Converting from zero coupon rates - Revision history

imported>Doug Williamson: Links ordering.

2022-07-01T20:42:47Z

Links ordering.

← Older revision		Revision as of 20:42, 1 July 2022
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	== See also ==		== See also ==
	* [[~~Zero coupon yield]]~~		* [[Bootstrap]]
	* [[Forward yield]]
	* [[Converting from forward rates]]		* [[Converting from forward rates]]
	* [[Par yield]]
	* [[Converting from par rates]]		* [[Converting from par rates]]
	* [[~~Bootstrap~~]]		* [[Coupon]]
			* [[Falling yield curve]]
			* [[Flat yield curve]]
			* [[Forward yield]]
			* [[Negative yield curve]]
			* [[No arbitrage conditions]]
	* [[Par bond]]		* [[Par bond]]
	* [[~~Coupon~~]]		* [[Par yield]]
			* [[Positive yield curve]]
			* [[Rising yield curve]]
	* [[Spot rate]]		* [[Spot rate]]
	* [[Yield curve]]		* [[Yield curve]]
	* [[Zero]]		* [[Zero]]
	* [[Zero coupon bond]]		* [[Zero coupon bond]]
	* [[~~Flat~~ yield ~~curve]]~~		* [[Zero coupon yield]]
	* [[Rising yield curve]]
	* [[Falling yield curve]]
	* [[Positive yield curve]]
	* [[Negative yield curve]]


	==Other ~~resources~~==		==Other resource==
	[[Media:2013_09_Sept_-_Simple_solutions.pdf\| The Treasurer students, Simple solutions]]		[[Media:2013_09_Sept_-_Simple_solutions.pdf\| The Treasurer students, Simple solutions]]

	[[Category:Financial_management]]		[[Category:Financial_management]]
	[[Category:Corporate_financial_management]]		[[Category:Corporate_financial_management]]

imported>Doug Williamson: Amend layout.

2021-03-25T16:30:40Z

Amend layout.

← Older revision		Revision as of 16:30, 25 March 2021
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	The zero coupon rate is also known as the [[zero coupon yield]], spot rate, or spot yield.		The zero coupon rate is also known as the [[zero coupon yield]], spot rate, or spot yield.



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	* Or as a [[synthetic]] deposit built using a forward contract, reinvesting intermediate principal and interest proceeds at a pre-agreed rate		* Or as a [[synthetic]] deposit built using a forward contract, reinvesting intermediate principal and interest proceeds at a pre-agreed rate
	* Or using a par investment, reinvesting intermediate interest to generate a total terminal cash flow		* Or using a par investment, reinvesting intermediate interest to generate a total terminal cash flow


	==Zero coupon rate to forward rate conversion==		==Zero coupon rate to forward rate conversion==
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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.)


	==Terminal cash is still the same==		==Terminal cash is still the same==
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	The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.		The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.


	==Present value is the same too==		==Present value is the same too==

imported>Doug Williamson: Categorise.

2018-02-22T12:09:10Z

Categorise.

← Older revision		Revision as of 12:09, 22 February 2018
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	==Other resources==		==Other resources==
	[[Media:2013_09_Sept_-_Simple_solutions.pdf\| The Treasurer students, Simple solutions]]		[[Media:2013_09_Sept_-_Simple_solutions.pdf\| The Treasurer students, Simple solutions]]

			[[Category:Financial_management]]
			[[Category:Corporate_financial_management]]

imported>Doug Williamson: Align headers.

2015-12-02T19:41:27Z

Align headers.

← Older revision		Revision as of 19:41, 2 December 2015
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	==~~Conversion~~==		==No arbitrage conversion principles==

	If we know the zero coupon rates (yield curve) for a given risk class and set of maturities, we can calculate both the [[forward yield]]s and the [[par yield]]s for the same maturities and risk class.		If we know the zero coupon rates (yield curve) for a given risk class and set of maturities, we can calculate both the [[forward yield]]s and the [[par yield]]s for the same maturities and risk class.
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	* Or as a [[synthetic]] deposit built using a forward contract, reinvesting intermediate principal and interest proceeds at a pre-agreed rate		* Or as a [[synthetic]] deposit built using a forward contract, reinvesting intermediate principal and interest proceeds at a pre-agreed rate
	* Or using a par investment, reinvesting intermediate interest to generate a total terminal cash flow		* Or using a par investment, reinvesting intermediate interest to generate a total terminal cash flow

			==Zero coupon rate to forward rate conversion==


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	(The one period forward rate f<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 forward rate f<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.)


