بهینه سازی سبد سرمایه گذاری وام مسکن با استفاده از برنامه نویسی احتمالی چند مرحله ای

We consider the dynamics of the Danish mortgage loan system and propose several models to reflect the choices of a mortgagor as well as his attitude towards risk. The models are formulated as multi-stage stochastic integer programs, which are difficult to solve for more than 10 stages. Scenario reduction and LP relaxation are used to obtain near optimal solutions for large problem instances. Our results show that the standard Danish mortgagor should hold a more diversified portfolio of mortgage loans, and that he should rebalance the portfolio more frequently than current practice.

1.1. The Danish mortgage market The Danish mortgage loan system is among the most complex of its kind in the world. Purchase of most properties in Denmark is financed by issuing fixed-rate callable mortgage bonds based on an annuity principle. It is also possible to raise loans, which are financed through issuing non-callable short term bullet bonds. Such loans may be refinanced at the market rate on an ongoing basis. The proportion of loans financed by short-term bullet bonds has been increasing since 1996. Furthermore, it is allowed to mix loans in a mortgage loan portfolio, but this choice has not yet become popular. Callable mortgage bonds have a fixed coupon throughout the full term of the loan. The maturities are 10, 15, 20 or 30 years. There are two options embedded in such bonds. The borrower has a Bermudan type call option, i.e. he can redeem the mortgage loan at par at four predetermined dates each year during the lifetime of the loan. When the interest rates are low the call option can be used to obtain a new loan with less interest payment in exchange for an increase in the amount of outstanding debt. The borrower has also a delivery option. When the interest rates are high this option can be used to reduce the amount of outstanding debt, in exchange for paying higher interest rate payments. There are both fixed and variable transaction costs associated with exercising any of these options. Non-callable short-term bullet bonds are used to finance the adjustable-rate loans. The bonds’ maturities range from 1 to 11 years and the adjustable-rate loans are offered as 10, 15, 20 or 30-year loans. Since 1996 the most popular adjustable-rate loan has been the loan financed by the one-year bond. From 2001, however, there has been a new trend, where demand for loans financed by bullet bonds with 3 and 5-year maturities has risen substantially. 1.2. The mortgagor's problem It is known on the investor side of the financial markets that investment portfolios should consist of a variety of instruments in order to decrease financial risks such as market, liquidity and currency risk while maintaining a fixed level of return. The portfolio is also rebalanced regularly to take best advantage of the moves in the market. The portfolio diversification principle and rebalancing is, however, not common in the borrower side of the mortgage market. Most mortgagors finance their loans in one type of bond only. Besides they do not always rebalance their loan when good opportunities for this have arisen. There are two major reasons for the mortgagors reluctance to better taking advantage of their options (that they have fully paid for) through the lifetime of the mortgage loan: 1. The complexity of the mortgage market makes it impossible for the average mortgagor to analyze all the alternatives and choose the best. 2. The mortgage companies do not provide enough quantitative advice to the individual mortgagor. They only provide general guidelines, which are normally not enough to illuminate all different options and their consequences. The complexity of the mortgage loan system makes it a non-trivial task to decide on an initial choice of mortgage loan portfolio and on finding a continuing plan to readjust the portfolio optimally. See e.g. Zenios, 1993 and Zenios, 1995, Nielsen and Zenios (1996a), Vassiadou-Zeniou and Zenios (1996), Zenios et al. (1998) and Zenios and Kang (1993). We assume in the following that the reader is familiar with the dynamics of a mortgage loan market such as the Danish one, as well as the basic ideas behind the mathematical modeling concept of stochastic programming. The Danish mortgagor's problem has been introduced by Nielsen and Poulsen (2004) (N&P). They use a two-factor term structure model for generating interest rate scenarios. They have developed an approximative pricing scheme to price the mortgage instruments in all nodes of the scenario tree and on top of it have built a multi-stage stochastic program to find optimal loan strategies. The paper, however, does not describe the details necessary to have a functional optimization model, and it does not differentiate between different types of risks in the mortgage market. The main contribution of this article is to make Nielsen and Poulsen's model operational by reformulating parts of their model and adding new features to it. We reformulate the Nielsen and Poulsen model in Section 2. In Section 3, we model different options available to the Danish mortgagor, and in Section 4, we model mortgagor's risk attitudes. Here, we consider both market risk and wealth risk. In the basic model we incorporate fixed transaction costs using binary variables. We use a non-combining binomial tree to generate scenarios in an 11 stage problem. This results in 51175 binary variables, making some versions of the problem extremely challenging to solve. Dupačová et al. (2003) and Heitsch and Römisch (2003) (Scenred) have modeled the scenario reduction problem (SRP) as a set covering problem and solved it using several heuristic algorithms. We review these algorithms in Section 5 and use them in our implementation to reduce the size of the problem and hereby reduce the solution times. Another approach to getting shorter solution times is proposed in Section 6, where we solve an LP-approximated version of the problem. In Section 7, we discuss and comment on our numerical results and we conclude the article with suggestions for further research in Section 8. We use GAMS (general algebraic modeling system) to model the problem and CPLEX 9.0 as the underlying LP and MIP solver. For scenario reduction we use the GAMS/SCENRED module (scenred manual). The obtained results show that the average Danish mortgagor would benefit from choosing more than one loan in a mortgage loan portfolio. Likewise he should readjust the portfolio more often than is the case today. The developed model and software can also be used to develop new loan products. Such products will consider the individual customer inputs such as budget constraints, risk profile, expected lifetime of the loan, etc. Even though we consider the Danish mortgage loan market, the problem is universal and the practitioners in any mortgage loan system should be able to use the models developed in this paper, possibly with minor modifications.

