Why is the Forward Rate Greater Than the Zero Rate and Yield?

[the4thamigo_uk]

I came across this question in chapter 4 of Hull 'Options Futures and other Derivatives'. I have the answer but I am not sure what the explanation is. Could anyone help?

The term structure of interest rates is upward sloping. Put the following in order of magnitude :

a) the 5 year zero rate
b) the yield on a 5 year coupon bearing bond
c) The forward rate corresponding to the period between 4.75 and 5 years in the future

The answer is c > a > b, but why?

 

[BWV]

C = the curve can be expressed as a product of forward rates, so for each 3 month interval, i = 1 to 20, the 5 year interest rate = ∏ (1+r_i). If the curve is upward sloping, i₂₀ > i₁

A = the zero has a longer duration than a coupon bearing bond so will have a higher yield than a similar maturity coupon bond

 

[the4thamigo_uk]

Thanks BWV.

The explanation for C makes a lot of sense now. I am not so clear about the explanation for A though. I can intuitively see that if I lock money away for a longer period I should expect a greater return, but how does the duration formula show this?

For the two bonds I have something like the following :

P_z = F/(1+R)^n
P_c = C(1/(1+r) + 1/(1+r)^2 + ... + 1/(1+r)^n) + F/(1+r)^n

Now P_z and P_c are not expected to be equal, and I can choose C to be anything I like, so I have complete flexibility to change P_c and C to give me an r > R or r < R.

 

[BWV]

forget about the duration formula for a second (although the duration of a zero is equal to its maturity while a coupon bond is always less)

so if you think about your formula for the coupon bond and the interest rate each coupon payment would command in the market if you sold it - i.e. if the bond pays semiannually the first coupon payment would be discounted at the 6 month interest rate, the second at the one year rate etc. the present value of all these payments makes up, along with the discounted value of the principal repayment the total of the bond value. The interest rate on the zero would be equal to the discount rate of the principal repayment at maturity which would be higher than the rate applied to any of the semiannual payments

 

[the4thamigo_uk]

Ok I got it now... thanks a lot for your help