Module 12: Bonds & Fixed Income

1. What Is a Bond?

If a stock is a claim on random future cash flows, a bond is a claim on (nearly) deterministic future cash flows. When you buy a bond, you are lending money to the issuer (a government or corporation) in exchange for a contractually specified stream of payments: periodic coupon payments and the return of principal at maturity.

A standard coupon bond has two types of cash flows:

• Coupon payments: Fixed dollar amounts paid at regular intervals (typically semi-annually). If a bond has a 5% coupon rate and $1,000 face value, you receive $25 every six months.
• Principal repayment: The face value (par value, typically $1,000) returned at maturity.

1.1 Bond Pricing Formula

The price of a bond is the present value of all its future cash flows, discounted at the market's required yield:

where:

• P = bond price
• C = coupon payment per period
• F = face value (par)
• y = yield per period (market discount rate)
• n = number of periods to maturity

The first term is a geometric series (the present value of an annuity) and the second is a simple present value of a lump sum. For a semi-annual bond, all quantities are halved or doubled appropriately: C = (coupon rate / 2) × F, y = annual yield / 2, n = years to maturity × 2.

2. Coupon Bonds vs. Zero-Coupon Bonds

There are two fundamental bond structures, and understanding their difference is critical:

2.1 Coupon Bonds

A coupon bond pays regular interest and returns principal at maturity. Its cash flow stream looks like a combination of an annuity (the coupons) and a lump sum (the principal).

2.2 Zero-Coupon Bonds

A zero-coupon bond pays no coupons at all. You buy it at a discount and receive the face value at maturity. US Treasury bills (T-bills) and Treasury STRIPS are common examples.

The zero-coupon bond is the purest fixed-income instrument: a single cash flow at a single point in time. Every coupon bond can be decomposed into a portfolio of zero-coupon bonds (one for each coupon and one for the principal). This decomposition is called stripping.

2.3 Comparison

3. Yield to Maturity (YTM) as Internal Rate of Return

The Yield to Maturity (YTM) is the single most important number associated with a bond. It is the discount rate that makes the present value of all future cash flows equal to the current market price:

In other words, YTM is the Internal Rate of Return (IRR) of the bond investment, assuming you hold to maturity and reinvest all coupons at the YTM rate.

3.1 Properties of YTM

• When price = par, YTM = coupon rate (the bond is priced "at par")
• When price < par, YTM > coupon rate (discount bond — higher yield compensates for lower price)
• When price > par, YTM < coupon rate (premium bond — lower yield because you paid more)
• Bond price and yield move in opposite directions — always. This is the fundamental relationship.

3.2 YTM Assumptions and Limitations

YTM makes two strong assumptions:

1. Hold to maturity: If you sell before maturity, your actual return differs from YTM because the sale price depends on market conditions.
2. Reinvestment at YTM: Coupon payments are assumed to be reinvested at the YTM rate. If rates fall, reinvestment income is lower — this is reinvestment risk.

3.3 The Yield Curve

The yield curve plots YTM against maturity for bonds of the same credit quality (typically US Treasuries). Its shape reveals market expectations about future interest rates:

4. Credit Spreads: The Price of Default Risk

Not all bonds are equally safe. A US Treasury bond is considered "risk-free" (the government can always print money to pay its debts). A corporate bond carries credit risk — the possibility that the issuer defaults. The market prices this risk through the credit spread:

This spread is the additional yield investors demand for bearing default risk. It is a direct market estimate of the risk premium.

4.1 Credit Spread Magnitudes

Note: 1 basis point (bp) = 0.01%. So a spread of 200 bps = 2.00%.

5. Credit Ratings as Ordinal Classification

Credit ratings assigned by agencies (S&P, Moody's, Fitch) are an ordinal classification of credit quality. They rank issuers by likelihood of default but do not provide exact probabilities.

5.1 The Rating Scale

5.2 The BBB/BB Cliff

The boundary between BBB- and BB+ is the most consequential threshold in fixed income. Many institutional investors (pension funds, insurance companies, central banks) are mandated to hold only investment-grade bonds. When a bond is downgraded from BBB- to BB+ ("fallen angel"), these investors are forced to sell, causing a sudden price drop.

6. Duration: The First Derivative

Duration is the single most important risk measure in fixed income. It answers the question: how sensitive is this bond's price to changes in interest rates?

6.1 Macaulay Duration

Macaulay duration is the weighted average time until a bond's cash flows are received, where the weights are the present values of each cash flow:

6.2 Modified Duration

While Macaulay duration is measured in years, modified duration directly measures price sensitivity:

Modified duration tells you the percentage change in price for a small change in yield:

This is a first-order Taylor approximation. The negative sign confirms that price and yield move in opposite directions.

6.3 Duration Properties

7. Convexity: The Second Derivative

Duration provides a linear approximation of price changes. But the price-yield relationship is a curve, not a line. The curvature is captured by convexity:

Convexity is always positive for standard (option-free) bonds, meaning the price-yield curve bows upward. This has a wonderful consequence for bondholders:

• When yields fall, the price increase is larger than duration alone predicts.
• When yields rise, the price decrease is smaller than duration alone predicts.

7.1 The Full Taylor Approximation

Combining duration and convexity gives us a second-order approximation of price changes:

This is remarkably accurate for yield changes up to about 200 basis points. For larger moves, you would need higher-order terms (third derivative, etc.) — but in practice, the second-order approximation is almost always sufficient.

7.2 Duration and Convexity by Bond Type

8. Python: Bond Pricing, Duration, and Convexity

8.1 Pricing a Coupon Bond

Pythonimport numpy as np

def bond_price(face, coupon_rate, ytm, years, freq=2):
    """
    Price a coupon bond.

