CHAPTER 9
Cost-at-Risk for the Domestic Debt
9.1 SUMMARY
Cost-at-Risk (CaR) is a supplementary measure used in the management of the interest-rate risk on the domestic central-government debt. CaR quantifies the risk on the debt and gives important input to the weighing of interest-rate risk against costs.
A distinction is made between absolute and relative CaR. Absolute CaR for a given year indicates the maximum costs with a probability of 95 per cent.
Relative CaR is the difference between absolute CaR and the average interest costs. Relative CaR thus indicates the maximum increase in the costs for a given year, with a probability of 95 per cent.
CaR is considered as a supplement to duration and the target for the shape of the redemption profile. CaR is used in risk management to e.g. assess the consequences of various issuing strategies for the risk on the debt.
9.2 BACKGROUND
The work on developing and incorporating CaR in the management of the domestic debt was initiated in 1997, cf. Chapter 7 of "Statens låntagning og gæld 1997" (available in English). The work on CaR reflects that a central element of government debt policy is to arrive at a suitable weighing of costs against risk, when the borrowing strategy is determined. The CaR definitions applied are set out in Box 9.1.
Simultaneously with this work a number of countries have initiated similar projects. In some countries the central government's total assets and liabilities are considered, rather than focusing exclusively on the debt. In Denmark, solely the risk and costs related to the government debt are considered.
Chart 9.2.1 presents an example of the distribution of the costs on a hypothetical debt portfolio. In the example, a normal distribution of costs is assumed around a mean value of DKK 5 billion, with a standard deviation of DKK 1 billion.
Box 9.1 CaR DEFINITIONS
Absolute CaR for a given year indicates the maximum costs with a probability of 95 per cent. Relative CaR is the difference between absolute CaR and the average interest costs. Relative CaR thereby indicates the maximum increase in costs for a given year, with a probability of 95 per cent. The starting point may also be other percentiles than 95 per cent, e.g. the 99 per cent percentiles, when considering more extreme situations.
Methodologically CaR is related to Value-at-Risk (VaR), which expresses the maximum decline in a portfolio's market value with a given probability over a given, typically relatively short, period. For both VaR and CaR the calculations to a high degree depend on the model used and the assumptions made.
The marked part of the right-hand "tail" in the distribution indicates the size of the costs in the 5 per cent of cases where the costs are highest. In this case it is thus found that with a probability of 95 per cent the costs will not exceed DKK 6.7 billion, equivalent to an absolute CaR of DKK 6.7 billion. With a mean value of DKK 5 billion, relative CaR in this case is DKK 6.7 less DKK 5 billion = DKK 1.7 billion.
In the CaR calculations risk is defined as the risk of an increase in the annual cost of the central-government debt. Focus is on the nominal costs, and the risk is assessed in relation to the central-government budget. It must be emphasised that the overall weighing of costs against risk takes due account of the other objectives and considerations of government debt policy.
Chart 9.2.1 ABSOLUTE AND RELATIVE CaR, DKK BILLION
The choice of risk measure reflects that what is perceived as the relevant risk is the risk that the central government – in a year with a high level of interest rates – will have to refinance a large proportion of the debt, which will result in rising costs and deterioration in the central-government budget. As described in Chapter 1, the aim is to limit the interest-rate and refinancing risk by spreading borrowing and thereby the debt across maturities. Even if the debt is spread across maturities the impact on the central-government budget of a rising level of interest rates can be considerable.
The central government is exposed to this risk because the interest rates for future borrowing are unknown. It follows that the future costs of the government debt are also unknown. In practice, the focus will usually be on rising costs and their impact on the central-government budget.
CaR is used as a supplement to duration and the shape of the redemption profile. The primary difference between CaR and the other measures is that, since the risk is quantified, CaR makes it possible to weigh costs against risk.
9.3 METHOD
Calculation of CaR is based on the future costs of the existing debt. On the basis of scenarios for the future interest rates and borrowing strategies possible future cost profiles related to the domestic debt are calculated. On the basis of a number of scenarios for future costs a probability distribution of the costs is found.
The input used can be divided into three categories. The first is the future interest rates. A large number of future interest-rate scenarios are chosen. The large number of interest-rate scenarios makes it possible to set out probability distributions.
The second category concerns the existing government debt. These inputs are a full description of the accrued future costs on the existing debt and the future payments on the debt.
The third category is assumptions concerning the government budget before interest and redemptions on the domestic debt, and the future distribution of borrowing on maturities.
Box 9.2 presents the method of calculating the borrowing requirement.
