Chapter 10
Cost-at-Risk
10.1 Summary
The interest-rate risk on the central-government debt is managed via duration and redemption profile. However, these
measures do not provide a basis for a quantification of the risk related to the central-government debt. In order to
evaluate this risk, Cost-at-Risk (CaR) analyses are used. CaR helps to quantify the risk on the basis of the estimated
future development in interest rates. This provides an important input to the assessment of the trade-off between
interest-rate risk and costs.
In CaR calculations, risk is defined as the risk of an increase in the annual costs of the central-government debt. The
central government is exposed to interest-rate and refinancing risks because the interest rates for future borrowing are
unknown. Thus the future costs are also unknown. A high level of interest rates in a given year will entail relatively
expensive refinancing of the central-government debt, and thereby higher interest costs in the following years.
In the CaR analysis interest-rate scenarios are generated, and the future costs of the domestic government debt are
calculated for each interest-rate scenario. On the basis of these computations, probability distributions of the central
government's future interest costs can be set up as a means to evaluate the interest-rate risk on the domestic
government debt.
In 2001 the CaR model will be subject to further development. The intention is for it to also be applied to the other
areas of central-government debt. There is also a need to expand the model with buy-backs. Another objective is to
work on the interest-rate input to CaR, since this is of crucial importance to the outcome of the results.
The first part of this Chapter is in general terms identical to Chapter 9 of Danish Government Borrowing and Debt
1999. Compared to the CaR model of 1999, the span of time covered has been extended from 5 to 10 years, and the
model includes krone-denominated interest-rate swaps. The last part of the Chapter presents new analyses of the
interest-rate input to the model.
10.2 Method
The primary risk related to the domestic government debt is unexpectedly high interest costs. In the CaR model, future
interest costs for the domestic government debt are calculated, and the interest-rate risk is quantified.
The basis for the calculation of CaR is data on the existing debt and the central government's expected future budget
surplus. Moreover future interest-rate scenarios are generated. Based on this input, future costs can be calculated.
The future costs of the debt are highly dependent on the future development in interest rates. The generation of a large
number of interest-rate scenarios provides a means to set up probability distributions of the future costs. The expected
costs in a given year are defined as the mean value of the future costs. Absolute CaR indicates the maximum costs with
a probability of 95 per cent, while relative CaR is the difference between absolute CaR and the mean value, cf. Box
10.1.
Box 10.1 CaR definitions
In the CaR model expected costs are defined as the mean value of the calculated future costs. 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 mean value. Relative CaR thereby indicates the maximum increase in costs in comparison to the mean
value for a given year, with a probability of 95 per cent. The evaluation can also be based on other percentiles than the
95th, e.g. the 99th percentile, when more extreme situations are considered.
In terms of methodology 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 interest-rate model used and the assumptions made.
The time span of the calculations is set at 10 years. Costs for each year are calculated. A period of 10 years gives an
insight on the risk in both the long and the short run.
The interest-rate input is a crucial factor in the calculation of CaR. There are many theoretical and empirical interest-rate
models. Here a one-factor model, developed by Cox, Ingersoll and Ross, also called the CIR model, is chosen[1].
This interest-rate model features the required stochastic structure and is fairly easy to handle in practice.
Quarterly future spot interest rates, also called spot rates, are generated, i.e. each interest-rate scenario contains 40
spot rates, equivalent to a period of 10 years. The method used to generate future spot rates is described in Box 10.2.
A total of 2,500 future spot-rate scenarios are generated.
Box 10.2 Generation of spot rates
The CIR model is a one-factor model in which the stochastic element is the spot rate. This means that the spot rate
determines the entire term structure to a given point in time.
The change in the spot rate is described by a stochastic process. The parameters in this process are respectively the
equilibrium value for the spot rate, the adjustment rate of the spot rate, i.e. the speed at which the spot rate moves back
to the equilibrium value, and the volatility of the spot rate.
The parameters in the CIR model are estimated on the basis of historical data. 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. The estimation is based on quarterly observations of the spot rate in 1987-2000. The result of the estimation
is: adjustment parameter=0.1670, equilibrium spot rate=7.20 per cent and spot-yield volatility=0.09845. The estimated
parameters are subsequently used to generate future spot rates.
