Optimal exercise strategies and valuation of prepayment options in Dutch mortgages

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U

NIVERSITY OF

A

MSTERDAM

M

ASTER

T

HESIS

Optimal Exercise Strategies and Valuation

of Prepayment Options in Dutch

Mortgages

Author:

M.A.VAN DERMEIJ

Supervisor: Prof. dr. H.P. BOSWIJK

Supervisor: drs. J.J. Fischer AAG (EY)

A thesis submitted in fulfillment of the requirements for the degree of Master of Science

in the

Amsterdam school of Economics Financial Econometrics

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Declaration of Authorship

I, M.A.VAN DER MEIJ, declare that this thesis titled, “Optimal Exercise Strategies and Valuation of Prepayment Options in Dutch Mortgages” and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a degree at this University.

• Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated.

• Where I have consulted the published work of others, this is always clearly attributed.

• Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed my-self.

Signed: Date:

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“Someone is sitting in the shade today, because someone planted a tree a long time ago” Warren Buffet

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University of Amsterdam

Abstract

Faculty of Economics and Business Financial Econometrics

Master of Science

Optimal Exercise Strategies and Valuation of Prepayment Options in Dutch Mortgages

by M.A.VAN DERMEIJ

Dutch regulations require financial institutions to account for the moneyness of pre-payment options in household mortgages. The available literature does not focus on the moneyness, uses suboptimal strategies, or only provides boundaries for the optimal solution. This thesis presents an algorithm to determine optimal strategies. The efficiency is enhanced by an improved search algorithm. Mortgages without standard redemption patterns (e.g. interest only mortgages) have a relatively high moneyness. Mortgages with standard redemption patterns benefit, by construction, less from the current low savings rate combined with the low outlook. The results appear robust to changes in parameters, but are sensitive to changes in calibration date and modeling assumptions.

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Introduction

The Netherlands has one of the largest mortgage value outstanding relative to the size of the national economy. In 2015, the notional value of these mortgages ex-ceedede638 billion, where the Dutch GNP was e586 billion. A material change in the value of mortgages is therefore also a material change in the value of the total as-sets held by Dutch banks and financial institutions. It is therefore important to value these assets properly. The Dutch Central Bank (DNB) wrote to the Dutch mortgage lenders on 20 March 2015 that they should take proper account of the moneyness of the prepayment option (DNB,2017a). Currently, only one of the fourteen mortgage lenders does this, making this research current and urgent.

The prepayment option gives the mortgage holder, the consumer, the right, but not the obligation, to yearly redeem a bit extra on his mortgage without penalty. Although the moneyness of the option is often not accounted for by Dutch banks and financial institutions, the associated risk of prepayment usually is. This prepayment risk is accounted for in different ways. Often, a certain percentage of the notional value of a mortgage is assumed to be prepayed every year.

Research on prepayments often considers prepayment behavior instead of val-uation of the prepayment option. The research on valval-uation is, unfortunately, not always applicable to the Dutch market. The research that is applicable to the Dutch market either uses suboptimal heuristics, or only provides boundaries for the valu-ation, and therefore needs improvement. A new technique is therefore required.

The main research question is: What is the moneyness of prepayment options in Dutch mortgages? To answer this question, a few sub-questions need to be an-swered. What research has been done and how should it be expanded? What are the available mortgage products and their specifications, e.g. contract rates, time to next interest rate reset, and how is the prepayment option currently taken into ac-count? How to determine exercise strategies? After these preliminary questions, a technique needs to be developed to value the prepayment option.

The rest of this thesis is organized as follows. In Chapter 2 the current literature on mortgage prepayment is reviewed.

In Chapter 3, I use an aggregate mortgage loan tape supplied by EY to study the current outstanding mortgage market in the Netherlands. The different mortgage types are explained and insight is provided in the frequency of the different types of outstanding mortgages.

Chapter 4 is dedicated to explaining how the prepayment option works and presents valuation methods for a single prepayment option and for a restricted set of options. The selection strategy is also explained.

We find that the prepayment option can be modeled as an interest rate swaption. To value this swaption, we need a model to simulate future interest rates, which is presented in Chapter 5. After calibration and transformation, the model is used for the valuation of the prepayment option. The transformation requires historical

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2 Chapter 1. Introduction EURIBOR and savings rates, provided by the ECB and DNB. The model is calibrated with swap prices, provided by Bloomberg.

In Chapter 6, the results of the valuations are presented. Also, the robustness of the results to the parameters and assumptions is presented and analyzed.

Chapter 7 concludes the thesis and gives suggestions for further research. Compared to existing research this thesis offers the following novelties: • a clear valuation of the moneyness of the option (as requested by DNB); • an algorithm that optimizes the exercise strategy for the interest only

mort-gage;

• a dynamic programming algorithm that optimally solves for the linear re-demption mortgage;

• inclusion of tax effects.

A drawback of focusing on the moneyness of the option, instead of on the pre-payment behavior, is that I assume that the funds necessary to make use of the mon-eyness of the option are available. In practice, funds may be limited. However, even then this research is relevant as the optimal strategy and valuation can be carried out by setting the remaining outstanding loan equal to the available funds. In the same manner this research is relevant outside of the Dutch market, e.g. in the United States, as it can be used to optimize the prepayment strategy when the available funds are limited.

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Chapter 2

Literature review

This chapter reviews the available literature on mortgage prepayment. It starts with the general distinction between exogenous and endogenous models, explained in the first section. The second section focuses on Dutch mortgages exclusively, as, due to common Dutch mortgage specifications, much research on the U.S. market is inapplicable. An important work on the pricing of mortgages, of which much served as an inspiration to this thesis, is the PhD thesis of Kuijpers (2004), discussed in the third section of this chapter.

2.1 Exogenous vs. endogenous modeling

The literature discusses two main model classes for assessing the risk of the pre-payment option. First, there are statistics-based exogenous models, which assess prepayment behavior empirically. These models determine which factors had the greatest influence on consumers’ decision-making process in opting to redeem their mortgage early. Second, there are endogenous, option-theoretical models, which examine the option’s value when it is exercised optimally.

The main benefit of an exogenous approach is that it reveals actual consumer behavior. Empirical results show that consumers do not exercise their options op-timally, and that their decision making tends to be informed by macro-economic factors such as changes in interest rates – and the resulting media attention – as well as by personal factors such as age and savings (Chau, Pretorius, and So,2002; Chernov, Dunn, and Longstaff,2016; Quercia, Pennington-Cross, and Tian,2016). A disadvantage of exogenous approaches is that the results might only prove valid in a historical context. In other words, previously observed consumer behavior might not accurately reflect current consumer behavior. An advantage of an endogenous model it that it remains relevant under changing circumstances: it is completely based in theory. The main drawback of an endogenous approach is that it does not accurately represent actors exhibiting less-than-optimal behavior.

Both exogenous and endogenous models are extensively applied to the U.S. hous-ing market (see Archer, Lhous-ing, and McGill (1996), McConnell and Singh (1994), and Stanton (1995)). However, the Dutch housing market is so dissimilar to that in the U.S., that directly applying American research results to questions relating to the Dutch situation would be nonsensical. In the U.S., early penalty free full mortgage redemption is the standard - in the Netherlands penalty free redemption is often lim-ited to only a part of the principal – usually 10 to 20%. This greatly complicates the valuation. (Though there are Dutch mortgage providers that now allow penalty-free full prepayment.1)

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4 Chapter 2. Literature review

2.2 Research on Dutch mortgages

Research on the Dutch mortgage market using exogenous models has been done by Bussel (1998), Hayre (2003), and Alink (2002). For an endogenous approach, see – to an extent – Bussel (1998) and Alink (2002).

