Converting from par rates and Loss relief: Difference between pages

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


The par yield is known as the Par rate, Swap rate or Swap yield.
Tax benefits to a taxpayer arising from having made losses.  


For example, an entitlement to offset tax losses against other taxable profits, to reduce the amount of tax payable.


'''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.
== See also ==
 
* [[Tax relief]]
 
* [[Terminal loss relief]]
<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.
 
 
(The one period par rate p<sub>0-1</sub> represents the identical deal to the one period zero coupon rate z<sub>0-1</sub>.  For this reason the rate is also identical = 2% per period.)
 
The conversion of par rates to zero coupon rates is sometimes known as '[[bootstrap]]ping', because the results of calculating each earlier zero coupon rate are used successively to calculate the next-longer maturity zero coupon rate.
 
 
 
<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
[[Category:Accounting,_tax_and_regulation]]
 
= 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]].
 
 
== See also ==
* [[Par yield]]
* [[Bond]]
* [[Bootstrap]]
* [[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]]

Revision as of 13:19, 10 March 2022

Tax.

Tax benefits to a taxpayer arising from having made losses.

For example, an entitlement to offset tax losses against other taxable profits, to reduce the amount of tax payable.


See also