Replicating a Bond’s Cash Flows to Infer Its No-Arbitrage Price

Summary

The document presents a cash-flow replication method for determining the price of a risk-free bond from two other bonds. Each bond is represented by its payments at the present, six-month, and one-year dates. The target bond pays nothing at the intermediate date and makes a final payment at maturity, so the solution seeks a combination of the two known bonds whose later cash flows match those of the target.

Under the assumption that risk-free investments over the same horizon should earn equivalent returns, the cost of that replicating combination gives the target bond’s no-arbitrage price. This is a linear-algebra view of discounting and replication that can be applied beyond the particular cash flows shown. The document states the matching procedure but does not work through the coefficients or calculate the resulting price; it also relies on the stated risk-free and equal-return assumptions.

Key ideas

• Represent each bond by its dated cash flows, including its purchase cost.
• Choose holdings in known bonds so their future payments match the target bond.
• The initial cost of the replicating portfolio determines the target’s no-arbitrage price.
• The method assumes risk-free cash flows with consistent returns over the horizon.

Tags

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


# 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

## Answer by nbbo2 (score 1, accepted)

https://quant.stackexchange.com/a/38021

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.