modeling and simulation of debt portfolios308385/fulltext01.pdf · assumption is that the firm has...

Modeling and Simulation of Debt Portfolios

Part One: Simulation of Debt Portfolios

Martin Andersson and Anders Aronsson

Part Two: The Underlying Model

Martin Andersson

Part Three: Arbitrage and Pricing of Interest Rate Derivatives

Anders Aronsson

November 2002

Abstract

In this paper a model is developed for simulating the performance for a firm’s debt portfolio with respect to income. The results from the simulations indicates that an efficient strategy for the debt portfolio should contain a dynamic refinancing strategy with a swap strategy that buys swaps whenever the current interest rate is higher that the previous one. The paper also presents a detailed description of the underlying model and its implementation. A discussion surrounding arbitrage issues and the pricing of interest rate derivatives completes the paper.

Part One

1. Introduction to Simulation of Debt Portfolios............................................................ 3

2. Theories on an Optimal Debt Portfolio....................................................................... 5

2.1 Risk ........................................................................................................................... 6

2.2 Criteria for an optimal debt portfolio........................................................................ 6

3. Portfolio Strategies........................................................................................................ 8

4 Strategy Simulation ..................................................................................................... 10

4.1 Starting Conditions ................................................................................................. 11

4.2 The Simulation Procedure ...................................................................................... 11

4.3 Basic Strategy Simulation....................................................................................... 11

4.3.1 Basic Portfolios ................................................................................................ 12

4.3.2 Results Basic Strategy Simulation. .................................................................. 16

4.3.3 Analysis of the Basic Portfolios....................................................................... 17

4.4 Combined Strategy Simulation............................................................................... 21

4.4.1 Combined Portfolios ........................................................................................ 21

4.4.2 Results Combined Strategy Simulation ........................................................... 23

4.4.3 Analysis of the Combined Portfolios ............................................................... 24

5 Recommendation on a Debt Portfolio Strategy......................................................... 26

6 Recommendations for Further Development............................................................ 26

7 The Underlying Model................................................................................................. 28

8 The Macroeconomic simulation model ...................................................................... 28

8.1 The stochastic process ............................................................................................ 29

8.2 Inflation................................................................................................................... 31

8.3 GDP growth ............................................................................................................ 32

8.4 Short interest rates .................................................................................................. 32

8.5 The Long Interest Rate, Spread and Yield.............................................................. 34

9 Financial Structure Simulation Model....................................................................... 35

9.1 Revenues................................................................................................................. 36

9.2 Revenues per square meter ..................................................................................... 37

9.3 Vacancy levels ........................................................................................................ 38

9.4 Running costs.......................................................................................................... 39

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9.5 Profit ....................................................................................................................... 40

9.6 Debt structure.......................................................................................................... 40

9.7 Implementation of Interest Rate Derivatives .......................................................... 43

10 Time Issues ................................................................................................................. 44

11 Arbitrage and Pricing of Interest Rate Derivatives................................................ 46

12 Arbitrage..................................................................................................................... 46

13 Risk Neutral Probability Measure ........................................................................... 48

14 The Martingale Measure........................................................................................... 51

15 The Term Structure Model ....................................................................................... 52

15.1 The Yield Curve Model and the Forward Rate..................................................... 54

16 Infinite Sample Spaces............................................................................................... 56

17 The Valuation of Interest Rate Derivatives ............................................................. 57

17.1 Valuation of the Plain Vanilla Swap..................................................................... 58

17.2 The Valuation of Caps .......................................................................................... 61

18 References................................................................................................................... 64

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1. Introduction to Simulation of Debt Portfolios The main purpose of this paper is to examine different strategies for a firm’s debt portfolio given an operating income that fluctuates with changes in macroeconomic variables. We have simulated eighty-seven different strategies for the debt portfolio through a stochastic model. These portfolio-strategies use different interest derivatives and alternative maturity structures to hedge against risk scenarios. Interest rate derivatives included in the debt portfolio are swaps, caps and caplets. A swap is an agreement to exchange one cash flow stream for another. The most common, and the one we use in this paper, is the plain vanilla swap, in which one party swaps a series of variable-level payments for a series of fixed-level payments. In our case this is the change of floating interest payments to fixed interest payments. A cap is a financial insurance contract, which protects the holder from having to pay more than a pre-specified rate. A cap is technically the sum of a number of caplets, which can only be used in a single period. For a thorough discussion on the valuation and application of interest derivatives, see part three. The stochastic model consists mainly of two parts: the first is a macroeconomic model and the second a firm replicating model. The macroeconomic model is constructed in such a way that it produces time series modeling inflation, real GDP and regime state, where inflation and real GDP are auto regressive processes and regime state is a stochastic variable simulated with a simple Markov chain that can vary between boom and recession. These variables determine in their turn the interest rate. The firm-replicating model generates cash flow streams dependent on the macroeconomic variables and the interest rate. This part of the model consists of two components, one that simulates the revenues and one that replicates the debt structure. The structure of the model is explained more precisely in part two. The portfolio strategies are to consider and hedge against the risk scenarios that can occur with changes in macroeconomic variables. These risks are largely connected to variations in interest rate, but also to the borrowing requirements. For simplicity, one can say that if we would have a complete positive correlation between the revenues and debt in macroeconomic variables and interest rate, the strategy selection would be trivial. This is unfortunately not the case and there is a complex relationship between debt and revenue. If we were able to identify this relationship a substantial part of the problem would be solved. The problem here is that we would be forced to study every strategy simulation in detail to recognize such a relation and this is a too time consuming job for anyone, thus instead we define risk parameters to indicate critical economic situations. The problem with risk is that it is somewhat of an unobservable variable and this fact enables us to use proxies for risk. Proxy variables of this kind might lack in precision, which means that we could miss some information about risk by choosing a certain proxy. This error source is hard to omit in these types of models and something we have to bear in mind while analyzing the results. Definitions of risk and risk proxies are stated and discussed in Section 2.1 and 2.2.

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To simulate the different economic situations that can occur we use a Monte-Carlo method.1 We have decided on a twenty year period that is simulated repeatedly a large number of times. In every simulation step mean value and standard deviation are calculated for certain key values and proxies. By letting the number of simulation steps be large, these values will in time converge to a steady state, where the difference between the current value and the previous is sufficiently small. A number of assumptions have been made to simplify the problem at hand. An important assumption is that the firm has already decided on its optimal capital structure2 taking all relevant considerations into account, and now the managers should decide on an optimal debt strategy. Due to lack of data in the revenue part, assumptions have been made on some parameters and will be explained more carefully in our model. We have in this work simulated the debt portfolio for a firm that acts on the Swedish real estate market. This is a large firm with investments mainly in the three large city regions. Revenues come from five separate sub areas; offices, stores, storages, private housing and others. The organization of part one in this paper is as follows: Section 2 introduces the theoretical background surrounding the problem, and discusses the risk aspects and its determinants. Further in that section we define the criterions for an optimal debt portfolio. Section 3 offers a deeper explanation of the strategies. Section 4 is divided into two simulation parts. The first part simulates the basic portfolios where we analyze the effect of the decision variables. The second part simulates the combined portfolios and analyzes these portfolios in an efficiency approach. In section 5 we decide on an efficient portfolio and formulate this as a recommendation. Section 6 discusses suggestions for future developments in this area. Section 7 presents concluding remarks.

1 For more information on the Monte Carlo method see Rebonato [17] and Lyuu [13]. 2 Capital structure: A firm’s mix of debt and equity, determined by its financial decisions. For further reading on capital structure, study Miller and Modigliani [14], Levy [12] and Titman and Wessels [21].

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2. Theories on an Optimal Debt Portfolio The existing literature surrounding optimal debt portfolio deals in first hand with the corporate debt maturity puzzle. A central question in this area has been: What determined a firm’s optimal debt maturity? Many have tried to answer this question and some have partially succeeded. Mitchell [15] finds that a firm is more likely to issue short-term debt if the firm is not traded on the New York Stock Exchange or in the S&P’s 400, and has convertible debt in its capital structure. She argues that her findings are consistent with the hypothesis that firms facing a high degree of information asymmetry choose short-term debt to minimize adverse selection costs. A study of debt issue by Guedes and Opler [8] found that firms with large investment grade are more likely to issue short-term debt and long-term debt, while firms with relative high growth prospects tend to issue only short-term debt. Barclay and Smith’s [2] finding somewhat supports the later conclusion. They found that smaller firms with more growth opportunities have a smaller proportion of debt maturing in more than three years. Diamond [7] predicted that rating was a relevant factor in debt maturity; he argues that firms with higher rating are more active participants in short-term credit markets, while lower-rated firms have a tendency to avoid short-term debt to minimize refinancing risk, the so-called funding risk. He also discovered that medium-rated firms tend to use bank debt. Hoven Stohs and Mauer [11] also found strong evidence that firms with high or very low bond rating have shorter average debt maturity. Maturity matching is a commonly accepted indicator of debt maturity. Maturity matching is when a firm matches its debt to coincide with its income or assets. Mitchell [15] could not find sufficient evidence to support this notion, but Hoven, Stohs and Mauer [11] found that proxies for maturity-matching hypotheses are generally significant determinants for the choice of debt maturity structure. A conclusion of great interest is the one of Brick and Ravid [6]. They argue that if the term premium, i.e. the difference between the implied forward interest rate and the future expected spot rate, is positive (sufficient negative) long-term (short-term) debt maturity structure is optimal. Other than this notion there is little that points to dependence between expectations on interest rate and the maturity of a debt structure. Agmon et al. [1] analyzed an efficient frontier in terms of the probability of bankruptcy and the expected debt repayment. According to their work, the solution for the choice problem, the optimal composition of the corporation debt portfolio, is based on a trade-off analysis between expected debt repayment and the probability of bankruptcy. They proposed that the special nature of bankruptcy risk makes the application of the mean-variance approach on this problem inadequate. Instead, they developed an efficient frontier, which balances the expected operating income minus debt repayment against the probability of bankruptcy. Telser [20] used another criterion, where he maximizes the expected rate of return, or profit, subject to a constraint of a minimum return with a predetermined probability. As we can see, risk is essential in determining an optimal debt portfolio. The next section defines the risks that we include in this paper.

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2.1 Risk In this paper we consider two different kinds of risks; the funding risk and the interest rate risk. Whenever a loan matures, there are two alternatives to repay this; either with own equity or with another loan. The first choice demands that you have sufficient equity whenever a loan matures and this leads to complicated maturity matching that both will be costly and time consuming. The second funding choice is the one we adapt in this work. The complication with this kind of refinancing strategy is that you at every time have to count on creditors will to supply equity. Problems involving repayment of loans are essentially the funding risk. Is it possible to hedge against this? This is a rather complex question that includes many aspects of a firm’s financial status and this will be discussed more closely later in this chapter. Fluctuation in interest rate is a very obvious risk to any firm that includes debt in its finance strategy, and possibly the most difficult to hedge against. Implications of a fluctuating interest rate are that the interest payments and profit will vary. These two risks constitute constraints that we have to address while constructing debt portfolios and formulate portfolio strategies.

2.2 Criteria for an optimal debt portfolio We believe that only having the criteria of minimizing the probability of bankruptcy to find an optimal debt portfolio, as the work of Roy [18] states, is insufficient, since we will lose the objective and importance of high profits and the firm’s specific preferences regarding risk. To only use the bankruptcy probability as a constraint is somewhat arbitrary as the accepted level of this type of risk, is set outside the optimization process. As a tool for comparing different portfolios with each other, we have decided upon a combination of Telser [20], Baumol [3] and Agmon et al.’s [1] work. It is our opinion that the standard deviation of the profit is a form of risk measure that we cannot exclude from the calculations. We use, unlike Agmon et al. [1], a profit measure consisting of the mean value minus the standard deviation of the profit. We formulate this in a definition: Definition 2.2.1 The expected least value of the profit is

( ) ( ) ( ) ( )( )( )22ssss PREPREPRELPRE −−= .

Here is a debt portfolio index. Moreover, ifs ( ) ( )21 LPRELPRE ≤ , we say that the profit series is better than and therefore portfolio 2 is preferred prior to portfolio 1 in the aspect of the least value of the profit.

2PR 1PR

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The interest rate risk is taken into account when the measure for probability of bankruptcy is calculated. When the interest rate rises to a level where profit becomes loss and if this loss increases to an unacceptable level, then this will contribute to the probability of bankruptcy. The bankruptcy measure is basically an indicator of how often the loss has reached this level of non-acceptance during a simulation period and not exactly a formal definition of bankruptcy-probability. To be able to distinguish a feasible set of efficient portfolios we also need to define this measure, and we formulate this as follows: Definition 2.2.2 Let be the pre-specified limit of maximum accepted loss, be the total number of periods and B the area of bankruptcy. Then the estimated probability of bankruptcy for portfolio

C N

η at time t is

( )N

CPRBP tt 0#

)( ,, <−= ηη

η .

In short, one can say that the nominator keeps track of the number of bankruptcies and the denominator the total number periods. is therefore the probability that bankruptcy occurs in a certain period. Now it is quite simple to realize that if , debt portfolio 2 is preferred before portfolio 1 in the sense of probability of bankruptcy.

η)(BP

( ) ( )21 BPBP >

The funding risk has in our model for simplicity been reduced to a problem of maturity. The debt maturity structure is the amount of money that matures in the first year, second year and so on at every given time. We have developed a function that supervises the debt maturity for the upcoming year in order to keep track of the amount of maturing loans. It allows us to define a maximum limit of maturity. To avoid complications with refinancing we have decided that the maturity of loans shall converge to a value equal to or less then 30 % of the total loan. While studying the specific firm’s annual report, which this paper treats, we draw the conclusion that this is a reasonable assumption. In the first part of the simulation, the supervising function is excluded, because the main objective is to study the decision variables and not the performance of a possible efficient portfolio. Suppose that we now have n different portfolios and that we can plot these in a least profit-probability of bankruptcy diagram. Then the efficient portfolios will lie on the left boundary of the total set of the simulated debt portfolios, the efficient frontier. It will also lie above point 1 in figure 2.1 because this is the point of minimum-probability of bankruptcy.

