imported>Doug Williamson |
imported>Doug Williamson |
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| The zero coupon rate is also known as the [[zero coupon yield]], spot rate, or spot yield.
| | #''UK.'' A corporation that is limited by shares. |
| | | #Similar arrangements in other countries. |
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| '''Conversion''' | |
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| 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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| The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves.
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| This means - for example - that the total cumulative cash flows from a two-year investment must be identical, whether the investment is built:
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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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| <span style="color:#4B0082">'''Example 1: Converting two-period zero coupon yields to forward yields'''</span>
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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-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:
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| £1m x (1 + z<sub>0-1</sub>) x (1 + f<sub>1-2</sub>) = £'''1.0608m'''
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| Using this information, we can now calculate the forward yield for 1-2 periods' maturity.
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| 1.02 x (1 + f<sub>1-2</sub>) = 1.0608
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| 1 + f<sub>1-2</sub> = 1.0608 / 1.02
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| f<sub>1-2</sub> = (1.0608 / 1.02) - 1
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| = 1.04 - 1
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| = '''0.04''' per period (= 4%)
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| 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.
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| 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.
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| Investing the same £1m in this synthetic two-periods maturity zero coupon instrument would return:
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| After one period: £1m x 1.02 = £1.02m
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| Reinvested for the second period at the pre-agreed rate of 0.04 per period for one more period:
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| = £1.02m x 1.04
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| = £'''1.0608m'''
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| ''This is the same result as enjoyed from the outright zero coupon investment, as expected.
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| == See also == | | == See also == |
| * [[Zero coupon yield]] | | * [[AG]] |
| * [[Bootstrap]] | | * [[Company]] |
| * [[Forward yield]] | | * [[GmbH]] |
| * [[Par yield]] | | * [[Plc]] |
| * [[Coupon]] | | * [[SA]] |
| * [[Spot rate]] | | * [[Limited company]] |
| * [[Yield curve]]
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| * [[Zero]]
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| * [[Zero coupon bond]]
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| * [[Flat yield curve]]
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| * [[Rising yield curve]]
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| * [[Falling yield curve]]
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| * [[Positive yield curve]]
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| * [[Negative yield curve]]
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