QIS 5 Risk-free interest rates Extrapolation method

Transcription

1 CEIOPS QIS 5 Rsk-free nterest rates Extrapolaton method Table of contents Introducton Lqud ponts of rsk-free nterest rate curve Extrapolaton method Determnaton of ultmate forward rate Transton to the equlbrum rate Technque for transton Speed of transton Allowance for lqudty premum Appendx A Estmaton of expected long term nflaton rate Estmaton of the expected real rate of nterest Appendx B Smth-Wlson technque... 6 Appendx C Relaton between spot and forward rates The Lqudty Premum - An Example... 5

2 Introducton For labltes expressed n any of the EEA currences, Japanese yen, Swss franc, Turksh lra or USA dollar, QIS5 provdes to partcpants rsk-free nterest rate term structures. The approprate rsk-free nterest rate term structure s n practce constructed from a fnte number of data ponts. Therefore, both nterpolaton between these data ponts and extrapolaton beyond the last avalable data pont of suffcent lqudty s requred. The purpose of ths paper s to descrbe how the extrapolated part of the nterest rate term structures for currences where the relevant rsk-free nterest rate term structures are provded n the spreadsheet ncluded n QIS5 package was set up. Lqud ponts of rsk-free nterest rate curve The non-extrapolated part of the rsk-free nterest rate curve for QIS5 purposes was delvered by the ndustry. The aspects of the rsk-free nterest rate term structure that had to be consdered were the selecton of the basc rsk-free nterest rate term structure, the method for adjustng nter-bank swaps for credt rsk, the assessment of the last lqud pont to enter the yeld curve extrapolaton and the dervaton of the lqudty premum. Detals on the determnaton of these can be found n the documentaton gven by CFO and CRO Forum. Ths data bass conssts of contnuously compounded spot rates. Furthermore, the rates have been derved from market data by fttng a smoothed regresson splne to market swap rates usng Barre & Hbbert's yeld curve fttng methodology. To use these rates n the extrapolaton tool, they had to be converted nto spot rates wth annual compoundng, as the extrapolaton tool expects spot rates wth annual compoundng as nput and delvers spot rates wth annual compoundng as output. Furthermore, the Smth-Wlson approach acheves both nterpolaton (for maturtes n the lqud end of the term structure where rsk-free zero coupon rates are mssng) and extrapolaton. A consstent approach for nter- and extrapolaton would be preferred by CEIOPS and we would thus recommend not to use already smoothed market data as a startng pont. Extrapolaton method For QIS5, macroeconomc extrapolaton technques are used to derve the extrapolaton beyond the last avalable data pont. The overall am s to construct a stable and robust extrapolated yeld curve whch reflects current market condtons and at the same tme embodes economcal vews on how unobservable long term rates are expected to behave. Macroeconomc extrapolaton technques assume a long-term equlbrum nterest rate. A transton of observed nterest rates of short-term maturtes to the assessed equlbrum nterest rate of long-term maturtes takes place wthn a certan maturty spectrum. Valuaton of techncal provsons and the solvency poston of an nsurer or rensurer shall not be heavly dstorted by strong fluctuatons n the short-term nterest rate. Ths s partcularly mportant for currences where lqud reference rates are only avalable for short term maturtes and smple extrapolaton of these short term nterest rates may 2

