Measuring Bond Portfolio Value At Risk: Us And Taiwan Government Bond Markets Empirical Research

Transcription

1 Measurng Bond Portfolo Value At Rsk: Us And Tawan Government Bond Markets Emprcal Research Thomas W. Knowles Stuart Graduate School of Busness Illnos Insttute of Technology, USA Ender Su Natonal Kaohsung Frst Unversty of Scence and Technology, Tawan Abstract Ths paper concerns value at rsk analyss of US and Tawan government bond portfolos. We explore four methods of modelng the yeld curve and ts rsk: key rate, three-factor model (level, slope, and curvature), prncpal component analyss (PCA), and structural equaton modelng (SEM). SEM has never been used prevously for analyzng bond rsk. We found that the data analyss methods obtaned smlar VaR fgures for the US and Tawan. For a one-day nvestment horzon, usng the key rate the values were $2.5 and $2.507, respectvely, and usng PCA the values were $2.504 and $ The model ft methods found smaller VaR fgure: usng the three-factor model the VaR was $ and $2.4902, for the US and Tawan, respectvely, and $2.494 and $2.498, usng SEM. The key rate, PCA and SEM all suggest that US has a slghtly hgher VaR value than Tawan. For US bond yeld rsk, the slope of the three-factor model and the medum term of the SEM exhbt hgher VaR senstvty whle for Tawan, the level of the three-factor model and the medum term of the SEM model rsk factors present hgher VaR senstvty. The three-factor model and SEM provde understandable method for nterest rate rsk management and forecastng, but because they tend to underestmate the rsk, data analyss methods such as key rate and PCA are needed to montor bond yeld curve rsk.. Introducton The value of a bond s subject to movement of the yeld curve. Many researchers such as Bloomberg, Standard & Poor, Morgan Stanley, and JP Morgan dsplay the US yeld curve chart movement every day on ther web stes. We can observe from those sources how the yeld curve flows wth changes n the US economy. Notably, the yeld curve flows dramatcally when the Federal Reserve Bank announces an nterest rate change. Typcally, the yeld curve wll be upflow when the US economy s strong and downflow when the US economy s weak. On the other hand, Tawan has become a WTO member, and we expect that the Tawan government bond market wll be traded actvely and lberally. Thus, certanly studyng the yeld curve of the Tawan bond market s mportant and necessary for the growth of the Tawan bond market rsk measure. It wll be nterestng to compare the US and Tawan bond market rsks wth respect to yeld curve model factors and economc factors. We wll apply four models: key rate, three factors--level, slope, and curvature, prncpal component analyss (PCA), and structural equaton modelng (SEM) to analyze and detal the yeld curve movement and ts rsk for the US and Tawan government bond markets. Wth respect to nterest rate rsk and value at rsk (VaR) emprcal study, J P Morgan n ts 996 year "techncal document" [6] has a detaled descrpton for the key rate varance-covarance value at rsk measurement. In realty, key rates play the mportant role for the standardzed rsk factors. As to the ndvdual bond or bond portfolo, we can just use J P Morgan s provded key rate parameters (varance-covarance matrx and key rates), and obtan the bond value at rsk easly. If the bond cash flow perods don't match the key rate perods used n RskMetrcs, then by the cash flow dstrbuton methods, e.g. duraton mappng or varance equalty, we dstrbute the unmatched bond cash flow to the key rate vertces,.e. the standardzed key rates. Smlarly, we can use the key rate parameters provded by J P Morgan to acqure the lnear value at rsk. In addton, key rate duraton (Ho [4]) also can be appled to the cash flow mappng and bond value at rsk measure. Golub and Tllman [4] have appled ths method for the dervaton of the bond value at rsk formulaton. [4] has derved the key rate factor block duraton that can be used for the VaR measure and VaR components of a bond portfolo. RskMetrcs or key rate duraton both retan the problem of dependence between dfferent key rates. In other words, how the relatonshps of key rate factors capture the real world yeld curve movement s the key pont for the key rate applcaton of bond value at rsk. However, the key rate factor dependence wll become an obstacle for nvestors snce they must consder how well the key rate factor relatonshps match the real yeld curve movement. Prncpal component analyss (PCA) s another data

2 analyss model lke the key rate method. Lke the three-factor model, prncple component analyss reduces the numerous yeld change components to a few major components and thus reduces the number of parameter estmates for the measure of the bond value at rsk. A few major component varances can explan almost 99% system varance and most mportantly the components are ndependent of each other. Sngh [9] compared the performances of the key rate duraton, the three-factors yeld curve model, and offered prncpal component analyss of the yeld curve rsk. Golub and Tllman [4] also dscussed usng the prncpal component duratons to measure the bond value at rsk. A new method of modelng the yeld curve s structural equaton modelng (SEM), whch encompasses many research subjects such as covarance structure analyss, latent varable analyss, and exploratory factor analyss. SEM has not been used prevously to model the yeld curve. Wth the help of LISREL and Amos, SEM has been appled n other contexts (e.g. Austn & Calderon [], Tremblay & Gardner [0]). In addton to the PCA exploratory factor analyss, we also wll adopt the SEM confrmatory factor analyss that wll search for the parsmonous unobserved latent factors to explan the observed known endogenous varables varances. Unlke PCA that requres that the factor components should be orthogonal, the SEM confrmatory factor wll allow some relatonshps among the latent factors. Wlson [4] presented the delta gamma approxmaton for nterest rate rsk measures of nonlnear dervatve securtes. Also, JP Morgan RskMetrcs has offered users varance-covarance matrx and the delta gamma method to calculate the nterest rate portfolo rsk measure. More recently, Khndanova and Rachev [7] and Bodurtha and Shen [2] gve portfolo value at rsk measures of bond optons and foregn currences. The major rsk measure process s to dstrbute the cash flow amount nto delta, gamma, and theta cash flows and then use the varance-covarance matrx of ther cash flows to measure the portfolo bond value at rsk. Vlaar [2] and Venkatesh [] proposed a GARCH type model to analyze the tme seres propertes of bond portfolos and test ther VaR estmates. 2. Research Methodology We prefer usng the factor nvolved models to explan the movement of the yeld curve and the value at rsk measures of the rsk factors (components), rather than use tme seres models such as GARCH type models to forecast the bond yeld volatltes and forgo the rsk factor measures. Therefore, we use four lnear models for our bond VaR measures: key rate duraton, three-factor model-level, slope, and curvature-, prncple component analyss, and structure equaton modelng. After modelng the factor s coeffcents, we can calbrate VaR and factor component VaR through the factor s coeffcent transformaton to gve the rsk measure. Because we fnd all the model parameters, we also wll be able to compare each model s descrpton of the yeld curve behavor. Whle each model wll explan the yeld curve n dfferent respects, we wll fnd the advantages and dsadvantages of each method of yeld curve modelng from dfferent aspects, such as the model factor representatons, senstvtes, correlaton, varance covarance, and the overall model sgnfcance tests. 2. Bond Portfolo VaR Measure Model I: Key Rate Duraton Structure and ts VaR TheRskMetrcs techncal document descrbes the constructon of the key rate duraton structure. Frst, t says the yeld curve would be dvded nto several knots or vertces accordng to preselected maturtes, months or years, named as key rates. Then, the actual cash flows of dfferent maturty bonds are dstrbuted nto the preselected key rates. There are two methods for dstrbutng the actual cash flow nto the preselected key rates. One s duraton mappng, and the other s cash flow mappng. We wll explan these later. When c s the dstrbuted key rate cash flow, r s the th key rate (=, 2 n), and V s the total portfolo poston, the key rate duraton can be defned as follows: k() = ( dv / dr )/ V = c /[( + r)...( + r)]* MDur/ V = v / V * MDur () where MDur s the th modfed duraton and s the th dstrbuted cash flow key rate. As to the VaR of a bond portfolo, we calculate the portfolo VaR functon (at a 95% confdence level) as n n p j p VaR =.65 v σ v σ corr(, j), (2) where v = j= and flow poston, and j v j are the th and j th key rate cash σ, σ and corr(, j) are the th and p p j j th key rate cash flow varance and correlaton. (Note: The p refers to prce and s not an exponent.) And, the frst dervatve of the bond prce (P) functon (proposed by Fsher[3]) s dp = MDur * P * dy. (3) Takng the varance of the bond sde equaton, we obtan p y σ = MDur * y * σ. (4) And, the VaR s n n y y VaR =.65 V k y σ k y σ corr(, j) (5) j j j = j= n n VaR =.65V kk cov( y, y ). (6) We know j j = j= V = k y and f we rewrte the above - n V = equaton n terms of matrx expresson, we can fnd VaR =.65 V kω k ', (7) where Ω s the varance-covarance matrx of the yeld change. RskMetrcs uses cash flow mappng for cash flows of maturtes not on the vertces. Under the prncple of volatlty equalty, the actual cash flows are dstrbuted nto the preselected key rates. The procedure s to fnd the v 2

