Bond Calculator. Spreads (G-spread, T-spread) References and Contact details

Transcription

1 Cbons.Ru Lt. irogovskaya nab., 21, St. etersburg hone: +7 (812) Bon Calculator Bon calculator is esigne to calculate analytical parameters use in assessment of bons. The tool allows calculating prices, accrue coupon interest, various types of bon yiels, uration, as well as moifie uration, curve, VB, making it possible to analyze volatility of the ebt market instruments an assess how bon price changes with the yiel. Software interface allows viewing key issue parameters an simulating them. It is also possible not only to analyze trae issues, but also to simulate bon cash flows an create user moels. Using the calculator Terms an Definitions Bon Classification Face Value, Lot of Multiplicity Minimum Denomination, Minimum Traing Lot Accrue Coupon Interest Calculating the Number of Days between Dates Designations Calculate Values Bon Yiel Yiel to Maturity Effective Yiel to Maturity Nominal Yiel to Maturity Simple Yiel to Maturity Yiel to Offer Yiel to the Next Coupon Current Yiel Ajuste Current Yiel Volatility, Duration, Convexity Years to Maturity Macaulay Duration Moifie Duration rice Value of Basis oint (VB) Convexity (Conv) Spreas (G-sprea, T-sprea) References an Contact etails Cbons.ru 1

2 Using the calculator To continue working with the calculator, you nee to loa the issue from the atabase or create a bon moel. Loaing Issues from Cbons Database 1. Enter either the issuer, or the issue registration number, or ISIN in the search bar, an click "Search". 2. Select a bon issue from the opene list an press "Loa". Creating a Bon Moel 1. Click the button "Create a Moel" an choose the type of the bon you want to create (coupon bon / iscount bon). 2. Fill in the issue parameters. Calculating Bon arameters The calculator allows computing analytical parameters either base on the known bon price, or base on the given yiel. rice of the bon is the input value by efault. To calculate bon parameters base on the given yiel, click "Calculate rice from Yiel". Bon price can be shown as a percentage of face value, or irectly in units of face value. You can make your calculations base on the known "net price" of the bon (price excluing ACI), or "irty price" (incluing ACI). By efault, calculations are mae from the net price shown as percentage of face value. Using the Constructor Moe In our calculator, you can eit parameters of an existing bon moel. To eit the bon cash flow (or create a new one), use the buttons unerneath it. Cbons.ru 2

3 With "Constructor" button, you can create a chart of coupon payments on the bon. Enter the perio, for which the cash flow will be fille, frequency of coupon payment (by efault, these parameters are fille base on the issue terms you have alreay entere), coupon rate an payment amount. ayment amount is optional - you can just click on "Automatic Calculation", an coupon payments will be calculate automatically. Use "A" button to a coupon payments to an existing coupon chart. Cash flow on the bon can be eite irectly in the table (you can change the ate, coupon rate, coupon payment an reemption amount). After eiting the cash flow parameters, select respective lines in the stream an click "Upate". The system will then recalculate the coupon payments base on your eit. To elete lines from the "Cash Flow" table, select the lines an click the "Delete" button. Cbons.ru 3

4 Using the calculator (Lite) The lite-version of the calculator is available for all website profiles via the menu item "Bon Calculator (Lite)". The functionality oes not require a flash-player support an works in all browsers. The basic version of the calculator allows computation of yiel by price an price by yiel for any ate. In the list of calculate inicators various types of prices, yiels, ACI an uration are isplaye. There is the function to moel simple coupon-bearing an iscount bons, which allows you to quickly assess the price or yiel of bons accoring to the input parameters. To continue working with the calculator, you nee to loa the issue from the atabase. Loaing Issues from Cbons Database 1. Enter either the issuer, or the issue registration number, or ISIN in the search bar. 2. Select a bon issue from the opene list an click. Calculating Bon arameters The calculator allows computing analytical parameters either base on the known bon price, or base on the given yiel. rice of the bon is the input value by efault. To calculate bon parameters base on the given yiel, click "Calculate rice from Yiel". Bon price can be shown as a percentage of face value, or irectly in units of face value. You can make your calculations base on the known "net price" of the bon (price excluing ACI), or "irty price" (incluing ACI). By efault, calculations are mae from the net price shown as percentage of face value. Cbons.ru 4