3 cause excessve volatlty. A macro-economc model meets the demands on a model that ensures relatvely stable results n the long term. There are some consderatons that have to be faced when specfyng the macroeconomc extrapolaton method for QIS5 purposes. These are examned further n the followng sectons. Determnaton of ultmate forward rate A central feature s the defnton of an uncondtonal ultmate long-term forward rate (UFR) for nfnte maturty and for all practcal purposes for very long maturtes. The UFR has to be determned for each currency. Whle beng subject to regular revson, the ultmate long term forward rate should be stable over tme and only change due to fundamental changes n long term expectatons. The uncondtonal ultmate long-term forward rate s determned for each currency by macro-economc methods. Common prncples governng the methods of calculaton should ensure a level playng feld between the dfferent currences. For all currences nterest rates beyond the last observable maturty - where no market prces exst - are needed. The most mportant economc factors explanng long term forward rates are long-term expected nflaton and expected real nterest rates. From a theoretcal pont of vew t can be argued that there are at least two more components: the expected long-term nomnal term premum and the long-term nomnal convexty effect. The term premum represents the addtonal return an nvestor may expect on rsk-free long dated bonds relatve to short dated bonds, as compensaton for the longer term nvestment. Ths factor can have both a postve and a negatve value, as t depends on lqudty consderatons and on preferred nvestor habtats. As no emprcal data on the term premum for ultra-long maturtes exsts, a practcal estmaton of the term premum s not undertaken for QIS5 purposes. The convexty effect arses due to the non-lnear (convex) relatonshp between nterest rates and the bond prces used to estmate the nterest rates. Ths s a purely techncal effect and always results n a negatve component. In order to have a robust and credble estmate for the UFR the assessment s based on the estmates of the expected nflaton and the expected short term real rate only. Makng assumptons about expectatons ths far n the future for each economy s dffcult. However, n practce a hgh degree of convergence n forward rates can be expected when extrapolatng at these long-term horzons. From a macro economcal pont of vew t seems consstent to expect broadly the same value for the UFR around the world n 00 years. Nevertheless, where the analyss of expected long term nflaton or real rate for a currency ndcates sgnfcant devatons, an adjustment to the long term expectaton and thus the UFR has to be appled. Therefore, three categores are establshed capturng the medum UFR as well as devatons up or down. Thus, the macro economcally assessed UFR for use n the QIS5 s set to 4.2 per cent (+/- percentage ponts) per anno. Ths value s assessed as the sum of the expected nflaton rate of annually 2 per cent (+/- percentage ponts) and of an expected short term return on rsk free bonds of 2.2 per cent per anno. Further detals on the estmaton of expected nflaton rate and expected real rate can be found n Appendx A. For QIS5 the followng UFR are used: 3

4 Category Currences Macro economcally assessed UFR JPY, CHF Euro, SEK, NOK, DKK, GBP, 4.2 USD, CZK, BGN, LVL, LTL, EEK, PLN, RON, HUF, ISK 3 TRY 5.2 Transton to the equlbrum rate Ths paragraph consders the ssue of how to extrapolate between the estmated forward rates and the uncondtonal ultmate forward rate. Technque for transton For QIS5 the Smth-Wlson method wll be used. If appled to observed zero coupon bond prces from the lqud market, ths method ensures that the term structure s ftted exactly to all observed zero coupon bond prces,.e. all lqud market data ponts are used wthout smoothng. If appled to the already smoothed market data that the ndustry has delvered for the lqud part of the term structure, the extrapolated term structure wll pass through all zero coupon market rates that are gven as nput. Furthermore, wth the Smth-Wlson approach both nterpolaton (for maturtes n the lqud end of the term structure, f rsk-free zero coupon rates are mssng) and extrapolaton can be acheved. The ndustry has already delvered rsk-free rates for all maturtes from one year up to the last lqud maturty (gven n whole years) for the currences n queston. Thus, for these currences, the Smth-Wlson approach wll be used only for nterpolaton n cases of non-nteger maturtes and for the extrapolaton beyond the last lqud data pont. It s a sophstcated approach that s stll easy to use, and gves both a relatve smooth forward rate and a smooth spot rate curve n the extrapolated part. Further detals on the Smth-Wlson technque can be found n Appendx B. Nevertheless, the lnear method has been also run n order to provde a knd of crosscheckng, avodng a full relance n a sngle method and enhancng the robustness of results provded by the Smth-Wlson approach. Speed of transton The speed of transton towards the UFR can be specfed by the maturty T2 at whch the forward rate curve reaches the UFR. A range for T2 between 70 and 20 years s consdered approprate. The forward rate curve s deemed to reach the UFR at T2, f the spread between the UFR and the annual forward rate at T2 - n absolute values - les wthn predefned lmts. These lmts are chosen as a threshold of 3 BP for QIS 5 purposes. The choce for the maturty at whch the ultmate forward rate wll be reached between 70 or 20 years has an mpact on the stablty of the yeld curve over tme. On the one hand the yeld curve follows a qute flat course beyond the maturty the UFR s reached. Therefore, the earler the UFR s reached the more stable s the yeld curve for long maturtes. On the other hand, t has to be consdered that the earler the UFR s reached, the more senstve wll the yeld curve be towards changes n the choce of the UFR. 4

