By the end of this chapter, students should be able to:
Fixed-income mathematics begins with one simple idea: a cash flow at a future date must be converted into value today.
If \(P(t,T)\) is the time-\(t\) price of one dollar paid with certainty at time \(T\), then a deterministic cash-flow stream
\[ (c_1,\ldots,c_n) \quad \text{paid at} \quad T_1<\cdots<T_n \]
has time-\(t\) value
\[ V_t = \sum_{i=1}^n c_i P(t,T_i). \]
This formula looks elementary, but it is the starting point for yield curves, bond pricing, duration, convexity, immunization, and interest-rate derivatives.
In equity-option pricing, the main random object was often the future stock price \(S_T\). In fixed income, the most important object is usually the term structure of interest rates. The price of a bond depends on cash flows at many different dates, so one interest rate is usually not enough.
For example, a two-year coupon bond has cash flows at several dates. A five-year coupon bond has a different collection of dates. A ten-year bond is exposed to longer-term discount rates. Therefore, fixed-income pricing naturally uses a curve indexed by maturity.
The central object in this chapter is the zero-coupon bond price
\[ P(t,T), \qquad 0\le t\le T. \]
It is the time-\(t\) price of one unit of currency paid with certainty at time \(T\).
The zero-coupon bond price \(P(t,T)\) is also called a discount factor. It tells us how much one future dollar at time \(T\) is worth at time \(t\).
Throughout most of this chapter, we work with deterministic cash flows and deterministic yield curves. Stochastic interest-rate models begin in Chapter 20.
A zero-coupon bond with maturity \(T\) pays exactly one unit at time \(T\) and no intermediate coupons.
At time \(t\le T\), its price is denoted by
\[ P(t,T). \]
By definition,
\[ P(T,T)=1. \]
If interest rates are positive, we usually expect
\[ 0<P(t,T)\le 1 \]
for \(T\ge t\). However, in markets with negative yields, \(P(t,T)\) may exceed \(1\). The mathematical definition still makes sense.
If a deterministic cash flow \(C\) is paid at time \(T\), its time-\(t\) present value is
\[ PV_t(C,T)=C P(t,T). \]
This is a linear rule in the cash flow \(C\).
If cash flows \(c_i\) are paid at maturities \(T_i\), then
\[ V_t = \sum_{i=1}^n c_i P(t,T_i). \]
This is the fixed-income version of the linear pricing principle from earlier chapters.
For a fixed valuation date \(t\), define
\[ \Pi_t(c_1,\ldots,c_n)=\sum_{i=1}^n c_iP(t,T_i). \]
Then \(\Pi_t\) is a linear functional on the vector space of deterministic cash-flow vectors.
This linear algebra viewpoint is useful. The curve \(P(t,T_i)\) gives the coordinates of the pricing functional.
Instead of quoting discount factors directly, markets often quote rates. For mathematical work, continuously compounded rates are especially convenient.
For \(t<T\), define the continuously compounded spot rate \(R(t,T)\) by
\[ P(t,T)=e^{-R(t,T)(T-t)}. \]
Equivalently,
\[ R(t,T)=-\frac{\log P(t,T)}{T-t}. \]
When \(t=0\), we often write
\[ P(0,T)=e^{-R(0,T)T}. \]
Suppose \(P(0,5)=0.82\). Then the five-year continuously compounded zero rate is
\[ R(0,5)=-\frac{1}{5}\log(0.82)\approx 0.03969. \]
So the continuously compounded five-year zero rate is approximately \(3.969\%\).
If the annually compounded spot rate is \(y(t,T)\), then
\[ P(t,T)=\frac{1}{(1+y(t,T))^{T-t}}. \]
Thus
\[ y(t,T)=P(t,T)^{-1/(T-t)}-1. \]
Continuously compounded rates and annually compounded rates are simply different coordinate systems for the same discount factor.
A spot rate discounts from today to a future date. A forward rate describes the implied rate between two future dates.
