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"Danish Government Borrowing and Debt 2001"

Chapter 9
Interest-Rate Models for Cost-at-Risk

9.1 Summary

Cost-at-Risk (CaR) supplements duration and redemption profile in the management of the interest-rate and refinancing risks on the government debt. CaR quantifies the risk on the basis of scenarios for the future development in interest rates. Using these interest-rate scenarios, the interest-rate risk can be determined as the maximum annual interest costs on the domestic government debt with a given probability, typically 95 per cent.

CaR is calculated in a model developed by Government Debt Management at Danmarks Nationalbank. In 2001 development work on the CaR model was focused on two areas. The model is now able to handle buy-backs of government securities, as well as swaps from kroner to euro. Moreover, alternative interest input to the CaR model has been modelled, since the CaR results are sensitive to the interest-rate model chosen.

So far, the Cox, Ingersoll and Ross (CIR) model has been used to generate interest-rate input to CaR. As the CIR model is relatively simple, it has difficulties in some areas to reproduce the interest-rate characteristics that are observed empirically. This Chapter presents two examples of alternative interest-rate models: a linear two-factor model and a forward-rate model. Simple statistical characteristics of the simulated interest rates from the three models are compared, and held against the equivalent characteristics of historical interest rates. The empirical results are provisional and should primarily be seen as an illustration of the issues relevant in a CaR context. This Chapter's two appendices present technical descriptions of the two models.

In 2002 the work on interest-rate models for CaR will continue. Moreover, it is the objective to expand CaR to include the foreign debt and the Social Pension Fund (SPF), so that CaR for the overall government debt can be calculated.

9.2 Method

Cost-at-Risk (CaR) is used as a supplement to duration and redemption profile to quantify the interest-rate and refinancing risks. In the CaR model, the future interest costs of the domestic government debt are calculated as expected interest costs, as well as the maximum future interest costs with a 95 per cent probability (absolute CaR) for each year up to a horizon of 10 years. On this basis, relative CaR can be determined as the difference between absolute CaR and the expected costs. Relative CaR thus indicates the maximum costs compared to the expected costs in a given year, and can also be interpreted as a measure of the uncertainty of the expected interest costs, cf. Box 9.1.

Box 9.1 CaR definitions

CaR is calculated in a spreadsheet model programmed in Excel, and developed by Government Debt Management at Danmarks National-bank. The work on the CaR model commenced in 1997. In 2001 the CaR model was further developed for modelling buy-backs and swaps from kroner to euro within the model's framework.

Elements in the CaR model
The basis for calculation of CaR is data concerning the existing debt, and thereby information on the redemption profile together with accrued costs and payments on the debt.

Assumptions of future budget balances are also included. The calculations of CaR include assumptions concerning the government-budget balances before interest costs for the individual years. The interest costs are calculated within the model.

Objectives for duration, redemption profile and outstanding volume in on-the-run issues are additional assumptions included.

Furthermore, the CaR model includes assumptions concerning the use of the government debt-policy instruments, i.e. the distribution of on-the-run issues in the 2-, 5- and 10-year segments, buy-backs, interest-rate swaps and swaps from kroner to euro.

Finally, the model includes input concerning the future development in interest rates. Given the information on the initial composition of the debt, assumptions concerning future budget balances, objectives for duration, redemption profile and outstanding volume, together with assumptions concerning the use of government debt-policy instruments, the interest costs for various scenarios of the development in interest rates can be calculated.

The generation of a large number of interest-rate scenarios provides a means to set up probability distributions for the future costs. 2,500 scenarios of the future development in the yield curve are generated. The yield curves in a scenario have a quarterly frequency. On this basis, the future annual costs are calculated. These scenarios are used to determine the expected costs as the average of the cost scenarios for each year, and absolute CaR as the 95th percentile of the cost scenarios.

