Classifications

• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR COUNTING
  • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
  • G06Q40/00—Finance; Insurance; Tax strategies; Processing of corporate or income taxes
  • G06Q40/04—Trading; Exchange, e.g. stocks, commodities, derivatives or currency exchange

Definitions

• This invention relates to a method and a data processing system for calculating the type, the number, and the volume of financial instruments for funding a loan with equivalent proceeds to a debtor, the loan being designed to be at least partially refinanced during the remaining term to maturity of the loan.
• the remaining term to maturity of the loan is also determined at the beginning of each period such that the debtor's payments on the loan during the entire term to maturity of the loan are within a band defined by a set of maximum and minimum limits which may be determined for each period, and such that the remaining term of the loan is within a band defined by a maximum limit and a minimum limit.
• a special financial instrument is determined which is designed to ensure that given maximum limits for payments on the loan and term to maturity are observed.
• the results of the method according to the invention may be used by a lender, e.g. a financing institution such as a mortgage credit institution, to ensure that such a loan is funded such that both interest rate risk and imbalances in the payment flows are prevented or minimized.
• a lender e.g. a financing institution such as a mortgage credit institution
• the lender may thus create a hedge between the lending and the funding.
• a characteristic feature of traditional loans with adjustable interest rates was a match between the term to maturity of the last maturing funding instrument and the period of time between interest rate adjustments, viz. 5 years. If this precondition is abolished, the way is paved for an, in principle, far wider range of opportunities as to funding and interest rate adjustment.
• the short-term interest rate is systematically minimum than the long-term interest rate, it will be possible to reduce the long-term borrowing costs for the borrowers. Furthermore, the borrowing costs may, as mentioned above, be reduced relative to callable bonds due to the absence of a call right and via increased liquidity and internationalization of sales.
• Whether it is possible to counter an interest rate adjustment by adjusting the remaining term to maturity of the loan depends on the determined maximum and minimum limits for payments on the loan and term to maturity, as well as on the extent to which the remaining debt of the loan is adjusted to the market rate at the time of the adjustment of the interest rate.
• the traditional loans with adjustable interest rates were characterized by the remaining debt of the loan being 100 per cent adjusted to the market rate every fifth year. Partly by allowing other frequencies with which the interest rate adjustment is performed, and partly by allowing only a partial adjustment of the interest rate of the remaining debt of the loan, larger changes in the interest rate on the loan than in the original structure will be compatible with the maximum and minimum limits for the payments on the loan. Therefore, it should be possible to combine a partial adjustment of the interest rate of the remaining debt of the loan, as well as other interest rate adjustment frequencies, with an adjustable term to maturity.
• the volume of the individual volumes of each of the financial instruments on the creditor side of the loan must be determined such that the market price of the financial instruments equals the volume of the loan on the debtor side.
• the debtor's interest rate on the loan must be determined such that the interest rate on the loan is based on the yield to maturity of the funding portfolio, said yield to maturity being given by the interest rate at which the present value of a future payment flow for funding instruments equals the remaining debt on the debtor's loan.
• This funding principle is, however, not compatible with the desire for issuing a range of e.g. 10 non-callable bullet bonds with terms to maturity of 1 to 10 years, and at the same time keeping the duration of the interest rate adjust - ment period at e.g. 1-2 years.
• the funding principle may e.g. be used in mark-to market pricing of loans and claims otherwise not traded.
• the principle it will be possible to determine a portfolio of traded financial instruments with an equivalent payment flow on the basis of which the loan or the claim may be priced in accordance with observed market prices.
• the funding principle may be applied to risk management of loans and claims, the principle being applicable to the determination of a hedge consisting of a port- folio of financial instruments, as well as to the pricing of such hedge.
• the trend has been towards a higher degree of attention being paid to financial risks, including the possibility of hedging these risks, so it is within this area in particular that the international interest in the funding principle is expected.
• Danish patent applications nos. 233/97, 308/97 and 770/97 concern this further development and relate to a method by which not only the above parameters may be determined, but by which requirements may also be laid down with respect to maximum (or minimum) payments on the loan for the debtor in one or more periods during the term to maturity of the loan, the term to maturity of the loan optionally being calculated as adjusted to these requirements.
• the method according to these patent applications it will be possible by the method according to these patent applications to lay down requirements with respect to the maximum (or minimum) term to maturity of the loan, and then calculate an adjusted payment on the loan.
• a first example is “step up” bonds.
• “Step up” bonds are long- term bonds for which the coupon rate changes periodically according to a predetermined pattern. Typically, the pattern is based on the structure of the forward rates. If the forward rate structure is rising, the coupon rate will typically also rise over time.
• the adjustment to the structure of the forward rate means that, in principle, "step up" bonds will carry the short-term interest rate initially. In periods with a rising yield curve, the debtor may thus gain an interest rate advantage comparable to the interest rate advantage of a LAIR.
• the changes in the coupon rate are combined with a call right.
• the debtor thus gets the possibility of prepaying the remaining debt at par in connection with the change in the coupon rate .
• the loan is thereby in the nature of a loan with a short-term interest rate combined with an option on the future interest rate, and the comparison to a LAIR combined with an option on the interest rate adjustments springs to mind.
• Step up bonds for funding mortgage loans is, as far as it is known, limited.
• “Step up” -bonds have mainly been used in the high-risk bond market, where the lower coupon is initially to secure the debtor's financial survival in the short run .
• Adjustable rate mortgages are loans in which the interest rate is pegged to an interest rate index optionally added an interest differential as a reflection of a credit risk or the like.
• the interest rate index may be e.g. a "treasury" -based index with a term to maturity of 1/2 year, 1 year, or 5 years.
• the interest rate on the loan is adjusted at fixed intervals typically of the same length as the interest rate index.
• the loan has characteristics in common with a LAIR I.
• a variant of the "adjustable rate mortgages" has as a facility a band in the interest rate index.
• the interest rate is bound upwards by a "cap”, whereas a "floor” sets a minimum limit for the interest rate.
• the interest rate on the loan will thus float within a band during the entire term to maturity of the loan.
• the present invention permits an appropriate and realistically practicable computerized calculation of the above parameters which are calculated in accordance with the above patent applications, as well as further calculation of pay- ments from (or to) a "payment guarantee instrument" of the above type.
• a "payment guarantee instrument” of the above type.
• the payment guarantee instrument is considered particularly convenient when apart from granting payments to the debtor in situations in which agreed maximum limits for payments on the loan and term to maturity are exceeded, it also receives payments from the debtor in situations in which payments on the loan and term to maturity would otherwise have fallen below their minimum limits. Therefore, this type of payment guarantee instrument is in particular the basis of the following explanation of the method according to the invention, even if it is understood that a payment guarantee instrument not designed to receive payments from the debtor could also be included and treated in the same way by the method according to the invention.
