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1. Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus rk 1 Introduction All Markit iBoxx indices follow a standard set of rules and calculation procedures This document outlines the calculation principles for all Markit iBoxx indices and the standard bond and index level analytics published for Markit iBoxx The annotations used in the formulae are attached in the Annotations section 2 Index Calculation 2 1 Price and Total Return Indices 2 1 1 Price Index Calculation All iBoxx indices are basket indices that express relative changes in value compared to the beginning of the respective period The composition and weightings of the index are adjusted at the beginning of each period Accordingly adjustments to index tracking portfolios are only needed at the end of each period Benchmark price index Cap Qu 5 PI Pl _ n 5 Ps s i t s Price index calculation for liquid indices For liquid indices that maintain cash at month ends in between quarterly rebalancing the price index includes the rebalancing cash YBSES NS VUES 1 1 5 Pg Figg Nig ESE t s i t s 14 8 The indices are based consolidated bid quotes Bonds not currently in the iBoxx universe enter the indices at the next rebalancing and are included in the index calculation at the beginning of t
2. Copyright 2014 Markit Group Limited All rights reserved www markit com 29 Index Calculus rk Please note that the estimated index ratio for a future cash flow may change from day to day if new CPI data is released The assumed inflation zz is usually estimated from the most recent CPI data and is the current one year inflation rate CPI fg CPI os A small number of countries use either a fixed inflation assumption or forecasted values which will be used instead of the above formula The inflation is assumed to remain constant throughout the life of the bond 5 3 Index Calculation The standard calculus formulae apply 5 4 Bond and Index Analytics The standard calculus formulae apply Copyright 2014 Markit Group Limited All rights reserved www markit com 30 Index Calculus rk 6 Appendix 6 1 Cash and Turnover Reinvestment Cost At the end of each month the proceeds from coupons received and from the sale of dropped or reduced weight bonds need to be reinvested in the indices At the same time new bonds will enter the indices These can be of two different types 1 new bonds entering the index family for the first time or 2 bonds migrating from one index to another e g due to rating changes A portfolio manager tracking a liquid index may attempt to replicate the index by reinvesting the proceeds from bonds sold at the bid price or coupon payments received into the index i e into bonds
3. fi mark Markit iBoxx Bond Index alculus 2014 RN VA i Index Calculus Ma rk Table of Contents 1e MPU OCU CUNO ca ooo ac ccc hae 4 2 1 cece a eaa ea sete cates D dE II Leti 4 24 Prneeand Total Return Indices 4 241 Price Index C alculation wiz iini ERE 4 2 1 28 Total Return Index 4 2 1 8 Total Return Index 9 5 2 14 Month to Date ted re RU eR RED peru 6 2 2 Gross Price and rer rh dede pente aa E Rari epe ea Re nce axes 6 pron ECI dJudic Siee ce es 6 PSAL END nee MACOS ssr S E AA S E 6 2 3 Rebalancing cost factor for liquid 7 2 4 Hedged and Unhedged Index Calculation ssesssssssssssseseeseeeee eene nennen nennen enne 7 2 5 Bond Cappingu stet teu Prado Dr Rex pea edu Ee 8 3 Bond Analytics enit eiie ect sac ok S
4. Before Ww Pi A fe M 32 Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma rk After w M The same applies to cash Solving for f and f leads to f w Before w M After ft P A It will be assumed that a bond increases its weight in the portfolio if its weight increased in the index w gt w gt f gt f The market value using index prices can be expressed using the new amount And using transaction prices M Yn nr Hn Please note that the index price and the transaction price of cash are equal to 100 Combining the formulas above Desa i em Or simplified A 6 5 i f and f be replaced in the previous formula o Y in a E Solving for gives 2 p un oe i l DE A PPt A i Since M gt it leads to Copyright 2014 Markit Group Limited All rights reserved www markit com 33 Index Calculus rk cost 1 We can now separate the cash from the bonds n E PP A 1 1 Weash W ia Ff tA cost 1 A cash T gt Ww The solution is independent of the individual portfolio In addition bid and offer prices amount outstanding before and after the rebalancing as well as the index market value before and after the rebalancing a
5. Dat C 2 Otherwise the accrued interest is calculated as follows M days scr FSD days c Dat C r days Dat C Dat C Dat C i Short first coupon days FSD 1 days Dat C 1 11 Coupon changes The following formula shows how to calculate accrued interest when a coupon change occurs during a period 22 Ca days Dat C C t days Dat C days Dat C i For the coupon payment in the period of the coupon change the following formula applies days 4c Dat C Dat C C days Dat C C C days c Dat C Dat C days cr Dat C Dat C P 3 1 3 30 360 The accrued interest calculation is based on ISDA standards For two given dates d1 m1 y1 and d2 m2 y2 1 If d1231 then set d1 30 Copyright O 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma rk 2 If d2231 and d1 30 then set d2 30 Afterwards the days between the two dates are calculated as follows dayS3 d1 ml1 yl d2 m2 2 y2 y1 360 m2 ml 30 d2 dl Accrued interest for the first coupon period n days o FSD t i CFA 360 U Accrued interest for the 274 and subsequent coupon periods The days between the settlement date and the last co
6. Group Limited All rights reserved www markit com 42 Index Calculus rk 8 Further Information Copyright 2014 Markit Group Limited All rights reserved www markit com For contractual or content issues please refer to Markit Indices Limited Walther von Cronberg Platz 6 60594 Frankfurt am Main Germany Tel 49 0 69 299 868 100 Fax 49 0 69 299 868 149 E mail iboxx markit com Internet indices markit com For technical issues and client support please contact iboxx markit com or Asia Pacific Europe USA Japan 81 3 6402 0127 General 800 6275 4800 1 877 762 7548 Singapore 65 6922 4200 Frankfurt 49 69 299 868 100 UK 44 20 7260 2111 Licences and Data iBoxx is a registered trademark of Markit Indices Limited Markit Indices Limited owns all iBoxx data database rights indices and all intellectual property rights therein A licence is required from Markit Indices Limited to create and or distribute any product that uses is based upon or refers to any iBoxx index or iBoxx data Ownership Markit Indices Limited is a wholly owned subsidiary of Markit Group www markit com Other index products Markit Indices Limited owns manages compiles and publishes the iTraxx credit derivative indices and the iBoxxFX Trade Weighted Indices 43
7. TRO Total Return index level after cost adjustment TRY Total Return index level before cost adjustment TR Local currency total return index level for bond i at time t TR Local currency total return index level at time t TR Local currency total return index level at the last rebalancing TR Total Return index level after rebalancing adjustment from the end of last month t s Date of last rebalancing Weight of bondi W Weight of bond i before rebalancing Wash Weight of cash in the index prior to the rebalancing Weight of bond i after rebalancing Weight of cash in the index after the rebalancing Duration weight of bond i at time t Wry Nominal weight of bond i at time t Fixed weight of bond i at the last rebalancing Wi Base market value weight of bond i at time t Wr Market value weight of bond i at time t Whee Market value weighting adjusted for cash of bond i at time t Variable indicating whether bond i entered the index at the last rebalancing t s during its ex dividend period 0 if the bond enters the index at the ex dividend period to ensure that the next coupon payment is excluded from the total return calculation 1 if a coupon payments not ex dividend b has not entered the index during an ex dividend period or c entered the index during a previous ex dividend period XD j The ex dividend factor of bond i at date j 1 i e one d
