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"Monetary review - 1st Quarter 2002"

Market Dynamics at Low Interest Rates

Louise Mogensen, Financial Markets

Introduction

In periods of low bond yields, e.g. autumn 2001, the effects of interest-
rate fluctuations can be self-reinforcing. To some extent these effects are due to options built into e.g. mortgage-credit bonds and interest-
rate guarantees by the pension sector. The effects are driven especially by the substantial market for US mortgage bonds, although the built-in options have also influenced the course of Danish interest rates.

Development in interest rates in 2001

In 2001, the development in long-term yields within the year was
affected by strong and rapid upward and downward movements around a generally low level. These rapid shifts were seen particularly in the autumn, cf. Chart 1.

Chart 1 10-year yields in the usa, germany and Denmark

Interest-rate fluctuations can be explained by factors such as the macroeconomic development, the general uncertainty arising after 11 September, and issuing-policy changes in the USA. There are also a number of dynamic effects, however, that are related to investors' reaction to the options built into several financial instruments. At low interest rates, a drop in interest rates as a consequence of the built-in options can trigger pressure to buy, causing interest rates to decline further. On the other hand, rising interest rates (from a low level) will exert pressure to sell, in turn causing interest rates to increase further.

Self-reinforcing effects and options

Many financial products contain option elements. This contributes to increasing volatility in the long-term yields at particularly low interest-
rate levels. Some of the movement in interest rates can thus be explained by the self-reinforcing effects triggered by the options.

Primarily two types of option are assumed to have contributed to stronger volatility in long-term yields in the autumn of 2001. Firstly, options as the right to early redemption (prepayment) of fixed-yield mortgage bonds have had an effect, especially in the USA, but to some extent also in Denmark. Secondly, the interest-rate guarantees offered by the Danish pension sector have contributed to greater volatility in Danish yields.

Options in callable bonds
Callable mortgage-credit bonds deviate from e.g. government bonds in that the borrower has the right to prepay the loan at par. This right is utilised when the borrower converts the loan. The right of prepayment corresponds to the holder of the mortgage-credit bond having sold a call option to the borrower^[1].

As a consequence of the conversion opportunity, the price-yield ratio for mortgage-credit bonds does not correspond to that for uncallable bonds, cf. Chart 2. The price of uncallable bonds increases when the general level of interest rates falls, while the price of callable mortgage-
credit bonds ceases to increase at around par value. This is due to the right of the borrower (home owner) to prepayment of the underlying loan, making the investor less willing to pay a price for the bond that is significantly above par.

The interest-rate risk, by which is meant the sensitivity of the bond price to fluctuations in interest rates, is measured by the duration^[2].

Chart 2 The theoretical price-yield ratio for bonds

When interest rates are high, the relation between price and yield for the two types of bond is almost identical, cf. Chart 2. On the other hand, if interest rates are falling, the price of the callable bond does not change as much as when interest rates are high. The duration therefore changes, and as the interest rate falls the duration of the callable bond approaches zero. Chart 3 presents the duration of various bonds on a change in interest rates.

The Chart shows that the duration of the uncallable government bond, 7 per cent 2024, is almost unchanged compared to the callable mortgage-credit bonds, irrespective of the level of interest rates. It is also seen that the sensitivity of the mortgage-credit bonds to changes in interest rates is dependent on their nominal yields. This is because when interest rates are falling high-coupon bonds are at risk of conversion before low-coupon bonds.

The changes in duration as a consequence of fluctuating interest rates is a measure of the convexity of the bond^[3]. Due to the conversion opportunity, mortgage-credit bonds usually show negative convexity, whereby the duration declines when interest rates decrease (the conversion probability increases).

Chart 4 presents a comparison of the changes in duration arising from fluctuations in the yields on various bonds.

Chart 3 Duration on fluctuations in interest rates

Chart 4 Change in duration at different interest rates (convexity)

Dynamic effect for callable mortgage-credit bonds
An investor holding mortgage-credit bonds will in many cases require that the portfolio's sensitivity to fluctuations in interest rates does not decline when interest rates fall, and on the other hand does not increase when interest rates rise. This can be attributed to several factors, e.g. that the investor is borrowing at fixed interest rates, that the investor requires a profit on a further drop in interest rates, or on the other hand does not wish to incur a higher loss on an increase in interest rates.