			==Zero coupon rates to par rates conversion==

	<span style="color:#4B0082">'''Example 2: Converting from zero coupon rates to par rates'''</span>		<span style="color:#4B0082">'''Example 2: Converting from zero coupon rates to par rates'''</span>
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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.)

	==Terminal cash is ~~indeed~~ still the same==		==Terminal cash is still the same==

	In theory, an investment of £1m in a par instrument should produce the same terminal cash flow as a zero coupon instrument or forward instruments. On these figures (forward rates of 2% and 4% for 0-1 and 1-2 periods maturity, respectively) we saw that was £1.0608m for an original investment of £1m.		In theory, an investment of £1m in a par instrument should produce the same terminal cash flow as a zero coupon instrument or forward instruments. On these figures (forward rates of 2% and 4% for 0-1 and 1-2 periods maturity, respectively) we saw that was £1.0608m for an original investment of £1m.
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	The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.		The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.

	==Present value is ~~also~~ the same==		==Present value is the same too==

	No-arbitrage pricing also says that the present value of the par instrument should be the same as the present value of the zero coupon and forward instruments.		No-arbitrage pricing also says that the present value of the par instrument should be the same as the present value of the zero coupon and forward instruments.

imported>Doug Williamson: Expand example heading.

2015-12-02T19:31:45Z

Expand example heading.

← Older revision		Revision as of 19:31, 2 December 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.)

	==Terminal cash is the same==		==Terminal cash is indeed still the same==

	In theory, an investment of £1m in a par instrument should produce the same terminal cash flow as a zero coupon instrument or forward instruments. On these figures (forward rates of 2% and 4% for 0-1 and 1-2 periods maturity, respectively) we saw that was £1.0608m for an original investment of £1m.		In theory, an investment of £1m in a par instrument should produce the same terminal cash flow as a zero coupon instrument or forward instruments. On these figures (forward rates of 2% and 4% for 0-1 and 1-2 periods maturity, respectively) we saw that was £1.0608m for an original investment of £1m.
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	The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.		The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.


	==Present value is also the same==		==Present value is also the same==

imported>Doug Williamson: Simplify heading title.

2015-12-02T19:30:34Z

Simplify heading title.

← Older revision		Revision as of 19:30, 2 December 2015
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	==Is ~~the~~ terminal cash the same in each case?==		==Is terminal cash the same in each case?==

	The no-arbitrage relationship says that making such a synthetic deposit should produce the identical terminal cash flow of £1.0608m.		The no-arbitrage relationship says that making such a synthetic deposit should produce the identical terminal cash flow of £1.0608m.
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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.)


	==Terminal cash is the same==		==Terminal cash is the same==

imported>Doug Williamson: Standardise headings with examples

2015-12-02T13:00:18Z

Standardise headings with examples

← Older revision		Revision as of 13:00, 2 December 2015
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	~~'''~~Conversion~~'''~~		==Conversion==

	If we know the zero coupon rates (yield curve) for a given risk class and set of maturities, we can calculate both the [[forward yield]]s and the [[par yield]]s for the same maturities and risk class.		If we know the zero coupon rates (yield curve) for a given risk class and set of maturities, we can calculate both the [[forward yield]]s and the [[par yield]]s for the same maturities and risk class.
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	~~'''~~Is the terminal cash the same in each case?~~'''~~		==Is the terminal cash the same in each case?==

	The no-arbitrage relationship says that making such a synthetic deposit should produce the identical terminal cash flow of £1.0608m.		The no-arbitrage relationship says that making such a synthetic deposit should produce the identical terminal cash flow of £1.0608m.
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	~~'''~~Terminal cash is the same~~'''~~		==Terminal cash is the same==