3 parameters â and w that keep the volatlty equal after mappng usng the followng equatons: ˆ σ = aˆσ + (- aˆ) σ 0 a (8) vertc j ˆ σ = w σ + 2 w(- w) ρ σ σ + (- w) σ, (9) vertc, j j j where â s the maturty weght of volatlty that can be estmated frst by lnear nterpolaton by maturty between th and jth cash flows (so, f the maturty s n the mddle of the maturtes, t s estmated as 0.5). w s the weght of cash flow mappng that can be solved for secondly by equaton (9) wth only w unknown. Thus, f we can estmate parameter â and solve for parameter w, then we can use above key rate duraton to measure bond cash flow VaR. 2.2 Bond Portfolo VaR Measure Model II: Level Slope Curvature Yeld Curve Modelng and ts VaR Nelson and Segel [8] derved the level (L), slope (S), and curvature (C) yeld curve model and Wllner [3] further appled ths model. The yeld curve model s used to explan the effects on the bond yeld curve due to the economc factor changes such as nflaton, the busness cycle, and nterest rate volatlty changes. We derve the model as follows: - m / τ (- e ) -mτ Y( L, S, C, m) = L+ ( S+ C) -Ce, (0) m / τ where m s the maturty and s the locaton parameter (hump or vertex). Further, we can defne X( m) = ( X2( m))/( m/ τ ) and ( ) m / τ X2 m = e ; thus the yeld curve functon becomes Y( L, S, C, m) = L+ X( m) S + ( X( m) X2( m)) C () We obtan the level, slope, and curvature duratons by takng the dervatve of the yeld curve functon to obtan dy dy dy dy = L + S + C (2) dl ds dc dy dy dy =, = X( m), = ( X( m) X 2 ( m )). (3) dl ds dc If the bond value s ct Bm ( ) =Σ t, (4) ( + Ym ( )) ( recall c t s the t th bond cash flow), the bond prce changes can be descrbed as db db = dy dy = db ( dy L+ dy S + dy C ) dy dl ds dc db = ( dl + X ( m ) ds + ( X ( m ) X 2( m )) dc ). (5) dy And, through the known three-factor coeffcents, we can derve the VaR as the followng equaton shown: 3 3 VaR =.65 v σ v σ corr( m, n), (6) m m n n m= n= where v m and v n are the cash flows attrbuted to the three factors. 2.3 Bond Portfolo VaR Measure Model III: Prncple Component Analyss and ts VaR Prncple component analyss s also a lnear structure model that tres to fnd the major lnear factors (usually three) to explan the varances of the research objectve dependent varables (often more than three). It s qute nterestng to compare those lnear models and fnd valuable consequences among those models such as the factor volatlty explanaton ablty, the factor relatonshps, and the factor representatons. We can obtan the egen vectors and values needed to obtan the prncple component factors by maxmzng the varance of the PCA factors after transformng the varance-covarance of dependent varables. Suppose that the key rate varance-covarance matrx s Σ, the egen vector s c, and the egen value s λ. Then, Σ c = λc, (7) The above equaton expressed n terms of matrx operatons s c... cn c... cn λ Σ =, (8) c n... c nn c n... c nn λ n Σ C = CΛ, (9) where C s called the characterstc matrx, and s orthogonal to tself; thus ' CC = I, (20) so that ' ' CΣ C = CCΛ =Λ. (2) Due to the prncple component factor ndependence, we thus could fnd the 95% confdence level would be.65 n f VaR = v σ, (22) 2 = where v s the th prncple component cash flow wth respect to the one bass pont change, σ f s the th varance of the prncple component,.e. the characterstc matrx. The v can be found through the characterstc matrx transformaton as the followng: v = pjc j, (23) j where c j s the j th element of the th characterstc component vector matrx and p j s the present value of the j th cash flow. 2.4 Bond Portfolo VaR Measure Model IV: Structure Equaton Model We adopt the SEM confrmatory factor model. If there are n set of measures wth m, m 2,, m n (q=m + m 2 + +m n ) varables n each set respectvely, then we can set X = Λ η + ε, (24) where X s the dfferent maturty yeld data vector wth order of q by one, η s the structure factor n by one vector, Λ s the structural coeffcent matrx of X on η wth order of q by n, and ε s the error term wth order of q by one. We note that confrmatory factor analyss needs 3

