Monetary Review - 2nd Quarter 2003

Peter Jayaswal, Financial Markets

Introduction and summary

Danmarks Nationalbank's portfolio of domestic securities comprises Danish bonds. Since long-term interest rates are normally higher than short-term rates, this normally yields higher returns than on very short-term placements. However, this strategy also involves a risk of capital losses when interest rates are rising. The bond portfolio thus reflects a long-term trade-off of yield against risk.

Danmarks Nationalbank's policy is to consider its overall risk taken as one.^[1] Risk management of the portfolio of domestic securities and the foreign-exchange reserve therefore overlap in many areas. For instance, interest-rate sensitivity is not assessed in isolation from respectively the portfolio of domestic securities and the foreign-exchange reserve, but on the contrary within a common overall frame of reference. There are, however, differences in the management of the two portfolios due to their different natures.

The foreign-exchange reserve may be required for intervention purposes at short notice, and the liquidity risk concerning the foreign-exchange reserve is held at a low level. The portfolio of domestic securities does not have the same function and can therefore comprise less liquid securities, if deemed favourable in view of a higher yield. Another difference is that the exchange-rate risk is irrelevant for the portfolio of domestic securities.^[2]

In other respects the securities portfolio is subject to more narrow limitations than the foreign-exchange reserve. Danmarks Nationalbank is a large player in the Danish bond market and also a unique player due to its role as central bank and banker to the central government. The aim is for the management of the securities portfolio to be neutral in relation to the Danish bond market.

Since monetary financing is not permitted, cf. Article 101 of the EU treaty, a number of considerations must also be taken into account in relation to purchase and sale of bonds issued by public authorities.

The securities portfolio includes callable mortgage-credit bonds whose characteristics must be taken into account in risk management. The non-linear relationship between price and yield when interest rates are low means that interest-rate sensitivity to significant changes in interest rates cannot be calculated merely by scaling the interest-rate sensitivity calculated on marginal movements in interest rates. In addition, the level of interest-rate sensitivity may change significantly when interest rates change.

Danmarks Nationalbank's management of the portfolio of domestic securities ensures that Danmarks Nationalbank knows the risks on the portfolio and that they are within the framework deemed to be acceptable. It also entails that Danmarks Nationalbank's special role as the Danish central bank is taken into account.

The limits and guidelines reflect that Danmarks Nationalbank is willing to incur a certain interest-rate risk in order to achieve revenue, but does not wish to assume any actual credit risk or dominate the market. Whether the portfolio of domestic securities complies with the limits and guidelines is monitored on a daily basis.

Portfolio of domestic securities

Danmarks Nationalbank has always held a portfolio of Danish bonds that have served various purposes over time. Previously, bonds were e.g. used to influence interest rates. The development in the securities portfolio since 1960 is shown in Chart 1.

Development in the securities portfolio	Chart 1

In the 1960s, Danmarks Nationalbank from time to time sought to counteract rising interest rates by purchasing bonds. The purchases also served to transfer back liquidity that was absorbed via the government budget surpluses during this period.

During the 1970s Danmarks Nationalbank adjusted its presence in the bond market, and no longer systematically sought to stabilise interest rates. However, strong fluctuations were still dampened when the currency situation allowed.

In 1986 Danmarks Nationalbank was once again active in the bond market. For a short period during the summer, Danmarks Nationalbank sought to moderate falling bond prices by purchasing bonds.

Apart from very short-term liquidity management operations in the market during the early 1990s, money-market operations in the bond market have not since been applied.

In view of the fixed-exchange-rate policy and the free movement of capital, interventions in the bond market are no longer effective or relevant in terms of affecting long-term interest rates, which to a large extent are determined by international conditions. Today, monetary policy is aimed solely at managing the short-term money-market interest rates in order to ensure a stable krone vis-à-vis the euro.^[3]

Danmarks Nationalbank's historical role in supporting ship credit

Historically, Danmarks Nationalbank has played a role in connection with political agreements to support various industries. This is no longer the case, and most of the agreements have been terminated. Although the acceptance of new commitments under the agreements ceased long ago, a few agreements concerning shipbuilding finance are of such long maturity that they will still affect Danmarks Nationalbank's balance sheet and portfolio management for some years to come.

