Bond Math
From one idea, a bond is fixed cash flows discounted at the market yield, to the whole rates desk: duration and DV01, the convexity that duration leaves out, the carry and roll-down a position earns for waiting, and the futures that trade off the cheapest bond to deliver.
Every bond question on a trading floor comes back to one sentence: a bond is a fixed set of cash flows, and its price is those cash flows discounted at the market yield. Raise the yield and the same payments are worth less, so prices and yields move in opposite directions. How much they move is duration; in dollars it is DV01, the number on the risk report; the bend duration leaves out is convexity. A financed position earns carry and roll-down while it waits, and a bond future is a forward on whichever bond is cheapest to deliver into it.
The conventions throughout. Annual coupons, annual compounding, prices per 100 of face, whole years to maturity and no accrued interest. Those keep every calculation doable by hand, which is how it will be asked. Where a recipe prints a rounded duration or price and asks you to work from it, the answer is built from the printed figure. The curve itself, steepeners and flatteners sized in DV01, is taught in FX & Rates, and the old short lesson's point about level, slope and curvature lives there now.
- Price and yield. Zeros and coupon bonds from the yield, the zero's yield from its price, which side of par, the coupon a syndicate needs.
- Duration. The average wait and the rate sensitivity, zeros against coupon bonds, a price change from duration, the yield move that costs a stated amount.
- DV01 and hedging. Dollars per basis point, a DV01-matched hedge and its ratio, the most face a limit allows, what a non-parallel move does to a hedged position.
- Convexity. The estimate with and without the bend, against the exact repricing; effective measures; the gain-loss asymmetry; barbell against bullet.
- Carry and roll-down. Coupon less repo, the bond repriced a year older, the breakeven shift and the breakeven repo, flat and inverted curves.
- Bond futures. Invoice and conversion factor, gross and net basis, implied repo, the cheapest to deliver, fair futures against repo, hedging in contracts.
Price from yield → how fast price changes with yield (duration) → in dollars (DV01) → the bend (convexity) → what the position earns while yields do nothing (carry and roll) → the forward on the bond (futures). Each chapter uses the one before it, and every one has a backwards question.
From one bond to a rates book
Each step points to the chapter that practices it.
Chapters
Price from yield, yield from price
11 minA bond is a fixed set of cash flows. Its price is those cash flows discounted at the market yield, and its yield is whatever rate makes the discounting land on the price. Everything else in bond math follows from that one sentence.
- Price a zero and a coupon bond from the yield, per 100 of face
- Read a zero's yield straight off its price
- Say which side of par a bond trades before calculating, from coupon against yield
- Find the coupon that makes a new issue price where the syndicate wants it
Duration: the average wait, and the rate sensitivity
12 minMacaulay duration is how long, on average, you wait for your money. Modified duration is how many percent the price moves for a one-point move in yield. They are the same idea, because the cash flows far out are the ones most exposed to the rate.
- Compute Macaulay and modified duration for a zero and for a coupon bond
- Estimate a price change from duration alone, in percent and in dollars
- Find the yield move that costs a stated amount
- Rank bonds by duration without calculating
DV01: dollars per basis point, and hedging with them
12 minTraders think in dollars per basis point, not in duration, because dollars add across a book. Match the DV01s of two bonds and a parallel move in yields cancels. What is left is a bet on the shape of the curve.
- Compute a position's DV01 and the P&L of a parallel yield move
- Size a DV01-neutral hedge in a shorter bond, and the hedge ratio
- Turn a DV01 limit into a maximum face, and read the duration a risk system is using
- Work out what a DV01-matched position makes when the curve does not move in parallel
Convexity: the bend duration leaves out
12 minDuration draws a straight line through a curve. The true price-yield relationship bends upward, so a bond gains more on a rally than it loses on an equal sell-off. That bend is worth paying for, and the market charges for it.
- Estimate a price change with duration alone and with duration plus convexity
- Compare both estimates with the exact repricing and see what each leaves out
- Read effective duration and convexity off three repriced values
- Explain the asymmetry between a rally and a sell-off, and why convexity costs yield
Carry and roll-down: the return from standing still
12 minA financed bond position earns money even if nothing happens: the coupon less the repo cost, and the price gain from ageing down an upward-sloping curve. Together they say how far yields can move against you before the trade loses.
- Compute carry on a repo-financed par bond, in dollars
- Compute roll-down by repricing the bond one year older on an unchanged curve
- Find the parallel shift in yields that wipes out a year of carry and roll, and the repo rate at which the trade breaks even
- Say what a flat or inverted curve does to each piece
Bond futures: the basis, the implied repo and the cheapest to deliver
13 minA bond future is a promise to deliver a government bond later at a price fixed now. Owning the bond against the future locks in a return, the implied repo rate; the bond with the highest one is the cheapest to deliver, and the future trades off it.
- Compute the invoice amount from the futures price and the conversion factor
- Split the gross basis into carry and net basis
- Compute the implied repo rate of a cash-and-carry, and compare it with actual repo
- Pick the cheapest-to-deliver bond and size a futures hedge off it