9.3 Interest-rate models

The calculation of the future interest costs for the debt is sensitive to the interest-rate model chosen to generate the future development in interest rates. So far, the interest input has been generated by an interest-rate model developed by Cox, Ingersoll and Ross, also called the CIR model^[1]. This interest-rate model is prominent in the finance literature, and it is implementable in practice. Since the model is relatively simple, it has difficulties in describing certain empirical characteristics of historical interest rates. For example, the CIR model implies that interest-rate volatilities decrease more with maturity than is observed empirically. This means that the CIR model tends to underestimate the volatility of the long-term yields. Moreover, the CIR model has certain limitations in terms of its ability to generate various types of yield curve. Therefore two alternative interest-rate models for CaR are considered, each in its own way based on the CIR model. The two models originate from different classes of interest-rate models, i.e. linear factor models and forward-rate models. These two concrete examples of alternative interest-rate models have been chosen as they are comprehensively described in the literature, and fulfil certain key criteria.

Criteria for choice of interest-rate model
The theoretical literature on interest-rate models is very extensive, and a number of different formulations of interest-rate models are available. On selecting an interest-rate model, it is appropriate to focus on the overall characteristics required of the model.

CaR modelling sets requirements of the interest-rate model's ability to describe the shape of the yield curve, as well as the dynamic of the development in interest rates over a longer horizon. This is because government securities are issued along the entire yield curve (up to 10 years), and CaR is calculated on the basis of simulated interest-rate developments over a long horizon. The emphasis is on the model's ability to give a good average picture of the structure of and the movements in the yield curve.

It is also important that the interest-rate model is relatively simple to interpret and use. The CaR model is already complex, with dynamic effects over a long horizon. It is important to restrict the complexity in order to ease implementation, including estimation and simulation of the interest-rate model. In this respect it is an advantage if the model has been subject to thorough theoretical analysis, and is generally known and used. This eases communication of the model.

Finally, the interest-rate model must be consistent in economic and financial terms. There are several aspects to this. The model must be able to generate interest rates that move within realistic levels. It must thus be ensured that the interest rates do not "explode". A key characteristic to ensure that the interest rates in the models do not explode is "mean reversion". This implies that the interest rates tend to move towards a given long-term or equilibrium level. Another requirement can be that negative interest rates should not occur. The requirement of the non-negative nominal interest rates can be interpreted as a requirement of non-arbitrage. Instead of placements at a negative interest rate, cash can be held at a zero rate of returns. The interest-rate models are required to be free of arbitrage within the model framework.

The CIR model
The CIR model belongs to the class of linear (affine) factor models. In the CIR model, any interest rate can be described as a linear function of a single stochastic factor, namely the short-term interest rate. The CIR model is a relatively simple model that fulfils the aforementioned criteria. Box 9.2 presents the key characteristics of the CIR model.

Box 9.2 The CIR model

A linear two-factor model
Like the CIR model, the first alternative interest-rate model is from the class of linear factor models. In these models, it is assumed that the yield curve and its development can be described by a limited number of factors. The term linear is used for these models because there is a linear relation between the interest rates and the factors.

By introducing additional factors to the CIR model, a more flexible interest-rate model is obtained, which can generate more varied types of yield curve.

The factors can be observable, e.g. the short-term interest rate and its volatility, or non-observable. In the latter case, the factors can be interpreted by analysing the co-variation between interest rates and factors.

A small number of factors has proved to give a good statistical description of the interest rates. The prominent factor models described in the literature typically have up to 3 factors. There are only few benefits to the statistical description of the interest rates from adding more factors, which would also increase the model's complexity. Here, a two-factor model developed by Longstaff and Schwartz (LS)^[2] is considered, cf. Box 9.3. The LS model can be viewed as an expansion of the CIR model, to which an extra factor has been added.

The two factors can be interpreted by showing their impact on the yield curve. The sensitivity of the interest rates to the two factors is summarised in the coefficients b₁(n) and b₂(n) in Box 9.3. Chart 9.3.1 shows the factors' impact on yields for maturities of up to 10 years. The effect of the first factor on the interest rates is by and large the same for all maturities. If the first factor increases by e.g. 1, all interest rates increase by around 1 percentage point. The first factor can be interpreted as a parallel shift effect on the interest-rate level.

Box 9.3 A two-factor model: the Longstaff and Schwartz Model

The second factor's effect on the interest rates decreases with maturity. The short-term interest rate is affected significantly more than e.g. the 10-year interest rate. The second factor thus affects the slope of the yield curve, and can be interpreted as a sloping effect on the yield curve.