5 Using the "Issue moel" To moel the issue enter the "Maturity", "Coupon rate", "The frequency of coupon paymwnts (per year)". At least one of the fiels "Current price" or "Yiel to maturity" is also require for calculation. ress the button "Calculate" to view all other calculating parameters. In the example we create the moel of shortterm zero-coupon bon with current price 100% an maturity 100 ays. Also we create the moel of 3-year coupon bon with current price 100% an coupon rate 10%. We use bon basis 365 ays per year to calculate all parameters. Cbons.ru 5

6 Terms an Definitions Bon Classification Bon is a security bearing an obligation of the issuer to pay its holer (lener) the face value or an equivalent in property at the en of the tenor. A bon may also inclue the holer's right to receive a percentage of the face value stipulate therein, or other property rights. Eurobon is a security issue in external (international, offshore) bon market with the following characteristics: international synicates act as the unerwriters; bons are simultaneously place with investors from ifferent countries; bons are issue outsie the jurisiction of any specific country an o not have to be registere. Securities issue both in the omestic an external markets are calle global bons. Foreign bons are bons issue in the omestic market of another country. Issuers of the foreign bon market are not officially registere in the country, where the bon is issue an trae. Depening on the sector: 1. Government 2. Municipal 3. Corporate Depening on the metho of interest payment: 1. Interest bearing bons: - coupon bons (bons with perioic coupon payments) - accrual bons (at maturity investors are pai the bon's face value together with the accrue coupon interest) 2. Zero-coupon bons (bons paying no coupon interest). Investors in zero-coupon bons, as a rule, earn on the ifference between the placement price an the face value. Depening on the metho of income generation: 1. Fixe permanent coupon bons 2. Fixe variable coupon bons 3. Floating rate bons 4. Inex-linke bons By metho of face value repayment: 1. Bons with reemption of face value in one payment in the en of tenor 2. Bons with repayment in installments istribute over time (amortization) Depening on early reemption terms: 1. Bons without an option of early reemption 2. Bons with a call option (reemption is initiate by the issuer). The issuer has the right to fully or partially repay the ebt before the maturity ate. 3. Bons with a put option, incluing: a) bons with an option of early reemption initiate by the investor The holer has the right for reemption of the bon at a preetermine price on the agree ate. b) bons with an option of resale (early repurchase) initiate by the investor (Russian analogue is the bon with irrevocable offer, which can be trae after the sale). Holer of such bons has the right to sell them back to the issuer at a preetermine price on an agree ate. Cbons.ru 6

7 Depening on the maturity: 1. Term bons 2. erpetual: - with a call option - with a put option Face Value Face value of a bon is par value set by the issuer an is usually inicate irectly on the security. The notion of outstaning face value applies to bons structure with amortization. It is a part of the face value remaining after partial repayments of par over the life of the bon. Analytical inicators on such bons are calculate base on the outstaning face value. Lot of Multiplicity Lot of multiplicity (enomination increment, traing lot increment) is the minimum number of securities at face value, with which settlement an epository operations are performe. Minimum Denomination Minimum enomination (minimum traing lot, minimum traing volume) is a parameter of a certificate bearer Eurobon. The borrower etermines the total size of the issue at face value, the lowest enomination an enomination increment Depositary can register trae an settlement transactions only if the amount of securities excees the minimum enomination (for example, USD 100,000) an is a multiple of the enomination (e.g., USD 1,000). Minimum Traing Lot Minimum traing lot is the minimum amount of securities at face value, available for sale. Cash flow on the bon is calculate from the minimum traing lot. Coupon Coupon is a perioic interest payment mae uring the life of the bon. Coupon is calculate as a percentage (per annum) of face value an/or an amount payable to bonholers. Accrue Coupon Interest Accrue coupon interest (ACI) is a value measure in monetary units, an characterizing the part of coupon income, which has "accrue" from the beginning of the coupon perio. Coupon on the bons is pai perioically, usually once every quarter, six months or a year. Accoringly, when one coupon is pai an the next coupon perio begins, the coupon begins to "accrue". On the coupon ue ate, investors receive a coupon payment for the respective coupon perio, an ACI is zero. Calculating this inicator is important ue to the fact that in most markets, bons are trae at so-calle "net price" excluing the ACI (there are exceptions, however: for example, in the bon market of Ukraine bons are quote at full price). Thus, in orer to get the full price payable by the bon buyer to the seller (also known as "gross" price), one nees to a ACI to the net price. t0 ti 1 A Ci ti ti 1 ACI may also be expresse as coupon rate in percentage points (usually these are the formulas given in issue prospectus), rather than the coupon size in monetary units. Then the ACI formula will be as follows: Cbons.ru 7

