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November 17th, 2009, 02:57 PM   #1
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Estimating the term structure from (insufficient) bond data

Hi everyone. This is my first topic and post in general, so please bear with me.

I'm trying to estimate the term structure of interest rates in a series of goverment bonds. More specifically it's a data set consisting of 9 bullet bonds, annual payment of coupons, with maturities (in years) 1,2,3,4,6,8,10,15,30.

I'm right at the beginning of this 'mission', and it occurs to me that I have a problem: 30 coupon dates and only prices for 9 bonds.

The usual approach in text books always involve at least as many bond prices as payment dates. In the case of more bond prices than payment dates, problems would usually also occur, if it resulted in a non-solveable system of m linear equations of n variables (discount factors - 'what we want from this data set'), with m>n. I know that approaches solving this type of problem usually involves statistical measures such as multiple regression. There's plenty of answers to find when surfing the internet on this problem.


But I can't seem to find any solution to my problem, that I have more payment dates (30) than bond prices (9), and needs to estimate the discount factors applying to each of these payment dates. More about the background of the problem:

.................................................. ..............
If we let
d(i) denote the discount factor for payment date i
P(j) denote the price of bond j
Cj(i) denote bond j's coupon payment for payment date i
Nj(m) denote bond j's principal payment at maturity (a coupon will be paid in addition to the principal at maturity)

and let "sum,i=t0,tn [ X(i) ]" be the sum function X(t0) + X(t1) + ... + X(tn)

Then we have

P(j) = sum,i=1,m [ Cj(i)d(i) + Nj(m)d(m) ] ( = Cj(1)d(1) + Cj(2)d(2) + ... + [ Cj(m)+Nj(m) ]d(m) )

Let the 9 bonds be ordered so that j=1 => m=1, ... , j=4 => m=4, j=5 => m=6, ... , j=8 => m=15, j=9 => m=30

Then we have

P(1) = [C1(1) + N1(1)]d(1) (a zero coupon bond actually)
P(2) = C2(1)d(1) + [C2(2) + N2(2)]d(2) = sum,i=1,2 [C2(i)d(i) + N2(2)d(2) ]
P(3) = C3(1)d(1) + C3(2)d(2) + [C3(3) + N3(3)]d(3) = sum,i=1,3 [ C3(i)d(i) + N3(3)d(3) ]
P(4) = C4(1)d(1) + ... + [ C4(4) + N4(4) ]d(4) = sum,i=1,4 [ C4(i)d(i) + N4(4)d(4) ]
P(5) = C5(1)d(1) + ... + [ C5(6) + N5(6) ]d(6) = sum,i=1,6 [ C5(i)d(i) + N5(6)d(6) ]
P(6) = C6(1)d(1) + ... + [ C6( + N6( ]d( = sum,i=1,8 [ C6(i)d(i) + N6(d( ]
P(7) = C7(1)d(1) + ... + [ C7(10) + N7(10) ]d(10) = sum,i=1,10 [ C7(i)d(i) + N7(10)d(10) ]
P( = C8(1)d(1) + ... + [ C8(15) + N8(15) ]d(15) = sum,i=1,15 [ C8(i)d(i) + N8(15)d(15) ]
P(9) = C9(1)d(1) + ... + [ C9(30) + N9(30) ]d(30) = sum,i=1,30 [ C9(i)d(i) + N9(30)d(30) ]
.................................................. ..............

The problem is how to find d(1), d(2), ... , d(30)

I suppose there exists many solution sets of discount factors d(1), d(2), ... ,d(30) - but how should optimum set be found/chosen, and why?
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