A New Paradigm in Mortgage Loan Advice Margrét S. Otterstedt Technical University of Denmark, DTU Copenhagen, Denmark
Kourosh M. Rasmussen Technical University of Denmark, DTU Copenhagen, Denmark [email protected]
Murat Kulahci Technical University of Denmark, DTU Copenhagen, Denmark Abstract—The Danish mortgage market has undergone considerable changes during the last 15 years. New and more complex variations of loan products have been introduced. Nevertheless, mortgage loan advice has remained, by large, unchanged. This paper addresses a study where a number of new refinancing rules are constructed with the help of a stochastic optimization model and the data mining method, CART. Keywords-component; stochastic optimization; mortgages loans; house hold refinancing advice;
I. INTRODUCTION The Danish mortgage market is one of the largest in Europe with a market value around 2000 billion DKK (approximately 270 billion EURO). It has increased substantially during the last decade, where it exceeded the Danish GDP in 2006. "A Danish mortgage bond is an instrument of debt secured against mortgages on a real property. It is a negotiable security and the price of the bond is determined in the open market". 1 A certain balance has to hold between the published bonds and loans. This is secured by the Balance principle. The principle imposes strict limitations to the difference allowed between the terms of a loan and the terms of the corresponding bond. The price of the bonds and the loan terms are therefore not determined by a mortgage institution but rather by the demand of the market. The balance principle can be seen as a risk management tool due to the fact that it shifts the market risk from the bank to the bondholder. The mortgage banks are relatively risk free institutions with the greatest risk factors being the credit risk and liquidity risk. The liquidity risk was hardly an issue before the introduction of the short adjustable-rate mortgages especially combined with initial interest only feature. Another unique property of the Danish mortgage system is the buy-back delivery option. The borrower is allowed to buy back the bond covering his loan at market price at any given time. This differs from all other mortgage markets where the borrower has to negotiate a certain price with the institutions in order to prepay his loan. Combination of the Balance 1
principle and the buy-back delivery option is what makes the Danish mortgage system so successful. It is characterized by great transparency within all the system, since information regarding rates and bond prices are easily accessible. Despite its high relevance, relatively few studies have been done on the subject of refinancing in the Danish market. In 2011, Rasmussen, Madsen and Poulsen published an extensive report  where focus was put on current refinancing advisory and possible improvements. By gathering information from different mortgage institutions they formalized the refinancing rules of thumb, which are unwritten standards throughout the market. Their research revealed that the existing rules of thumb yield in most cases little or no gain at all for the borrower. They suggest a model-based advisory framework for refinancing and show that following the models considerably enhances the refinancing gains. In practice, however, few borrowers or even advisors, are keen on using model-based refinancing strategies, that they consider as being black boxes. They prefer a rule-based approach where they can make a refinancing decision directly based on observations of market parameters, rather than using models as an intermediary. The objective of this study is therefore to improve current market rules of thumb by using intuitions on how the model-based strategies of Rasmussen, Madsen and Poulsen (2011) work. In other words, the results from these models were analyzed in order to find the main drivers of refinancing from a model viewpoint. A number of economic indicators were introduced and refinancing strategies were constructed based on the observations on these economic indicators. II. REFINANCING RULES OF THUMB A high percentage of Danish home owners choose to refinance their mortgage loans before maturity. More than 150.000 Danish households obtained a reduction in the average mortgage payments after refinancing their mortgage loan in 2009.2 Fig. 1 shows the percentage of mortgage bonds 2
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The coupon rate of the new mortgage loan is at least 2% lower than of the existing one.
The price of the new loan is greater than 95.
In case more than one option is available the one resulting in a greater decline in first year's payment after tax should be chosen.
Refinance from a lower to a higher coupon if: The outstanding debt is reduced by at least 10%.
In case more than one option is available the one resulting in the greatest reduction in outstanding debt should be chosen.
No special rules exist when going from a fixed rate to an adjustable-rate mortgage or vice versa. The mortgage banks are often reluctant to recommend adjustable-rate loans since they may become very risky if interest rate increases. Nevertheless, adjustable-rate mortgages cover two thirds of the Danish mortgage market . III. Figure 1. Ord inary and extra -ordinary redemption of bonds throughout the period 1995 to 2012.
redeemed during the years 1995 to 2012. In over 20 quarters, up to 5%-16% of the entire amount of outstanding Danish mortgage bonds were refinanced. This is mainly due to the buy-back delivery option. The graph in Figure 1 is consistent with the refinancing rules of the market. However, in near all these cases, refinancing is taking place from a higher to a lower coupon. When making a refinancing decision, the borrower is faced with three alternatives:
Refinance to lower rates in order to reduce the monthly payment
Refinance to higher rates in order to reduce the outstanding debt
Refinance to a different loan type, for instance from a fixed rate to an adjustable rate mortgage or vice versa.