5 In lght of ths, CEIOPS provdes a set of extrapolated nterest rate curves for dfferent choces of T2, namely 70, 90 and 20 years. Nevertheless, there s no fxed, predefned maturty where the UFR s deemed to be arrved at n the Smth-Wlson approach, but the speed of convergence to the uncondtonal ultmate forward rates has to be set. Therefore, n the Smth-Wlson technque the speed of transton as defned above has to be translated nto a convergence parameter α (alpha). Thomas, who was lookng at Australan term structures, ftted ths parameter emprcally to α = 0. as t ensured sensble results and economcally approprate curves n most cases. For QIS5, n a frst step the default value for alpha s 0.. Only f the extrapolated rates devate from the UFR at the predefned tme T2 by more than 0.03% (.e. 3 BP), then alpha s recalbrated n a second step, such that the spread les wthn ths threshold. Ths ensures that the extrapolated curve s suffcently close to the chosen UFR at T2. Allowance for lqudty premum For every currency, the lqudty premum s allowed for n the rsk-free nterest rate curve up to a cut-off pont. Past that cut-off pont a phasng-out perod of 5 years for the s appled. There are two alternatves for the applcaton of the lqudty premum to the basc rskfree nterest rate curve. The frst alternatve s to nterpret the as a spot rate and adjust the rsk-free basc term structure (.e. the basc zero coupon rates) wth ths premum. The second alternatve would be to nterpret the as a forward rate and adjust the forward rates wth the. An example of these two possble optons can be found n Appendx C. The man dfference between adjustng the spot rates and adjustng the forward rates s the mpact on dscountng cash flows beyond maturtes for whch a lqudty premum n assets s deemed to exst and s deemed to be measurable:. In the frst alternatve, the s restrcted to the dscountng of cash flows of maturtes for whch a lqudty premum for assets can be assessed. The method therefore fulfls the requrement that no lqudty premum should be ncluded n the extrapolated part of the nterest rate curve, as formulated n pont I 6 c) (page 8) of the Task Force report and as reterated n the Commsson draft mplementng provsons, Artcle IR6(3): No llqudty premum shall be appled to the extrapolated part of the relevant basc rsk-free nterest rate term structure. Furthermore, the method s n lne wth the measurement methods as presented n the Task Force report. The common feature n all three methods mentoned s that the s estmated from the spreads of the yelds on corporate bonds over the yelds of government bonds for gven maturtes. The CDS Negatve-Bass method deduces from these spreads the prces of the Credt Default Swaps on the corporate bonds n order to get the. In the Structural Model Method the actual computed spreads are decreased by theoretcal credt spreads, computed wth methods from opton prcng theory. In the Covered Bond Method the s computed drectly as spread between the yelds of two bonds (other nstruments) whch are equal n all, except lqudty. Mchael Thomas, Eben Maré: Long Term Forecastng and Hedgng of the South Afrcan Yeld Curve, Presentaton at the 2007 Conventon of the Actuaral Socety of South Afrca 5