    Parameters
    ----------
    face : float        Face value (par), typically 1000
    coupon_rate : float Annual coupon rate (e.g., 0.05 for 5%)
    ytm : float         Annual yield to maturity
    years : float       Years to maturity
    freq : int          Coupon frequency (2 = semi-annual)

    Returns
    -------
    float : bond price
    """
    n = int(years * freq)
    c = face * coupon_rate / freq     # coupon per period
    y = ytm / freq                     # yield per period

    # Present value of coupons (annuity formula)
    if y == 0:
        pv_coupons = c * n
    else:
        pv_coupons = c * (1 - (1 + y) ** (-n)) / y

    # Present value of face value
    pv_face = face / (1 + y) ** n

    return pv_coupons + pv_face

# Example: 10-year, 5% coupon bond at various yields
print("--- Bond Price vs. Yield ---")
print(f"{'Yield':>8} {'Price':>10} {'Premium/Discount':>18}")
for y in [0.03, 0.04, 0.05, 0.06, 0.07]:
    p = bond_price(1000, 0.05, y, 10)
    status = "Par" if abs(p - 1000) < 0.01 else ("Premium" if p > 1000 else "Discount")
    print(f"{y:>7.1%} ${p:>9.2f}   {status}")

8.2 Computing Duration and Convexity

Pythondef bond_analytics(face, coupon_rate, ytm, years, freq=2):
    """
    Compute price, Macaulay duration, modified duration, and convexity.
    """
    n = int(years * freq)
    c = face * coupon_rate / freq
    y = ytm / freq

    # Build cash flow vector
    cash_flows = np.full(n, c)
    cash_flows[-1] += face  # last period includes principal
    periods = np.arange(1, n + 1)

    # Discount factors
    disc = (1 + y) ** (-periods)

    # Price
    pv = cash_flows * disc
    price = pv.sum()

    # Macaulay duration (in periods, then convert to years)
    mac_dur_periods = np.sum(periods * pv) / price
    mac_dur = mac_dur_periods / freq

    # Modified duration
    mod_dur = mac_dur / (1 + y)

    # Convexity
    convexity_periods = np.sum(periods * (periods + 1) * pv) / (price * (1 + y) ** 2)
    convexity = convexity_periods / freq ** 2

    return {
        'price': price,
        'macaulay_duration': mac_dur,
        'modified_duration': mod_dur,
        'convexity': convexity,
    }

# Compare different bonds
bonds = [
    ("2Y 4% Coupon",   1000, 0.04, 0.045, 2),
    ("5Y 4% Coupon",   1000, 0.04, 0.045, 5),
    ("10Y 4% Coupon",  1000, 0.04, 0.045, 10),
    ("30Y 4% Coupon",  1000, 0.04, 0.045, 30),
    ("10Y Zero-Coupon",1000, 0.00, 0.045, 10),
    ("30Y Zero-Coupon",1000, 0.00, 0.045, 30),
]

print(f"{'Bond':>18} {'Price':>8} {'Mac Dur':>9} {'Mod Dur':>9} {'Convex':>9}")
print("-" * 56)
for name, face, cpn, ytm, yrs in bonds:
    a = bond_analytics(face, cpn, ytm, yrs)
    print(f"{name:>18} ${a['price']:>7.2f} {a['macaulay_duration']:>8.2f}y "
          f"{a['modified_duration']:>8.2f}  {a['convexity']:>8.1f}")

8.3 Taylor Approximation vs. Exact Price Change

Pythondef taylor_vs_exact(face, coupon_rate, ytm, years, yield_change):
    """
    Compare first-order (duration), second-order (duration + convexity),
    and exact price changes for a given yield shock.
    """
    analytics = bond_analytics(face, coupon_rate, ytm, years)
    p0 = analytics['price']
    dur = analytics['modified_duration']
    cvx = analytics['convexity']

    # Exact new price
    p1 = bond_price(face, coupon_rate, ytm + yield_change, years)
    exact_change = (p1 - p0) / p0

    # First-order approximation (duration only)
    first_order = -dur * yield_change

    # Second-order approximation (duration + convexity)
    second_order = -dur * yield_change + 0.5 * cvx * yield_change ** 2

    return exact_change, first_order, second_order

# Test with a 30-year bond for various yield shocks
print("--- 30Y 4% Coupon Bond, YTM = 4.5% ---")
print(f"{'dY (bps)':>10} {'Exact':>10} {'Dur Only':>10} {'Dur+Cvx':>10} {'Dur Err':>10} {'D+C Err':>10}")
for bps in [-200, -100, -50, -25, 25, 50, 100, 200]:
    dy = bps / 10000
    exact, fo, so = taylor_vs_exact(1000, 0.04, 0.045, 30, dy)
    print(f"{bps:>+8}   {exact:>+9.2%}  {fo:>+9.2%}  {so:>+9.2%}  "
          f"{abs(fo - exact):>9.4%}  {abs(so - exact):>9.4%}")

9. Risk Decomposition in Fixed Income

A bond investor faces multiple sources of risk. Understanding and decomposing these is essential for portfolio management:

10. Types of Bonds

11. Summary

This module has established the mathematical foundations of fixed-income analysis through a statistical lens:

With stocks (random cash flows) and bonds (deterministic cash flows) under our belt, we are ready for the most mathematically rich asset class: options and derivatives. In Module 13, we will see how truncated distributions, partial derivatives, and stochastic calculus converge to create the Black-Scholes framework.