	In theory, an investment of £1m in a par instrument should produce the same terminal cash flow as a zero coupon instrument or forward instruments. On these figures (forward rates of 2% and 4% for 0-1 and 1-2 periods maturity, respectively) we saw that was £1.0608m for an original investment of £1m.		In theory, an investment of £1m in a par instrument should produce the same terminal cash flow as a zero coupon instrument or forward instruments. On these figures (forward rates of 2% and 4% for 0-1 and 1-2 periods maturity, respectively) we saw that was £1.0608m for an original investment of £1m.
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	~~'''~~Present value is also the same~~'''~~		==Present value is also the same==

	No-arbitrage pricing also says that the present value of the par instrument should be the same as the present value of the zero coupon and forward instruments.		No-arbitrage pricing also says that the present value of the par instrument should be the same as the present value of the zero coupon and forward instruments.
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	===Other resources===		==Other resources==
	[[Media:2013_09_Sept_-_Simple_solutions.pdf\| The Treasurer students, Simple solutions]]		[[Media:2013_09_Sept_-_Simple_solutions.pdf\| The Treasurer students, Simple solutions]]

imported>Doug Williamson: Link with The Treasurer.

2015-11-22T22:13:18Z

Link with The Treasurer.

← Older revision		Revision as of 22:13, 22 November 2015
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	* [[Positive yield curve]]		* [[Positive yield curve]]
	* [[Negative yield curve]]		* [[Negative yield curve]]


			===Other resources===
			[[Media:2013_09_Sept_-_Simple_solutions.pdf\| The Treasurer students, Simple solutions]]

imported>Doug Williamson: Rationalise number and unit layout

2015-11-20T10:12:53Z

Rationalise number and unit layout

← Older revision		Revision as of 10:12, 20 November 2015
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	£1m x 1.029951<sup>2</sup>		£1m x 1.029951<sup>2</sup>

	= £'''1.~~0608m~~'''		= £'''1.0608'''m


	Under no-arbitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of one period's maturity, together with a forward contract for the second period - for reinvestment at the forward market yield of '''f<sub>1-2</sub>''' per period, as follows:		Under no-arbitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of one period's maturity, together with a forward contract for the second period - for reinvestment at the forward market yield of '''f<sub>1-2</sub>''' per period, as follows:

	£1m x (1 + z<sub>0-1</sub>) x (1 + f<sub>1-2</sub>) = £'''1.~~0608m~~'''		£1m x (1 + z<sub>0-1</sub>) x (1 + f<sub>1-2</sub>) = £'''1.0608'''m


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	= £1.02m x 1.04		= £1.02m x 1.04

	= £'''1.~~0608m~~'''		= £'''1.0608'''m


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	0.030995 + 1.029803		0.030995 + 1.029803

	= £'''1.~~0608m~~''' (as before)		= £'''1.0608'''m (as before)


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	£1.0608m x 1.029951<sup>-2</sup>		£1.0608m x 1.029951<sup>-2</sup>

	= £'''1.~~0000m~~'''		= £'''1.0000'''m


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	Time 2: £1.029803 x 1.029951<sup>-2</sup> = 0.97078		Time 2: £1.029803 x 1.029951<sup>-2</sup> = 0.97078

	Total = £'''1.~~0000m~~'''		Total = £'''1.0000'''m

imported>Doug Williamson: Round intermediate workings appropriately.

2015-11-15T11:20:24Z

Round intermediate workings appropriately.

← Older revision		Revision as of 11:20, 15 November 2015
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	The present values of these cash flows are:		The present values of these cash flows are:

	Time 1: £0.029803 x 1.02<sup>-1</sup> = 0.~~02921862745~~		Time 1: £0.029803 x 1.02<sup>-1</sup> = 0.02922

	Time 2: £1.029803 x 1.029951<sup>-2</sup> = 0.~~09707804583~~		Time 2: £1.029803 x 1.029951<sup>-2</sup> = 0.97078

	Total = £'''1.0000m'''		Total = £'''1.0000m'''