6 Nevertheless, ths method nduces forward rates (mpled by the adjusted rsk-free term structure) that mght not behave very smooth n the 5 year phasng-out perod. For a hgh and hgh maturtes, the forward rates mpled by the adjusted rsk-free term structure can even become negatve durng ths perod. 2. In the second alternatve the has an mpact on the dscountng of cash flows of all maturtes even beyond maturtes for whch a lqudty premum n assets s deemed to exst. Ths s due to the fact that spot rates are a knd of average of forward rates, and thus spot rates mplctly contan the lqudty adjustments on the forward rates that enter the average. For ths method the term structure s smoother, and the mpled forward rates stay postve durng the 5 year phasng-out perod. Nevertheless, ths method s not consstent wth the requrement that no lqudty premum should be ncluded n the extrapolated part of the nterest rate curve, as formulated n pont I 6 c) of the Task Force Report and as reterated n the Commsson draft mplementng provsons Artcle IR6(3). After havng dscussed the pros and cons of the ssue, the extrapolaton team decded to attach the utmost weght to the condton that no lqudty premum should be ncluded n the extrapolated part of the nterest rate curve, as requred n pont I 6 c) (page 8) of the Task Force report, and hence to mplement the as an adjustment of the spot rate for QIS5. In ths case no lqudty premum would be allowed for n the dscountng of the cash flows beyond maturtes for whch a lqudty premum n assets s deemed to exst. 6

7 Appendx A Estmaton of expected long term nflaton rate The expected nflaton should not solely be based on hstorcal averages of observed data, as the hgh nflaton rates of the past century do not seem to be relevant for the future. The fact s that n the last 5-20 years many central banks have set an nflaton target or a range of nflaton target levels and have been extremely successful n controllng nflaton, compared to prevous perods. Barre Hbbert 2 propose to assess the nflaton rate as 80 per cent of the globally prevalng nflaton target of 2 per cent per anno and 20 per cent of an exponentally weghted average of hstorcal CPI nflatons when modellng the term structure n ther Economc Scenaro Generator. When they assess the hstorcal nflaton average of the man economes they stll compute a hgh level as of December 2007 (they assess an expected global nflaton rate of 2.4 per cent per anno) but wth a strong downward trend over the sample of data they consdered. In order to have a robust and credble estmate for the UFR, the standard expected long term nflaton rate s set to 2 per cent per anno, consstently to the explct target for nflaton most central banks operate wth 3. Nevertheless, based on hstorcal data for the last 0-5 years and current nflaton, two addtonal categores are ntroduced to capture sgnfcant devatons ether up or down n the expected long term nflaton rate for certan countres. Table shows nflaton data for the OECD-countres n the perod Table : Inflaton OECD Countres Prce ndces (MEI) : Consumer prces - Annual nflaton Data extracted on 5 Mar 200 3:35 UTC (GMT) from OECD.Stat Measure Frequency Percentage change on the same perod of the prevous year Annual Tme Country Australa Austra Belgum Canada Czech Republc Denmark Fnland France Germany Greece Hungary Iceland Steffen Sørensen, Interest rate calbraton How to set long-term nterest rates n the absence of market prces, Barre+Hbbert Fnancal Economc Research, September Also the European Central bank ams at an annual nflaton just below 2 per cent. 7