On the basis of this input the annual costs for the selected period related to the chosen strategy are calculated for each of the interest-rate scenarios. Then the mean value, and absolute and relative CaR are calculated for each year of the period considered. The structure of the procedure is presented in Chart 9.3.1.
Chart 9.3.1 STRUCTURE OF CaR CALCULATIONS
By nature CaR is a concept which is particularly applicable to long-term strategic considerations. On the other hand, there is a natural requirement to consider the more short-term effect of a given borrowing strategy. The horizon for the calculations is therefore set at 5 years. It is assumed in the calculations that borrowing takes place 4 times a year, once every quarter.
Box 9.2 BORROWING REQUIREMENT AND COST CONCEPTS
The calculations are based on the central-government budget balance before interest and redemptions on the domestic debt. After deducting the interest expenditure on the domestic debt from this figure, the net borrowing requirement is found. An increase in the level of interest rates and thereby in interest expenditure will increase the net borrowing requirement, leading to higher borrowing and thereby higher interest expenditure. On the other hand, a lower level of interest rates implies a decreasing net borrowing requirement, lower borrowing and thereby lower interest expenditure. Changes in the level of interest rates thus have an immediate impact on the net borrowing requirement and thereby on borrowing, which subsequently affects the interest costs in the following years.
In the CaR calculations accrued costs are applied, equivalent to the concept of costs used in the central-government accounts. When calculating the borrowing requirement the actual payments are used.
Box 9.3 CREATION OF TERM STRUCTURES
The model used for simulation of the interest-rate input for the CaR calculations is based on 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, p. 385-407. The stochastic element in the model is the spot interest rate. Changes in the spot interest rate are described using the following stochastic process:
where r(t) is the spot interest rate at time t, is the equilibrium value for the spot interest rate, is the volatility of the spot interest rate, is the speed at which the spot interest rate moves back to the equilibrium value , and W(t) is a stochastic process – a "Wiener process". If the development in the spot interest rate is described by this stochastic process, it can be shown that no negative spot interest rates are possible in the simulation and that the variation in the spot interest rate is greater, the higher the spot interest rate. In the calculations made =0.072, =0.167 and =0.099. The values for , and are based on estimation using quarterly observations of the spot interest rate from estimated zero-coupon interest rates. The method applied is presented in Overbeck, L., and Rydén, R., 1997, Estimation in the Cox-Ingersoll-Ross Model, Econometric Theory, Vol. 13, p. 430-461.
In the model the spot interest rate tomorrow depends on the spot interest rate today. Generally, if the spot interest rate is above the long-term level , this will draw towards a lower interest rate, and vice versa if the spot interest rate is below (mean reversion).
The yield to maturity for a zero-coupon bond with a given time to maturity is calculated on the basis of the value of the spot interest rate. The calculation is among other things based on an assumption concerning the investors' degree of risk aversion.
An interest-rate scenario thereby consists of 20 yield curves, 4 for each year. One interest-rate scenario results in one profile of future costs. 2,500 future cost profiles are calculated, with one cost profile for each interest-rate scenario. A term structure model is used to generate the future term structures. It is described in Box 9.3.
The output from the calculations is thus 2,500 future cost profiles for the domestic government debt. On the basis of these cost profiles the mean value, and absolute and relative CaR, are calculated.
Characteristics of the applied interest rates
Chart 9.3.2 presents the average of the simulated yield curves and the average of the actual yield curves for the period 1987-99. The simulated interest rates are lower than the average
of the actual interest rates for 1987-99. It is also seen that the level of the simulated interest rates corresponds to the average level of the actual interest rates for the period 1992-99, and
that the slope of the curves is by and large the same.
Chart 9.3.2 AVERAGE ACTUAL AND SIMULATED YIELD CURVES
The simulation is based on the spot interest rate at the beginning of January 2000. Chart 9.3.2 shows a rising yield curve, i.e. the yield for short maturities is lower than the yield for longer maturities. This indicates that on average it is less expensive to borrow at a short maturity. On the other hand, however, the fluctuations in the short-term interest rates are normally greater than for longer maturities.
Chart 9.3.3 QUARTERLY VOLATILITY CONVERTED TO ANNUAL RATES, 1987-99
The fluctuations in interest rates, also called the volatility, can be measured by the standard deviation over time for the various maturities. Chart 9.3.3 shows that the volatility of the actual interest rates is higher than the volatility of the simulated interest rates for longer maturities. This probably means that the calculations shown slightly overestimate the difference in the various borrowing strategies. The historical observations thus show that borrowing in short-term securities on average takes place at lower, but more volatile, interest rates, while borrowing in long-term securities on average takes place at higher, but less volatile interest rates.
