Yield Curve

The yield curve is a graphical representation of the different interest rates at various maturities.

There are different types of yield curves: spot, forward, par etc. No matter what type of yield curve, it represents different interest rates (at different maturities) at a single point of time.

Slope of the Yield Curves

In developed markets, yield curves are most commonly upward sloping with diminishing marginal increases in yield for identical changes in maturity; that is, the yield curve “flattens” at longer maturities. Because nominal yields incorporate a premium for expected inflation, an upward-sloping yield curve is generally interpreted as reflecting a market expectation of increasing or at least level future inflation (associated with relatively strong economic growth). The existence of risk premiums (e.g., for the greater interest rate risk of longer-maturity bonds) also contributes to a positive slope.

An inverted yield curve is somewhat uncommon. Such a term structure may reflect a market expectation of declining future inflation rates (because a nominal yield incorporates a premium for expected inflation) from a relatively high current level. Expectations of declining economic activity may be one reason that inflation might be anticipated to decline, and a downward-sloping yield curve has frequently been observed before recessions.

A flat yield curve typically occurs briefly in the transition from an upward-sloping to a downward-sloping yield curve, or vice versa.

A humped yield curve, which is relatively rare, occurs when intermediate-term interest rates are higher than short- and long-term rates.

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Daily Treasury Yield Curve

Par Curve

The par curve is a hypotenetical yield curve for coupon-paying Treasury securities that assumes all securities are priced at par. It represents the yields to maturity on coupon-paying government bonds, priced at par, over a range of maturities. In practice, recently issued (“on the run”) bonds are typically used to create the par curve because new issues are typically priced at or close to par.

The par curve is important for valuation in that it can be used to construct a zero-coupon yield curve. The process makes use of the fact that a coupon-paying bond can be viewed as a portfolio of zero-coupon bonds. The zero-coupon rates are determined by using the par yields and solving for the zero-coupon rates one by one, in order from earliest to latest maturities, via a process of forward substitution known as bootstrapping (a statistical method for estimating a sample distribution based on the properties of an approximating distribution).

Forward Rate

A forward rate is an interest rate that is determined today for a loan that will be initiated in a future time period. It can be think of as the spot rate in the future.

The term structure of forward rates for a loan made on a specific initiation date is called the forward curve.

Forward rates and forward curves can be mathematically derived from the current spot curve.