8 Ireland Italy Japan Korea Luxembourg Mexco Netherlands New Zealand Norway Poland Portugal Slovak Republc Span Sweden Swtzerland Turkey Unted Kngdom Unted States G OECD - Europe OECD - Europe excludng hgh nflaton countres OECD - Total OECD - Total excludng hgh nflaton countres Table shows that two OECD-countres had nflaton above 5 percent n 2009: Iceland (2 percent) and Turkey (6.3 percent). Durng the last 5 years, Turkey has been categorsed by OECD as a hgh nflaton country 4. Turkey s nflaton target s also hgher (5-7.5% for the perod ) than n other countres. Based on ths data bass, Hungary and Iceland are possble canddates for the hgh nflaton group. However, devatons to the average nflaton rate are far more moderate than those for Turkey. Furthermore, these countres are expected to jon the Euro sooner or later (and thus have to fulfl the convergence crtera). Therefore, Hungary and Iceland are classfed n the standard nflaton category. Japan, havng deflaton n the perod snce 994, s an obvous canddate for the low nflaton -group. Swtzerland can also be seen as an outler. Ths s due to the fact that hstorcally relatvely low nflaton rates can be observed and that Swtzerland s partcular attractve n the nternatonal fnancal markets (exchange rate condtons, lqudty, save haven 5...). For these reasons, lower nflaton assumptons are appled for the Swss francs. The estmate covers one-year nflaton rate years from now. It s arbtrary to say whether the nflaton dfferences we see today and have seen the last 5 years wll persst 00 years nto the future. However, hstorcal evdence and current long term nterest rates ndcate that t s reasonable to have three groups of currences wth dfferent nflaton assumptons. The standard nflaton rate s set to 2 per cent per anno. To allow for devatons up and down to the standard nflaton rate, an adjustment to the estmate of +/- percentage pont s appled for the hgh nflaton group and the low nflaton group respectvely. Ths adjustment of percentage pont wll be appled to the Why are Returns on Swss Francs so low? Rare events may solve the puzzle. Peter Kugler, Weder d Mauro 8

9 estmated nflaton rate for outlers based on dfferences n current long term nterest rates (30Y), observed hstorcal dfferences between the average nterest rate and dfferences n short term nflaton expectatons. The followng groupng s used for the estmated expected long term nflaton rate: Standard nflaton rate set to 2%: Euro-zone, UK, Norway, Sweden, Denmark, USA, Poland, Hungary, Iceland, Czech Republc, Bulgara, Latva, Lthuana, Estona and Romana Hgh nflaton rate set to 3%: Low nflaton rate set to %: Turkey Japan, Swtzerland* * combned effects Estmaton of the expected real rate of nterest We expect that the real rates should not dffer substantally across economes as far out as 00 years from now. Elroy Dmson, Paul Marsh and Mke Staunton provde a global comparson of annualzed bond returns over the last 0 years (900 to 2009) for the followng 9 economes: Belgum, Italy, Germany, Fnland, France, Span, Ireland, Norway, Japan, Swtzerland, Denmark, Netherlands, New Zealand, UK, Canada, US, South Afrca, Sweden and Australa 6. 6 Credt Susse Global Investment Returns Yearbook 200, To be found at 9

10 Fgure : Real return on bonds Source: Dmson, Marsh and Staunton Credt Susse Global Investment Returns Yearbook n % Aus Ble Can Den Fn Fra Ger Ire It Jap Neth NZ No SouthA Spa Sw e Sw UK USA Fgure shows that, whle n most countres bonds gave a postve real return, sx countres experenced negatve returns. Mostly the poor performance dates back to the frst half of the 20 th century and can be explaned wth tmes of hgh or hypernflaton 7. Aggregatng the real returns on bonds for each currency 8 to an annual rate of real return on globally dversfed bonds gves a rate of.7 per cent. In an earler publcaton, the same authors compared the real bond returns from the second versus the frst half of the 20 th century for the followng 2 economes: Italy, Germany, France, Japan, Swtzerland, Denmark, Netherlands, UK, Canada, US, Sweden and Australa 9. The average real bond return over the second half of the 20 th century was computed as annually 2.3 per cent (compared to -. percent for the frst half of the 20 th century). 7 German hypernflaton n 922/923, n Italy an nflaton of 344% n 944, n France 74% n 946 and n Japan 37% n Average where each return s weghted by ts country s GDP. 9 Elroy Dmson, Paul Marsh and Mke Staunton: Rsk and return n the 20th and 2th, Busness Strategy Revew, 2000, Volume ssue 2, pp -8. See Fgure 4 on page 5. The artcle can be downloaded at: 3D %26rep%3Drep%26type%3Dpdf+Rsk+and+return+n+the+20th+and+2th+Centures&hl=n o&gl=no&sg=ahietbqbxwuxzno6vvlqkv0kz63lkhb0g 0