For \(t<T_1<T_2\), the forward discount factor from \(T_1\) to \(T_2\) observed at time \(t\) is
\[ F_D(t;T_1,T_2)=\frac{P(t,T_2)}{P(t,T_1)}. \]
This is the price, at time \(T_1\), of one unit paid at time \(T_2\), locked in at time \(t\).
The continuously compounded forward rate \(F(t;T_1,T_2)\) is defined by
\[ \frac{P(t,T_2)}{P(t,T_1)} = \exp\{-F(t;T_1,T_2)(T_2-T_1)\}. \]
Therefore,
\[ F(t;T_1,T_2) = -\frac{\log P(t,T_2)-\log P(t,T_1)}{T_2-T_1}. \]
If \(t=0\) and \(P(0,T)=e^{-R(0,T)T}\), then
\[ F(0;T_1,T_2) = \frac{R(0,T_2)T_2-R(0,T_1)T_1}{T_2-T_1}. \]
This identity is fundamental: forward rates are slopes of the function \(T\mapsto -\log P(0,T)\).
A forward rate is not the expected future spot rate unless additional assumptions are imposed. It is an arbitrage-free rate implied by today’s discount curve.
The instantaneous forward rate is obtained by shrinking the forward interval.
Assume \(T\mapsto P(t,T)\) is differentiable. Define
\[ f(t,T)=-\frac{\partial}{\partial T}\log P(t,T). \]
Then
\[ P(t,T)=\exp\left(-\int_t^T f(t,u)\,du\right). \]
This representation is especially important for the Heath–Jarrow–Morton framework in Chapter 21.
At \(t=0\),
\[ R(0,T)=\frac{1}{T}\int_0^T f(0,u)\,du. \]
Thus the zero rate is an average of instantaneous forward rates.
The following interactive example shows how a simple parametric zero-rate curve determines discount factors and instantaneous forward rates.
Use the sliders to change the level, slope, and curvature of a zero-rate curve. The discount curve and forward curve update automatically.
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0.8%
A coupon bond pays a sequence of coupons and a final principal payment. Suppose the bond has face value \(F\), coupon rate \(q\) per year, payment frequency \(m\) per year, and maturity \(T\). The coupon payment is
\[ C=\frac{qF}{m}. \]
If the payment dates are
\[ T_1,T_2,\ldots,T_n=T, \]
then the cash flows are
\[ c_i=C \quad (1\le i<n), \qquad c_n=F+C. \]
The arbitrage-free price is
\[ B_0=\sum_{i=1}^{n} c_i P(0,T_i). \]
The symbol \(B_0\) here denotes the bond price, not the money-market account. Context matters.
Consider a three-year annual coupon bond with face value \(100\), coupon rate \(5\%\), and discount factors
\[ P(0,1)=0.9700, \qquad P(0,2)=0.9300, \qquad P(0,3)=0.8900. \]
The cash flows are \(5,5,105\). Therefore
\[ B_0=5(0.9700)+5(0.9300)+105(0.8900)=102.95. \]
The yield to maturity is the single constant rate that reproduces the observed bond price when applied to all cash flows.
For annual compounding, the yield \(y\) solves
\[ B_0=\sum_{i=1}^n \frac{c_i}{(1+y)^{T_i}}. \]
For continuously compounded yield, the yield \(y_c\) solves
\[ B_0=\sum_{i=1}^n c_i e^{-y_cT_i}. \]
Yield to maturity compresses the entire cash-flow profile and entire discount curve into one number. It is useful for quoting and comparison, but the mathematically primitive objects are discount factors or zero rates.
For fixed cash flows \(c_i>0\), the price as a function of yield is
\[ B(y)=\sum_{i=1}^n c_i(1+y)^{-T_i} \]
under annual compounding. Differentiating,
\[ B'(y)= -\sum_{i=1}^n T_i c_i(1+y)^{-T_i-1}<0. \]
Thus bond price decreases as yield increases.
