Tell apart the spot curve, the par curve and the forward curve
The idea
The spot curve gives the yield on a single payment at each future date. The par curve gives, for each maturity, the coupon rate at which a bond would price at par, and it is derived from the spot curve. The forward curve gives the rates implied today for borrowing over future periods. The three are different presentations of the same information and can be derived from one another. When the spot curve slopes upward the par curve lies below it and the forward curve above it, because each forward rate must be high enough to pull the average up.
A practice question
What does the par curve show?
- The yield on a single payment at each future dateWhy not: That is the spot curve, made of zero-coupon rates.
- The coupon rate at which a bond of each maturity would price at par — correct
- The rate implied today for borrowing at a future dateWhy not: That is the forward curve.
It answers the issuer's question directly: what coupon would a new bond of this maturity need to sell at face value.
A harder one
When the spot curve slopes upward, where does the forward curve lie relative to it?
- Below itWhy not: That is where the forward curve sits when the spot curve slopes *downward*. The relationship follows the direction of the slope.
- On top of itWhy not: They coincide only when the spot curve is flat, since the wedge between them is created entirely by slope.
- Above it — correct
Each spot rate is a geometric average of the forwards up to that date. For the average to be rising, each new forward must exceed it.
The hardest tier
An analyst has only the spot curve. What else can they construct from it?
- Nothing further without market quotesWhy not: The three curves are one set of discount factors presented three ways, so any one determines the others.
- Both the par curve and the forward curve, by arithmetic alone — correct
- The par curve, but not the forward curveWhy not: Forward rates follow directly from consecutive spot rates — arguably the easier of the two derivations.
The three are presentations of one set of discount factors, so any one of them determines the others.
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