p1.t4.415 spot and forward rates question3

[Xin Zhan]

the price of a four year zero coupon bond is 86, the price of a similar five year bond is 79.51. if we assume all rates are expressed er annum with semi-annual counpounding, which is nearest to the one-year implied forward rate from year four to year five.

d) 8%

the d(4) = 86/100, d(5) = 79.51/100.
f(4,5) = (0.86/0.7951 - 1) *2 = 16.32%? why do you need to sqrt(86/79.51)? question 2 from the same question set didn't use sqrt?

 

[ShaktiRathore]

Hi
Price of 4 yr zero=86=100/(1+s1/2)^8 where d(4)=1/(1+s1/2)^8=86/100
Similarly, d(5)=1/(1+s2/2)^10=79.51/100
(1+s1/2)^8*(1+1 yr forward rate/2)^2=(1+s2/2)^10
(1yr forward rate/2+1)^2=d(4)/d(5)=(1+s2/2)^10/(1+s1/2)^8=86/79.51==>1+1yr foward rate/2=sqrt(86/79.51)=1.04=>1 yr forward rate/2=.04=>1 yr forward rate=.08, its a semiannually compounded foward rate.this is a semi annually compounded thats why there is square root,if it was simply annual compounded rate being asked no square root comes.we would have used 1+1yr forward rate ibstead of (1+1yr forward rate/2)^2,so 1+1yr forward rate=86/79.51 in annual case,in general for n times compunding we have nth root. If continous compunded forward rate was asked n>>inf (1+s1/2)^8*(1+1 yr forward rate/n)^n=(1+s2/2)^10 => e^(1 yr forward rate)*1=(1+s2/2)^10/ (1+s1/2)^8=86/79.51== >1 yr forward rate=ln(86/79.51).
Thanks