9.4 CAR FOR SELECTED BORROWING STRATEGIES
The following reviews a number of CaR calculations for various borrowing strategies. Calculations are made for 5 different borrowing strategies. Strategies are considered where borrowing is in various combinations of 2-, 5- and 10-year fixed-rate bullet loans, equivalent to the strategy applied today. Two of the scenarios whereby borrowing in respectively 100 per cent 10-year and 100 per cent 2-year securities is assumed are included to show the entire scope. The comments on the results below focus primarily on the three other scenarios.
In all scenarios a constant outstanding amount of Treasury bills of around DKK 45 billion is assumed, and all strategies are evaluated using the same simulated interest rates and the same assumptions regarding the balance of the central-government budget before interest and redemptions on the domestic debt during the period.
In all strategies the same distribution of borrowing is assumed across maturity segments during the period considered. Table 9.4.1 presents the distribution of borrowing in the various strategies.
For all scenarios it is assumed that the budget surplus before interest and redemptions on the domestic debt is DKK 60 billion in 2000 and DKK 45 billion in the following years.
Table 9.4.1 DISTRIBUTION OF BORROWING IN SCENARIOS
| Per cent | 10-year | 5-year | 2-year |
| Basic scenario | 40 | 20 | 40 |
| Short-term | 20 | 35 | 45 |
| Long-term | 45 | 35 | 20 |
| 100% 10-year | 100 | 0 | 0 |
| 100% 2-year | 0 | 0 | 100 |
Chart 9.4.1 MEAN VALUE OF COSTS
For comparison, the budget surplus before interest and redemptions on the domestic debt is expected to be almost DKK 55 billion in 1999, while in the last 4 years it has been between DKK 35 and DKK 75 billion. Section 9.6 presents the consequences of altering these budgetary assumptions. No further use of interest-rate swaps or buy-backs is included in the scenarios.
Chart 9.4.2 ABSOLUTE CaR
Chart 9.4.3 RELATIVE CaR
The duration for the three scenarios in the longer term is around 3.75 years for the basic scenario, 3.25 years for the scenario with short-term borrowing, and 4 years for the scenario with long-term borrowing.
Chart 9.4.1 shows that, with the chosen assumptions, it is reasonable, by and large, to expect unchanged costs of almost DKK 45 billion per year up to 2004. It is also shown that the differences between the expected costs for the three scenarios increase over time.
It is important to be aware that both the applied interest-rate input, the structure of borrowing and the assumptions concerning the central government's budget have an impact on the mean value of the costs.
Chart 9.4.2 shows that for all strategies absolute CaR is rising from just over DKK 46 billion in 2000 to almost DKK 57 billion in 2004. The absolute CaR shows that the increase in relative CaR dominates the decrease in the expected costs for all strategies. Chart 9.4.3 presents relative CaR. The detailed results for the basic scenario are shown in Table 9.4.2.
Table 9.4.2 CaR VALUES FOR BASIC SCENARIO
| DKK billion | 2000 | 2001 | 2002 | 2003 | 2004 |
| Mean value | 44.9 | 42.9 | 44.0 | 42.0 | 44.6 |
| Absolute CaR | 46.2 | 46.3 | 49.0 | 48.3 | 52.7 |
| Relative CaR | 1.3 | 3.4 | 5.0 | 6.3 | 8.1 |
Chart 9.4.4 RELATIVE CaR AND MEAN VALUE IN 2004, DKK BILLION
Assessment of results
The above results show clear differences in the values of both absolute and relative CaR for the three strategies. It is also seen that the risk of very high costs is small for all strategies.
One underlying factor is that as a consequence of the conducted policy to equalise the redemption profile, the debt is spread across maturity segments. All else being equal, this entails a
lower risk.
The differences in the expected costs for the three scenarios are modest in the first couple of years, but increase during the period. On the basis of the calculations made it must be assessed that for realistic borrowing strategies one can expect unchanged nominal costs of around DKK 45 billion in the period up to 2004.
On consideration of the weighing of costs against risk it is seen that a mean value which is DKK 1 billion lower on average during the period considered leads to an increase in relative CaR of around DKK 1.4 billion. Chart 9.4.4 presents the relation between relative CaR and the mean value for 2004.
9.5 STRESS TEST
To investigate the robustness of the results it is important to conduct a "stress test" based on extreme interest-rate input. Table 9.5.1 presents the costs of the aforementioned basic scenario with this type of interest-rate input.