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E(LPR) 2

4

5 3

1

P(B)

Figure 2.1 The efficient frontier. On the line between point 1 and 2 lays the efficient portfolios. One of these will be the optimal portfolio for the firm, regarding to its risk preferences. The point 3 dose not represent an efficient portfolio since P(B)5<P(B)3 and E(LPR)4>E(LPR)5.

An efficient portfolio will be defined as follows: Definition 2.2.3 If P is an efficient portfolio, then there exists no other portfolio Q such that

( ) ( )QP LPRELPRE ≤ ∧ ( ) ( )QP BPBP ≥ . Remark 2.2.1 A portfolio is efficient if and only if it lies on the efficient frontier. This follows directly from the definition of an efficient portfolio. 3. Portfolio Strategies There are several ways to formulate the strategies for debt portfolios in models of this kind. We have chosen three main strategies; refinancing strategy, swaps and caps. With each strategy, a number of decision variables are accounted for. These variables will determine the outcome of the portfolio strategy. The first one is the refinancing strategy, stating how the borrowing requirement is refinanced across the yield curve in each period. In simulation models, refinancing strategies are often static, i.e. the borrowing requirement is distributed over different maturities according to a fixed pattern (see debt structure in part two). However, they could also be dynamic. A dynamic refinancing strategy means for example that if the long interest rate drops below some point, issue in debt with longer maturities or vice versa. We have considered both the static and the dynamic refinancing strategy in this model.

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Decision variables for this strategy are: Maturity Structure: The outstanding debt in each period is repaid by taking new loans. The decision variables for this strategy are how the borrowing requirement is to be distributed over different maturities in each period. Refinancing Dynamics: As mentioned above, there are two ways to refinance; static or dynamic. Static means that maturing loans are refinanced according to a fixed pattern and dynamic that the portfolio uses several different patterns depending on the actual interest rate. When interest rate crosses a certain limit, the refinancing pattern is changed. The second strategy would be the use of swaps. A swap, as mentioned in the introduction, is an agreement to exchange one cash flow stream for another. The risk associated with the plain vanilla swap, which we use, is placed solely on the party that swaps to the variable-level payments, in this case the counterpart of the firm (since the firm already has variable coupon payments and is interested in swapping them to a fixed rate). This risk motivates the counterpart to charge a higher fixed rate in exchange for the variable one. Buying swaps is only useful when the interest rate is sufficiently low. A straightforward strategy would therefore be to buy swaps whenever the interest rate is below a certain limit.3 We call this strategy static. An alternative approach is to buy swaps when the previous rate is below the current while underneath a certain rate as specified above. This approach is called the dynamic swap. It means that the economic cycle might be on its way to a boom with high interest rates when the firm buys the swap. Moreover, we made an assumption that the firm is not allowed, by the creditors, to buy longer swaps than four years. Decision variables for swap strategies are: Swap Limit: The Swap limit for each maturity length is based on the short rate and the yield curve. For example, to investigate if we are to buy a swap for a loan with a maturity of ten years, we compare the given ten- year rate, from the macroeconomic model, with the one we get from the swap limit plus the long-term spread for the ten year interest rate (in our model 0.74 percent). If this limit is above the given ten-year interest rate, we swap the loan for a given number of periods i.e. the swap length. Swap Length: This decision variable determines how long the loan at hand should be swapped. For example, if we decide on a swap length of 16 periods, then every loan that is swapped is going to have a fixed rate during this period. If the loan itself is shorter than 16 periods we only swap it until the credit is repaid. 3 We also considered a strategy with different swap length at different interest rate. For example if the rate is below 5.5 % but above 4.5 % we only swap for 4 periods and if the interest rate falls under 4.5 % we swap an even longer period. The relevance of this strategy became questioned when we started the simulations and we discovered that the result was not satisfying. Naturally this effect is somewhat self-redundant due to the maturity of the portfolio. All the loans with shorter maturity are, in a strategy where one swaps the same length under only one interest rate limit, going to mature in a cumulative manner and therefore recreate some of the dynamics we wanted to create with a “interest rate window” where we swap a shorter period.

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Swap Dynamics: We have decided upon two different Swap Dynamics if the interest rate is below the swap limit. As mentioned above, these are the static or the dynamic approach. The static approach always swaps the loan when the interest rate is under the current swap limit and the dynamic method only swaps the loan if the interest rate is under the swap limit and if the current interest rate is higher than the previous. In the third strategy, we consider the use of caps and caplets. These can be used in two ways; at the beginning of a loan or when a swap runs out. Another strategy could be to buy the cap whenever the interest rate drops below some specified level. The interest rate would then be capped for one quarter in the future. We also consider capping a loan more than one period. Such derivatives are commonly known as caps and are portfolios of caplets. Since caps and caplets become very expensive when the interest rate is high, we have chosen to buy caplets and caps only when the interest rate is relatively low. The same alternative approach used for swaps can also be applied for caps, that is, whenever a switch from recession to boom is about to take place. When combining caps with swaps, we naturally never buy a cap when the coupon rate is already fixed. Our main strategy with these derivatives is to insure periods of relative acceptable levels of interest rate when we issue in a loan or whenever a swap is running out. Decision variables for cap and caplet strategies are: Cap Limit: This limit works in the same manner as the swap limit, and we invest in a cap or caplet if the current interest rate is below the cap limit. Also in this case, all maturities have their own limit, depending on the yield curve. Cap Rate: The cap rate is the insurance level, which means that you never have to pay more than this pre-specified rate. Execution Date: When a cap is bought one has to decide when it’s going to be used. For this we have the variable Execution date. If we want to cap a loan in five periods we simply set this variable to five. Cap Share: This variable decides how large part of a loan that should be capped. Cap Length: Similar to the decision variable swap length, this variable decides how many periods a loan should be capped from the execution date. Cap Dynamics: Also this variable has the exact same objective as its swap counterpart. The static version always invests in caps or caplets if the interest rate is below the cap limit and the dynamic approach has the additional condition that the current interest rate has to be higher then the previous. 4 Strategy Simulation The strategy simulation is divided into two parts. The first part contains the strategies described above simulated separately. By separately we mean that with each portfolio, only one decision variable is varied, all other variables being constant. In the second part of the simulation the strategies that performed the best during the first part are simulated in combination. The exact portfolios and their performance will be shown for both

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separated and combined strategy. The results will then be discussed in the remainder of this section. We start by describing the starting conditions for the strategies.

4.1 Starting Conditions Starting all the simulations with the firm’s actual portfolio would make the costs and risks rather similar during the first part of the simulation. Also, there might be unwanted transition costs when shifting from one strategy to another. Since the aim of the simulation is to analyze long-term differences between debt portfolios, rather than transition costs when changing strategy, it would be optimal to let all strategies start in a portfolio fulfilling the specified strategy from the beginning. In that way, we can be sure that differences in costs between any two strategies are due to different debt structures, and not transition costs. This is almost true in our model. Since we use a refinancing scheme for the debt, the loan and maturity structure do not start in its limit-value but rather in the refinancing vector (see debt structure in part two). The vector lies very close to the limit-values from the start of the simulation and will approach the final structure quite fast. The structures shown in the preceding sections are the average structure during a 20-year period. The sum of the debt outstanding at the start of the simulation is set to 22 billions SEK. All macroeconomic variables are initially set to its average value.

4.2 The Simulation Procedure To go through all possible economic scenarios, it is not sufficient to simulate only a 20 year-period. That is why we have chosen to use the so-called Monte Carlo simulation. We let a 20 year-cycle be repeatedly simulated a large number of times. Each cycle will have its specific mean value and variance and so on. By letting the number of rounds become sufficiently large, all calculated values will converge to a certain limit. To improve the accuracy we simulate all the portfolios in each simulation round in the same simulation of macroeconomic variables. This means that every portfolio act in the same economic scenario. We can in this way directly distinguish differences between portfolios, even though we are not performing any statistical tests.

4.3 Basic Strategy Simulation In the basic strategy simulations we only use the strategies separately as mentioned above. Regarding the refinancing strategy we simulate both static and dynamic portfolios but with predetermined durations. We have chosen to construct five vectors with different maturities, which we have called extreme-short (X-short), short, medium, long and extreme-long (X-long). These vectors will act as building blocks for all portfolios

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regarding the refinancing strategy. They will be used through the entire strategic simulation process and will never be changed or altered along the way.

Table 4.1

Duration 3

month 6

month 1

year 2

year 3

year 4

year 5

year 6

year 7

year 8

year 9

year 10

year

X-short 40 15 15 10 10 10 0 0 0 0 0 0 Short 40 10 10 10 10 10 10 0 0 0 0 0

Medium 20 10 10 10 10 10 10 10 5 5 0 0 Long 20 0 10 0 10 10 10 10 10 10 5 5

X-long 0 0 0 10 10 10 10 10 10 10 15 15

The number in each maturity length, indicate the percentage of new loans to be taken of the outstanding debt in each step of the simulation. This does not imply that the given percentage in each maturity length above will be the actual percentage of loans lying in that maturity length, after convergence (see starting conditions). While simulating the swap and cap strategies, we decided to only use the medium length static refinancing strategy, and this due to the large numbers of permutations it would render if we would simulate all static and dynamic strategies with different swap lengths, interest rate limits and dynamic swap strategies. We are aware of the fact that we omit several portfolios and that the conclusions for further simulations may not be completely satisfying. But the time aspect, which will be discussed in a later section, has forced us to do this simplification. Caplet strategies are also applied on the medium static refinance strategy and in the same manner of excluding permutations as in the swap case. Since one of the criteria for an efficient portfolio is that not more than 30 % is to mature every year, the loan and maturity structure of the portfolios is of importance. The structures calculated will be the average structure during a 20-year period. They will be plotted along with the portfolios. The results that this part of the simulation produces are then used to construct combined portfolio strategies, which will be explained in section 4.3.2.

4.3.1 Basic Portfolios This section presents the construction of the “Basic Portfolios” and their parameters. In portfolio 1 to10, the refinancing strategy is applied, first five static and then five dynamic. These are used as benchmark portfolios and will act as a comparison measure. For the first five portfolios the vectors above will be used separately. The loan and maturity structure for these are shown in the table below

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Maturity Structure Static Refinancing

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 Years

Percent

X-Short

Short

Medium

Long

X-Long

Table 4.2

Loan Structure Static Refinancing

05

101520253035

1 2 3 4 5 6 7 8 9 10 Years

Percent

X-Short

Short

Medium

Long

X-Long

Table 4.3 For the dynamic refinancing strategy we take combinations of the five vectors, with different interest rate limits. The limit that makes the decision of which vector to use in each step is based on the ten-year interest rate. So whenever the ten-year interest rate is below a certain limit, loans are refinanced according to a vector with longer maturities and vice versa. The decision variable chosen for this strategy is positioned in an interval around the average of the ten-year interest rate.

Table 4.4 Decision variables

Refinancing Strategy X-Short Short Medium Long X-Long

Portfolio

6 Dynamic 1 r<0.0474 * 0.0474<r<0.0674 * 0.0674<r 7 Dynamic 2 r<0.0574 * * * 0.0574<r 8 Dynamic 3 * r<0.0474 0.0474<r<0.0674 0.0674<r * 9 Dynamic 4 * r<0.0574 * 0.0574<r *

10 Dynamic 5 r<0.0374 * 0.0374<r<0.0774 * 0.0774<r

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Maturity Structure Dynamic Refinancing Strategies

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 Year

Percent

Dynamic 1

Dynamic 2

Dynamic 3

Dynamic 4

Dynamic 5

Table 4.5

Loan Structure Dynamic Refinancing Strategies

02468

101214161820

1 2 3 4 5 6 7 8 9 10 Year

Percent

Dynamic 1

Dynamic 2

Dynamic 3

Dynamic 4

Dynamic 5

Table 4.6 For swaps and caps, the decision variable concerning the interest rate limit is based along the yield curve and not the ten-year interest rate. This is because we regard each loan in each maturity separately and not as a vector of loans where one can only have one decision in every time-step. The number in the table indicates the limit for swapping a three-month loan (three-months loans are never swapped though). For the maturities that can be swapped, the limit follows an imaginary yield curve with the average spread between the three-month and the ten-year interest rate (0.74 %). For example, if the number in the table were 5 %, the limit to swap the ten-year interest rate would be 5.74 % and the maturities down to the six-months loan would lay along the constructed yield curve described in The Model. An alternative to the static swap is the dynamic swap, as we have chosen to call it. For

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the dynamic swap, yet another criterion has to be fulfilled in order to swap a loan. The interest rate in the previous step has to be below the current. Otherwise it works the same way as the static swaps. It is reasonable to let the limit for swapping lie around the average of the interest rate. We test swap-lengths from 4 quarters up to 20, even though we believe that the creditors are not willing to let borrowers swap their loans for 5 years. To set a maximum swap-length limit we have chosen 16 quarters, or 4 years.