• the invention relates to a method for determining, by means of a first computer system, the type, the number, and the volume of financial instruments for funding a loan, determining the term to maturity and payment profile of the loan, and further determining the payments on a payment guarantee instrument designed to ensure that the payments on the loan and the term to maturity of the loan do not exceed predetermined limits, and from which instrument payments are made to the debtor in situations in which the maximum limits for payments on the loan and term to maturity would otherwise have been exceeded, the loan being designed to be at least partially refinanced during the remaining term to maturity of the loan,
• a third set of data specifying a desired/intended refinancing profile, such as one or more point (s) in time at which refinancing is to take place, and the amount of the remaining debt to be refinanced at said point (s) in time, and/or said third set of data specifying a desired/intended funding profile, such as a desired/intended number of financial instruments applied for the funding together with their type and volumes,
• one or more recalculations being made if necessary, including if necessary, selection of a new number of the financial instruments stored under (e) , storing, in a memory or a storage medium of the computer system, after each recalculation the recalculated profile of the interest rate on the loan, - the recalculated term to maturity profile, the recalculated payment profile, the recalculated remaining debt profile, and the selected financial instruments with their calculated volumes, until all the conditions stated under (b) and (d) have been fulfilled, and the payments on the payment guarantee instrument optionally being calculated in accordance with (gl) , and the recalculated payments being stored in a memory or a storage medium of the computer system after each recalculation,
• LAIR III Lians with Adjustable Interest Rates III
• the data may be output to a display or a printer.
• the memories applied may be ⁇ ⁇ to KJ H H
• Step (f) may also be carried out at an arbitrary stage in the sequence unless it is chosen, as is often preferred, to have the computer calculate a first guess at a profile of the interest rate on the loan, and either a first term to maturity profile or a first payment profile, in which case step (f ) will definitely follow step (e) .
• Another example of data being either input or guessed/calculated is the desired/intended payment or the desired/intended term to maturity under (b) (iv) ; if no initial value thereof has been input/stored, the computer system is conveniently designed to "guess" or calculate a value according to an established rule, e.g. as an average of the values stored under (b) (i) and (b) (ii) .
• the period mentioned under (b) (i) is preferably a refinancing period, which will therefore normally be a default in the computer system but, in principle, this period may be any period desired by the debtor, said period normally being input together with the mentioned limits.
• the requirement with respect to the maximum permissible difference in balance is linked to a period which, depending on the legislation or the practice which is to form the basis in connection with the calculations, may be a calendar year, a year not following the calendar year but comprising the time of a payment to the creditor, or another period either comprising or not comprising the time of a payment to the creditor. In Denmark a strict balance requirement must be fulfilled per calendar year.
• the require- ment with respect to maximum permissible difference in balance is, according to the current Danish rules of mortgage loans, given by a strict balance, i.e. no appreciable difference in balance occurs or, to put it differently, the difference is practically zero.
• the method according to the invention may also be used where a certain difference in balance is tolerated or perhaps even desired, this tolerance or this positive difference in balance then being stored as part of the data set in (d) .
• both the requirement with respect to the difference in proceeds, the requirement with respect to the difference in interest rates as well as the requirement with respect to the difference in balance may be specified in different ways.
• Data may e.g. be input, specifying a direct maximum permissible difference in balance between, on the one hand, the sum of the market price ⁇ LO t NJ H H
• the calculation method according to the present invention is also applicable in situations in which the input data specifies that more than one debtor payment on the loan will be made within one creditor payment period.
• the input data specify that full refinancing of the remaining debt is performed at the end of a predetermined period which is shorter than the term to maturity of the loan, and in a second important instance, the input data specify that refinancing of the remaining debt is performed with a fixed annual fraction.
• the method according to the invention may be applied for determining the number and the volume of the financial instruments, the term to maturity and the payment profile in the situation in which the loan is to be calculated for the first time, i.e. in the first funding situation, as well as in the situation in which a refinancing is to be calculated.
• the expression funding thus covers both “new funding” and "refinancing” .
• information concerning the type, the number, and the volume of the financial instruments which have not yet matured at the time of refinancing is included in the calculations in the refinancing situation. This information is often stored in the computer system from the previous calculation, but inputting this information is evidently within the scope of the invention.
• parameters under (a) - (f) are parameters which are related to the funding situation in question, so that for the case in which a refinancing is calculated, they are naturally related to the remaining debt of the loan as the volume of the loan and to the remaining term to maturity of the loan as the term to maturity of the loan.
• the result of the method according to the invention as defined above is usually at least one set of data which may be applied in the next funding situation, whether this situation is the first funding period of the loan, or a later refinancing situation.
• term to maturity profile is related to a term to maturity being calculated by the method according to the invention, as mentioned above, usually for each funding or refinancing period.
• expression term to maturity profile refers to the series of terms to maturity which being assigned to the refinancing period at each calculation in connection with a refinancing.
• profile of the interest rate on the loan is similarly related to a calculation of the interest rate on the loan being performed by the method according to the invention, usually for each funding or refinancing period.
• expression profile of the interest rate on the loan refers to the series of interest rates on the loan being assigned to funding periods at each calculation in connection with the refinancing.
• the calculations are performed with financial instruments which are not directly interest-bearing, first, a calculation is conveniently performed of the expected payment flows such that a calculation of an internal interest rate may be performed, causing the payment flow or flows or the likely payment flow or flows to be expressed in parameters corre- sponding to the above-mentioned parameters for interest- bearing claims, primarily a yield to maturity.
• a calculation is conveniently performed of the expected payment flows such that a calculation of an internal interest rate may be performed, causing the payment flow or flows or the likely payment flow or flows to be expressed in parameters corre- sponding to the above-mentioned parameters for interest- bearing claims, primarily a yield to maturity.
• the data being stored as characteristics of the instruments in section (a) above may be data directly defining the financial instruments in question, and the computer system may be adapted to perform a conversion into parameters characterizing an interest-bearing claim according to predetermined principles.
• the procedure is similar as the same payment flows may be expressed by corresponding interest-bearing instruments, the characteristics of which may then be stored as stated in section (e) , or the computer system may preferably be adapted to perform a conversion into parameters characterizing an interest-bearing claim according to predetermined principles. It will be understood that in each individual ) ⁇ NJ NJ H ⁇ >
• repayment profile In accordance with common practice, the expressions "repayment profile”, “remaining debt profile” and “payment profile” specify the development over time in repayments, remaining debt and payments on the loan, respectively.
• the repayment profile may follow the annuity loan principle as well as the serial loan principle. In addition, any arbitrary placing in time of the repayments is naturally pos- sible.
• the repayment profile may be determined either on the basis of the interest rate on the loan applying at the time in question, or on the basis of the original interest rate on the loan, or on the basis of an arbitrarily determined interest rate.
• ncing profile and "funding profile” respectively specify the type, the number, and the volume of the financial instruments applied for the funding.
• the expression may be used about the desired or intended funding profile which is input and stored under (c) , and which might not be fulfilled, as well as about the accurate funding profile which is the result of the calculations following application of the method.
• ncing profile specifies at which points in time and with which amounts the loan is to be refinanced.
• the desired/intended refinancing profile stored as a second set of data under (c) above may be rewritten as a funding profile, viz. as a number of financial instruments with their type and volumes.