One of the agreements gives opportunities for shipbuilding loans via the issue of index-linked bonds. At the time of issue, Danmarks Nationalbank purchased the bonds at par, and in the event of early redemption Danmarks Nationalbank is under an obligation to return the underlying bonds.

The most recent agreement is from 1993 and allowed Danish Ship Finance to borrow in dollars against issue of bonds nominally denominated in kroner. Danmarks Nationalbank acquired the bonds at market value, and receives dollars and pays kroner for the term of the loan. In connection with early redemption under this agreement Danmarks Nationalbank is also under an obligation to return the underlying bonds. Loans still exist under both agreements, and the last loan will be terminated by 2012 at the latest.

The portfolio of domestic securities today

Via the securities portfolio Danmarks Nationalbank incurs an interest-rate risk that contributes to its revenue. Asset diversification also helps to minimise the credit risk. Chart 2 shows the securities portfolio by issuer. Most of the portfolio is placed in government bonds and mortgage-credit bonds.

Portfolio of domestic securities at market value by issuer	Chart 2

Note: Distribution as at 16 April 2003.

The mortgage-credit bonds are distributed almost evenly between uncallable and callable bonds. The latter are the traditional long-term mortgage-credit bonds with maturities of up to 30 years. When a bond is callable, the borrower is entitled to repay the loan at par value throughout its maturity. This is taken into account in risk management, cf. below.

The distribution of the securities portfolio by coupon rate is shown in Chart 3. There are a number of bonds with a high coupon rate, for which the market price, at the current level of interest rates, is above par. This means that a capital loss is incurred solely because the bonds draw close to par value as they approach maturity.^[4] On the other hand, they yield high returns. The coupon-rate distribution thus affects the breakdown of Danmarks Nationalbank's overall result on interest income and capital losses/gains.

Portfolio of domestic securities at market value by coupon rate	Chart 3

Note: Bonds with a non-integer coupon are included under the nearest integer. These constitute an insignificant share of the securities portfolio. Distribution as at 16 April 2003.

The management of the securities portfolio should be neutral in relation to the Danish bond market. Danmarks Nationalbank's securities portfolio constitutes only 2 per cent of the total Danish bond market, cf. Table 1.

Danmarks Nationalbank's percentage share of the bond portfolio, end of 4th quarter 2002			Table 1
Kr. billion, nominal value	Volume in circulation	Portfolio of securities	Percentage
Government bonds	579.3	15.1	3
Mortgage-credit bonds	1,473.6	17.7	1
Municipal bonds	33.3	0.7	2
Ship-credit bonds	43.8	1.2	3
Other bonds	5.6	0.5	9¹
Total	2,135.6	35.2	2
Note: Only bonds issued in kroner on the Copenhagen Stock Exchange. Source: Statistics Denmark and own figures. ¹ Reflects a large portfolio of bonds issued by the Fisheries Bank of the Kingdom of Denmark.

Compared to other market participants Danmarks Nationalbank's securities portfolio is not particularly large, and average trading in the portfolio is negligible compared to the daily trading volume on the Copenhagen Stock Exchange.

The banks may borrow bonds from the securities portfolio against payment of a fee and provision of collateral. This may be relevant for a bank which has a shortfall in a bond series for a brief period in connection with settlement of a trade.

Risks

The bond portfolio yields a higher expected return than short-term placements, but also entails a number of risks. These include market risks, i.e. how interest rates develop in the market, which bond series will be converted, how liquidity in the market develops, etc. In addition there is credit risk, i.e. the risk that the borrower will not service the loan as agreed. The credit risk not only includes losses on non-performing loans, but also the risk of a wider interest spread between the credit product and a risk-free asset. This risk is known as the spread risk.

Bond portfolios also entail a number of other risks. There is a settlement risk on trades, as well as legal risks, and a model risk on calculating key ratios, etc.

Each risk category may have many subcomponents, and the individual risks cannot be viewed in isolation. Reducing one risk will often entail increasing another.