The LS model can generate more varied yield curves than the CIR model. While the CIR model can generate only one type of yield curve for a given level of the short-term interest rate, the two-factor model can generate a number of different types of yield curve. Chart 9.3.2 illustrates the various types of yield curve that can be generated at a short-term interest-rate level of around 5 per cent. The short-term interest rate is equal to the sum of the two factors in the model, cf. Box 9.3. This means that when the value of the first factor is e.g. 3 per cent, the value of the second factor is 2 per cent.

A forward-rate model
The second class of model studied is the forward-rate model that can be attributed to Heath, Jarrow and Morton (HJM), 1992.^[3] The focus of this model is on forward rates and the development in the entire forward yield curve. This approach exploits the fact that a direct formulation in time-dependent forward rates provides a suitable method to model interest-rate volatilities along the entire yield curve.

When this model framework is implemented in practice, certain simplifications of the general framework formulated by Heath, Jarrow and Morton are necessary. The actual model chosen was developed by P. Ritchken and L. Sankarasubramanian, and is referred to as HJM-RS.

In the HJM-RS model, the simplification lies in an assumed structure of interest-rate volatility. It is thus possible to describe the movement in the entire yield curve using only two state variables. The two state variables in the model are respectively the spot rate and a volatility factor, cf. Box 9.4. The volatility of the forward rate in the HJM-RS model is assumed to be a function that varies with the remaining maturity of the forward rate and the volatility of the spot rate.

Box 9.4 A Forward-rate model: The HJM-RS model

In contrast to the CIR model, the HJM-RS model takes into account the historical development in interest rates in the simulation process via the drift term in the spot rate process. The volatilities of the spot rates in the models are closely resemblant.

Chart 9.3.3 presents an example of an interest-rate scenario. The development shown contains only increasing yield curves. The HJM-RS model can generate yield curves of different shapes, e.g. inverse and humped yield curves.

Overview of the three interest-rate models
Table 9.3.1 summarises some key characteristics of the three interest-rate models.

Table 9.3.1 Key characteristics of the three interest-rate models

Data for estimation of interest-rate models
In the estimation of the interest-rate models' parameters, zero-coupon yields based on Danish government securities are used. CaR is normally calculated on the basis of interest rates that go back to 1987. On simulations over a long period, e.g. 10 years, it is reasonable to use a long estimation period. This captures possible developments in interest rates over a long period in the parameter estimations. A long estimation period also improves the precision of the parameter estimations.

The period since 1987 contains intervals of highly volatile short-term interest rates, especially in 1992 and 1993, cf. Chart 9.3.4, and there is relatively large variation in the level of interest rates over the period.

The period 1994-2001 has been chosen in this Chapter to illustrate the significance of the choice of estimation period to the CaR results. 1994 is chosen as the commencement year of the estimation period because interest rates in 1994 were at around today's level. Both the level and the fluctuation in interest rates were higher in the preceding period, 1987-1993, than in the chosen estimation period. There are thus signs of a change of level between the periods 1987-1993 and 1994-2001. The estimation of the interest-rate models on the basis of data from the period 1994-2001 will thus imply a lower level of interest rates and lower interest-rate volatility than the estimation based on data since 1987.

9.4 Comparison of interest input

In this section, simple statistical characteristics of simulated quarterly CIR, LS and HJM-RS interest rates are compared with the equivalent characteristics for historical zero-coupon yields from the period 1994-2001, i.e. the interest rates which are the basis for estimation of the three models. The time frame of the interest-rate simulations is 10 years, and 2,500 scenarios of the course of the interest rates are simulated. Comparison is made of the average, the volatility (standard deviation) and the autocorrelation in the simulated and historical interest rates.

The average interest rates determine the expected interest costs in the CaR model. The volatility of the interest rates is essential to the level of CaR, and thereby the interest-rate risk. The autocorrelation in the interest rates reflects their tendency to stay at the same level, i.e. the degree to which the interest rate "tomorrow" depends on the interest rate "today". The greater the autocorrelation, the greater the tendency for the interest rate to stay at the same level. The autocorrelation adds an extra dimension to the comparison of simulated and historical interest rates.