Table 4.7

Decision Variables Refinancing

Strategy Swap Limit

Swap Length Swap Dynamics

Portfolio

11 Medium 0.03 16 Static 12 Medium 0.04 16 Static 13 Medium 0.05 16 Static 14 Medium 0.06 16 Static 15 Medium 0.07 16 Static 16 Medium 0.05 4 Static 17 Medium 0.05 8 Static 18 Medium 0.05 12 Static 19 Medium 0.05 20 Static

20 Medium 0.04 16 Dynamic 21 Medium 0.05 16 Dynamic 22 Medium 0.06 16 Dynamic

Caplets are bought for each and every loan with a specified execution date and cap rate every time the interest rate drops below a certain level. The same thing discussed for swaps, concerning limits following an imaginary yield curve, applies for caplets as well. Also, dynamic cap strategy is simulated, similar to the one for swaps. Execution dates are varied from two up to eight quarters. Longer dates means higher costs, so we discarded caps with longer execution dates than two years. The cap rates tested are six, seven and eight percent. Higher cap rates would do no good and lower would cost too much. Portion of the loans to cap are simulated as well.

Table 4.8 Decision Variables

Refinancing Strategy

Cap Limit

Cap Rate

Execution Date

Cap Share

Cap Length

Cap Dynamics

Portfolio

23 Medium 0.03 0.07 6 1 1 Static 24 Medium 0.04 0.07 6 1 1 Static 25 Medium 0.05 0.07 6 1 1 Static 26 Medium 0.04 0.06 6 1 1 Static 27 Medium 0.04 0.08 6 1 1 Static 28 Medium 0.04 0.07 2 1 1 Static 29 Medium 0.04 0.07 4 1 1 Static 30 Medium 0.04 0.07 8 1 1 Static

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31 Medium 0.04 0.07 6 0.25 1 Static 32 Medium 0.04 0.07 6 0.5 1 Static 33 Medium 0.04 0.07 6 0.75 1 Static 34 Medium 0.04 0.07 6 1 2 Static 35 Medium 0.04 0.07 6 1 4 Static 36 Medium 0.04 0.07 6 1 8 Static 37 Medium 0.04 0.07 6 1 16 Static 38 Medium 0.03 0.07 6 1 6 Dynamic 39 Medium 0.04 0.07 6 1 6 Dynamic 40 Medium 0.05 0.07 6 1 6 Dynamic 41 Medium 0.04 0.07 6 1 4 Dynamic 42 Medium 0.04 0.07 6 1 8 Dynamic

In order to form the combined portfolios in the second part of the simulation we have to select the values of the decision variables that performed the best during this part.

4.3.2 Results Basic Strategy Simulation.

Table 4.9

Portfolio Refinancing

Strategy E(Least-Profit)

108 SEK E(Profit) 108 SEK

Std(Profit) 107 SEK

E(Interest Cost) 108 SEK

E (Fixed Rate length) quarter

Prob (Bankruptcy)‰

1 X-Short 2.2814 3.1463 8.6493 2.9319 1.0000 3.053

2 Short 2.2343 3.0989 8.6465 2.9793 1.0000 3.488 3 Medium 2.1704 3.0350 8.6456 3.0433 1.0000 4.088 4 Long 2.1240 2.9886 8.6462 3.0897 1.0000 4.763 5 X-Long 2.1045 2.9692 8.6473 3.1090 1.0000 5.000

6 Dynamic 1 2.1744 3.0337 8.5931 3.0445 1.0000 3.888 7 Dynamic 2 2.1792 3.0327 8.5349 3.0456 1.0000 3.775 8 Dynamic 3 2.1556 3.0080 8.5232 3.0703 1.0000 3.938 9 Dynamic 4 2.1763 3.0340 8.5772 3.0443 1.0000 3.863

10 Dynamic 5 2.1650 3.0210 8.5599 3.0573 1.0000 3.913

Table 4.10




Std(Profit) 107 SEK




11 Medium 2.2409 3.1160 8.7504 2.9623 2.4710 3.738

12 Medium 2.3054 3.1444 8.3907 2.9338 4.2865 3.200

13 Medium 2.3108 3.0960 7.8512 2.9823 6.4211 2.388

14 Medium 2.2583 3.0426 7.8425 3.0357 7.4698 2.375 15 Medium 2.2040 3.0069 8.0284 3.0714 7.8757 3.088

16 Medium 2.1825 3.0385 8.5605 3.0398 1.8088 4.038 17 Medium 2.2249 3.0588 8.3397 3.0194 3.2932 3.550 18 Medium 2.2717 3.0772 8.0552 3.0010 4.8906 2.900 19 Medium 2.3394 3.1105 7.7103 2.9678 7.7085 2.025

20 Medium 2.2808 3.1388 8.5799 2.9394 3.2925 3.400 21 Medium 2.3189 3.1358 8.1689 2.9424 5.2152 2.588

22 Medium 2.2873 3.0819 7.9463 2.9963 6.6735 2.350

16

Table 4.11




Std(Profit) 107 SEK




23 Medium 2.1706 3.0351 8.6446 3.0432 1.0000 4.075 24 Medium 2.1720 3.0349 8.6289 3.0418 1.0000 4.038 25 Medium 2.1692 3.0257 8.5647 3.0357 1.0000 3.888

26 Medium 2.1801 3.0389 8.5881 3.0337 1.0000 4.013 27 Medium 2.1704 3.0347 8.6428 3.0432 1.0000 4.088

28 Medium 2.1704 3.0350 8.6455 3.0433 1.0000 4.088

29 Medium 2.1705 3.0347 8.6422 3.0431 1.0000 4.088

30 Medium 2.1753 3.0357 8.6033 3.0397 1.0000 3.975

31 Medium 2.1709 3.0350 8.6409 3.0429 1.0000 4.075

32 Medium 2.1713 3.0350 8.6365 3.0425 1.0000 4.075

33 Medium 2.1717 3.0350 8.6325 3.0422 1.0000 4.050

34 Medium 2.1731 3.0349 8.6174 3.0410 1.0000 4.000 35 Medium 2.1748 3.0342 8.5945 3.0394 1.0000 3.975 36 Medium 2.1774 3.0314 8.5399 3.0359 1.0000 3.863 37 Medium 2.1714 3.0202 8.4877 3.0300 1.0000 3.563

38 Medium 2.1705 3.0351 8.6451 3.0432 1.0000 4.075 39 Medium 2.1716 3.0352 8.6361 3.0424 1.0000 4.050 40 Medium 2.1722 3.0319 8.5968 3.0388 1.0000 3.938

41 Medium 2.1739 3.0352 8.6132 3.0408 1.0000 3.988

42 Medium 2.1771 3.0345 8.5738 3.0381 1.0000 3.850

4.3.3 Analysis of the Basic Portfolios In this section the results from the basic simulation will be discussed and interpreted. We will with the information from this part of the simulation select the values of the decision variables that are going to proceed to the simulations of the combined strategies. From the first five portfolios we can see that interest costs drops the shorter the maturity of the portfolio. It is a rather logical result since the yield curve has a positive slope in the long run. This implies that for an optimal debt portfolio regarding only the maturity structure, the firm should try to have as short duration as possible, but because of the funding risk, we chose to have a restriction on maturating loans as discussed above. The choice of restriction for the refunding is a matter of preference for the firm in question.

17

The dynamic refinancing strategy without swaps and caps differs little from the static since all that matters are the interest costs i.e. the shorter maturity the cheaper. But when combining the strategy with interest derivatives, especially the swap, it makes a significant difference, as we shall see below.

P(B)

MaturityX-Short Short Medium Long X-Long

Figure 4.1 a, b The performance of portfolios with different maturity structures.

X-Short Short Medium Long X-Long Maturity

E(LPR)

Depending on risk preferences, the limit of buying swaps should either be at five or six percent as figure 4.2 shows. The higher limit you choose, the more loans you swap and the more secure you are during periods with high interest rates. The result implies on the other hand that there is a trade-off between a high expected least-profit and a low probability of bankruptcy and therefore one has to rely on risk preferences to be able to separate the strategies. We chose three different dynamic swap limits, compared to five for the static case and this because we believe those to be the only relevant ones for the dynamic strategy. While simulating the combined strategies we will use a swap limit of five percent for the static approach and swap limits five- and six percent for the dynamic mode.

E(LPR)

Simple Swap Dynamic Swap

0.03 0.04 0.05 0.06 0.07 0.03 0.04 0.05 0.06 0.07 Swap Limit

P(B)

Figure 4.2 a, b The simulated Swap rates for both simple and dynamic Swap strategies.

Swap Limit

As we can see in figure 4.3, the longer swap the higher least profit and the lower probability of bankruptcy. This is a strong conclusion and we will use swap-length 16 consistently in the combined strategy simulation.

18

E(LPR)

Swap Length 4 8 12 16 20

P(B)

Figure 4.3 a, b Expected least-profit and probability of bankruptcy plotted against Swap length.

Swap L 4 8 12 16 20 ength

We tested three relevant limits for the static and the dynamic caplet. They proved to perform quite similarly for a cap-limit of five percent. For the case when the cap-limit is six percent, the dynamic swap performed much better regarding our definition for an efficient portfolio. We will therefore use the cap limit of five percent for both and only the limit of six percent for the dynamic strategy.

E(LPR)

Simple Cap Dynamic Cap

0.03 0.04 0.05 0.03 0.04 0.05 Cap Limit

P(B)

Figure 4.4 a, b The Cap limit plotted against expected least-profit and probability of bankruptcy for both simple and dynamic Cap/Caplet strategies.

Cap Limit

Cap rates higher than eight percent are hardly ever executed and cap rates lower than six percent are too expensive. These cases are therefore not included in simulation. From Figure 4.5, we can see that a cap rate of six percent performs best in combination with a cap limit of four.

19

Figure 4.6 can be interpreted as the shorter time until execution date, the more seldom the option is actually executed. This is intuitively a strong conclusion and we will consequently pertain an execution date of eight quarters in following simulation.

E(LPR)

0.06 0.07 0.08 0.06 0.07 0.08 Cap Rate

P(B)

Figure 4.5 a, b Performance of three portfolios with different cap rates keeping other variables constant.

Cap Rate

The idea of only capping a part of the loan instead of all of it proved to be unsuccessful as we can see in figure 4.7. In the next part, the whole part of the loans will be capped.

E(LPR)

2 4 6 8 2 4 6 8

P(B)

Figure 4.6 a, b Different Execution date plotted against the two criteria for an efficient debt portfolio.

Execution date

E(LPR)

0.25 0.5 0.75 1 0.25 0.5 0.75 1 Cap Share

P(B)

Figure 4.7 a, b If we chose the Cap/Caplet share one we maximizes the least profit and minimizes the probability of bankruptcy.

Cap Share

20

Our last decision variable is the cap, a sum of caplets. The performance of the cap, consisting of different amounts of caplets, is shown in Figure 4.8. The choice here is straightforward. We will use caps with cap length eight from now on.

E(LPR)

Simple Cap Dynamic Cap

1 2 4 8 16 Cap Length 1 2 4 8 16

P(B)

Figure 4.8 a, b With respect to Cap length the simple Cap/Caplet strategy performs better than the dynamic strategy.

Cap Length

Combinations of cap limit five and cap rate seven percent as well as cap limit four and cap rate six percent will be the strategies to use in the proceeding portfolios.

4.4 Combined Strategy Simulation From the previous simulation we observed which strategies that performed the best and in this section these strategies are combined to more advanced strategies. It should be pointed out though, that combinations of the best portfolio strategies, does not imply that they will perform the best together in every given situation. It is rather guidance for us, which strategies might perform well when combined and which will not. Some of the decision variables from the basic strategy are now set as constants according to our strategy in which some values are now excluded. In some cases all three strategies are used together and in others only two of them. Swaps have proven to be an essential part in every successful debt portfolio and will be included in all portfolios in this section.

4.4.1 Combined Portfolios The portfolios are as follows.

21

Table 4.12 Decision Variables

Refinancing Strategy

Swap Limit

Swap Dynamics Cap Limit Cap Rate Cap Dynamics

Portfolio

43 Medium 0.05 Static 0.04 0.06 Static 44 Medium 0.05 Static 0.05 0.07 Static 45 Dynamic 1 0.05 Static 0.04 0.06 Static 46 Dynamic 1 0.05 Static 0.05 0.07 Static 47 Dynamic 3 0.05 Static 0.04 0.06 Static 48 Dynamic 3 0.05 Static 0.05 0.07 Static 49 Medium 0.05 Static 0.04 0.06 Dynamic 50 Medium 0.05 Static 0.05 0.07 Dynamic 51 Dynamic 1 0.05 Static 0.04 0.06 Dynamic 52 Dynamic 1 0.05 Static 0.05 0.07 Dynamic 53 Dynamic 3 0.05 Static 0.04 0.06 Dynamic 54 Dynamic 3 0.05 Static 0.05 0.07 Dynamic

55 Medium 0.05 Dynamic 0.04 0.06 Dynamic 56 Medium 0.05 Dynamic 0.05 0.07 Dynamic 57 Dynamic 1 0.05 Dynamic 0.04 0.06 Dynamic 58 Dynamic 1 0.05 Dynamic 0.05 0.07 Dynamic 59 Dynamic 3 0.05 Dynamic 0.04 0.06 Dynamic 60 Dynamic 3 0.05 Dynamic 0.05 0.07 Dynamic 61 Medium 0.06 Dynamic 0.04 0.06 Dynamic 62 Medium 0.06 Dynamic 0.05 0.07 Dynamic 63 Dynamic 1 0.06 Dynamic 0.04 0.06 Dynamic 64 Dynamic 1 0.06 Dynamic 0.05 0.07 Dynamic 65 Dynamic 3 0.06 Dynamic 0.04 0.06 Dynamic 66 Dynamic 3 0.06 Dynamic 0.05 0.07 Dynamic 67 Medium 0.05 Dynamic 0.04 0.06 Static 68 Medium 0.05 Dynamic 0.05 0.07 Static 69 Dynamic 1 0.05 Dynamic 0.04 0.06 Static 70 Dynamic 1 0.05 Dynamic 0.05 0.07 Static 71 Dynamic 3 0.05 Dynamic 0.04 0.06 Static 72 Dynamic 3 0.05 Dynamic 0.05 0.07 Static 73 Medium 0.06 Dynamic 0.04 0.06 Static 74 Medium 0.06 Dynamic 0.05 0.07 Static 75 Dynamic 1 0.06 Dynamic 0.04 0.06 Static 76 Dynamic 1 0.06 Dynamic 0.05 0.07 Static 77 Dynamic 3 0.06 Dynamic 0.04 0.06 Static 78 Dynamic 3 0.06 Dynamic 0.05 0.07 Static 79 Medium 0.05 Dynamic * * * 80 Medium 0.05 Dynamic * * * 81 Dynamic 1 0.05 Dynamic * * * 82 Dynamic 1 0.05 Dynamic * * * 83 Dynamic 3 0.05 Dynamic * * * 84 Dynamic 3 0.05 Dynamic * * * 85 Medium 0.05 Dynamic * * * 86 Dynamic 1 0.05 Dynamic * * * 87 Dynamic 3 0.05 Dynamic * * *

