Bonds: The Quiet Giant of Finance
Equities get the headlines, but bonds run the world. The global bond market is worth over $130 trillion — significantly larger than the equity market. Governments fund themselves with bonds. Companies raise capital with bonds. Central banks conduct monetary policy through bonds. And the interest rates embedded in bond prices affect everything else — mortgages, business loans, currency values, and derivatives pricing.
For quants, fixed income is one of the richest areas of work. Bond pricing is inherently mathematical, and the models used — yield curve construction, interest rate derivatives, credit risk — require serious quantitative chops.
What Is a Bond?
A bond is a loan, securitised. The issuer (borrower) promises to make regular interest payments (coupons) and return the face value (par) at maturity.
A typical UK gilt might look like: £100 face value, 4% annual coupon, maturing in 10 years. The holder receives £4 per year for 10 years, then £100 back at the end.
Bond Pricing
A bond's price is the present value of all future cash flows:
[ P = \sum_{t=1}^{T} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T} ]
where ( C ) is the coupon, ( F ) is the face value, and ( y ) is the yield to maturity.
If you know sigma notation and discounting, you already understand bond pricing. It is a direct application of those concepts.
Yield to Maturity (YTM)
The yield to maturity is the single discount rate that makes the present value of all cash flows equal to the current market price. It is the bond's effective interest rate.
Finding YTM requires solving a polynomial equation — there is no closed-form solution for bonds with multiple coupons. In practice, numerical methods (Newton-Raphson, bisection) are used. Optimisation strikes again.
When people talk about "interest rates rising," they usually mean bond yields are going up, which means bond prices are going down. This inverse relationship is fundamental.
Duration: Sensitivity to Rate Changes
Duration measures how sensitive a bond's price is to changes in interest rates. It is essentially the weighted average time until you receive your cash flows:
[ D = \frac{1}{P} \sum_{t=1}^{T} t \cdot \frac{C_t}{(1+y)^t} ]
Modified duration translates this into a price sensitivity:
[ \frac{\Delta P}{P} \approx -D_{\text{mod}} \cdot \Delta y ]
If a bond has a modified duration of 7 and yields rise by 1%, the price drops by approximately 7%. This is a first-order approximation — it is the bond equivalent of delta in options.
Convexity
Duration is a linear approximation. Convexity adds the second-order correction:
[ \frac{\Delta P}{P} \approx -D_{\text{mod}} \cdot \Delta y + \frac{1}{2} \cdot C_{\text{conv}} \cdot (\Delta y)^2 ]
This is a Taylor expansion applied to bond pricing. Convexity is always positive for standard bonds, meaning the duration approximation always understates the actual price change — prices rise more than expected when yields fall, and fall less than expected when yields rise. Bond investors like convexity.
The Yield Curve
The yield curve plots bond yields against maturity. It is one of the most closely watched indicators in finance.
Normal curve (upward sloping): longer maturities have higher yields — investors demand a premium for locking up money longer.
Inverted curve: short-term yields exceed long-term yields — historically a reliable predictor of recession. The Bank of England publishes daily yield curve data.
Flat curve: similar yields across all maturities — the market is uncertain about future rates.
Spot Rates and Forward Rates
Spot rates are the yields on zero-coupon bonds of different maturities. Forward rates are implied future interest rates derived from the spot curve:
[ (1 + s_2)^2 = (1 + s_1)(1 + f_{1,2}) ]
Building a spot rate curve from observed bond prices is called bootstrapping — a core skill in fixed income quant work.
Credit Risk in Bonds
Not all bonds are created equal. A UK gilt has essentially zero credit risk (the government prints the currency). A high-yield corporate bond might have a very real chance of default.
The credit spread is the extra yield investors demand for taking on credit risk:
[ \text{Credit spread} = \text{Corporate yield} - \text{Government yield} ]
Modelling credit spreads, default probabilities, and recovery rates is a major area of quant work, particularly in banking. The credit risk module explores this further.
Fixed Income Quant Work
Some of the most mathematically interesting quant work happens in fixed income:
- Yield curve construction: building consistent, arbitrage-free curves from messy market data
- Interest rate derivatives: pricing caps, floors, swaptions using models like Hull-White or SABR
- Mortgage-backed securities: modelling prepayment risk (the borrower might pay back early)
- Relative value: finding bonds that are cheap or expensive relative to the curve
Getting Practical
import numpy as np def bond_price(face, coupon_rate, ytm, years): """Price a fixed-rate bond.""" coupon = face * coupon_rate pv_coupons = sum(coupon / (1 + ytm)**t for t in range(1, years + 1)) pv_face = face / (1 + ytm)**years return pv_coupons + pv_face price = bond_price(100, 0.04, 0.05, 10) print(f"Bond price: £{price:.2f}") # Trades below par (coupon < yield)
Want More?
Fixed income is deep — far deeper than a single blog post can cover. Quantt takes you from basic discounting through to yield curve modelling and duration management, with Python exercises at every step.
For supplementary reading, Investopedia's fixed income section is well-structured, and the Bank of England's explainer on bonds is clear and authoritative.
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