The comparison of historical and simulated interest rates should be seen primarily as an illustration of the significance of the choice of different interest-rate models to the simulation of interest rates in CaR.

Chart 9.4.1 shows the average historical and simulated yield curves. The average of the historical yield curves is higher than for the model-generated interest rates from the CIR and LS models. This is related to the fact that the historical average interest rates with maturities of up to 10 years range from 4.5 to 6.5 per cent, while the starting point for the generation of interest rates in the models is a yield structure with interest rates from 3 to 5 per cent, equivalent to the current level of interest rates. This means that the simulated interest rates in the first periods of the simulation on average lie somewhat below the level of the average historical interest rates. Over time, the simulated interest rates will (on average) move up towards the level of the average historical interest rates.

Of the simulated interest rates, the average interest rates from the HJM-RS model are highest for maturities of up to around 4 years. Hereafter, the interest rates generated from the CIR model are highest. The average interest rates in the LS model are lowest for maturities of up to 10 years. It is thus expected that the average interest costs calculated in the CaR model are lowest when the generated yields from the LS model are used.

The volatility of the interest rates is shown in Chart 9.4.2. The volatility of the historical interest rates increases with maturity, while the volatility of the simulated interest rates from the three models decreases with maturity, as is normally observed empirically. Moreover, the volatility of the simulated interest rates is below the volatility of the historical interest rates for all maturities. The volatility of the short-term interest rate is closest to the volatility of the historical interest rates. Hereafter the distance between the volatility of the historical interest rates and of the simulated interest rates increases with maturity.

The volatility decreases most strongly for the CIR model. The structure of the volatility curve for the CIR model is related to the fact that CIR is a single-factor model in which all interest rates are a fixed function of the short-term interest rate. When the model is estimated on the basis of the volatility of the short-term interest rate, the volatility of the long-term interest rates will be functionally determined within the model.

The volatility of the long-term interest rates, in the LS and HJM-RS models, reflects the volatility of the historical interest rates better. An interest input with a spot-rate volatility at the same level as the CIR model, but higher volatility for the long-term interest rates will, other things being equal, imply higher CaR figures.

While the volatility of the historical interest rates increases with maturity, the volatility of the historical interest-rate changes decreases with maturity. This characteristic is reflected by all of the simulated interest rates, cf. Chart 9.4.3.

Chart 9.4.4 shows the autocorrelation coefficients for interest rates with maturities of up to 15 years. Interest rates generally show high autocorrelation, i.e. the interest rates tend to remain at the same level from one period to the next. The historical interest-rate autocorrelations show that short-term yields are less autocorrelated than long-term yields.

The autocorrelation in the interest-rate simulations of the HJM-RS model are generally closest to the autocorrelation for the historical interest rates. The LS model can generate interest rates where the autocorrelation increases with maturity, as can be observed empirically. The level of autocorrelation is, however, somewhat below the historically observed level, especially for short maturities.

9.5 Comparison of CaR results

On the basis of yield curves generated by respectively the CIR, LS and HJM-RS models, CaR figures for a borrowing scenario with 40-20-40 per cent issues in the 2-, 5- and 10-year segments respectively are calculated.

The mean value of interest costs is shown in Chart 9.5.1. For all of the models, the interest costs decrease with the horizon for the calculations. This can be attributed to the assumed future budget surpluses, and thereby the decreasing debt.

The interest costs calculated on the basis of the three interest-rate models are relatively close to each other. The interest input from the LS model gives the lowest interest costs. This result reflects the fact that the average yield curves for the LS model are lowest for maturities of up to 10 years, i.e. in the segments in which issues take place.

Absolute CaR depends on the interest-rate volatility and the mean value of the interest rates in the simulations. The higher the volatility and average interest rates, the greater absolute CaR will be. Absolute CaR based on the HJM-RS model tends to lie highest, cf. Chart 9.5.2. The average interest rates in the HJM-RS model are higher than for the LS model, while the interest-rate volatility is around the same level for the two models. The interest-rate volatility is higher for the HJM-RS model than for the CIR model, while the average interest rates are at the same level in the two models.

