Every concept, endless questions
Each concept is a small financial model. Read how it works, then draw a real interview question from it: forwards, backwards or what-if, with fresh numbers and a worked solution every time.
Forward runs a concept the usual way. Inverse runs it backwards, which is what separates understanding from memorizing.
Bond Math
6 conceptsPrice from yield, yield from price
A bond is a fixed set of cash flows, so its price is just those cash flows discounted at the market yield — and its yield is whatever rate makes that discounting land on the price. Because the cash flows never change, price and yield move in opposite directions: a higher discount rate makes the same payments worth less. When the coupon is above the yield the bond pays more than the market requires, so it trades above par; when it is below, below par. A zero-coupon bond is the cleanest case — one payment, so the yield can be read straight off the price.
Duration
Duration answers two questions that turn out to be the same. Macaulay duration is the average time you wait for your money, each cash flow weighted by its share of the price. Modified duration is how many percent the price moves for a one-point move in yield. They are linked because a cash flow far in the future is discounted many times over, so it is the most sensitive to the rate. A zero has one cash flow, so its duration is its maturity; coupons pay some money back early and pull the average — and the rate sensitivity — below maturity.
DV01 and hedging
Traders do not think in duration; they think in dollars per basis point. DV01 turns a bond’s rate sensitivity into the number that matters on a risk report: how much the position makes or loses when yields move one basis point. It is also what makes hedging work. A 2-year note moves far less per dollar of face than a 10-year bond, so hedging one with the other face-for-face leaves most of the risk on. Match the DV01s instead and a parallel move in yields cancels — but a move that is not parallel does not, which is why a duration-hedged position is really a bet on the shape of the curve.
Convexity
Duration draws a straight line through a curve. The true price-yield relationship bends: as yields rise, each extra basis point costs a little less, because the price it is working on has already shrunk; as yields fall, each basis point earns a little more. That bend is convexity. It makes a long bond position win more on a rally than it loses on an equal sell-off, so it is worth paying for — and the market does charge for it, with a lower yield. For small moves it hardly matters; for the 100–200 bp moves that happen in a bad year it is the difference between a good estimate and a bad one.
Carry and roll-down
A bond position earns money even if nothing happens, in two separate ways. Carry is the coupon you collect less the cost of financing the bond in repo. Roll-down is the price gain from ageing: on an upward-sloping curve, a 10-year bond becomes a 9-year bond a year later, and 9-year yields are lower, so the same bond is worth more. Together they are the return from standing still — and they set how far yields can move against you before the trade loses money. On an inverted curve roll-down turns into a cost, and when repo is above the coupon, so does carry.
Bond futures
A bond future is a promise to deliver a government bond at a fixed price later. The short can choose from a basket of bonds, so the exchange scales each one by a conversion factor to make them roughly comparable — and whichever is still cheapest after scaling gets delivered. Owning the cash bond and selling the future locks in a return: you pay the cash price, earn the coupon, and get the invoice price at delivery. That return is the implied repo rate. If it beats the rate you can borrow at, the future is rich; the bond that offers the highest implied repo is the cheapest to deliver, and the future trades off it.