The second derivative is
\[ B''(y)=\sum_{i=1}^n T_i(T_i+1)c_i(1+y)^{-T_i-2}>0 \]
when payment times are integer years. Thus the price-yield curve is convex.
import numpy as np
from math import exp, log
face = 100.0
coupon_rate = 0.05
maturity = 3
payment_times = np.array([1, 2, 3], dtype=float)
cash_flows = np.array([5, 5, 105], dtype=float)
discount_factors = np.array([0.9700, 0.9300, 0.8900])
price = np.dot(cash_flows, discount_factors)
price102.95def bond_price_from_ytm(y, times, cfs):
return np.sum(cfs / (1 + y) ** times)
def ytm_bisection(target_price, times, cfs, low=-0.9, high=1.0, tol=1e-12, max_iter=200):
for _ in range(max_iter):
mid = 0.5 * (low + high)
mid_price = bond_price_from_ytm(mid, times, cfs)
if abs(mid_price - target_price) < tol:
return mid
# Price is decreasing in y.
if mid_price > target_price:
low = mid
else:
high = mid
return 0.5 * (low + high)
ytm = ytm_bisection(price, payment_times, cash_flows)
ytms_percent = 100 * ytm
ytms_percent3.9382180647929816The yield to maturity is the constant annual-compounded rate that reproduces the price computed from the term structure.
Duration measures first-order sensitivity of a bond price to interest-rate changes. It also has a useful interpretation as a weighted average maturity.
Suppose a bond has positive cash flows \(c_i\) at times \(T_i\), and its price under a constant continuously compounded yield \(y\) is
\[ B(y)=\sum_{i=1}^n c_i e^{-yT_i}. \]
Define present-value weights
\[ w_i(y)=\frac{c_i e^{-yT_i}}{B(y)}. \]
Then \(w_i(y)>0\) and
\[ \sum_{i=1}^n w_i(y)=1. \]
The Macaulay duration under continuous compounding is
\[ D_{\mathrm{Mac}}(y)=\sum_{i=1}^n T_i w_i(y). \]
Thus duration is the present-value-weighted average payment time.
Differentiate the price:
\[ \frac{dB}{dy} = -\sum_{i=1}^n T_i c_i e^{-yT_i}. \]
Divide by \(B(y)\):
\[ -\frac{1}{B(y)}\frac{dB}{dy} =\sum_{i=1}^n T_i\frac{c_i e^{-yT_i}}{B(y)} =D_{\mathrm{Mac}}(y). \]
Therefore,
\[ \frac{dB}{B}\approx -D_{\mathrm{Mac}}\,dy. \]
For continuously compounded yields, Macaulay duration is also the modified duration.
For annual compounding, if
\[ B(y)=\sum_{i=1}^n c_i(1+y)^{-T_i}, \]
then the Macaulay duration is
\[ D_{\mathrm{Mac}}(y)=\sum_{i=1}^n T_i\frac{c_i(1+y)^{-T_i}}{B(y)}, \]
and the modified duration is
\[ D_{\mathrm{mod}}(y)=\frac{D_{\mathrm{Mac}}(y)}{1+y}. \]
The first-order approximation is
\[ \frac{\Delta B}{B}\approx -D_{\mathrm{mod}}\Delta y. \]
Duration gives a linear approximation. Convexity improves the approximation by adding a second-order term.
For continuously compounded yield,
\[ B(y)=\sum_{i=1}^n c_i e^{-yT_i}. \]
The second derivative is
\[ \frac{d^2B}{dy^2}=\sum_{i=1}^n T_i^2 c_i e^{-yT_i}. \]
Define convexity by
\[ C(y)=\frac{1}{B(y)}\frac{d^2B}{dy^2} = \sum_{i=1}^n T_i^2 w_i(y). \]
Then Taylor’s theorem gives
\[ \frac{\Delta B}{B} \approx -D_{\mathrm{Mac}}\Delta y+ rac12 C(\Delta y)^2. \]
Convexity is useful because bond-price curves are typically convex in yield. For a given magnitude of yield change, positive convexity makes price gains from yield decreases larger than price losses from yield increases, relative to a purely linear approximation.