8. Value of any coupon payment received from bond i at the first payment date If none the value is 0 GI Gross price index at date t Gross price index at the last rebalancing before t IC Coupon income index at date t IC Coupon income index at the last rebalancing before t IN ncome index at date t IN ncome index at the last rebalancing before t IR Redemption income index at date t IR Redemption income index at the last rebalancing before t IR Index ratio applicable to bond i on the calculation date IR Index ratio based on the most recently published CPI level on the calculation date applicable to t IR Index ratio applicable to the cash flow at t for bond i estimated as of the calculation date t IV ndex market value at time t Woe Hedged index market value at the rebalancing Hedged portion of the index market value at time t y a sedResidual T IV HedgedPorion Unhedged portion of the index market value at time t Copyright 2014 Markit Group Limited All rights reserved www markit com 38 Index Calculus IXR 1 1 5 return MDU MDU MDU MDPU MDPU Copyright 2014 Markit Group Limited All rights reserved www markit com mark Local currency index return level at time t Local currency index return level at the last rebalancing can apply to both total return and price Hedged index retu
9. bonds without options For sinking funds and amortizing bonds the average life is used as a measure for the expected remaining life of the bonds The average life is calculated as the sum of the distances to each future redemption cash flow weighted by the percentage redeemed at the respective date The redemption payments are not discounted Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma rk gt Ry Linj LF jot 2 5 jot Amortizing bonds with options For callable putable bonds the yield to worst calculation determines the expected redemption path of the bond All outstanding redemption payments are assumed to be made on the worst date determined by the yield to worst calculation The average life is then calculated as per the formula above Fully redeemed bonds The expected remaining life for fully redeemed bonds is O 3 3 3 Average Redemption Yield The yield of a bond at time t is calculated as follows P F 1 t j l For bonds during an ex dividend period the future cash flows for the yield calculation exclude the current coupon payment The Newton iteration method is used to solve the equation for y F is a redemption factor that is relevant for sinking funds amortizing bonds and unscheduled full redemptions For other bond types F always equals 1 The yield is set to 0 if HF is 0 The true yield is calculated as follow
10. days Dat C C Dat C B 360 360 3 1 5 5 252 The calculation follows the convention for bonds that accrue interest only on business days The distance between two calendar days is expressed as the number of days that are not public holidays and that do not fall on a Saturday or Sunday The convention is mainly used for Brazilian bonds A standard year is supposed to contain 252 business days regardless of the actual number of business days The accrued interest for standard bonds is calculated as ____bdays Dat C t E una bdays Dat C Dat C 100 Ja 1 100 FA For bonds the coupon of which is paid at maturity bdays FSD t onn 252 A 14 100 FA 100 i 3 2 Bond Returns Daily and month to date bond returns are calculated for all bonds The formulae below are valid for daily and month to date calculations the daily returns read t 1 instead of t s Daily local bond return CV MV LCR Month to date local bond return CV LCR 1 11 8 3 3 Bond Analytics 3 3 1 Yield to worst Calculation For callable and putable bonds a yield to worst calculation is necessary to determine when a bond is likely to be redeemed For European options the yield to worst calculation is performed at each exercise date For American options the yield to worst calculation is performed at the first futur
11. for additional risk associated with holding the bond Markit calculates the benchmark spread as the difference between the yield of the bond and the benchmark bond Selection criteria for a benchmark bond are Government bond is selected as an approximation of a default free bond The difference between maturities of a bond and the benchmark bond is the smallest in comparison to other alternatives Copyright 2014 Markit Group Limited All rights reserved www markit com 18 Index Calculus fma rk 6 0 SS Benchmark Spread Q 4 0 ec 99 3 0 e e e e 2 096 e e 1 0 0 0 05 25 45 65 85 10 5 12 5 14 5 16 5 18 5 20 5 22 5 24 5 26 5 28 5 30 5 Benchmark Yield Curve Bond Yield Annual Benchmark Spread The annual benchmark spread of a bond i at time f is BMS ih Y5 Zye 1 Note that for benchmark bonds BMS7 0 Semi annual Benchmark Spread The semi annual benchmark spread of a bond i at time f is 0 5 Lt BM i t Note that for benchmark bonds BMS 0 3 4 2 Spread to Benchmark Curve Spread to benchmark curve can be defined as a premium above the yield on a default free bond necessary to compensate for additional risk associated with holding the bond The default free yield to maturity is found by a linear interpolation of two benchmark bonds with maturities being just above and just below the time to maturity of a bond Selection criteria for benchmark bonds
12. has already been fully redeemed CV Cy ae Coupons 2 Diga Fa Nig t s lt j lt t CV Redemption R RP FA F it ni i j i j tij i t s i t s t s lt j lt t Generally it is assumed that there is only one coupon payment and one redemption payment per calculation period The XD factor only applies for the first coupon payment in the given period In situations where this is not the case special cash payments are dealt with as follows CV cis i t i i t s x G FA in Vu 2 R RP mun 22964 tl lt j lt t t s jet The different adjustment factors F ntt are used for sinking funds amortizing bonds pay in kind bonds and unscheduled full redemptions For other bond types F always equals 1 The cash payment of all bonds in an index is calculated as follows CV i l Benchmark total return Index The calculation for the local currency total return index is below The total return index can be expressed in terms of market values and cash MV CV MV 1 5 TR TR Total return index calculation for liquid indices The main difference between the various liquid index methods is the frequency of the cash investment in the money market There are two main varieties of liquid indices with cash Copyright O 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma
13. i at time t Amount invested for bond i 37 Index Calculus fma rk Amount invested per bond after the rebalancing rog Amount invested per bond before the rebalancing F The product of the redemption adjustment and the pay in kind adjustment factors for bond i at date t The product of the redemption adjustment and the pay in kind adjustment factors for sinking funds amortizing and pay in kind bonds of fully redeemed bond i at date j 1 i e one day before Pus The product of the redemption adjustment and the pay in kind adjustment factors for bond i at the last rebalancing p Capping factor for bond i Capping factor for bond i at the last rebalancing FA Flat of accrued flag of bond i and date t 0 if the bond is trading flat of accrued 1 otherwise FA s Flat of accrued flag of bond i at the last rebalancing 0 if the bond is trading flat of accrued 1 otherwise FA Flatof accrued flag of bond i and date t that is valid on date t 0 if the bond is trading flat of accrued 1 otherwise FSD First settlement date EX UST Spotexchange rate at t rebalancing FX 2 Spot exchange rate at 1 5 last rebalancing FX eer Forward exchange rate at t s for the period t s t C Coupon payment received from bond i between the day of the payment and month end If none the value is set to 0 G Value of any coupon payment received from bond i at time t If none the value is 0 a