22

4.4.2 Results Combined Strategy Simulation Table 4.13




Std(Profit) 107 SEK




43 Medium 2.2946 3.0767 7.8216 2.9977 6.3797 2.038 44 Medium 2.2703 3.0563 7.8593 2.9954 6.3797 1.963 45 Dynamic 1 2.2995 3.0757 7.7614 2.9991 6.5389 1.900 46 Dynamic 1 2.2725 3.0544 7.8191 2.9962 6.5389 1.888 47 Dynamic 3 2.3049 3.0834 7.7851 2.9913 6.5146 1.888 48 Dynamic 3 2.2796 3.0631 7.8351 2.9875 6.5146 1.850





73 Medium 2.2545 3.0504 7.9590 3.0191 6.6385 1.925

74 Medium 2.1975 3.0158 8.1836 3.0204 6.6385 2.163

75 Dynamic 1 2.2585 3.0509 7.9244 3.0183 6.7511 1.913 76 Dynamic 1 2.2018 3.0177 8.1593 3.0182 6.7511 2.150 77 Dynamic 3 2.2646 3.0579 7.9327 3.0116 6.7429 1.850 78 Dynamic 3 2.2082 3.0245 8.1632 3.0113 6.7429 2.063


85 Medium 2.3109 3.0983 7.8733 2.9967 6.3797 2.163 86 Dynamic 1 2.3209 3.1018 7.8085 2.9932 6.5389 2.013

87 Dynamic 3 2.3259 3.1091 7.8315 2.9859 6.5146 2.000

23

4.4.3 Analysis of the Combined Portfolios For the analysis of the combined portfolios, we will use the tool for efficient portfolios, developed in section 2.2, to be able to sort out the efficient portfolios. An interesting fact is that almost all portfolios in the combined strategies performed better than the portfolios in the first part of the simulation, even though the restraining function on the maturity, described under The Model in part two is implemented. It means that all of the combined portfolios are actually “good” in comparison to the basic portfolios, in line with the definition 2.3.2 for an efficient portfolio. The portfolios are plotted in the Least Profit-Probability of Bankruptcy diagram in figure 4.9.

C

E(LPR)

2.36

2.34 79

2.32 55

2.3 67

2.28

2.26

2.24 72 70

B68 2.22

78 762.2 A 74

P(B)1.750 1.875 2.000 2.125 2.250 2.375 2.500

The total set of combined portfolios plotted in a least-profit probability of bankruptcy diagram. Figure 4.9

We can by using definition 2.2.3 see that there are only seven portfolios which fulfill the demand for an efficient portfolio. This implies that all other portfolios can be discarded as inefficient portfolios. Even though inefficient, these portfolios can be used to analyze certain effects caused by a decision variable. If we study the set of portfolios simulated in the combined strategy section, there are six strategies that perform significantly worse than the rest of the portfolios. We find these in area A in figure 4.9. All these portfolios share a number of attributes, such as they all have a static cap strategy and at the same time a dynamic swap strategy. They also have the same cap limit and cap rate, five and seven percent. We know that caps and caplets are very expensive and it is possible, that using this interval is more costly than the other, included in the simulation. It is likely that the interest rate does not climb above seven percent frequently enough to motivate the purchase of caps and caplets up to an interest rate of five percent. Another reason why these portfolios have such low efficiency may be related to their cap and swap dynamics. The static cap strategy in combination with a dynamic approach when to issue in swaps, might lead to unnecessary capping because of

24

our main cap strategy, to cap in the beginning of a loan and when a swap is running out. See caps and caplets in the strategy section three. Area B includes portfolios with the highest probability of bankruptcy. Similar for these portfolios are that they all have a medium refinancing strategy and they all have swap limit of five percent. In the basic strategy simulation part we observed a close relationship between high probability of bankruptcy and a relatively low swap limit. We can once again recognize such a connection and we strongly believe that the high probability of bankruptcy for these portfolios can be related to this result. The medium refinancing strategy appears to perform worse than its dynamic counterpart. We can observe this by studying cases where the portfolios have the same conditions except for the refinancing strategy, see table 4.13. Moreover, the strategy of dynamic three is better than dynamic one in all cases except for two, where dynamic one has a lower probability of bankruptcy. Now we shall concentrate on the seven portfolios that made the cut into the efficient frontier. The two portfolios mentioned above, where dynamic one is the refinancing strategy, lay on the efficient frontier as portfolio 64 and 82. All of the other portfolios have a dynamic three strategy. As suspected, the medium strategy failed to place itself on the efficient frontier.

E(LPR)

64

65 82

84 53

2.34 83

2.33

87 2.32

2.31

2.3

2.29

2.28

2.27

P(B) 1.750 1.875 2.000 2.125 2.250 2.375 2.500

The total set of combined portfolios plotted in a least-profit probability of bankruptcy diagram. Figure 3.10

All of the portfolios on the efficient frontier, except for two, use a dynamic approach for the interest derivatives i.e. while being below a specified limit a derivative is only bought if the previous rate is below the current. Thus, if a firm consequently uses the dynamic derivative strategy instead of static, it is more likely they will make more money. This might not be a surprising result since the firm does not buy and pay for derivatives while

25

the interest payments are still decreasing but only when the economy might be on its way to a period with high interest rate. It is quite obvious that portfolio strategies, which exclude the use of cap and caplets, achieve very good results in this type of model. It should be noted that these efficient portfolios, are specific for this firm in replication and may not be the same for other firms. 5 Recommendation on a Debt Portfolio Strategy We have chosen portfolio 84 as our recommendation for an optimal debt portfolio. We base this on the fact that there is a relatively large increase in expected least-profit and a fairly little add to the probability of bankruptcy if we move along the efficient frontier line from portfolio 64, which has the lowest probability of bankruptcy. The difference between these two portfolios regarding expected profit is approximately 4.1 millions SEK quarterly. On the other hand if we go from portfolio 83, which has the highest least profit, there is significant reduction in probability of bankruptcy at the cost of an acceptable low decrease in least-profit. The difference in probability of bankruptcy between portfolio 83 and 84 is 0.688 ‰ which is almost 52 of the total probability of portfolio 84. Thus, an efficient portfolio in this case is one that only uses swaps and does this in a dynamical fashion. Swap limit is set to six percent and it fixes the rate for sixteen periods, or depending on the length of the loan. This portfolio will have a dynamic refinancing strategy where it switches from a medium maturity structure to a short if the interest rate exceed six percent and to a long maturity structure if it drops below four percent according to the vectors described in section 4.3. Caps or caplets are not included in this portfolio and we can only confirm that it is hard to find a good strategy for these derivatives. 6 Recommendations for Further Development This section discusses possible development and extensions surrounding all three parts of the paper. The initial portfolio in the model could be improved in the sense of letting it start by fulfilling its defined maturity structure. The portfolio would in such a case have a continuity of maturing loans in short as well as long interest rates from the start of the simulation. The restriction problem of thirty percent would become a constraint instead of a function that moderates the effect. If a thorough regression analysis for the revenues and the vacancies could have been done, then more precise relationships between different variables for extreme economic scenarios could have been found. This is due to lack of data. Instead of a two state Markov chain for the regime, a more advanced structure could be developed. There might be more sophisticated strategies to employ than the ones we have used.

26

More interest rate derivatives, such as the swap option, the floor et cetera could be implemented. If accurate correlation between profit and macroeconomic variables could be found then optimization process could be described as a pure mathematical problem.

27

7 The Underlying Model The main objective of this part is to describe and explain the stochastic model used for simulation in the previous part. It will be divided into two sections, where the macroeconomic simulation model will be treated in the first and the financial simulation model in the second. To construct a macroeconomic model, the number of variables to use as driving processes must be decided. The two most obvious ones, the growth rate and the inflation, are chosen. The growth rate is governed by a regime switching process that shifts from recession to boom in a cyclical manner. The inflation is independent of both the growth rate and the regime switching process. The reason for this is that one can assume that the central bank is successful in its task to stabilize inflation. Consequently, inflation will be stable around the inflation target irrespective of economic development. The three-month and the ten-year rate are evolved from these two driving processes with an additional stochastic disturbance factor. A function representing the yield curve determines rest of the interest rates between the three-month and the twenty-year rate. The computation of maturity structure and interest payments in the debt structure, are constructed as a complex series of loops and matrix calculations. Loan matures and is refinanced into new loans according to a certain vector as the maturity structure changes. All coupons are paid with each respective interest rate to finally arrive at the total cost of interest. Interest derivatives are bought according to the strategy of the portfolio and have to be accounted for. Assumptions will be made and motivated carefully. Implementation of all the different parts of the model will be fully described and the economic and financial cycle illustrated. For the macroeconomic model, some parts are influenced by the model that the Swedish National Debt Office developed to analyze the long-term costs and possible risk scenarios for central government debt portfolios.4 The debt portfolio simulation model is a self-developed model, which simulates a single firm’s debt and revenues. The organization of The Model is as follows; Section one describes the macroeconomic model and its contents with a thoroughly description of technicalities of the stochastic process. Section two explains how the debt structure and the firm’s income and expenses are implemented and developed. 8 The Macroeconomic simulation model In order to make this model work the economy had to be assumed to be stable with cyclical swings between booms and recessions and that the central bank is successful in

4 Bergstöm, P. and Holmlund, A. [4] develop this model. In this paper we are only using the model to simulate variables for the Swedish economy, and we don’t simulate an exchange rate.

28

its goal to stabilize the inflation, on average. A simple scheme over the model is presented in the figure below.

Short Interest Rate Model

All Rates Between 3 month and 20 Years

The Long Interest Yield Curve

Spread

GDPInflation

Economic Cycle

Figure 8.1 Scheme of the Macroeconomic Model. The dependence between the included function is here graphically displayed.

The economic cycle determines the regime state, recession or boom. This influences the calculations of real GDP growth and the spread between the short and the long interest rate. As mentioned above is the inflation calculated free from dependence to other variables. From inflation and GDP the short interest rate is derived. Moreover the spread function determines the long interest rate, here referred as the ten year rate. The yield curve has been extended so that one is able to obtain all the rates between three-month and twenty-year. Interest rates with maturities over ten years are only included for the cost evaluation of interest-rate derivatives.

8.1 The stochastic process A simple but very flexible way of modeling macroeconomic time series is to use autoregressive processes. It is a discrete process where each step is in quarters. For a majority of the variables in the macroeconomic model a stationary first order autoregressive process is used. The AR(1) process can be written as

ttt yy εβα ++= −1 , ),0( 2σε Nt ∈ where α is a constant, β the autoregressive parameter which describes the effect of a unit change in on and 1−ty ty ε the random shock which is assumed to be an IID (Independent Identically Distributed) normal variable. Since β is positive and less than

29

one, we have a positive dependence on the past and yet a stationary process comparing to if β would be greater than one where it would be an evolutionary process.

<

The expected value of for this discrete differential equation, s periods into the future, can be calculated with the following geometric series:

y

∑−

=+ +=

1

0][

s

it

sist yyE ββα .

As s tends to infinity we will get the long-term equilibrium. For geometric series with

1β and the formula converges to: ∞→s

βα−

=+ 1][ styE .

The growth process β is allowed to shift between two different values. This is modeled by a two state Markov chain. The states represent booms and recessions and will capture the economic cycle in the Swedish market. The growth function is influenced by an unobserved discrete random variable, s. This variable denotes the prevailing regime at the time and alters the growth function accordingly. The AR(1) processes for such a function will have the following appearance:

ttst yy εβα ++= −1 ( )et N σε ,0~ RBs ,∈ .

In practice, this means that the AR(1) process has two sets of parameters depending on whether the economy is in a boom or a recession. The shift in β is due to the transition probabilities in the Markov chain. A Markov chain is a discrete process with the lack of memory. It only depends on which state it is currently located in when the decision where it will move is decided. The probability of shifts is presented in the matrix below:

=

RRRB

BRBB

pppp

P ,

where is the probability that a boom state will follow a boom state and and are the probabilities of shifts in states. Here

BBp BRp RBp

=+ BRBB pp 1=+ RBRR pp .

In every step of the simulation a Bernoulli random number is generated to decide whether a shift in state is to take place or not.

30

The parameters of all the AR(1) processes, as well as the transition probabilities can be estimated on historical data using the Hamilton [9] procedure. The expected duration of each state is calculated by:

BBB p

D−

=1

1 and RR

R pD

−=

11 .

By transforming these durations to probabilities of being in a certain state and then substituting them as weights into the long-term equilibrium formula above we will get the expected value of : g

−

−−

−+

−

−−

−=

RRRBB

BB

BRRBB

RRt pp

ppp

pyβα

βα

121

121

.