11 Fgure 2: Real bond returns: frst versus second half of 20 th century* Source: Dmson, Marsh and Staunton (ABN- Ambro/LBS) ,6 4,7,8 3,2 2,3,6 0,9 2,4,2,2,0,4,6,6 3,9 2,, 2,8 3,,6,5 0 n % -2 -, ,2-6 -6,5-8,2 Ger Fra It Jap AVG UK Can Neth US Den Aus Sw Swe Frst half-century Second half-century * Data for Germany excludes AVG = Average In lght of the above data, 2.2 per cent s an adequate estmate for the expected real nterest rate. Appendx B Smth-Wlson technque The Smth-Wlson approach s a macroeconomc method: a spot (.e. zero coupon) rate curve s ftted to observed bond prces wth the macroeconomc ultmate long term forward rate as nput parameter. In ts most general form, the nput data for the Smth-Wlson approach can consst of a large set of dfferent fnancal nstruments relatng to nterest rates. We wll lmt the nput to zero coupon bond prces, and wll only put down the formulae for ths smple case. In other words: we assume that n the lqud part of the term structure the rsk-free zero coupon rates for all lqud maturtes are gven beforehand. Our task s to assess the spot rate for the remanng maturtes. These are both maturtes n the lqud end of the term structure where rsk-free zero coupon rates are mssng (nterpolaton) and maturtes beyond the last observable maturty (extrapolaton). Let s assume that we have market zero coupon rates for J dfferent maturtes: u, u 2, u 3, and so on. The last maturty for whch market data s gven s u J. The market prce P(t) for a zero coupon bond of maturty t s the prce, at valung tme

12 t 0 = 0, of a bond payng at some future date t. Dependng on whether the market data spot rates are gven as contnuously compounded rates R ~ or as rates R wth annual compoundng, the nput zero bond prces at maturtes u j are: P ~ ) = exp( u * R ) for contnuously compounded rates, and ( u j j u j u j u j -u j P ( u j ) = ( + R u j ) for annual compoundng. ~ The relaton between the two rates s gven through R = ln( + R ). u j u j Our am s to assess the functon P(t) for all maturtes t, t > 0. From the defnton of the ~ prce functon P t) = exp( t * R ) for contnuously compounded rates and ( t -t P ( t) = ( + ) for annual compoundng, we then can assess the whole rsk-free term R t structure at valung date t 0 = 0. General on extrapolaton technque Most extrapolaton methods start from the prce functon, and assume that the prce functon s known for a fxed number of say J maturtes. In order to get the prce functon for all maturtes, some more assumptons are needed. The most common procedure s to mpose n a frst step - a functonal form wth K parameters on the prce functon P 0. These functonal forms could be polynomals, splnes, exponental functons, or a combnaton of these or dfferent other functons. In some of the methods, n a second step, the K parameters are estmated va least squares at each pont n tme. In other methods K equatons are set up from whch the K parameters are calculated. The equatons are set up n a manner that guarantees that P has the features desred for a prce functon: A postve functon, wth value at tme t=0, passng through all gven data ponts, to a certan degree smooth, and wth values convergng to 0 for large t. Smth-Wlson approach Smth and Wlson 23 proposed a prcng functon (here reproduced n a restrcted form, only for valung at pont t 0 =0) of the followng form: P J UFR* τ ( τ ) = e + ς * W ( τ, u ), τ = 0 wth the symmetrc Wlson - functons W ( τ, u ) defned as : 0 In ther respectve models, Svensson for nstance mposes a parametrc form wth 6 parameters and Nelson- Segel one wth 4 parameters. BarreHbbert use cubc splnes n the lqud part of the term structure and Nelson-Segel for the extrapolaton part. 2 Smth A. & Wlson, T. Fttng Yeld curves wth long Term Constrants (200), Research Notes, Bacon and Wodrow. Referred to n Mchael Thomas, Eben Maré: Long Term Forecastng and Hedgng of the South Afrcan Yeld Curve, Presentaton at the 2007 Conventon of the Actuaral Socety of South Afrca 3 Andrew Smth: Prcng Beyond the Curve dervatves and the Long Term (200), presentaton to be found at 2

