Staff Memo N O 2005/11. Documentation of the method used by Norges Bank for estimating implied forward interest rates.

Transcription

1 N O 005/ Oslo Noveber 4, 005 Staff Meo Departent for Market Operations and Analysis Docuentation of the ethod used by Norges Bank for estiating iplied forward interest rates by Gaute Myklebust

2 Publications fro Norges Bank can be ordered by e-ail: or fro:norges Bank, Subscription service, P.O.Box. 79 Sentru N-007 Oslo, Norway. Tel , Fax Publications in the series Staff Meo are available as pdf-files on the bank s web site: under "Publications". Staff Meos present reports on key issues written by staff ebers of Norges Bank, the central bank of Norway - and are intended to encourage coents fro colleagues and other interested parties. Views and conclusions ressed in Staff Meos can not be taken to represent the views of Norges Bank. 005 Norges Bank The text ay be quoted or referred to, provided that due acknowledgeent is given to source. Publikasoner fra Norges Bank kan bestilles over e-post: posten@norges-bank.no eller ved henvendelse til: Norges Bank, Abonneentsservice Postboks 79 Sentru 007 Oslo Telefon , Telefaks Utgivelser i serien Staff Meo er tilgengelige so pdf-filer på under «Publikasoner». Staff Meo inneholder utredninger so inngår i bankens arbeid ed sentrale problestillinger. Hensikten er å otta koentarer fra kolleger og andre interesserte. Synspunkter og konklusoner i arbeidene representerer ikke nødvendigvis Norges Banks synspunkter. 005 Norges Bank Det kan siteres fra eller henvises til dette arbeid, gitt at forfatter og Norges Bank oppgis so kilde. ISSN online only ISBN online only

3 Docuentation of the ethod used by Norges Bank for estiating iplied forward interest rates By Gaute Myklebust, Departent for Market Operations and Analysis Introduction Forward interest rates are an iportant indicator for onetary policy. They are coonly used, both by central banks and arket participants, to gain inforation about arket ectations of future interest rates. Forward rates in the short ter are traded directly in the arket, but for longer horizons they ust be derived iplicitly fro the yield curve. In this paper we will briefly coent on the ethod used by Norges Bank for estiating iplied forward interest rates. Previously, Kloster 000 has described this in ore detail. However, the Bank has ade soe adustents to the procedure described in Kloster s article. Hence, this paper is an update and a suary of the ethod currently used. The paper is structured as follows: Firstly, we will coent on the selection of instruents. There are in general two alternative types of instruents suitable for estiating iplied forward rates in the Norwegian arket. One is the arket for governent securities, while the other is the swap arket. Secondly, we will give a brief description of the estiation procedure, which is based on the Svensson ethod or the extended Nelson and Siegel ethod. Finally, we will coent on the adustents we ake for risk preia. There are a nuber of potential risk preia in the estiated forward rates. However, the corrections we ake are only inor, and relate to the credit risk eleent in the forward rates derived fro the swap arket. Interest rate data In the short ter it is coon to use the arket for forward rate agreeents FRAs when deriving interest rate ectations. This arket is regarded as the ost liquid part of the oney arket in Norway. Contracts are traded with settleent on the third Wednesday - the so-called IMM dates in March, June, Septeber and Deceber. The FRAs are of 3, 6 and onths aturity and settle against NIBOR prevailing two days before the IMM date. Cobining these contracts provides 3-onth FRAs with settleent on the first 7 IMM dates, where the 7 th IMM date has settleent 8- onths ahead.