4 to be done to fnd the approprate factor connectons wth the observable varables. 3. Yeld Curve Model Applcatons For US and Tawan Government Bonds The US government bond market s the most effcent nterest rate market n the world. Many fnancal nsttutons-banks, nsurance, and nvestors- make a large number of nterest rate transactons to nvest, hedge, speculate or arbtrage nterest rate changes. US treasury blls, treasury notes and treasury bonds are the most traded securtes and have dfferent maturtes. By ssung them, the US government can fnance ts budget defct or refnance ts old debt. Snce 99 the Tawan government bond market has adopted bd-ask prce auctons, and over tme the bond market n Tawan wll become more lqud and effcent. The tradng volume has ncreased snce Research Data The US Treasury bond yeld curve data can be acqured from the web database of the US Federal Reserve Bank. However, the yeld data of dfferent maturty Tawan government bonds would be dffcult to obtan snce the Tawan bond market s not as actve and publc. We can fnd the bond data from Aremos databank or the R.O.C. Over The Counter Securty Exchange databank. For the US government bond yeld data, we use 3-month, 6-month, -year, 2-year, 3-year, 5-year, 7-year, 0-year, 20-year, and 30-year nterest rates because those maturtes matched Rskkmetrcs key rate maturtes. We used maturtes of 3-month, 6-month, -year, 3-year, 5-year, 7-year, and 0-year for the Tawan government bond yeld data. However, Tawan's long-term bonds have been traded sporadcally. There s not enough data n Aremos and t has questonable ntegrty. Thus, we must nterpolate Tawan's long-term data for 5-year, 7-year, and 0-year maturty bond data. We prmarly use splne nterpolaton methods to form the tme seres of ts bond data. Unfortunately, there s lttle data for longer terms such as 5 years and more; thus we forgo terms longer than 5 years for Tawan. The research perods for US bond yeld data are from /4/999 to /4/2005 and from 2/04/999 to /4/2005 for Tawan bond yeld data. Ths yelds a total sample sze of 502 observatons for each country s bond yeld data. 3.2 Key Rate Duraton Value at Rsk Frst, we wll use the key rate duraton method to measure the bond value at rsk. Due to the wde range of dfferent bond maturtes, we use RskMetrcs' key rates and use maturtes of 3 months, 6 months, year, 2 years, 3 years, 5 years, 7 years, 0 years, 20 years, and 30 years as our research key rates. Furthermore, JP Morgan RskMetrcs has provded ts US bond market ndex. Thus, we wll use ths ndex as the base nvestment amount for the measure of bond value at rsk. In Table, we present the key rate correlaton/varance-covarance matrx for the computaton of the VaRs for US and Tawan bond yeld data. Table Key Rate Correlaton/Varance-Covarance Matx (page of 2) I. US Bond Yelds Maturty Spot Rate Stdev CM3m CM6m CMY CM2Y CM3Y CM5Y CM7Y CM0Y CM20Y CM30Y CM3m 2.370%.929% 3.72E E E E E E-04.85E-04.37E E-05.47E-04 CM6m 2.60%.853% E E E E E-04.8E-04.34E E-05.42E-04 CMY 2.870%.844% E E E E-04.85E-04.38E E-05.44E-04 CM2Y 3.240%.740% E E-04 2.E-04.8E-04.36E E-05.40E-04 CM3Y 3.40%.558% E-04.9E-04.64E-04.24E E-05.27E-04 CM5Y 3.70%.232% E-04.3E-04.00E E-05.02E-04 CM7Y 3.970%.068% E E E E-05 CM0Y 4.230% 0.823% E E E-05 CM20Y 4.760% 0.596% E E-05 CM30Y 4.640% 0.867% E-05 Note: the correlaton s n the lower trangle and the varance covarance s n the upper trangle. II. Tawan Bond Yelds Maturty Spot Rate Stdev CM3m CM6m CMY CM3Y CM5Y CM7Y CM0Y CM3m 0.996%.66% 2.7E E E E E E E-04 CM6m 0.972%.69% E E E E E E-04 CMY.698%.54% E E E-04 2.E E-04 CM3Y 3.28%.62% E E E E-04 CM5Y 3.323%.47% E E-04 2.E-04 CM7Y 4.487%.48% E E-04 CM0Y 3.096%.49% E-04 Note: the correlaton s n the lower trangle and the varance covarance s n the upper trangle. It shows there are standard devatons greater than % from 3-month to 7-year maturtes n US bond data and hgher correlatons between adjacent maturtes (the values just below the dagonal) n both US and Tawan bond data. Comparng the countres bond data, shorter-term nterest rates of US bonds have hgher volatltes whereas longer-term nterest rates of Tawan bonds have hgher volatltes. On the other hand, the bond ndex (.e. the supposed nvestment postons among dfferent maturtes) doesn't need to be mapped 4

