Tuckman: Par rates and DV01 / Duration / Convexity

[cAse113]

Hi everyone,

I am keen to know how Tuckman produces the graph where he shows the price - par rate relationship for the three securities in "One-Factor Risk Metrics and Hedges". Until now I've only seen price - YTM relationship graphs (usually authors explain duration etc. with this relationship as in the CFA ...).

As I understand it, par rates (swap rates) are basically derived from a given term structure of spot rates applied to a specific security - i.e. we set the coupon rate of this security to this par rate - such as the price is equal to par.

Is my assumption correct that Tuckman uses the following price - par rate function for these graphs:

P = 1+ [c - C(T)]/2 * A(T)
where c=coupon, C(T)=par rate, A(T)=annuity factor i.e. sum of all discount rates derived from the given spot rates

And what does a shift in this par rate such as +1bps imply with regards to the underlying term structure of spot rates?

Thanks a lot!

 

[ShaktiRathore]

Hi
Yes par rate is the coupon rate of a fixed income asset such that present values of the coupons and the par value discounted at spot rate curve equals the par value.
Yes par rate is basically same as swap rates , we set the price of fixed income asset as equal to the par value and derive the coupon rate(par rate) using spot rate curve which would be same as the par yield(swap rate) derived by setting the coupon as par rate ans setting the price at par , because here fixed income asset is priced at par and its known at par the yield of fixed income asset is equal to the coupon rate the par rate is same as swap rate. also visit: https://forum.bionicturtle.com/threads/par-yield-ytm-and-spot-rate.1355/#post-4975
Did you yourself guessed at formula P = 1+ [c - C(T)]/2 * A(T) ,which i cant comment to be correct or not i am not sure.May be David can help.Hope this helps.
I guess a shift in this par rate such as +1bps would imply an upward shift of +1bps in par yield which in turn would imply an upward shift in the underlying term structure of spot rates.
Thanks

 

[cAse113]

Thanks ShaktiRathore for your answer!

I got the formula from Tuckman chapter "Spot, Forward, and Par Rates" (p. 184 of the GARP book) ...