This demo compares the exact bond price with the duration-only approximation and the duration-plus-convexity approximation.
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10y
4.00%
import numpy as np
def bond_cash_flows(face=100, coupon_rate=0.05, maturity=10, frequency=1):
n = int(maturity * frequency)
times = np.arange(1, n + 1) / frequency
coupon = face * coupon_rate / frequency
cashflows = np.full(n, coupon)
cashflows[-1] += face
return times, cashflows
def bond_price_continuous(y, times, cashflows):
return np.sum(cashflows * np.exp(-y * times))
def duration_convexity_continuous(y, times, cashflows):
pv = cashflows * np.exp(-y * times)
price = np.sum(pv)
weights = pv / price
duration = np.sum(times * weights)
convexity = np.sum(times**2 * weights)
return price, duration, convexity
times, cfs = bond_cash_flows(face=100, coupon_rate=0.05, maturity=10)
y = 0.04
price, duration, convexity = duration_convexity_continuous(y, times, cfs)
price, duration, convexity(107.42329348357868, 8.184247052782528, 75.53052412429821)for dy in [-0.01, -0.005, 0.005, 0.01]:
exact_price = bond_price_continuous(y + dy, times, cfs)
linear_approx = price * (1 - duration * dy)
convex_approx = price * (1 - duration * dy + 0.5 * convexity * dy**2)
print(
f"dy={dy:+.3%} | exact={exact_price:8.4f} | "
f"duration={linear_approx:8.4f} | duration+convexity={convex_approx:8.4f}"
)dy=-1.000% | exact=116.6341 | duration=116.2151 | duration+convexity=116.6208
dy=-0.500% | exact=111.9223 | duration=111.8192 | duration+convexity=111.9206
dy=+0.500% | exact=103.1272 | duration=103.0274 | duration+convexity=103.1288
dy=+1.000% | exact= 99.0245 | duration= 98.6315 | duration+convexity= 99.0372Bootstrapping means extracting discount factors from observed bond prices.
Suppose annual zero-coupon or coupon-bond prices are available. If the one-year discount factor \(P(0,1)\) is known, then a two-year coupon bond price can be used to solve for \(P(0,2)\).
For example, suppose a two-year annual coupon bond has coupon \(C\), face value \(F\), and price \(B_0\). Its pricing equation is
\[ B_0=C P(0,1)+(F+C)P(0,2). \]
Therefore,
\[ P(0,2)=\frac{B_0-C P(0,1)}{F+C}. \]
More generally, if a bond maturing at \(T_n\) has cash flows \(c_1,\ldots,c_n\) and earlier discount factors are known, then
\[ P(0,T_n) = \frac{B_0-\sum_{i=1}^{n-1}c_iP(0,T_i)}{c_n}. \]
This is a triangular linear system. The maturities must be ordered so that each new instrument introduces one new discount factor.
# Instruments with annual payments, ordered by maturity.
# Each row: observed price and cash flows at years 1,2,...,n.
instruments = [
{"price": 97.0874, "cashflows": [100.0]}, # 1-year zero
{"price": 99.1400, "cashflows": [4.0, 104.0]}, # 2-year 4% coupon
{"price": 101.2500, "cashflows": [5.0, 5.0, 105.0]},
{"price": 102.6000, "cashflows": [6.0, 6.0, 6.0, 106.0]},
]
dfs = []
for inst in instruments:
price = inst["price"]
cfs = np.array(inst["cashflows"], dtype=float)
known_value = np.dot(cfs[:-1], np.array(dfs)) if len(dfs) > 0 else 0.0
new_df = (price - known_value) / cfs[-1]
dfs.append(new_df)
dfs = np.array(dfs)
zero_rates = -np.log(dfs) / np.arange(1, len(dfs) + 1)
list(zip(range(1, len(dfs) + 1), dfs, zero_rates))[(1, 0.970874, 0.029558582241568588),
(2, 0.9159279230769232, 0.043908802000602466),
(3, 0.8744380036630037, 0.044724626834553576),
(4, 0.8116279286750984, 0.052178314937704855)]This calculation is simple because each instrument adds one new maturity. Real bond bootstrapping must handle day-count conventions, accrued interest, coupon frequencies, holidays, bid-ask spreads, and interpolation.
