@MASTERSTHESIS\{IMM2004-03130,
    author       = "K. M. Rasmussen",
    title        = "Portef{\o}ljeoptimering for danske realkreditl{\aa}n",
    year         = "2004",
    school       = "Informatics and Mathematical Modelling, Technical University of Denmark, {DTU}",
    address      = "Richard Petersens Plads, Building 321, {DK-}2800 Kgs. Lyngby",
    type         = "",
    note         = "Supervised by Professor Jens Clausen",
    url          = "http://www2.compute.dtu.dk/pubdb/pubs/3130-full.html",
    abstract     = "Form{\aa}let med projektet er at udvikle et matematisk analysev{\ae}rkt{\o}j til r{\aa}dgivning af realkreditkunder - l{\aa}ntagere - s{\aa}ledes at de, til at begynde med, v{\ae}lger det optimale l{\aa}n eller en optimal kombination af l{\aa}n, og dern{\ae}st plejer l{\aa}neportef{\o}ljen mest hensigtsm{\ae}ssigt gennem l{\aa}neperioden. Til det form{\aa}l vil vi - l{\ae}seren og jeg - betragte en sammenkobling af matematisk finansiering og operationsananlyse. Inden for matematisk finansiering m{\o}der vi l{\aa}nrelaterede beregninger, diskret rentetr{\ae} udvikling og prisfasts{\ae}ttelse af obligationer og optioner p{\aa} obligationer. Inden for operationsanalyse anvender vi multi-stadie heltal stokastisk optimering.

English summary:

The Danish mortgage loan system is among the most complex of its kind in the world. Both adjustable-rate and fixed-rate loans are available. It is also possible to mix loans. As for the fixed-rate loans there are both call and delivery embedded options, which give the mortgagor the possibility to refinance the loan in both directions. When the interest rates are low the call option will be used to obtain a new loan with less interest payment. When the interest rates are high the delivery option can be used to reduce the amount of outstanding debt. A mortgagor with an adjustable-rate loan can buy a series of caps to hedge against possible rate increase.

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. There exists as of today no functional optimization model to advise the individual mortgagor on his choice of loan.

In this report we will develop and solve a realistic model for the optimal debt management problem of the Danish private mortgagor. The thesis is inspired by the research work of Nielsen and Poulsen (N\&{P,} [18]). Some aspects of N\&P model have been reformulated and extra constraints have been added, making the model more realistic. Both market risk and shape risk have been explicitly taken into consideration. Fixed transaction costs have been modeled by use of binary variables.

An effective algorithm to price a series of caps on an adjustable rate loan is outlined. Likewise ways to use the optimization model for risk analysis regarding the price of these caps have been explained. The multi stage stochastic integer program describing the mortgagors problem has been implemented in {GAMS}. Scenario reduction algorithms (GAMS/{SCENRED} module) have been applied to the problem with promising results.

The results obtained in this project show that the average Danish mortgagor should take better advantage of his options for mixing loans in his loan portfolio. Likewise he should readjust the portfolio more often than is the case today. The developed model and software in this project can be used as an analytical tool to develop new loan products, where an optimal portfolio of mortgage loans is chosen initially and then it is readjusted according to an optimal strategy given the individual customer input."
}