Zero coupon yield - Revision history

imported>Doug Williamson: Links ordering.

2022-07-01T20:35:54Z

Links ordering.

← Older revision		Revision as of 20:35, 1 July 2022
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	== See also ==		== See also ==
	* [[Bootstrap]]		* [[Bootstrap]]
			* [[Converting from zero coupon rates]]
			* [[Coupon]]
			* [[Falling yield curve]]
			* [[Flat yield curve]]
	* [[Forward yield]]		* [[Forward yield]]
			* [[Negative yield curve]]
			* [[No arbitrage conditions]]
	* [[Par yield]]		* [[Par yield]]
	* [[~~Coupon~~]]		* [[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]]
	* [[Rising yield curve]]
	* [[Falling yield curve]]
	* [[Positive yield curve]]
	* [[Negative yield curve]]
	* [[Converting from zero coupon rates]]

	[[Category:Financial_management]]		[[Category:Financial_management]]
	[[Category:Corporate_finance]]		[[Category:Corporate_finance]]

imported>Doug Williamson: Categorise.

2018-02-22T12:25:14Z

Categorise.

← Older revision		Revision as of 12:25, 22 February 2018
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	* [[Negative yield curve]]		* [[Negative yield curve]]
	* [[Converting from zero coupon rates]]		* [[Converting from zero coupon rates]]

			[[Category:Financial_management]]
			[[Category:Corporate_finance]]

imported>Doug Williamson: Remove the material transferred to the new page Converting from zero coupon rates.

2015-11-15T09:06:02Z

Remove the material transferred to the new page Converting from zero coupon rates.

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-<span style="color:#4B0082">'''Example 1: Cash flows from 3-period zero coupon instrument'''</span>
+<span style="color:#4B0082">'''Example: Cash flows from 3-period zero coupon instrument'''</span>
 The zero coupon yield for the maturity 0-3 periods is 2% per period.
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 If we know the zero coupon yield, we can calculate both the [[forward yield]] and the [[par yield]] for the same maturities and risk class.
-The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves. This means that the total cumulative cash flows from a two-year investment must be identical, whether the investment is built:
+The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves.

imported>Doug Williamson at 08:12, 15 November 2015

2015-11-15T08:12:44Z

← Older revision		Revision as of 08:12, 15 November 2015
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	* [[Positive yield curve]]		* [[Positive yield curve]]
	* [[Negative yield curve]]		* [[Negative yield curve]]
			* [[Converting from zero coupon rates]]

imported>Doug Williamson: Colour change of example headers

2015-11-13T15:25:48Z

Colour change of example headers

← Older revision		Revision as of 15:25, 13 November 2015
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	'''Example 1: Cash flows from 3-period zero coupon instrument'''		<span style="color:#4B0082">'''Example 1: Cash flows from 3-period zero coupon instrument'''</span>

	The zero coupon yield for the maturity 0-3 periods is 2% per period.		The zero coupon yield for the maturity 0-3 periods is 2% per period.
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			<span style="color:#4B0082">'''Example 2: Converting two-period zero coupon yields to forward yields'''</span>
	'''Example 2: Converting two-period zero coupon yields to forward yields'''

	Periodic zero coupon yields ('''z''') are:		Periodic zero coupon yields ('''z''') are:
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	Under no-~~abitrage~~ pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of one periods 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 periods 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>) = £'''1.0608m'''

imported>Doug Williamson: Add new example.

2015-11-13T15:11:57Z

Add new example.

← Older revision		Revision as of 15:11, 13 November 2015
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	The cash returned at Time 2 periods in the future, from investing £1m at Time 0 in a zero coupon instrument at a rate of 2.9951% per period is:		The cash returned at Time 2 periods in the future, from investing £1m at Time 0 in a zero coupon instrument at a rate of 2.9951% per period, is:

	£1m x 1.029951<sup>2</sup>		£1m x 1.029951<sup>2</sup>
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	Under no-abitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of ~~two~~ periods maturity, at the market yield of '''z<sub>0-2</sub>''' per period, as follows:		Under no-abitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of one periods 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>0-2</sub>)<~~sup~~>2</~~sup~~> = £'''1.0608m'''		£1m x (1 + z<sub>0-1</sub>) x (1 + f<sub>1-2</sub>) = £'''1.0608m'''


	Using this information, we can now calculate the ~~zero coupon~~ yield for ~~two~~ periods' maturity.		Using this information, we can now calculate the forward yield for 1-2 periods' maturity.

