochastic interest rate models (like Vasicek or CIR models), where these deterministic rates become random variables that evolve over time.