imported>Doug Williamson |
imported>Doug Williamson |
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| '''Example 1: Cash flows from 3-period zero coupon instrument''' | | <span style="color:#4B0082">'''Example: Cash flows from 3-period zero coupon instrument'''</span> |
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| 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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| 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. | | 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. |
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| 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. |
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| * '[[Outright]]' from a two-year zero coupon investment
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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
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| * Or using a par investment, reinvesting intermediate interest to generate a total terminal cash flow
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| '''Example 2: Converting two-period zero coupon yields to forward yields'''
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| Periodic zero coupon yields ('''z''') are:
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| z<sub>0-1</sub> = 0.02 per period (2%)
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| z<sub>0-2</sub> = 0.029951 per period (2.9951%)
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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:
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| £1m x 1.029951<sup>2</sup>
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| = £'''1.0608m'''
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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:
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| £1m x (1 + z<sub>0-2</sub>)<sup>2</sup> = £'''1.0608m'''
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| Using this information, we can now calculate the zero coupon yield for two periods' maturity.
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| (1 + z<sub>0-2</sub>)<sup>2</sup> = 1.0608
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| 1 + z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup>
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| z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup> - 1
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| = '''0.029951''' per period (= 2.9951%)
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| 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.
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| 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.
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| Investing the same £1m in a two-periods maturity zero coupon instrument at a rate of 2.9951% per period would return:
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| £1m x (1.029951)<sup>2</sup>
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| = £'''1.0608m'''
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| ''This is the same result as enjoyed from the forward investments, as expected.'''''Example 3'''
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| Now using the zero coupon rates ('''z'''), the par rates ('''p''') can also be calculated in turn.
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| The periodic zero coupon yields ('''z''') are:
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| z<sub>0-1</sub> = 0.02 per period (2%)
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| z<sub>0-2</sub> = 0.029951 per period (2.9951%)
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| The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:
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| p<sub>0-n</sub> = (1 - DF<sub>n</sub>) / CumDF<sub>n</sub>
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| ''Where:''
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| p<sub>0-n</sub> = the par rate for maturity n periods, starting now
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| DF<sub>n</sub> = the discount factor for 'n' periods maturity, calculated from the zero coupon rate (z<sub>n</sub>)
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| 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>)
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| ''Applying the formula:''
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| p<sub>0-2</sub> = (1 - DF<sub>2</sub>) / CumDF<sub>2</sub>
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| p<sub>0-2</sub> = (1 - 1.029951<sup>-2</sup>) / (1.02<sup>-1</sup> + 1.029951<sup>-2</sup>)
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| = 0.029803 (= 2.9803% per period)
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| This is the theoretical fair (no-arbitrage) market price for the par instrument.
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| 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.
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| In theory, such an investment - again of £1m - should produce the same terminal cash flow. Let's see.
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| Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, assuming an initial investment of £1m:
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| Interest coupon at Time 1 period = £1m x 0.029803 = £<u>0.029803</u>m
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| Principal + interest at Time 2 periods = £1m + 0.029803m = £'''1.029803'''m
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| 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.
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| So the Time 2 proceeds from the reinvested coupon received at Time 1 are:
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| £0.029803 x 1.04
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| = £'''0.030995'''m at Time 2
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| The total terminal value at Time 2 periods is:
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| 0.030995 + 1.029803
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| = £'''1.0608m''' (as before)
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| | This is illustrated on the page [[Converting from zero coupon rates]]. |
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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]]
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| * [[Rising yield curve]]
| | [[Category:Financial_management]] |
| * [[Falling yield curve]]
| | [[Category:Corporate_finance]] |
| * [[Positive yield curve]]
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| * [[Negative yield curve]]
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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 zero coupon yield is equal to the current market rate of return on investments in zero coupon bonds of the same maturity.
Example: Cash flows from 3-period zero coupon instrument
The zero coupon yield for the maturity 0-3 periods is 2% per period.
This means that a deposit of £1,000,000 at Time 0 periods on these terms would return:
£1,000,000 x 1.023
= £1,061,208 at Time 3 periods.
(No intermediate interest is payable.)
An application of zero coupon yields is the pricing of zero coupon bonds.
The zero coupon yield is also known as the Zero coupon rate, spot rate, or spot yield.
Conversion
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 is illustrated on the page Converting from zero coupon rates.
See also
