Could anyone please explain why Par Curve has the flattest shape of 3 curves : par curve, spot curve and forward curve?
It is really just mathematical.
If you have spots
1yr 5%
2yr 10%
When we calculate the 2 year par rate it is going to between the 2 but closer to the furthers value, ie. 10%
Assumingf $1 par
1 = C/(1 +s1) + (C + 1)/(1+s2)^2
The par value is going to unfluence the weighting much more than the coupons (unless we have exceptionally high coupon rates)
You still have some weightin due to the coupon at year 1 so par rate < year2 spot
But Coupon and Par at year 2 has much more weight in the calculation so par > spot 1
Don’t go any further unless you like algebra
If we re-arrange the formula above
C = [1 - 1/(1 + s2)^2] / [(1+s1) + (1+s2)^2)
Where C is the par rate
Assume initially a flat curve we get s1 = s2 = par
As s2 gets larger (assuem s1 stays the same) the numerator gets larger and denominator smaller increase the par rate. But the effect on the C is not the same magnitude as the the chnage in S2 as the numerator includes 1/(1+s1) giving a weighting the shorter/lower spot rate.
What we will also notice is that as the spot curve gets steeper relatively speaking the par curve gets faltter. This is the par value is discounted at a higher and higher rate it has less weight in the calculation.
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Thank you so much
The spot curve is something of a moving average of the forward curve. Averages are less volatile than the underlying data.
The par curve is something of a moving average of the spot curve. Averages are still less volatile than the underlying data.
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