The expected value and the economic cycle for the growth process lies around nine years. It means that it takes about nine years until a cycle is completed. Now on to describing how the variables in the macroeconomic model are constructed. First the structure of each time series process is explained and then their parameterization.

8.2 Inflation The inflation is modeled as an auto regressive process as described in the previous section:

ttt εφπδπ ++= −1 ( )πσε ,0~ Nt .

Inflation, π , is not effected by the regime because of the assumption made earlier that the central bank is successful in stabilizing the inflation. The inflation will therefore be stable around the inflation target irrespective of the development of the economy. In the equation 00025.0=δ , 95.0=φ , and 00062.0=πσ . The annual average of inflation is 2.0 %. As pointed out by Bergstöm et al. [4], empirical estimates of the equation on historic data, suggests that the inflation process is fairly persistent, with a φ -parameter value of 0.95.

31

8.3 GDP growth Unlike inflation, the real GDP growth is assumed to be dependent of regime i.e. the cycle in the economy. As for inflation, growth is also an AR(1) process.:

ttst ggt

εµρ ++= −1 ( )et N σε ,0~ ., RBs =

tg is the percentage growth rate of real GDP, and B and R represents the two states boom and recession. The change in regime in this model is captured by the constant sρ ,

Bρ when the economy is in a boom state and Rρ for recession. Parameter values in the equation is 00045.0=Bρ , 00028.0−=Rρ , 95.0=µ and 00094.0=εσ . The probability for the economy to stay in a boom state is 95 % and consequently 5 % to change state to recession. On the other hand, if the economy is in recession there is an 80 % probability that it will stay there and a 20 % probability that it will change to boom. The Markov chain, described in section 1.1.2, models the changes in regime. Annual growth rate is 2.5 %. The inflation rate and GDP growth rate are in this model independent of each other and the reason is, as we mentioned above, the central banks main task to stabilize the inflation.

8.4 Short interest rates From empirical data, dependence between short rate and the macroeconomic variables, inflation and GDP, is found. The reason for modeling the coupon-bearing short interest rate, that is, the rate where you are obliged to pay coupons every quarter, is due to simplification in implementation in the model i.e. zero-coupon rate is only needed when a caplet is used while the bearing rate is used all the time calculating the debt costs. The link between growth, inflation and the short interest rate is modeled according to the Taylor rule. The Taylor rule is a monetary policy rule developed by John Taylor [19]5 in the 90’s, to stabilize the inflation, using the short interest rate as a tool. Thus

),ln(ln)( ***tt

et

et

Tt YYri −+−++= λππθπ

where i is the Taylor interest rate, T

t*r the equilibrium real interest rate, the expected

annual inflation rate andY and Y is the real respectively potential GDP. Real GDP is calculated as follows:

etπ

t*

t

5 The Taylor rule argues that the central bank should focus and respond in the following way: First, the policy should respond to changes in both real GDP and inflation. Second, the policy should not try to stabilize the exchange rate, an action that frequently interferes with the domestic goals of inflation and output stability. Third, the interest rate rather than the money supply should be the key instrument that is adjusted. More about the Taylor rule for the interested reader in Macroeconomics by Blanchard, O. [5].

32

)1(1

0j

tjt gYY +∏= = .

To compute the real GDP, a so-called HP-filter is used. The Hodrick-Prescott filter is a smoothing method widely used among empirical macroeconomists to obtain a smooth estimate of the long-term trend components of a series. Technically, the HP-filter is a two-sided linear filter that computes the smoothed series Y of Y by minimizing the variance of Y around , subject to a penalty that constrains the second difference ofY . That is, the filter chooses to minimize:

*t t

t*

tY *t

[ ]∑ ∑=

−

=−+ −−−+−

T

t

T

ttttttt yyyyyy

1

1

2

2*1

***1

2* )()()( λ .

Still, the expected value of the Taylor rate converges to

** π+= riTt ,

since the expected value of the last two terms are zero as t tends to infinity. If the central bank were to follow the Taylor rule completely, the interest rate would swing sharply from one period to another. Therefore, the central bank practices some kind of interest rate smoothing to hedge against these switches with a certain time lag. Again the autoregressive function is used to adjust the move of the Taylor rate to a more realistic one. Thus

tTtttt iiii εςρκ +−−+= −−− )( 111 , ),0( 2σε N∈

whereς is technically the speed adjustment of the short interest rate towards the Taylor rate. The parameters are chosen so that, in equilibrium the short rate interest rate is equal to the Taylor rate. Parameterization for all variables is shown in table 2.1.

Table 8.1

Parameters Numeric Value

θ 0.5 λ 0.5 κ 0.0025 ρ 0.95 ζ 0.1

σ 0.00145

33

8.5 The Long Interest Rate, Spread and Yield The long interest rate is modeled in a similar fashion as GDP. It’s also a regime switching auto regressive function where the term η indicates the prevailing regime. That is,

ttist SSt

εφη ++= −+ 1 ( )εσε ,0~ Nt RBs ,= .

tS is here the spread between the short interest rate and the ten-year interest rate. To determine the spread, the function forecasts the regime i periods ahead in a perfect manner. Yield curve spreads are modeled so that recessions are associated with flatter yield curve compared to the booms. The period forecast in the model is in this case six months. Parameters in the equation is 002.0=Bη , 001.0−=Rη , 98.0=φ and

0017.0=εσ . The average spread is 0.74%. The interest rates between the three-month rate, , and the spread, is constructed in the following manner:

mtI , tS

tmttmt SYII ,1,, += .

whereY is a number between zero and one, such that a convex yield curve can be obtained, similar to a real life yield curve. An exponential function seemed to be a suitable solution to achieve the characteristic appearance of a yield curve. Thus,

mt ,

( )ων /)1(

, 1 −−−= pmt eY ,

where p is the maturity in quarters and ν =1.0528 andω =13, constants intuitively and empirically found. Figure 2.2 shows the appearance of the yield curve. Yt,m

1.0 0

3 month 10 years Maturity Figure 8.2 Yield Function.

34

It should be noted that for interest rates longer than ten years, Y , becomes greater than one and will stagnate towards the twenty year rate at approx 1.15. This function is used for all spreads, included negative.

mt ,

9 Financial Structure Simulation Model As macroeconomic variables change, so will also vacancies, prices and debt costs. Via the simulation of the macroeconomic variables for eighty quarters, or twenty years, the firm’s revenues and debt can be simulated. In each quarter, decisions are made for the maturity of the loans to borrow and whether to use a derivative or not. The simulation keeps track of all the cash flows, such as running costs, interest payment and profit, and calculates risks and rating for the firm in replication. The model is divided into two sections; one for the revenue and one for the debt. A scheme of the financial model is presented in figure 3.1, where growth rate, inflation and debt strategy act as building blocks for the whole procedure.

35

Inflation GDP

Interest Rate Revenues/m2 Vacancy LevelsInterest-Rate Derivatives

t-1 Debt Revenues

Exogenous Factors

Profit

Interest Payments

Loan StructureRating

Total Earning

Running Cost

Debt Strategy

Figure 9.1 Scheme of the Financial Structure Simulation Model. The scheme shows the dependence between the different functions included in the model. The scheme includes the exogenous given factors from the macroeconomic model. These are inflation, GDP and interest rate. The interest rate is the actual term structure, where all coupon-rates between three month and twenty years. The revenues are estimated by the inflation and GDP. The debt strategy determines the use of interest-rate derivatives. Depending on the interest rate, or rather the term structure, the debt strategy specifies a loan structure. This loan structure together with the interest-rate derivatives governs interest payments. Interest payments subtracted from the total earnings make out the profit. The total earnings and the interest payments also determines the next periods rating which in its turn decides on the costs of the interest-rate derivatives. In this scheme, such functions as loan maturity, fixed interest rate evaluation, probability of bankruptcy and real estate values are excluded. The reason for this is that these functions unlike the ones included in the scheme are not functions internationalized in the decision chain, merely functions that give valuable information on the performance of the debt strategy.

Interest payments as a whole are then computed. Notice that the costs of the derivatives depend on the performance of the firm in the previous quarter. Vacancies, revenues and running costs are calculated separately to finally become total earnings. Earnings minus interest payments result in profit. The following of this section describes how the revenue, vacancy levels, running cost and debt portfolio of the actual firm is simulated and implemented.

9.1 Revenues The revenues have been divided into five different posts. These posts have substantial differences in earnings, running costs and vacancy levels. Consider for instance that one of these posts acts on a highly regulated market, such as private housing, then that post

36

will differ significantly from a post in the free market, such as offices and stores, which acts without regulation. This motivates to do this separation. Because of the scarce data available, more or less intuitive assumptions had to be made. Some of these assumptions have been discussed with the company and approved as acceptable. Functions for revenues per square meter, running costs and vacancy levels are described in the following subsections.

9.2 Revenues per square meter Office and Store areas The data provided by the firm did not include enough information for a complete regression analysis for all possible economic scenarios. Therefore, assumptions had to be made about the revenues, especially in extreme economic situations, such as a very high inflation. One assumption is that inflation higher than three percent and lower than seven percent is just an extreme boom in the economy due to the central bank’s failure to stabilize the inflation and not a collapse in the economy. Inflation higher than seven percent, on the other hand, indicates accordingly to our assumptions, a collapse in the economy, whereas the revenues drop. The dependence between inflation, growth rate and the revenue are modeled as a polynomial of the third degree, in order to capture all possible situations. Thus

dttdtdtdtdddt GDPInflInflInflR ,,43

,32

,2,1, εββββα +++++= , SOd ,∈ where is the revenues per square meter for the specific post and the inflation in period . Here, indicates the post, Offices and the post, Stores. The parameters for respective area are:

dtR ,

ttInfl

o s363=Oα , 8.312=Sα , 9.1927,1 =Oβ , 7.1155,2 =Oβ , 734.9,3 =Oβ ,

1679,1 =Sβ , 9.830,2 S =β , 614.9,3 =Sβ , ( )734.9,0~,O Ntε and ( )614.9,0~ N,Stε .

37

Revenue/m2

E(R/m2)

E(Inflation) Inflation

Figure 9.2 Revenues per square meter plotted against the inflation rate. Revenues have the same appearance for office and store areas but they have different mean values. The expected inflation is 0.5 percent per quarter.

Private Housing Proceeds from private housing only depend on inflation and just in linear form. This is due to the regulations on this market. The contract is designed so that they only vary with the inflation. Thus

dttdddt IR ,, εβα ++= Hd ∈ , where 194=Hα , 491=Hβ and ( )16.2,0~, NHtε . Storage and Others The data supplied no evidence that either of these post would have any dependents with inflation or GDP. These posts have for simplicity been simulated with a constant and a disturbance factor as follows:

dtddtR ,, εα += OthStd ,∈ , with 186=Stα , 150=Othα , ( )04.11,0~, NSttε and ( )32,19,0~, NOthtε

9.3 Vacancy levels The data has unfortunately been limited in this area, and therefore have vacancy levels been approximated in agreement with the company. No distinction has been made

38

between the two posts offices and stores. The reason for this is because in the information that has been available, these two posts have been on aggregated form and not separated from each other. The post storage has been simulated in a comparable fashion with but with different parameters. Vacancy for these posts has been simulated as the function:

( )( )4arctan −−= tdddt GDPV φδβ d StSO ,,∈ , with 7, =SOβ , 5.7=Stβ , 5.1, =SOδ , 85.1=Stδ , 2.1, =SOφ and 1.1=Stφ .

Vacancy

E (V)

E(GDP) GDP(t-4)

Figure 9.3 Vacancy levels plotted against the percentile GDP growth. Vacancy levels are identical for office and store areas and the vacancy level for storage have the same appearance but with different mean value vacancy. Expected growth rate of GDP is 0.64 percent per quarter.

Figure 9.3 shows how the vacancy varies with growth rate for offices and stores. The tangent function was the most appropriate function for vacancies for these posts, but different posts may make the function tilt a bit. Since vacancies are slow in their adjustment to the growth rate, this function will be lagged four periods, or one year. The remaining two posts, private housing and others, have been simulated as a constant and a disturbance factor as follows:

tdtV εα += OthHd ,∈ . Here, 025.0=Hα , 0669.0=Othα and ( )014.0,0~,, NOthHtε . Critic can be directed to this for being unscientific, but lack of information in this area forced to make these assumptions.

9.4 Running costs

39

To estimate the firms running costs, information from a previous work that estimated an average cost per square meter for the five different posts, has been used. This information was obtained from the firm’s annual report and therefore cannot be declared here as a reference. The running cost for the different posts has the following appearance:

dttdtdt InflInflRC ,*

,, )1( εα +−+= , where is the long-run average inflation in quarters. *Infl

9.5 Profit The amount of money that the firm makes every quarter is the revenue minus total costs of interest derivatives, coupon payments and running costs. For this part, all cash flow streams are put in vectors and simple vector addition is used:

ttttt RCDCRP −−−= , where P is the profit, R the revenue, C the coupon payments, the cost of derivatives and the running costs. Profits are added, or withdrawn if the profit is negative, into an equity-pool in every step. This pool is set to a certain level at the start of the simulation and will never exceed that limit during simulation. So, if the profit is high enough to make the equity-pool exceed that level, this money will be assumed spent on dividends and the likes that the model ignores. On the other hand, if the pool drops to a specific low critical level, the firm and its strategy are assumed to be close to bankruptcy and the scenario will be flagged. A flag indicates a great loss for the firm and is therefore used as one of the risk measures in the main work. The bankruptcy probability is calculated as the total amount of flags divided by the number of simulation runs.