The model has mean reversion characteristics, i.e. the spot rate will tend to move towards the equilibrium level.
A zero-coupon yield curve can be calculated from each spot rate. As there are 40 spot rates in each interest-rate
scenario, and a total of 2,500 interest-rate scenarios, the CaR calculations are based on 100,000 yield curves.
The remaining data input to the CaR model is information on the existing debt, i.e. accrued costs and payments on the
debt. Assumed values for the central government's budget surplus before interest costs 10 years forward are also
included.
Finally, values are assumed for the structure of borrowing, e.g. that 40-20-40 per cent of respectively 2-, 5- and
10-year government securities are issued per year, and that the outstanding volume in the Treasury bill programme is
constant.
On the basis of the above input annual costs are calculated 10 years forward. The output from the calculations is thus
2,500 scenarios of future annual costs of the domestic debt. Based on these scenarios, the mean value and the 95th
percentile of the costs are calculated, and thereby absolute and relative CaR.
10.3 Domestic interest-rate swaps
In 1998 the central government introduced interest-rate swaps in Danish kroner as a new instrument of domestic
government debt policy. An interest-rate swap where for instance a floating 6-month interest rate is paid and a fixed
10-year interest rate is received in isolated terms implies a reduction of the duration of the debt. This should be viewed
against the fact that this type of interest-rate swap is an alternative to short-term borrowing. A shorter duration of the
debt results in a higher risk, but lower expected interest-rate costs. A more detailed review of domestic interest-rate
swaps is given in Chapter 8 of Danish Government Borrowing and Debt 1998.
In the modelling of interest-rate swaps in CaR a fixed spread between the zero-coupon yield curve and the swap curve
is assumed. The swap rate is determined so that the swap has a value of zero at the time of entering into the contract.
The method to calculate the swap rate is presented in Box 10.3.
Box 10.3 The swap rate
The swap rate is calculated by the following formula, cf. Danish Government Borrowing and Debt 1998, p. 103:
where c is the swap rate, d(0,t) is the discounting factor between the time 0 and t, and t(i,F) is the time of payment of
the i'th payment of the swap's fixed-rate leg out of a total of f payments, t(f,F)=T.
The swap rate is determined exclusively by the aforementioned discounting factors and these can be calculated with the
help of the swap curve, which is known at the time 0. In the calculations a fixed spread between the zero-coupon yield
curve and the swap curve is assumed.
The outstanding amount in domestic interest-rate swaps was DKK 21 billion at end-2000. Chart 10.3.1 presents the
distribution of the costs in 2001 with and without the existing interest-rate swaps. It is shown that the cost distribution
with interest-rate swaps has thicker tails. In other words, the risk of extreme costs is greater with swaps than is the case
without swaps. In addition to this the mean value of the cost distribution with swaps is lower. This shows that the
existing interest-rate swaps have entailed lower borrowing costs due to shorter duration, but on the other hand a higher
interest-rate risk.
Chart 10.3.1 Distribution of costs with and without swaps in 2001
10.4 CaR for selected borrowing strategies
The following reviews the CaR calculations for four different borrowing strategies. For all borrowing strategies a fixed
distribution of borrowing on maturity segments from 2001 to 2010 is assumed. A constant outstanding amount in
Treasury bills and in interest-rate swaps is assumed. Table 10.4.1 presents the distribution of borrowing on maturity
segments.
Table 10.4.1 Borrowing strategies
| Per cent |
2-year |
5-year |
10-year |
| Basic scenario |
40 |
20 |
40 |
| 50-0-50% borrowing |
50 |
0 |
50 |
| 100% 2-year |
100 |
0 |
0 |
| 100% 10-year |
0 |
0 |
100 |
The same interest-rate input is used in each borrowing strategy. It is also assumed that the central-government budget
surplus before interest costs on the domestic debt is unchanged. The data is based on the Ministry of Finance's
medium-term budget forecast and on average shows a net cash surplus of DKK 24 billion per year.