The level of a risk can to a large extent be managed via the choice of instruments and counterparties. Typically, expected income from the investment should be weighed against willingness to assume risk. For instance, the credit risk may be limited by investing in government bonds rather than mortgage-credit bonds, although government bonds are also expected to give a lower yield.^[5]

To quantify the individual risks it is necessary to assess how much the risk factors affect the portfolio, as well as the likely outcomes of the factors. The relationship is as follows:

The consequences of the various risk factors can be expressed by their sensitivity. For instance, the sensitivity to changes in interest rates expresses the price effect of a change in interest rates, but not the size of the losses that are normally to be expected. This risk must be linked to probabilities of changes in risk factors, in this case interest rates.

It is not necessarily expedient to find a single expression of the risk. In many cases it is useful to operate with several risk measures that illustrate the risk from several perspectives. This might e.g. be different Value-at-Risk (VaR) calculations and stress scenarios, cf. below. It could also be more verbal consideration of possible consequences in different situations, and their probability. The primary risk in connection with the securities portfolio is the interest-rate risk, including the conversion risk associated with callable bonds.

Interest-rate sensitivity
There are several ways of determining a bond's sensitivity to interest-rate fluctuations. For instance, the duration indicates a bond's relative sensitivity to interest-rate changes, i.e. the percentage change in the bond's price on a marginal change in interest rates.^[6] A bond's interest-rate sensitivity can also be expressed as the Base-Point Value, which indicates the absolute price change in price points on a change in interest rates of 0.01 per cent (1 base point). A third measure is the krone duration (equivalent to the invested amount in kroner multiplied by the duration), which indicates the change in the market value of a bond as a consequence of a 1-percentage-point change in interest rates.

For uncallable bonds the calculations are straightforward, and the interest-rate sensitivity is affected only slightly by the level of interest rates. Calculations for callable mortgage-credit bonds must take into account the borrower's option to redeem the loan prematurely at par, and the interest-rate sensitivity will typically be sensitive to the level of interest rates.

Callable bonds
A callable bond can be seen as a purchased uncallable bond and a sold option entitling the borrower to purchase the bond at par. The conversion risk is that the value of the call option is sensitive to volatility, and also that the interest-rate sensitivity is highly dependent on the level of interest rates.

The sensitivity to volatility reflects that the value of the call option is highly dependent on the volatility of the interest rate. The greater the interest-rate fluctuations, the higher the probability that it will be pos-sible to exercise the option, and the higher the value of the call option. This also means that if the option is sold at a given volatility level, a loss will be incurred if the volatility subsequently increases, since the value of the option thus appreciates. Consequently there is a volatility risk in connection with callable mortgage-credit bonds.

Interest-rate sensitivity varies strongly with the level of interest rates. The closer the price of a callable mortgage-credit bond to its par value, the greater the probability that the call option is exercised, and the higher the value of the option. So the price increases less and less as interest rates fall, i.e. interest-rate sensitivity decreases, and it is hard for a callable bond to rise much beyond par. This relationship is illustrated in Chart 4.

The effect of the option element on pricing	Chart 4

Note: Yields and prices for the period 4 May 2000 to 16 April 2003. The price of the mortgage-credit bond may exceed par because transaction costs, tax, notice periods, etc. must be taken into account.

Interest-rate sensitivity, read as the inclination of the curves, is almost constant for the government bond. For the mortgage-credit bond, however, interest-rate sensitivity declines significantly as interest rates fall, i.e. the curve shows negative convexity, cf. below. At a very low level of interest rates there may even be a positive relation between yield and price, i.e. the bond's interest-rate sensitivity is negative.^[7]

Model for calculation of key ratios for callable bonds
The option element means that there is uncertainty related to the future payment series for a mortgage-credit bond. It is therefore very difficult to determine key ratios such as interest-rate sensitivity. This requires a model for when homeowners will convert their housing loans, and depends not only on the rate of interest, but also on the alternative, i.e. the terms of the new loan.^[8] In addition come the costs of the conversion, and there may also be psychological elements, e.g. people convert after having heard or read about it in the media.

There are several models for calculating key ratios for callable mortgage-credit bonds, and some financial analysts have developed their own models. To calculate the interest-rate sensitivity on the callable mortgage-credit bonds in the securities portfolio, Danmarks Nationalbank applies an external calculation model that is also used by other players.