Bussel (1998) describes both an exogenous and an endogenous valuation ap-proach. The endogenous approach Bussel applies is a simplified version that only takes into account two basic decision-making factors in modeling consumer behav-ior while not necessarily striving for optimal behavbehav-ior. One of these factors is that mortgages are redeemed in full if the net present value of future cash flows exceeds the outstanding debt. Van Bussel’s exogenous approach fails due to a lack of avail-able data regarding the Dutch situation.

Hayre (2003) provides an empirical, exogenous explanation for prepayment be-havior. His approach does not include any tax effects.

Alink (2002) examines which factors are decisive in leading consumers to re-deem their mortgage early. He finds that – among other factors – refinancing stim-uli, the age of the mortgage, the loan-to-foreclosure value, the distribution channel, and whether the mortgage in question is a first or second mortgage all impact the decision-making process. Alink stresses the fact that, according to the literature, the most important variable among these are the refinancing stimuli (refinancing stimuli relate to the economic-rational exercise of the prepayment option, which is the sub-ject of the present thesis). One disadvantage Alink identifies when it comes to the use of theoretical, endogenous option pricing models, is that – as mentioned above – consumers often do not exercise options optimally, making optimal-call strategies interesting in theory but ineffective in practice, as they do not accurately model the actual risk run by the mortgagee.

2.3 Mortgage valuation and the term structure of interest rates

Kuijpers (2004) describes a method used to determine fair mortgage rates in the Netherlands – the fair rate is the contract rate at which the mortgagee breaks even under risk neutral probabilities. For the simulation of interest rate paths he uses a single-factor model, favoring the Black Derman Toy (BDT) model especially, as it uses a short rate with lognormal distribution, as well as mean reversion (Black, Derman, and Toy,1990). Kuijpers – like Hayre – did not take tax effects into consid-eration.

In determining the fair rate, Kuijpers distinguishes between mortgages without a standard redemption pattern (e.g. interest only mortgages, savings mortgages, investment mortgages) and mortgages with a standard redemption pattern (e.g. an-nuity redemption mortgages, linear redemption mortgages). Mortgages without a standard redemption pattern can be valued optimally in an efficient manner. There are two possible scenarios in this case: either there are more prepayment options than years till mortgage maturity (N > M ), or there are more years till mortgage maturity than prepayment options (N < M ). In cases where N > M , the fair rate can be determined by splitting a mortgage up into American callable bonds, with the bonds being valued using a lattice method. If N < M , a choice must be made which options to exercised. Kuijpers optimizes this decision-making process by applying a dynamically programmable strategy tree.

For mortgages with a standard redemption pattern, the valuation of the partial prepayment option is path-dependent, according to Kuijpers. As a result of this,

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2.4. Conclusion 5 the valuation trees no longer recombine. These non-recombining trees grow expo-nentially as the number of periods increases, meaning there is no efficient way of arriving at an optimal solution. Kuijpers describes a linear programming (LP) prob-lem in which the upper and lower bounds of the fair rate can be determined. The upper bound of the fair rate is determined by the interest rate for interest only mort-gages, as long as the underlying term structure slants upwards. The lower bounds is determined by a strategy in which the partial prepayment option is exercised when it would be optimal to exercise a full prepayment option.

The nodes in the tree in which a full prepayment is optimal form a set which is lo-cated below a certain boundary: the full prepayment boundary. Kuijpers’ strategy is to execute partial prepayment when the interest rate falls below the full prepayment boundary. Furthermore, he recommends the initial use of a lattice, with a new sub-lattice being formed at every intersection point below the full prepayment boundary. This still requires a calculation time of O(T(2+K)), with T being the number of peri-ods and K the total number of prepayment options to be exercised, but is faster than the non recombining tree.

2.4 Conclusion

Concluding, I note that the work of Alink (2002) and Hayre (2003) provide exoge-nous techniques for the valuation of the financial risk associated with prepayments, but not the moneyness of the option. The endogenous work of Bussel (1998) and Kuijpers (2004) (via transformation) provide a way of valuing the moneyness of the option, but are insufficient for mortgages with standard redemption patterns as they either use suboptimal heuristics or only provide boundary values.

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Chapter 3

Mortgages, prepayments and the

Dutch market

This chapter gives an overview of the current mortgage market in the Netherlands. We discuss the different types of mortgages that are present in the current portfo-lio of Dutch financial institutions (FIs). Next, we mention the prepayment option (PPO), and describe how the related risk is currently modeled by many FIs. In the last section an aggregated sample of several Dutch mortgage financiers is analyzed to find what mortgage types and specifications (mortgage interest rate and time to interest rate reset [TIRR]) are frequently found in the portfolios.

3.1 Mortgage types

The mortgage types present in the Dutch market can be divided in those with reg-ular redemption schemes and those without, the interest only combinations. The remaining outstanding mortgage debt over time for the different mortgages types is depicted in Figure3.1.

3.1.1 Regular redemption schemes

The regular redemption scheme mortgages require the lendee to repay a part of the notional value of the mortgage to the lender at set times and amounts. These schemes ensure that the notional value of the mortgage is completely redeemed at maturity.

Linear redemption mortgage

A linear redemption (LR) mortgage is a contract type in which the lendee redeems the notional value of the mortgage at a constant rate, thereby decreasing the remain-ing loan, and interest payments, linearly over time.

Annuity redemption mortgage

An annuity redemption (AR) mortgage is a construction in which the lendee re-deems the notional value of the mortgage such that the total monthly payment (con-sisting of redemption plus interest payment) stays constant over time. The total cost of interest payments on an AR mortgage is higher than that on a LR mortgage, as the remaining outstanding loan decreases slower in the beginning of the mortgage lifetime. AR mortgages do have the benefit that the total payment in the beginning of the mortgage lifetime is lower than that of a LR mortgage.

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8 Chapter 3. Mortgages, prepayments and the Dutch market

FIGURE3.1: The remaining outstanding mortgage debt over time for the different mortgages types, for a 30 year repayment scheme

3.1.2 Interest only combinations

Interest only combinations have long been popular in the Netherlands, because of the tax-deductibility of interest paid on mortgage loans. There is a variety of prod-ucts that are a combination of an interest only mortgage and a form of capital accu-mulation to redeem the notional value of the mortgage at maturity. Due to changes in regulation, the tax deductibility of interest paid on interest only mortgages is only allowed for mortgages that were issued before the first of Januari 2013. Therefore, the interest only mortgages and varieties are exiting the Dutch market. This process will take years, as many of these mortgages have maturities of 30 years.

Interest only mortgage

An interest only (IO) mortgage is a construction in which the lendee does not redeem the notional value of the mortgage, but only pays interest over the notional value.

Life mortgage

A life mortgage combines an IO mortgage with a life insurance. The capital accu-mulated in the life insurance is used to repay the notional value of the mortgage at maturity.

Savings mortgage

A savings mortgage is a combination of an IO mortgage with a savings account on which the lendee is obliged to add fixed amounts of capital such that at maturity of the IO mortgage, the notional value of the mortgage can be redeemed by the funds accumulated on the savings account. This way the interest paid on the mortgage stays high (which is tax efficient for the consumer).

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3.2. Prepayment 9

Investments mortgage

An investment mortgage is similar to a savings mortgage with the difference that the money is invested, e.g. in stocks, bonds and indices, instead of saved. The expected return on investments is higher, but makes full redemption at maturity uncertain.

Hybrid mortgage

A hybrid mortgage is a combination of a savings mortgage and an investment mort-gage. In this type of mortgage the consumer can choose along the lifetime of the mortgage what percentage of the capital is saved and what percentage is invested.