14. in his portfolio to the new weights of the bond in the index Any bonds that need to be sold will be sold at the bid price while bonds purchased are bought at the offer price If the pricing of a bond in the index deviates from the prices that the investor has to use he will incur cost The following table gives a summary by region Summary by type bid ask indices dee Portfolio Index New Old Region Description price Price Portion Portion 1 Bond drops out Bid Bid 0 2 2 Bond does not need to be purchased Bid Bid f f No 2 Bond has to be purchased Ask Bid f f Yes 3 New bond to a liquid index Ask Ask f 0 No Summary by type mid indices D ipti Portfolio Index Old oun price Price Portion Portion 1 Bond drops out Bid Mid 0 f Yes 2 2 Bond does not need to be purchased Mid Mid d f No 2 Bond has to be purchased Ask Mid f f Yes 3 New bond to a liquid index Ask Mid 0 Yes 2 Region of all bonds with a weight increase in the portfolio f gt f For the change in amount outstanding during the rebalancing The amount invested per bond after the rebalancing can be stated as 2 M i Similarly f be calculated using the amount outstanding of the bond and index portfolio market value before the rebalancing The weighting per bond before and after rebalancing can be described as follows A fo
15. rk i Liquid indices with quarterly or semi annual rebalancing and monthly cash accrual For non rebalancing months the cash is invested into the money market until the following month end CASH CASH days yq t 5 0 ii Liquid indices with cash accrual until the end of the month In addition to the investment of cash from month end to month end according to the version i above some indices also invest coupon and redemption payments until the end of the month at the money market rate of the payment date 1 1 CASH CV cv 1 days yay i t CASH 1 daysyu t 5 1 lt cash is added to the standard formula for the indices CASH TR TR B MV CASH 2 1 4 Daily and Month to Date Returns Daily index returns are calculated for all Markit iBoxx benchmark indices according to the following formula Rei A 1 TR 1 1 Month to date index returns are calculated as follows 2 2 Gross Price and Income Indices 2 2 1 Gross Price Indices The gross price index represents the portion of the total return that is due to movements of the dirty price of the constituent bonds Benchmark Gross Price Indices The benchmark gross price index is calculated as follows MV GI GI_ 1 8 Liquid Gross Price Indices The liquid gross price index is calculated as follows MV 1 XR CASH CV XR M
16. that need to be purchased at the ask price In the case of bonds already in the index the portfolio manager will purchase additional notional of the bond on the ask side but the same bond is valued in the index at the bid or in the case of a mid price index the mid thus resulting in a tracking cost The following rules can be established for an index tracking portfolio 1 Buying and selling only takes place on the rebalancing date cash is reinvested and no cash is added to or taken from the portfolio 2 During the bond substitution at the end of a month the complete proceeds of a bond are invested in a new bond 3 If no buying and selling takes place the index and the portfolio are marked to market using the same price as the index either bid or mid depending on the index methodology and no cost is incurred 4 Forthe iBoxx indices that use bid prices the following re balancing scenario applies All bonds in the index are valued at their bid prices except new bonds that enter at their ask price Therefore selling or buying a new bond does not incur any cost If a trader must purchase additional notional of an existing bond he will incur costs in the form of the bond s bid ask spread 5 Forthe iBoxx indices that use mid prices the following re balancing scenario applies All bonds in the index are valued at their mid prices Therefore selling bonds does incur a mid bid spread and purchasing bonds incurs a mid ask s
17. to the calculation date Cash payment of all bonds at time t Cash payment of bond i at time t Coupon payments of all bonds i at time t Redemption payments of all bonds at time t Coupon payment of bond i at time t Redemption payment of bond i at time t Annual convexity of bond i at time t Convexity of bond i at time t Semi annualized convexity of bond i at time t Average semi annualized portfolio convexity at time t Average semi annualized portfolio convexity for bond i at time t Average annual portfolio convexity at time t Average annual portfolio convexity for bond i at time t Average semi annualized convexity at time t Average semi annualized convexity for bond i at time t Average annual convexity at time t Average annual convexity for bond i at time t Duration of bond i at time t Day of date 1 2 Date of coupon change Date of coupon payment Date of the next coupon payment Fictitious coupon date one exact coupon period before the first coupon payment date Fictitious coupon date one exact coupon period prior to Dat C Function to calculate the number of days between two dates for the ACT ACT day count Day count fraction using the actual number of days in the period Day count fraction between dates t s and t according to the prevailing money market day count Dirty Price Average portfolio duration at time t Average portfolio duration for bond i at time t Average duration at time t Average duration for bond
18. LC hb Y SW APt Semi annual Spread to LIBOR Curve The spread to benchmark of a bond i at time f is 0 SLC Y YswAp Lt 3 4 4 Z Spread The Z spread is a measure of the spread the investor would realize over the entire benchmark zero coupon curve if the bond is held to maturity The Z spread is calculated as the spread that will make the present value of the cash flows of respective bond equal to the market dirty price when discounted at the benchmark spot rate plus the spread The spread is found iteratively using the Newton method In general constant spread s over the spot curve z L for a bond at time fon an annual basis is calculated iteratively by using the Newton method 1 2 Z 5 Benchmark Zero Coupon Curve Benchmark zero curve z L also known as spot rate curve is calculated form dirty prices of defined benchmark bonds The following equation is solved using the Nelder Mead Simplex Method 12 min ds A p Lez t In the figure below the estimated zero curve is compared to annual yields of the benchmark bonds Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma rk 5 0 4 596 4 096 3 596 3 096 2 596 2 0 1 5 1 0 0 5 Spot Rate e Annual Yield 0 0 Z Spread Over LIBOR Curve The Z spread over Libor Curve is a measure of the spread that the investor would realize over the entire ICAP curve constructe
19. V _ CASH GI GI 2 2 2 Indices The income indices measure the portion of the index return that is due to actual cash payments Interest payments are represented in the coupon income index redemptions in the redemption income index and the total of these in the income index Copyright O 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma rk Income indices are set to 0 at the beginning of each calendar year Benchmark Income Indices The benchmark coupon income index is calculated as follows Coupons GL _ The benchmark redemption income index is calculated as follows Redemptiots IR IR_ GI 1 5 The benchmark income index is calculated as follows CV Coupons CV Redemptions IN 2 IN 61 E MV t s Or simplified IN IR Liquid Income Indices The liquid coupon income index is calculated as follows CV XR CASH CASH CV IC IC 4 GI MV CASH The liquid redemption income index is calculated as follows CV Redemptiors IR IR GI CASH The liquid income index is calculated as follows CV XR CASH CASH CV MV CASH IN IN 1 2 3 Rebalancing cost factor for liquid indices At the end of each month after the rebalancing the index level of most liquid indices is adjusted to account for the cost occurr