8 A (%) C i(%) t 0 t B i1 For zero-coupon bons, ACI inex is not calculate. Calculating the Number of Days between Dates Days calculation metho etermines the formula use to calculate the notional number of ays between the starting an ening ates of the ACI perio, an the notional number of ays in a year (calculation basis). The choice of metho affects the iscount value when calculating analytical parameters of the bon. For Russian bons, the generally use metho is Actual/365F; for Ukrainian bons, we usually use methos 30/360 or Actual/365F; 30E/360 is the most commonly use metho for Eurobons. 30/360 Methos Starting ate: D1.M1.Y1 (ay.month.year) Ening ate D2.M2.Y2 (ay.month.year) Difference between the ates (Day count) = (Y2-Y1)*360+(M2-M1)*30+(D2-D1) 30/360 German (other names: 30E/360 ISDA) Source: 2006 ISDA Definitions (Section 4.16(h)) D1 an D2 ajustment rules: if D1=31, then D1=30 if D2=31, then D2=30 if D1 is the last ay of February, then D1=30 if D2 is the last ay of February, then D2=30 The last ay of February: February 29 in any leap year, February 28 in any non-leap year. 30/360 ISDA (30/360) (other names: Bon Basis, U.S. Municipal) Source: 2006 ISDA Definitions (Section 4.16(f)) D1an D2 ajustment rules: if D1=31, then D1=30 if D2=31 an D1=30, then D2=30 30/360 US (other names: 30U/360, 30US/360) 1 D1 an D2 ajustment rules: if D1=31, then D1=30 if D2=31 an D1=31, then D2=30 if D1 is the last ay of February, then D1=30 if D1 is the last ay of February an D2 is the last ay of February, then D2=30 Last ay of February: February 29 in any leap year, February 28 in any non-leap year. 30E+/360 1 D1 an D2 ajustment rules: if D1=31, then D1=30 if D2=31, then D2.M2.Y2 is the first ay of the following month ((D2=1; Y2=Y2+integral part((m2+1)/12); M2 = ((M2 +1) mo 12) remainer of iviing (M2+1) by 12) 30E/360 (other names: 30/360 Eurobon, 30/360 ISMA, 30/360 European, 30S/360 Special German, Eurobon Basis) Cbons.ru 8