3.2 Prepayment

Mortgages usually have the option for prepayment, such that each year the mort-gage holder can redeem an extra fraction of his mortmort-gage without penalty. When the interest the holder pays on the mortgage is higher than the interest he receives on his savings account, it is, in general, beneficial for the mortgage holder to prepay. (One should also take into account tax regulations etc.) The benefit of this action is the moneyness of the prepayment option. Now that the current rate on savings accounts is only marginally above zero, e.g. ING bank pays 0.10% on savings up to e75,0001_{, and mortgages that have been set 20 years ago might require 6% interest,}

it is often wise for mortgage holders to prepay.2

3.3 Current state of the industry

Prepayment risk is accounted for in different ways. The Constant Prepayment Rate (CPR) assumes that a fixed fraction of the future cash flows is lost yearly due to prepayments. The Public Securities Administration (PSA) designed a prepayment model for the US. In the PSA model the prepayment rate increases in the first 30 months at 0.2% starting from 0.2%, whereafter it remains at 6%. PSA modeling is uncommon in the Dutch industry.

Swaption modeling of the prepayment risk decomposes the mortgage into a set of callable bonds and a non-callable coupon bond. Monte Carlo modeling provides a way of estimating the value of the option by generating interest rate paths and calculating the value of the option for those paths.

Currently prepayment risk in the Netherlands is, by almost all financial institu-tions, modeled via CPR. CPR rates were usually set by lenders at 6%-8% in the past 2 years. The predicted prepayment rates were about 1 percentage point off compared to the actual prepayment rates.

3.4 Composition of the Dutch market

I conduct research on loan tapes, provided by EY, of many financial institutions that are active in the Dutch mortgage market. This section gives insight in the composi-tion of the different mortgages and their market share.

1 https://www.ing.nl/particulier/sparen/vergelijk-alle-spaarrekeningen-en-spaarrentes/index.html

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Mortgage type Frequency Percentage Value weight

Interest Only 292,000 73% 66%

Linear Redemption 12,000 3% 4%

Annuity Redemption 96,000 24% 30%

TABLE3.1: Distribution of the mortage types

FIGURE3.2: Frequency of contract rate and TIRR for IO mortgages

The aggregated sample consists of over 700,000 mortgages, of which 400,000 are IO, LR, or AR. Most of the remaining 300,000 mortgages are life-, savings-, or hybrid contracts.

For the remainder of this thesis I focus on three standard forms: IO, LR, and AR. The distribution of these mortgage types is displayed in Table3.1 (where the numbers are rounded to nearest percentage).

This section shows how the TIRR and mortgage contract rates are distributed per mortgage type.

3.4.1 Interest only mortgages

IO mortgages have long been popular because of their maximal use of tax benefit. Due to changing tax regulations, this benefit has ceased over time, though still a large portion of the current Dutch mortgages consists of this type.

In Figure3.2we see a high frequency at a TIRR of 5 years, with a contract rate of around 5%. This is high compared to the current rate which is about 1.5% − 2.0%.3

3.4.2 Linear redemption mortgages

LR mortgages are not very popular in the Netherlands. In Table3.1 it is depicted that they make up only 3% of the selected sample.

As seen in Figure 3.3, LR mortgages have, on average, a longer TIRR than IO mortgages. Many have TIRRs of 9 or 20 years. The average rate is also lower (de-creasing the value of PPO) for this type of mortgage.

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3.4. Composition of the Dutch market 11

FIGURE3.3: Frequency of contract rate and TIRR for LR mortgages

FIGURE3.4: Frequency of contract rate and TIRR for AR mortgages

3.4.3 Annuity redemption mortgages

The classical AR mortgage appeals to consumers more than the LR mortgages. This may be due to the higher tax benefit found in these mortgages in the first years of the contract, but also to the fact that the payments remain constant over time, which consumers may find comfortable.

The shape of the distribution of rate and TIRR of AR mortgages seems very sim-ilar to that of LR mortgages, as is shown in Figure3.4.

3.4.4 Selection of mortgages

Based on the graphs in this section, I select three representative mortgage specifica-tion for each mortgage type, presented in Table3.2. The value of the prepayment option of these mortgages are analyzed in Chapter6.

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Mortgage type Maturity Mortgage rate

Interest Only 5 5% Interest Only 8 3% Interest Only 19 2.5% Linear Redemption 4 4% Linear Redemption 7 3% Linear Redemption 19 2.5% Annuity Redemption 5 5% Annuity Redemption 8 3% Annuity Redemption 19 2.5%

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Chapter 4

Valuation of the prepayment

options

This chapter presents a technique to value the PPO. An FI needs to know his ex-pected losses, caused by the possible exercise of PPOs, hereafter called PPO risk. To determine this risk, he must know which individual PPOs a rational consumer exer-cises - the consumer’s optimal strategy. To determine this strategy, the values of the PPOs are required.

In the first section of this chapter I explain the relation between PPOs and swap-tions. To make the modeling comprehensible, I use a set of assumptions, presented in Section4.2. Section4.3provides the valuation methods for individual PPOs. The optimal strategy is presented in Section 4.4. This chapter ends with a method to determine the total PPO risk an FI bares on a mortgage contract.

Because of the tax effect, explained in Section5.2.3, the consumer effectively pays a different mortgage rate than the FI receives. Therefore, the prepayment values dif-fer between provider and consumer. In the determination of the consumer’s strat-egy we therefore work with the effective mortgage rate r∗m(defined in Equation5.7), while for the FI we value the prepayment by the contract mortgage rate rm. The val-uation formulas presented in this chapter are identical for the consumer as for the FI, so where one reads rmfor the valuation of the FI, one should also read r∗mfor the consumer.

To determine the present value and the strategies, we use a notation that corre-sponds to a trinomial tree. This tree is discussed in detail in Chapter5. For now it suffices to know that the interest rate can follow different paths on a tree. The tree consists of nodes, labeled (j, i), and branches between the nodes. j represents the level in the tree and i the time period. Each node (j, i), except the terminal nodes, has 3 child nodes (k −1, i+1), (k, i+1) and (k +1, i+1). Here the level k corresponds to the most probable child node.

4.1 Introduction on relation PPO - Swaption

To determine the total PPO risk of a certain contract for an FI, note that this risk is determined by the consumer’s strategy combined with the value of the option. The consumer’s strategy maximizes his own profit and thereby maximizes the institu-tion’s loss. To illustrate the modeling of PPO risk, I provide an example:

Consider a consumer that has an IO mortgage with a remaining maturity of 4 years, remaining loan RL = 20, 000, initial notional value N = 100, 000 and prepay-ment size 10% · N = 10, 000.

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14 Chapter 4. Valuation of the prepayment options In the first year the consumer chooses whether to exercise the PPO available that year. The relevant factors are: What cash flows follow from exercising the PPO and what are the further consequences of exercising this PPO?

If the consumer exercises the PPO (prepaying with money from his savings ac-count) this lowers his savings by e10,000. He therefore receives rs· e10,000 less interest on his savings account every year. Also, the remaining mortgage debt de-creases bye10,000, implying that he spends r_m∗·e10,000 less on mortgage interest payments every year. Now assume that the savings rate, rs, is a floating rate and fu-ture savings rates are uncertain. Further assume that the mortgage interest rate, rm, is fixed for the remainder of the mortgage lifetime. The prepayment is then a fixed-floating interest rate swap, in which the fixed rate is the mortgage rate, the fixed-floating rate is the savings rate, the notional is the size of the prepayment and the maturity is the remaining lifetime of the mortgage. The prepayment on an IO mortgage can be seen as a receiver swap. I.e. because of the prepayment you now effectively re-ceive the fixed mortgage rate and pay the floating savings rate. The present value of the PP is then equal to the PV of the swap with the before mentioned specifi-cations. The PPO is therefore equal to a swaption. For the valuation of a swap the standard pricing formulas (as in Hull (2006)) apply. For a more detailed explanation, see AppendixC.