Chart 9.5.3 shows cost distributions in 2006 for respectively CIR, LS and HJM-RS interest input. Flatter and broader curves imply a greater probability of extreme costs, and thereby a greater interest-rate risk. The flatter and broader curves for the cost distribution from the LS and HJM-RS models compared to the CIR model's relatively centred cost distribution reflect the greater interest-rate volatility implied by these two models. The lower average costs for the LS model mean that the distribution is displaced to the left compared to the distributions for the two other models.

9.6 Further work

The work on alternative interest input to CaR is still in its preliminary phase. The empirical results for the alternative interest-rate models to the CIR model are thus provisional, and should be seen primarily as an illustration of the issues involved in modelling interest input to CaR.

In 2002, work will continue on alternative interest-rate models to the CIR model within the model frameworks presented in this Chapter.

CaR calculations are made only for the domestic government debt. The objective is to include the development in all components of the government debt in the CaR calculations. Including the foreign debt in the CaR calculations e.g. entails that interest rates and exchange-rate risk related to the foreign debt must be modelled within the CaR model.

9.A Appendix: A linear two-factor model 

The Longstaff and Schwartz (LS) two-factor model is described below.^[4] First, pricing of financial assets with the stochastic discount factor is described. Formulating interest-rate models within this framework ensures a consistent model free of arbitrage. Then the processes for the two factors in the model are examined. The model can be derived on the basis of these two elements. Finally, estimation and simulation of the interest-rate model are described.

A formulation in discrete time is chosen for the presentation of the linear two-factor model. This formulation may be less elegant, but is also less mathematically and technically demanding compared to the formulation in continuous time that is typically used in the literature.

Pricing of bonds
A key theoretical result in finance is that, in a world without arbitrage, there is a positive stochastic variable, M_t+1, so that for every asset, i, with price P_i, the following pricing equation applies:

where P_i,t is the price at time t ("today"), P_i,t+1 is the price at time t+1 ("tomorrow") and M_t+1 is the stochastic discount factor. E_t is the expectation at time t when all available information "today" is included when the expectation of "tomorrow" is formed. The equation states that the price of an asset "today" is given by the expectation of the discounted future payments. For an n period (zero-coupon) bond at price P_tⁿ , with price P_t+1^n-1 "tomorrow", the following applies:

Box 9.A.1 gives a more detailed interpretation of the key equation for pricing financial assts.

Box 9.A.1 Calculation rules and interpretation of the pricing equation

The process for the factors in the model
The two factors follow a stochastic process equivalent to the process that applies to the short-term interest rate in the CIR model. The processses are formulated in discrete time. A model in discrete time means that the model generates interest rates at discrete intervals, e.g.1 month or 1 quarter. Box 9.A.2 describes the relation between a discrete and a continuous formulation.

Box 9.A.2 Continuous and discrete time formulation

The two stochastic factors that drive the interest rates in the model, denoted z₁ and z₂, can be described by first-order autoregressive processes using the following formula:

where the parameters j, q and s are constant, subject to the following restrictions: 0<j<1 and s, q>0. e is the stochastic shock to the process which is independently, normally distributed over time with mean value 0 and variance 1. By calculating expressions of the mean value, variance and autocorrelation for the processes, cf. Table 9.A.1, the individual parameters in the processes can be interpreted.

Table 9.A.1 Characteristics of the processes for the factors

The mean value (unconditional expectation) of the factor q can be interpreted as the long-term or equilibrium level which the factor will tend to move towards over time.

1-j measures the rate by which the distance between the current level of the factor and the long-term level is reduced, i.e. the strength of "mean reversion". If the factor is e.g. 1 percentage point above the long-term level, and j = 0.9, so that 1-j = 0.1, in the next period the factor will tend to move 0.1 percentage point towards the long-term level, making the distance 0.9 percentage point. If no stochasticity had been involved in the process, the factor would make this movement with certainty.

On the other hand, the first-order autocorrelation coefficient j indicates the tendency for the factor to have the same value tomorrow as today. The closer j is to 1, the greater the tendency for the factor "tomorrow" to have the same value as the factor "today". j is thus a measure of the sluggishness of the process.