4 For longer horizons, however, the forward rates ust be estiated using other instruents. There are in general two alternative types of instruents for estiating iplied forward interest rates in the Norwegian arket. One is the arket for governent bonds and Treasury bills; the other is the swap arket, including the interbank NIBOR, FRA and interest rate swap arket. The yields on Treasury bills and governent bonds contain no credit risk. Iplied forward rates derived fro these instruents can therefore be interpreted as ectations concerning the future key rate the sight deposit rate, assuing no other types of risk preia in the forward rates. There are, however, relatively few outstanding Treasury bills and governent bonds in the Norwegian arket, and the volue is by international standards sall 3. As a consequence, the pricing of governent securities ay be considerably influenced by variation in supply and deand. Historical erience suggests that this ay be the case. 4 In addition, Treasury bills are often used for cash anageent purposes. Their yields ay therefore contain less accurate inforation about ectations of future interest rates in the near ter. Since the purpose of estiating iplied forward rates is to derive interest rate ectations, this suggests that governent securities are not the best suited. In the estiations we therefore use rates fro the swap arket. This arket is considered to be ore liquid than the Norwegian arket for governent securities. The input data consist of oney arket rates NIBOR with aturities, 3, 6 and onths and the fixed rates on NIBOR-related interest rate swaps with aturities in the range of to 0 years. Iplied forward rates fro this arket can be interpreted as ectations of future oney arket rates, again assuing no risk preia in the forward rates. An advantage in using these instruents is that the nuber of observations along the yield curve increases. 5 Hence, the estiation will be ore accurate. A disadvantage is that the instruents contain credit risk. However, since all Norwegian Interbank Offered Rate There are norally four and five outstanding Treasury bills and governent bonds respectively. The bills are auctioned at each IMM date with aturity of one year, while the bonds are auctioned with a twoyear frequency. The shortest bond atures in January 007, while the longest atures in May On Septeber the outstanding volue in governent securities was NOK 05.7 billion. 4 See Hein 003.

5 instruents are related to NIBOR, they will contain a coon credit risk eleent. Using these instruents will therefore provide a consistent yield curve. The sae arguent applies when coparing the estiated 3-onth forward rates against the FRA rates. As both curves are related to NIBOR, they will norally show the sae. Any differences should then be due to the estiation ethod. The estiation ethod In the estiation of iplied forward interest rates we use the estiation ethod proposed by Svensson 994, 995. This is an extended version of the ethod developed by Nelson and Siegel 987 and is therefore often referred to as the extended Nelson and Siegel ethod. The Svensson ethod is a paraetric ethod, where the entire yield curve is described by a single set of paraeters representing the long-run level of interest rates, the slope of the curve and hups in the curve. The input in the estiation procedure is the yield to aturity on the different instruents entioned and their corresponding coupon payents and aturity dates. While NIBOR rates can be interpreted as zero-coupon bonds, an interest rate swap can be interpreted as a par bond. The coupon rate will then equal the swap rate. Svensson s function has six paraeters. Once these have been estiated, the function provides continuously copounded zero-coupon rates for all aturities and forward rates for all settleent dates and aturities. Pricing the synthetic bonds based on NIBOR and interest rate swaps with the estiated zero-coupon or forward rates results in a set of estiated yields on these bonds. The paraeters in Svensson s function are deterined so that the su of squared yield errors between the estiated yields and the observed yields in the arket is iniised. This proble is solved by axiu likelihood estiation. An alternative to iniising yield errors would be to iniise price errors. As pointed out by Svensson 995 however, iniising price errors soeties results in fairly large yield errors for instruents with short aturities. This is due to the fact that prices are very insensitive to yields for short aturities. Miniising yield errors 5 The total nuber of securities used is 3, copared to a total of 9 available governent securities. 3

6 gives a better fit for short aturities, while the two procedures see to perfor equally well for long aturities. Also for onetary policy purposes the focus is on interest rates, so it akes sense to iniise errors in the yield diension rather than in the price diension. For a ore technical description of Svensson s ethod and the estiation procedure, see the Appendix. Chart shows the observed yield curve, estiated zero-coupon curve and the iplied forward interest rate curve on 0 August 005. Chart : Observed yields, estiated zero-coupon and iplied forward rates. 0 August Forward rates Estiated zerocoupon rates Observed yields aug 05 aug 07 aug 09 aug aug 3 aug 5.0 There are also other estiation techniques available. Spline-based odels are an alternative to the paraetric ethods developed by Nelson and Siegel and Svensson. For a description of these odels and a coparison of paraetric and spline-based odels, see Anderson and Sleath 999. Risk preia in estiated forward rates The purpose of estiating iplied forward interest rates is to derive interest rate ectations in the arket. However, the existence of risk preia in forward rates ay cause the forward rates to deviate fro the true interest rate ectations in the arket. The risk preiu includes preia for differences in credit quality and liquidity considerations, and a ter preiu arising fro interest rate uncertainty and 4