Concerning the further consequences of exercising a PPO, note that exercising the PPO lowers the remaining mortgage loan frome20,000 to e10,000. The PPOs in year 2, 3 and 4 (each of sizee10,000) can then not all be exercised as the remaining loan (e10,000) is insufficient to exercise more than one PPO of e10,000. So exercising a PPO in year 1 reduces the number of allowed prepayments in the subsequent years. The total value of the PPOs in the mortgage contract is therefore not simply equal to the sum of the values of the individual PPOs, as it is not allowed to exercise all four PPOs. The valuation problem of the total PPOs is therefore not solvable by the standard formulas, as in Hull (2006), alone. We need to determine the strategy by which a consumer selects the PPOs he wants to exercise in all scenarios.

In year 1 he is only allowed to exercise the PPO of year 1. Nonetheless, an in-telligent consumer determines the present value of all four PPOs to determine his strategy. In our example the consumer may only exercise two PPOs in total. So in year 1 the consumer determines which PPOs have the highest PV. If one of the two of the highest valued PPOs is that which expires in year 1, then the consumer exercises the PPO of year 1. If not, e.g. the PPOs of year 2 and 3 have a higher PV than the PPO of year 1, then the consumer does not exercise the PPO in year 1.

In year 2, and the other subsequent years, the consumer must also choose whether he wishes to exercise the PPO of that specific year. A difficulty is that we do not know in year 1 what the ranking of the values of the PPOs of year 2, 3 and 4 will be in year 2. The ranking depends on the interest rate reality at year 2 and the expectations of year 3 and 4 at year 2. Two possible expected interest rate paths, starting from year 2, are presented in Figure4.1. In the first interest rate path, Figure4.1(A), we see that the current savings interest rate, rs, in year 2 is 8%, and the consumer expects the savings rate to decrease in year 3 and 4 to 3%. In that example the PV of the swap in year 2 has a negative value of 1 ∗ (4% − 8%) + 2 ∗ (4% − 3%) = −2% of the size of the prepayment (simplified estimate), where the swaps of the year 3 and 4 have a posi-tive value (as visible in green in graph (A)). In the second interest rate path, Figure

4.1(B), the savings rate is initially low, creating a positive value for the swap of year 2, but the savings rate is expected to move back above the mortgage rate, implying a negative value for the swaps of year 3 and 4. In scenario (A), the consumer does

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4.2. Assumptions 15 (A) years rate rm rs 2 3 4 3% 4% 8% (B) years rate rm rs 2 3 4 0% 4% 6%

FIGURE4.1: Interest rate scenarios starting in year 2

not exercise the swaption of year 2 as it is not the most valuable swap, in scenario (B) the consumer exercises the swap as it is the most valuable swap.

As the valuation of the entire set of PPOs is done in year 1, the dependency of the value of the total of PPOs on a consumer’s actions in different future scenarios has to be accounted for. A valuation method therefore consist of the valuation methods of the individual swaps, combined with a model to determine the future scenarios in which the consumer makes decisions, and a strategy that the consumer applies in these different scenarios.

4.2 Assumptions

To make the modeling of the value of the prepayment and the optimal strategy com-prehensible and to clearly define the framework in which we operate, certain as-sumptions are made and described in this section.

4.2.1 Assumptions on consumers

Concerning consumer behavior and their characteristics I assume the following. • Consumers behave rationally in the choice of their prepayment strategy, i.e.

they exercise an option when that results in maximizing the present value of their cash flows.

• Consumers always have sufficient funds available for prepayment. If, in real-ity, consumers do not have the money in their savings account, then it could also be made available by withdrawing mortgage equity1(in which case there should be made some adjustments to the rates in the determination of the strat-egy).

• Consumers prepay the maximum allowed (penalty free) amount, if they choose to prepay. As when an option is preferred over others, it is optimal to fully ex-ploit that option.

• The income of a consumer equals 1/5th of the notional value of his mortgage. That corresponds to the maximum initial mortgage loan one generally receives from a lender (information based on requesting banks in the Netherlands on information on maximum loan value for different levels of income).2 In prac-tice the ratio can be lower, e.g. 1/4 or 1/3, as income generally increases over 1_{’Overwaarde’in Dutch}

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16 Chapter 4. Valuation of the prepayment options time while the notional value of the mortgage remains the same. How the results differ as the ratio changes is investigated in Section6.3.

• The relevant discount factor for the consumers is equal to the savings rate. That itself presumes that consumers typically evaluate the value of income next year to the value they receive on a savings account.

• Consumers only consider prepayment in December of each year. This is in line with the research of Alink (2002) on prepayment behavior. This is plausible as consumers are inclined to review their financial situation at the end of each year so to make use of the capital tax advantage that ends at December 31st. (Lowering your savings before December 31st implies that you do not have to pay taxes over the amount you prepaid.)

4.2.2 Assumptions on modeling

Concerning the modeling procedure I make the following assumptions.

• The horizon, T , of the prepayment strategy optimization is set equal to the TIRR. This is plausible as at the next interest rate reset the new mortgage rate should be fair, e.g. if the interest rate has loweres over time, then the new mortgage rate at the reset date is lower as well. Though, after the refinancing there still is a positive prepayment expected value. Properly accounting for this would involve modeling the mortgage rate setting of the lenders. I do not investigate that in this thesis.

• I model the rate movement on a yearly basis, as consumers decide in December of each year whether to prepay. Each node in the interest rate tree corresponds to a December. At node t = T , there are still twelve months of mortgage dura-tion. Increasing the number of steps in a year would increase the similarity of the valuation in the tree compared to valuation with the differential equation. A disadvantage of increasing the number of periods in a year is the computa-tional power it requires to compute the strategies.

• The payout of a prototypical swap is at the end of the period for which the swap is set up. The floating rate that determines the payout, is determined at the beginning of the period. So at i = 1 the rate is determined that is used at the pay out at i = 2. For the consumer, I also apply this convention. Thus, the yearly benefit of a prepayment option is received in full at the end of the year.

4.3 Value of a single prepayment

In this section the value of a single prepayment is determined for the different mort-gage types. The value for the FI is calculated using rm, the value for the consumer is calculated using r∗_m(the tax adjusted mortgage rate).

4.3.1 Value of a prepayment on interest only mortgages

The present value at node (j, i) of a prepayment on an IO mortage with exercise date l (P P IOl), which is effectively a receiver fixed-floating swap with exercise date l(Sl), is given by

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4.4. Optimal strategy for exercising prepayment options 17 P Vj,i(P P IOl) = P Vj,i(Sl) =            e−rs(j,T )∆t(T )_[r m− rs(j, T )] · ∆t(T ) · P P, if i = T e−rs(j,i)∆t(i)_[r m− rs(j, i)] · ∆t(i) · P P if i < T +e−rs(j,i)∆t(i)Ph=k+1 k−1 p(j, h, i) · P Vh,i+1(Sl), (4.1) that is, the P Vj,i of a swap at the end of the TIRR, T, is equal to the discounted value of the difference between the fixed and the floating leg of the swap for the duration of the last period, ∆t(T ), on the notional P P (Hull,2006; Brigo and Mer-curio, 2007). In the non-terminal nodes, i < T , the P Vj,i of the swap is the value increment in period i plus the discounted expected value one period ahead, (i + 1), with p(j, i, h) the probability of moving form node (j, i) to (h, i + 1).