• An indication of a desired annual interest rate adjustment percentage of 100 may e.g. be rewritten into the loan being O )
• an immediate result of the calculations may indicate that the date of maturity does not coincide with the date of maturity of the last maturing financial instrument considered. It is natur- ally possible to apply such a result but in a preferred embodiment, the date of maturity of the loan is corrected such that it corresponds to the date of maturity of the last maturing financial instrument.
• the correction comprises determining whether the term to maturity is to be round up to a creditor payment date (a date of maturity of a financial instrument) or be round down to the preceding creditor payment date (a date of maturity of a financial instrument one period earlier) .
• the adjustment of the date of maturity may preferably be performed as follows:
• the set of data under (c) specifies that calculations are to be performed for the case in which full refinancing of the remaining debt is to be performed periodically with a predetermined period which is shorter than the term to maturity of the loan, and the remaining term to maturity of the loan is shorter than the period which according to (c) elapses between two successive interest rate adjustments, and the remaining term to maturity does not correspond to the maturity of the last maturing financial instrument selected under (h) , but it is desired that the loan matures at the same time as the maturity of the last maturing financial instrument selected under (h) , the term to maturity may conveniently be determined by the method according to the invention as
• the calculation of the payments on the payment guarantee instrument is conveniently performed on the basis of an interest rate on the loan which is recalculated such that the limits for payments on the loan as well as term to maturity are observed, and wherein either resulting differences in the payments on the debtor side and the payments on the financial instruments or resulting differences in the market price of sold financial instruments and the funding demands correspond to the payments on the payment guarantee instrument.
• the funding demand is defined at the disbursement of the loan as the volume of the loan and at the adjustment of the interest rate of the loan as the amount at which the requirement with respect to maximum permissible difference in balance is fulfilled in the year immediately preceding.
• the payments on the payment guarantee instrument correspond to the differences in the market price of sold financial instruments and the funding demand resulting from the recalculation, the volume of the financial instru- ments being determined such that the requirement with respect to maximum permissible difference in balance is fulfilled.
• this embodiment of the method extends to a series of recalculations in the outer loop, each of these recalculations normally occasioning a series of LO ⁇ NJ NJ ⁇ > H
• the finan- cial instruments applied for the refinancing may e.g. be calculated in the inner model in the same way as the financial instruments applied for the initial funding, in other words, it would be possible to perform a new calculation according to the method of the volume of financial instru- ments for funding a new loan, the volume of the new loan corresponding to the amount to be refinanced.
• the inner loop it may be specified in the input data corresponding to the refinancing profile that a partial refinancing of the remaining debt is to be performed.
• a solution may be found to the volume of the financial instruments constituting the volume, if it has been input e.g. that refinancing is to be performed periodically with a predetermined period which is shorter than the term to maturity of the loan.
• a solution may also be calculated if it is specified that periodic refinancing of a fraction of the remaining debt of the loan is to be performed, the denominator of the fraction corresponding to the whole number of years of the financial instrument having the longest term to maturity when the loan was obtained.
• the selected period may be e.g. 1 year, but other periods such as 2, 4, 5, 6 or 10 years may be selected.
• periods corresponding to a whole number of months e.g. 2, 3, 4 and 6 months may be selected.
• the refinancing in the inner loop may further comprise a funding by use of additional funding for the financial instruments or funding volumes remaining at the time of the refinancing.
• additional funding and new refinancing instruments are also designated as the addition to the volume of the financial instruments.
• the calculation method according to the present invention will also provide a solution to the volumes of the additions to the financial instruments applied for the refinancing.
• data comprising possible new refinancing instruments within the range of selected financial instruments must be input.
• the proceeds criterion may e.g. be given as a requirement with respect to the difference between, on the one hand, a funding demand given by the balance requirement and, on the other hand, the sum of the market price of the addition to the financial instruments.
• the issue of new financial instruments, as well as additional issue of financial instruments already applied may be made in connection with a refinancing.
• it will also be possible to repurchase the financial instruments already applied but this involves a number of inconveniences, inter alia, an additional depreciation risk on the part of the borrower and problems pertain- ing to the mortgages, for which reason repurchase is not effected in practice. Therefore, according to a preferred embodiment of the method, the volume of the additions to the financial instruments " will be calculated in consideration of the volumes of the previously applied financial instruments remaining at the time of refinancing.
• the payment guarantee instru- ment has a price or a value of zero. This may be achieved by the desired/intended term to maturity of the loan being input under (b) (iii) and/or the limits for the payments on the loan and/or the limits for the term to maturity are established such that the present value of the payments on the payment guarantee instrument is zero.
• the calculation of the present value of the payments on the payment guarantee instrument may conveniently be performed by use of a stochastic yield curve model.
• the stochastic yield curve model is preferably calibrated to a yield curve which is determined at the time of calculation.
• the stochastic yield curve model is conveniently formulated in discrete time and implemented in a yield curve lattice, appropriately in e.g. a trinomial lattice according to Hull & White (references to Hull & White in the present text com- prise: "On derivatives. A compilation of articles by John LO to t H ⁇ >
• the recalculations in this part of Type F comprise one or more interest rate iterations, each interest rate iteration comprising calculating and storing, in a memory or a storage medium of the computer, data specifying a new interest rate on the loan which is preferably based on the previous interest rate on the loan and the calculated interest rate adjustment, calculating and storing, in a memory or a storage medium of the computer, data specifying a new payment profile and remaining debt profile for the debtor, said payment profile and remaining debt profile being calculated in consideration of the new interest rate on the loan, the volume, the term to maturity, and the repayment profile of the loan as input under (a) , and the refinancing profile and/or the funding profile input under (b) , and calculating and storing, in a memory or a storage medium of the computer system, data specifying a new set of volumes of the financial instruments applied for the funding.
• the interest rate iteration is preferably performed applying a numerical optimization algorithm or by "grid search” .
• Examples of numerical optimization algorithms are a Gauss- Newton algorithm, a Gauss algorithm, a Newton-Ramphson algorithm, a quadratic hill climbing algorithm, a quasi- ⁇ NJ NJ H H
• either the first element of the indicator function has the value zero, or the sum of the elements in the indicator function is strictly less than 2, in each of which cases only one coefficient is calculated in the polynomial function such that the resulting range of adjustments to the volumes of the financial instruments fulfil the requirement with respect to maximum difference in proceeds; the resulting adjustment of the interest rate will be determined by a residual calcula- tion in accordance with the requirement with respect to maximum permissible difference in balance.
• the above-mentioned analytical method for determining the function coefficients in the polynomial function is a method which is easy to calculate and hence time-saving.
• the function coefficients may also be calculated by iteration as discussed in the now immediately subsequent sections.
• Type P the recalculations of all or some of the data mentioned in (g) and (h) , and/or one or more function coefficients for the function representing the shifted level remaining debt profile, and/or the interest rate on the loan in the inner loop are performed by iteration carried out by applying numerical optimization algorithms or by grid search.
• the calculation of the new financial instruments may preferably be based on the difference in balance for the periods in which the corresponding, previously found financial instruments do not fulfil the requirement with respect to maximum permissible difference in balance .
