Interest-Rate Risk Management – New Initiatives
11.1 Summary
The central-government debt is managed via strategic benchmarks for the distribution of borrowing by maturities and for the interest-rate exposure of the portfolio. The strategic benchmarks are determined on the basis of a long-term analysis of costs and risk. During the year the strategic benchmarks may be deviated from within fixed limits.
In 2003, a number of new initiatives were introduced within this framework. The foreign portfolio has been incorporated in the Cost-at-Risk (CaR) model, which is used to quantify interest costs and risk. The model now comprises the domestic and foreign debt and the swap portfolios.
The key strategic benchmark for the interest-rate exposure is the duration target for the central-government debt. Interest-rate fixing has been introduced as a supplementary exposure measure. The interest-rate fixing, calculated at a given time, is the amount in the debt portfolio for which a new interest rate is to be fixed in the following year. In addition to redemptions on the debt due within one year, the measure includes floating-rate debt and the interest-rate-swap portfolio for which a new rate of interest is to be fixed within one year. The interest-rate-swap portfolio now contributes significantly to the exposure. In order to relate the interest-rate exposure to the macroeconomic development, the interest-rate fixing as a ratio of GDP is applied as a measure of the central government's "real" exposure.
In addition, a new risk measure, conditional CaR, has been introduced. Conditional CaR quantifies the risk in the individual years seen over the same time horizon – e.g. the risk in 2006 seen from the beginning of 2005, and the risk in 2007 seen from the beginning of 2006. In other words, risk in a given future year is conditional on the development in the period up to that year.
Finally, a new estimation method has been introduced for the model used for interest-rate simulation in the CaR model, where model interest rates are fitted to forecasts of the future development in interest rates.
11.2 The framework for management of interest-rate risk on central-government debt
The overall objective of the government debt policy is to cover the central government's financing requirement at the lowest possible long-term borrowing costs, subject to a prudent degree of risk. This objective implies weighing costs against risk in the strategic planning. To support the choice of a suitable strategy, an analysis is thus performed of the trade-off between costs and risk.
A strategy is formulated as strategic benchmarks for borrowing and the interest-rate exposure of the portfolio. The strategic benchmarks are used as guiding points in the subsequent implementation of the strategy.
Issuance strategy vs. risk profile
The issuance strategy for government securities is determined on the basis of an objective to ensure liquid series in the internationally most important maturity segments. A liquidity premium is thereby achieved, which contributes to reducing the central government's borrowing costs. Building-up of the individual bond series to a sufficient amount reduces the scope for shifting borrowing from one maturity segment to another. The possibilities of influencing the risk profile via issuance are therefore limited.
The risk profile of the portfolio can, however, be managed independently of the issuance strategy by means of interest-rate swaps and buy-backs. Interest-rate swaps are used to shift the interest-rate exposure from one maturity segment to another, while buy-backs are used to smoothen the redemption profile and thus the size of the interest-rate exposure from year to year.
Exposure vs. quantification
The risk profile for a given strategy is characterised by the portfolio's exposure measures, which comprise the duration and the size of the amount in the portfolio for which a new rate of interest is to be fixed from year to year.
The choice of a given strategy is made on the basis of quantification of costs and risk for different levels of the exposure measures. Quantification implies that the strategies are compared in terms of their consequences for the probability distribution of interest costs in the individual years over a long horizon. The future development in interest rates and budget are the central risk elements. Box 11.1 outlines the use of interest-rate simulations.
| Interest-rate simulations | Box 11.1 | ||
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The CaR model's quantification of risk on the central government's interest costs is based on 2,500 simulated scenarios for the future development in interest rates using the CIR1 interest-rate model. Two approaches are taken to estimation of the interest-rate model. One is based on the historical development in interest rates. This method is now supplemented with estimations based on the interest-rate forecasts from the Ministry of Finance. The purpose of estimating the CIR model on the basis of the historical development in interest rates is to recreate the empirical characteristics of the historical interest rates in the simulated interest rates, including interest-rate volatility and level. The choice of the historical estimation basis has a major impact on the simulated interest rates, so different historical periods are applied. The model is also fitted to the forecasts of future interest rates from the Ministry of Finance. This means that the simulated average interest rates are in accordance with the interest-rate forecasts from the Ministry of Finance.
where r(t) is the spot rate at the time t, θ is the equilibrium value for the spot rate, σ scales the volatility of the spot rate, and κ is the speed at which the spot rate returns to the equilibrium value θ. W(t) is a stochastic process – a so-called Wiener process where the increase, dW(t), has a mean of 0 and a variance of dt. In addition to these parameters, the model includes the parameter λ, which is called the price of risk and determines the average slope of the yield curve. Given these parameters, the zero-coupon rate for all maturities is given as a linear function of the spot rate.
