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

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Discussion Overview

The discussion revolves around the relationship between forward rates, zero rates, and yields on coupon-bearing bonds, particularly in the context of an upward sloping term structure of interest rates. Participants explore the reasoning behind the ordering of these rates and the implications of duration on bond pricing.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the term structure of interest rates is upward sloping and presents a question regarding the ordering of the 5-year zero rate, the yield on a 5-year coupon-bearing bond, and the forward rate for a specific future period.
  • Another participant explains that the curve can be expressed as a product of forward rates, suggesting that if the curve is upward sloping, the forward rate for later periods will be greater than for earlier ones.
  • It is proposed that the zero rate has a longer duration than a coupon-bearing bond, which could imply a higher yield for the zero rate compared to the coupon bond.
  • A participant expresses uncertainty about how the duration formula supports the idea that longer durations yield greater returns, despite intuitively understanding the concept.
  • There is a discussion about the present value calculations for zero and coupon bonds, highlighting that the interest rate applied to the principal repayment of a zero bond is higher than the rates applied to the coupon payments of a coupon bond.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the relationship between duration and yield, with some points of agreement on the upward sloping nature of the term structure, but no consensus is reached on the implications of duration for bond pricing.

Contextual Notes

Participants mention specific formulas for bond pricing and the relationship between interest rates and present value, but there are unresolved aspects regarding the application of duration in this context.

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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?
 
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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+ri). 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
 
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.
 
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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
 
Ok I got it now... thanks a lot for your help
 

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