• the range of financial instruments determined under (e) is selected among a number of available financial instruments. It will be understood that this number of instruments may, if desired, be input to a data base in the computer system or be available via a network, and that the determination may, if desired, be performed automatically or semi-automatically by means of the computer system according to predetermined criteria or functions.
• the invention also relates to a data processing system, such as a computer system for determining the type, the number and the volume of financial instruments for funding a loan, determining the term to maturity and payment profile of the loan, and further determining the payments on a payment guarantee instrument designed to ensure that the payments on the loan and the term to maturity of the loan do not exceed predetermined limits, and from which instrument payments are made to the debtor in situations in which the maximum limits for payments on the loan and term to maturity would otherwise have been exceeded, the loan being designed to be at least partially refinanced during the remaining term to maturity of the loan, requirements having been laid down stipulating that the term to maturity of the loan is not longer than a predetermined maximum limit nor less than a predetermined minimum limit, - debtor's payments on the loan are within predetermined limits,
• said data processing system comprising
• (b) means, typically input means and a memory or a stor- age medium, for inputting and storing a second set of data specifying
• (c) means, typically input means and a memory or a storage medium, for inputting and storing a third set of data specifying a desired/intended refinancing profile, such as one or more poin (s) in time at which refinancing is to take place, and specifying the amount of the remaining debt to be refinanced at said point (s) in time, and/or said third set of data specifying a desired/intended funding profile, such as a desired/intended number of financial instruments applied for the funding together with their type and volumes,
• (d) means, typically input means and a memory or a storage medium, for inputting and storing a fourth set of data comprising a maximum permissible difference in balance within a predetermined period, a maximum permissible difference in proceeds and, optionally, a maximum permissible difference in interest rates equivalent to the difference between the interest rate on the loan and the yield to maturity of the financial instruments applied for the funding and, optionally, the payment guarantee instrument,
• (e) means, typically input means and a memory or a storage medium, for determining and storing a fifth set of data specifying a selected number of financial instruments with inherent characteristics such as the type, the price/market price, and the date of the price/market price
• (f) means, typically input means and a memory or a storage medium, for determining and storing a sixth set of data representing a first profile of the interest rate on the loan and either a first term to maturity profile or a first pay- ment profile of the loan
• (g) means, typically calculating means and a memory or a storage medium, for calculating and storing a seventh set of data representing a first term to maturity profile or a first payment profile (depending on what was determined under (f ) ) corresponding to interest and repayments for the debtor and a first remaining debt profile, said term to maturity profile or payment profile, as well as the remaining debt profile, being calculated on the basis of - the volume and repayment profile of the loan as input under (a) , the set of data input under (b) , the refinancing profile and/or the funding profile input under (c) - and the profile of the interest rate on the loan and either the payment profile or the term to maturity profile established under (f ) ,
• (gl) means, typically calculating means and a memory or a storage medium for, if necessary/desired, calculating and storing an eighth set of data representing payments (positive, zero or negative) on the payment guarantee instrument, the requirements with respect to maximum permissible difference in balance and maximum permissible difference in proceeds, as well as the limits for payments on the loan and term to maturity, always being fulfilled,
• (h) means, typically calculating means and a memory or a storage medium, for selecting a number of financial instruments among the financial instruments stored under (e) , and calculating and storing a ninth set of data specifying these selected financial instruments with their volumes for use in the funding of the loan, said ninth set of data being calculated on the basis of the payment profile established under (f) or calculated under (g) and - the remaining debt profile calculated under (g) , the payments on the payment guarantee instrument optionally calculated under (gl) , the refinancing profile input under (c) and/or the funding profile input under (c) , - the set of data input under (b) , the requirements input under (d) , and in the case of a refinancing where financial instruments from a previous funding have not yet matured, the type, the number and the volume of these instruments,
• said means being adapted to perform, if necessary, one or more recalculations, including, if necessary, selecting a new number of the financial instruments stored under (e) , said means further being adapted to store, after each recalculation, - the recalculated profile of the interest rate on the loan, the recalculated term to maturity profile, the recalculated payment profile, the recalculated remaining debt profile, and - the selected financial instruments with their calculated volumes, until all the conditions stated under (b) and (d) have been fulfilled, and the means further being adapted to optionally recalculate the payments on the payment guarantee instrument in accordance with (gl) , and store, after each recalculation, the recalculated payments in the memory or the storage medium,
• a computer system which may be applied for the method according to the present invention may comprise means for inputting and storing the data necessary for the calculations.
• the input means may comprise a keyboard or a mouse, a scanner, a microphone or the like but may also comprise means for carry- ing out electronic inputting via a storage medium or via a network.
• the storage media may be electronic memories such as ROM, PROM, EEPROM or RAM, or storage media such as tapes, discs or CD-ROM.
• the system comprises calculating means adapted to perform the calculations necessary for the implementation of the method.
• the calculating means may typically comprise one or more microprocessors.
• the system may be a computer system programmed such that the system is capable of performing the calculations necessary for the implementation of the method according to the invention.
• the system is capable of performing the calculations necessary for the implementation of the method according to the invention.
• there may be different embodiments of the system meaning that these different embodiments are adapted to perform the calculations specified in the different embodiments of the method accord- ing to the invention mentioned above and in the claims.
• the example section contains - apart from preferred examples of the method according to the invention - a description of a number of preconditions for the invention, and of a number of preferred applications of the method according to the invention, and of the results obtained by the method.
• the expression "bonds" is used about a financial instrument in the ordinary meaning of the word.
• the expression covers all types of interest-bearing and non-interest-bearing claims, including financial instruments and bonds .
• Figure 1 shows an example of a lattice structure and tree structure of a binomial model. A tighter structure is achieved in the trinomial models .
• Figure 2 shows the connection between the continuous structure and the lattice structure.
• the average value for the interest rate is determined by the initial yield curve to which the model is calibrated. This appears from the figure by the graph (1) .
• pricing is performed solely on the basis of this graph.
• the connection is illustrated in principle. No calculations form the basis of the figure.
• Figure 3 shows an example of the dynamic adjustment of the lattice structure.
• Figure 4 shows an example of the yield curve for r* prior to the calibration to the initial, observed yield curve.
• Figure 5 shows the possible impact of the recalculation of the interest rate on the loan on the payment profile and remaining debt profile of the loan.
• Figure 6 shows the calculation of probabilities in the lattice by means of Bayes ' rule.
• the shown lattice is calibrated to a flat yield curve.
• Figure 7 shows the flow diagram of the model for a LAIR III type F.
• Figure 8 shows the determination of the next value of the interest rate on the loan in the iteration according to the Gauss-Newton algorithm.
• Figure 9 shows the flow diagram of the model for a LAIR III type P .
• Figure 10 shows an example of the adjustment of the trend function.
• the volumes shown are not calculated.
• the loan could be a LAIR type P20,0 in which extreme yield curve has resulted in the volume of the bond with a term to maturity of four years being disproportionate, for which reason the trend function "breaks".
• Figure 11 shows the possible pattern of the payments on the financial instrument.
• the payments are positive due to the high interest rate.
• the low interest rate implies negative payments on the instrument.
