Converting from par rates - Revision history

imported>Doug Williamson: Categorise.

2018-02-21T12:42:24Z

Categorise.

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	* [[Converting from zero coupon rates]]		* [[Converting from zero coupon rates]]
	* [[Converting from forward rates]]		* [[Converting from forward rates]]

			[[Category:Corporate_financial_management]]

imported>Doug Williamson: Standardise headings with examples

2015-12-02T12:55:10Z

Standardise headings with examples

← Older revision		Revision as of 12:55, 2 December 2015
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	~~'''~~Conversion~~'''~~		==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.		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.

imported>Doug Williamson: Typo. Add previously missing close bracket.

2015-11-16T13:13:57Z

Typo. Add previously missing close bracket.

← Older revision		Revision as of 13:13, 16 November 2015
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	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 - (0.029803 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> = ( 1.029803 / (1 - 0.0292186) )<sup>(1/2)</sup> - 1

	z<sub>2</sub> = 1.0608<sup>(1/2)</sup> - 1		z<sub>2</sub> = 1.0608<sup>(1/2)</sup> - 1

imported>Doug Williamson: Send the link to the fuller related page Bootstrap.

2015-11-16T08:49:16Z

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← Older revision		Revision as of 08:49, 16 November 2015
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	(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 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 '[[~~bootstrapping~~]]', because the results of calculating each earlier zero coupon rate are used successively to calculate the next-longer maturity zero coupon rate.		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.

imported>Doug Williamson: Identify the calculation explained on linked page.

2015-11-15T21:53:32Z

Identify the calculation explained on linked page.

← Older revision		Revision as of 21:53, 15 November 2015
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	~~This~~ calculation is explained in more detail on the page [[Converting from zero coupon rates]].		The calculation of forward rates from zero coupon rates is explained in more detail on the page [[Converting from zero coupon rates]].

imported>Doug Williamson: Spacing.

2015-11-15T20:45:55Z

Spacing.

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	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.		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.

imported>Doug Williamson: Explain 'bootstrapping'.

2015-11-15T20:44:39Z

Explain 'bootstrapping'.

← Older revision		Revision as of 20:44, 15 November 2015
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	The par rate is equal to the fixed coupon rate payable on a ‘[[par bond]]’.		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.		The par yield is known as the Par rate, Swap rate or Swap yield.
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	(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 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 '[[bootstrapping]]', because the results of calculating each earlier zero coupon rate are used successively to calculate the next-longer maturity zero coupon rate.

imported>Doug Williamson: Tidy up.

2015-11-15T11:14:44Z

Tidy up.

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	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.		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.)

imported>Doug Williamson: Tidy up.

2015-11-15T11:08:41Z

Tidy up.

← Older revision		Revision as of 11:08, 15 November 2015
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	The periodic par yields ('''p''') are:		The periodic par yields ('''p''') are:

	p<sub>0-1</sub> = 0.02 per period (2%)		p<sub>1</sub> = 0.02 per period (2%)

	p<sub>0-2</sub> = 0.029803 per period (2.9803%)		p<sub>2</sub> = 0.029803 per period (2.9803%)


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	p<sub>n</sub> = the par rate for maturity n periods, starting now		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 ~~maturity~~, calculated from the zero coupon rates (z<sub>1</sub> to z<sub>n-1</sub>)		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>)


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	It is the ~~calculated~~ 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 be repaid at Time 2.		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.

imported>Doug Williamson: Create the page.

2015-11-15T11:05:06Z

Create the page.

New page

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:

p0-1 = 0.02 per period (2%)

p0-2 = 0.029803 per period (2.9803%)

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

zn = ( (1 + pn) / (1 - pn x CumDFn-1) )(1/n) - 1

''Where:''

zn = the zero coupon rate for maturity n periods

pn = the par rate for maturity n periods, starting now

CumDFn-1 = the total of the discount factors for maturities 1 to 'n-1' periods maturity, calculated from the zero coupon rates (z1 to zn-1)

''Applying the formula:''

zn = ( (1 + pn) / (1 - pn x CumDFn-1) )(1/n) - 1

z2 = ( (1 + p2) / (1 - p2 x CumDF1) )(1/2) - 1

z2 = ( (1 + 0.029803) / (1 - 0.029803 x DF1) )(1/2) - 1

z2 = ( 1.029803 / (1 - (0.029803 x 1.02-1) )(1/2) - 1

z2 = ( 1.029803 / (1 - 0.0292186)(1/2) - 1

z2 = 1.0608(1/2) - 1

= 0.029951 (= 2.9951% per period)

It is the calculated 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 be repaid at Time 2.

'''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 '''f1-2''' is set out below:

f1-2 = ( 1.0299512 / 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 ==
* [[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]]