The interpretation of s is related to the volatility of the process. The greater the value of s, the higher the volatility measured as e.g. the variance of the factors.

The square root in the stochastic term of the process for the factors means that q (mean value of z) is included in the expression of the unconditional variance of z. In the same way, the conditional variance, z_ts², depends on the factor's current level. So when the interest rate is close to 0, stochastic shocks will lead to minor changes in the interest rate, and the probability of negative factor values is very small.

j is also part of the determination of the unconditional variance. The greater the value of j, the greater the variance. The intuition is that the greater the value of j, the smaller the tendency for the factor to remain around a given long-term level. An alternative perception is that the factor approaches a "random walk" when j is close to 1 (the greater the value of j).

In the two-factor model, both factors follow processes of the type described in this section. If there had been only one factor, this factor would be the short-term interest rate, and the model would correspond to the CIR model in discrete time.

The Longstaff-Schwartz model in discrete time
By connecting the above individual elements, the overall two-factor model can be described fully, based on the following equations:

The pricing equation and the processes for the two factors are described above. The process for the stochastic discount factor is the new element.^[5] It is assumed that e₁ and e₂ are independently distributed. Assumptions and restrictions for the parameters and the stochastic shocks correspond to the above section. An extra type of parameter is introduced in the process for the stochastic discount factor, l₁, called the price of risk, cf. the description below.

On the basis of this equation system, the interest rate at a given maturity can be determined as a linear function of the two factors. The solution principle for the model is outlined briefly in Box 9.A.3. The continuously compounded interest with maturity n at time t is denoted as y_tⁿ, and the following linear relation applies:

 and

where for n=0,1 the following applies:

Box 9.A.3 Derivation of expressions of interest rates

In this model, the short-term interest rate is given by the sum of the two factors, and the conditional variance of the short-term interest rate is a linear combination of the two factors:

The CIR model can be obtained as a special case of the model by setting q₂, j₂ and s₂ to zero.

l₁, also called the price of the interest-rate risk, can be interpreted on the basis of the equation in Box 9.A.1., which determines the expected additional yield on long-term bonds compared to a single-period bond:

where r_t+1ⁿ and r_t denote the continuously compounded, single-period yield to respectively an n-period and a single-period bond. Inserting the two-factor model's expression of the yield gives:

The last term is an adjustment arising because logarithmic yields and interest rates are used, cf. Box 9.A.1. The first term is the interesting one in economic terms. It expresses the risk premium achieved from placement in bonds at long maturity in a single period compared to investment in single-period bonds with a certain yield over a period. The risk premium is proportional to l₁, which must be negative to achieve a positive risk premium. This will be the case in a world with risk-averse investors. l₁ is called the price of the risk, because a greater (numerical) value of l₁ entails higher average additional returns on long-term (high-risk) bonds compared to safe investment in a single-period bond. The slope of the yield curve thus tends to increase with (the numeric value of) l₁.

The risk premium is only dependent on the first factor, so that only one price is connected to the volatility originating from the first factor. This assumption originates from Longstaff and Schwartz.

Estimation of interest-rate models
The parameters in the model are estimated using the Generalised Method of Moments (GMM), cf. Box 9.A.4.

Box 9.A.4 GMM estimation

The parameters are estimated on the basis of monthly zero-coupon interest rates for government securities accrued continuously since 1994. Quarterly data are used in the CaR model, but estimation based on monthly interest rates provides more information on the interest process than if estimation is based on quarterly interest rates. The estimation assumes that the long-term level of the short-term interest rate (1-month interest rate) is given by the average of the short-term interest rate over the estimation period. This gives an anchor for the interest-rate level in the estimation. It is also assumed for simplification purposes that q₁=q₂. The other parameters are estimated on the basis of the spread to the 1-month interest rate together with the variance in interest rates and interest-rate spreads for selected maturities. Autocorrelation of the 1-month interest rate is also included in the estimation.^[6]

The estimation gives the following parameter estimates: q₁=q₂=0.00183, j₁=0.987, j₂=0.748, s₁=0.00467, s₂=0.00839 and l₁=-2.958. Annualised in per cent, the value of q₁ and q₂ is around 2.2 per cent, implying that the long-term level q₁+q₂ of the short-term interest rate (the 1-month interest rate) is around 4.4 per cent, equivalent to the average 1-month interest rate for the estimation period. The price of risk, l₁, has the expected negative sign.