20. al index spread to benchmark curve is calculated as follows Y SBC w Index of bonds SBC 4 w Index of indices i l Semi annual Index Spread to Benchmark Curve The semi annual index benchmark spread is calculated as follows gt SBC wp Index of bonds SBC gt SBC w Index of indices i l 4 2 9 Index Spread to LIBOR Curve The index spread to Libor curve is calculated as follows Copyright 2014 Markit Group Limited All rights reserved www markit com at Index Calculus wp Index of bonds i l SLC wp Index of indices i l 4 2 10 Index Z Spread The index Z spread is calculated as follows 3 Z Spread w Index of bonds Z Spread Z Spread w Index of indices same formula applies for the index Z spread over LIBOR 4 2 11 Index OAS Index of bonds OAS 4 7 gt OAS w Index of indices i l 4 2 12 Index Asset Swap Spread gt ASW wp Index of bonds i l ASW 4 Y ASW w Index of indices i l Copyright 2014 Markit Group Limited All rights reserved www markit com mark 28 Index Calculus fma rk 5 Inflation Linked Index Calculations The calculation of the inflation linked indices follows the calculation principles for the standard bond indices Each inflation linked index is calculated in two versions one version without adjusting for inflation real and one ve
21. are Government bonds are selected as an approximation of a default free bond difference between maturities of a bond and the benchmark bonds is the smallest in absolute terms comparison to other alternatives Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus rk 6 Spread to Benchmark Cune e 496 o s E oa a 5 3 e e 2 o 1 Benchmark Yield Bond Yield Interpolated Yield 0 0 5 45 8 5 12 5 16 5 20 5 24 5 28 5 32 5 Annual Spread to Benchmark Curve The annual spread to benchmark of a bond i at time f is SBC a a ee Note that for benchmark bonds SBC 0 Semi annual Spread to Benchmark Curve The semi annual benchmark spread of bond i at time f is SBC n Y Yipois i t Note that for benchmark bonds SBC 0 3 4 3 Spread to LIBOR Curve Spread to Libor curve can be defined as a premium above the yield on Markit SWAP curve constructed from Libor rates and ICAP swap rates necessary to compensate for additional risk associated with holding the bond 5 0 4 4 0 4 3 0 4 2 0 1 0 0 0 15 20 25 3 0 35 40 45 50 55 60 65 70 7 5 80 85 90 95 Bond Yiled 9 ICAP Curve Copyright 2014 Markit Group Limited All rights reserved www markit com 18 Index Calculus fma rk Annual Spread to LIBOR Curve The spread to benchmark of a bond i at time f is S
22. ay before j 0 if the bond enters the index at the ex dividend period to ensure that the next coupon payment is excluded from the total return calculation 1 if a coupon payments are not ex dividend b has not entered the index during an ex dividend period or c entered the index during a previous ex dividend period XR Rebalancing flag It is linked to whether an index rebalancing occurs on the day It is 1 on calculation days where the index re balances and zero elsewhere XR applies to full rebalancings as well as partial rebalancings e g month ends between quarters for liquid indices Y Yield of bond i at time t i t Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus yl y2 Y Y Y adit Y BM i t i t mark Year of date 1 2 Annualized yield of bond i at time t Semi annualized yield of a bond at time Annual benchmark yield of bond i at time t Semi annual benchmark yield of bond i at time t Annualized yield of the interpolated benchmark of bond i at time t Semi annualized yield of the interpolated benchmark of bond i at time t Annualized value of Markit SWAP curve at time t Semi annualized value of Markit SWAP curve at time t Semi annual yield of bond i at time t 1 month interest rate for cash at the last rebalancing the function constructed by natural splines with defined knots is the Z spread of a bond i at time t Copyright 2014 Markit
23. ayable at t C Coupon payment CASH Cash at time t CASH Cash at the previous trading day CASH Cash at the end of the last month Cash flow of bond i in the jth period CF Cash flow of bond i at date t quoted to a notional of 100 Cash flow of a bond CO Average coupon at date t CO Average coupon for bond i at time t cost Cost factor Value of the next coupon payment of bond i during an ex dividend period because the next coupon is separated from the bond during the ex dividend period Outside the ex dividend period the value is 0 CP Value of the next coupon payment of bond i at the last rebalancing during an ex dividend 1 4 8 period because the next coupon is separated from the bond during the ex dividend period Outside the ex dividend period the value is 0 Copyright 2014 Markit Group Limited All rights reserved www markit com 36 Index Calculus CPI CPI o CPI s CV CV Coupons 6 Redemptiors MN 22 CX CX CX CXPU CXPU CXPU CXU CXU CXU CXU f D 41 42 Dat C C Dat C Dat C days datel date2 convention days days t s t DP DPU DPU DU DU t i t Copyright 2014 Markit Group Limited All rights reserved www markit com mark CPI level on the calculation date CPI level on the base date of bond i CPI level one year prior
24. d Index Return t s t LCY LCY CCY LCY CCY FX 1 8 LCY CCY LCY CCY xR FX FX Generally the hedged index is calculated by multiplying the month to date hedge return with the hedged index level from the last rebalancing IXR Hedged Multicurrency Index Return IXR IXR Yow t t LCY LCY CCY LCY CCY INR CU 24 pX Vee 1 1 5 LCY LCY CCY n IXR p For additional information please reference chapter 6 2 Calculation of the hedged index returns in terms of market values and cash 2 5 Bond Capping Weight limits caps can be applied in order to prevent a single bond or groups of bonds from dominating an index A group of bonds can be represented by a country or a market sector At the regular rebalancing dates the index composition and weighting is reviewed and weight caps are applied if necessary For market value weighted indices the weight of a bond is derived from its market value in relation to the overall index market value In the following the term class refers to either a single bond or groups of bonds In general two basic procedures are applied Pro Rata Capping If one or more classes account for more than the defined maximum weight their weight is reduced to the cap limit by reducing the amount outstanding of all the bonds in the class The excess weight is distributed pro rata to the remaining classes S
25. d from Libor rates and ICAP swap rates if the bond is held to maturity The Z spread is calculated as the spread that will make the present value of the cash flows of respective bond equal to the market dirty price when discounted at the ICAP rate plus the spread The spread is found iteratively using the Newton method In general Z spread s over Markit SWAP curve z L for a bond at time fon an annual basis is calculated iteratively using the Newton method B Py 1 2 B Si ij y j l 3 4 5 Option Adjusted Spread Similar to Z spread OAS is the spread over the benchmark zero coupon curve realized if the bond is held until maturity The major difference is the interest rate volatility assumption used in OAS Due to the fact that interest rate changes can affect the cash flows of the security with embedded option the following relationship cab be highlighted Z spread OAS Option cost Constructing Binomial Interest Rate Tree Two components are required to determine the interest rate tree Benchmark Zero Coupon Curve Empirical Volatility Empirical volatility is annualized daily standard deviation of percentage change in daily yields from their mean The number of observations equals to the number of trading days in one year period minus 5 of observations from each tail of the distribution of percentage change in daily yields The daily standard deviation is annualized by multiplying it by the square ro