5 nto key rates snce we use JP Morgan s nterest rate ndex structure to analyze our key rate duraton and value at rsk. Therefore, wth the key rate varance-covarance and VaR(%) on hand and accordng to the bond ndex nvestment, we fnd our bond portfolo value at rsk. Table 2 contans the value at rsk fgures of US and Tawan bond yelds based upon the JP Morgan bond ndex. Suppose that our bond ndex nvestment s $00 (n $mllon); we estmate the one-day VaR accordng to equaton (2) n the US and Tawan government bond markets as about $2.5 and $2.507 (n $mllon), respectvely, for a one-day nvestment tme horzon. We see that the US bond yeld has bond VaR amounts greater than Tawanese bonds even consderng the dfferences between the duraton, the key rate volatlty, and the correlaton of the key rates. Though, we should note that the second order effect of the bond yeld- convexty- would affect bond rsk measure as well. Notably, n the US bond market, the dversfcaton effect on the VaRs show the bond portfolo don t have lower rsk even though the bond maturty spread becomes larger or when more of the bonds wth dfferent maturtes are nvolved n the bond nvestment. We see the bond ndex poston has a VaR of $2.5, greater than that for postons, 2 and 3. But, poston 4, consstng of only two maturtes (3 month and 30 year) wth a larger maturty dfference, has a larger VaR, $ Ths ndcates that the convexty and dversfcaton effects are very small among US bond yeld data. On the contrary, n Tawan, the bond ndex poston has a lower VaR value than poston and a hgher VaR value than poston 2, whch ndcates that convexty and dversfcaton effects exst n Tawan bond yeld data. Table 2 JP Morgan Bond Index Investment Value at Rsk (page of 2) (daly base, 95% confdence level) () US Government Bonds Maturty Spot Rate VaR(%) Bond Index Weghts Key Rate Duraton poston poston2 poston3 poston4 CM3m 2.37% 3.72% $ CM6m 2.6% 3.048% $ CMY 2.87% 3.033% $ CM2Y 3.24% 2.862% $ CM3Y 3.4% 2.563% $ CM5Y 3.7% 2.026% $ CM7Y 3.97%.757% $ CM0Y 4.23%.355% $ CM20Y 4.76% 0.980% $ CM30Y 4.64%.426% $ Total = $00.00 Duraton= Undversfed VaR= $2.534 $2.279 $2.407 $2.462 $2.949 VaR= $2.5 $2.275 $2.393 $2.427 $2.928 Dversfcaton Effect= $0.023 $0.004 $0.04 $0.035 $0.02 Dversfcaton Effect (%)= 0.92% 0.6% 0.58%.43% 0.7% Table 2 Contnued ( page 2 of 2)(2) Tawan Government Bonds Maturty Spot Rate VaR(%) Bond Index Weghts Key Rate Duraton Poston Poston2 Poston3 Poston4 CM3m 0.996%.66% $ CM6m 0.972%.69% $ CMY.698%.54% $ CM3Y 3.28%.62% $ CM5Y 3.323%.47% $ CM7Y 4.487%.48% $ CM0Y 3.096%.49% $ Total = $00.00 Duraton= Undversfed VaR= $2.553 $2.663 $2.58 $2.577 $2.629 VaR= $2.507 $2.663 $2.48 $2.52 $2.589 Dversfcaton Effect= $0.046 $0.000 $0.038 $0.065 $0.040 Dversfcaton Effect (%)=.79% 0.00%.49% 2.54%.53% What f the bond cash flow cannot match our key rate maturty? Then, a cash flow mappng wll be needed, usng ether varance equalty or duraton equalty. Consder an example portfolo consstng of three knds of US bonds: 3% coupon -year bonds, 5% 3-year bonds, and 6% 5-year bonds. We nvest a total amount of mllons n the portfolo. Accordng to the cash flow mappng (by prncple of equalty varances), we can obtan cash flow allocaton as shown n Table 3-. The weght of the nterest rate volatlty s the base for 5

6 the cash flow dstrbuton between 3-year and 5-year key rates. Therefore 50.68% of the 4-year cash flow wll be allocated nto the 3-year key rate and 49.32% nto the 5-year key rate. Fnally, the VaR of the, 3, and 5-year bond portfolo s computed as $8.055 or 2.35%, whch s less than JP Morgan s bond ndex VaR of 2.5%, because of the lower volatltes and correlatons. Table 3- Bond Portfolo Cash Flow Dstrbuton on the Key Rates by Varance Equalty and ts Value at Rsk Maturty Spot Rate VaR(%) CM3m 2.37% 3.72% CM6m 2.6% 3.048% yr 3yr 5yr Cash Flow Present Key Rate Present Cash Flow Cash Flow CashVaR Correlaton CMY 2.87% 3.033% $03 $5 $6 $4 $0.82 $0.82 $ CM2Y 3.24% 2.862% $5 $6 $ $0.36 $0.36 $ CM3Y 3.4% 2.563% $05 $6 $ $0.07 $04.02 $ CM4Y 3.7% 2.026% $6 $6 $5.28 $0.00 $0.00 CM5Y 3.97%.757% $06 $06 $89.87 $92.20 $ CM7Y 4.23%.355% Total= $348 $37.39 $37.39 $8.9 CM0Y 4.76% 0.980% Vol Wetght = 50.68% CM20Y 4.64%.426% Undversfed VaR= $8.94 CM30Y 2.37% 3.72% VaR = $ %VaR= 2.35% Dversfed Effect= $0.358 (.658%) Table 3-2 Key Rate Cash Flow Dfference Effect upon Value at Rsk Cash Flow Dfference Between and 5 yr Key Rates $.00 $3.00 $5.00 $7.00 $9.00 $.00 $3.00 $5.00 VaR $8.07 $8.09 $8.4 $8.2 $8.30 $8.42 $8.57 $8.73 %VaR 2.38% 2.325% 2.338% 2.359% 2.386% 2.420% 2.46% 2.50% Dverfed VaR $0.066 $0.067 $0.069 $0.07 $0.073 $0.076 $0.077 $0.078 Key Rate Cash Flows CMY $5.00 $7.00 $2.00 $27.00 $35.00 $45.00 $57.00 $7.00 CM2Y $.00 $.00 $.00 $.00 $.00 $.00 $.00 $.00 CM3Y $.00 $.00 $.00 $.00 $.00 $.00 $.00 $.00 CM4Y $6.00 $6.00 $6.00 $6.00 $6.00 $6.00 $6.00 $6.00 CM5Y $05.00 $03.00 $99.00 $93.00 $85.00 $75.00 $63.00 $49.00 Total Cash Flow $ $ $ $ $ $ $ $ Cash Flow Dfference $0.00 $4.00 $22.00 $34.00 $50.00 $70.00 $94.00 $22.00 Table 3-2 shows the cash poston dfference effect upon the bond portfolo VaR number. If there s more weght on cash nvestment n the 3-month maturty bond, the VaR fgure of the bond portfolo wll ncrease more rapdly because of the hgh volatlty of the 3-month bond yeld and the low volatlty of the 5-year bond yeld. 3.3 Level, Slope, and Curvature Duraton Value at Rsk We frst revew the yeld curve term structure change for both countres bond data. Fgure presents the US bond yeld curve change from early 999 to early 2005 accordng to equaton (0). It shows the level, slope, and curvature movements of the yeld curve as tme passes. We fnd that n md-999, US nterest rate structure exhbted a hgher level for the term structure, and relatvely lower structural slope and curvature. There s some evdence showng that from early 999 to late 200, the US seemed to have modest economc growth. The US Federal Reserve rased ts dscount nterest rate hopng to cool the economy. Contrary to md-999, we fnd that n md-2003, the US nterest rate structure exhbted a tendency to a lower level and a hgher slope. In early 2005, the US nterest rate structure has a lower level, slope and curvature. The term structure movement tells us snce early 2002, the US economy s gradually weakenng. Especally, the hgher slope n the prevous perod has dropped and become nearly flat at present. We consder the level of the nterest rate term structure as reflectve of the nflaton stuaton, the slope of the nterest rate term structure reflectve of the economc 6