4.3.2 Value of a prepayment on linear redemption mortgages

A prepayment on a LR mortgage can be expressed as a combination of swaps. When a PPO on a LR mortgage is exercised, that reduces the remaining notional value to RL − P P . Lowering the remaining notional value changes the amortization scheme. The regular payments on the mortgage are adjusted such that the mort-gage is completely redeemed at maturity at a constant redemption rate. The regular payments at t = i + 1, ..., T , when there is no option exercise at time i, are

Yearly redemption|¬Ex = RLj,i(¬Ex)

T − i . (4.2)

Here, the index ¬Ex means that the PPO is not exercised. RLj,i(¬Ex) is the re-maining loan at node (j, i) when the PPO, exercisable at time i, is not exercised. RLj,i(Ex)is the remaining loan at node (j, i) when the PPO exercisable at time i is exercised. These regular payments, the yearly redemptions, are swaps. One prepay-ment (for LR mortgages) causes a series of swaps, the first one, a receiver swap, at time i = l with a notional of P P , followed be a series of payer swaps (in value equal to minus a receiver swap) at every regular payment in time i = l + 1, ..., T with

Notional of swap l = RLj,i(¬Ex)

T − i −

RLj,i(Ex)

T − i (4.3)

= RLj,i(¬Ex) − RLj,i(Ex)

T − i (4.4)

= P P

T − i. (4.5)

The total value of a prepayment on a LR mortgage at every node (j, i) with exer-cise date l, expressed in swaps with a notional of P P is

P P LRl= Sl− 1 T − l X m>l Sm. (4.6)

4.4 Optimal strategy for exercising prepayment options

In the previous section, individual prepayments were valued by considering them as swaps. The valuation of the total set of options in a contract is more difficult.

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18 Chapter 4. Valuation of the prepayment options One cannot simply sum the present values of the individual prepayment, as it is not always possible to exercise all swaptions. E.g. when you have a remaining TIRR of 20 years, and the remaining loan is 90% of the initial amount/notional, then you cannot prepay 20 times a portion of 10% of the notional. For IO mortgages, you can exercise 9 swaptions (9=90%/10%). For the LR mortgage you can exercise even less as the remaining loan also diminishes through regular payments.

4.4.1 Optimal strategy for exercising PPOs on interest only mortgages

The optimal strategy for IO mortgages is determined as follows. At every node (j, i) and all possible states of RS, you know the present value of all PPOs, but only choose whether to exercise P P IOigiven rs. To make that choice, you rank all remaining PPOs by their present value.3 When the present value of P P IOlis among the top rs PPOs, you exercise it. The procedure is displayed in Algorithm1.

Algorithm 1:Determining the PPO exercise strategy for IO mortgages

1 for rs=1,. . . ,RS do 2 for i=1, . . . , T do

3 for j=T-(i-1),. . . , T+(i-1) do

4 if P Vj,i(P P IOi) ∈ maxrs,m≥i{P Vj,i(P P IOm)}then

5 Stratj,i(rs) = 1

6 else

7 Stratj,i(rs) = 0

This algorithm creates a recombining 3D tree. For the decision to exercise the PPO on (j, i, rs) = (4, 1, 5), it is irrelevant whether other PPOs were exercised at i = 3or at i = 2. This tree, named Strat (for strategy) indicates whether a PPO is exercised in each node of the 3D tree. (Stratj,i(rs) = 1if exercised and 0 otherwise).

4.4.2 Optimal strategy for exercising PPOs on linear redemption mort-gages

Unlike prepayment strategies for IO mortgages, prepayment strategies for LR mort-gages are not suited for representation in a recombining 3D tree as prepayments and regular redemption schemes cause a great increase in the number of states in the state space of RL, which are also not equidistant. Therefore, we turn to a dynamic programming approach.

I combine strategy determination and valuation of the total set of PPOs in one algorithm, which is discussed in Section4.5.2.

4.5 Value of the total set of PPOs

In this section a valuation technieque for the set of PPOs is presented, given the optimal strategy for IO mortgages and the dynamic programming structure for LR mortgages.

3_{The consumer determins the strategy, therefore the present value is calculated using r}∗ m.

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4.5. Value of the total set of PPOs 19

4.5.1 Value of the total set of PPOs on interest only mortgages

Combining the 2D value tree for individual PPOs with the 3D strategy tree, results in a valuation of the total set. That is, multiplying the binary indices of Strat with the option values, gives the value of the PPOs that you exercise.

The value of the set of options at (j, i, RS) = (0, 1, N/P P ) is found by setting P Vj,T(Stratj,T(RS)) = Stratj,T(RS) · P Vj,T(P P IOT) (4.7) and recursively working back to i = 1, by

P Vj,i(Stratj,i(RS)) =Stratj,i(RS) · P Vj,i(P P IOi)+ e−rs(j,i)·∆T (i)

k+1 X

h=k−1

p(j, h, i)P Vh,i+1(Stratj,i(RS − Stratj,i,RS)) (4.8)

4.5.2 Value of the total set of PPOs on linear redemption mortgages

The value of the set of PPOs is dynamically calculated, in Equation4.9, by consider-ing in each node, whether the present value of exercisconsider-ing the option and thus reduc-ing RL is higher than the present value of not exercisreduc-ing the option and maintainreduc-ing the RL.

Bj,i(RLj,i) = max[P Vj,i(P P LRl)+ e−rs(j,i)·∆T (i)

k+1 X

h=k−1

p(j, h, i)Bj,i(RLj,i− P P ),

0 + e−rs(j,i)·∆T (i)

k+1 X

h=k−1

p(j, h, i)Bj,i(RLj,i)]

(4.9)

There are two stopping criteria, presented in equation4.10, one for when there is no remaining loan left and one for when the time horizon is reached.

if RL =0

Bj,i(RLj,i) = 0 if i = T

Bj,T(RLj,T) = max[P Vj,T(P P LRT), 0]

(4.10)

As the consumer determines the strategy, he considers the present value using r∗_m. The present value for the FI is simultaniously calculated using rm.

This solution method is highly computationally demanding, as the optimization does no longer take place in a recombining tree. The number of calculations grows exponentially with the horizon T . Some computational easing can be achieved in reusing calculation results from similar calculations. E.g. when we have determined that it is optimal to exercise a PPO in i = T − 1 when RL = 1/2 · N/P P , then it is also be optimal to exercise that option for all scenarios where RL > 1/2 · N/P P .

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20 Chapter 4. Valuation of the prepayment options

Increasing optimization efficiency

The distribution of RL over the PPOs is a scarcity problem, so when Stratj,i(x) = 1, then Stratj,i(y) = 1 ∀y > x. There must be some threshold zj,i, such that ∀RL > zj,i, Stratj,i(RL) = 1and ∀RL < zj,i, Stratj,i(RL) = 0. Finding zj,idecreases the number of scenarios to analyze in Equation4.9. zj,i might be found analytically, or, as done in this section, by a search algorithm.

I construct a matrix of lower bounds

¯zj,i, and one of upper bounds ¯zj,i. The initial values of

¯zj,i and ¯zj,i, presented in Equation 4.11, are set as follows. When P Vj,i(P P LRl) < 0, exercise is never optimal, so

¯zj,i = ¯zj,i = ∞. In terminal nodes, there are no other PPOs to consider, so exercise, given that P Vj,i(P P LRl) > 0, is always optimal

-¯zj,i = ¯zj,i = 0. In all other nodes, exercise is not possible when RL < 0, so

¯zj,i= 0, and there is no further information to assume anything about z, so ¯zj,i= ∞. ¯zj,i=     

∞ for (j, i)|P Vj,i(S(LR)i) < 0, 0 for (j, i)|i = T,

0 for all other (j, i).

