imported>Doug Williamson |
imported>Doug Williamson |
| Line 1: |
Line 1: |
| The par rate is equal to the fixed coupon rate payable on a ‘[[par bond]]’.
| | 1. |
|
| |
|
| | The lowest quarter of an ordered distribution divided into four equal parts in terms of the number of items. |
|
| |
|
| The par yield is known as the Par rate, Swap rate or Swap yield.
| |
|
| |
|
| | 2. |
|
| |
|
| '''Conversion'''
| | The boundary between the lowest quarter - as defined above - and the remaining three quarters of the distribution. |
| | |
| 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.
| |
| | |
| | |
| <span style="color:#4B0082">'''Example 1: Converting from par rates to zero coupon rates'''</span>
| |
| | |
| Given par rates ('''p'''), the zero coupon rates ('''z''') can also be calculated.
| |
| | |
| | |
| The periodic par yields ('''p''') are: | |
| | |
| p<sub>1</sub> = 0.02 per period (2%)
| |
| | |
| p<sub>2</sub> = 0.029803 per period (2.9803%)
| |
| | |
| | |
| The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:
| |
| | |
| z<sub>n</sub> = ( (1 + p<sub>n</sub>) / (1 - p<sub>n</sub> x CumDF<sub>n-1</sub>) )<sup>(1/n)</sup> - 1
| |
| | |
| | |
| ''Where:''
| |
| | |
| z<sub>n</sub> = the zero coupon rate for maturity n periods
| |
| | |
| p<sub>n</sub> = the par rate for maturity n periods, starting now
| |
| | |
| CumDF<sub>n-1</sub> = the total of the discount factors for maturities 1 to 'n-1' periods, calculated from the zero coupon rates (z<sub>1</sub> to z<sub>n-1</sub>)
| |
| | |
| | |
| ''Applying the formula:''
| |
| | |
| | |
| z<sub>n</sub> = ( (1 + p<sub>n</sub>) / (1 - p<sub>n</sub> x CumDF<sub>n-1</sub>) )<sup>(1/n)</sup> - 1
| |
| | |
| z<sub>2</sub> = ( (1 + p<sub>2</sub>) / (1 - p<sub>2</sub> x CumDF<sub>1</sub>) )<sup>(1/2)</sup> - 1
| |
| | |
| z<sub>2</sub> = ( (1 + 0.029803) / (1 - 0.029803 x DF<sub>1</sub>) )<sup>(1/2)</sup> - 1
| |
| | |
| z<sub>2</sub> = ( 1.029803 / (1 - (0.029803 x 1.02<sup>-1</sup>) )<sup>(1/2)</sup> - 1
| |
| | |
| z<sub>2</sub> = ( 1.029803 / (1 - 0.0292186)<sup>(1/2)</sup> - 1
| |
| | |
| z<sub>2</sub> = 1.0608<sup>(1/2)</sup> - 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.
| |
| | |
| | |
| <span style="color:#4B0082">'''Example 2: Converting from zero coupon rates to forward rates'''</span>
| |
| | |
| 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<sub>1-2</sub>''' is set out below:
| |
| | |
| f<sub>1-2</sub> = ( 1.029951<sup>2</sup> / 1.02 ) - 1
| |
| | |
| = 0.04
| |
| | |
| = 4% per period.
| |
| | |
| | |
| This calculation is explained in more detail on the page [[Converting from zero coupon rates]].
| |
|
| |
|
|
| |
|
| == See also == | | == See also == |
| * [[Par yield]] | | * [[Inter quartile range]] |
| * [[Bond]] | | * [[Quartile deviation]] |
| * [[Bootstrap]] | | * [[Upper quartile]] |
| * [[Coupon bond]]
| |
| * [[Forward yield]]
| |
| * [[Market yield]]
| |
| * [[Par]]
| |
| * [[Swap spread]]
| |
| * [[Yield curve]]
| |
| * [[Zero coupon yield]]
| |
| * [[Flat yield curve]]
| |
| * [[Rising yield curve]]
| |
| * [[Falling yield curve]]
| |
| * [[Positive yield curve]]
| |
| * [[Negative yield curve]]
| |
| * [[Converting from zero coupon rates]]
| |
| * [[Converting from forward rates]]
| |
