Fixed Income Basics: Yield, Duration, and Convexity
We are leaving the world of options for a bit and entering the world of fixed income. This is the world of bonds, interest rates, and cashflows. Chapter 13 of Wilmott’s book is a self-contained introduction that does not require anything from earlier chapters. If you have ever wondered what a yield curve is or why bond traders care about something called “duration,” this is the post for you.
The Basic Instruments
Let me walk through the main building blocks.
Zero-coupon bond: The simplest possible bond. You pay some amount today, and you get a fixed amount (the principal) back at a specific date in the future. No payments in between. If the principal is $1 paid in 10 years, you might pay 60 cents for it today. The difference is your return.
Coupon-bearing bond: Same idea, but you also get periodic payments (coupons) along the way. A bond with a 4% coupon and $1 principal pays 2 cents every six months, plus the $1 at maturity. You can think of a coupon bond as a bundle of zero-coupon bonds, one for each coupon payment and one for the principal.
Money market account: Your regular bank account. Interest accumulates at a rate that changes over time. Flexible (you can withdraw anytime) but unpredictable in terms of what rate you will earn.
Floating rate bond: Similar to a money market account but structured as a bond. The coupon payments vary based on some reference rate, typically LIBOR (London Interbank Offer Rate).
Forward rate agreement (FRA): Two parties agree that a specific interest rate will apply to a specific principal over some future period. Basically locking in an interest rate for the future.
Repo: Short for “repurchase agreement.” You sell a security and agree to buy it back at a fixed price on a fixed date. The difference in price implies an interest rate (the repo rate). Repos are everywhere in the bond market.
STRIPS: Stands for “Separate Trading of Registered Interest and Principal of Securities.” Someone takes a normal coupon bond, strips it into pieces, and sells each coupon and the principal separately as zero-coupon bonds. This creates zero-coupon bonds with longer maturities than those issued directly.
Bonds Around the World
Different countries have their own naming conventions. In the US: bonds under one year are called bills (usually zero-coupon), 2 to 10 years are notes, and over 10 years are bonds. The UK calls its government bonds gilts. Japan has JGBs (Japanese Government Bonds). Foreign bonds get fun names too: non-US bonds in the US market are called Yankee bonds, and yen bonds from non-Japanese issuers are Samurai bonds.
Clean Price, Dirty Price, and Day Counting
When you see a bond price in the newspaper, it is the clean price, which does not include interest that has built up since the last coupon payment. The actual price you pay is the dirty price = clean price + accrued interest.
Accrued interest is calculated based on how many days have passed since the last coupon, but “how many days” depends on which convention you use: Actual/Actual (count real calendar days), 30/360 (pretend every month has 30 days), or Actual/360 (real days but only 360 in a year). This sounds ridiculous but it matters when real money is involved.
Measuring Yield
With so many different bond structures out there, how do you compare them? This is where yield comes in.
Current yield is the simplest measure: annual coupon divided by market price. A bond paying 4 cents per year that costs 88 cents has a current yield of 4/88 = 4.55%. It is quick and dirty. It ignores the principal repayment, the time value of money, and any capital gains.
Yield to maturity (YTM) or internal rate of return (IRR) is much better. You find the single interest rate that, when used to discount all future cashflows (coupons plus principal) back to today, gives you the current market price.
Mathematically, if the bond price is V, the principal is P, and coupons C_i are paid at times t_i, the YTM is the value y that satisfies:
V = sum of [C_i * e^(-y*(t_i - t))] + P * e^(-y*(T - t))
This has to be solved iteratively since there is no closed-form solution for y. For a five-year bond with 3% semi-annual coupons trading at 96 cents, the YTM works out to about 6.84%.
The yield to maturity is the right measure if you plan to hold the bond to maturity.
The Yield Curve
Plot the yield to maturity against time to maturity for a set of bonds, and you get the yield curve. This is one of the most important charts in all of finance. It tells you what interest rates the market expects for different time horizons.