¯ zj,i =     

∞ for (j, i)|P Vj,i(S(LR)i) < 0, 0 for (j, i)|i = T,

∞ for all other (j, i).

(4.11) For all iterations where Stratj,i(l)is requested, check if RL <

¯zj,i(the PPO is not exercised), or RL > ¯zj,i(the PPO is exercised). If

¯zj,i < RL < ¯zj,i both scenarios of Equation4.9are considered. If

P Vj,i(P P LRl)+e−rs(j,i)·∆T (i) k+1 X

h=k−1

p(j, h, i)Bj,i(RLj,i− P P ) >

0 + e−rs(j,i)·∆T (i)

k+1 X

h=k−1

p(j, h, i)Bj,i(RLj,i),

(4.12)

then Stratj,i(RL) = 1and the upper bound is updated, ¯zj,i = RL. If not, Stratj,i(RL) = 0and the lower bound is updated,

¯zj,i = RL. The complete algorithm, combining the value of the set of options, the strategy and the threshold updating is presented by Algorithm2.

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4.6. Conclusion 21

Algorithm 2:Determining the strategy for PPO on LR mortgages

1 if RL = 0 then 2 B(j, i, RL) = 0; 3 if i = T then 4 B(j, i, RL) = max{P Vj,T(P P LRT), 0}; 5 if RL < ¯zj,ithen 6 Stratj,i(RL) = 0; B(j, i, RL) = 0 + e−rs(j,i)∆t(i)Ph=k+1 k−1 p(j, i, h) · B(h, i + 1, RL);

7 else if RL > ¯zj,ithen

8 Stratj,i(RL) = 1; B(j, i, RL) =

P Vj,i(P P LRi) + e−rs(j,i)∆t(i)Ph=k+1k−1 p(j, i, h) · B(h, i + 1, RL − P Pef f ect);

9 else

10 if Value if not exercised > Value if exercised then 11 Stratj,i(RL) = 0;

¯zj,i = RL;

12 B(j, i, RL) = 0 + e−rs(j,i)∆t(i)Ph=k+1

k−1 p(j, i, h) · B(h, i + 1, RL);

13 else

14 Stratj,i(RL) = 1; ¯zj,i = RL;

15 B(j, i, RL) =

P Vj,i(P P LRi)+e−rs(j,i)∆t(i)Ph=k+1k−1 p(j, i, h)·B(h, i+1, RL−P Pef f ect);

4.6 Conclusion

The determination of the exercise strategy is a scarcity problem of the distribution of the remaining loan over the different PPOs. The optimal strategy consists of ex-ercising the currently exercisable swap when it is in the set of the currently highest valued swaps. The size of this set is simple for IO mortgages, but complicated for regular redemption mortgages. The search algorithm for finding these strategies is improved in speed by updating the thresholds on the remaining loan for which the strategy is already known. The valuation of PPOs is dependent on future interest rates. The modeling of these rates is done in Chapter5. The results of the valuation, based on the strategy and the interest rates, are shown in Chapter6.

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23

Chapter 5

Interest rate models

In this chapter the savings rate, rs, is modeled. We expose how to calculate different savings rate scenarios based on a transformation of a short rate model fitted to EURI-BOR, taking into account the relation to EURIBOR and tax regulations. We further discuss how the parameters of the EURIBOR rate can be calibrated and estimated and whether market calibration or historical estimation is preferred for our studies.

5.1 Analyses of the relation EURIBOR - savings rate

For the modeling of the savings rate, which is done by a tree method, we need a parameter set and quoted term structures. For the calibration of the parameters, a set of derivatives is required which are not traded for the savings rate. Therefore, I resort to modeling EURIBOR via a tree method and transform the modeled scenarios to savings rate scenarios. Considering the graphs of EU RIBOR6mand the savings rate, as presented in Figure 5.1, note that the shape of the graphs are somewhat alike, though EURIBOR is smoother. By transforming the EU RIBOR6m linearly, we approximate the savings rate, with the following model:

rs(t) = φ + χ · EU RIBOR6m(t) + . (5.1)

For estimating the constant adjustment of the EU RIBOR6mto the savings rate I fit Equation5.1 on the data set starting January 2003 up to December 2016. The savings rate is taken from the DNB database (DNB,2017b). The EU RIBOR6mrates are taken from the ECB database (ECB, 2017). I find that φ = 0.3463% and χ = 0.0840. The fitted savings rate is displayed in Figure5.2.

This assumption on the relation between EURIBOR and the savings rate has a practical necessity, but in reality the relation between rs(t)and EU RIBOR6m(t)is not this simple. E.g. rs does not change as frequently as EURIBOR does. Also rs responds faster to decreases in EURIBOR then to increases, as banks can plausibly explain a decrease in rsto their customers, while not being inclined to increase rates fast. The three large banks in the Netherlands effectively set rs in a competition setting. When the other banks are not increasing their rates, there is no incentive for consumers to re-allocate their savings. So far, banks are not inclined to increase their rates to pursue a larger market share. Alternatively, the relation can be modeled by using a nonlinear transformation, or by a distributed lag, assuming that banks set the savings rate depending on the recent history of EU RIBOR6m.

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24 Chapter 5. Interest rate models

FIGURE5.1: EU RIBOR6mand savings rate

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5.2. Adjusting mortgage and savings rate for tax effects 25

5.2 Adjusting mortgage and savings rate for tax effects

5.2.1 Mortgage interest tax deduction

Comparing the savings rate, rs, with the mortgage rate, rm, without taking into account the tax effect of outstanding mortgage debt and of capital savings would not represent the evaluation that a consumer makes. In the Netherlands, interest paid on mortgage debt is tax deductible from income tax. As our mortgage loan tapes do not include the taxable income of the lendee, I assume that the income is equal to 1/5 of the notional value of the mortgage. I assume this ratio as it is more or less the maximum loan size a consumer could receive from a lender.

Income(N ) = 1

5 · N (5.2)

The relation between income and taxation level is depicted in Equation5.3. These ranges and taxation levels are for the year 2017 (Belastingdienst, 2017a). The de-picted rates are marginal tax rates, e.g. only the amount of income above€19,981 is taxed at 40.8%. As interest paid on mortgage debt is tax deductible, only this up-per regime is relevant for the valuation. I assume that the entire interest payment is within the highest regime applicable to a specific consumer. In reality, the interest payment may be such that half of the payment is within the 40.8% regime and half is within the 36.55% regime.

T axincome(N ) =      36.55% ifN < 99 905 ←→ Income < 19 981, 40.8% ifN ∈ (99 905, 335 355) ←→ Income ∈ (19 981, 67 071), 52% ifN > 335 355 ←→ Income > 67 071. (5.3) The mortgage rate, adjusted for mortgage interest tax deduction, for the con-sumer is

˜

rm= (1 − T axincome(N )) · rm (5.4)

E.g. for a notional value of€300,000, a taxable income of €60,000 is presumed. Corresponding to a taxation of 40.8%. If the mortgage contract rate is 5% then ˜rm = (1 − 0.408) ∗ 5% = 2.96%.

5.2.2 Taxation of capital

In the Netherlands, capital assets are taxed. The tax regime on savings changes in 2017 (Belastingdienst,2017b). From then on, consumers pay the taxes as in Equation

5.5. T axcapital=      0% for savings < 25 000, 0.86% (2.871% · 30%) for savings ∈ (25 000, 75 000), 1.38% (4.6% · 30%) for savings ∈ (75 000, 975 000). (5.5)

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26 Chapter 5. Interest rate models So the effective savings rate is

rs= rsavings contract rate− T axcapital (5.6) E.g. for a savings amount of €50,000 and a savings contract rate of 0.5%, the effective savings rate rsis 0.5% − 0.86% = −0.36%.

