On the efficient utilisation of duration

In this article we present a new approach to estimate the change of the present value of a given cashflow pattern caused by an interest rate shift. Our approximation is based on analysing the evolution of the present value function through a linear differential equation. The outcome is far more accurate than the standard approach achieved by a Taylor expansion. Furthermore, we derive an approximation formula of second order that produces nearly accurate results. In particular, we prove that our method is superior to any known alternative approximation formula based on duration. In order to demonstrate the power of this improved approximation we apply it to coupon bonds, level annuities, and level perpetuities. We finally generalise the approach to a non-flat term structure. As for applications in insurance, we estimate the change of the discounted value of future liabilities due to a proportional shift in the set of capital accumulation factors. These findings are of particular importance to capital adequacy calculations with respect to interest rate stress scenarios that are part of regulatory solvency requirements.