9 Source: 2006 ISDA Definitions (Section 4.16(g)) D1 an D2 ajustment rules: if D1=31, then D1=30 if D2=31, then D2=30 Actual Methos Actual/360 (other names: Act/360, French) Source: 2006 ISDA Definitions (Section 4.16(e)) Number of ays in the perio is calculate as the ifference between the ates without any ajustments, base on 360- ay year. Calculation basis = 360. Actual/365A (other names: Actual/365 Actual) Source: The Actual-Actual Day Count Fraction (1999)(Section 2 (с)) Number of ays in the perio is calculate as the ifference between the ates without any ate ajustments. Calculation basis = 366, if the leap ay (February 29) falls on the perio, otherwise ACI calculation basis = 365. Actual/365F (other names: Actual/365 Fixe, English) Source: 2006 ISDA Definitions (Section 4.16()) Number of ays in the perio is calculate as the ifference between the ates without any ate ajustments. Calculation basis = 365. Actual/365L (other names: Actual/365 Leap year) 1 Number of ays in the perio is calculate as the ifference between the ates without any ate ajustments. Calculation basis = 366, if the en ate of the perio falls on a leap year, otherwise ACI calculation basis = 365. Actual/Actual (other names: Act/Act, Actual/Actual (ISDA)) Sources: 2006 ISDA Definitions (Section 4.16(b), The Actual-Actual Day Count Fraction (1999)(Section 2 (a)) Fractional number of ays = (Number of ays in the perio, which falls on a leap year) / (number of ays in the perio, which falls on a non-leap year) / 365. Actual/Actual (ISMA) (other names: Actual/Actual (ICMA)) Sources: 2006 ISDA Definitions (Section 4.16(c), ISMA Rule Book (Rule (iii)), The Actual-Actual Day Count Fraction (1999)(Section 2 (b)) In this metho, all coupon payments are always the same size. ACI is the same for every ay of the coupon perio. Size of the coupon payment is equal to the annual coupon rate ivie by payment frequency per year an multiplie by face value of the bon. Number of ays in the perio is calculate as the ifference between the ates without any ate ajustments. Fractional number of ays = Number of ays in the perio / ((number of ays in the current coupon perio)*(number of payments per year)). Actual/364 - instance Actual/Actual (ISMA), when the coupon perio is 91 or 182 ays. Use for some short-term securities. Calculation basis = 364. NL/365 (other names: Actual/365 No Leap year, NL 365) 1 Number of ays in the perio is calculate as the ifference between the ates without any ate ajustments. 1 is eucte from the number of ays in the perio, if the leap ay (February 29) falls on this perio. Calculation basis = 365. Cbons.ru 9

10 Calculation basis - Notional number of ays in the year. Fractional number of ays means the number of ays in the perio ivie by the number of ays in the year (ACI calculation basis). Depens on the ACI Calculation Metho applie. Designations arameter Definition Y effective yiel, % p.a. Y n nominal yiel, % p.a. Y s simple yiel, % p.a. CY current yiel, % p.a. ACY ajuste current yiel, % p.a. A accrue coupon interest, ACI, units of face value "net price" of the bons excluing ACI (Clean rice), units of face value (%) "net price" of the bons excluing ACI (Clean rice), % of face value +A, "gross price" of the bons incluing ACI (Dirty rice), units of face value C (%) coupon rate, % p.a. C i size of i-th coupon payment, units of face value N face value of the bon, units of currency N (%) face value of the bon, % N i the i-th payment of the ebt face value (incluing reemption of principal uner offer, amortization payments, full repayment), units of face value n coupon frequency m number of coupon payments t i reemption ate of the i-th coupon, face value etc. t 0 current ate or calculation ate t m maturity ate B number of ays in a year taken for calculation purposes, calculation basis D uration (Macaulay), ays (years) MD moifie uration Tm years to maturity VB price value of a basis point Conv convexity 1 we use prospectuses, expert opinions an site eltaquants.com to escribe the metho Cbons.ru 10

11 Calculate Values Bon Yiel Yiel to Maturity Yiel to maturity is the rate of return on investments in bons, provie that the investor hols them until maturity. It is usually shown as percentage per annum. Yiel to maturity can be calculate either incluing reinvestment of coupon payments over the course of the year (effective yiel), or excluing reinvestment of coupon payments over the course of the year (nominal yiel). It shoul be note that yiel to maturity is only an ESTIMATE of the return investors will get from the bon, as calculation of the yiel to maturity takes into account reinvestment of coupons at the same interest rate. In reality, this assumption can not be true, which is why the actual yiel will iffer from the estimate yiel to maturity. However, yiel to maturity is the most frequently use metho of assessing bons. Effective Yiel to Maturity Effective yiel to maturity (YTM) is yiel to maturity calculate base on reinvestment of coupon payments uring the year at the rate of the initial investment. Effective yiel to maturity is an internal rate of return on the bon cash flow. Effective yiel is the root of the equation: A m i1 C i N (1 Y) an represents a iscount rate, at which the "irty" price of a bon is equal to the current cost of payment flows on the bon. The calculator computes the effective yiel using Newton's metho (also known as the tangent metho). Effective yiel of zero-coupon bons is calculate with the equation (a special case of the equation to calculate the effective yiel when A = 0 an C i = 0): i ti t B 0 N (1 Y) t m t B 0 Methoologically, effective yiel is a more correct measure than the nominal yiel. However, nominal yiel is traitionally in wier use in many evelope bon markets. In Russia, effective yiel is use more commonly, while in Ukraine both nominal an effective yiels are in use. Cbons.ru 11