The calculation of callable mortgage-credit bonds thus entails a model risk, since the key ratios calculated depend to a high degree on the model assumptions. The calculations must therefore be assessed critically, and the key ratios for a bond may vary in the market as a result of different models and assumptions.

Despite model risk, calculations that take the option element into account do, however, give a far better picture of the risk associated with a bond than if the bond is merely seen as uncallable. Chart 5 illustrates the interest-rate sensitivity of a callable mortgage-credit bond, calculated as the change in price on a change in interest rates by 0.01 per cent. The bond is seen from two points of view: as uncallable, and taking into account the call option.

Interest-rate sensitivity of a callable mortgage-credit bond	Chart 5

Note: Interest-rate sensitivity is calculated as the price change on a change in interest rates by 0.01 per cent.

If the bond is viewed as uncallable, interest-rate sensitivity increases slightly as a consequence of falling interest rates. In mid-April 2003, interest-rate sensitivity was 0.10 price points, i.e. the price would fall by 0.10 price point on an increase in interest rates by 0.01 per cent at the given level of interest rates. This is, however, highly misleading. Chart 4 showed that the interest-rate sensitivity of a callable bond declined when interest rates were very low. This is shown by the option-adjusted interest-rate sensitivity. Low interest rates have led to many conversions in the bond series, and interest-rate sensitivity has declined significantly. In mid-April 2003 the option-adjusted interest-rate sensitivity was 0.01 price point, i.e. only 1/10 of the non-option-adjusted interest-rate sensitivity.

Convexity
The falling interest-rate sensitivity when interest rates are declining cor-responds to a bend in the price-yield curve, cf. also Chart 4. This is known as "negative convexity". The non-linearity means that the sensitivity measure cannot merely be scaled to different interest-rate changes. The calculation of the capital loss on a change in interest rates is only "correct" for a marginal change. If the calculated interest-rate sensitivity is merely multiplied by a larger change in interest rates, the actual capital loss is underestimated when interest rates increase, while the capital gain on a drop in interest rates is overestimated.^[9] The more the curve bends, the greater the error. To calculate the precise interest-rate sensitivity for a given change in interest rates the corresponding price change must be calculated explicitly.

Another consequence of the convexity is that for two bonds with the same interest-rate sensitivity at the given level of interest rates, the change in interest-rate sensitivity on a change in interest rates may differ. How sensitive the interest-rate sensitivity of a bond is to interest-rate changes depends on the bend of the price-yield curve.^[10]

In Chart 6 the theoretical price for three callable bonds is calculated at different changes in interest rates. At the zero point, i.e. at the given level of interest rates on the calculation date, the theoretical price may in principle be compared with the actual price on the same day, in order to assess the pricing of the bond in the market.

Theoretical prices calculated on changes in interest rates	Chart 6

All three curves show the observed relationship between the interest rate and the price of callable bonds (negative convexity). At the given level of interest rates, the 5% bond will "behave" like an uncallable bond on minor changes in interest rates, but on large decreases in interest rates the bond will be "threatened by conversion", and the price will be less sensitive to interest-rate changes. The interest-rate sensitivity for the 6% bond is very low at the given level of interest rates, where the price is above par and the bond is "threatened by conversion". On a large increase in interest rates, interest-rate sensitivity will increase significantly, since the bond will then be less threatened by conversion. The interest-rate sensitivity of the 7% bond is negative. This reflects that the bond is so threatened by conversion at the given level of interest rates that a further decline in interest rates entails a lower price. This relationship is best illustrated via an "S curve", cf. Chart 7.

Option-adjusted interest-rate sensitivity on changes in interest rates	Chart 7

In Chart 7 the option-adjusted interest-rate sensitivity is calculated for various changes in interest rates. At the zero point, the interest-rate sensitivity of the bonds at the given level of interest rates on the calculation date can be found. For the 5% and 6% bonds, this is respectively 0.06 and 0.01 price points. The interest-rate sensitivity for the 7% bond is negative, since the risk that the bond is converted at par is very high at lower interest rates, cf. also Chart 6.