Simulation of the model
On the basis of the parameters estimated for monthly data, monthly interest rates are simulated using the equation for the linear relation between interest rates and factors, and the processes for the two factors. Quarterly yield curves are extracted from the monthly interest-rate scenarios used in the CaR calculations.

9.B Appendix: Forward-rate models

This Appendix presents forward-rate models. First, the basic concept for modelling of forward rates proposed by Heath, Jarrow and Morton (HJM) in 1992^[7] is described. This is followed by a description of the simplification by Ritchken and Sankarasubramanian (RS)^[8] of the HJM model to make it implementable. Finally, the generation of interest rates in practice is described.

Heath-Jarrow-Morton
In 1992, Heath, Jarrow and Morton proposed the modelling of forward rates rather than the traditional modelling of spot rates. Within the HJM framework movements in the entire forward yield curve are modelled simultaneously. The direct formulation in forward rates makes it possible to specify the volatility of the interest rates more flexibly than in traditional interest-rate models.

The forward rate is defined in the model as an interest rate applying at a future point in time. Under HJM, the movement in the forward rate is modelled using the following diffusion process:

where f(t,T) is the forward rate observed at time t, applying at time T where T³t. In other words f(t,T) is the spot rate at time T observed at time t. m_f(t,T) and s_f(t,T) are respectively the drift and volatility of the forward rate, and dW(t) is a normally distributed Wiener increment, cf. Box 9.2. Drift indicates the movement in the forward rate if there had been no stochasticity in the process.

Once the forward rate is specified, expressions of spot rate, zero-coupon prices, etc., can be derived.

To ensure an arbitrage-free market under the model, the following condition between drift and volatility, known as the HJM drift condition, must hold in a risk-neutral market, cf. Box 9.B.1:

Box 9.B.1 The HJM drift condition

where drift and volatility can be values, parameters or functions. The HJM drift condition shows that volatility is the key element in the model. Once volatility is known, drift can be calculated, and thereafter the change in the forward rate can be determined.

In its general form the HJM model requires specification of volatilities for all forward rates along the yield curve (infinite in continuous time), in order to simulate the entire yield curve. On practical implementation of the model, the relationship between volatilities for various maturities is simplified.

The HJM-RS model
The P. Ritchken and L. Sankarasubramanian model (HJM-RS model) from 1995 is a simplification of the HJM model. In the HJM-RS model volatility is assumed to have a functional structure, which means that the model can describe movements throughout the yield curve with only two state variables, in contrast to the general HJM's infinite state variables.

It is assumed that the volatility of the forward rate is a function that depends on the remaining maturity for the forward rate and the volatility of the spot rate, i.e. the following volatility restriction:

where s(r(t))^g is the volatility of the spot rate. s, k and g are fixed parameters and T-t is the remaining maturity for the forward rate f(t,T). Given this function, the drift of the forward rate can be calculated using the HJM drift condition, and the diffusion equation for the forward rate can be derived. Then expressions of the spot rate and zero-coupon prices can be derived.

A spot rate observed at time t corresponds to a forward rate observed at time t, applying at time t, i.e. r(t)=f(t,t). With the RS volatility restriction, the spot rate follows the process:

where f(t) is the integrated volatility factor, cf. below. k, s and gare parameters from the volatility restriction. For g=½ the spot-rate volatility is the same as in the CIR model, while the drift term of the spot rate deviates from the CIR model since it contains information concerning the development in the interest rate up to time t. The integrated volatility factor has the following expression:

The development in the entire yield curve can be described using the spot rate and the integrated volatility factor in the HJM-RS model. An implementable interest-rate model is thus obtained.

With the parameters from the RS volatility restriction, zero-coupon prices can be calculated by the following expression:

where t and T indicate the time index for the required zero-coupon price. P(0,t) and P(0,T) are known zero-coupon prices observed at time 0. This expression is derived on the basis of the zero-coupon formula described in Box 9.B.2.