• Figure 12 shows the pricing of the financial instrument in each node according to the backward induction principle.
• Figure 13 shows the flow diagram of the model for quoting the limits for payments on the loan and term to maturity.
• Figure 14 shows the flow diagram of the model for Type F * in the case in which the limits for payments on the loan and term to maturity are compatible, payments on the financial instrument thus not being necessary.
• Figure 15 shows the flow diagram of the model for Type F * in the case in which the limits for payments on the loan and term to maturity are incompatible, payments on the financial instrument thus being necessary.
• Figure 16 shows the alternative modelling of a LAIR III type P.
• Figure 17 shows an example of the initial adjustment of the trend function.
• the underlying yield curve is important for the pricing of financial claims (claims are to be taken in a broad sense as securities, debt and financial instruments) .
• the yield curve is an expression of the interest rate of different claims as a function of a selected characteristic feature.
• the selected characteristic feature is the remaining term to maturity or duration of the claim, and thus it is the horizontal yield curve.
• the vertical yield curve is an expression of the interest rate of claims with identical terms to maturity but with different credit risks, liquidity, or the like.
• Zero-coupon rates express the interest rate of a claim with only one payment in the entire term to maturity of the claim. Claims with different cash flows may thus be seen as different portfolios of zero-coupons, and once the zero-coupon yield curve has been determined, a pricing of any known cash flow is possible.
• the finance theory has many different suggestions as to the modelling of the yield curve.
• the different suggestions deviate by including, in different ways, factors, such as volatility, observed market prices etc.
• the models are widely different in their degree of operationability, which should be considered fairly important for this purpose.
• Prior to the presentation and description of the selected model aspects of the selection of yield curve model are explained without this developing into a review of recent yield curve theory, however.
• the stochastics are introduced in such a way that the interest rate will approach an (equilibrium) level in the long term, whereas the short-term movement may fluctuate quite significantly around the long-term trend.
• This modelling of the stochastics seems plausible assessed on the basis of economic principles.
• the disclosure of more or less irrelevant information may influence the formation of interest rates solely because economic agents predict the reaction of other agents etc.
• the interest rate will converge towards an equilibrium level which is not affected by the irrelevant information. Realization of the equilibrium level requires, however, that no new information is revealed for a period of time, and will thus not necessarily occur.
• Ito processes One category of stochastic processes fulfilling the above-mentioned properties is the so-called Ito processes .
• the Ito process is formulated by (1.1) .
• dZ-(t) is a so-called Wiener process.
• the process is also termed a generalized Brownian motion
• the Wiener process is to be seen as the counterpart to a random walk in continuous time, and is thus a random walk in continuous time.
• the Wiener process complies with the following equation.
• the Wiener process, and thus also the Ito process, are also characterized by being Markovian.
• a stochastic process X t is said to be Markovian if
• a Markovian process has no memory. Only the immediately preceding value x t _ ⁇ is crucial for the value of the process in the current period x t , whereas all other preceding values of the process are immaterial.
• the finance theory has accepted this challenge in the so-called multiple factor models which include factors such as inflation, the interest rate in other countries, or similar factors in the stochastic process for the interest rate. In the nature of the case, also the included other factors are described by a stochastic process.
• a prominent example is Heath, Jarrow and Morton (1991) "Bond Pricing and the structure of interest rates: A new methodology for contingent claims valuation” Working paper Cornell University.
• the multiple factor models have obvious theoretical advantages, but suffer from the weakness of not being Markovian. This means that in practice, the models may be operationalized only to a small extent, which is a central property for this purpose. Therefore, the modelling is limited to a one-factor model in the following.
• Stochastic yield curve models are generally divided into two categories .
• the first category consists of equilibrium models.
• the basis of these models is of a microeconomic nature.
• the yield curve is determined in accordance with the preferences of the agents so as to provide a balance in the capital markets.
• the preference structure should reflect the degree of risk aversion of the agents, said risk aversion traditionally causing opinions to differ.
• the advantage of the equilibrium models is that as soon as the preference structure is described, all claims may be priced. The use of parameters in the models is thus limited.
• An example of the equilibrium model is the CIR model (Cox, Ingersoll and Ross 1985) "A theory of the term structure of interest rates", Econometrica 53) and the Vasicek model (Vasicek (1977) "An equilibrium characterization of the term structure", Journal of Financial Economics 5).
• the problems related to modelling the preference structure of the agents have led to the development of a new category of models, the so-called no arbitrage models.
• the no arbitrage models are characterized in that the modelling of the future interest rate is calibrated to an observed initial yield curve and optionally to a volatility structure.
• the modelling of the future yield curve is thus no arbitrage, as no possibilities of arbitrage occur between the observed prices and the claim prices fixed in the model.
• Hull and White (1996) / (1994a) (Hull and White (1996) is a collection of previously published articles. (1996) / (1994a) refers to Hull and White's article from 1994 that is included in the collection of articles. This reference is used henceforth) adduce an argument against the diffusion coefficient being time-dependent.
• the volatility structure proves to develop very differently from the traditional perception of the volatility.
• the future volatility structure is particularly sensitive to the initial estimate of the volatility of claims with a long term to maturity.
• Hull & White compare the time-dependent diffusion coefficient with an excessive parameterization of the model and conclude, on this basis, that the most reliable results are obtained with a value of ⁇ (t,x) which is not time-dependent . It is preferred that the following recommendation is followed in a method according to the invention. 1.1.4. Requirements with respect to the modelling of the yield curve derived from the financial instrument
• the most important requirement is that the modelling is to be performed in discrete time.
• the financial instrument is characterized in that the payments on the instrument are dependent on the other variables on the debtor and funding sides of the loan, said variables being dependent of the yield curve. This implies that the payments are determined at each adjustment of the interest rate for the period up to the next adjustment of the interest rate on the basis of the yield curve. This pattern cannot immediately be described in a model in continuous time .
• the transition to discrete time means that the continuous process for the interest rate must be approximated by the discrete expression
• Hull and White have developed a general frame in which different, originally continuous yield curve models may be made discrete and be implemented in a trinomial lattice. Further, the model frame distinguishes itself by being more operational than other discrete yield curve models. Thus, there are good arguments in favour of following Hull and White's (1996) approach to a stochastic modelling of the yield curve in discrete time 1.2
• Hull and White's very general model frame permits an implementation of a number of yield curve models.
• the process for the interest rate may be set up as in (1.6)
• Vasicek 's (1977) model appears. This model is characterized in that the diffusion is not dependent in the level of the interest rate.
• the relatively simple model has a number of favourable analytical properties (see e.g. Hull and White (1996) / (1990) ) , primarily in relation to the pricing of European options .
• Vasicek 's model One disadvantage of Vasicek 's model is that the negative interest rates are not excluded, but will occur in the model with a positive predictability.
• the occurrence of negative interest rates constitutes a technical problem, as an argumentation based on arbitrage arguments can hardly be extended to a situation with negative interest rates.