13 W ( τ, u ) = e * α*max( τ, u ) α*mn( τ, u ) α*mn( τ, u ) { α * mn( τ, u ) 0.5* e *( e e )} UFR*( τ + u ) The followng notaton holds: - J = the number of zero coupon bonds wth known prce functon - u, =, 2, J, the maturtes of the zero coupon bonds wth known prces - τ = term to maturty n the prce functon - UFR = the ultmate uncondtonal forward rate, - α = mean reverson, a measure for the speed of convergence to the UFR - ζ = parameters to ft the actual yeld curve The so called kernel functons K (τ) are defned as functons of τ: K ( τ ) = W ( τ, u ), τ > 0 and =,2, J They depend only on the nput parameters and on data from the nput zero coupon bonds. For each nput bond a partcular kernel functon s computed from ths defnton. The ntuton behnd the model s to assess the functon P(t), from whch we am to calculate the term structure, as the lnear combnaton of all the kernel functons. Ths remnds of the Nelson-Segel method, where the forward rate functon s assessed as the sum of a flat curve, a sloped curve and a humped curve, and the Svensson method, where a second humped curve s added to the three curves from Nelson-Segel. The unknown parameters needed to compute the lnear combnaton of the kernel functons, ζ, =, 2, 3 J are gven as solutons of the followng lnear system of equatons: P( u ) = e P( u ) = e P( u 2 J ) = e UFR* u UFR* u2 UFR* uj J = J = J = ς * W ( u, u ) ς * W ( u, u )... ς * W ( u In vector space notaton ths becomes: 2 J, u ) Ρ = Ε +W *ζ, wth: Ρ = ( P( u), P( u2 ),... P( u J )) T (The superscrpt T denotng the transposed vector) Ε = ( e, e,... e UFR* u UFR* u2 UFR* uj T ς = ( ς, ς,... ς,) and 2 T, ), 3

14 W = (W(u, u j )) =, J, j=, J, a JxJ-matrx of certan Wlson functons From ths notaton we see at once that the soluton (ζ, ζ 2, ζ 3, ζ J ) s easly calculated by nvertng the JxJ-matrx (W(u, u j )) and multplyng t wth the dfference of the P-vector and the E-vector,.e. ζ = W *( P E), We can now plug these parameters ζ, ζ 2, ζ 3, ζ J nto the prcng functon and get the value of the zero coupon bond prce for all maturtes τ, for whch no zero bonds were gven to begn wth: P J UFR* τ ( τ ) = e + ς * W ( τ, = u ), τ>0 From ths value t s straghtforward to calculate the spot rates by usng the defnton of ~ the zero coupon bond prce. The spot rates are calculated as R * ln( τ = ) for τ P( τ ) contnuous compounded rates and R ( τ = ) f annual compoundng s used. Appendx C τ P( τ ) Relaton between spot and forward rates The rsk-free spot rate for a gven maturty T can be nterpreted as the yeld of a rsk-free zero coupon bond wth maturty T. Forward rates are the rates of nterest mpled by spot rates for perods of tme n the future. The relaton between spot and forward term structures can best be llustrated by the followng formulae, the frst for annually compounded spot rates and the second for contnuously compounded spot rates: Annual compoundng: ( + R T ) T = ( + =... = T T 2 RT ) ( + FR( T, T )) = ( + RT 2) ( + FR( T 2, T ) )( + FR( T, T )) = ( + FR(0,) ) ( + FR(,2) ) ( + FR(2,3) )... ( + FR( T 2, T ) ) ( + FR( T, T )), where R T, (R T- ) denotes the spot rate for maturty T, (T-), whle FR(,+) denotes the annual forward rate for the perod from year end to year end +, for =0,, 2,. T. Contnuous compoundng: e ~ ~ ~ T* RT ( T )* RT FRc( T, T ) ( T 2)* RT 2 = e = e e = e FRc(0,) e FRc(,2)... e FRc( T 2, T ) e e FRc( T, T ) FRc( T 2, T ), e FRc( T, T ) =.. R ~ ( R ~ where T T ) s the contnuously compounded basc spot rate for maturty T, (T-), whle FRc(,+) denotes the annual contnuous forward rate for the perod from year end to year end +, for =0,, 2,. T. 4