Chart 10.4.1 shows that the expected costs for all strategies are decreasing during the period. The reason is the
estimated government budget surpluses in the period considered, entailing a lower financing requirement and thereby a
declining debt. The expected costs in 2001 are approximately DKK 41 billion for all scenarios.
Chart 10.4.1 Mean value of costs
The costs are relatively stable towards changes in borrowing strategies in the short run (2001 and 2002). The difference
between the borrowing strategies becomes more apparent over time as the effects are accumulated. This reflects that it
takes time to change the risk profile of the debt when the debt is distributed evenly on maturities. The difference
between the borrowing scenarios is due mainly to the differing interest costs on various loan segments. These
differences arise because normally the yield curve is increasing.
Table 10.4.2 shows that the borrowing strategy with the most short-term borrowing strategy gives the lowest expected
costs, but in return the highest interest-rate risk.
Table 10.4.2 Selected car data
| DKK billion |
2001 |
2002 |
2005 |
2010 |
| Mean value |
|
| Basic scenario |
41.2 |
40.2 |
34.9 |
27.3 |
| 50-0-50% borrowing |
41.2 |
40.1 |
34.4 |
26.0 |
| 100% 2-year |
41.2 |
39.9 |
31.7 |
17.3 |
| 100% 10-year |
41.3 |
40.3 |
37.0 |
31.2 |
| Relative CaR |
|
| Basic scenario |
2.3 |
4.9 |
8.1 |
8.3 |
| 50-0-50% borrowing |
2.3 |
5.0 |
8.2 |
7.6 |
| 100% 2-year |
2.5 |
5.6 |
11.0 |
13.1 |
| 100% 10-year |
2.1 |
4.3 |
6.0 |
6.5 |
Chart 10.4.2 presents the relation between expected costs and relative CaR for 2005 for the four borrowing strategies.
It is shown in the Chart that the costs cannot be reduced without increasing the risk. The relation between expected
costs and relative CaR in 2005 can be approximated linearly. Relative CaR increases by approximately DKK 1 billion
when the expected costs are reduced by DKK 1 billion.
Chart 10.4.2 Relation between mean value and relative CaR in 2005, dkk billion
Chart 10.4.3 presents selected cost distributions for the basic scenario in the period considered. It shows that the
curves flatten out over time. This reflects the uncertainty concerning the future interest rates. Flatter curves imply that the
probability of costs around the mean value is reduced, while the probability of more extreme costs is increased.
Chart 10.4.3 Cost distributions for the basic scenario
Furthermore, the Chart shows that the mean value (the expected costs) is reduced over time. As stated, the reason is
the estimated government budget surpluses in the period considered. A government budget surplus entails a lower
financing requirement, leading to a reduction of borrowing and thereby lower interest costs over time.
10.5 Interest-rate input and stress test
The CaR calculations require a large amount of input and assumptions concerning interest rates, financing requirements,
etc. Changes to these variables entail changes in the CaR results. In the following the model's current interest-rate input
is described and compared to historical interest rates. The characteristics of some of the more extreme interest-rate
scenarios are also illustrated.
Characteristics of the applied interest rates
The applied interest rates are generated in the CIR model. The parameters are estimated on the basis of historical
interest-rate data from the period 1987-2000. The generation of future interest rates is based on the level of interest
rates at the time of calculation.
Chart 10.5.1 presents the structure of the generated yield curves in 2005. The zero-coupon interest rates in the average
yield curve range from 6 to 8 percentage points. Furthermore, the percentiles show that the short-term interest rates are
far more volatile than the long-term interest rates, i.e. short-term borrowing is less expensive, but the fluctuations in the
annual costs will be greater.
In Table 10.5.1 the historical yield curves are compared with the simulated yield curves. The yield curves are
categorised under two main categories, i.e. normal curves and inverse curves.