12 Nominal Yiel to Maturity Nominal yiel to maturity is the yiel to maturity, which oes not inclue reinvestment of coupon payments uring the year. If the paper is being place at par, at the time of placement nominal yiel will be equal to the coupon rate. For example, a bon with semiannual coupons of 10% woul have a nominal yiel to maturity of 10%, while the effective yiel woul be 10.25%. Nominal yiel is calculate using effective yiel an base on the following equation: Y 1 m n m Y (1 ) Nominal yiel to maturity is a commonly use inicator in financial markets of most evelope countries. This is largely ue to traition, has to o with relative simplicity of this inicator. In Russia, nominal yiel is official in calculation of yiels in the short-term government bon (GKO) market an is generally accepte in the promissory note market. For a zero-coupon bon, nominal yiel to maturity is calculate with the formula: Y n N 1 t i B t 0 Simple Yiel to Maturity Simple yiel to maturity is the yiel to maturity, which oes not take into account reinvestment of coupon payments uring the year. It is calculate from the ratio: Y n ( C i N ) i ( покупки) ( покупки) t i B t 0 Yiel to Offer Yiel to put (put option) is the rate, at which the iscounte value of cash flows receive before the expecte ate of compulsory re-purchase by the issuer, as well as the put price on that ate (as per the scheule), together are equal to the price of the bon. Yiel to call (call option) is the rate, at which the iscounte value of cash flows receive before the expecte ate of possible re-purchase by the issuer, as well as the call price on that ate (as per the scheule), are together equal to the price of the bon. In our calculator, yiel to put an yiel to call are esignate as "yiel to offer". In contrast to yiel to maturity, yiel to offer takes account only of those payments (incluing reemption of face value), which will be mae before the offer ate. Metho for calculation of effective, simple an nominal yiels to offer is similar to computation of the respective yiels to maturity. Cbons.ru 12

13 Only yiel to put (an not yiel to maturity) is calculate for bons with a put/call option, which has not been exercise, an with a cash flow that has not been set until maturity ate. In this case, calculations are mae to the ate of the last known coupon preceing the expecte offer ate. Yiel to the Next Coupon Yiel to the next coupon is the interest rate, at which the iscounte value of cash flows receive before the ate of the next coupon, are in aggregate equal to the price of the bon. In calculations, it is assume that the remaining outstaning face value is repai on the ate of the next coupon. Such yiel is calculate for bons, for which only the next coupon rate is known, an the subsequent cash flow is not efine. Current Yiel Current yiel (CY) is the bon yiel base on the current coupon perio only. It is assume that the net price of the bon will remain unchange uring this perio. The calculator uses the following formula to etermine the current yiel: C CY *100 For example, let us assume that the bon price is 90% of face value, with an annual coupon of 9% p.a. In this case the current yiel will be 9/90 = 10%. An the yiel to maturity of such bons will unoubtely be higher, since the price will ten to face value as the tenor ecreases. In contrast, for bons traing above par the current yiel will be higher than the yiel to maturity, as potential reuction in prices will not be taken into account. With this in view, current yiel is not the best inicator of the bon's investment appeal. Thanks to its simplicity, however, this value is often calculate as an aitional parameter. Ajuste Current Yiel Ajuste current yiel is the yiel on a bon that takes into account possible purchase of bons at a premium or a iscount. The calculator uses the following formula to etermine the ajuste current yiel: 100 ACY CY T m (%) Cbons.ru 13