Last years trend has been to refinance to a lower coupon or from a fixed rate to an adjustable rate mortgage loan. The benefits of these refinancing are easily seen by the borrower since it most often results in an immediate reduction in monthly/quarterly payments. Currently there are no unanimous market rules of when a refinancing should take place. Despite that fact, various institutions all tend to give resembling advice. After interviewing several different mortgage banks, Rasmussen, Madsen and Poulsen (2011) discussed the common criteria for refinancing based on advice given by the different mortgage banks. Based on the criteria they formalized the so called rules of thumb: Refinance from a higher to a lower rate coupon if:
The following section describes the model, used for building new refinancing rules, followed by a comparison between the performance of the refinancing rules of thumb and the results from the optimization model. A. The model The stochastic optimization model was formulated by Rasmussen and Zenios (2007) and used in the analysis performed by Rasmussen, Madsen and Poulsen (2011). The risk assessment used is the conditional value at risk (CVaR). The optimization model is a single-stage mixed integer stochastic program. At time t0 all information regarding the loans are available. The model can start with one or more existing loans. At each time node the model can choose to hold or refinance existing loan. The analysis is performed on 38 alternative simulated periods starting either at the year 1995 or 2000 and having a horizon of 10 years. Following is a description of the stochastic model. Parameters: λ: Risk attitude, going from 0 to 1. If λ=0 the model will minimize the period cost without taking risk into consideration. If λ=1 the model will only minimize the risk (CVaR). Pt 0,i : Price of publishing a bond for loan i at time t0. K t 0 ,i : Price of redeeming a bond for loan i at time t0. Since fixed-rate bonds can always be bought back at the price of 100 then K t 0,i = min(1, Pt 0,i ) for callable fixed rate bonds and K t 0,i = Pt 0,i for the non-callable ones. I t 0 ,i : Outstanding debt for loan i at time t0. p s: The probability of scenario s taking place.
Electronic copy available at: http://ssrn.com/abstract=2192464
Os: Accumulated period cost for loan i for scenario s when the outstanding debt at the beginning equals 1 DKK. c: Variable transaction cost as a percentage of the loans outstanding debt. cf: Fixed transaction cost for a refinancing. M: A big number which forces a binary variable to have the value 1 whenever a refinancing takes place. α: Confidence level for Conditional Value at Risk – we use 95% throughout the paper. Variables:
y t 0 , i : Published bonds to finance loan i at time t0 .
x t 0 , i : Redeemed bonds for loan i at time t0. Z t 0 ,i :Binary variable which takes value 1 if loan i is
published or redeemed at time t0 and 0 otherwise. ξ:Value-at-Risk (VaR) with a confidence interval of 100α%. ξ+s: The positive deviation above the VaR for scenario s. Objective Function: The objective function is to find the minimum cost given a certain risk, where the risk attitude is measured with the parameter λ. The cost is calculated as the average of the total period cost for all scenarios multiplied by the published bonds. min. (1 - ) i
p s y t ,i O is CVaRy; 0
Constraints: Equation 2 describes the balance between the outstanding debt, publishedand redeemed bonds.
zt 0 , i I t 0 , i xt 0 , i y t 0 , i
The new outstanding debt must equal the redeemed loan plus all transaction costs.
P t ,i y t ,i K t ,i y t ,i cy t ,i x t ,i c f Z t ,i 0
yt ,i Ois ξ 0
Equation 4 defines the binary variable z t , i , where M is a 0 big number. The constraint forces Z ,i to take the value 1 if a t0
bond is published and 0 otherwise. MZ t0 ,i y t0 ,i
Equations 5 and 6 calculate the Conditional Value at Risk. The previous finds the deviation for scenarios that take value above the Value at Risk (VaR) and the latter calculates CVaR by finding the weighted average of these deviations.
p s s s CVaRy;α
1 Equation 7 states the boundary of variables. Redeemed and published bonds as well as the positive deviation from VaR cannot take value below zero and Z t 0 ,i is defined as a binary variable. s 0, y t 0 ,i , x t 0 ,i , Z t 0 ,i 0,1
z t 0 ,i : Outstanding debt for loan i at time t0.
