# 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 '18 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. – noob2 Feb 1 '18 at 22:02
• Wow great explanation. Thanks again, I really appreciate it. – Jack Feb 1 '18 at 22:03