Normally the yield curve slopes upward: longer-term bonds have higher yields because there is more uncertainty and you need compensation for locking up your money longer. Sometimes it inverts (short-term rates higher than long-term), which historically has been a recession signal.
Duration: How Sensitive Is Your Bond to Rate Changes?
As interest rates change, bond prices move. Duration tells you how much.
The Macaulay duration is a weighted average of the times to each cashflow, where the weights are the present values of those cashflows:
Duration = (1/V) * sum of [t_i * C_i * e^(-yt_i)] + (T * P * e^(-yT)) / V
It has units of time and represents the “average life” of the bond. A zero-coupon bond has a duration equal to its maturity. A coupon bond has a duration shorter than its maturity because the coupons pull the average earlier.
Why does this matter? Because for small changes in yield:
Change in price ~ -Duration * Change in yield * Price
If a bond has a duration of 7 years and yields rise by 1%, the bond price drops by approximately 7%. Duration is the bond equivalent of delta in options.
You can use duration for hedging too. If you hold two bonds and set their quantities in inverse ratio of their durations (one long, one short), you are hedged against small parallel shifts in the yield curve.
Convexity: The Second-Order Effect
Duration is a linear approximation. For larger yield changes, you need the next term in the Taylor expansion. That is convexity.
Convexity = (1/V) * d^2V/dy^2
Here is why convexity matters. Imagine two bonds with the same price and the same duration at the current yield. They look identical for small rate moves. But if one has higher convexity, it will be worth more when yields move significantly in either direction. Higher convexity is like free insurance: you benefit more from big moves.
Wilmott shows a nice picture of two bonds with equal price and duration but different convexities. Bond A (higher convexity) always wins. For any yield change, A has a higher price than B. This is why traders like convexity, and why it connects to the absence-of-arbitrage arguments we will see later.
Forward Rates and Bootstrapping
The yield to maturity has a problem: it is not consistent across instruments. Two five-year bonds with different coupon structures can have different yields. You cannot say there is one interest rate for five-year money.
Forward rates fix this problem. Instead of one rate that applies from now to maturity, forward rates specify what rate applies over each future interval. They are interest rates for specific future periods that are consistent across all instruments.
If you have zero-coupon bond prices Z(t; T) for all maturities T, the forward rate for time T is:
F(t; T) = -d/dT [ln Z(t; T)]
In practice, you build the forward curve through bootstrapping. Start with the shortest-maturity bond, extract the rate for that period. Move to the next bond, use the rate you already found for the first period, and solve for the rate in the new period. Keep going.
This works with coupon bonds too, though it requires more assumptions when you have fewer data points than unknowns. And in practice, swaps are used more often than bonds for bootstrapping because the swaps market is more liquid with more available maturities. (We will cover swaps in the next post.)
Between data points, you need to interpolate. Piecewise constant forward rates are the simplest approach. Some people use continuous forward rates, cubic splines, or other methods. Wilmott notes that the “correct” way to join the dots has been debated endlessly. If you need a rate for 2.5 years and your nearest bonds are at 2 and 3 years, you are making assumptions no matter what.
The Bigger Picture
Wilmott is honest about the limitations of this chapter’s framework. Everything here assumes interest rates are deterministic, either constant or known functions of time. That works for simple bonds and for comparing instruments, but it cannot handle complex products where the randomness of interest rates is the whole point.
For simple bond analysis, duration and convexity are used far more in practice than any fancy stochastic model. They are fast, intuitive, and good enough for most purposes. But for pricing interest rate derivatives, options on bonds, callable bonds, and the like, we need stochastic models. Those come in later chapters.
The key thing to remember: yield, duration, and convexity are tools for analyzing bonds, not for pricing derivatives. They help you understand how sensitive your positions are and how to hedge them against rate changes. When you hear a bond trader say “I’m long duration,” now you know what they mean.
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