7 investor risk aversion. Risk preia are not observable and need therefore to be estiated. However, this is a difficult task as the preiu consists of different eleents, and ay also be tie varying. The instruents used to estiate the iplied forward rates are all based on or related to NIBOR. They will therefore contain a coon credit risk eleent. But it is likely that the credit risk in forward rates increases with aturity. If so, this will contribute to an upward bias in the estiates of the ected interest rates derived fro iplied forward rates. A coparison of the forward rates derived fro the swap arket with the rates derived fro the governent arket is an easy way to gain inforation about the size of the credit risk preiu at different horizons on the forward curve. The forward rates based on governent securities contain no credit risk, and all else equal, any difference should then be due to the credit risk in the swap arket. The average 6 difference between these two forward curves indicates that the credit risk sees to increase with settleent up to -4 years, and reains fairly constant thereafter. The Bank s approach for correcting for increasing credit risk is based on this average. It iplies that the credit risk preiu in 3-onth forward rates is constant the first six onths and increases by -4 basis points 7 after -4 years. Fro 4 to 0 years we ake no further adustents. Table shows the adustents we ake for the 3-onth iplied forward interest rates for different ties to settleent. Table : Increasing credit risk in 3-onth forward rates for different ties to settleent Tie to settleent 6 onths onths years 4 years Credit risk 0 bps 4 bps bps 4 bps 6 In the period January August 005 5

8 Chart shows the iplied forward interest rate curve, with and without adusting for increasing credit risk preia. Chart : Forward interest rates and adusted rates Forward rates Adusted forward rates aug 05 aug 07 aug 09 aug aug 3 aug 5.0 This ethod for adusting for credit risk is very siple and obviously there are possible sources of error. Firstly, the credit risk preiu ay vary through tie, and using an average can therefore be isleading. However, the period used to estiate the average is fairly short and the calculations are updated on a regular basis. This should therefore reduce soe of the potential errors resulting fro tievarying credit risk. Secondly, the spread between yields on governent bonds and swap rates - the swap spread - will also reflect other factors. As noted earlier, the yields on governent bonds ay be considerably influenced by variation in supply and deand. Consequently, the swap spread will be affected, and part of it ay be due to factors other than differences in credit risk. However, taking an average will probably reduce soe of this potential error. The forward rates will also be affected by a tievarying ter preiu. As the size of the preiu is very uncertain and therefore difficult to estiate, we do not correct for 7 A basis point is 0.0 percentage point. 6

9 this. 8 The estiated forward rates adusted for increasing credit risk will therefore norally be interpreted as ected future oney arket rates. The discussion so far has been liited to deriving a path for the ected future oney arket rates. To derive a siilar path for the ected key rate, we need to ake an adustent for the difference between the 3-onth oney arket rate and the key rate. Assuing there are no ectations of changes in the key rate the next three onths, this difference is norally around 5 basis points. But as for the other preia discussed, this spread ay also vary through tie. Conclusions In this paper we have described the ethod used by Norges Bank for estiating iplied forward interest rates. We have also described the adustents we ake for risk preia in the estiated forward rates. In short, the procedure is as follows: Forward interest rates are calculated on the basis of oney arket rates NIBOR with aturities, 3, 6 and onths and the fixed rates on NIBORrelated interest rate swaps with aturities in the range of to 0 years. The estiation technique used is the paraetric ethod developed by Svensson. The estiated iplied forward rates are adusted soewhat downwards due to the fact that the credit risk in iplied forward rates is likely to increase with the tie to settleent. We ake no further corrections for other types of potential risk preia. The adusted forward interest rate curve is norally interpreted as a path for the ected oney arket rate. To derive a path for the ected key rate the sight deposit rate, we adust the oney arket curve downwards by 5 basis points. 8 However, the Bank has done soe work in this area. See Valseth 003 and Myklebust 005 for a detailed description of this work. 7

10 Appendix This appendix describes in further detail the estiation ethod used by Norges Bank when estiating iplied forward interest rates. We consider the paraetric forward interest rate function f, proposed by Svensson 994,995 where denotes the reaining aturity and the paraeter vector to be estiated. The corresponding spot interest rate function can be written as the average of the instantaneous forward rates with settleent between 0 and. s, = = 0 f, d In the estiation we use NIBOR with aturities, 3, 6 and onths and the fixed rate on interest rate swaps with aturities in the range of to 0 years. These are converted into synthetic bonds, in the way that NIBOR rates are interpreted as zerocoupon bonds and the interest rate swaps are interpreted as par bonds. The coupon rates will then equal the swap rates. For a given trading date let there be n synthetic bonds c,, y, p, =,.., n represented by their coupon rates c, tie to aturity, observed yield to aturity y and the observed price p. The coupon bonds have annual coupon payents, and we index the payents by the sequence k, k =,., K, where K denotes the nuber of coupon payents for bond which equals the aturity. The estiated price of a coupon bond P can be written as the su of prices of a sequence of zero-coupon discount bonds related to each coupon payent and the face value of the bond noralised to, each priced with the discount function s, d, =. 00 8