26. e iBoxx yield to worst calculation date and any ensuing iBoxx yield to worst calculation date iBoxx yield to worst calculation dates are Coupon and or redemption payment date inside an exercise period Start and end of an exercise period Date on which the exercise price changes Copyright O 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma rk The final redemption date The first future iBoxx yield to worst calculation date is The next iBoxx yield to worst calculation date on or after the current settlement date plus any notice period if the latter exists or the next iBoxx yield to worst calculation date after the current settlement date The yield of the bond is calculated according to the general yield formula for each relevant call put date and the final maturity date The following procedure determines when the bond is most likely to be redeemed 1 The call date with the lowest yield to call is determined 2 The put date with the highest yield to put is determined 3 The determined yield to call yield to put and the yield to maturity are compared yield to call is lower than the yield to maturity then the bond is assumed to be called Ifthe yield to put is higher than the yield to maturity then the bond is assumed to be put 4 The expected redemption path is Bond is assumed Bond is assumed to Expected redemption date to called be put No No Maturity date Y
27. e issue in 96 of par of bond i in the jth period Index return for bond i Daily index return Month to date index return Total return of sub index i from the last rebalancing t s to t Hedged return at time t Value of the real monetary unit on the calculation date Redemption price of a redeemed portion of bond i at date t Redemption price of a redeemed portion of bond i in the jth period Average semi annual portfolio yield at time t Average annual portfolio yield at time t Average semi annual yield at time t Average semi annual yield for bond i at time t Average annual yield at time t Time since last rebalancing Annual spread to benchmark curve of bond i at time t Semi annual spread to benchmark curve of bond i at time t Settlement date Annual spread to LIBOR curve of bond i at time t Annual spread to LIBOR curve of bond i at time t Markit SWAP curve rate at the next coupon payment day 40 Index Calculus fma rk t Time of calculation t Date of the coupon payment t in the same month as the settlement date t but before or att Date of a cash flow t Calculation date for which most recently published CPI is valid 10 Base date of an inflation linked bond t y 1 One year prior to the calculation date ti Next coupon payment after the settlement date t t2 Next but one coupon payment after the settlement date t ti Date ti the date of the i th cash flow TR Total return index level at time t
28. ed in rebalancing the index For most of the index adjustments between two rebalancing dates the adjustment cost should be zero and no adjustment will be necessary There are two approaches to capture the cost The cost is either fixed e g 2bps or takes into account the actual transaction costs P P _ WCASH TR Zr p Wi TR WCASH PI Z i T Wi PI Pi cost 1 costs cost 1 5 5 y Lewes wt a Lane WCASH TR 2i VT Wi TR CASH PIT Ai pT Wi PI 1 1 Hence at each rebalancing the index level is adjusted as follows TRee TRES 1 cost pifinal 1 cost qqideal 1 cost For a detailed explanation of the formula please refer to the Appendix 2 4 Hedged and Unhedged Index Calculation Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus rk Currency hedging is applied to the index constituents on each monthly rebalancing At the rebalancing day the position is fully hedged using one month forwards During the month the index will be partially hedged with the bond market value fluctuations since rebalancing remaining unhedged The below formulae apply for both Total Return and Price Return indices Unhedged Index Return xR TR FX l 1 5 Unhedged Multicurrency Index Return n FX LCY CCY 1 i LCY LCY CCY L i t s FX 5 i l Hedge
29. erage Time to pL ea se eas 27 4 2 8 Index Spread to Benchmark 27 4 2 9 Index Spread to LIBOR 22 04440 essen ente nennns nnn ns sinn taste nn en nsi tnn 27 4 2 10 Index Z Spread eii det estes Dee de fa een Dru n EQ 28 LTEM IS S8 EDT 28 4 2 12 Index Asset Swap Spread tata Ee aea esee ERREUR es e X RE REL ru a ERE REEL e a arua 28 5 Inflation Linked Index Calculations eieesseeeeeeeeeeeeeeeeeee nennen nnne nn nnn ansia sitne nnn nnns satanas nnne nnns 29 Bal Calculating nominal data iicet ett eee nae Fe DE Rae xa DEA 29 MEE stum 29 Copyright 2014 Markit Group Limited All rights reserved www markit com 2 Index Calculus fma rk 532 2 a a 5 4 Bondiand Index Analytics XEP CIDG C E 0 0 E E 400X 0040 0 S0n0 rm 6 1 Cash and Turnover Reinvestment Cost 6 2 Calculation of the hedged index returns in terms of market values and 7 Annotallons II I IM 8 2
30. es No Call date with lowest yield to call No Yes Put date with highest yield to put Yes Yes Earlier of the call and put date The expected redemption date is used in the calculation of the expected remaining life and in the other bond analytics calculations 3 3 2 Average Expected Remaining Life The calculation depends on the bond type and takes into account the day count convention of the bond Bullet bonds without options For plain vanilla bonds the expected remaining life of the bond is its time to maturity calculated as the number of days between the rebalancing and its maturity Bullet bonds with options For callable putable bonds a yield to worst calculation determines when the bond is expected to redeem This date is the expected redemption date The expected remaining life is calculated as the number of days between the rebalancing and the expected redemption date Hybrid capital For hybrid capital bonds with call dates i e perpetual bonds or other callable dated hybrid capital bonds the first call date is always assumed to be the expected redemption date The expected remaining life is calculated as the number of days between the rebalancing and the expected redemption date For non callable fixed to floater bonds the expected remaining life is calculated as the number of days between the rebalancing and the conversion date i e the date on which the bond turns into a floating rate note Amortizing
31. f indices the market value weight is equal to the current weight of the sub index in the overall index as of the last rebalancing DU 15 D 11 51 1 1 5 i SDU on gp we i l 1 1 5 1 1 1 5 4 2 Analytics As noted above in section 3 1 Bond Analytics the analytics for securities that are traded flat of accrued are assumed to be zero and are not accounted for in the overall index analytics 4 2 1 Average Yield The average annual yield is calculated by weighting the yield of each bond with the corresponding market capitalization and duration of the respective bond wi Index of bonds RY lt t B n Y RY wi Index of indices average semi annualized yield is calculated as follows Y wp Index of bonds RYS 7 Y RYS wi Index of indices i l The average portfolio yield is calculated by adjusting the portfolio for the cash portion MV RY Index of bonds MV CV RYP t x RYP w Index of indices average semi annualized portfolio yield is calculated as follows Index of bonds RYS MV RYPS RYPS Index of indices i l Copyright 2014 Markit Group Limited All rights reserved www markit com E Index Calculus rk 4 2 2 Index Benchmark Spread The index benchmark spread is calculated for all indices The annualized index benchmark s
32. he next period using the closing ask prices from the last trading day of the previous period 2 1 2 Total Return Index Components Nominal Value The nominal value of the index is the sum of the individual bond nominal values and is calculated as follows NV YR Lu N 1 1 5 Market Value The market value of a single bond at time t is calculated as follows MV A XD s CP FA Ba Nias uL The capping factor will be normally be 1 unless in cases where capping is applied 1 1 5 The market value of the index is the sum of the market values of all bonds at time and is calculated as follows MV i l Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus rk Base Market Value The base market value is the market value of the bond calculated at the rebalancing date t s it also does not take cash payments into account The base market value of a single bond at time t is calculated as follows BMV i FA DL FSE 1 1 5 1 1 5 1 1 5 The base market value of the index is the sum of the base market values of all bonds and is calculated as follows BMV gt BMY i l i t s 2 1 3 Total Return Index Calculations Cash Payment The cash payment for a single bond at time t is the sum of all coupon and scheduled redemption payments since the last index rebalancing plus the redemption value if the bond