The effect on interest-rate sensitivity to fluctuations in interest rates is not the same for all three bonds. If interest rates rise by 0.5 percentage points, the interest-rate sensitivity for the 7% bond will remain almost unchanged, while the interest-rate sensitivity for the 5% and 6% bonds will increase, since they will be less threatened by conversion. Particularly the interest-rate sensitivity for the 6% bond will increase significantly if interest rates rise since, at the given level of interest rates, the bond is highly sensitive to the level of interest rates.

Interest-rate risk
In order to quantify the interest-rate risk it is necessary to determine not only the interest-rate sensitivity for uncallable and callable bonds, but also a number of probabilities concerning the development in interest rates. The expected distribution of the interest rates can be determined in several ways. The distribution may be simulated on the basis of historical observations or artificially produced observations. The distribution may also be assumed to adhere to a statistical distribution function whereby parameters are estimated on the basis of historical data.

There are several risk measures that perceive risk from different angles. A much-used risk measure is Value-at-Risk (VaR) calculations. VaR is an estimate of the maximum loss which, at a chosen probability, can be expected under normal market conditions. On the other hand, the VaR calculation does not indicate the size of the losses under extremely negative circumstances. This may be illustrated via stress tests indicating the expected worst-case losses.^[11]

Management of the portfolio of domestic securities

In its risk management of the portfolio of domestic securities, Danmarks Nationalbank observes the prevailing standards and methods applied by other financial players.

As a consequence of Danmarks Nationalbank's special role as the Danish central bank, the investment policy and portfolio management must also take certain elements into account. For instance, the credit risk on non-performing loans must be held at a very low level. On the other hand, the securities portfolio comprises interest-rate risk, conversion risk, liquidity risk and spread risk, reflecting the revenue target.

Danmarks Nationalbank's actual interest-rate sensitivity	Table 2
Capital losses in kr. billion on a general increase in interest rates by 1 percentage point	End-March 2003
Foreign-exchange reserve	1.4
Portfolio of domestic securities	0.9
Total	2.3

Target for the interest-rate sensitivity of the securities portfolio
The primary loss risk on the securities portfolio derives from the interest-rate sensitivity relating to bonds. An interest-rate-sensitivity target is therefore a significant element of risk management. Danmarks National-bank applies a duration target, the krone duration, which indicates the change in the portfolio's market value in kroner on a 1-percentage-point change in interest rates.^[12]

Danmarks Nationalbank's risks are perceived as one, and the interest-rate sensitivity for respectively the foreign-exchange reserve and the domestic securities portfolio are thus not considered in isolation. Table 2 shows the actual krone duration at end-March 2003 for respectively the foreign-exchange reserve and the securities portfolio.

The interest-rate-sensitivity target is determined on the basis of an assessment of Danmarks Nationalbank's total financial risks and reflects a long-term trade-off of yield against risk. The actual krone duration deviates from the target within certain fixed limits.

Target of the interest-rate sensitivity of the krone duration
As stated above, the interest-rate sensitivity of callable bonds is sensitive to the level of interest rates. How much a portfolio's interest-rate sensitivity changes when interest rates fluctuate depends on the structure of the portfolio at the given level of interest rates.

Since Danmarks Nationalbank manages interest-rate sensitivity according to a krone duration target, it may be necessary to restructure the portfolio as a consequence of the development in interest rates. In order to avoid major restructuring and to manage the conversion risk, there is a ceiling for the sensitivity of the krone duration to changes in interest rates.

Chart 8 shows changes in the krone duration of the securities portfolio in connection with changes in interest rates for two given days. The krone duration at the zero point was kr. 882 million on 31 March 2003, i.e. if interest rates rose by 1 percentage point on 31 March, the market value of the securities portfolio would have fallen by kr. 882 million. On 1 July 2002 the krone duration was higher, and an increase in interest rates by 1 percentage point on 1 July would have entailed higher capital losses than on 31 March. In other words, the interest-rate sensitivity has declined.

Krone duration of the securities portfolio on changes in interest rates	Chart 8

It can also be seen that the interest-rate sensitivity has become less sensitive to changes in interest rates. On an increase in interest rates by 1 percentage point, the krone duration would have increased to kr. 1,014 million on 31 March 2003, while the krone duration would have increased to kr. 1,381 million on 1 July 2002.

The fact that the krone duration has become less sensitive to changes in interest rates reflects, inter alia, that the securities portfolio included fewer callable mortgage-credit bonds at end-March 2003 compared to the summer of 2002.

