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Qstn Ppr Prblm,Chptr-14,

P

Prerna

Member
1.Why the forward rates increase more rapidly wid term than the spot rates?(frm November 2006,India ppr)
n
2.Why the gross redemption yield from the four-year bond is lower than
the one-year forward rate up to time 4, f3,1.......where f3,1 is 5.5024% & GRY frm 4-yr. bond is 4.544%.(frm September 2007,UK ppr)

Not able to sum up exactly,the reasoning given for 1. tht accumulatn factors related to spot rates r geometric averages of accumulatn factors related to forward rates.

n for 2. tht the bond can be seen to be a series of zero coupon bonds (1 year, 2years etc.) each with lower yields than the forward rate.The GRY from the bond is,in effect,an average of spot rates that are themselves a weighted average of earlier forward rates.Please explain me dese reasonings...
 
Q1 (As well as being on the Indian November 2006 paper, this was also on the UK September 2000 paper)

In this question, you calculate the spot rates first and then use these to calculate the forward rates.

However, in general, you can do it the other way round i.e. given the forward rates, you can use them to calculate spot rates.

If you do this to calculate a 2-year spot rate, say, you use:

1+y2=sqrt[(1+f0,1)(1+f1,1)]

This is the accumulation factor for the 2-year spot rate in terms of the accumulation factors for the 1-year forward rates starting at time 0 and time 1. This is the same as the process of taking a geometric average of the forward rate accumulation factors.

So, the spot rates are a (geometric) average of the forward rates. If the forward rates increase, the spot rates (an average of the forward rates) increase more slowly. This is a consequence of the averaging process - for example, just thinking about arithmetic averages (means):

Mean of 2,4 = 3
Mean of 2,4,8 = 4.667
Mean of 2,4,8,16 = 7.5

While the numbers in the series double each time, the mean less than doubles. So the average increases more slowly than the terms it is averaging.

Q2.

As above, spot rates are an average of the forward rates. Another consequence of this is that any spot rate must lie between the highest and lowest forward rate used in its calculation (as it's an average).

Now, the GRY is the annual effective interest rate over the period of an investment. In this question, we can view the four year bond as a series of zero coupon bonds (one maturing after 1 year, one after 2 years, one after 3 years and one after 4 years). The GRY for each of these zero coupon bonds is just the spot rate for the appropriate term, so the GRY for the overall bond will be a (weighted) average of the GRYs for the zero-coupon bonds it is comprised of. In other words, the GRY for the overall bond is a (weighted)average of spot rates.

If the GRY is the average of the spot rates, it must be lower than the highest spot rate. The highest spot rate must in turn be lower than the highest forward rate (as the spot rate is an average of forward rates). Hence the GRY is lower than f3,1 (the highest forward rate).
 
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