			1.02 x (1 + f<sub>1-2</sub>) = 1.0608

	(1 + z<sub>0-2</sub~~>)<sup>2</sup~~> = 1.0608		1 + f<sub>1-2</sub> = 1.0608 / 1.02

	~~1 + z~~<sub>0-2</sub> = 1.0608~~<sup>(~~1/2)~~</sup>~~		f<sub>1-2</sub> = (1.0608 / 1.02) - 1

	~~z<sub>0-2</sub>~~ = 1.~~0608<sup>(1/2)</sup>~~ - 1		= 1.04 - 1

	= '''0.~~029951~~''' per period (= ~~2.9951~~%)		= '''0.04''' per period (= 4%)


	This is the market ~~zero coupon~~ rate which we would enjoy if we were to make a ~~two~~-~~year~~ deposit, ~~putting~~ our money into the deposit ~~today~~.		This is the market forward rate which we would enjoy if we were to pre-agree today, to make a one-period deposit, committing ourselves to put our money into the deposit one period in the future.


	The no-arbitrage relationship says that making such a deposit should produce the identical terminal cash flow of £1.0608m. Let's see if that's borne out by our calculations.		The no-arbitrage relationship says that making such a synthetic deposit should produce the identical terminal cash flow of £1.0608m. Let's see if that's borne out by our calculations.


	Investing the same £1m in a two-periods maturity zero coupon instrument ~~at a rate of 2.9951% per period~~ would return:		Investing the same £1m in this synthetic two-periods maturity zero coupon instrument would return:

	£1m x (1.~~029951)<sup>2</sup>~~		After one period: £1m x 1.02 = £1.02m

	~~= £'''1~~.~~0608m'''~~		Reinvested for the second period at the pre-agreed rate of 0.04 per period for one more period:

			= £1.02m x 1.04

	~~''This is the same result as enjoyed from the forward investments, as expected.'''''Example 3'''~~		= £'''1.0608m'''

	~~Now using the zero coupon rates ('''z'''), the par rates ('''p''') can also be calculated in turn.~~


	~~The periodic zero coupon yields ('''z''') are:~~

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

	~~z<sub>0-2</sub> = 0.029951 per period (2.9951%)~~


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

	~~p<sub>0-n</sub> = (1 - DF<sub>n</sub>) / CumDF<sub>n</sub>~~


	~~''Where:''~~

	~~p<sub>0-n</sub> = the par rate for maturity n periods, starting now~~

	~~DF<sub>n</sub> = the discount factor for 'n' periods maturity, calculated from the zero coupon rate (z<sub>n</sub>)~~

	~~CumDF<sub>n</sub> = the total of the discount factors for maturities 1 to 'n' periods maturity, again calculated from the zero coupon rates (z<sub>1</sub> to z<sub>n</sub>)~~


	~~''Applying the formula:''~~

	~~p<sub>0-2</sub> = (1 - DF<sub>2</sub>) / CumDF<sub>2</sub>~~

	~~p<sub>0-2</sub> = (1 - 1.029951<sup>-2</sup>) / (1.02<sup>-1</sup> + 1.029951<sup>-2</sup>)~~

	~~= 0.029803 (= 2.9803% per period)~~


	~~This is the theoretical fair (no-arbitrage) market price for the par instrument.~~

	It is the calculated rate of interest payable on a two-year investment on par rate terms. This means that 2.9803% interest will be paid on the amount of the original investment, at Times 1 and 2 periods. In addition, the original investment will be repaid at Time 2.