Credit risk on the domestic securities portfolio
Danmarks Nationalbank has chosen to invest in bonds that are subject to a very small credit risk. Most of the portfolio is placed in government bonds and mortgage-credit bonds. Danish mortgage-credit bonds are assessed by the rating agencies to entail a very low credit risk, and are typically rated AA and upwards.^[13] Nonetheless, a limit has been set to the volume of mortgage-credit bonds in the securities portfolio.

In order to limit the credit risk at issuer level, there is a ceiling on an individual issuer's share of the total market value of the securities portfolio. There is also a ceiling on Danmarks Nationalbank's ownership share of one issuer's bonds as a proportion of the issuer's total volume of outstanding bonds. The latter also helps to support the neutrality of Danmarks Nationalbank in the Danish bond market.

Rules for purchase of bonds issued by public authorities
Under the EU Treaty, all EU member states must conduct sound and sustainable government budget policies. Therefore the central banks of the member states may not finance public expenditure either directly by lending to the central government, or indirectly by purchasing a disproportionately large volume of bonds issued by public authorities.^[14] Rules have therefore been laid down for the central banks' purchases of such securities.

Danmarks Nationalbank may not purchase government bonds directly from the central government, i.e. in the primary market. In the ordinary bond market, i.e. in the secondary market, Danmarks Nationalbank is free to purchase and sell government bonds on equal terms with other players, subject to the limitation that Danmarks Nationalbank may not buy a disproportionately large volume of securities issued by public authorities.

[1] See Ib Hansen and Christian Ølgaard, Danmarks Nationalbank's Risk Management, Danmarks Nationalbank, Monetary Review, 2nd Quarter 2000.

[2] Management of the foreign-exchange reserve is described in Peter Kjær Jensen, Management of the Interest-Rate Risk on the Foreign-Exchange Reserve, Danmarks Nationalbank, Monetary Review, 1st Quarter 2001.

[3] Cf. Danmarks Nationalbank's Use of Monetary-Policy Instruments, Danmarks Nationalbank, Monetary Review, 1st Quarter 2003.

[4] This effect on the price is known as mathematical price adjustment. Likewise, there is a mathematical capital gain if the price is below par.

[5] The individual instruments can also be seen as packages of various risk factors which can be purchased or sold via derivatives. Derivatives are not used for the portfolio of domestic securities.

[6] The interest-rate sensitivity is calculated by the inclination (tangent) of a point on the price-yield curve. The percentage price change on a change in interest rates thus applies only marginally around the point, unless the price-yield curve is linear. The more the curve bends, the less precise the scaling of price sensitivity to major changes in interest rates, cf. the section on convexity.

[7] At low interest-rate levels, the option element may have a self-reinforcing effect on interest rates. See Louise Mogensen, Market Dynamics at Low Interest Rates, Danmarks Nationalbank, Monetary Review, 1st Quarter 2002, for details.

[8] With the increasing use of adjustable-rate loans, where the underlying bonds are uncallable, the refinancing options have been expanded and the conversion behaviour has presumably changed. Conversion of mortgage-credit bonds is described in detail in Ulrik Knudsen, Conversions of 30-Year Mortgage-Credit Bonds During the Last 10 Years, this Monetary Review, page 61.

[9] For an uncallable bond the curve bends the other way. In other words, convexity is positive and the capital loss is overestimated when interest rates rise and the capital gain underestimated when interest rates fall.

[10] In mathematical terms interest-rate sensitivity corresponds to the first derivative on the price-yield curve, while the sensitivity of the interest-rate sensitivity to changes in interest rates corresponds to the second derivative on the curve.

[11] See Morten Malle Høyer, Value-at-Risk as a Measure of Danmarks Nationalbank's Market Risk, Danmarks Nationalbank, Monetary Review, 3rd Quarter 2002 for a detailed review of this topic.

[12] The target does not indicate the size of the potential losses, cf. above. This is mainly assessed via Value-at-Risk and stress tests.

[13] Ratings are described in Thomas Enevoldsen and Ole Bøgemark, Ratings in 2002, Danmarks Nationalbank, Monetary Review, 1st Quarter 2003.