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Management according to strategic benchmarks
The analysis forms the basis for determining strategic benchmarks for the distribution of borrowing on various maturities, the volume of buy-backs and the portfolio duration over a long horizon. For management purposes the strategic benchmarks are, however, determined for one year at a time, after which an annual update of the risk analysis may lead to a revision of the strategy.
The strategic benchmarks are used as guiding points in the ongoing portfolio management. For 2004 the strategic benchmarks comprise the following:
- Financing of the domestic borrowing requirement via a 40-20-40 per cent distribution on 2-, 5- and 10-year bullet loans, respectively. The 2- and 5-year on-the-run issues are ultimately built up to at least DKK 35 billion, while the 10-year securities are ultimately built up to at least DKK 60 billion.
- Financing of the foreign borrowing requirement via a 5-year euro bullet loan.
- A zero net-financing contribution from the Treasury bill programme.
- A duration target (calculated at a fixed discount rate) within 3 years +/- 0.25 years.
- Guiding points for buy-backs of domestic securities maturing after 2004.
Management according to the strategic benchmarks ensures that the ongoing transactions keep the cost and risk profile of the portfolio on track – also in the longer term. The ex-ante objective is to disperse borrowing and swap transactions over the year to smoothen exposure to day-to-day volatility.
The strategic benchmarks are not used as automatic pilots. The development in interest rates over the year may justify a change in the exposure of the portfolio, and likewise shifts may occur in the relative attractiveness of the various instruments. To enable a response to a given development, a limited scope around the strategic benchmarks is determined, which may be used in the ongoing management. The duration may e.g. be changed within a fixed band, and issues in the various maturities may be larger or smaller within fixed limits. Using this scope does not reflect short-term tactical positioning or an attempt to "beat" the market by exploiting diverging expectations regarding future market developments.
11.3 Exposure and risk measures
Exposure measures
The duration of the portfolio is used as a strategic benchmark for the interest-rate exposure. In addition, the interest-rate fixing, i.e. the amount in the debt portfolio for which a new interest rate is to be fixed in the following year is compiled.
Duration is a measure of the average fixed-interest period for the portfolio. It is the central exposure target since it contains information on the trade-off between costs and risk. All other things being equal, shorter duration will reduce the average costs, but entail higher interest-rate risk.
As an average measure, the duration contains no information on the dispersion of the portfolio's interest-rate exposure over time. The duration target is therefore supplemented with a measure for interest-rate fixing. The interest-rate fixing at a given time comprises the redemptions due within the coming year, as well as the amount of the floating-rate debt and the swap portfolio, for which a new rate of interest is to be fixed within one year. [1] The interest-rate fixing is thus an ex-ante measure which captures the development in the interest-rate exposure over time for a given strategy. The actual interest-rate exposure in a given year will also depend on the size of the budget balance for the year. The analysis of the central government's future interest-rate risk therefore includes budget estimates as an important element which influences the determination of duration and interest-rate fixing. All other things being equal, it is appropriate to smooth out the exposure from year to year in order to avoid fixing a new rate of interest on a relatively large proportion of the portfolio in a year when interest rates are extraordinarily high.
The interest-rate fixing as a percentage of GDP is an indicator of the central government's "real" exposure. Higher GDP can be expected to render the government less sensitive to the development in interest rates, e.g. as a consequence of a larger taxation base. The interest-rate fixing as a percentage of GDP is applied as a supplementary exposure measure.
Cost and risk measures
Costs and risk for a given strategy are quantified by simulating the distribution of the annual interest costs over a long horizon. On the basis of the distributions, various cost and risk measures are calculated. Chart 11.3.1 shows a simulated distribution of interest costs for 2006 for the central government's domestic and foreign debt and illustrates various cost and risk measures.
| Simulated distribution of interest costs for 2006, domestic and foreign debt | Chart 11.3.1 |
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The expected costs are calculated as the mean of the distribution, which is DKK 32.6 billion in 2006. The interest costs do not have a normal distribution, so the median is a relevant supplementary cost measure not affected by extreme observations. Absolute CaR is the 95th percentile of the distribution, i.e. the maximum costs in a given year with a probability of 95 per cent. Relative CaR is calculated as the difference between absolute CaR and the mean. Relative CaR is the maximum amount, with a probability of 95 per cent, by which costs in a given year will exceed the expected costs, and thus expresses the uncertainty of the expected costs. In 2006 relative CaR is DKK 5.8 billion. Tail CaR is the mean of the costs, given that costs are higher than absolute CaR. Tail CaR thereby expresses the expected interest costs if costs exceed the 95th percentile.