• the problem is manageable. Firstly, the likelihood of negative interest rates will be limited by a realistic determination of the parameters of the model. Secondly, a situation has arisen in practice, in which the interest rates have been 0 (zero) or negative as a consequence of imperfections of the market. At the same time, the model - like most other models - paves the way for very high interest rates. The focus on the possibility of negative interest rates with merely a minor probability may therefore seem exaggerated.
• Hull and White's method for implementing the model contains a facility which reduces the possibility of negative interest rates.
• the probability of negative interest rates may be reduced to 0 (zero) at a second specification of ⁇ . If a value within the interval ]0.1[ is assigned to ⁇ , the effect of the diffusion of the interest rate process will grow drastically if the interest rate approaches 0 (zero) . With the probability of 1, the interest rate will thus be increased by the diffusion before it assumes a negative value.
• the value V2 is assigned to ⁇ , and thus the CIR model do not allow negative interest rates with a positive probability.
• the CIR model has an obvious theoretical advantage.
• the formulation of the diffusion results in the model being difficult to implement. To Hull and White this is a crucial argument in favour of applying Vasicek' s model as a basis, said argument being the one currently preferred to follow in a method according to the invention.
• the idea underlying the implementation of the model in a trinomial lattice is that the lattice is to reflect the development in the underlying continuous interest rate process .
• a continuous distribution of the adjustment to the interest rate will exist for every t, the distribution, the average, and the variance being determined by the continuous process for the interest rate.
• the continuous distribution is approximated in the lattice by a discrete distribution consisting of an increasing number of nodes . From each node there are three branching possibilities : up, middle, and down. The probabilities of each of these results are determined in each node such that the process in the discrete lattice develops (approximately) in the same way as the underlying continuous process. However, a difference will always occur as a consequence of the transition from a continuous to a discrete distribution.
• the lattice there will be a maximum and a minimum limit for the adjustment to the interest rate for a given value of t. This follows from the distribution being discrete. Hence, the lattice will not span an interval of adjustments as wide as will the continuous distribution.
• Hull and White do not only adjust the probabilities to the drift and volatility of the process, but also to the branching structure.
• the adjustment of the branching structure to the drift of the process is introduced by the parameter
• the minimum limit will fluctuate. It is not a foregone conclusion that the minimum limit is determined at a positive interest rate ( 0). Thus, the branching procedure does not preclude for certain the occurrence of negative interest rates in the lattice.
• the adjustment of the lattice structure has the favourable feature that the range of the lattice is limited.
• the model thus obtains a higher degree of operationability, as it is not necessary to operate with extreme interest rates, which is also uninteresting in practice.
• the adjustment constitutes a (minor) theoretical problem for the pricing.
• the adjustments of the branching structure and the probabilities are performed such that the interest rate process still corresponds to the underlying continuous distribution.
• the adjustment will result in an imbalance in the determination of the payments. Whether this imbalance affects the pricing is difficult to assess. Hull and White's model frame is applied in many situations, and e.g. in the pricing of options that must be considered very sensitive, indeed, to these imbalances. Thus, the consensus is that these imbalances may be ignored.
• Hull and White begin by considering the continuous interest rate process for dr*, which appears by setting ⁇ (t) and the initial value of r to 0 (zero) in (1.7) .
• ⁇ (t) is the step size in the lattice.
• the step size may be determined arbitrarily in consideration of the claim that is being priced. The determination of the step size is discussed in more detail under the pricing of the financial instrument in section 3.
• each node may be described at the point in time and at the interest rate, i.e. (t,r) .
• the fixed values for both ⁇ t and ⁇ r * open the prospect of a more appropriate notation based on the adjustments.
• Each node may thus be described by (g,h), where g denotes the number of periods elapsed and h denotes the number of up results. This gives the following relations
• P 0 , P m and P n The probabilities of up, middle, or down branching are denoted P 0 , P m and P n .
• Three requirements may be laid down with respect to the probabilities. Formally, it is the fulfilment of these three requirements which makes P 0 , P m and P n eligible for being perceived as probabilities.
• the average of the adjustment in each node in the discrete lattice is to correspond to the average of the underlying continuous process r*(t)E. This may be formalized to
• the probabilities may be found as the solution to three equations with three unknown quantities.
• the probabilities may thus be found as the solution to the matrix equation
• k may assume the values ⁇ -1,0,1 ⁇ in the lattice. If the possible values for k are inserted in (1.19) to (1.21), the actual probabilities for the branching are deduced, at the same time applying that
• the only factor lacking in the determination of the lattice structure is the determination of a value for k.
• h For a sufficiently large value of h - and thus for a high interest rate - the drift downwards of the process is so strong that one or more of the probabilities is/are immediately to be assigned a negative value in order to fulfil the requirement given in ( 1.13 ) .
• E means that it will always assume negative values for a>0. This means that the maximum limit for h (h-, ax ) is to be found among the negative values of x. The maximum limit for h is given by a integer value fulfilling
• the calibration to an initial yield curve is performed by a new lattice being formed as a displacement in the vertical plane of the old lattice.
• the displacement is determined such that the lattice prices zero-coupon bonds in accordance with the observed zero-coupon rates constituting the initial zero-coupon yield curve.
• the displacement of the lattice is introduced by the parameter ⁇ * g which is time-dependent but not level-dependent. For a given g, the displacement of all nodes will thus be identical, making it possible to determine the interest rate in each node as the interest rate in the previous lattice plus oc g .
• the state (g * +l,h * ) is obtainable in three ways from the nodes (g * ,h * -l), (g * ,h * ) and (g * ,h * +l).
• the argument is based on the normal branching but may be immediately extended to situations in which the node is obtainable in less or more ways.
• the probability of (g * +l,h * ) being realized is given by q(h * -l,h * ) .
• the interest rate is given by r(g * ,h * -l) .
• the expected value of the payment 1 in (g * +l,h * ) is thus (g * ,h * -l) in the node.
• ⁇ 0 is to be determined on the basis of the zero-coupon bond with a maturity of l ⁇ .
• the zero-coupon bond is assumed to have the observed price
• R(.) is the observed zero-coupon rate.
• the bond is to have the price
• the lattice has thus been calibrated to the initial yield curve.
• One example of the possible appearance of the lattice is shown in figure 3.
• the results of the model The interest rate modelled above is defined over a period of ⁇ t . It is thus a short-term interest rate, depending on the exogenous selection of the step size in the lattice, but not as short as in continuous models in which the instant interest rate is modelled.
• the modelled interest rate may be sufficient for the pricing of claims in which the payments are independent of the prevailing yield curve.
• the modelled interest rate is applied to the discounting of the payments via the lattice, causing the present value, and hence the price, to be determined.
• the payments may be dependent on the yield curve .
• the payments are determined as the loss or gain in proceeds in connection with the bond funding of the loan, provided the payments on the loan and the term to maturity are within the allowed limits, cf. section 2. Since the loan may be funded by bonds with a maximum term to maturity of 11 years, the payments on the instrument will consequently to a considerable extent depend on the interest rate curve, and not just on the ⁇ t interest rate. In each node in the lattice, therefore, a yield curve, and not just an interest rate, is to be determined.
• Hull and White (1996 )/( 1996) deduce an expression for the yield curve, which may be calculated in each individual node.
