Optimal liquidity reserve with funding liquidity risk

Abstract

In this article, we take the funding liquidity risk into account when determining the optimal liquidity reserve ratio for a commercial bank. A simple continuous-time model is developed, and a discrete-time model is built as a benchmark. We find that compared with the continuous-time model, the discrete-time model would generate a higher reserve ratio when the illiquid project has low haircut rate, and a lower reserve ratio when the project has high haircut rate.

Keywords:

• optimal reserve ratio
• funding liquidity risk
• liquidity crisis
• continuous-time model

JEL Classification:

• G21
• G33

Acknowledgements

This study is supported by the National Science Foundation of China (No. 71001071, 70831004).

Notes

1 The central bank is always the lender of last resort for providing the commercial bank with liquidity, however a haircut is usually imposed on the collateral, see e.g. Hull (2009). According to the report by IMF (2007), the haircuts were widened during the subprime crisis.

2 is the amount of new withdrawals if it is positive, and is the amount of new deposits if it is negative.

3 Under the assumption that the profit is zero in case of bankruptcy, the true objective function should be given as which however is difficult to calculate. Given that in the optimization model, the bankruptcy probability is constrained to be quite close to zero, is a good approximation to the true objective function.

4 Under the assumption that the project due after T, the commercial bank will be faced with funding liquidity risk at T.