In the analysis of costs and risk for a given strategy, the annual distributions of interest costs are simulated 10 years ahead. The development in costs and risk over time depends on a number of factors, including the development in the size of the portfolio and its composition. Another key factor is the calculation horizon, since it is more difficult to predict the level of interest rates 10 years ahead than 1 year ahead. This is materialised as a larger spread in the simulated interest rates the wider the horizon, cf. Chart 11.3.2. The spread of the simulated interest-cost distributions thus also widens.
| Simulated yield to maturity for 5-year bullet loan | Chart 11.3.2 |
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A new risk measure, conditional CaR (relative and absolute), has been introduced to remove this horizon effect from the risk calculation and supplement the analysis. The aim is an assessment of the central government's interest-rate risk from year to year for different strategies that is not affected by an increasing calculation horizon. Conditional CaR makes it easier to compare the risk development from year to year because the risk is measured over the same horizon.
Relative CaR for 2006 is DKK 5.8 billion, while relative CaR for 2007 increases to DKK 7.5 billion. These figures show the uncertainty of the expected costs in the respective years, seen from the present. Conditional CaR may supplement the analysis by e.g. measuring the risk in 2006 seen from the beginning of 2005 and the risk in 2007 seen from the beginning of 2006. These calculations give an indication of the uncertainty associated with the ongoing interest-rate budgeting in the Budget Review from the Ministry of Finance.
Conditional CaR can be calculated in different ways. For example, if we wish to calculate the interest-rate risk in 2006, seen from the beginning of 2005, the portfolio and interest-rate developments may first be projected in accordance with one expected scenario up to the beginning of 2005. On the basis of the expected scenario, a number of stochastic scenarios up to and including 2006 can then be calculated, capturing the risk over this horizon.
Since the development until the beginning of 2005 is unknown, conditional CaR may alternatively be calculated on the basis of the portfolio and level of interest rates at the beginning of 2005 for various stochastic scenarios until this point in time. Chart 11.3.3 illustrates such a conditional relative CaR calculation for total interest costs for the domestic and foreign debt in 2006. The distribution is based on the calculation of 500 scenarios of the difference between the actual simulated interest costs in 2006 and the costs we would have expected in the 1st quarter of 2005 for each scenario. The expected costs in 2006 seen from the 1st quarter of 2005 are calculated by switching off the stochastics from the development in interest rates after the 1st quarter of 2005 in the scenario calculations. [2]
| Distribution of the difference between simulated interest costs in 2006 and expectations seen from the 1st quarter of 2005 (conditional relative CaR) | Chart 11.3.3 |
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Conditional relative CaR calculated as the 95th percentile of the distribution is DKK 4.3 billion, i.e. with a probability of 95 per cent interest costs will increase by less than DKK 4.3 billion in 2006 compared to the expectation in the 1st quarter of 2005. Compared with relative CaR for a given year, the risk expressed as conditional relative CaR is reduced, since the calculation horizon is reduced.
11.4 Scenarios for the central government's interest-rate exposure and risk
This section illustrates a number of scenarios for the development in the central government's exposure and risk on the domestic and foreign portfolios for selected strategies.
All strategies assume that the domestic borrowing requirement is
financed via a 40-20-40 per cent distribution on 2-, 5- and 10-year bullet loans, respectively, and that domestic securities are bought back to smoothen the annual redemptions. The foreign borrowing requirement is financed via 5-year euro bullet loans. The development in the portfolio's exposure target is determined by adapting the interest-rate-swap strategy. Interest-rate swaps from fixed to floating interest rates shorten the duration and increase interest-rate fixing because the rate of interest on the variable leg is refixed every 6 months.
The development in the interest-rate fixing at constant duration [3]
Charts 11.4.1 and 11.4.2 show the development in the interest-rate fixing in absolute figures and as a percentage of GDP. The interest-rate fixing has increased relatively strongly in recent years, and at end-2003 it was DKK 275 billion. The increase has taken place concurrently with a duration reduction from 4.4 years at end-1998 to 3.2 years at end-2003. This development should be seen against the background of a decline in central-government debt as a percentage of GDP from 49.4 per cent in 1998 to 37 per cent in 2003. This has led to a strategic decision to reduce the duration.
| Interest-rate fixing in absolute figures, year-end | Chart 11.4.1 |
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| Interest-rate fixing as a percentage of gdp, year-end | Chart 11.4.2 |
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Charts 11.4.1 and 11.4.2 also show the future development in the interest-rate fixing, assuming two different strategies based on fixed durations of 3.25 and 2.75 years, respectively.