15 The Lqudty Premum - An Example In Annex A of the Task Force report a possble proxy for the observable n fnancal markets s gven. Applyng the smplfed formula gven a n per annum bps relatve to swap was computed for EUR, GBP and USD at gven dates. The most recent estmated for Euro was 59 bps as per End September In accordance wth prncple #3 of pont I 4 Task Force report (page 3), the addton of a lqudty premum shall be lmted to maturtes where an addtonal return can be earned. Let us assume that for the Euro-zone approprate nstruments wth maturtes up to 30 years are avalable and that the porton of the observed n fnancal markets correspondng to (re-)nsurance oblgatons s 00%. Followng the nstructons n the Task Force report we compute the gven n Table below. If the from ths example s appled as spot rate adjustment (frst alternatve n 3.3), then the rates n Table can be added to the spot rates from the basc rsk-free term structure. A cash flow wth maturty 26 years wll thus be dscounted wth the basc rskfree rate for maturty 26 years ncreased by 47 bps. A cash flow wth maturty 33 years wll accordngly be dscounted wth the basc rsk-free rate for maturty 33 ncreased by o bps. As can be seen, the wll have no mpact on the dscountng of the cash flows wth maturtes of 30 years and from 30 years onwards. Table : Lqudty premum 00%, EUR, as of Maturty n years n bps Maturty n years n bps Maturty n years n bps Maturty n years n bps Maturty n years n bps Maturty n years Source: Task Force on the Illqudty Premum, Report. Ceops-SEC-34/0, March 200 n bps If, on the other hand, the n Table s appled as forward rate adjustment (second alternatve n 3.3), then we have to add the s to the forward rates mpled by the rsk- 5

16 free basc spot rate term structure. Wth these (adjusted) forward rates we can compute the adjusted spot rate curve. We are nterested n the effect of the on the adjusted spot rate curve, f ths method (second alternatve n 3.3) s chosen. The best way to see the dfference s to calculate the spreads between the adjusted and the unadjusted spot rates, and compare them to the values gven n table. We want to keep our example smple. Due to the followng remark we can avod assumng an explct basc spot rate term structure, as t s possble to assess the spread drectly by applyng the method gven below n pont 2. For contnuously compounded spot rates, we can see at once from the formula gven n 6. that method and method 2 descrbed below gve the same adjusted spot rate term structure:. Add the s to the basc rsk-free forward rates and apply the formula from Compute spot adjustment rates from the s (usng formula n 6. wth the as forward rates) and add these to the basc spot rate term structure. The spot rate adjustments computed from the rates n Table (usng method 2), are presented n Table 2. Cash flows wth maturty 26 years wll have to be dscounted wth the basc rsk-free rate for maturty 26 years ncreased by 59 bps, cash flows wth maturty 33 years wth the basc rsk-free rate for maturty 33 ncreased by 48 bps, and so on. Even n the dscountng of cash flows as far out as 20 years from now, a 3 bps adjustment due to the lqudty premum has to be taken nto account. Table 2: Lqudty premum spot rates adjustment derved from the rates n Table Maturty n years n bps Maturty n years n bps Maturty n years n bps Maturty n years n bps Maturty n years n bps Maturty n years n bps 6