Box 9.B.2 Calculation of zero-coupon prices and yields

Estimation of the model's parameters
The parameters s, k and g in the HJM-RS model used in the simulation of interest rates are estimated on the basis of historically observed interest data from 1994 to 2001. The Generalised Method of Moments (GMM), cf. Box 9.A.4, is used for the estimation.

The GMM estimation of the parameters is based on the formula for zero-coupon prices of the HJM-RS model. Historically observed zero-coupon prices are inserted as input to this formula. The parameters are estimated on the basis of monthly zero-coupon yields for government securities for the period 1994-2001. The estimation gives s=0.02861, k=0.08889 and g=0.4077.

Generating yield curves in the HJM-RS model
With the above equations for the spot rate r(t), the integrated volatility factor f(t), and zero-coupon prices P(t,T), future yield curves can be generated. On generating interest rates in discrete time (e.g. quarterly interest rates) the equations must be re-written.

In discrete time, the required generation period can be divided into a number of equal-length intervals, e.g. n intervals. The interval length is Dt, and can e.g. be a month or a quarter.

The zero-coupon price in discrete time is denoted as P(i,j), where i is the observation time index, and j is the time index for expiry of the bond.

In the same way, the spot rate and the integrated volatility factor are denoted by r(i) and f(i) for i=0, 1, …, n in the discrete version of the model. The change in the spot rate Dr(i) over the time interval Dt can be simulated by the following equation^[9]:

where Z(i) is a normally distributed variable with mean value 0 and variance Dt. The spot rate in the following period is thus calculated as the spot rate in the current period plus the change in the spot rate, r(i+1)=r(i)+Dr(i+1).

The integrated volatility factor has the following expression in discrete time:

Zero-coupon prices can be re-written to:

Future yield curves can be simulated from the above three equations in discrete time. Input to the simulation is the current term structure and the estimated parameters k, s and g from the HJM-RS model, cf. Box 9.B.3.

Box 9.B.3 Simulation of interest input for CaR

Footnotes

^[1] Cox, J. C., Ingersoll, J. E. and Ross, S. A., 1985, A Theory of the Term Structure of Interest Rates, Econometrica, vol. 53, no. 2, pp. 385-407.

^[2] See Appendix 9.A for a description of the model.

^[3] See Appendix 9.B for a description of the model.

^[4] The description is based on Longstaff, F. and E. Schwartz, 1992, Interest Rate Volatility and the Term Structure: A two-factor General Equilibrium Model, Journal of Finance, Vol. 47, pp. 1259-1282. Longstaff and Schwartz formulate the model in continuous time. The discrete formulation in this Chapter is based on Campbell, J.Y., A.W. Lo, and A.C. MacKinlay, 1997, The Econometrics of Financial Markets, Chapter 11, Princeton University Press, Princeton, NJ and Backus, D., S. Foresi and C. Telmer, 1998, Discrete-Time Models of Bond Pricing, Working Paper Stern School of Business, New York University.

^[5]  For a more detailed interpretation of the process for the stochastic discount factor see the background articles to this Chapter.

^[6] The method is inspired by Backus, D., S. Foresi, A. Mozumdar and L. Wu, 2001, Predictable changes in yields and forward rates, Journal of Financial Economics, vol. 59, pp. 281-311.

^[7] Heath, D., Jarrow, R. & Morton, A. (1992), Bond Pricing and the Term Structure of Interest Rates, Econometrica 60:1, pp. 77-105

^[8] P. Ritcken and L. Sankarasubramanian, Volatility Structure of Forward Rates and the Dynamics of the Term Structure, Mathematical Finance, Vol. 5, No. 1 (January 1995), pp. 55-72.

^[9] The method follows Ritchken and Sankarasubramanian (1995). In Ramaswamy, S., 1997, Global Asset Allocation in Fixed Income Markets, Working Paper No. 46, BIS, there is an example of use of the model to simulate interest rates over a horizon of 2 years in order to determine the composition of a bond portfolio.

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Version 1.0 March 2002 Nationalbanken.
Published by Danmarks Nationalbank March 2002, http://www.nationalbanken.dk/