DRC

9.6 Debt structure To simulate the debt portfolio, some kind of algorithm is needed. We have chosen to regard each loan separately and not as a portfolio of loans. This means, all decisions made, are made for each loan in every step of the simulation. In each step, a refinancing vector of length forty is used to take loans, where each element in the vector indicates the proportion of the loan to be taken in every maturity length. The maximum maturity of loans is ten years, or forty quarters. This is because of the reasonable assumption that creditors are not willing to grant loans with longer maturities than ten years. A matrix is first created with zeros in every row and column. The amount of loans to be taken in each step and maturity length is placed in this matrix. The matrix is built one

mnA ,

40

row at a time in a double-loop process, with the first loop determining every time-step, up to and the second every maturity length,

in j up to maturity m , where is the

total number of quarters in a simulation run and = 40 is the maximum maturity length. Which refinancing vector to use, is decided at the start of every time-step, depending on the interest rate at the given time.

80=nm

x,1 2

j M' ...1

∑=

m

1ijE ,

[ ]imimi xxTDA ,...,...1, = ,

where TD is the outstanding debt (TD is the total debt of the firm) and i 1 x the proportion of the outstanding debt that is to be refinanced in different maturities. A three-dimensional matrix M is constructed so that all different maturity loans has a two dimensional matrix on its own, with a one as a diagonal and the rest zeros. The diagonal is moved one step to the right with each maturity length longer. For every time step we go through all the maturities

mnn ,,

ij and place them in a temporary matrix . That is, nm,E

jnnnnj AE ,...1,,...1...1, = .

The sum of the values in in the i :th column are the outstanding debt for the next time step. Thus

E

+ =j

iTD 1 .

The process is then repeated until the loan structure, is complete, that is for all i up to 80. The matrix lay as a foundation for the whole debt structure. More precisely, it describes the amount of money to be issued in certain maturity at a given time. The matrix is graphically presented below to enhance the reader’s understanding.

mnA ,

mnA ,

=

=

40,801,80

40,11,1

40,802,801,80

40,22,21,2

40,12,11,1'

80

2

1

,

.................................

...

...........................

..

..

.

.

.

.

.

XX

XX

xxx

xxxxxx

TD

TDTD

A mn

41

Further, a loan taken in step will have coupon payments each quarter until its maturity date,

iji where + j is the maturity in quarters for the loan. To obtain the coupons that

have to be paid in every step and in all maturities, yet another three-dimensional matrix had to be constructed . It has the same analogy as but with ones wherever coupons have to be paid instead of only when they expire. The sum of all coupons are aggregated into a matrix, according to which rate it is associated with in the following way

mnnN ,,

nB ,

mnnM ,,

m

∑=

=n

ijnijijn NAB

1,...1,,,...1 mj ...1= ,

where is a vector and therefore the sum indicates a vector addition. Each vector in the matrix is then multiplied componentwise with each respective interest rate vector and summed to achieve the total amount of coupon payments

in the following manner:

jniji NA ,...1,,

jnB ,...1 mnB ,

jnR ,...1

nC ...1

∑=

=m

jjnjnn RBC

1,...1,...1...1 .

One can therefore say that each element in C is the amount of interest payment that has to be paid in that quarter.

n

Since a portfolio in the simulation does not start in perfect convergence for the maturity structure, there will be occasions when more than thirty percent of the loans mature in any given year. This effect becomes even stronger when using the dynamic refinancing strategy compare to the static. It happens mostly in extreme booms, when interest rate is high a longer period of time and the portfolio tries to issue in short loans in order to reduce its interest costs at the same time as a large portion of longer loans are about to mature. This effect is ignored in the first part of the simulation because the objective is to study the portfolios decision variables separately and in comparison with each other rather than considering it to be an actual possible efficient portfolio, as described in the definitions for an efficient portfolio in the main work. For the portfolios in the combined strategy, each portfolio is allowed to exceed this level, but not by much and especially not for a longer period of time. If the portfolio would fulfill the assigned strategy from the beginning, this case would never occur for the static refinancing strategy and only in extreme booms for the dynamic. Anyhow, for the program to restrain this effect, some kind of moderation is needed. To be able to develop such a function, the maturity structure has to be computed i.e. how much and when loans mature. The maturity structure is also useful for comprehensive reasons.

42

The three-dimensional matrix , from the creation of the loan structure above, is used to evaluate the proportion of loans that mature in

mnnM ,,

τ quarters from time in the following way:

t

∑=

=40

1,,,...1,...1

ττττ nnnn MAG .

The sum of the first four quarters divided by the total outstanding debt equals the proportion of maturing debt in the first year; the sum of the next four quarters divided by the total outstanding debt equals the proportion of loans that mature in the second year and so on. Thus, for the first year

∑=

=4

1,...1...1

iinn GH .

Calculations are made for vectors, where each element in the vector indicates the proportion at the actual time . t Whenever more than thirty percent matures in the upcoming year, the program decides to issue in longer maturities despite of what the main portfolio strategy wants. So, whenever this value exceeds this limit, it will start decreasing until it finally drops below. To be able to make decisions for the different strategies for every loan in every given time-step until it matures, an advanced procedure had to be developed, that can alter rates (swap and cap rates) and loans along the way. Unfortunately, the only way this could be done was through a triple loop, which is a quite time-consuming process. The computation of the matrices , , C and that required loops could fortunately constructed inside the triple loop.

mnA , mnB , n mnG ,

The procedure starts with evaluating all loans taken in time-step one until they expire. Decisions will be made whether to buy a swap or a cap and rates will be fixed according to that strategy. The costs for the caps are summed into a vector in the same manner as for the coupon payments. The procedure continuous to time-step two, and evaluates all the loans taken in that period and so on.

9.7 Implementation of Interest Rate Derivatives The only time the zero coupon rate is needed is when a derivative has to be evaluated. This motivates therefore the simulation of coupon bearing rates in every period instead of the zero coupon rates. The discount factors and forward rates will therefore only be computed when a derivative is used. Whenever a fixed coupon rate is preferred by the portfolio strategy prior to the variable rate, to hedge against the interest rate risk, a swap is used (see part three). When the swap

43

is bought, the loan in effect is indexed with a number, the length of the swap, which in every step of the simulation is reduced by one until it hits zero, where a new swap can be bought. The computation of a swap’s cost is divided into two parts; by considering the forward rates until the end of the swap (detailed description in part three) and by evaluating the rate of the firm. The latter is done by a function quite arbitrarily called rating. This function does not calculate the firm’s formal rating, but there is a close resemblance to it. The function calculates the interest rate coverage, i.e. how large the operating income is with respect to your interest rate payments. This value is called EBITDA (Earnings Before Interest Taxes Depreciation and Amortization) and is calculated in every time step. The value is compared with the value from the previous step and if the two differ sufficiently much the rating changes. Then rating calls the function base-points and from the current rating this function calculates the cost for the swap u (explained in part three). The cost is not measured in a certain percentage of the loan, as one might think, but as an additional percentage to the current level of interest rate. The cost, in percentage of the rate, is therefore independent of the amount that is borrowed. More on swaps in part three.

k

Another way to hedge against interest rate risk is the cap. The costs associated with a cap are relatively high and the use of it is therefore limited. When a cap is executed, the loan in effect is indexed by the number of caplets bought, in the same manner as for swaps. The pricing of caps is quite complicated and will be thoroughly explained in part three. If a portfolio includes both of the derivatives, the swap will always have a higher priority. It means that, if the criterion for buying a derivative is fulfilled for both the swap and the cap, then the swap will be taken and not the cap. The reason for this is that swaps have proven to be more efficient and less costly than its counterpart. 10 Time Issues The model is implemented in the program Matlab6. There were two main reasons why we chose this program. The first reason was the simple implementation and the second reason was the powerful tools for matrix operation that Matlab features. The simple implementation helped us to keep down to construction time of the model. Even though, this part of the project took approximately two and a half month. This time includes not only the actual coding, but also the initial studying to get to know the problem at hand. A large part of the information is stored in matrices and the program would have been extremely loop-intensive if we would have used another programming language. By using Matlab we were able to keep the loops to a minimum. The main drawback is that when we actually have to loop an algorithm, then Matlabs weak side is exposed. Loops are generally slowly executed in Matlab. For instance, at one point in every simulation step we are forced to carry out a triple loop to fill a three dimensional matrix with

6 Version student edition 5.3.

44

information and this is the individual most time consuming operations next to the valuation of the interest derivatives. To simulate a single portfolio for a period of twenty years with quarterly steps, a thousand times, takes approximately a half an hour. To simulate all the 87 portfolio strategies would take about twenty hours. Due to the fact that we divide the basic and the combined strategy simulations, one cannot give an exact value for the total time. The basic simulation consumes lesser time then the combined strategies since they don not use both derivative strategies for a single portfolio. The simulation time could be cut down significantly if we would chose to execute some calculations in the programming language C++. The reason is the superior class and function based platform of C++. But the developing time would increase unacceptably for this project.

45

11 Arbitrage and Pricing of Interest Rate Derivatives This part of the paper will treat the valuation of the interest-rate derivatives included in the model from the preceding section. For a correct valuation of the derivatives, the conditions for no-arbitrage opportunities must first be derived. The fundamental theorem of asset pricing states that if the number of trading periods T is finite, then there is no arbitrage opportunities if and only if there exists a risk neutral probability measure. It then suffices for us to show that such a probability measure exists in our model and there by avoiding all arbitrage opportunities. This is done by first introducing the concept of risk neutral probability measure in a single period model. Further the conditions for no-arbitrage in the single period model are extended to hold even for the multi-period model, and the intuitive motive for this is that the term structure in our model cannot be defined in a single period model, for obvious reason. The fact that we have a model with infinite sample space makes the problem of identifying a risk neutral measure a little bit more difficult as we shall see in later sections. But there are ways to circumvent this inconvenient fact. After a thorough discussion and explanation of the risk neutral probability measure in the single and multi period cases, the term structure model and the yield curve approach is introduced. These concepts refer to the way one chooses to build the model. With the yield curve approach we are able to avoid arbitrage opportunities in an elegant and fairly easy way provided the existence of a risk neutral probability measure. When shown that the model is free from arbitrage opportunities, it is possible to proceed with the valuation of the interest-rate derivatives. This organization of this paper is as follows: Section 2 introduces an initial discussion about arbitrage and what it really is. The section will also feature three simple examples of arbitrage. In Section 3 the risk neutral probability measure is discussed. Section 4 elaborates on the Term Structure model and it will also treat the Yield Curve model. The infinite sample space case is briefly presented in Section 5. In Section 6, which is the last section, the valuation of the interest-rate derivatives is computed. 12 Arbitrage Under what condition can a derivative’s exact value be determined before the expiration date? To answer this question, we turn to the concept of arbitrage, which is very important in option pricing. Arbitrage is a transaction whereby a certain positive cash flow is produced in the future on an investment requiring zero outlay today. Or, one can say, if zero money is invested today produces a positive inflow with certainty in the future, we have an arbitrage transaction. The market itself does everything in its power to avoid arbitrage possibilities, since all actors on the market prefer a higher gain before a lower. Arbitrage exists in real life and

46

people make money on it, but only for a shorter period of time. As soon as the arbitrage is discovered it will cease to exist since all actors will try to take advantage of it. In theory though, arbitrage transactions are not possible. To see this, suppose you can buy a stock for price 1 dollar today and obtain 1.2 dollars with certainty one year from now. If you can borrow money at the risk free rate of ten percent annually, you will have a money machine. Everyone would want this money machine. The increased demand for this type of stock will push its market price up until equilibrium is restored and the following condition is fulfilled:

09.11.12.1

0 ≈=S .

The stocks certain rate of return must equal the risk less interest rate, so that no excess profit can be made by borrowing and investing the borrowed money in the stock. At price S , the possibility of arbitrage vanishes. In equilibrium, arbitrage transactions are not available because the market forces will have eliminated them.

0

With options one can create a portfolio composed of the option and the underlying stock such that this portfolio’s return is certain. Let us first assume that such a portfolio can be created and show how the option price is determined. We demonstrate this with the use of call options. For example, suppose we create a portfolio by buying one share of stock and selling two call options on that stock. This portfolio’s total investment will be

00 2CSV −=

at time . When you sell a call, you receive , which decreases the portfolio’s initial outlay. If the return on this portfolio at date

0=t 0C1=t is R with certainty, then by the no-

arbitrage argument in equilibrium, we must have

rCS

RVR

+=−

= 12 00

where r is the risk less interest rate, corresponding to the investment period, in this case the time interval between and 0=t 1=t . If we know R , and0S r , then C can be determined, this is called the option’s equilibrium price. Any deviation from this value implies that arbitrage transactions are available.

0

Another classical case of arbitrage is the relationship between forwards and spot rates. Suppose there exists two bank accounts with one and two years maturity and yields of and . If we leave

1y

2y K in the two-year bank account, it will grow to at the end of two years. Alternatively, we might place the same amount

22 )1( yK +

K in the one-year account and simultaneously make arrangements that the proceeds, which will be , will be lent for one year starting a year from now. That loan will accrue interest at a prearranged rate of . The rate is the forward rate of investing money for a year, a year from now and decided today. The final amount of money we receive at the end of

)11( yK +

2,1f 2,1f

47

two years under this compound plan is . We now have two different ways of investing

)1)(1( 211 fyK ++

K for two years. Since both are available, they must be equal. Thus

1)(1() 12 fyK ++=

11

)1(

1

22

2 −++

=y

y

Q Ω

0> Ω∈ω] 0* =nS =n

)1( 2,12yK +

or

,1f .

