Modelling of asset allocation in banking using the mean-variance approach

Abstract

Bank asset management mainly involves profit maximization through invest- ment in loans giving high returns on loans, investment in securities for reducing risk and providing liquidity needs. In particular, commercial banks grant loans to creditors who pay high interest rates and are not likely to default on their loans. Furthermore, the banks purchase securities with high returns and low risk. In addition, the banks attempt to lower risk by diversifying their asset portfolio. The main categories of assets held by banks are loans, treasuries (bonds issued by the national treasury), reserves and intangible assets. In this mini-thesis, we solve an optimal asset allocation problem in banking under the mean-variance frame work. The dynamics of the different assets are modelled as geometric Brownian motions, and our optimization problem is of the mean- variance type. We assume the Basel II regulations on banking supervision. In this contribution, the bank funds are invested into loans and treasuries with the main objective being to obtain an optimal return on the bank asset port- folio given a certain risk level. There are two main approaches to portfolio optimization, which are the so called martingale method and Hamilton Jacobi Bellman method. We shall follow the latter. As is common in portfolio op- timization problems, we obtain an explicit solution for the value function in the Hamilton Jacobi Bellman equation. Our approach to the portfolio prob- lem is similar to the presentation in the paper [Hojgaard, B., Vigna, E., 2007. Mean-variance portfolio selection and efficient frontier for defined contribution pension schemes. ISSN 1399-2503. On-line version ISSN 1601-7811]. We pro- vide much more detail and we make the application to banking. We illustrate our findings by way of numerical simulations.