Table 10.5.1 Shape of yield curves
| Type of yield curve |
Historical yield curves |
Simulated yield curves |
| Per cent |
| Normal curves |
|
| Flat |
0 |
3 |
| Ordinary |
26 |
34 |
| Steep |
17 |
33 |
| Hump-shaped |
33 |
|
| Normal curves, total |
76 |
70 |
| Inverse curves |
|
| Flat |
0 |
0 |
| Ordinary |
2 |
8 |
| Steep |
4 |
8 |
| Hump-shaped |
18 |
14 |
| Inverse curves, total |
24 |
30 |
| Total |
100 |
100 |
| Note: |
A normal (inverse) curve is monotonously increasing (decreasing). A hump-shaped curve
is not monotonously increasing or decreasing. A normal hump-shaped curve is defined as first decreasing and then increasing. The absolute difference between the spot and 10-year interest rate is for a flat, ordinary and steep curve respectively <1, 1-4 and >4 percentage points. The simulated curves are generated by the CIR process, while the empirical curves are estimated via the Nelson-Siegel method. The CIR process cannot generate normal hump-shaped curves. |
The Table shows that the yield curves in the simulated data reveal approximately the same distribution for respectively
normal and inverse curves as in the historical interest-rate data. This means that at the overall level the CIR model is in
accordance with the empirical findings.
Comparison of the curves at the detailed category level, however, shows that hump-shaped yield curves represent a
large proportion of the historical interest-rate data. A problem with the CIR model istherefore that it generates relatively
few yield curves of this type. This can result in erroneous assessment of the risk in the short-term segment.
Moreover Chart 10.5.1 shows that volatility in the generated long-term interest rates is relatively small compared to
volatility in the short-term interest rates. This corresponds to observed data. However, compared to the observed data
the CIR model underestimates volatility in the long-term interest rates. This may lead to underestimation of the risk on
the long-term segments.
Chart 10.5.1 The generated yield curves
The above shows that continued work on interest-rate input to the CaR calculations is necessary, including work on
alternative term-structure models, e.g. multi-factor models.
Stress test
The 95th cost percentile is applied to the risk evaluation. Chart 10.5.2 shows the spot-rate scenario which results in an
interest cost equivalent to the 95th percentile in 2010. For comparison, two other spot-rate scenarios are also shown.
One leads to the 99th percentile in the distribution of costs in 2010. The other is the spot-rate scenario in which the
highest spot rates occur (the stress scenario).
Chart 10.5.2 Stress scenarios for spot rates
The spot-rate scenario resulting in the 95th percentile in 2010 lies at around 5 percentage points from 2001 to 2007,
after which the spot rates rise to 25 percentage points towards the end of the period. The two other spot-rate scenarios
show greater volatility, and on average at a higher interest-rate level. The cost scenarios in Table 10.5.2 correspond to
the spot-rate scenarios in Chart 10.5.2.
Table 10.5.2 Costs in stress interest-rate scenarios, basic scenario
| DKK billion |
2001 |
2002 |
2003 |
2005 |
2007 |
2008 |
2010 |
| 95th percentile in 2010 |
40.9 |
39.9 |
38.6 |
31.6 |
36.3 |
35.5 |
35.6 |
| 99th percentile in 2010 |
41.7 |
45.4 |
51.2 |
52.0 |
48.3 |
48.9 |
42.6 |
| Stress scenario |
42.1 |
43.4 |
41.6 |
50.5 |
70.3 |
64.9 |
51.5 |
The historical development in interest rates in the period 1987-2000 for respectively the spot rate, the 2-year and the
10-year rates are presented in Chart 10.5.3. Comparison of the historical spot rates, shown in Chart 10.5.2, with the
simulated spot rates shows that for a short period the historical spot rates have been at the same level as the interest
rates in the spot-rate scenario of the 95th percentile.
Chart 10.5.3 Historical zero-coupon interest rates, quarterly observations
10.6 Conclusion
A key element of government debt policy is to find an appropriate trade-off between expected costs and risk. CaR is a
supplement to duration and redemption profile in the management of the domestic government debt, and is of particular
use because it quantifies the interest-rate risk.
The CaR results indicate that the risk associated with the domestic government debt in its present structure is relatively
low. The primary reason is that the debt is distributed evenly on maturities.
In future, the work will focus on including the effect of buy-backs in the CaR model, and on developing different types
of interest-rate input. Moreover, the present CaR calculations solely include domestic debt. The objective is for the
CaR model in due course to include the other areas of the central-government debt.
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, p. 385-407.
| 