14 Volatility, Duration, Convexity To unerstan the price volatility as reaction to interest rates, one nees a tool to measure volatility. Depenence of the price on the yiel in case of a yiel change only: a convex curve. Key factors affecting the calculation: coupon (cash flow) an tenor. rices may also change ue to a shift in perceptions of creit quality, approaching maturity ate, changes in market rates. Objective: being able to quickly assess the price change following the change in yiel. Years to Maturity This parameter represents the time (in years) remaining until maturity of the bon. Macaulay Duration Macaulay uration is an estimate of the average tenor of payment flows on the bon, taking into account iscounting the cost of certain payments. Thus, uration will always be less than or equal to the term to maturity of the bon, an it will be equal to the term to maturity of zero-coupon bons only. The formula for calculating of uration is as follows: D m i1 ( t i Ci N t ) ( ti t (1 Y) A 0 0 i ) B Duration is a measure of the bon price elasticity to the interest rate, an characterizes the risk of changes in bon prices following a change in interest rates. From this view point, uration can be conceive of as: - minor change in bon gross price, Y - minor change in bon yiel, - percentage change in bon gross price, (1 Y) 1Y - percentage change in bon yiel. D (1 Y) :, where 1 Y From the formula it follows that: Y D 1 Y This formula is use for approximate calculation of the relative price change base on given change in yiel an given uration.. Cbons.ru 14

15 Using only uration when calculating the relative price change oes not give a very accurate estimate of the percentage change in the bon price. The more the yiel to maturity changes, the less accurate the estimate will be. The error of result occurs because the uration is a linear estimate of the percentage change in bon price. Irrespective of the yiel change, the tangent line always lies below the price/yiel curve, thus uration always unervalues the actual bon price. Duration is usually measure in years, but in the Russian an Ukrainian markets, it is often specifie in ays. Let us assume that the bon tenor is 3 years, annual coupon of 10%, effective yiel is 10% p.a., an the bon is traing at par. Then the uration of this paper will be: 100*1/(1.1) 100*2/(1.1) D *3/(1.1) 2.74 Duration properties: 1. Macaulay uration of a zero-coupon bon remains the same until maturity. When yiel to maturity changes, its uration remains unchange. 2. The higher the rate, the lower the cost of later payments compare to the short-term ones, an the smaller the uration; an visa versa - the lower the rate, the longer the uration of the payment flow. 3. When yiel to maturity grows, uration ecreases; when yiel to maturity shrinks, uration grows. 4. The longer the time to maturity, the greater the uration. However, longer tenor of the bon oes not automatically mean an increase in uration. 5. The higher the coupon frequency, the shorter the uration, as more payments are scheule closer to the starting point. 6. Regarless of the coupon size, uration of a coupon bon, increasing the time until maturity tens to a limit equal to 1 1 Y Duration not only shows the average tenor of payment flows on the bon, but is a goo measure of price sensitivity to changing interest rates. The higher the uration, the greater the volatility of interest rates in Cbons.ru 15 3

16 relation to price changes. The phrase "bon uration is three years" means that the bon in question has the same price sensitivity to interest rates changes as a three-year zero-coupon bon. Moifie Duration Moifie uration is an inicator characterizing reaction of the bon's price to changes in the yiel to maturity, linear approximation. In terms of mathematics, it is the first erive function of price from yiel. It is important to note that moifie uration shows volatility of the full price incluing ACI. It is the value, by which the "irty" price changes when the yiel changes by 100 bp. Moifie uration is connecte to the uration value through the following formula: D MD 1 Y ' ( ) y In terms of erivatives: MD In case of small values, the following equality applies: MD* Y Moifie uration is the relative change in the bon price occurring when the yiel changes by one percent, provie that values of the expecte cash flows on the bon remain the constant when the yiel changes. Moifie uration provies a more accurate estimate of the percentage change in bon price, compare to Macaulay uration. Characteristics: 1. Moifie uration of a zero-coupon bon is less than the time before its maturity. In this case, Tm the moifie uration equals MD 1 Y 2. Moifie uration ecreases as the yiel to maturity grows, an increases when the yiel shrinks. Let us assume that the moifie uration is 4, the bon is traing at a price of 90% with a yiel of 8%, ACI is zero. How oes the price change, if the yiel rises to 8.5% (a change of 0.005). The price change can be calculate as follows: -4*0.005*90=-1.8. Thus, the bon price will fall by 1.8 to 88.2%. rice Value of Basis oint (VB) In contrast to the moifie uration, which is a relative value, VB inicates the absolute value of a bon price change following the change in the yiel by one basis point. MD ( A) % VB Let us assume that the moifie uration is 4, the bon is traing at a price of 90% with a yiel of 8%, ACI is zero VB Cbons.ru 16