33. hould another class exceed the cap limit afterwards its weight is reduced and redistributed accordingly Step wise weight reduction If one or more classes account for more than the defined maximum weight their weight is reduced by reducing the weight of the smallest bond If as a consequence this bond is removed from the index portfolio the process is continued with the next smallest bond Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus rk The iBoxx index guides provide detailed information on which applicable capping method is used for those indices which are subject to weight restrictions Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus mark 3 Bond Analytics The following section defines how bond analytics are calculated for the securities within the index With the exception of section 3 2 Bond Returns all analytics for securities traded flat of accrued are assumed to be zero 3 1 Accrued Interest The following day count conventions are taken into account when calculating the iBoxx Indices ACT 360 ACT 364 ACT 365 ACT ACT 30 360 30E 360 BUS 252 The following conventions are necessary for bonds that pay a coupon at around month end Non EOM Bonds with this convention pay on the same calendar day each month i e 30 June and 30 December No Leap Year Bonds that do not pay interest on 29 February Other b
34. ices i l 4 2 5 Average Convexity The calculation method for average convexity is similar to that previously described for average duration and average modified duration except that duration modified duration is replaced by convexity gt CX wit Index of bonds CXU 4 7 Y CXU wi Index of indices i l The average semi annualized convexity is calculated as follows gt CX wit Index of bonds CXU 4 Y CXU wi Index of indices i l The average portfolio convexity is calculated as follows SCX wit Index of bonds CXPU 1 gt CXPU wj Index of indices average semi annualized portfolio convexity is calculated as follows Copyright 2014 Markit Group Limited All rights reserved www markit com 26 Index Calculus fma rk 5X cx Index of bonds 247 gt Index of indices i l 4 2 6 Average Coupon The average coupon is nominal weighted For bonds with a multi coupon schedule the current coupon is included C wi Index of bonds 24 Index of indices i l 4 2 7 Average Time to Maturity The calculation method for average time to maturity is similar to the previously described calculation A weighting is carried out in accordance with the amount outstanding wj Index of bonds i l LFU LFU w Index of indices i l 4 2 8 Index Spread to Benchmark Curve Annual Index Spread to Benchmark Curve The annu
35. ion Adjusted 20 3 46 Asset Swap Spread esie nere pereat e saat suyagceduany tases 21 4 Index Anali GS esien 23 41 Weightings for index 2224 nn tr trii snnt nnne 23 4 1 1 Nominal value 23 4 1 2 Base market value weighting eene ANNEN nnn nene rennen nns 23 4 1 3 Market value weightlhg 23 4 1 4 Duration adjusted market value 24 4 2 Madex Anay 32 Tm 24 4 21 24 4 2 2 Index Benchmark Spread 25 42 3 Average Duration eese repere rint 25 4 2 4 Average Modified 25 425 ie ene cages 26 426 Average COUPON eve dl 27 4 2 7 Av
36. n fixed and floating payments Given the frequency of fixed rate payments the floating rate payment frequency is determined as follows 1 Fixed rate paid yearly Floating rate paid semiannually 2 Fixed rate paid semiannually Floating rate paid quarterly 3 Fixed rate paid quarterly Floating rate paid monthly 4 else Fixed frequency Floating frequency Fixed Rate Payments Floating Rate Payments DF E Fixed DF ne prene 360 SWAP 41 SWAP 1 Copyright 2014 Markit Group Limited All rights reserved www markit com 21 Index Calculus rk Fixed Rate Payments Floating Rate Payments T T DEC 4 Principal DEP 2 TET Desine 1 1 Given the present value of fixed and floating rate payments the asset swap spread is calculated as follows P DP PV Floating ASW Copyright 2014 Markit Group Limited All rights reserved www markit com 22 Index Calculus rk 4 Index Analytics 4 1 Weightings for index analytics There are four different weighting concepts for index analytics depending on the specific analytical value being calculated 4 1 1 Nominal value weighting For an index of bonds the nominal weight is the share of each bond s notional in the aggregate notional of the index Cap w F po 1 4 8 t x n 25 E CASH i l For an index of indices the nominal weight is equal t
37. o sacs an eh iet cust uate cys souudeed eactavesseuuensueetueyys 10 Sk Accmied Interest oce irte tei ee ee 10 9 1 1 360 ACT 364 865 Lire rtt terrd Coetu a ur Eau a RED Peak 10 912 epo r ed 11 919 11 Ou EMEUSEI MM 12 252 E A 13 13 39 Dm 13 3 3 1 Yield to worst 13 3 3 2 Average Expected Remaining Life 14 3 3 3 Average Redemption 4 4 0000 15 eae T i ia 15 3 3 5 Modified Durat 15 9 3 6 COMVOXIY EETTTP 16 3 4 Bond Spread Analytics 16 941 Benchmark 16 3 4 8 Spread to Benchmark Curve esssssssssesssseee eene enne nne nennen nnns entente reset nsi en nennen 17 3 4 8 Spread to LIBOR Curve seniii ercran inea aeaaeae aas aeaa aa aana aiara EE ENAKA 18 Ur MEMO 1e E A T 19 3 4 5 Opt
38. o the fixed weight of the sub index in the overall index as of the last rebalancing N _ Fix Wit Wi t s 4 1 2 Base market value weighting For an index of bonds the base market value weight is the share of each bond s base market value in the aggregate base market value of the index MV 1 1 5 i l BMV Wi For an index of indices the base market value weight is equal to the fixed weight of the sub index in the overall index as of the last rebalancing Fix 11 8 Wi W 4 1 3 Market value weighting For an index of bonds the market value weight is the share of each bonds market value in the aggregate market value of the index MV 1 1 i t For an index of indices the market value weight is equal to the current weight of the sub index in the overall index as of the last rebalancing Fix MV _ 1 Fest Wi ts n Fix 1 V esa Wi t s i l The bond market value weighting adjusted for cash is not applicable for indices of indices For indices of bonds it is calculated as Copyright 2014 Markit Group Limited All rights reserved www markit com 23 Index Calculus fma rk 4 1 4 Duration adjusted market value weighting For an index of bonds the duration adjusted market value weight is the adjusted share of each bonds market value in the aggregate adjusted market value of the index D YD i l For an index o
39. of all bonds in the index at time t Base market value of bond i at the last rebalancing Base market value of the index at the last rebalancing Market value of bond i referring to transaction prices Market value of bond i referring to index price Number of bonds number of future cash flows in the index Adjusted amount issued of bond i at date t Notional of bond i at the last rebalancing a Notional amount outstanding of bond i at the last rebalancing 39 Index Calculus OAS 1 5 Floating RYS RYS RY t Lf S SBC SBC SD SLC SIG SWAP Copyright 2014 Markit Group Limited All rights reserved www markit com mark b Fictitious nominal of bond i substitutes c Zero 0 for dropped bonds Inflation adjusted notional for bond i on the calculation date Nominal value at date t Assumed annual inflation on the calculation date is the OAS of a bond i at time t Clean price of bond i at time t Index price of bond i Nominal clean price for bond i on the calculation date Portfolio price of bond i Real clean price for bond i on the calculation date Closing price of bond i on the last trading day of the previous month Price index level at time t Closing price index level on the last calendar day of the previous month Present value of fixed payments Present value of floating payments Redeemed portion of the issue in of par of bond i at date t Redeemed portion of th