For this thesis I suppose that the relevant regime is 0.86%. I choose as I do not have any data on the savings of consumers and find the corresponding amount of savings intuitively plausible.

5.2.3 Total taxation

The decision to exercise the prepayment option only depends on the spread r∗m− rs, so to keep the tax adjustments easy, I adjust the mortgage rate for both taxations. The effective mortgage rate becomes

r_m∗ = (1 − T axincome(N )) · rm+ T axcapital (5.7)

5.3 Short rate models

I model the future interest rates with a short rate model, as is the industry standard. More advanced models predict interest rates more accurately, but given that interest rate modeling is not the main objective of this thesis I do not use those.

5.3.1 Number of factors when using a short rate model

A decision is made about the number of factors included in the model. In Kuijpers (2004) one-factor models appeared a solid choice. Multi-factor models did not de-crease the pricing error. Driessen, Klaassen, and Melenberg (2003) state that the Hull White model does not improve below a 8.5% prediction error for swaptions when using more than a one-factor model.

Fan, Gupta, and Ritchken (2007) find that the prices of swaptions calculated by one-factor models perform equally well as those calculated by two-, three- or four-factor models. For hedging purposes a four-four-factor model improves the results. An overview of the performance of different one- and two- factor models can be found in Hull and White (1990), James and Webber (2000), and Jarrow (2002).

An overview of different short rate models, e.g. Merton, Ho&Lee, Black Derman Toy, Black Karasinski, Vasicek, Hull White, Cox-Ingersoll–Ross, can be found in Hull (2006).

5.3.2 Choice of short rate model

Within the set of one-factor short rate models, there are several options. For this thesis, I want a model that incorporates negative interest rates and is fitted to a term structure. I use the one-factor Hull White Vasicek (HWV) model as it suffices and is widely used in the industry.

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5.4. The Hull White Vasicek model 27

5.4 The Hull White Vasicek model

The HWV model, as described in Hull and White (1994) assumes that the short rate moves satisfying the diffusion equation

dr(t) = [θ(t) − α · r(t)] dt + σ dW (t), (5.8) where θ is the long run parameter, α is the mean-reversion parameter, σ is the volatility and Wtis a standard Brownian motion. This process is a mean-reverting Ornstein-Uhlenbeck process. I represent the equation by constructing a trinomial tree. As shown by Hull and White (1990), a trinomial tree is capable of fitting both the expected drift of the short rate and the instantaneous forward volatility.

5.4.1 Building the tree

The discretization procedure which I follow matches that of Brigo and Mercurio (2007). The long run parameter θ(t) is re-parameterized to a(t), by a(t) = θ(t)/α. We build a tree that exactly matches the moments of Equation5.8. These moments are given by:

E{r(t)|Fs} = r(s)e−α(t−s)+ a(t) − a(s)e−α(t−s), Var{r(t)|Fs} = σ2 2α h 1 − e−2α(t−s) i . (5.9)

By subtracting the term structure component, a(t), from r(t), we have x(t) = r(t) − a(t). In discrete time the formulas are then given by Equation5.10. In which node xj,iis the value of x at level j and time i. The equations transform to:

E{x(ti+1)|x(ti) = xj,i} = xj,ie−α∆ti =: Mj,i, Var{x(ti+1)|x(ti) = xj,i} =

σ2 2α h 1 − e−2α(∆ti)i_{=: V}2 i , (5.10)

We set xj,i = j∆xi and ∆xi = Vi−1 √

3. This means that in step i, we have j different states, with vertical distance between the nodes ∆xi. From every node (j, i) branches are built to three nodes in time (i + 1). The three child nodes are centered around node k. For every (j, i), k is determined by

k = round Mj,i ∆xi+1 , (5.11)

where round(b) returns the integer closest to b. Thus, k is chosen such that the center child node is as close as possible to the expected value of xj,i. This also means that not all child nodes have three parent nodes, even though all parent nodes have three child nodes, as can be seen in Figure5.3. In this figure k(−1, 1) = (0, 2), therefore the three child nodes of (−1, 1) are (1, 2), (0, 2) and (−1, 2). Furthermore node (−2, 2) does not have any parent node.

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28 Chapter 5. Interest rate models (2,2) (1,1) (1,2) (0,1) (0,1) (0,2) (-1,1) (-1,2) (-2,2) Pu P m P_d

FIGURE5.3: Example HWV trinomial tree

Next the transition probabilities are defined. Start by defining ηj,k= Mj,i− xi+1,k for each j, used in the probabilities:

Pu = 1 6 + η_j,k2 6V_i2 + ηj,k 2√3Vi , Pm= 2 3 − η_j,k2 3V_i2, Pd= 1 6 + η_j,k2 6V2 i − ηj,k 2√3Vi , (5.12)

where Puis the probability of moving up, Pmis for middle, and Pdfor moving down.

5.4.2 Re-entering the term structure

In the first stage of the tree building we only modeled the Ornstein-Uhlenbeck pro-cess x(t). In this part we reintroduce the a. a1 is directly extracted from the market discount rate.

a1 = ln(P (0, t1))/t1, (5.13)

where P (0, t) is the market discount rate for maturity t; the price that that you pay in the market at time t to receive one unit of currency at time T . So P (t, T ) = e−R(t,T )τ (t,T ). In which R(t, T ) is the continuously compounded spot interest rate prevailing at time t for maturity T and τ (t, T ) is the time difference between t and T in years. For i = 2, ..., T we recursively build Qj,i+1, the Arrow-Debreu prices, and

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5.5. Calibrating the interest rate model 29

ai, the term structure component. This gives: Qj,i+1=

X

h

Qh,iq(h, j) exp(−(ai+ h∆xi)∆ti), (5.14)

wherein q(h, j) is the probability of moving from node (h, i) to node (j, i+1). Finally, ai = 1 ∆ti ln P jQj,iexp(−j∆xi∆ti) P (0, ti+1) . (5.15)

5.5 Calibrating the interest rate model

Calibration of the model can be done either by calibrating to traded instruments (caps, swaptions, etc.) resulting in a risk neutral parameter set, or to historical in-terest rate data, resulting in a real world parameter set, P. In the determination of the optimal strategy, the actor is the consumer. As this consumer does not have full access to the market of financial instruments, risk neutral probabilities do not determine his decisions. He wants to maximize his expected payoff by real world probabilities.

An FI wants to model its expected loss by the risk neutral probability measure, Q. Three reasons for that are: The FI wants to know his risk by measures that he already uses in his portfolio. The FI has access to complete capital markets where he can hedge his risk. Constructing a probability measure by itself may induce expectations about the future that are very different from what the market expects.

Therefore, the scenarios and their probabilities will be calculated risk neutral. In these interest rate scenarios on Q the consumer determines if he exercises his PPO. The consumer makes his decision on his probability measure, which might not be Q, as the conditions for risk neutral valuation do not apply to him, and will turn to modeling his own scenarios under P.

Properly accounting for both probability measures requires building an interest rate tree for the FI to determine the possible scenarios and the transition probabili-ties between the nodes, and then determining on all nodes (j, i) a new tree for the consumer to determine the optimal strategy. Every individual consumer tree would require its own parameter set and expected term structure. I will not construct ways to determine these parameters in all nodes.

Instead, I make a simplification of the problem. I assume that both the FI and the consumer use the risk neutral probability measure. As explained hereafter, this simplification should not considerably change the results and if the strategy under Q would differ from that under P, then the expected loss for the FI under Q will not be smaller than the expected loss under P.