7 7% 6% Interest Rate 5% 4% 3% 2% % 0% Interest Rate Maturty Actual /4/05 Forecasted Prevous 0/3/04 Prevous /05/04 Prevous 4/5/03 Prevous 4/22/02 Prevous /27/02 Prevous 4/22/99 Fgure US Bond Yeld Term Structure Movement 7.000% 6.000% Interest Rate 5.000% 4.000% 3.000% 2.000%.000% 0.000% Interest Rate Maturty Actual 4/0/05 Forecasted Prevous 26/05/00 Prevous 27/05/04 Prevous 25/06/0 Prevous 07/06/02 Prevous 07/0/03 Prevous 26/07/99 Fgure 2 Tawan Bond Yeld Term Structure Movement busness cycle, and the curvature of the nterest rate term structure reflectve of economc volatlty. Clearly n the US term structure movement, the level and slope have a postve relatonshp,.e. when the level s hgh, the economy becomes stronger, and when the level s hgh, the slope wll lkely also have a hgh level, and the economy s gong to get stronger. On the other hand, n Tawan, recently short-term nterest rates have came down more than n the US. Investors such as banks, nsurers and other fnancal nsttutes antcpate that the economy wll be stagnant for some future years, as people n Tawan prefer to nvest and do busness n foregn countres because of ther cheaper labor and product materal. Fgure 2 dsplays the slghtly negatve relatonshp between level and slope for Tawan bond data. Recently, n Tawan, the economy s becomng weaker and the level drops deeply wth the longer term nterest rate gong downward slghtly and thus the slope becomes hgher, not lower. Ths mght be explaned because as the economy grows, people wll prefer longer term borrowng and short term nvestment snce people predct a lower short rate and the hgher long rate wll go down, and as the economy weakens, people do the opposte n the US bond market. In Tawan, people wll predct an ncreasng long-term rate as the economy grows and react 7

8 dfferently than people n the US. Next, to measure the bond value at rsk by the three-factor yeld curve model, we have estmated the three factors' parameters. Table 4 provdes the level, slope, and curvature correlatons, and varance covarance nformaton computed by usng factor transformatons wth the whole perod yeld curve data. As to the bond value at rsk measure, we must fnd the duraton to specfy the level, the slope, and the curvature, lke the key rate duraton. Snce the level s fxed at one, we can clearly recognze t has the duraton equal to our cash flow. The slope and the curvature formed by our three-factor model would need to be estmated and optmzed accordng to equaton (0). Usng the least squares forecastng error method, we obtan Table 5 that shows both the US and the Tawan estmates of the three-factor model parameters. We obtan the optmal tau (τ) by mnmzng the sum of the squared errors where the forecasts come from equaton (0). The τ s our optmal term structure vertex (hump), the level remans one, and the slope and the curvature are estmated by τ. For the US nterest rate term structure τ s estmated as around.786 years and for Tawan t s estmated around years as shown n Table 5. The longer term of the vertex n Tawan partly results from the data s shorter nterest rate term structure and partly from ts hgher volatlty and curvature components. We used the three-factor yeld curve model to approxmate the yeld curve term structure of 0/5/2005 and we then obtan a term structure yeld curve forecast as shown n Fgure for US bond yeld data and n Fgure 2 for Tawan bond yeld data. Table 4 Level Slope Curvature Correlatons () US Bond Yelds Correlaton Level Slope Curvature Level.000 Slope Curvature Varance-Covarance Level Slope Curvature Level 2.659E-05 Slope 4.787E E-04 Curvature 5.45E E E-04 (2) Tawan Bond Yelds Correlaton Level Slope Curvature Level Slope Curvature Varance-Covarance Level Slope Curvature Level Slope Curvature () US Bond Yeld Table 5 Level, Slope, Curvature Parameter Estmates Optmal τ Pos Maturty Level Slope Curvature Cash Flow 0.25 CM3m $ CM6M $3.9 CMY $7.9 2 CM2Y $ CM3Y $ CM5Y $ CM7Y $ CM0Y $ CM20Y $ CM30Y $.68 Total= $00.00 (2) Tawan Bond Yeld Optmal τ Bond Index Maturty Level Slope Curvature Cash Flow 0.25 CM3m $ CM6M $3.55 CMY $ CM3Y $ CM5Y $ CM7Y $ CM0Y $6.43 Total= $00.00 Wth the three factor parameters and ts varance-covarance, usng equaton (6), we then can estmate the VaR fgures of bond yeld change for both the US and Tawan, $ and $2.4902, respectvely as 8

9 shown n Table 6. The lnear level, slope, and curvature three factor model, whch uses fewer factors, has the advantage of model parsmonousness but provdes weaker estmates of the bond data nonlnear effects of the convexty and dversfcaton. Therefore, unlke the VaR estmaton results of the key rate method that nvolved more rsk factors, the VaR of the US bond data s less than the VaR of the Tawan bond data. Table 6 provdes the VaR senstvty measures of the three factors by computng ts VaRdelta and VaRbeta. As the table shows, n the US the slope of the bond data has the hghest VaR delta and component whle n Tawan the level has the hghest VaR delta and component.. Ths means that the most VaR senstve s the slope factor for US whereas t s the level factor for Tawan. Ths descrpton corresponds to our forgong economc ntroducton that n US the slope s weakenng, and the level s gong down sgnfcantly n Tawan. () US Bond Yeld L,S,C Covarance Level Slope Curvature Table 6 Level, Slope, Curvature Duraton Value at Rsk LSC Cash Flow VaRDelta VaR Component VaRBeta Level 2.659E E E-05 $00.00 $0.007 $ % Slope 4.787E E E-04 $52.78 $ $ % Curvature 5.45E E E-04 $23.84 $0.099 $ % (2) Tawan Bond Yeld L,S,C Covarance Level Slope Curvature LSC Cash Flow Total = $ % VaR = $ VaRDelta VaR Component VaRBeta Level $00.00 $ $ % Slope $96.35 ($0.5085) ($ ) % Curvature $3.52 ($0.5822) ($2.0477) % Note: LSC cash flows are obtaned by key rate cash flows transformatons from the factor coeffcents. Total = $ % VaR = $ Prncpal component duraton value at rsk Prncpal component analyss uses the characterstc value of the key rate varance-covarance matrx and correlatons and then transforms the structure of varables nto several prmary components, whle mantanng the orgnal total varance or maxmzng the transformaton total varance. After varable transformaton by PCA, the prmary components wll be ndependent of each other and normally there wll be two or three prmary components accountng for most of the varable varance. Compared to the key rate and three-factor yeld curve model, the PCA has the advantage of factor ndependence; factor ndependence s helpful for nvestment analyss snce we don't need to care about the consequental factor relatonshp rsk. However, the prmary components are not easy to descrbe. Therefore, PCA has been consdered as a theoretcal tool. On the other hand, unlke the three-factor model that fnds the optmal τ by mnmzng the forecast error, PCA fnds ther egen values and vectors by maxmzng the score varance. Thus, PCA wouldn't be used for the tme seres predcton but used for a system analyss that uses the whole nformaton for the parameter estmates such as the egen value and vector computaton. By equaton (8), we obtan the egen values and egen vectors of the varance and covarance of the key rates as shown n Table 7. Table 8 shows that three of the components account for 99.80% of the term structure varance for US bond yelds and 98.9% of the term structure varance for Tawan bond yelds. Further usng the PCA cash flow and egen values only wthout ts varance and covarance structure, we can measure the PCA bond yeld VaR fgure for both US and Tawan bond yeld snce PCA are ndependent of each other. Then, Table 8 uses the PCA cash flow comprsed of the key rate cash flows and the egen vectors, and egen volatlty to measure the bond yeld VaR accordng to equaton (22) for both the US and Tawan. We obtan the US bond yeld data VaR of $2.504 and the Tawan bond yeld data VaR of $ The hgher US bond yeld PCA VaR fgure means ts nterest rate system rsk s more than Tawan's. However, the nonlnear effects cannot be dscovered by PCA, and thus PCA tends to overestmate the VaR of the Tawan s bond yeld data that exhbts convexty and dversfcaton effects, the same as key rate method mentoned prevously. Besdes, the PCA treats the tme data as cross-sectonal data and therefore can t be used to forecast the term structure or the condtonal VaR. 9

