1           Par Yield Curve

  Yield Curve   Yield to maturity is the single discount rate that equates the present value of a bond’s cash flows to its market price. A yield curve is a graph of bond yields against their maturities.

  Par Yield Curve    Par yield curve is a curve constructed for theoretical bonds whose prices equal par. While on-the-run Treasury bonds are generally issued with price near par, the on-the-run yield curve often resembles the par yield curve.

    2           Spot Rate Curve

The spot rate is the discount rate of a single future cash flow such as a zero-coupon bond. The spot rate curve is the graph of the relationship between spot rates and maturity.

    3           Forward Rate Curve

The forward rate is the future yield on a bond. For example, the yield on a three-month Treasury bill six months from now is a forward rate. The forward rate curve is the graph of the relationship between forward rates and maturity.

    4           Relationship between Different Curves.

Therefore, forward rates measure the marginal reward for lengthening the maturity of the investment by certain future period. Spot rates can be seen as (geometric) averages of one or more forward rates. Par rates are averages of one or more spot rates. Thus, par curves have the flattest shape of the three curves.

If the spot curve is upward sloping, the spot curve lies above the par curve, and the forward rate curve lies above the spot curve. If the spot curve is downward sloping, the par curve is highest and the forward curve lowest. The three curves are identical only if they are horizontal.

    5           Example

5.1          Practice in Handbook

Suppose that the yield curve is upward sloping. Which of the following statements is true?

A.       The forward rate yield curve is above the zero-coupon yield curve, which is above the coupon-bearing bond yield curve.

B.        The forward rate yield curve is above the coupon-bearing bond yield curve, which is above the zero-coupon yield curve.

C.        The coupon-bearing bond yield curve is above the zero-coupon yield curve, which is above the forward rate yield curve.

D.       The coupon-bearing bond yield curve is above the forward rate yield curve, which is above the zero-coupon yield curve.

Answer: A

As the coupon yield curve is an average of the spot, zero-coupon curve, it has to lie below the spot curve when it is upward sloping. Moreover, if the spot curve is upward sloping, the forward curve has to be above the spot curve.

Reference:

1)         Salomon Brothers, Understanding the Yield Curve

2)         Bruce Tuckman, Fixed Income Securities

3)         Handbook

4)         Schweser Study Notes