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Frequently a company wants to match its assets and liabilities. However, perfect matching is not practical due to fluctuations in interest rates. So they hedge their risk using immunization. This can be achieved using present value matching, duration matching, and greater convexity for assets. In particular:

  • $PV_{A}(i_0) = PV_{L}(i_0)$
  • $\frac{d}{di} PV_{A}(i)|_{i_0} = \frac{d}{di} PV_{L}(i)|_{i_0}$
  • $\frac{d^2}{di^2} PV_{A}(i)|_{i_0} > \frac{d^2}{di^2} PV_{L}|_{i_0}$

Could we add any extra conditions so that the risk is reduced even more(e.g. could be look at $\frac{d^3}{di^3} PV_{A}(i)$ etc..)?

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You are likely to get some answers on the sister StackExchange site devoted to Quantitative Finance: quant.stackexchange.com –  Andrey Rekalo May 10 '11 at 23:19

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