A parallel yield shift assumes every maturity changes by the same amount. This is often too simple. In reality, short rates, medium rates, and long rates can move differently.
Let a bond price be written as a function of discount rates at different maturities:
\[ B=\sum_{i=1}^n c_i e^{-R_iT_i}. \]
The sensitivity to the rate \(R_j\) at maturity \(T_j\) is
\[ \frac{\partial B}{\partial R_j} = -T_jc_j e^{-R_jT_j}. \]
The normalized sensitivity is
\[ -\frac{1}{B}\frac{\partial B}{\partial R_j} = T_j\frac{c_j e^{-R_jT_j}}{B}. \]
These maturity-specific sensitivities are the clean mathematical prototype of key rate durations.
In practice, key rate duration is defined relative to chosen benchmark maturities and an interpolation scheme. The idea is to measure exposure to localized curve movements rather than only to parallel movements.
Duration can be used to match the sensitivity of assets and liabilities.
Suppose a liability \(L\) is due at time \(T_L\). Under continuous compounding, its present value is
\[ PV_L(y)=L e^{-yT_L}. \]
Its duration is exactly
\[ D_L=T_L. \]
An asset portfolio with present value \(PV_A(y)\) and duration \(D_A(y)\) is first-order immunized against small parallel yield changes if
\[ PV_A(y)=PV_L(y) \]
and
\[ D_A(y)=D_L. \]
The first condition matches value. The second condition matches first-order interest-rate sensitivity.
Convexity provides second-order protection. If value and duration are matched and the asset portfolio has larger convexity than the liability, then for small yield changes the asset value tends to stay above the liability value.
Classical duration immunization assumes parallel shifts, deterministic cash flows, no default, no transaction costs, and stable curve dynamics. It is a useful first-order mathematical model, not a guarantee.
# Match a 7-year zero-coupon liability using two zero-coupon bonds.
# Asset 1 matures in 3 years, asset 2 matures in 12 years.
y = 0.04
liability_face = 1000.0
liability_time = 7.0
asset_times = np.array([3.0, 12.0])
liability_pv = liability_face * np.exp(-y * liability_time)
liability_duration = liability_time
# Let x1 and x2 be present values invested in the two zero-coupon assets.
# Conditions: x1 + x2 = liability_pv
# 3 x1 + 12 x2 = 7 liability_pv
A = np.array([[1.0, 1.0], asset_times])
b = np.array([liability_pv, liability_duration * liability_pv])
asset_pvs = np.linalg.solve(A, b)
asset_faces = asset_pvs * np.exp(y * asset_times)
asset_pvs, asset_faces(array([419.87985636, 335.90388509]), array([473.41321609, 542.84567029]))def pv_zero(face, T, y):
return face * np.exp(-y * T)
for dy in [-0.02, -0.01, 0.0, 0.01, 0.02]:
yy = y + dy
asset_value = sum(pv_zero(face, T, yy) for face, T in zip(asset_faces, asset_times))
liability_value = pv_zero(liability_face, liability_time, yy)
surplus = asset_value - liability_value
print(f"yield={yy:.2%} | asset={asset_value:8.3f} | liability={liability_value:8.3f} | surplus={surplus:8.3f}")yield=2.00% | asset= 872.861 | liability= 869.358 | surplus= 3.503
yield=3.00% | asset= 811.398 | liability= 810.584 | surplus= 0.813
yield=4.00% | asset= 755.784 | liability= 755.784 | surplus= 0.000
yield=5.00% | asset= 705.391 | liability= 704.688 | surplus= 0.702
yield=6.00% | asset= 659.659 | liability= 657.047 | surplus= 2.612The constructed portfolio matches present value and duration at the base yield. Because the asset portfolio has a different convexity than the liability, surplus changes nonlinearly as yields move away from the base value.