Remark 12.1 Due to the possible confusion between the indexes and the potencies, the forward rate will in remaining sections be denoted as or in general . 2,1f

21f

jif

Hence the forward rate is determined by the two yields. If these two methods of investing money did not return the same amount, then there would be an opportunity to make arbitrage profit. It is important to notice that the theory of arbitrage excludes transaction costs such as credibility fees and other related costs for arranging a deal. 13 Risk Neutral Probability Measure An investor is said to be risk neutral if that person is indifferent between a certain return and an uncertain return with the same expected value. Risk neutral investors therefore only care about expected return and not on the risk at hand. The expected rates of return of all securities must be the risk-less rate when the investors are risk neutral. For this reason, the probability measure is called the risk neutral probability measure and we form a definition for this below:

QQ

Definition 13.1: A probability measure on is said to be a risk neutral probability measure if

(i) ( )ωQ for all and (ii) [∆QE , 1, 2, …, N

where indicates the n:th security discounted with the risk less interest rate. Here is the total sample space and

*nS Ω

iω is one of finitely many possible outcomes, i.e. Q assigns a positive probability to every possible outcome. The notation [ ]XEQ means the expected value of the random variable X under the probability measureQ .

nS is the n:th risky security in the model and the notation in (ii) should be thought of as

48

[ ] ( ) ( )[ ] ( )[ ] ( )0101 *****nnQnnQnQ SSESSESE −=−=∆

so [ ] 0* =∆ nQ SE is equivalent to

( )[ ] ( )01 **nnQ SSE = , =n 1, 2, …, N

Note that because the discounted initial value is the initial value of the security. The interpretation of the equation is that all risky securities should have the same return, as mentioned above. Definition 13.1 is true for the single period case where we only have an initial date t and a terminate date

( ) ( )00*nn SS =

0= 1=t where trading is allowed. Here follows an explanation and interpretation of Definition 13.1. Consider the set

*: GXRXW K =∈= for some trading strategy H . W is here thought of as a set of all the risky securities, i.e. random variables, and as the discounted possible wealth at time

WX ∈1=t , or all possible outcomes when the initial

value of the investment is zero. The gains process is a random variable that describes the total profit or loss generated by the portfolio between times zero and one. indicates the discounted value of the growth. The trading strategy

describes an investor’s portfolio as carried from time zero to one.

G*G

( NHHH ,...,, 10= )H Note that W is actually a linear subspace of KR and for any WXX ∈21 , scalars and

also has . a

b WbX ∈+ 21aXConsider another set such that A

0: >∈= XRXA K which is the strictly positive orthant of KR . For the case of 2=K , is the strictly positive quadrant. It is fairly obvious that there exists an arbitrage opportunity if and only if W ø, since the corresponding trading strategy

A

≠∩ A H will generate discounted growth greater than zero with probability one. The orthogonal subspaceW is defined as ⊥

WXYXRYW K ∈∀=⋅∈=⊥ ,0: .

The inner product of X and Y is denoted as:

)()(...)()( 11 KK YXYXYX ωωωω ++=⋅ .

49

50

If one considers the geometric picture in figure 13.1, it is easy to believe that ≠∩ AW ø implies the existence of a vector in ⊥W such that all points except for the origin is strictly positive along this ray. On this ray there is a point where all components sum to one, called a probability measure P . Thus

0,...0,1...: 121 >>=+++∈= KKK XXXXXRXP .

It means that =∩ AW ø if and only if ≠∩⊥ PW ø. Since WSn ∈∆ * for all n , it follows that any element of the set PW ∩⊥ is actually a risk neutral probability measure.

If Q is a risk neutral probability measure, then for any WG ∈* the following must hold:

[ ] [ ] 01

*

1

** =∆=

∆= ∑∑

==

N

nnQn

N

nnnQQ SEHSHEGE ,

where PWQ ∩∈ ⊥ . Moreover, if we let M be the set of all risk neutral probability measures, we have that

PWM ∩= ⊥ . From the geometric intuition used above, it is easy to understand that =∩ AW ø if and only if ≠M ø. It is the same as: There are no arbitrage opportunities if there exists a risk neutral probability measureQ .

A

M

P

W

W

X(ω1)

X(ω2)

Figure 12.1 Geometric interpretation of the risk neutral probability measure.

The underlying simplification in this section is that the model has to be a single period model. The problem becomes slightly more complicated when the single period model is extended into a multi period model. And this will be the topic for the next section. 14 The Martingale Measure The risk neutral valuation approach is sometimes referred to as using an equivalent martingale measure. If the no-arbitrage condition shall be extended to the multi-period model, then there must exist a suitable measure such that the expectation of the future value of relative prices must equal its value today. Apart from technical conditions, this is exactly the requirement for a stochastic process to be a martingale. To start from the beginning, Harrison and Kreps [10] showed that in a world where interest rates are zero and there are no arbitrage opportunities, there exists a unique equivalent martingale measure under which the price of any non-income-producing security equals its expected future price7. It is easy to see that the equivalent martingale measure is consistent with the risk-neutral measure that is presented in the previous section. When interest rates are zero, the risk neutral measure result tells us that we can set the drifts of all securities equal to zero and value the securities by using a zero discount rate. Setting the drifts equal to zero creates the martingale measure. Using a zero discount rate means that the price today equal the expected future price. The equivalent martingale measure theory can be extended to a world where interest rates are nonzero. Define V as the value of an instrument that is worth one dollar and is invested at each moment in time at the instantaneous risk-free interest rate. Consider the change of units of measurement so that the prices of all other securities are defined in “units ofV ” rather than in dollar. That would mean that if the dollar price of a security were P , the price in units of V is VP . With this change, we move to a situation where interest rates are zero and the equivalent martingale measure argument is valid. More formally, X is said to be a martingale if

[ ] tttQ XXE =ℑ+1 , under a probability measure Q and ℑ being the flow of information gathered up until now. In other words, if a process is a martingale the best guess as to its expectation over the next time step, contingent on all the information up to time t , is its value at time . This property also extends over as many time steps within the trading horizon as desired:

t

[ ] ttjtQ XXE =ℑ+ 0>∀j

The function states that the security price deflated by the discount factor from time t to

tXjt is a martingale under the risk neutral probability measure . A martingale is a

zero-drift stochastic process and this result can be rephrased as: If no-arbitrage is to be + Q

7 For more on this see J. M. Harrison and D. M. Kreps [10].

51

allowed, then one must be able to construct a probability space such that relative prices are martingales. This requirement of zero drift might seem surprising, since it seems to imply no expected return of an asset, but the martingale requirement is rather equivalent to imposing the same return from all assets. Martingales are often used as models of fair gambling games. It originates from the famous gambling strategy to double the stake as long as you lose and leave whenever you win. In short, the strategy is called a martingale. In the model in part two each discounting is made a single step at a time once the interest rate structure is determined, and then backward discounted in a recursive evaluation process. Because of this association with martingales, the risk-neutral probabilities are often termed martingale probabilities. However, in preceding sections it will be referred to as the risk neutral probability measure. 15 The Term Structure Model The procedure for modeling the interest rate is described in the macroeconomic model. This model can be viewed upon as a securities market model. Here the securities in question are fixed income securities such as coupon bearing bonds. We call this market a term structure model. Three conditions are required for the securities market model to be a term structure model. First, the model has to be a multi period model. The second condition is that the interest rate r must be a strictly positive, predictable process, so that the interest rate r , for borrowing and lending over the period (t-1, t] is known at time t-1. Here we make an initial definition: Definition 15.1 A bank account process is a sequence NtBB t ...0: == , where 10 =B and i are random variables. The bank process will be distinguished from the other securities because its time

iB N...1∈Nt ...1= prices ( )ωtB will be assumed to be strictly

positive for all Ω∈ω where Ω is the probability space, i.e. 01 >− −tt BB . Since the term structure model will feature several interest rates, r will be called the spot interest rate as well as the risk less interest rate. The specific nature of the interest rate simulation in the previous part of this paper, would have forced us to derive the zero coupon bearing interest rate from the simulated coupon bearing interest rate and then from this obtain the spot rate. The third condition is that the zero-coupon or discount bonds are included among the risky securities. The price of the zero-coupon bond is called Z and will be defined for all maturities ς where T≤≤ ς1 . If the maturity date of a zero-coupon bond isT , then the t-price will be denoted as Z . If is today, then the t-price for T

t t ς>t is not defined. Hence

52

there is a collection Tt

tt

tt ZZZ ,...,, 21 ++

Q

of zero coupon bond prices in each time . This collection is called the term structure of zero coupon bond prices.

t

[ ttsQs BZBE= |ςς

( ) (sst rB += + 1...1 1

ς=

] ς0

s

The term structure model must be free from arbitrage opportunities, so there must be a risk neutral probability measure under which the discounted prices of the zero coupon bonds are martingales. Moreover, there must exist an arbitrage free transformation from the coupon bearing interest rate to the zero-coupon bearing interest rate. In other words, there must exist some probability measure with Q ( ) 0>ωQ for all Ω∈ω such that, for everyς ,

]sZ ℑ , ς≤≤≤ ts0 (15.1) In equation 15.1, )trB + where is the zero-coupon bearing interest rate and . is as before, the information known to the investors up until time . If we were to take

ir1=ς

ςZ sℑ st , we can see that zero-coupon bonds must satisfy the relationship

[ ( ) ( ) [ ]ssQssQs rrEBBEZ ℑ++=ℑ= + |1...11| 1 ςςς , ≤≤ s

given any risk neutral probability measure Q . But as mentioned above we need to “transform”, the existing coupon bearing interest rate to the zero-coupon interest rate. This is done in the following fashion. We start by formulating the discount factor for the single period case,

τcss

ss r

D1,

1

11

+

+

+=

where is the coupon rate from to c

ssr 1, + s 1+s and τ is the fraction of the year between the :th and the (s )1+s :th reset, therefore equal to 21 or 41 for a semi-annual or quarterly discount factor. The coupon bearing rate is the interest you have to pay the creditors when borrowing money. This rate is a floating rate and the coupons will therefore vary from one period to another. Also note that there exists one specific coupon rate for each maturity length. will mean the discount factor between the and ls

sD + ls + . If a loan has length l , the following recursive relationship must hold;

∑=

++

+ =+l

k

kss

clss

lss DrD

1, 11 τ .

This equality can be rewritten as;

53

( ) 111

1,, =++ ∑

−

=

++

++

l

k

kss

clss

lss

clss DrDr ττ ,

or

( ) ∑−

=

++

++ −=+

1

1,, 11

l

k

kss

clss

lss

clss DrDr ττ . (15.2)

To compute all following discount factors, equation 15.2 is solved for , ending up in the recursive function,

lssD +

τ

τ

clss

l

k

kss

clss

lss r

DrD

+

−

=

++

+

+

−=

∑,

1

1,

1

1. (15.3)

The following relationship for the zero-coupon bearing interest rate of length , compounded annually, is the well known

l zlssr +,

( ) τlzlss

lss

rD

+

+

+=

,11

and if we solve this for we obtain z

lssr +,

111

, −

=

++

τl

lss

zlss D

r .

By letting we get the desired spot rate, mentioned above. Now we can express the zero-coupon bond in an alternative way, which will include the discount factor and indirectly, also the coupon bearing interest rate,

1=l

[ ]ςς

sQs DEZ = .

The term structure described in this section can be developed further into a yield curve model, which will be the topic of the next section

15.1 The Yield Curve Model and the Forward Rate An expanded approach to build the term structure model is called the yield curve approach. Unlike the method mentioned above, one regards the whole term structure as a

54

state of a stochastic process. Instead of modeling one interest rate process as above, you directly specify the whole term structure and how it evolves in time. This is the approach we chose when constructing our model. The term structure in our model consists of the coupon bearing interest rate or equivalently the prices of the coupon bearing bonds with maturity up to ten years. Even this way of constructing the model may introduce arbitrage opportunities. Consider the following equation

[ ]ssQsss ZEZZ ℑ= ++ |1

1 ςς , s T≤< ς (15.4) where is today. For the absence of arbitrage, it is necessary and sufficient that there exists a probability measure Q such that the equation 15.4 is satisfied for all indicated

and

s

s ς . What this equation really states is that the price of a zero coupon bond between and s ς must be the same as combining a bond with initial date and maturity date

with a future bond with initial date s

1+s 1+s and maturity dateς , i.e. these to investments must give the same result.8 The forward interest rate and the forward bond prices are in our model derived from the discount factor. As we shall see, the result that is derived below will agree with equation 15.4. Because the coupon bearing rate is simulated we choose here to define the forward rate with respect to the coupon rate r . The present value of a coupon bearing bond with face value 1 and τ payments a year, equals the discounted value of the cash flow stream

∑=

++

+ +=j

k

kssjss

jss DrD

1,1 τ . (15.5)

An alternative bond investment with equal present value is (and must be, if the condition stated in equation 15.4 should hold for our model)9:

∑ ∑= +=

++

++

++

+ ++=i

k

j

ik

ksis

jsis

kssiss

jss DfDrD

1 1,1 ττ where . (15.6) ji <

We now have two different ways to invest that will give the same return. These two should be equal, since both of them exist. Thus if we use equation 15.5 and 15.6 we have

∑ ∑ ∑= = =

++

++

++

++ +=

j

k

i

k

j

ik

ksis

jsis

kssiss

kssjss DfDrDr

1 1,, τττ

+1

. (15.7)

8 For a more thorough deduction of the equation 15.4, see Pliska [16]. 9 It is fairly obvious that equation 15.4 and 15.7 are not exactly the same, but the transformation part between the zero coupon bond and the coupon bearing bond have here been omitted. The reader can for his or hers own verification, do this transformation with the information from the previous part where we via the discount factor go between the zero coupon interest rate and the coupon bearing interest rate.

55

The main objective of this paper is to clarify how the interest rate derivatives were priced. To do this we need the forward rate, and if we solve equation 15.7 for and if we add and subtract 1 to the denominator on the right hand side we get:

jif

∑

∑ ∑

+=

++

= =

++

++

++

−+−= j

ik

ksis

j

k

i

k

kssiss

kssjss

jsis

D

DrDrf

1

1 1,, 11

τ

ττ,

and further,

∑

∑∑

+=

++

=

++

=

++

++

++

−−

−

= j

ik

ksis

D

j

k

kssjss

D

i

k

kssiss

jsis

D

DrDrf

jss

iss

1

1,

1, 11

τ

ττ

444 8444 76444 8444 76

.