Table 9.5.1 MEAN VALUEs FOR BASIC SCENARIO WITH VARYING INTEREST-RATE INPUT
| DKK billion | 2000 | 2001 | 2002 | 2003 | 2004 |
| Simulated interest rates | 44.9 | 42.9 | 44.0 | 42.0 | 44.6 |
| 1988-93 | 46.8 | 48.2 | 52.1 | 51.7 | 56.4 |
| 1990-94 | 47.9 | 49.1 | 52.7 | 51.7 | 51.5 |
| 1995-99 | 45.8 | 42.6 | 42.3 | 38.3 | 38.0 |
To illustrate the sensitivity of the results, the future costs are calculated on the basis of historical interest-rate scenarios.
The course of interest rates for the period 1988-93 includes a currency crisis at the end of the period. Using the development in interest rates for this period as input, an impression is gained of the impact on borrowing costs of high interest rates in a given year, which is also reflected in significantly rising costs from 2003 to 2004, cf. Table 9.5.1.
The period 1990-94 was characterised by a general decrease in interest rates and a currency crisis. Using the development in interest rates for this period as input, a test is made of the impact on costs of a period with declining interest rates and general interest-rate unrest. When using this interest-rate input the calculated costs are at the same level as for the calculations where the interest rates for 1988-93 are used, although there is no marked increase from 2003 to 2004.
In 1995-99 interest rates were stable and showed a downward trend. Using this interest-rate development as input, an impression is gained of what the level of costs would be, given historically very low interest rates. It is – not surprisingly - seen that considerably lower costs are obtained with this interest-rate input in comparison to the other calculations.
9.6 BUDGET SENSITIVITY
Below, the sensitivity of the CaR calculations to changes in the size of the central government's budget balance is considered. Two different budget situations are compared, with a strategy for borrowing which is equivalent to the aforementioned basic scenario, i.e. a strategy whereby 40 per cent is borrowed in the 10-year segment, 20 per cent in the 5-year segment, and 40 per cent in the 2-year segment, while the outstanding amount of Treasury bills is kept constant at DKK 45 billion.
For the first budget situation a budget surplus before interest and redemptions on the domestic debt of DKK 60 billion is assumed in 2000, and thereafter DKK 45 billion for all subsequent years. This means that the same results are obtained as for the basic scenario described above.
Chart 9.6.1 MEAN VALUE
For the second budget situation a budget surplus of DKK 60 billion in 2000 is assumed, and thereafter DKK 0 billion. This budget situation generally corresponds to a situation with a deficit of 3 per cent of GDP on general-government finances, if the budget of local government and social security funds is at equilibrium. This corresponds to just fulfilling the EU Treaty's requirements of the general-government budget deficit. As stated above, the equivalent historical figures for the last 4 years have been in the range of DKK 35-75 billion, which clearly shows that the assumption of a value of 0 is a significant deviation from the level in recent years.
Chart 9.6.2 ABSOLUTE CaR
Chart 9.6.3 RELATIVE CaR CaR
Chart 9.6.1 presents the development in costs for the two budget situations. It is seen that the results show significant differences. Chart 9.6.2 presents the development in absolute CaR. In the scenario with a low budget surplus absolute CaR increases to DKK 70 billion in 2004.
Chart 9.6.3 presents the development in relative CaR. The difference in relative CaR increases in step with the impact over time of the differences in the amount of borrowing.
9.7 APPLICATION OF CaR
The work in recent years on developing calculation methods and introducing CaR as a measure for the management of the interest-rate risk on the domestic debt has yielded a large body of experience.
The general assessment is that CaR is a valuable supplement to duration and redemption profile. In particular the CaR calculations have contributed to valuable discussions with regard to the choice of borrowing strategy, where focus has been on weighing costs against risk.
Experience has shown that this type of calculation is particularly sensitive to the interest-rate input used. Moreover, the interest-rate input chosen implies implicit selection of expectations of the future development in interest rates. Therefore neither absolute nor relative CaR are objective risk measures.
The calculations performed show that a decrease in mean value by DKK 1 billion gives an increase in relative CaR in the range of DKK 1.0-2.0 billion. It is also the impression that absolute CaR for the domestic debt is relatively low, since for a number of years emphasis has been on a smooth redemption profile in view of the refinancing risk.
CaR statistics are reported at quarterly meetings of the Ministry of Finance, the Ministry of Economic Affairs and Danmarks Nationalbank.
At present the CaR calculations solely comprise the domestic debt of the central government. The long-term objective is to expand the CaR model to comprise the total central-government debt. Moreover, work will be continued on developing different types of interest-rate input as a supplement to that used today.
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Version 1.0 March 2000 Nationalbanken.
Published by Danmarks Nationalbank March 2000, http://www.nationalbanken.dk