• the deduction goes via Ito's lemma which is too comprehensive to be reviewed herein. Therefore, the focus is solely on Hull and White's deduction of the expression.
• the zero-coupon rate from time t to T may be calculated on the basis of the relationship between the interest rate at time 0 of zero-coupon bonds with maturity at time t and T, respectively. This implies that it is impossible to calculate zero-coupon rates stretching further forward than the initial yield curve which typically corresponds to the length of the lattice.
• Hull and White (1996) / (1994a) recommend a very general, but not very operational principle. According to Hull and White, the parameters of the model are to be determined such that the volume of
• the parameterization of the model is a question of determining values for a and ⁇ which are both crucial for the volatility of the interest rates.
• ⁇ determines the volatility of the short interest rate. In the long term, the drift will, as already mentioned, dominate the diffusion and hence the part of the volatility originating from ⁇ . ⁇ should therefore be estimated on the basis of the observation of the volatility in the short end of the interest rate spectrum.
• a determines how fast the interest rate approaches the equilibrium level which is determined in the model by the initial yield curve.
• the relationship between the volatility of the short interest rate and the volatility of the long interest rate is decisive for the determination of a. Therefore, a should be determined on the basis of observations of this relationship.
• the basis of the pricing of the financial instrument in a LAIR III is a modelling of the future interest rate.
• the future interest rate will be decisive for the volume of the payments on the financial instrument.
• the future interest rate will be decisive for the present value of the future payments, and hence for the price of the instrument.
• the modelling of the future interest rate is thus an important part of the method according to the invention.
• the applied model is Hull and White's extended Vasicek model which has a number of theoretical as well as practical advantages .
• the interest rate in the model is considered stochastic. Perfect predictability should be precluded for obvious reasons. Therefore, a reliable modelling of the interest rate must involve stochastics.
• the stochastics is introduced in the model via the general Ito process which has a number of favourable characteristics. It follows from the Ito process that the interest rate i the short term will fluctuate about an equilibrium level which is approached by the interest rate in the long term. And the Ito process is Markovian, which permits the implementation of the model in a discrete interest rate lattice.
• Hull and White's extended Vasicek model belongs to the category of no arbitrage models. This means that the model may be calibrated to an initial, observed yield curve. Thus, provision is made for the theoretical prices of the model being in accordance with the observed market prices. The calibration also implies that an estimation of the risk aversion of the agent is unnecessary.
• Hull and White's model frame is characterized by a high degree of operationability .
• the model frame permits the setting up of continuous stochastic processes for the interest rate in discrete time.
• the model may be implemented in a discrete trinomial lattice. The very possibility of implementation in a discrete lattice is essential in the determination of the payments on the financial instrument.
• the construction of the interest rate lattice constitutes the most important part of the model.
• the lattice structure is adjusted to the drift in the stochastic process on a current basis such that the lattice structure reflects the initial yield curve.
• the lattice structure may also be made dynamic so that a modelling of extremely low or extremely high interest rates is avoided.
• An essential result of the model is the deduction of the yield curve in each node in the lattice .
• the payments on the financial instrument depend not only on the short-term interest rate but on the entire yield curve. Therefore, the result is a prerequisite for the determination of the payments of the instrument.
• a LAIR III comprises a financial instrument combined with a LAIR II or a LAIR I.
• a LAIR I may be perceived as a special case of a LAIR II. In the situation in which identical maximum and minimum limits are determined for the term to maturity, these limits will be fixed during the entire term to maturity of the loan. Thus, said LAIR II degenerates to a LAIR I.
• the model for a LAIR III consists of a model for a LAIR II, as well as an extension managing the financial instrument.
• the section contains a more verbal review of central aspects of the model.
• an adjustable term to maturity including the determination of the term to maturity, the term to maturity concept, etc. are explained.
• it is discussed in the section how the limits for the term to maturity and the payments on the loan are determined in consideration of the debtor's costs pertaining to the financial instrument.
• Section 2.2 contains a general description of how to hedge the limits of the loan in combination with a LAIR II. This is may be done in different ways, each of which is discussed before the established method is described.
• section 2.3 the implementation of the model in the lattice is explained.
• the model set up calculates the debtor and the funding sides of one interest rate adjustment period at a time. Therefore, the calculations in the model are to be performed in each node which coincides with an interest rate adjustment, which requires e.g. a determination of the input to the model in each node. A method for determining input is thus deduced in the section.
• Section 2.4 contains a review of the adjustment of the interest rate on a LAIR.
• the adjustment of the interest rate on a LAIR divides the product into two types of loans, LAIR type f and type p, respectively, with very different characteristics with regard to the determination of the bond volumes, etc. Consequently, a distinction must be made in the modelling between these two types of loans.
• appendix A contains the modelling of a variant of type f which occurs with certain structures of input.
• appendix B contains an alternative method for modelling type P.
• the interest rate on the loan will rise and fall in line with the interest rate level at the time of the interest rate adjustment.
• a falling interest rate is not a problem for the debtor (the interest rate risk of the remaining debt of the loan not being taken into account) , and, therefore, a falling interest rate does not give rise to considerations regarding the product.
• the adjustment to a higher interest rate level constitutes a potential problem for the debtor.
• a rising interest rate may influence the loan in two ways .
• the payments on the loan may increase.
• a LAIR with an adjustable term to maturity is characterized in that the payments on the loan float within a band defined by a set of maximum and minimum limits for the payments on the loan.
• the payments fluctuate as the loan is interest rate adjusted to the prevailing market rate. The limits are denoted
• J 0
• J M
• the fluctuation within the band is ensured by a correction of the term to maturity of the loan when the payments on the loan would otherwise have fluctuated outside the band.
• the possible corrections are defined on an interval defined by a maximum and a minimum limit for the term to maturity.
• the limits for the term to maturity are denoted L max and L m ⁇ n , L generally denoting the term to maturity.
• No requirements in the model stipulate that the term to maturity is in integer years or payment periods.
• the possible corrections of the term to maturity are defined on a continuous interval limited by L max and L m ⁇ . This is necessary if it is to be possible at each adjustment of the interest rate on the loan to calculate a payment which is within a relatively narrow band.
• the term to maturity is corrected such that the loan matures on 1 January at the same time as the underlying bonds.
• the subsigns are maintained in the notation in order not to lose generality.
• the notation also opens up the prospect of having the serial loans being covered by LAIR III.
• the payment profile of a serial loan is decreasing, for which reason fixed limits for the payments on the loan would produce inconvenient results.
• the term to maturity concept of a LAIR with an adjustable term to maturity paves the way for several possible interpretations of limits for the term to maturity.
• the adjustable term to maturity means that for a loan there will be a sequence of terms to maturity given by
• the pricing of the financial instrument will depend on the initial term to maturity of the loan, the maximum and minimum limits for the term to maturity, and the maximum and minimum limits for the payments on the loan.
• the model must be so flexible that an arbitrary determination of all variables within the legislative and credit policy framework is possible.
• a favourable solution would be that the financial instrument has the price 0 (zero) at the disbursement of the loan, causing the variables of the loan to be interdependent .
