Yields-to-maturity for zero-coupon government bonds could be analyzed for a full range of maturities, which is called the government bond spot curve (or zero curve). Government spot rates are assumed to be risk-free.

## Spot Curve

The spot curve is upward sloping and flattens for longer times-to-maturity. Longer-term government bonds have usually higher yields than shorter-term bonds. The hypothetical spot curve is ideal for analyzing the maturity structure because it meets the “all other things being equal” assumption.

## Par Curve

The par curve differes from the spot curve in that it is a sequence of yields-to-maturity such that each bond is priced at par value. The par curve is obtained from the spot curve. All bonds on the par curve are supposed to have the same credit risk, periodicity, currency, liquidity, tax status, and annual yields. Between coupon payment dates, the flat price (not full price) is equal to par value.

### Obtaining Par Rates from Spot Rates

Since the par curve is a sequence of yields-to-maturity such that each bond is priced at par value, then the formula to obtain par rates is the following:

\(100=\frac { PMT }{ { (1+{ Z }_{ 1 }) }^{ 1 } } +\frac { PMT }{ { (1+{ Z }_{ 2 }) }^{ 2 } } +…+\frac { PMT + Principal }{ { (1+{ Z }_{ N }) }^{ N } } \)

Where *Z _{N}* = the spot rate at time

*N*

#### Example

Assuming that the 1-year and 2-year spot rates on government bonds are respectively 5.25% and 5.75%:

- The 1-year par-rate is 5.250%.

$$100=\frac { PMT+100 }{ 1.0525 } ;\quad PMT=5.25$$

- The 2-year par-rate is 5.736%.

$$100=\frac { PMT }{ 1.0525 } +\frac { PMT+100 }{ { 1.0575 }^{ 2 } } ; \quad PMT=5.736$$

## Par Curve

The forward curve is a series of forward rates, each having the same time frame. We will talk in length about forward rates in the next learning objective.

QuestionThe yield curve derived from a sequence of yields-to-maturity on zero-coupon bonds is called the:

A. Par curve and all bonds on this curve are supposed to have the same annual yields

B. Flat curve and all bonds on this curve are supposed to have the same liquidity and similar tax status

C. Forward curve and all bonds on this curve are supposed to have the same periodicity

SolutionThe correct answer is A.

The par curve is a sequence of yields-to-maturity such that each bond is priced at par value. All bonds on the par curve are supposed to have the same credit risk, periodicity, currency, liquidity, tax status, and annual yields.

*Reading 44 LOS 44i: *

*Define and compare the spot curve, yield curve on coupon bonds, par curve, and forward curve*