	~~In theory, such an investment - again of £1m - should produce the same terminal cash flow. Let's see.~~


	~~Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, assuming an initial investment of £1m:~~


	~~Interest coupon at Time 1 period = £1m x 0.029803 = £<u>0.029803</u>m~~

	~~Principal + interest at Time 2 periods = £1m + 0.029803m~~ = £'''1.~~029803~~'''m


	~~The coupon receivable at Time 1 period is reinvested at the pre-agreed forward rate of 4% (0.04) for the maturity 1-2 periods.~~

	~~So the Time 2 proceeds from the reinvested coupon received at Time 1 are:~~

	~~£0.029803 x 1.04~~

	~~= £'''0.030995'''m at Time 2~~


	~~The total terminal value at Time 2 periods is:~~

	~~0.030995 + 1.029803~~

	~~= £'''1.0608m''' (as before)~~

			''This is the same result as enjoyed from the outright zero coupon investment, as expected.

imported>Doug Williamson: Punctuation.

2015-11-13T14:57:06Z

Punctuation.

← Older revision		Revision as of 14:57, 13 November 2015
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	The rate of return on an investment today, for a single cashflow at the final maturity of the instrument. No intermediate interest is payable or receivable. There are no interest coupons, hence the name 'zero coupon'.		The rate of return on an investment today, for a single cashflow at the final maturity of the instrument. No intermediate interest is payable or receivable. (There are no interest coupons, hence the name 'zero coupon'.)

	The zero coupon yield is equal to the current market rate of return on investments in zero coupon bonds of the same maturity.		The zero coupon yield is equal to the current market rate of return on investments in zero coupon bonds of the same maturity.

imported>Doug Williamson: Link with Coupon page.

2015-11-13T14:56:48Z

Link with Coupon page.

← Older revision		Revision as of 14:56, 13 November 2015
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	The rate of return on an investment today, for a single cashflow at the final maturity of the instrument. No intermediate interest is payable or receivable.		The rate of return on an investment today, for a single cashflow at the final maturity of the instrument. No intermediate interest is payable or receivable. There are no interest coupons, hence the name 'zero coupon'.

	The zero coupon yield is equal to the current market rate of return on investments in zero coupon bonds of the same maturity.		The zero coupon yield is equal to the current market rate of return on investments in zero coupon bonds of the same maturity.
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	The cash returned at Time 2 periods in the future, from investing £1m at Time 0 in ~~the~~ zero coupon instrument at a rate of 2.9951% per period is:		The cash returned at Time 2 periods in the future, from investing £1m at Time 0 in a zero coupon instrument at a rate of 2.9951% per period is:

	£1m x 1.029951<sup>2</sup>		£1m x 1.029951<sup>2</sup>
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	* [[Forward yield]]		* [[Forward yield]]
	* [[Par yield]]		* [[Par yield]]
			* [[Coupon]]
	* [[Spot rate]]		* [[Spot rate]]
	* [[Yield curve]]		* [[Yield curve]]

imported>Doug Williamson: Improve run-in.

2015-11-13T14:34:22Z

Improve run-in.

← Older revision		Revision as of 14:34, 13 November 2015
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	The rate of return on an investment today, for a single cashflow at maturity of the instrument.		The rate of return on an investment today, for a single cashflow at the final maturity of the instrument. No intermediate interest is payable or receivable.

	~~Equal~~ to the current market rate of return on zero coupon bonds of the same maturity.		The zero coupon yield is equal to the current market rate of return on investments in zero coupon bonds of the same maturity.

imported>Doug Williamson: Add more examples.

2015-11-13T13:13:49Z

Add more examples.

@@ Line 4: / Line 4: @@
-'''Example'''
+'''Example 1: Cash flows from 3-period zero coupon instrument'''
 The zero coupon yield for the maturity 0-3 periods is 2% per period.
@@ Line 25: / Line 25: @@
 '''Conversion'''
-If we know the zero coupon yield, we can calculate both the [[par yield]] and the [[forward yield]] for the same maturities and risk class.
+If we know the zero coupon yield, we can calculate both the [[forward yield]] and the [[par yield]] for the same maturities and risk class.