Investors that manage the interest-rate risk on their investments seek to take into account that a change in interest rates can change duration quite considerably. The duration of long-term mortgage-credit bonds can thus "disappear" when the interest rate decreases. In order to maintain the interest-rate risk at a given level investors must buy duration using other instruments. They can do this via uncallable bonds, e.g. government bonds, or mortgage-credit bonds that are not at risk of conversion, leading to further downward pressure on the level of interest rates. The dynamic effect is thus initiated.

How much extra duration the investor must buy on a drop in interest rates depends on the portfolio's convexity. The greater the numerical convexity, the more duration the investor will have to buy when interest rates fall. The convexity is strongly dependent on the level of interest rates, cf. Chart 4. If interest rates change from a level far from the nom-
inal yield on the bond, this does not significantly affect duration, but the change in duration increases significantly when interest rates fluctuate at a level close to the bond's nominal yield.

The development in the duration of an index of Danish callable mortgage-credit bonds is presented in Chart 5. For comparison, the yield on 10-year Danish government bonds is shown. The duration takes the conversion probability, i.e. the built-in option, into account, and it is clear that duration decreases when interest rates fall.

In 1998, the year of the highest number of Danish conversions so far (seen over one year), duration increased considerably after the conversions had taken place. This is because borrowers converted their old loans to newly-issued long-term fixed-yield bonds at a price below par, which were therefore subject to a lower conversion risk. In connection with the surge of conversions in autumn 2001, duration did not increase in the same way as in 1998, which can be attributed to such factors as borrowers to a high degree converting to the short-term interest-
adjustment loans, rather than long-term callable loans.

The drop in interest rates thus has a self-reinforcing effect due to the change in the duration of the callable bonds, and investors' wish to maintain duration at a given level. On the other hand, an increase in interest rates, particularly from a low level, can also generate self-
reinforcing effects, since a sudden increase in the duration of the callable bonds leads to sale of duration (and thereby bonds), which exerts further upward pressure on interest rates.

Chart 5 Option-adjusted duration (oad) for index of mortgage-credit bonds

Part of the significant, rapid upward and downward fluctuations in interest rates in the autumn of 2001 can therefore be assumed to be related to dynamic effects derived from the option element in callable bonds. The callable US bonds affected the international development in interest rates. In Denmark it could be noted how the yield spread narrowed between the mortgage-credit bonds with low nominal yields that were at least risk of conversion, and government bonds.

Interest-rate guarantees in the pension sector
The fact that Danish pension and life assurance companies have guaranteed a minimum interest rate to their customers also constitutes a type of built-in option. The option entails that the companies' obligations gain higher market value in step with falling interest rates. To avoid the risk of insolvency, the companies therefore have a requirement to own bonds – or other instruments – that appreciate when interest rates fall.

The effect can be illustrated as shown in Chart 6, based on the pension companies' balance sheets. The value of the liabilities is compiled as the net present value of the company's pension and insurance liabilities.

Chart 6 Example of pension company liabilities on fluctuations in interest rates

If the level of interest rates is close to or lower than the issued interest-
rate guarantee, the obligations on the liabilities side will correspond to a high degree to a long-term fixed-yield bond with a yield requirement equivalent to the guaranteed interest rate (the liabilities side gains high duration). On the other hand, when interest-rate levels are high (above the issued interest-rate guarantee) the liabilities will correspond to a short-term bond at low duration.

The asymmetry as a consequence of the interest-rate guarantees causes the value of the obligations to increase significantly, and often by more than the assets, when the interest rate is falling, with a consequential risk of mismatch between assets and liabilities, and thereby a reduction of the companies' reserves. If the asset side is to correspond to the appreciation on the liabilities side, the asset portfolio is required to have a high duration, in order to achieve an equivalent capital gain when interest rates are falling.

Dynamic effect at low interest rates
In periods of falling interest rates the pension companies will require duration due to fear of insolvency (mismatch between obligations on the liabilities and asset sides), and because to a great extent pension companies own mortgage-credit bonds, and are thereby subject to the effects described in the section on options in mortgage-credit bonds.

The demand for duration on the asset side exerts further downward pressure on the interest rate, and the dynamic effect is established.

Conclusion

In periods with low interest rates built-in options in financial instruments will lead to self-reinforcing effects, increasing the volatility in the interest rates. Options are known especially from callable mortgage-
credit bonds and interest-rate guarantees in pension and life assurance companies.

Footnotes

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