We have developed a functional optimization model that can be used as the basis for a quantitative analysis of the mortgagors decision options. This model in conjunction with different term structures or market expert opinions on the development of bond prices can assist market analysts in the following ways. Decision support: Instead of calculating the consequences of the single loan portfolios for single interest rate scenarios, the optimization model allows for performing “what-if” analysis on a higher level of abstraction. The analyst can provide the system with different sets of information such as the presumed lifetime of the loan, budget constraints and risk attitudes. The system then finds the optimal loan portfolio for each set of input information. Product development : Traditionally, loan products are based on single bonds or bonds with embedded options. In some mortgage markets such as the Danish one it is allowed to mix bonds in a mortgage portfolio and there are even some standard products which are based on mixing bonds. The product P33P33 is, for example, a loan portfolio where 33% of the loan is financed in 3-year non-callable bonds and the rest in fixed-rate callable bonds. These mixed products are currently not popular since the rationale behind exactly this kind of mix is not well argued. The optimization model gives the possibility to tailor mixed products that, given a set of requirements, can be argued to be optimal for a certain mortgagor. The greatest challenge in solving the presented models is on decreasing the computing times. We have experimented with scenario reduction (Dupačová et al., 2003, Heitsch and Römisch, 2003 and GAMS/SCENRED, 2002) (scenred) and we have suggested an LP approximation method to reduce the solution times while maintaining solution quality. It is, however, an open problem to develop tailored exact algorithms such as decomposition algorithms (see Birge, 1985 and Birge and Louveaux, 1997) to solve the mortgagors problem. Another approach for getting real time solutions is to investigate different heuristic algorithms or make use of parallel programming (see Nielsen and Zenios, 1996b and Ruszczynski, 1988) to solve the problem. Integration of the two disciplines of mathematical finance and stochastic programming combined with use of the state-of-the-art software has a great potential, which has not yet been realized in all financial markets in general and in mortgage companies in particular. There is a need for more detailed and operational models and high performing easy to use accompanying software to promote use of the mathematical models with special focus on stochastic programming.