Yield based formulas for DUR and CONV

[Molnár András]

Dear David,

In the FRM books I have learned about the yield based Price, Duration and Convexity and DV01.
Could you help me or maybe send me the general yield based Duration and Convexity formulas, where the settlement date other than coupon payment dates? This kind of Price formula is in the FRM book.

Thanks and regards,

András Molnár

 

[ShaktiRathore]

Hi,
please visit the following link:
http://forum.bionicturtle.com/threa...rations-lets-settle-this-now.6012/#post-18668

thanks

 

[Molnár András]

Dear ShaktiRathore,

Thanks for your answer, but i couldnt find the equations on that forum. The conversation was about the Duration only. And there was no yield based formula on that.

What i would like to know and search, is an equation on MDUR, DV01 and Convexity expressed with a closed formula through the yield.
For example, the closed yield-based formula on Bond Price is:
P= (1+y/k)^(1-t) * [C/y*(1-1/((1+y/k)^kT))+1/((1+y/k)^kT)] where "t" is the period to the next coupon payment date; "k" is the number of coupon payment in a year.

Could you or anybody else send me this kind of closed formula for MDUR, CONV and DV01?

Regards,

András

 

[Molnár András]

Dear David,

Could you please help me in the above theme?

Thanks,

András

 

[ShaktiRathore]

Hi,
Price of bond=P=C/(1+y/k)^1 + C/(1+y/k)^2 + ...... + C+F/(1+y/k)^T
applying GP, a=C/(1+y/k)^1 and r=common ratio=1/(1+y/k)
Sum of above GP= a(1-r^T)/(1-r)=[C/(1+y/k)]*(1-(1/(1+y/k))^T)/(1-1/(1+y/k))=[C/(1+y/k)]*(1-(1/(1+y/k))^T)/(y/k/(1+y/k))=[C]*(1-(1/(1+y/k))^T)/(y/k)=(kC/y)*(1-(1+y/k)^-T)
Hence price of bond=P=(kC/y)*(1-(1+y/k)^-T)+F*(1+y/k)^-T
Duration is given by D= 1/P*(dP/dy)
dP/dy=d/dy((kC/y)*(1-(1+y/k)^-T)+F*(1+y/k)^-T)=d/dy(kC/y)*(1-(1+y/k)^-T)+(kC/y)*d/dy(1-(1+y/k)^-T)+d/dy(F*(1+y/k)^-T))
dP/dy=(-kC/y^2)*(1-(1+y/k)^-T)-(kC/y)*(T/k*(1+y/k)^-(T+1))-F*1/k*T*(1+y/k)^-(T+1)
dP/dy=(-kC/y^2)*(1-(1+y/k)^-T)-(C/y)*(T*(1+y/k)^-(T+1))-F*1/k*T*(1+y/k)^-(T+1)
dP/dy=(-kC/y^2)+(1+y/k)^-(T+1)*[(-kC/y^2)*(1+y/k)-(C/y)*(T)-F*1/k*T]
dP/dy=(-kC/y^2)+(1+y/k)^-(T+1)*[(-kC/y^2)*(1+y/k)-(C/y)*(T)-F*1/k*T]
So D=1/P*dP/dy={(-kC/y^2)+(1+y/k)^-(T+1)*[(-kC/y^2)*(1+y/k)-(C/y)*(T)-F*1/k*T]}/{(kC/y)*(1-(1+y/k)^-T)+F*(1+y/k)^-T} which is rather ugly formula but yes we can derive like this way.

Similarly the convexity is the rate of change of duration w.r.t the yield for dD/dy which would produce the closed formula for convexity(try out yourself).

check for formula validity for perpetual bond T-->infinity
D(as T-->infinity)={(-kC/y^2)+0*[(-kC/y^2)*(1+y/k)-(C/y)*(T)-F*1/k*T]}/{(kC/y)*(1-0)+0}
D(as T-->infinity)={(-kC/y^2)+0/{(kC/y)*(1)}
D(as T-->infinity)={(-kC/y^2)/{(kC/y)}
D(as T-->infinity)=-1/y which is nothing but the duration of a perpetuity bond.

Also DV01 is the dollar change in value of bond with 1 basis point(.01% change in yield)
DV01=D*.0001*P=1/P*(dP/dy)*.0001*P=(dP/dy)*.0001
DV01={(-kC/y^2)+(1+y/k)^-(T+1)*[(-kC/y^2)*(1+y/k)-(C/y)*(T)-F*1/k*T]}*.0001

thanks