40. onds that pay at month end always pay on the last calendar day on the month i e 30 June and 31 December In addition some bonds may pay interest on the first or last business day of the month If such a bond is identified the relevant coupon interest payment schedule is used to calculate accrued interest and coupon payments for this bond Coupon Date Formulae for Different Day Count Conventions The table below summarises the different formulae used for calculating accrued interest under different scenarios The table includes the section reference where the formulae are defined Day Count Normal Coupon Normal First Coupon Long First Coupon Short First Coupon ACT n Section 3 1 1 Section 3 1 1 Same formula as Same formula as Normal Coupon Normal Coupon ACT ACT Section 3 1 2 Section 3 1 2 Section 3 1 2 Section 3 1 2 30 360 Section 3 1 4 Section 3 1 4 Section 3 1 4 Section 3 1 4 30E 360 Section 3 1 5 Section 3 1 5 Same formula as Same formula as Normal Coupon Normal Coupon BUS 252 Section 3 1 6 Section 3 1 6 N A N A 3 1 1 ACT n ACT 360 ACT 364 ACT 365 The accrued interest calculation is based on ISMA standards In addition to the normal coupon dates the calculation also takes into account odd first coupons coupon changes during a period and ex dividend details The calculation is the same regardless of whether the basis is 360 364 or 365 days Accrued interest for the 274 and subsequent coupon pe
41. ot of 252 trading days in a year Copyright 2014 Markit Group Limited All rights reserved www markit com 20 Index Calculus rk OAS Calculation The OAS is calculated as the spread that will make the present value of the cash flows of the respective bond equal to the market dirty price when discounted at the benchmark spot rate plus the OAS spread Given spot rates and empirical volatility the interest rate tree is derived using the iterative search method 9 568 5 383 a 2 717 3 553 1 319 0 147 0 622 0 490 0 182 Interest rate tree of 2 5 bond with semi annual coupon payments After the binomial interest rate tree is derived the theoretical price of the bond is found by conventional backward induction of future cash flows The OAS is found iteratively by using the Newton method such that adding the spread to every node of the interest rate tree would make the present value of the cash flows equal to the market dirty price of the bond 3 4 6 Asset Swap Spread Markit SWAP curve constructed from Libor rates and ICAP swap rates is central to the process of asset swap spreads calculation As soon as the curve is defined the present value of fixed and floating payoffs is calculated and the asset swap spread is determined The curve is interpolated to account for fixed and floating payoffs dates Asset Swap Spread Calculation To evaluate asset swap spread one needs to distinguish betwee
42. pread Liquid Index Cost Factor Rebalancing Scenario of Markit iBoxx Liquid Indices Bonds that Bonds that Bonds that drop out of remain in the enter the the index 1 index 2 index 3 Bonds in the four different regions may be characterized as follows Regioni Bonds that leave the liquid indices at the rebalancing 2 Bonds that remain in the liquid indices Region 3 Bonds that newly enter a liquid index at the rebalancing The time before the rebalancing is denoted with The time after the rebalancing is denoted with The market value of the portfolio before the rebalancing without ex dividend periods is given as the sum of the market value of all bonds plus cash Copyright 2014 Markit Group Limited All rights reserved www markit com 31 Index Calculus rk M Y P 4 f i l Cash is treated as a bond with a price of 100 and accrued interest of 0 Since no cash is added or taken from the portfolio at the rebalancing the assumption of no further cash addition leads us to the following equation M That is the market value of the portfolio before the rebalancing equals the market value after rebalancing using transaction prices Cost means the relative difference between market value of the portfolio using transaction prices to the portfolio valued with index prices M M 1 cost The investor has to rebalance his index tracking portfolio by adjusting the weights of each bond
43. pread at time t is gt BMS w Index of bonds BMS 5 gt BMS wp Index of indices i l The semi annualized index benchmark spread at time t is gt BMS w7 Index of bonds BMS gt BMS w7 Index of indices i l 4 2 3 Average Duration The average duration is weighted by the market capitalization of the respective bonds D wy Index of bonds 4 1 n Index of indices i l The average portfolio duration is calculated as follows D wi Index of bonds DPU 4 Y DPU wi Index of indices i l 4 2 4 Average Modified Duration The calculation method for average modified duration is similar to that previously described for average duration except that duration is replaced by modified duration Annual Modified Duration Y MD wi Index of bonds MDU 4 U w Index of indices i l The average semi annualized modified duration is calculated as follows Copyright 2014 Markit Group Limited All rights reserved www markit com 25 Index Calculus rk Index of bonds i l MDU U w1 Index of indices i l The average modified portfolio duration is calculated as follows 3 MD wit Index of bonds MDPU 1 Y MDPU wit Index of indices i l The average semi annualized modified portfolio duration is calculated as follows gt MD wit Index of bonds MDPU 4 Y MDPU wi Index of ind
44. re known at the time of the rebalancing so the cost percentage can be calculated using data known at the time of the rebalancing 6 2 Calculation of the hedged index returns in terms of market values and cash The local currency index market value is the sum of the market values of all the components MV plus any cash that has been paid since the last rebalancing IV MV Cash The rebalancing index base market value is the sum of all components in the index at the rebalancing BMV IV BMV Hedged indices are calculated by hedging the value of the index from one rebalancing to the next Hedging therefore occurs once a month at the rebalancing During the month the index value consists of a hedged portion and an unhedged portion The unhedged part of the index is caused by the performance of the bonds Hed IV BMV HedgedPorion __ LCY CCY IV BMV FX t t s t hedgedResidual LCY CCY 1V UnhedgedResidual Cash BMV _ 2 The hedge performance can therefore be written as HedgedPorton UnhedgedResidual We quy inn ez Hed LCY Y vie ge ICC Rewriting using local currency market values and cash MV Cash BMV BMV H 1 8 1 1 BMV FX Eee Rearranging MV Cash Ex Oe BMV pee BMV FX 1 5 1 5 oz Copyright 2014 Markit Group Limited All rights reserved www ma
45. riods days TU t i t C FA n Accrued interest for the first coupon period days FSD t n A i t C FA Coupon changes Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus rk The following formula shows how to calculate the accrued interest when a coupon change occurs during a period days Dat C days Dat C C t m J a D l G Fa a n n For the coupon payment in the period of the coupon change the following formula applies C days acr Dat C Dat C C days b n 3 1 2 ACT ACT The accrued interest calculation is based on ISMA standards In addition to the normal coupon dates the calculation also takes into account odd first coupons coupon changes during a period and ex dividend details Accrued interest for the 274 and subsequent coupon periods dasar DWC D days Dat C Dat C i t tst Accrued interest for the first coupon period days FSD t days c FSD Dat C 111 Long first coupon The coupon payment is divided into a short and a normal coupon period If the settlement date lies between the first settlement date and the following fictitious coupon date accrued interest is calculated as follows des days cr FSD 1 days Dat C