First, why the simplification will not considerably change the strategy of the con-sumer. The strategy for the consumer, Stratj,i(RS), is a binary operator. Therefore, it might not matter so much whether the strategy is determined under P or Q. E.g. when under P: P Vj,i(P P LRi) = 1050and P Vj,i(P P LRi+1) = 950then you prefer exercising swap i, i.e. Stratj,i(RS) = 1, when under Q: P Vj,i(P P LRi) = 1000 and P Vj,i(P P LRi+1) = 925then you still prefer exercising swap i, i.e. Stratj,i(RS) = 1. Although the change in measure might induce a change in PV, this does not imply a change in Stratj,i(RS). Only when P: P Vj,i(P P LRi) > P Vj,i(P P LRi+1), but Q: P Vj,i(P P LRi) < P Vj,i(P P LRi+1)(or < and >) will the use of Q instead of P have an impact on the results.

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30 Chapter 5. Interest rate models Second, this simplification will not lower the expected loss the FI faces: When the consumer optimizes his strategy under Q instead of P, then the consumer will make such decisions as to maximize the expected loss of the FI. I.e. the FI has his own expected loss under Q, which the consumer maximizes by optimizing the profit of the consumer under Q. When I assume that the consumer optimizes under Q, but the consumer in reality optimizes under P, then the expected loss under Q will not be higher than if the consumer had optimized over Q.

This simplification of the problem should not have a great impact on the calcu-lated risks, and in the cases in which there is an impact, this impact will not cause underestimation of the risk of the FI.

5.6 Risk neutral calibration

The value of a swap is a function of the term structure (and the swap rate); indepen-dent of other variables, such as mean reversion or volatility. The HWV model can therefore not be calibrated to swap prices, because the swap prices do not contain any information about the parameters σ and α. The HWV model assumes the term structure to be exogenous. For the calibration of the model, I therefore need non linear derivatives, like swaptions.

The risk neutral parameters that are found when calibrating to swaptions differ depending on the swaptions that are used. I therefore calibrate the model for each mortgage contract to its own appropriate set of swaptions. For the calibration of the model, I use 3 swaptions with strike = rm as the mortgage interest rate with 3 different swaption maturities (3/10 TIRR, 5/10 TIRR, 7/10 TIRR) and 3 tenors (7/10 TIRR, 5/10 TIRR, 3/10 TIRR). In doing so I use parameters for the interest rate tree that are close to those used to value a relevant prepayment option. This procedure implies that I re-estimate the model parameters for each different rmand TIRR.

I then value each of the three swaptions with the tree, varying the parameters, α and σ. The optimal estimates for α and σ are determined by minimizing the average squared difference between the swaption price provided by the tree, Stree(a, σ, α), and the swaption price in the market Smarket(a). With these ˆσ and ˆα I build the interest rate tree for the valuation of the prepayment option. This results in:

{ ˆα, ˆσ} = arg min α,σ ( 1 3 3 X a=1

[Stree(a, σ, α) − Smarket(a)]2 )

. (5.16)

5.7 Historical estimation

Two parameters need to be estimated, σ and α. I estimate the mean reversion by estimating an autoregressive model with one lag (AR(1)) on the interest rates:

rt= β + φ · rt−1+ t. (5.17)

The φ that is found here is used to calculate the mean reversion of the rate by α = 1 − φ. For the entire available data set on EU RIBOR6m, starting in January 1994,

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5.8. Conclusion 31 we find that α = 0.0055. The volatility is then calculated by

ˆ t= rt− ( ˆβ + ˆφ · rt−1), ˆ σ = v u u t 1 T T X t=1 ˆ 2 t. (5.18)

The volatility found is ˆσ = 0.0273.

5.8 Conclusion

This chapter shows the following. The savings rate can be fitted to EU RIBOR6m. EU RIBOR6mderivatives can be used to calibrate the interest rate tree. The effective mortgage rate for the consumer depends on the applicable taxation of income and capital. I use the one-factor HWV model as it allows for negative rates and can be fitted to the term structure. Calibrating the model to traded instruments is preferred to fitting to historical interest rate data. The calibration to the swaptions is done such that the MSE of the misfitting is minimized. The results of this method are shown in Chapter6.

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Chapter 6

Results

The results of the methods, developed in Chapter 4 and 5, are presented in this chapter. Section6.1shows the fit and the parameter estimates of the trees, calibrated to the different swaptions, as proposed in Section5.6. In the subsequent section, the value of the set of PPOs for the mortgages selected in Chapter3is shown. Section

6.3 exposes the robustness of the valuation under changes in the parameters and assumptions.

6.1 Tree calibration

When calibrating the tree for the selected mortgages, as in Table3.2, I find that the parameters of the tree can be set such that the difference between the market price and the model price for a single swaption is almost zero. The difference between the two is less then 1 euro for swaptions that have values over 1 million euro.1 _{In which}

case the fitting of the swaption appears to be solely done by optimizing the volatil-ity. When calibrating the trees to three swaptions simultaneously, by the procedure described in Section5.6, I find the parameters as presented in Table6.1. The fit of the individual model prices to the market price is now no longer 100%, as one would expect given the different implied volatilities observed in the market. The full cal-ibration results for all the swaptions are presented in AppendixA. Notable is that the relative difference between the model price and the market price is consistently higher for the third swaption in each set. This can be explained by noting that these swaptions have the longest maturity and the shortest tenor, which in the relevant market have the lowest price.

As the specifications, i.e. mortgage rate and TIRR, for the selected IO and AR mortgages are the same, the results are only printed for the IO mortgages. As, later in this chapter, the results will be considered with market data as on 01-01-2016, I also present the calibration of those trees to the corresponding swaptions.

6.2 Value of the PPO

The value of the set of PPOs for the selected mortgages, based on a tree calibrated to both α and σ, is presented in Table6.2. The mortgage used for these calculations has some fixed specifications. It has an initial loan of N =200,000, a remaining loan of RL=100,000 and an allowed prepayment size of 10% of the initial loan (20,000).

Notable is that PPOs on IO mortgages have a relatively high value compared to those on LR mortgages. This may be due to the fact that, as described in Section

4.3.2, a prepayment on a LR mortgage is not only a long position in a receiver swap, 1_{The swaptions are all a notional of 10 million euro.}

Optimal exercise strategies and valuation of prepayment options in Dutch mortgages

U

NIVERSITY OF

A

MSTERDAM

M

T

Optimal Exercise Strategies and Valuation

of Prepayment Options in Dutch

Mortgages

Declaration of Authorship

Abstract

Contents

Chapter 1

Introduction

Chapter 2

Literature review

2.1

Exogenous vs. endogenous modeling

2.2

Research on Dutch mortgages

2.3

Mortgage valuation and the term structure of interest rates

2.4

Conclusion

Chapter 3

Mortgages, prepayments and the

Dutch market

3.1

Mortgage types

3.2

Prepayment

3.3

Current state of the industry

3.4

Composition of the Dutch market

Chapter 4

Valuation of the prepayment

options

4.1

Introduction on relation PPO - Swaption

4.2

Assumptions

4.3

Value of a single prepayment

4.4

Optimal strategy for exercising prepayment options

4.5

Value of the total set of PPOs

4.6

Conclusion

Chapter 5

Interest rate models

5.1

Analyses of the relation EURIBOR - savings rate

5.2

Adjusting mortgage and savings rate for tax effects

5.3

Short rate models

5.4

The Hull White Vasicek model

5.5

Calibrating the interest rate model

5.6

Risk neutral calibration

5.7

Historical estimation

5.8

Conclusion

Chapter 6

Results

6.1

Tree calibration

6.2

Value of the PPO