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Michael Brenndoerfer

Integral Calculus and Differential Equations › Present Value and the Yield Curve

Michael BrenndoerferOctober 28, 20257 min read

Quantitative FinanceAll chapters

Part 1 of 15Mathematical Foundations

Chapter 7Integral Calculus and Differential Equations › Present Value and the Yield Curve

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An important application of integration in finance is computing present values when interest rates vary over time. In the real world, interest rates are not constant. Short-term rates typically differ from long-term rates, creating a term structure that may be quoted as par yields, zero or spot rates, forward rates, or discount factors. The integration formula below applies directly to the instantaneous forward curve. A continuously compounded zero rate maps directly to a discount factor, while par yields must first be transformed into zero rates or forward rates.

A specified instantaneous forward curve determines the discount factors used in these cash-flow models. Slope alone does not determine a discount factor: the upward-sloping forward curve and flat curve below have identical ten-year discount factors, even though they differ at other maturities. The integral uses the full curve up to the chosen maturity.

Continuous Discounting with Time-Varying RatesLink Copied

For a time-zero term structure, let $f(0,t)$ denote the instantaneous forward rate for maturity $t$ . We can no longer apply $e^{-rT}$ with one constant rate. Instead, the discount factor from time $0$ to time $T$ integrates the forward curve across maturities:

D(0, T) = \exp\left( -\int_0^T f(0,s) \, ds \right)

where:

$D(0, T)$ : discount factor from time $0$ to time $T$
$f(0,s)$ : time-zero instantaneous forward rate for maturity $s$
$s$ : dummy variable indexing maturity
$\int_0^T f(0,s) \, ds$ : accumulated continuously compounded forward rates through maturity $T$

The corresponding continuously compounded $T$ -maturity spot, or zero, rate is the maturity average

R(0,T)=-\frac{\log D(0,T)}{T}=\frac{1}{T}\int_0^T f(0,s)\,ds.

Thus an instantaneous forward rate, a maturity-specific spot rate, and a future short rate are distinct objects. The same exponential formula can integrate a deterministic future short-rate path, but such a path is a model assumption rather than the observed time-zero spot curve.

Discount Factor

The discount factor $D(0, T)$ represents the present value of $1 to be received at time $T$ . For $T>0$ , a positive accumulated forward rate gives $D(0,T)<1$ . At time zero the factor is one; negative accumulated rates can produce a factor above one.

Consider a time-zero instantaneous forward curve that starts at 3% and increases linearly to 5% at a 10-year maturity:

f(0,t) = 0.03 + 0.002t

This is an illustrative upward-sloping term structure, not a causal account of why the curve slopes upward. The discount factor for a 10-year cash flow is:

D(0, 10) = \exp\left( -\int_0^{10} (0.03 + 0.002t) \, dt \right)

Computing the integral:

\int_0^{10} (0.03 + 0.002t) \, dt = \left[ 0.03t + 0.001t^2 \right]_0^{10} = 0.3 + 0.1 = 0.4

Therefore:

D(0, 10) = e^{-0.4} \approx 0.6703

A cash flow of $1,000 in ten years has a present value of approximately $670.32. Its maturity-specific zero rate is $R(0,10)=10^{-1}\int_0^{10}f(0,s)\,ds=4\%$ , so the flat-4% factor agrees exactly at this maturity: $e^{-R(0,10)10}=e^{-0.4}$ . This identity holds for any integrable forward curve; linearity makes the average easy to calculate. One maturity average does not determine discount factors at other maturities.

In[15]:

Code

import numpy as np
from scipy import integrate


def instantaneous_forward(t):
    """Time-zero forward curve: f(0,t) = 3% + 0.2% per year."""
    return 0.03 + 0.002 * t


def discount_factor(T):
    """Compute discount factor from 0 to T under time-varying rates."""
    integral, _ = integrate.quad(instantaneous_forward, 0, T)
    return np.exp(-integral)


# Calculate discount factors for various maturities
maturities = np.array([1, 2, 5, 10, 20, 30])
discount_factors = np.array([discount_factor(T) for T in maturities])

Out[16]:

Console

Discount Factors Under Upward-Sloping Yield Curve:
---------------------------------------------
Maturity (years)     Discount Factor PV of $1000    
---------------------------------------------
1                    0.969476        $969.48        
2                    0.938005        $938.00        
5                    0.839457        $839.46        
10                   0.670320        $670.32        
20                   0.367879        $367.88        
30                   0.165299        $165.30

The discount factors decrease with maturity because the accumulated rate increases. Relative to a flat 4% curve, the time-varying factor is higher for $0<T<10$ , equal at $T=10$ , and lower for $T>10$ ; a global claim that its decline always accelerates would be incorrect.

Out[17]:

Visualization

Linear instantaneous forward curve increasing from 3% to 9%, with the zero rate increasing from 3% to 6%. — The instantaneous forward curve f(0,t) and its maturity-dependent zero-rate curve R(0,T).

Discount factors under f(0,t) = 3% + 0.2% times t and a flat 4% rate, crossing at the 10-year maturity. — Compared with a flat 4% curve, the time-varying discount factor is higher for maturities between 0 and 10 years, equal at 10 years, and lower after 10 years.

Pricing a Bond with Continuous CouponsLink Copied

Consider a bond paying continuous coupon at rate $c$ per year on a principal of $P$ , maturing at time $T$ . Coupon bonds normally pay on discrete dates. A continuous-coupon stream is a simplifying model and can approximate a sufficiently frequent payment schedule when the coupon timing and discount curve are smooth. The present value combines the stream of coupons and the principal repayment:

PV = \int_0^T c \cdot P \cdot D(0, t) \, dt + P \cdot D(0, T)

where:

$PV$ : present value of the bond
$c$ : continuous coupon rate (per year)
$P$ : principal (face value)
$T$ : maturity time
$D(0, t)$ : discount factor from time $0$ to time $t$

The first term captures the present value of all coupon payments: at each instant $t$ , the bond pays $c \cdot P$ per year, and this payment is discounted back to today using $D(0, t)$ . The second term is the present value of the principal repayment at maturity.

With our upward-sloping instantaneous forward curve:

In[18]:

Code

def bond_price(principal, coupon_rate, maturity):
    """Price a bond with continuous coupons under time-varying rates."""

    # Present value of coupon stream
    def coupon_pv_integrand(t):
        return coupon_rate * principal * discount_factor(t)

    coupon_pv, _ = integrate.quad(coupon_pv_integrand, 0, maturity)

    # Present value of principal
    principal_pv = principal * discount_factor(maturity)

    return coupon_pv + principal_pv


# Price a 10-year bond with 4% coupon
principal = 1000
coupon_rate = 0.04
maturity = 10

price = bond_price(principal, coupon_rate, maturity)

Out[19]:

Console

Bond Parameters:
  Principal: $1,000
  Coupon rate: 4.0%
  Maturity: 10 years

Bond Price: $1,005.54
  Coupon PV: $335.22
  Principal PV: $670.32

The bond is slightly above par, at about $1,005.54. Its cumulative average forward rate $R(0,t)=0.03+0.001t$ is below 4% for every $0<t<10$ , equaling 4% at maturity. Thus each earlier coupon has a larger discount factor than under the flat-4% continuous-coupon bond, while principal has the same factor. The instantaneous forward itself is below 4% only during the first five years.