The development in the interest-rate fixing for these strategies is calculated according to different assumptions regarding the development in the central government's budget and GDP. The base scenario is based on medium-term projection of the development in the budget and GDP from the Ministry of Finance. The alternative scenario assumes that the primary budget surplus is reduced by almost DKK 17 billion on average from 2005 onwards, thus increasing the central government's borrowing requirement, and that the rate of GDP growth halves.
A constant duration of 3.25 years gives a fairly stable absolute level of interest-rate fixing for a number of years, assuming that the base scenario is realised. If the duration is reduced to 2.75 years, the interest-rate fixing, in absolute terms and relative to GDP, increases in the short term for both scenarios. The reason is that a duration of 2.75 years requires a relatively large volume of interest-rate swaps from fixed to floating rate. This increases the portfolio amount for which a new rate of interest is to be fixed within a short horizon.
| Trade-off between costs and risk for two strategies based on durations of 3.25 and 2.75 years, respectively | Chart 11.4.3 |
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Trade-off between costs and risk at durations of 3.25 and 2.75 years, respectively
The higher risk at a duration of 2.75 years compared to 3.25 years should be seen in the light of the expected reduction in interest costs. Chart 11.4.3 illustrates quantification of costs and risk for the 2.75 strategy compared to the 3.25 strategy based on the base budget scenario. The difference in relative CaR, i.e. the maximum increase in costs compared to expected costs with a probability of 95 per cent, is widening steadily. On the other hand, the expected interest costs will be almost DKK 0.4 billion lower on average if the duration is reduced from 3.25 years to 2.75 years.
Duration development at constant "real" exposure
In the above, the strategies were defined by a constant duration. Alternatively, the strategies may be defined by a certain course for the interest-rate fixing. The interest-rate fixing can be managed by adapting the strategy for interest-rate swaps and buy-backs, whereby the duration is residually determined. As a portfolio measure, the interest-rate fixing, calculated at a given time, summarises the size of the portfolio's interest-rate exposure over the coming year. The actual interest-rate exposure and risk also depend on the central government's current budget surplus or deficit, which respectively reduces or increases the borrowing requirement. The budget scenarios entered in the CaR model will thus affect the levels for the interest-rate fixing and duration determined on the basis of the CaR analysis.
| Duration at constant interest-rate fixing as a percentage of GDP, year-end | Chart 11.4.4 |
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Within government debt it is relevant to measure interest-rate exposure and risk relative to the macroeconomic development. All other things being equal, higher GDP entails a lower debt burden and strengthening of the government's ability to absorb interest-rate risk. The interest-rate fixing as a ratio of GDP gives an intuitive measure of the central government's "real" exposure. A fixed level for the interest-rate fixing as a ratio of GDP means that the interest-rate exposure is increased in step with GDP growth.
Chart 11.4.4 shows two scenarios for the duration development where the swap strategy has been determined with a view to ensuring a constant ratio between the interest-rate fixing calculated at year-end and GDP equivalent to the level of 19 per cent in 2003. The duration is calculated for two different scenarios for development in GDP and the budget, corresponding to the scenarios applied in Chart 11.4.1.
In the base scenario, based on medium-term projection by the Ministry of Finance, there is scope for reducing the duration over time without increasing the "real" exposure from year to year in the form of the interest-rate fixing as a share of GDP. In the alternative scenario a less favourable budget development is assumed, as well as a lower rate of GDP growth. A reduced budget surplus increases the borrowing requirement and increases the interest-rate fixing over time. Together with the lower rate of GDP growth, this means that there is less scope for transacting interest-rate swaps if the interest-rate fixing as a share of GDP is to be maintained at the current level. In the event that the alternative scenario for GDP and budget is realised, the duration must thus be extended slightly over time to maintain the "real" exposure.
[1] The interest-rate fixing is a gross measure which only comprises the liabilities of the central-government debt portfolio. In reality the interest-rate exposure is reduced by the portfolio's assets, i.e. the balance of the central government's account with Danmarks Nationalbank, and the part of the portfolio of the Social Pension Fund which is to be reinvested in a given year.
[2] Given the rate of interest in the 1st quarter of 2005 for each scenario, the CIR model is applied to determine the expected future development in interest rates, cf. the equation in Box 11.1. Thereby the expected interest costs for the debt can be determined for each scenario.
[3] As from 2004 a methodological change in the duration calculation has been introduced, cf. Chapter 6. The change implies that duration at end-2003 is reduced by around three months compared to the previous calculation. The new calculation method has not yet been implemented in the CaR model. The durations reported in this section have therefore been calculated using the old method.