11 Hence K P = c d, d,, =,..., n k = k K We note that we can characterise each bond either by the observed triplet c,, p or the triplet c,, y, replacing the price p of the bond with the bond s yield to aturity y. Fro the coupons c, =,., n and the indexed sequence of payents k, k =,., K we can then use the present value function and estiate a corresponding price p of bond. P K c,, y = k K = k y y c 00 00, =,..., n Alternatively, when we know the observed price p on bond, we can estiate the yield to aturity by solving for y in the above equation using a standard Newton- Raphson algorith. Likewise, this relationship between y and p can be used in the paraetric case when we derive the discount function fro the forward interest rate function f,. Hence the estiated yield to aturity for bond denoted Y can then be coputed fro the present value function P K = k K k = Y Y c 00 00, =,..., n also using a standard Newton-Raphson algorith. The observed yield to aturity y is assued to differ fro the estiated yield to aturity Y by a norally distributed error ter ε ~ Niid0,,, i. e. σ ε 9

12 0 n Y y,...,, = = ε We use the ethod proposed in Svensson 994, 995 and estiate the following forward rate function, with paraeters = 0,,, 3,, : = 3 0, f This relationship is also denoted as the extended Nelson-Siegel forward rate function. The paraeters 0, and ust be positive. Svensson s function is a su of four coponents. The first coponent is a constant, 0, and is the horizontal asyptote of the forward rate function. It ay be interpreted as the constant level the forward rates converge towards in the long ter. The second coponent, -/, is onotonically decreasing or increasing, if is negative towards zero when the ter to aturity is increasing. When the ter to aturity approaches zero, the forward rate approaches the constant 0. This ust obviously be non-negative to ensure non-negative forward interest rates. The third coponent generates a hup or a U shape, if is negative on the curve as a function of the ter to aturity. Finally, the fourth coponent generates an additional hup or U shape. Thus, the function allows for two hups or U shape. The second additional hup or U shape is also the difference between the Svensson ethod and the Nelson and Siegel ethod. It can be shown that the corresponding spot interest rate function can be ressed as: = 3 0, s The paraeters in the forward rate function are estiated by solving the following axiu likelihood estiation proble:

13 = ln : n y Y n Max ε ε σ πσ inserting the following MLE: = = n y Y n σ ε for σ ε. The continuously copounded spot and forward interest rates that are derived fro the equations above for a given, are finally transfored into annually copounded interest rates, i.e. = 00, 00, s s a = 00, 00, f f a

14 References Anderson, N & Sleath, J. 999: New estiates of the UK real and noinal yield curves, Bank of England Quarterly Bulletin Noveber 999 BIS 999: Zero-coupon yield curves: Technical Docuentation, Bank for International Settleents, March 999 Hein, J. 003: Liquidity and scarcity in the Norwegian governent bond arket, Econoic Bulletin, 4/003, Norges Bank Kloster, A. 000: Estiating and interpreting interest rate ectations, Econoic Bulletin, 3/000, Norges Bank Myklebust, G. 005: Risikopreier i det norske rentearkedet, Penger og Kreditt, 3/005, Norges Bank Nelson, C. R. & Siegel, A. F. 987: Parsionious odelling of yield curves, Journal of Business, Vol 60 Svensson, L. E. O. 994: Estiating and interpreting forward interest rates: Sweden , International Monetary Fund, IMF Working Paper, 994/4 Svensson, L. E. O. 995: Estiating forward interest rates with the extended Nelson & Siegel Method, Quarterly Rewiev, 995:3, Sveriges Riksbank Valseth, S. 003: Renteforventninger og betydningen av løpetidspreier, Penger og Kreditt, /003, Norges Bank

15 Gaute Myklebust: Docuentation of the ethod used by Norges Bank for estiating iplied forward interest rates Staff Meo 005/