46. rkit com Index Calculus rk The first part of the return is the unhedged performance of the index and the second part is the hedge effect LCY CCY LCYICCY 1 Cash Fx joe HS t BMV px ewm FX EE Copyright 2014 Markit Group Limited All rights reserved www markit com 35 Index Calculus rk 7 Annotations Change in amount outstanding of bond at time i A Accrued interest for bond i Accrued interest of bond i at time t Accrued interest of bond i at the last rebalancing Aj Nominal accrued interest for bond i at date t Ai Real accrued interest for bond i at date t ASW Asset swap spread of a bond i at time t bdays Business Days BMS Annualized index benchmark spread at time t BMS Semi annualized index benchmark spread at time t Annualized benchmark spread of bond i at time t BMS Semi annualized benchmark spread of bond i at time t BMV Base market value of the index at the rebalancing BMV Base market value of bond i at the rebalancing C Annual Coupon C Next coupon before the coupon change C Next coupon after the coupon change Next coupon payment Current coupon of bond i at time t Ca Coupon payment in the period of the coupon change pou Annual coupon of bond i Nominal coupon payment for bond i payable at t as of calculation date Real coupon payment for bond i p
47. rns at time t can apply to both total return and price return Unhedged index returns at time t can apply to both total return and price return Time in years for bond i between date t and the jth cash flow Time in coupon periods for bond i between date t and the jth cash flow Time difference in coupon periods between t and j Number of days between floating rate payments Daily local index return for bond i at time t Daily local index return for bond i at time t Expected remaining life of bond i at time t average life for amortizing bonds and sinking funds Average expected remaining life at time t Average expected remaining life for bond i at time t Number of coupon payments per year Month of date 1 Market value of portfolio before rebalancing Modified duration of bond i at time t Average annualized modified duration at time t Average annualized modified duration for bond i at time t Average semi annualized modified duration at time t Average semi annualized modified duration for bond i at time t Average annualized modified portfolio duration at time t Average annualized modified portfolio duration for bond i at time t Average semi annualized modified portfolio duration at time t Average semi annualized modified portfolio duration for bond i at time t Market value of portfolio after rebalancing based upon index prices Market value of portfolio after rebalancing based upon transaction prices Market value of bond i at date t Market value
48. rsion including the past and projected future inflation adjustments this chapter real values without inflation adjustments are denoted with the superscript and nominal values with the superscript e g P for real price and P for nominal price The standard calculus formulae apply to the corresponding real and nominal index and analytics calculations For the index calculation in real terms all prices accrued interest and projected future cash flows used need to be without inflation adjustments and for the nominal calculations all data needs to include the inflation adjustment 5 1 Calculating nominal data Nominal prices accrued interest as well as coupon and redemption payments are calculated by multiplying the real values with the applicable index ratio I CP AS IR IR The current market capitalization of a bond is affected by the real face amount issued the real price of the bond and by any adjustments due to past inflation Similarly the inflation adjusted amount outstanding is equal to the product of the index ratio multiplied by the unadjusted amount outstanding NS Ni IR 5 2 Index Ratio The index ratio captures the inflation adjustment due on a specific calculation date or a future cash flow date The current index ratio JR captures the inflation adjustment due on the calculation date The index ratio is derived from the corresponding country specific infla
49. s Y Y m The periodic yield can be transformed into the annual yield ys 1 4 Y lu semi annualized yield is calculated as follows 2 i 3 3 4 Duration The duration of a bond at time t is calculated as follows a 1 ig 1 Yis 1 1 j L Aty SCF Le M Je P 2 i j i t j l D i t 3 3 5 Modified Duration The modified duration of a bond at time t is calculated as follows 1 1 Y In the same way the annual modified duration can be expressed as D i Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma rk 1 as The semi annualized modified duration is calculated as follows 5 1 MDU D 2 3 3 6 Convexity The convexity of a bond at time t is calculated as follows ae 27 z M5 A cene OE xz 42 The convexity is set to O if F is 0 Using the annualized yield the annualized convexity of a bond can be calculated as if 1 V CX CX 1 yos MD 1 4 1 Yait ym i m Similarly using the semi annualized yield the semi annualized convexity of a bond can be calculated as 2 2 4 m j y 2 CX MD t 1 l 2 9 2 3 4 Bond Spread Analytics 3 4 1 Benchmark Spread The benchmark spread can be defined as a premium above the yield on a default free bond necessary to compensate
50. tion index CPI There are two main versions of the index ratio For most countries the index ratio is calculated by dividing the current CPI level by the CPI level from the base date which may be the interest accrual date of the bond or a fixed date CPI IR CDL The base date is bond specific and therefore the index ratio is bond specific as well 1 0 index ratio is rounded to a country specific number of decimal places Since the CPI is only released monthly or quarterly and is only available in arrears the CPI level CPI applicable to a certain calculation date refers to CPI data n months prior to the calculation date and may be interpolated between two neighboring CPI levels The specific procedure varies from country to country For Chile Colombia Costa Rica Mexico Uruguay use a real monetary unit RMU that serves as the index ratio The real monetary unit is derived from the CPI changes but is the same for all bonds IR RMU The index ratio is not bond specific but applies equally to all issued inflation linked debt from this country For future cash flows R will be estimated to capture the additional inflation adjustment expected to occur between the calculation date t and the cash flow date t The index ratio for future cash flows is estimated from the index ratio calculated from the most recently released CPI data by adding the projected future inflation to it IR IR l 2 0 tt i t t
51. upon date are calculated as follows days Dat C _ t a 360 C FA ia Accrued interest for irregular first coupon dates days Dat C CFA Coupon changes The following formula shows how to calculate the accrued interest when a coupon change during a period occurs a days44 Dat C Dat C C ome days44 Dat C C t 360 360 i For the coupon payment in the period of the coupon change the following formula applies days Dat C Dat C C days44 Dat C C Cig 360 Cat 360 e 3 1 4 30E 360 The accrued interest calculation is based on ISMA standards If one of the dates t or Dat C is dated on the 31 it is set to 30 The days between the settlement date and the last coupon date are calculated as follows days Dat C _ t y2 y1 360 m2 ml 30 42 dl Normal coupon date DAC FA 360 5 First coupon date FSD t ELEM C FA 360 U Coupon changes The following formula shows how to calculate the accrued interest when a coupon change occurs during a period d days Dat C Dat C C C4 days Dat C C t LC FA 360 360 Copyright 2014 Markit Group Limited All rights reserved www markit com Index Calculus fma rk For the coupon payment in the period of the coupon change the following formula applies C days Dat C Dat C C Ch

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