10 Table 7 EgenValues and EgenVectors of the Varance-Covarance Interest Rate Term Structure (page of 2) () US Bond Yeld Egen Values and Egen Vectors Egen Value λ Egen Vol Egen Vector.9738E E E E E E E E E E E E E E E E E E E E E E E E E E-03 Sum of λ= a a= a aj= (2) Tawan Bond Yeld Egen Values and Egen Vectors Egen Value λ Egen Vol Egen Vector.639E E E E E E E E E E-03 Sum of λ= a a= a aj= () US Bond Yeld Maturty Spot Rate VaR(%) Bond Index EgenValueλ PC Vol Table 8 PCA Cash Flows and VaR % Vol Explaned % Vol Cummulated PCA Cash Flow (v ) 2 v*λ CM3m 2.37% 3.72% $ E % 96.58% 96.58% CM6m 2.6% 3.048% $ E % 2.89% 99.48% CMY 2.87% 3.033% $ E % 0.34% 99.82% CM2Y 3.24% 2.862% $ E % 0.2% 99.93% E-04 CM3Y 3.4% 2.563% $ E % 0.03% 99.97% E-05 CM5Y 3.7% 2.026% $ E % 0.02% 99.98% E-06 CM7Y 3.97%.757% $ E % 0.0% 99.99% E-07 CM0Y 4.23%.355% $ E % 0.00% 00.00% E-06 CM20Y 4.76% 0.980% $ E % 0.00% 00.00% E-08 CM30Y 4.64%.426% $ E % 0.00% 00.00% E-07 Total= $ % VaR= $2.504 (2) Tawan Bond Yeld Maturty Spot Rate VaR(%) Bond Index EgenValue λ PC Vol % Vol Explaned % Vol Cummulated PCA Cash Flow (v ) 2 v*λ CM3m % 2.727% $ E % 95.44% 95.44% CM6M % 2.773% $ E % 2.26% 97.70% CMY.6975% 2.535% $ E %.22% 98.9% CM2Y 3.279% 2.663% $ E % 0.49% 99.40% CM3Y % 2.424% $ E % 0.35% 99.75% CM5Y % 2.440% $ E % 0.20% 99.95% CM7Y % 2.456% $ E % 0.05% 00.00% Total= $ % VaR= $2.506 Note: PCA cash flows are obtaned by key rate cash flows transformaton from egen vectors. 0

11 3.5 Structural Equaton Yeld Curve Model Prncpal component analyss explores the data reducng process to fnd the prmary component varances that maxmze the total varances after varable transformaton. Unlke PCA that explores the prmary components wth entre factor loadng estmates, structural equaton modelng (SEM) tres to control the specfcatons of the constructs wth partal factor loadng estmates that are the measurement model of the SEM,.e. the construct or factor measurements. Thus, n our SEM analyss of the nterest rate term structure, we wll subjectvely try to fnd the controlled constructs that can be measured by our ndcators- the structural key rates wth dfferent maturtes. We wll use LISREL as our SEM solver. The four steps nvolved are as follows: Step One: Fnd Construct Measurement Consderng our 0 key rates for the US bond data rangng from 3 months to 30 years and the modfcaton ndces obtaned from LISREL, we suggest splttng the term structure nto short-term, medum-term, and long-term constructs as Table 9-(a) shows for the SEM confrmatory factor analyss. Construct one has 3-month, 6-month, and -year maturtes; construct 2 has 2-year, 3-year, 5-year and 7-year maturtes; construct 3 has 0-year, 20-year, and 30-year maturtes. As for Tawan bond yeld, after examnng the modfcaton ndces, ch-square, and root mean square resduals by addng and deletng factors, we suggest usng three constructs consstng of the key rates rangng from 3 months to 0 years: hence construct one has 3-month and 6-month maturtes, construct two has -year and 2-year maturtes, and fnally construct three has 3-year, 5-year, and 7-year maturtes as shown n Table 9-(b). (a) US Constructs Exogenous Indcator Table 9 Confrmatory Factor Analyss SEM measure model Notatons Exogenous Constructs Error (Key Rates) X(CF3m) λ ξ δ X2(CF6m) λ 2 ξ δ 2 X3(CFy) λ 3 ξ δ 3 X4(CF2y) λ 2 ξ 2 δ 4 X5(CF3y) λ 22 ξ 2 δ 5 X6(CF5y) λ 32 ξ 2 δ6 X7(CF7y) λ 42 ξ 2 δ 7 X8(CF0y) λ 3 ξ 3 δ 8 X9(CF20y) λ 23 ξ 3 δ 9 X0(CF30y) λ 33 ξ 3 δ 0 Step Two: Data Input SEM tres to search the approprate relatonshps (loadngs) of the factors and ndcators. Therefore, the data nput for the SEM model estmaton s the varance-covarance decomposton of the covarance matrx or the correlaton matrx. Though the correlaton has been used as the measure unt, we should use the covarance as the nput data for the varance research needed for our value at rsk analyss. Step Three: Optmal Model Estmaton Lke multvarate data analyss methods such as lnear multvarate regresson, multvarate logt or probt regresson, and even the exploratory factor analyss and prncpal component analyss, the parameters of the SEM can be estmated by ordnary least squares (OLS) or maxmum lkelhood estmaton (MLE). Step Four: Testng Model Estmates Snce MLE s one of the estmaton methods, the lkelhood rato ch-square (χ 2 ), Wald statstcs, and Lagrange multpler can be used to measure the model ft. Other approaches provded by LISREL are the goodness of ft ndex and the root mean square error. Table 0 (b) Tawan Constructs Exogenous Indcator Exogenous Constructs Error (Key Rates) X(CF3m) λ ξ δ X2(CF6m) λ 2 ξ δ 2 X3(CFy) λ 3 ξ δ 3 X4(CF2y) λ 2 ξ 2 δ 4 X5(CF3y) λ 22 ξ 2 δ 5 X6(CF5y) λ 3 ξ 3 δ 6 X7(CF7y) λ 23 ξ 3 δ 7 shows the confrmatory construct factor loadngs for each ndex (observable) measure. The t statstcs of the loadngs and the R 2 of the construct measures exhbt hgh sgnfcance of the coeffcents and good varance explanaton of the measure model. In Table 0, the ch-square values are very large numbers for US, and for Tawan, so the model does not ft the covarance of the bond data very well. Nevertheless, our am s to fnd the approprate factor loadngs and sgnfcances for the varance and covarance decomposton. Fnally to compute the value at rsk for the nterest rate term structure of the SEM, we use LISREL to estmate the correlatons of the three factors as shown n Table. In addton, the varances of the three factors also can be obtaned by bond data transformaton nto factor data and computng factor data varances. Wth the correlaton of the factors and ther estmated varances, we can calbrate the SEM VaR fgure as done n the PCA model and the three-factor model- level, slope, and curvature. We descrbe the mathematcs of the SEM VaR calbraton as follows:

12 (a) US Bond Yeld t Statstcs CM3m = *Factor, Errorvar.= , R 2 = ( ) (0.0000) CM6M = *Factor, Errorvar.= , R 2 =.0025 ( ) (0.0000) CMY = *Factor, Errorvar.= , R 2 = ( ) (0.0000) CM2Y = *Factor2, Errorvar.= , R 2 = ( ) (0.0000) CM3Y = *Factor2, Errorvar.= , R 2 =.0020 ( ) (0.0000) CM5Y = *Factor2, Errorvar.= , R 2 = ( ) (0.0000) CM7Y = 0.005*Factor2, Errorvar.= , R 2 = ( ) (0.0000) CM0Y = *Factor3, Errorvar.= , R 2 =.0068 ( ) (0.0000) CM20Y = *Factor3, Errorvar.= , R 2 = ( ) (0.0000) CM30Y = *Factor3, Errorvar.= , R 2 = ( ) (0.0000) Note:. Ch-Square=2233.8(df=32), P value= , and RMSE= Numbers n parenthess are standard devatons. Table 0 Confrmatory Factor Loadng Tests (b) Tawan Bond Yeld t Statstcs CM3m = 0.066*Factor, Errorvar.= , R 2 =.0043 ( ) (0.0000) CM6M = *Factor, Errorvar.= , R 2 = ( ) (0.0000) CMY = *Factor2, Errorvar.= , R 2 = ( ) (0.0000) CM3y = *Factor2, Errorvar.= , R 2 = ( ) (0.0000) CM5y = *Factor3, Errorvar.= , R 2 = ( ) (0.0000) CM7Y = *Factor3, Errorvar.= , R 2 = ( ) (0.0000) CM0Y = *Factor3, Errorvar.= , R 2 = ( ) (0.0000) Note:. Ch-Square=675.6 (df=), P value= , and RMSE= Numbers n parenthess are standard devatons. Table SEM Confrmatory Factor Correlaton (a) US Factor Correlaton (Covarance) (b) Tawan Factor Correlaton (Covarance) Factor Factor2 Factor3 Factor Factor2 Factor3 Factor Factor Factor (0.0024) Factor (0.0004) Factor (0.0062) (0.009) Note: In each factor block, standard devaton s n parenthess and t statstcs s at the bottom. P/ P = P/ P f P/ P f2, (25) where P/ P s the total percentage prce change (bond yeld) and P/ P f s the change n prce due to factor (the SEM factors), σ 2 ( P/ P) = factor cash flow* Ω *factor cash flow' (26) Factor (0.0022) (0.005) where Ω s the factor varance-covarance matrx and factor cash flow s the nvestment poston allocated to the specfc factor. Table 2 shows the factor cash flows for three factors (short, medum, and long term) accordng to our factor loadngs. Table 3 presents the VaR estmates wth the confrmatory factor SEM method. 2

13 (a) US Bond Yeld Factor Cash Flows Factor Coeffcents Maturty Factor Factor2 Factor3 Key Rate Cash Flow CM3m $.73 CM6M $3.9 CMY $7.9 CM2Y $3.85 CM3Y $9.75 CM5Y $9.77 CM7Y $5.79 CM0Y $4.28 CM20Y $4.04 CM30Y $.68 Total= $00.00 Factor Factor2 Factor3 Factor Cash Flow Note: Factor cash flows are obtaned usng key rate cash flows transformaton by the factor coeffcents. Table 2 SEM Confrmatory Factor Cash flows (b) Tawan Bond Yeld Factor Cash Flows Factor Coeffcents Maturty Factor Factor2 Factor3 Key Rate Cash low CM3m $.92 CM6M $3.55 CMY $9.90 CM2Y $35.39 CM3Y $2.95 CM5Y $0.86 CM7Y $6.43 Total= $00.00 Factor Factor2 Factor3 Factor Cash Flow Table 3 Value at Rsk wth SEM Confrmatory Factor Estmate (a) US Bond Yeld VaR Factor Covarance CFA Factor Factor2 Fctor3 (Correlaton) Cash Flow VaRDelta VaR Component VaRBeta Factor $0.42 $.5993 $ % Factor $.04 $.6397 $ % Factor $0.073 $.5609 $ % Total= $.535 $ % VaR= $2.494 (b) Tawan Bond Yeld VaR Factor Covarance CFA Factor Factor2 Factor3 (Correlaton) Cash Flow VaRDelta VaR Component VaRBeta Factor $0.09 $.6253 $ % Factor $0.870 $.6405 $ % Factor $0.563 $.6296 $ % Total= $.523 $ % VaR= $2.498 We should note that the three factors of the SEM have entrely dfferent descrptons from the three factors of the level, slope, and curvature model. The level, slope, and curvature model uses all of the key rates as ther factor constructs, but the SEM uses only part of the key rates as ther factor constructs as seen n Table 5 and Table 9. As for the model ft, the data model ft of the SEM performs neffectvely and also t s not a good tme seres predctor, whereas the level, slope, and curvature factor model fnds ts optmal ft by the root mean square forecast errors and thus t s a better tool for yeld predcton. For the Tawan bond data, the three-factor SEM VaR measure s approxmately the same ($2.498) as the level, slope, and curvature VaR measure ($2.4902) n Table 6; both of the factor models- SEM and level, slope, and curvature- models measure part of the total varance gnorng the resdual varances. However, for the US bond data VaR, the three-factor SEM VaR measure has a slghtly dfferent result than from the level, slope, and curvature due to the constrants of the constructs set by the SEM. In addton, usng SEM, we see hgher VaR senstvty of the medum-term yeld for both the US and Tawan bond data. Usng the level, slope, and curvature model, we observe hgher VaR senstvty of the slope for US and the level for Tawan. Ths can be explaned by the hgher factor correlatons between medum and both the short and long-term yelds and the hgher cash poston allocated n the medum-term yeld. On the other hand, whle both the SEM and three-factor model have some degree of correlaton between ther factors, the PCA components are ndependent of each other and have a dvergent VaR fgure ($2.504 for US and $2.506 for Tawan) from the factor models-sem ($2.494 for US and $2.498 for Tawan) and three-factor model ($2.464 for US and $2.490 for Tawan). Ths dvergence can be explaned by the resduals of the model ft of the factor models, such as 3

