# How to work out bond price given other bond prices?

I'm stuck on the following problem from a financial maths course, and was wondering whether anybody would be able to help me. I don't really know where to begin.

The following risk free bonds are available.

A. Matures and pays £1 in 6 months, plus a final interest payment of 4p. Costs 102p.

B. Matures in a year and pays £1. Also gives interest payments of 2.5p in 6 months and another 2.5p in a year. Costs 101p.

C. Also matures in a year and pays £1, but gives one final interest payment of 6p.

Assuming all risk free investments over a given period should give the same return, what should C cost?

Thanks,

Jack

Represent each bond by a vector having 3 elements (now, 6 months hence, 1 year hence):

Bond A: [-102 104 0]

Bond B: [-101 2.5 102.5]

Bond C: [-X 0 106]

Now find a linear combination of A and B such that the last two entries match C (i.e. second entry is 0 and third entry is 106). The first entry in the linear combination gives the desired price X.

• Thank you very much for your reply. Do you mind explaining roughly why this works?
– Jack
Feb 1, 2018 at 21:58
• If a suitable combination of Bond A and Bond B can be found that has exactly the same cash flows as Bond C, that combination, being equivalent to Bond C, must sell for the same price. Feb 1, 2018 at 22:02
• Wow great explanation. Thanks again, I really appreciate it.
– Jack
Feb 1, 2018 at 22:03