B. Optimization Results Throughout the article the assumption is made that a borrower is holding a bond loan of 2.000.000 DKK with 30 years till maturity, that gives both principle and interest rate payments four times per year,. After 10 years the loans are redeemed. Description and generation of the scenarios used in this analysis are explained in the report of Rasmussen, Madsen and Poulsen (2011). Fig. 2 shows the refinancing gains and losses acquired with the different strategies for the period 2000 to 2010. The figure was made by firstly implementing an issue and hold strategy, where the model was forced to issue a fixed rate loan with the price closest to 100. The issue and hold strategy was run for 250 different interest rate scenarios. Afterwards the different strategies were implemented and for each scenario, their results compared to the one of the issue and hold. From that information, their distribution functions were plotted. This was done for the purpose of comparing the robustness of the different strategies. The wait & see model is a so called ex post analysis which finds the optimal possible solution for a certain past time period. It gives a higher limit to the possible refinancing gains for a strategy. The stochastic model is both shown for risk neutral preferences, as well as for risk averse. The risk neutral strategy profits more in average than the risk averse model. However, the left tale of the distribution is substantial if compared to the other strategies. Hence, all further research is solely based on the risk averse model. The rules of thumb perform significantly worse than the model based strategies for the 250 scenarios. The same applies for historical data for the period 2000-2010, where the rules of thumb perform poorly and would have recommended one refinancing, from a 7% coupon to 5% which would have led to an increased period cost of 0.02%. IV. RULE BASED REFINANCING Despite the fact how well the optimization model is performing, it is improbable to be used in practice. As stated before, practitioners are reluctant to base their consultancies on model-based strategies since the models are considered to be so called black boxes. They might perform well, however the reason behind their behavior is not fully understood. It is lacking the transparency which explains why a refinancing is reasonable at one point but not at another.
possible. However this maximum size tree will most likely overfit the data hence possess low prediction power. To overcome this problem, the tree is then pruned akin to backward elimination method in linear regression. The “best” tree can be obtained using for example cross validation. Hence in constructing a classification tree, three major issues have to be addressed: 1) Determining the splitting rule at each node and a measure of impurity. 2) Determining the level of pruning and deciding on the best tree 3) Determining the class assignment rule for each terminal node
Figure 2. Co mparison of the crystal ball, stochastic optimization models (risk averse and risk neutral) and the rules of thumb during the period 2000 to 2010.
To gain a better insight of when and why the model chooses to refinance, the data mining method, "Classification and Regression Trees", was used. The outcome from the method was then used to construct a new refinancing strategy, based on the results from the optimization model. A. Statistical test set up/ CART CART algorithm is a non-parametric modeling method originally introduced by Breiman et al. (1984). For categorical (nominal or ordinal) responses classification trees and for continuous responses regression trees are used. The idea is to form “decision trees” by recursive binary splits of the dataset where the impurity is minimized in each split. The impurity is measured in terms of misclassification for the classification tree which will be the main focus in this paper since the response is a categorical variable with binary outcomes: refinance or keep the current loan. The decision for splits is taken based on explanatory factors which could be either categorical or continuous. The tree begins at the root node. From there the variable which leads to the best split is chosen. The best split is defined as the split of the data which minimizes the impurity (or maximizes homogeneity). The process relies on binary recursive partitioning so at each decision point CART will look at a variable X ≤ c to check whether the statement is TRUE or FALSE and the data is split into two groups accordingly. The root nodes will then give rise to child nodes and child nodes give rise to own children. The last partitioning ends at the terminal node. The classification is then defined by following the path from the root node to the terminal node. The procedure can be divided into three stages; tree growing, tree pruning and optimal tree selection. Within the first stage a tree is grown to the maximum size where all possible splits of a node are considered. This is done for the purpose of gaining as much knowledge about the data as
The training data is defined as a matrix X which has M number of variables xj and N numbers of observations and the classification vector Y which consists of K classes with N number of observations. Let xjopt notate the best possible splitting value for the variable xj and np represent the parent node and nl and nr respectively the left and right child node. Let´s assume that Pl and Pr stand for the probability of node nl and nr. The impurity function i(n) defines the maximum homogeneity of the child nodes. Since the impurity of the parent node np is constant the change in impurity can be calculated as:
i(n) i( n p ) Pl i( nl ) P r i( n r )
Hence at each node the CART model will seek to maximize the ∆i(n), that is:
arg max i( n p ) Pl i( n p ) P r i( n r )
x j xopt j , j 1,..., M
There are many ways of defining the impurity function i(n). Gini is the default splitting rule for CART since it so often gives the best split. It defines the impurity function i(n) as: i(n)
pk n pl n
where k,l represent indices of class K , p(k|n) is the conditional probability of k given that the time period is n. Therefore the change in impurity can be calculated as: i(n)
p 2 k n p Pl p 2 k nl P r p 2 k n r
which gives the optimization problem: arg max x j xopt j , j 1,..., M
p 2 k n p Pl
p 2 k nl P r
p2 k nr K
One of the main advantages of CART is interpretability. A proper CART analysis often yields small enough decision trees that can be easily interpreted and applied. Once the tree is obtained, it can also be programmed quite easy through a succession of “if” statements in any coding language. This is the primary reason for applying CART in this study. It is expected that the final decision tree helps explaining the decisions of refinancing taken by the stochastic optimizer under different scenarios. In general, CART has other
advantages as it can handle large data sets with different types of responses such as categorical or continuous. In many classical statistical modeling approaches this can be an issue particularly for parametric models. Moreover missing values and outliers are easily dealt with in CART algorithm compared to many other statistical methods through for example surrogate splits.
∆ Payment: The relative difference between the alternative payment and the existing payment.
∆ Effective Rate: The difference between the effective rate of the alternative bond and the existing bond.
∆ Coupon: The difference between the alternative coupon and the existing coupon.
There are also some general concerns regarding the CART algorithm. First of all, there is a lack of smoothness in predictions made by a CART model. However this is of particular concern for continuous responses and not much of an issue in our case. Another issue is related to recursive nature of tree based decision tools in capturing additive structures in the relationship between the inputs and the response.
Long Term Rate: Interest rate for a 30-year investment.
Short Term Rate: Interest rate for a 1-year investment.
∆ Interest Rate: The difference between the long term and short term rate.
Alternative Price: Price of the new loan.
Existing Price: Price of the loan currently held by the borrower.
Horizon: Time till maturity. The 10 years are divided into quarters, so the total horizon at time 1 consists of 40 quarters.
Type of Refinancing: The data is classified whether the refinancing is taking place to a higher or a lower coupon.
Breakeven: When a refinancing takes place from a higher to a lower coupon, the breakeven is the time it takes until the reduction in payment starts compensating for the increase in the outstanding debt. When a refinancing takes place from a lower to a higher coupon, the breakeven is the time where the reduction in debt no longer compensates for the increased payments.
This is also not expected to be a source of concern in our case. What needs to be considered in our case in terms of drawbacks of CART algorithm, however, is its potential instability. Due to its hierarchical structure, CART model is often sensitive to changes in the data set. Omissions from the original data or addition of new data can result in considerably different decision tree as a change in the splitting rule at a top node affects all the splitting rules at the following nodes. In order to take into account this potential instability issue in our study, we specifically focus on paths on the decision tree that led to refinancing decision rather than considering the decision tree as a whole. Then each of these refinancing paths is analyzed in practical terms and only the paths that make intuitive and practical sense are retained. Therefore our primary goal was not necessarily to use CART decision model as a whole to make refinancing decisions but to use the paths defined by that decision tree in order to find alternative rules of refinancing to the financial rules of thumbs used in the mortgage market. B. CART analysis In order to give CART as broad input as possible all possible economic indicators with possible impact on the choice of refinancing were formulated. At each quarter the idea was to calculate the above mentioned indicators, whether or not a model refinancing actually took place. So at the quarters where no refinancing took place an alternative loan was chosen, which ought to represent the most likely refinancing option and a fictitious refinancing calculated based on that loan. At each of these nodes, where no refinancing took place, the alternative loan chosen was the one with the price closest to and below 100. So at every quarter, data was collected and the quarter labeled 0 if no refinancing took place and 1 otherwise. The indicators that were chosen are shown below:
∆ Outstanding Debt: The relative difference between the alternative debt and the existing debt.