By using the equation 15.5 we obtain the following formula for the forward rate;

∑+=

++

++++

−= j

ik

ksis

jss

issjs

is

D

DDf

1τ

.

We have now developed the interest rates that are needed for the pricing of the derivatives. We can draw the conclusion that our model satisfies the equation 15.4 and therefore is free of arbitrage opportunities if and only if there exists a risk neutral probability measure . By deriving the future interest rate as described in this section we ensure that the zero coupon bond price process is a martingale. Now it suffices to show that the other condition for the existence of a risk neutral measure holds in our model (remember from earlier sections that this was

Q

( ) 0>iQ ω for all i , this means that we need a finite sample space so that the probability measureQ assigns a positive value to every possible outcome iω ). Our yield curve has the opportunity to assume any form, and this puts us in a position where we have an infinite sample space and this complication is the subject of the next section. 16 Infinite Sample Spaces If there were not to be any arbitrage opportunities in a security market model, one had to have a risk neutral probability measure. This fact enforced the model to hold for two

56

conditions, where the first was that there had to be finite sample spaces where one could define a strictly positive mass for every state Ω∈ω . The second condition was (and still is) that every stochastic price process had to be martingales. If these two conditions were crucial for the existence of a risk neutral probability measure, we would be in trouble due to our infinite sample spaces. Now, it turns out that the condition of finite sample spaces can be relieved if the number of trading periods in the model is finite. Thus, if the number of trading periods ∞<T and if the second condition from above, i.e. if

[ ] ttst ZZE =ℑ+ | , for all s 0, ≥t holds, then there exists a risk neutral measure. To establish this result demands a relative technical discussion surrounding the concept σ -fields or σ -algebras. If a collection of subsets ℑ of the total sample space fulfills certain conditions, this collection of subsets is called aσ -algebra. By using this concept in order to model the information known to investors and by introducing the concept of almost surely one can verify that the result above holds. This is though an all too broad subject to be included in this paper and one can only refer to relevant literature for the interested reader.10 17 The Valuation of Interest Rate Derivatives Interest rate derivatives are often more difficult to value than equity derivatives. There are a number of reasons for this, such as: The probabilistic behavior of an individual interest rate is much more complicated than that of a stock price or for example an exchange rate. For the valuation of many products it is necessary to develop a model describing the probabilistic behavior of the entire yield curve (as we have done in our case). The volatility of different points on the yield curve may not be the same. We have in our model for simplicity assumed the volatility to be the same over the whole yield curve. Another complication is that the interest rates are used to discount as well as for the payoff from the derivatives. This section will feature the valuation of the two interest rate derivatives that we used in the simulation of the debt portfolios, the plain vanilla swap and the cap.

10 For more information read Pliska [16] and Rebonato [17].

57

17.1 Valuation of the Plain Vanilla Swap A plain vanilla interest-rate swap is an agreement whereby two parties undertake to exchange, at known dates in the future, a fixed for a floating set of payments.

Interest rate

Time Start of Swap End of Swap

Figure 17.1 Borrower’s effective interest rate with a floating-rate loan and an interest rate swap.

The payments are often referred to as the fixed and floating rate of a swap. The fixed leg is made up by payments U t

τLXU t =

where is the notional principal of the swap, L τ is the fraction of the year between the :th and the ( ):th reset (in our case equal to 0.25 for a quarterly swap) and t 1+t X is the

fixed rate contracted at the outset to be paid by the fixed-rate payer at each payment time. For a plain vanilla swap each fixed rate payment U occurs at the end of the accrual period . If we denote the price of discount bond maturing at time

t

1+t T by , the present value of each fixed payment U is given by:

TtZ

t

1)( += t

tt ZLXUPV τ .

As for the floating leg, each payment , also occurring attH 1+t , is given by

τtt LRH = ,

where is the tR τ -period coupon-bearing rate. Similar as for the fixed leg, the present value at time is t

1)( += tttt ZLRHPV τ . (17.1)

58

The magnitudes of the fixed-leg payments are known at time zero and the present value can therefore be associated to them, while for the floating-leg payments, is yet unknown and we do not know what present value to associate them to.

tR

Let us consider the following strategy: suppose we purchase at time 0=t , a discount bond maturing at timeT , and sell a discount bond maturing at timeTZ 0 1+T 1

0+TZ . The

notional principle is set to one. At time T the resulting portfolio will have a value V as 1T

111 1 ++ −=−= T

TTT

TTT ZZZV ,

which, assuming simple compounding over the period T to 1+T , is equal to

ττ

τ T

T

TT R

RR

V+

=+

−=11

111 .

From equation 17.1, the payer of the floating leg will have to make a payment at time

1+T of present value V at time 2T T equal to

ττ

τT

TTTTT R

RZRV+

== +

112 .

This implies that the payoff from the floating leg is equivalent to the long/short strategy from above. So, at time zero, the commitment to pay in the floating leg and the strategy of holding a bond Z and shorting a bond must therefore have the same value:

TR1

0+TT

0 Z

1

01

00++ =− T

TTT ZRZZ τ .

To avoid arbitrage, one can value the floating leg of a swap by setting the unknown coupons equal to the value TR

τ1/ 1

00 −=

+TT

TZZ

R ,

but this equation is simply the well-known definition of a simple compounded forward rate . Therefore, in order to avoid arbitrage, the a priori unknown cash flows in the floating leg must be set equal to the projected forward rates. Notice that the quantities

in equation 17.1 are stochastic variables and possess certain variance. It is only the present value of each floating reset and the strategy of long/short bonds that has no variance, where a purely deterministic approach is required. The equilibrium swap rate is then defined as the fixed rate

1+ttf

)( tHPV

X , such that today’s present values of the fixed and floating legs are the same:

59

∑ ∑ ∑∑= = =

+++

=

===T

t

T

t

T

t

tttt

tT

tt ZLfHPVZLXGPV

0 0 0

10

110

0)()( ττ ,

where T is the length of the swap. The equilibrium swap rate will therefore be

∑

∑

=

+

=

++

= T

t

t

T

t

ttt

ZL

ZLfX

0

10

0

10

1

τ

τ. (17.2)

In other words, one can see this as a weighted average of the projected forward rates. This can be seen more clearly by setting

∑=

+

+

= T

t

t

t

t

ZL

ZLw

0

10

10

τ

τ.

Equation 6.2 can then be rewritten as

∑=

+=T

t

ttt fwX

0

1 .

By the way the equilibrium rate has been obtained it follows that entering a swap today has zero cost for both parties i.e. they have undertaken to exchange legs of identical value. After an equilibrium swap has been entered, the swap itself will in general no longer have zero value, since interest rates fluctuate and are not likely to follow its implied forward curve. This means that, in general, either party will lose or gain when entering an equilibrium swap. For the payer of the fixed rate the present value of the swap at time will be given by i

∑ ∑= =

+++ +−=i

t

i

t

tt

tt

tti ZLfZLXPV

0 0

111 ττ

where are now the forward rates calculated from the discount curve at time . To get a better insight into the equilibrium rate, the numerator in equation 17.2 can be expanded:

1+ttf i

( )∑∑ ∑=

++

= =

+++ −=

−

=T

t

tttT

t

T

t

t

t

ttt ZZLZ

ZZ

LZLf0

100

10

0 0

10

0

10

1

1

τττ .

60

Notice that the frequency of the plain vanilla swap does not affect the value of the floating leg. The equilibrium swap rate is therefore given by the ratio of two portfolios of discount bonds with constant principals 1=L :

∑=

+

+−= T

t

t

t

ZL

ZX

0

10

101

τ

When dealing with plain vanilla swaps, pricing can be completely reduced to a suitable manipulation of pure discount bonds, or zero coupon bonds. The present value of a bond, paying coupons n X at regular intervals every τ years (quarter in our case) until final maturity at time T is given by

∑=

+=T

t

TtT ZZXPV

000 100τ

What has been referred to before as the equilibrium rate is actually called the par coupon when using bond conventions, which can be defined as that particular coupon that prices the bond today exactly at par. In the model in part two, credit considerations are taken into account. This is simply done by adding a small rate u to k X in the following manner:

kT

t

t

t

uZL

ZX +

−=

∑=

+

+

0

10

101

τ.

The size of u depends on which state of rating the firm is currently in. This means that the lower the firm is rated , the higher penaltyu .

k kk k

17.2 The Valuation of Caps A cap is a collection of caplets. A caplet, in turn, is a contract that pays at time 1+T the difference between the specified spot rate at timeT , and a strike priceTR K , multiplied by the year fraction τ if this difference is positive, and zero otherwise. Thus

[ ]τ0,KRMaxC TT −=

61

where C is the return from the derivative. Figure 6.2 shows a series of caplets and how they ensure a payer of the floating leg not to have interest payments higher than a certain amount.

T

Capped Interest rate

Figure 17.2 Borrower’s effective interest rate with a floating-rate loan and an interest rate cap.

Interest rate

Time

The no-arbitrage arguments presented in this paper show that, for valuation purposes, the unknown future value of the rate must be set equal to today’s implied forward. If, in addition, one accepts the lognormal assumption for the forward rates, one arrives at the Black model. This is indeed the case for the model in part two, since with probability one at all times and typical lognormal distribution, see figure 17.3.

0>jif

fX(x)

x 0

Figure 17.3 The density function of the lognormal distribution. The lognormal distribution is not defined for x≤ 0

The Black formula refers to the valuation formula for a call option where the underlying variable is assumed to be lognormal with zero drift. The same terminology will therefore apply irrespective of whether the variable is forward rate or a forward bond price. The absence of arbitrage constitutes to a drift less forward rate. In the Black model, a caplet can be seen as a call option on a forward rate:

62

[ ] 121

1 )()( ++ −= Tt

TTt ZhKNhNfC ,

where denotes the cumulative distribution and )( ihN

tT

tTK

f

h

TT

−

−±

=

+

σ

σ )(21ln 2

1

2,1 .

The volatility σ is the percentage volatility of the forward rate.

Let us now consider a call expiring at time T and struck at τX+1

1 on a discount bond

maturing at time τ+T . It means that the cap will be executed if XR > where R , is the rate to cap and X the cap rate. At option expiry its payoff will be given by

τττ

ττ

τ

+−

+=

+−= + 0,

11

11max0,

11max

XRXZPayoff T

T .

But since τX+1

1 is known at the outset, the payoff at time T of the call on the discount

bond can be written as

[ ]τττ

ττ

τττ

RRX

XXRRXPayoff

+−

+=

++

−=

10,)(max

110,

)1)(1()(max .

Notice that a put on the same discount bond, expiring and paying at timeT , is equivalent to a T -expiry caplet on the τ -period rate. Given the market practice of pricing caps using the Black formula, there is a one-to-one correspondence between the prices and the volatilities. One can therefore say that the market expresses its views about the volatility of the underlying forward rates via a complete set of cap prices.

63

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2. Barclay, M.J. and Smith, C.V. Jr. “The Maturity Structure of Corporate Debt.” Journal of Finance 50 (June 1995), p. 609-631.

3. Baumol, W.J. “An Expected Gain-Confidence Limit Criterion for Portfolio Selection.” Management Science (October 1963)

4. Bergström, P. and Holmlund, A. ”A Simulation Model Framework for Government Debt Analysis” Technical Report, Riskgäldskontoret” (2000)

5. Blanchard, O. “Macroeconomics” Second edition, Prentice-Hall (2000) 6. Brick I.E. and Ravid S.A. “Interest Rate Uncertainly and the Optimal Debt

Maturity.” Journal of Financial and Quantitative Analysis 26 (March 1991), p. 63-81

7. Diamond, D.W. “Debt Maturity Structure and Liquidity Risk.” Quarterly Journal of Economics 106 (August 1991a), p. 709-737.

8. Guedes, J. and Opler, T. “The Determinants of the Maturity of New Corporate Debt Issues.” Working Paper. Columbus: Ohio State University (1994).

9. Hamilton, J. “A New Approach to the Economic Analysis of nonstationary Time Series and the Business Cycle.” Econometrica (1989)

10. Harrison, J.M. and Kreps, D.M. “Martingales and Arbitrage in Miltiperiod Securities Markets.” Journal of Economic Theory, 20 (1979) p.381-408.

11. Hoven Stohs, M. and Mauer, D.C. “The Determinants of Corporate Debt Maturity Structure.” The Journal of Business 69 (July 1996), p. 279-312.

12. Levy, H. “Principles of Corporate Finance.” South-West Collage Publishing (1998).

13. Lyuu, Y. “Financial Engineering and Computation.” Cambridge University Press (2002).

14. Miller, M.H. and Modigliani, F.F. “The Cost of Capital Corporation Finance, and the Theory of Investment.” American Economic Review (June 1958).

15. Mitchell, K. “The Call, Sinking Fund and Term-to-Maturity Features of Corporate Bond: an Empirical Investigation.” Journal of Financial Quantitative Analysis 26 (June 1991), p.201-222.

16. Pliska, S.R. “Introduction to Mathematical Finance, Discrete Time Models” Blackwell Publishers (1997)

17. Rebonato, R. “Interest-Rate Option Models.” John Wiley and Sons Ltd. June (1997).

18. Roy, A.D. “Safety First and the Holdage of Assets.” Econometrica 20 (1952) 19. Taylor, J. “Inflation, Unemployment, and Monetary Policy.” MIT Press (1998) 20. Telser, L. “Safety First and Hedging.” Review of Economic Studies 23 (1955-

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