Choose cash-flow dates \(T_1,\ldots,T_n\). A deterministic cash-flow stream is a vector
\[ c=(c_1,\ldots,c_n)^\top\in\mathbb R^n. \]
The discount vector is
\[ p=(P(0,T_1),\ldots,P(0,T_n))^\top. \]
The price is the dot product
\[ \Pi(c)=p^\top c. \]
If we observe \(m\) traded bonds, collect their cash-flow vectors as columns of a matrix
\[ C=\begin{bmatrix} | & | & & |\\ c^{(1)} & c^{(2)} & \cdots & c^{(m)}\\ | & | & & | \end{bmatrix}\in\mathbb R^{n\times m}. \]
If their market prices are
\[ b=(b_1,\ldots,b_m)^\top, \]
then absence of deterministic cash-flow arbitrage requires the existence of a discount vector \(p\) such that
\[ C^\top p=b. \]
If the system has no positive solution \(p\) consistent with the market conventions, the quoted prices contain an arbitrage or inconsistency.
This is the fixed-income analogue of the state-price approach from Chapters 3 and 6.
# Four bonds, four annual cash-flow dates.
C = np.array([
[100.0, 4.0, 5.0, 6.0],
[ 0.0, 104.0, 5.0, 6.0],
[ 0.0, 0.0, 105.0, 6.0],
[ 0.0, 0.0, 0.0, 106.0],
])
prices = np.array([97.0874, 99.1400, 101.2500, 102.6000])
# Prices satisfy C.T @ discount_factors = prices.
discount_factors_linear = np.linalg.solve(C.T, prices)
zero_rates_linear = -np.log(discount_factors_linear) / np.arange(1, 5)
discount_factors_linear, zero_rates_linear(array([0.970874 , 0.91592792, 0.874438 , 0.81162793]),
array([0.02955858, 0.0439088 , 0.04472463, 0.05217831]))The triangular bootstrapping formula is a special case of solving a linear system.
Ask an AI tool:
Explain why a discount factor is more fundamental than a yield to maturity when pricing a coupon bond. Use a three-year coupon bond as the example. Then identify one way your explanation could be misleading.
A strong answer should say that the bond price is a linear combination of discount factors, while yield to maturity is the single internal rate that reproduces that price for a particular bond.
Ask an AI tool:
A student writes: “Duration is the derivative of bond price with respect to yield.” Correct this statement precisely for both continuously compounded and annually compounded yield conventions.
A strong answer should distinguish \(dB/dy\), \(-(1/B)dB/dy\), Macaulay duration, and modified duration.
Ask an AI tool to review your bootstrapping code for the following issues:
This chapter introduced the mathematical foundations of fixed-income pricing.
The central objects are discount factors:
\[ P(t,T). \]
They price deterministic future cash flows by the linear formula
\[ V_t=\sum_i c_iP(t,T_i). \]
Zero rates and forward rates are alternative descriptions of the same discount curve:
\[ P(t,T)=e^{-R(t,T)(T-t)} \]
and
\[ f(t,T)=-\frac{\partial}{\partial T}\log P(t,T). \]
A coupon bond is just a portfolio of zero-coupon bonds. Its yield to maturity is an internal rate, not the primitive pricing input. Duration and convexity describe how bond prices respond to yield changes:
\[ \frac{\Delta B}{B} \approx -D\Delta y+ rac12 C(\Delta y)^2. \]
Bootstrapping converts market bond prices into discount factors by solving a triangular linear system. Immunization uses value and duration matching to control first-order interest-rate risk.