# Converting from par rates

The par rate is equal to the fixed coupon rate payable on a ‘par bond’.

The par yield is known as the Par rate, Swap rate or Swap yield.

## Conversion

If we know the par yield, we can calculate both the zero coupon yield and the forward yield for the same maturities and risk class.

**Example 1: Converting from par rates to zero coupon rates**

Given par rates (**p**), the zero coupon rates (**z**) can also be calculated.

The periodic par yields (**p**) are:

p_{1} = 0.02 per period (2%)

p_{2} = 0.029803 per period (2.9803%)

The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:

z_{n} = ( (1 + p_{n}) / (1 - p_{n} x CumDF_{n-1}) )^{(1/n)} - 1

*Where:*

z_{n} = the zero coupon rate for maturity n periods

p_{n} = the par rate for maturity n periods, starting now

CumDF_{n-1} = the total of the discount factors for maturities 1 to 'n-1' periods, calculated from the zero coupon rates (z_{1} to z_{n-1})

*Applying the formula:*

z_{n} = ( (1 + p_{n}) / (1 - p_{n} x CumDF_{n-1}) )^{(1/n)} - 1

z_{2} = ( (1 + p_{2}) / (1 - p_{2} x CumDF_{1}) )^{(1/2)} - 1

z_{2} = ( (1 + 0.029803) / (1 - 0.029803 x DF_{1}) )^{(1/2)} - 1

z_{2} = ( 1.029803 / (1 - (0.029803 x 1.02^{-1}) )^{(1/2)} - 1

z_{2} = ( 1.029803 / (1 - 0.0292186) )^{(1/2)} - 1

z_{2} = 1.0608^{(1/2)} - 1

= 0.029951 (= 2.9951% per period)

2.9951% per period is the rate of interest payable on a two-period zero coupon investment. This means that 2.9951% interest will be paid on the amount of the original investment, rolled up and compounded at the end of two periods. In addition, the original investment will also be repaid at Time 2.

(The one period par rate p_{0-1} represents the identical deal to the one period zero coupon rate z_{0-1}. For this reason the rate is also identical = 2% per period.)

The conversion of par rates to zero coupon rates is sometimes known as 'bootstrapping', because the results of calculating each earlier zero coupon rate are used successively to calculate the next-longer maturity zero coupon rate.

**Example 2: Converting from zero coupon rates to forward rates**

Given the calculated zero coupon rates (**z**), the forward rates (**f**) can also be calculated in turn.

A short-form calculation of the forward rate **f _{1-2}** is set out below:

f_{1-2} = ( 1.029951^{2} / 1.02 ) - 1

= 0.04

= 4% per period.

The calculation of forward rates from zero coupon rates is explained in more detail on the page Converting from zero coupon rates.