14 SEM and the level, slope, and curvature model. When we use data analyss models that do not use any estmaton procedure to ft the model lke the factor models, we fnd that the PCA VaRs ($2.504 for US and $2.506 for Tawan) have smlar estmates as the key rate method VaRs ($2.5 for US and $2.507 for Tawan). 4. Concluson Key rate duraton applcaton s the prmary methodology used by JP Morgan for ts bond portfolo value at rsk measure. In realty, although there exsts the ssues of capturng the real world key rate varance-covarance structure and yeld curve movement, many nterest rate rsk managers stll prefer to use t for ts smple and detaled descrptons of real factor defntons. However, for nterest rate rsk researchers, the unstable key rate varance-covarance and correlaton structure and the unknown yeld curve current movement often lmt ts use for long term nterest rate rsk management. The three factor--level, slope and curvature--nterest rate model, mght not ft the general needs of nterest rate rsk management. It tends to underestmate the VaR fgure. Nevertheless, because of the dstnctve characterstcs of the three factors and ts parsmonous features, ths yeld curve model should have some advantages as well. Especally, when consderng the economc changes of nflaton, busness cycle, and economc volatlty, the three factors--level, slope, and curvature model would perform well at descrbng the yeld curve movement. For nstance, we can explan the three-factor model effect upon the bond portfolo rsks such as the barbell and bullet bond portfolos. When there s a hgher term structure level or curvature change, the hgher cost barbell bond would have a lower bond value at rsk snce t has a larger maturty dfference (.e. convexty) and a lower correlaton between maturtes. When there s a hgher term structure slope change, the bullet bond would have a lower bond value at rsk. Emprcally, the US bond data exhbts a hgher slope rsk whereas Tawan bond data reveal a hgher level rsk accordng to the level, slope, and curvature model. If we want to analyze only two or three specfc component nterest rates and forgo the component nterest rate varance-covarance and correlaton structures, PCA s a good method of VaR estmaton owng to ts factor ndependences and varable reducton. However, the explanaton of the components s not easy for researchers, not to menton nvestors. Some researchers tend to explan ts components as the level, slope, and curvature but the PCA factors should be ndependent and the level, slope, and curvature would lkely have some knd of correlaton among them. Concernng SEM, t s prmarly used for the confrmatory factor analyss. In addton to the exploratory factor analyss that searches for the optmal factor loadng by achevng the maxmum lkelhood or least square objectve, SEM needs the actual percepton of the researchers n the feld of factor constructs (measurements) to fnd the approprate measurement and structural models. Although n ths research, SEM does not ft the data model well (hgh ch-square and small p value), we just amed to fnd the factor loadngs (varance decomposton) and use the covarance and correlaton as the key nput data for SEM analyss. Compared to three-factor model, SEM has done equally well for the VaR measure despte the model s dsappontng ft to the data. Nonetheless, n practce, researchers want to construct good and easly nterpreted factors for nvestor rsk management as n our case: the short, medum, and long term VaR measures that can be recognzed panlessly, and VaR components as well as senstvty analyss. Emprcally, the US bond data and Tawan bond data have hgher medum term VaR senstvtes, and US bond data VaR s slghtly more than Tawan bond data VaR accordng to SEM. In sum, prncpal component analyss and factor models such as the level, slope, and curvature model and SEM are parsmonous models that use fewer varables to explan the data change behavor whle key rate uses more rsk varables. In partcular, the rsk factors of the SEM and level, slope, and curvature model are easly understood and useful for the nterest rate rsk analyss. However, both of the factor models tend to underestmate the VaR fgures slghtly because of dong the model ft and bypassng the resdual varances. On the other hand, the data analyss models such as the key rate and PCA methods perform the varance/covarance decomposton and combnaton to fnd out the bond yeld rsk and rsk components. References [] Austn and R.F. Calderon. (996), Theoretcal and Techncal Contrbutons to Structural Equaton Modelng: An Updated Annotated Bblography, Structural Equaton Modelng 3: [2] Bodurtha, Jr., James N. and Q Shen. (994), Hstorcal and Impled Measures of Value at Rsk: The DM and Yen Case, Workng Paper, McDonough School of Busness, Georgetown Unversty. [3] Fsher, L. (966), An Algorthm for Fndng Exact Rates of Return, The Journal of Busness 39/, January, pp. -8. [4] Golub, W. and Tlman, M. (997), Measurng Yeld Curve Rsk Usng Prncpal Components Analyss, Value at Rsk, and Key Rate Duratons, The Journal of Portfolo Management. [5] Ho, T.S.Y. (992), Key Rate Duratons: Measures of Interest Rate Rsks, The Journal of Fxed Income. New York: Vol. 2, Iss. 2; p. 29 (6 pages). [6] JP Morgan/Reuters. (996), RskMetrcs - Techncal Document. See [7] Khndanova, Irna N. and Svetlozar T. Rachev (2002), VALUE AT RISK: RECENT ADVANCES, Workng Paper, Unversty of Calforna, Santa Barbara and Unversty of Karlsruhe, Germany. [8] Nelson, R. and Segel, F. (987), Parsmonous Modelng 4