In the following discussion breakeven is simplified by disregarding discount factors and calculated as: Breakeven
Alternative Debt Existing Debt Alternative Payment Existing Payment
where the payment is calculated on a yearly basis. Table 1 shows how the signal of breakeven is defined, e.g. would a 100.000 DKK decrease (-100.000) in outstanding debt and 20.000 DKK increase (+20.000) in payment give a breakeven value of -5 years. As well, if a refinancing leads to a decrease in both the outstanding debt and the payment, the breakeven would have a value of -∞. TABLE 1. DEFINIT ION OF SIGNALS FOR BREAKEVEN
∆ Outstanding Debt ∆ Yearly Payment
Figure 3. CART model results when refinancing to a lower fixed rate loan based on data from the risk averse model for the period 2000 t o 2010.
Breakeven ≤ 4.99 Horizon > 7.5
(47% refinancing out of 36 quarters) (92% refinancing out of 14 quarters)
A. Assumptions The first step in the stochastic optimization analysis was to solve the model, previously described, for 38 different scenarios. At time 1 the model was obliged to publish the bond with price closest to 100. At the last quarter all bonds must be redeemed giving rise to 39 quarters where a possible refinancing could take place. For this reason, the total data for each of the three analysis consisted of 1482 quarters. B. Fixed rate loan analysis 1) Refinancing to a lower coupon The total number of refinancing from a higher coupon to a lower one, given by the 38 scenarios, was 119. However, the alternative loans gave the possibility of 512 fictitious refinancing. The total data analyzed consisted therefore of 631 quarters. Fig. 3 shows the results from the CART model where the most significant paths are highlighted with the color red and green. A path was considered significant in case it was logical from a refinancing point of view and if it included enough data to assume the path was not random. Below are the statistics from following the red and green paths. CART Path 1 (Red): Alternative Price > 96.9 (49% refinancing out of 191 quarters) Horizon > 15.5 (84% refinancing out of 91 quarters) Alternative Price > 97.4 (96% refinancing out of 68 quarters) CART Path 2 (Green): Alternative Price > 96.9 (49% refinancing out of 191 quarters) Horizon ≤ 15.5 (18% refinancing out of 100 quarters)
For path 1, in 96% of all quarters where the above given criteria hold, the risk averse model refinanced to an alternative loan. Given all refinancing taking place from a higher to a lower coupon within the model, 55% is explained simply by following this particular path. It should be noted that the horizon is classified as quarters so 15.5 quarters represent approximately 4 years. The rule implies that as long as the price of the new loan is above 97.4 and the horizon is greater than 4 years then it is reasonable to refinance to a lower coupon. Path 2 implies that if the horizon is short one must look at the breakeven and see whether the refinancing will pay off before maturity of the loan. Combined, the two paths, red and green, are able to explain 66% of all refinancing made by the risk averse model to a lower coupon. Other paths found by CART seemed to have relatively small significance. 2) Refinancing to a Higher Coupon The data analysis for refinancing from a lower to a higher coupon consisted of a total 851 quarters. The risk averse model made 85 refinancing in all the 38 scenarios and the alternative loans gave the opportunity for 766 fictitious refinancing. In this case, it was necessary to categorize the data, which was used as input into CART. The data was then divided into defined intervals. The reason for the necessity of categorization, is that when the gap between different values is too large, it may encumber CART in finding a logical pattern within the data. Table 2 gives an example of how the values of breakeven were divided into categories. Other indicators were categorized in a similar manner.
Figure 4. Co mparison of refinancing gain for the rules of thumb, risk averse model and a new strategy constructed of route 1 to 3 fro m results given by CART. Based on data from 2000 to 2010. TABLE 1. . BREAKEVEN BEFORE AND AFT ER CAT EGORIZAT ION
Categorized Breakeven Value
The most significant route found by CART after the categorization is given by:
Figure 5. Co mparison of refinancing gain fo r the ru les of thumb, the CA RT strategy and a new refined strategy given by a sensitivity analysis of the CART model. Based on data from 2000 to 2010.
