Modeling and Simulation of Debt Portfoli
os
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
9.5 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
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.
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].
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.
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
(
)
(
)
(
( )
2(
(
)
)
2)
s s s s E PR E PR E PR LPR E = − − .Here is a debt portfolio index. Moreover, ifs E
(
LPR1)
≤E(
LPR2)
, 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.2
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 C PR B P( ) = # η,t − η,t <0 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 ) (B P
( )
B 1 P( )
B 2 P >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
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
(
LPR)
P E(
LPR)
QE ≤ ∧ P
( )
B P ≥P( )
B Q.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.
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.
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
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
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
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
0 5 10 15 20 25 30 35 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
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
0 2 4 6 8 10 12 14 16 18 20 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
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
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 (Bankruptcy)‰ Prob
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 Portfolio Refinancing Strategy E(Least-Profit)
108 SEK E(Profit) 108 SEK Std(Profit) 107 SEK E(Interest Cost) 108 SEK length) quarter E (Fixed Rate (Bankruptcy)‰ Prob
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
Table 4.11 Portfolio
Refinancing Strategy
E(Least-Profit)
108 SEK E(Profit) 108 SEK Std(Profit) 107 SEK E(Interest Cost) 108 SEK length) quarter E (Fixed Rate (Bankruptcy)‰ Prob
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.
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)
Maturity X-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.
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.
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.
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 1 2 4 8 16 Cap Length
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.
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 * * *
4.4.2 Results Combined Strategy Simulation Table 4.13
Portfolio Refinancing Strategy E(Least-Profit) 108 SEK E(Profit) 108 SEK Std(Profit) 107 SEK E(Interest Cost) 108 SEK length) quarter E (Fixed Rate (Bankruptcy)‰ Prob
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 49 Medium 2.3036 3.0837 7.8013 2.9995 6.3797 2.075 50 Medium 2.2871 3.0691 7.8198 2.9979 6.3797 1.975 51 Dynamic 1 2.3115 3.0855 7.7399 2.9984 6.5389 1.925 52 Dynamic 1 2.2932 3.0700 7.7677 2.9971 6.5389 1.863 53 Dynamic 3 2.3162 3.0924 7.7615 2.9914 6.5146 1.913 54 Dynamic 3 2.2989 3.0776 7.7872 2.9893 6.5146 1.838 55 Medium 2.3171 3.1287 8.1162 2.9558 5.1605 2.375 56 Medium 2.3026 3.1153 8.1267 2.9541 5.1605 2.288 57 Dynamic 1 2.3258 3.1316 8.0573 2.9537 5.3175 2.200 58 Dynamic 1 2.3100 3.1177 8.0770 2.9520 5.3175 2.125 59 Dynamic 3 2.3287 3.1359 8.0720 2.9492 5.2931 2.200 60 Dynamic 3 2.3136 3.1225 8.0888 2.9469 5.2931 2.150 61 Medium 2.2817 3.0735 7.9179 3.0101 6.6385 1.875 62 Medium 2.2596 3.0564 7.9683 3.0107 6.6385 1.875 63 Dynamic 1 2.2913 3.0789 7.8756 3.0055 6.7511 1.788 64 Dynamic 1 2.2675 3.0610 7.9354 3.0064 6.7511 1.763 65 Dynamic 3 2.2955 3.0839 7.8845 3.0003 6.7429 1.775 66 Dynamic 3 2.2724 3.0666 7.9420 3.0006 6.7429 1.775 67 Medium 2.2885 3.1023 8.1381 2.9664 5.1605 2.375 68 Medium 2.2314 3.0687 8.3733 2.9612 5.1605 2.400 69 Dynamic 1 2.2918 3.1005 8.0866 2.9680 5.3175 2.263 70 Dynamic 1 2.2360 3.0691 8.3307 2.9609 5.3175 2.300 71 Dynamic 3 2.2968 3.1069 8.1008 2.9618 5.2931 2.250 72 Dynamic 3 2.2411 3.0750 8.3389 2.9549 5.2931 2.225 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 79 Medium 2.3208 3.1398 8.1895 2.9552 5.1605 2.475 80 Medium 2.2918 3.0899 7.9813 3.0050 6.6385 1.938 81 Dynamic 1 2.3320 3.1446 8.1259 2.9504 5.3175 2.300 82 Dynamic 1 2.3024 3.0964 7.9398 2.9985 6.7511 1.788 83 Dynamic 3 2.3350 3.1492 8.1416 2.9458 5.2931 2.288 84 Dynamic 3 2.3070 3.1019 7.9493 2.9930 6.7429 1.800 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
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 B 68 2.22 78 76 2.2 74 A 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
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
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 almost2 5of 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.
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.
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.
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
GDP Inflation
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
t t
t y
y =α +β −1 +ε ε N(0,σ2),
t ∈
where α is a constant, β the autoregressive parameter which describes the effect of a unit change in yt−1 on and yt ε the random shock which is assumed to be an IID (Independent Identically Distributed) normal variable. Since β is positive and less than
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 i t s i s t y y E α β β .As s tends to infinity we will get the long-term equilibrium. For geometric series with
1
β and s→∞the formula converges to:
β α − = + 1 ] [yt s E .
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:
t t s
t y
y =α +β −1+ε εt ~ N
(
0,σe)
s∈B,R.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:
= RR RB BR BB p p p p P ,
where is the probability that a boom state will follow a boom state and and are the probabilities of shifts in states. Here
BB p pBR pRB = + BR BB p p pRR + pRB =1.
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.
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:
BB B p D − = 1 1 and RR R p D − = 1 1 .
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 − − − − + − − − − = R RR BB BB B RR BB RR t p p p p p p y β α β α 1 2 1 1 2 1 .
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:
t t
t δ φπ ε
π = + −1+ εt ~ N
(
0,σπ)
.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 δ =0.00025, φ =0.95, and σπ =0.00062. 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.
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.:
t t s t g g = ρ t +µ −1 +ε εt ~N
(
0,σe)
s=B, R. 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 constantρs,
B
ρ when the economy is in a boom state and ρRfor recession. Parameter values in the equation isρB =0.00045,ρR =−0.00028, µ =0.95 and σε =0.00094. 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 ) ( * * * t t e t e t T t r Y Y i = +π +θ π −π +λ −
where iTis the Taylor interest rate,
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: e t π 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].
