Converting from par rates: Difference between revisions
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The periodic par yields ('''p''') are: | The periodic par yields ('''p''') are: | ||
p<sub> | p<sub>1</sub> = 0.02 per period (2%) | ||
p<sub> | 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 | 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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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. | |||
Revision as of 11:08, 15 November 2015
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:
p1 = 0.02 per period (2%)
p2 = 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, 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)
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.
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.