strategy tested was a combination of paths 1,2 and 3 found by CART. The strategy was solved for 250 scenarios and a graph plotted, based on the same methodology as in Fig. 2. The results are displayed on Fig. 4. As can be seen the new strategy clearly over performs the rules of thumb. The left tail is significantly smaller and the right tail even exceeds the results from the risk averse model. A sensitivity analysis of the CART strategy revealed that the strategy would though improve by removing the constraint from Route 2 which states that no refinancing should be made for a horizon less than 7.5 quarters. This is reasonable since that route already puts constraints on possible losses when refinancing close to maturity by introducing the breakeven. The optimal strategy found from the sensitivity analysis is shown on fig. 5. This new refined CART strategy consists of the following rules:
CART Path 3: Breakeven ≤ -10 years (83% refinancing out of 102 quarters) Put together, the three paths found in the fixed to fixed analysis are able to explain 75% of all refinancing made by the risk averse model. One point to be aware of, when using CART, is that it does not always capture the most intuitive path. For instance, in path 3, CART recommended as well, that the existing price should be greater the 90, since a high percentage of the data did have a price greater than 90. However, from a refinancing point of view, that sort of constraint would be without any reason. Therefore, an expert’s knowledge is at all times required, to analyze and fully understand the results from CART. 3) Testing After gaining a better intuition from CART of which indicator to study the new strategy had to be tested. First
Rule 1: To a lower coupon: Alternative price < 0.98 Horizon < 3 years Rule 2: To a lower coupon: Horizon ≤ 3 years Breakeven ≤ 3 years Rule 3: To a higher coupon: Breakeven ≤ -10 years The refined CART strategy, found with the sensitivity analysis, will be used in all following discussion as the final fixed to fixed strategy. The average number of refinancing for
Figure 6. Changes in outstanding debt, payment and effective rate for the risk averse model and the refined CA RT strategy for h istorical data during the period 2000 to 2010.
Figure 7. Changes in outstanding debt, payment and effective rate for the risk averse model and the rules of thumb for historical data during the period 2000 to 2010.
the 10 year period is 4.1 meaning that the loan should be refinanced approximately every 2.5 years. That is considerably higher than for the rules of thumb which in average refinance 1.6 times or approximately every 6 years. Fig. 6 shows how the refined CART strategy would have performed on historical data from the year 2000 to 2010. It makes 4 refinancing throughout the 10 year period and lowers the effective rate by almost 1%. At the same time where the rules of thumb increase the effective rate of the loan by nearly 0.02%, as can be seen on fig. 7.
The main constraint in the rules of thumb, when refinancing to a higher coupon, specifies that the outstanding debt should be reduced by more than 10%. For only 11% of the refinancing made by the risk averse model did this criteria hold.
Fig. 8 shows the effect of the refined CART strategy when rule 3 is neglected, thereby only allowing refinancing to a lower coupon. The distribution curve is shifted to the left, hence increasing risk substantially. The figure is a good demonstration of the importance of exploiting both refinancing opportunities related to increasing and decreasing market fluctuations in order to decrease the cost of the loan. VI. DISCUSSION In the fixed to fixed analysis it is interesting to see the importance of the alternative price when refinancing to a lower coupon. The market rules of thumb recommend no refinancing unless the difference in coupon is greater than 2% and the alternative price is higher than 95. The CART model showed no sign of the coupon difference being a significant factor. In fact in only 5% of all model refinancing did the coupon difference exceed 1%.
Figure 8. Co mparison of the refined CART strategy and a refined CA RT strategy where only refinancing to a lo wer coupon is allo wed. Based on data from 2000 to 2010.
Path 3 found by CART was able to explain 100% of all refinancing made by the risk averse model to a higher coupon by solely using the term breakeven. VII. CONCLUSION In this paper, new strategies for mortgage refinancing are presented based on intuitions gained from the optimization model of Rasmussen, Madsen and Poulsen (2011). These strategies are shown to over perform the existing rules of thumb, widely known in the market. It has been found that these rules of thumb are too conservative and thereby yield little or no refinancing gains. The new strategies propose that the existing loans should be redeemed in average every 2.5 years while the existing rules of thumb recommend refinancing in average every 6 years. Not all borrowers may wish to refinance their loan so frequently. Nevertheless, the possible gains resulting in using a more aggressive strategy have been demonstrated. Many households have the preference to refinance only to a lower coupon since they have difficulties accepting a temporary increase of their monthly payments in spite of
considerable reduction in outstanding debt. Mortgage institutions are also reluctant to recommend refinancing to a higher coupon. However, consistent with the findings of Rasmussen, Madsen and Poulsen (2011), the results of this paper recommend a refinancing strategy based on a combination of up and down refinancing which can result in a considerable reduction of the total cost of the loan. REFERENCES 
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