• the initial term to maturity of the loan may be applied for the determination of a level for the payment initially.
• the positive price of hedging the maximum limit for the payments on the loan will be significantly higher (numerically) than the negative price of hedging the minimum limit.
• this price may be conveniently fixed as the initial payment on the loan.
• the minimum limit for the term to maturity is fixed conveniently as the initial term to maturity.
• the minimum limits provide, to the greatest extent possible, the possibility of determining a maximum limit for the payments on the loan at a relatively low level. This possibility is supported by a determination of the maximum limit for the term to maturity at a level as high as possible within the legislative and credit policy framework.
• the maximum limit for the payments on the loan is determined unambiguously on the secondary condition that the price of the instrument is 0 (zero) .
• the directions imply that the maximum limit will be determined at the lowest possible level.
• the model is to allow that the limits are determined according to other directions determined by the debtor. Therefore, the limits will be considered exogenous.
• the above method for determining the limits is modelled.
• the financial instrument is to prevent the payments on the loan from fluctuating outside the band defined by the maximum and minimum limits.
• a cap/floor approach Firstly, the hedging of the limits for the payments on the loan may be performed by a cap/ floor approach.
• the financial instrument is defined directly by the payments which is outside the band in the model for an adjustable term to maturity.
• the model may be used with adjustable as well as fixed terms to maturity - i.e. as a further development of both a LAIR I and a LAIR II.
• the model will be a pure further development of either an underlying LAIR I or LAIR II, which renders implementation of the model relatively easy, as the underlying LAIR I or LAIR II may, in principle, be calculated in an existing model.
• the legislative and fiscal conditions mean that most conveniently, the financial instrument exists only on the debtor side of the loan. On the basis of the payments from the underlying LAIR corrected by the payments on the financial instrument, a new debtor side is thus to be calculated.
• the recalculated debtor payment is to be smaller than on the underlying loan. This is achieved by reducing the interest rate.
• the new interest rate on the loan may be determined in a relatively simple way as the interest rate which in the expression for the calculation of an annuity produces a payment corresponding to the maximum limit given the term to maturity.
• the reduced remaining debt at the end of the interest rate adjustment period expressed by the difference between (4) and (5) in the figure gives rise to imbalances. Bonds corresponding to the remaining debt of the underlying loan mature on the bond side, whereas the interest rate adjustment amount on the debtor side is calculated on the basis of the reduced remaining debt.
• the imbalance may be equalized by the reduction in the remaining debt also being covered by the payments of the financial instrument. However, this produces a very distorted development in the payments, the fixed payments for each payment date in the interest rate adjustment period thus being supplemented with a large payment at the end of the period.
• the imbalance may be equalized by the repayment profile being fixed.
• the fixing may be performed at the disbursement of the loan, which is a model that has previously been applied for floating rate mortgage credit loans. However, the model does not allow an adjustable term to maturity and must, on this basis, be precluded.
• the fixing may be performed at the beginning of each interest rate adjustment period so that e.g. the repayment profile of the underlying loan is maintained. This model is difficult to comprehend for the debtor and will result in either the payments or the interest on the loan not being constant during the interest rate adjustment period.
• the approach implies that the funding side of the loan is made artificially large in certain cases.
• the approach implies that bonds are issued, the payments of which are covered by the financial instrument. This does not represent a problem but must be considered a less than optimum solution from a theoretical point of view.
• the approach implies that the payments on the financial instrument are not defined directly by he payment profile of the underlying loan, but instead as the necessary reduction of the volume of underlying bonds ensuring that the payments on the loan are within the band defined by the maximum and minimum limits.
• the approach is based on the development of the adjustment of the interest rate on a LAIR.
• a funding demand arises as a result of the adjustment of the interest rate on the loan at the end of the preceding interest rate adjustment period.
• the underlying volume of bonds matures fully or partially
• the market price of the sold bonds influences the interest rate on the loan in the interest rate adjustment period. At a low market price, the extent of the necessary sale of bonds increases. The larger volume of bonds causes larger payments on the bond side and thereby also on the debtor side, the balance principle having to be respected. The interest rate on the loan is thus increased.
• the mechanics of the adjustment of the interest rate on a LAIR means that a reduction in the volume of the payments, so that a maximum limit for the payments on the loan is observed, is obtainable by means of a smaller sale of bonds at the beginning of the interest rate adjustment period.
• a loss in proceeds will thereby occur at the beginning of the period in the relationship between, on the one hand, the nominal value of the mature bonds and, on the other hand, the market price of the sold bonds. According to the approach, this loss in proceeds defines the payments from the financial instrument.
• a gain in proceeds may occur if an increased volume of bonds is sold to enhance the volume of the payments such that a minimum limit for the payments on the loan is observed.
• the gain in proceeds constitutes the payments to the financial instrument.
• the financial instrument formed by losses and gains in proceeds is comparable to a collar.
• the hedging of the maximum limit by means of covering the loss in proceeds corresponds to the debtor having a long position in put options with an exercise price corresponding to the interest rate on the loan when the maximum limit has been observed.
• the hedging of the minimum limit corresponds to a short position in a call option with an exercise price corresponding to the interest rate on the loan when the minimum limit is binding.
• the composition of the bond portfolio underlying the loan must be known at all future interest rate adjustments. If the distribution of bonds in the individual years is known, it naturally follows that it will be impossible to take positions in options on the individual bonds .
• the model for calculating the debtor and funding sides of the loan must be implemented in the trinomial lattice set up in section 1.
• the payments of the debtor and bond sides are calculated in each node on the basis of the yield curve in the node in question.
• the debtor side as well as the bond side depend not only on the yield curve. A number of variables determined at the preceding interest rate adjustment will also affect the current interest rate adjustment.
• the volume of the remaining debt and the interest rate adjustment amount at the end of the preceding interest rate adjustment will affect the current interest rate adjustment.
• the term to maturity in the preceding interest rate adjustment is also to be used in the calculation of the current interest rate adjustment. Basically, the term to maturity is not changed provided the maximum and minimum limits for the payment are not exceeded.
• the bond volumes from the preceding interest rate adjustment and the associated coupon rates are to be input to the model because of the rolling movement in the bond funding, cf. section 2.4.
• the said inputs must be determined as an expected value of the nodes from which the current node may be reached with a positive probability.
• input are determined by a projection through the lattice in which the probabilities of the different branching structures are weighted.
• the projection is complicated by the fact that there is not necessarily a one-to-one correlation between the step size in the lattice ( ⁇ t) and the length of the interest rate adjustment periods.
• ⁇ t the step size in the lattice
• V* the length of the interest rate adjustment periods.
• the interest rate adjustment periods may last up to 10 years for which reason there may be many nodes between each interest rate adjustment.
• the projection may thus follow a forward induction method comparable to that applied in section 1.
• the vector x(g,h) is defined.
• the elements in the vector are constituted by the above input variables, i.e.
• x ⁇ remaining debt and interest rate adjustment amount at end of period; term to maturity of the loan; bond volumes at end of period, coupon rates associated to the volumes ⁇

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