17 If the price is 90% of par, estimate change in the bon's monetary value following the change in its yiel by one bp is currency units (per 100 nominal currency units). Convexity (Conv) Convexity is an inicator of the curve-shape relationship between the bon price an yiel, which shows how the moifie uration changes when the yiel shifts by 100 bp. It gives a much better approximation of price change ue to yiel change. In terms of erivatives: Conv m i1 ( ) Conv ( Ci Ni ) t( t 1) t2 (1 Y) A " y t t, where B t i 0 rice approximation: 1 MD* Y Conv( Y ) 2 2 Use of moifie uration an convexity allow a rather accurate estimation of the percentage change in the bon price ue to a significant change in the yiel to maturity. Convexity properties: 1. Value of convexity grows along with ecreasing yiel to maturity, an shrinks along with a raise in the yiel. 2. With uration grows, convexity grows faster than uration. This is a consequence of the quaratic epenence of convexity on uration. 3. At a given value of yiel to maturity an time of reemption, the value of convexity is greater for bons with lower coupon. 4. For the given level of yiel to maturity an coupon, convexity increases along with the tenor. t( t 1) Conv (1 Y) 2 Y 5. For a zero-coupon bon 2 6. Convexity of a perpetual bon is 2 Let us assume that the bon tenor is 3 years, annual coupon of 10%, effective yiel is 10% p.a., an the bon is traing at par. Then the value of convexity for such bon will be equal to: 100*1*2/(1.1) Conv 3 100*2*3/(1.1) *3*4/(1.1) How will the price change, if the yiel grows to 11% (change of 0.01), MD = 2.5. The price change can be calculate as follows: (-2.5* *22.4*(0.01) 2 )*100 = - 2.4%. Thus, the bon price will fall 2.4% to 97.6%. Cbons.ru 17

18 Spreas (G-sprea, T-sprea) The current version of the calculator allows computing G-spreas an T-spreas. G-sprea is calculate as the ifference between the issue yiel an the yiel for the point on G-curve with the same uration. G-sprea can only be calculate for Russian ruble-enominate bons. Results of G- spreas computation are publishe aily in the ruble bon traing results of the Traing Floor Quotes section. The archive of spreas is calculate starting from *Zero coupon yiel curve for government securities (G-curve) is a zero coupon yiel curve efine base on bon transactions in the market of short-term zero-coupon government bons (GKO) an feeral loan bons (OFZ). Zero coupon yiel curve for government securities is calculate by the Moscow Exchange. G-curve is calculate in real time following transactions with bons inclue in the calculation ata base. T-sprea is calculate as the ifference between the issue yiel an the yiel on government securities of the USA, Great Britain an Germany in the corresponing issue currency an with comparable moifie uration (the calculations are base on the effective yiels only). The value is compute only for issues in USD, EUR, GB. "Benchmark T-sprea" fiel isplays the issue, against which the T-sprea is compute on the ay of calculation. Issues with floating coupon rate an kins of issues like STRIS are exclue from the total amount of benchmarks. In the search for a benchmark for T-sprea calculation, ata from Cbons Valuation floor is use. Results of T-spreas computation are publishe aily in the USD bon traing results of the Traing Floor Quotes section. The archive of spreas for issues in USD is calculate starting from 2013, for issues in EUR, GB from September Cbons.ru 18

19 References 1. O.V. Lomatize, M.I. Lvova, A.V. Bolotin Basic Course of Securities Market. - Moscow: KNORUS, F. Fabozzi, S. Mann. The Hanbook of Fixe Income Securities, seventh eition, volume Moscow: "I.D. Williams", A.N. Burenin Duration an Convexity in Bon ortfolio Management. - Moscow: Scientific an Technical Society name after Vavilov, 2009 Contact etails Konstantin G. Vasilyev, artner, Hea of Department at Cbons, h.d. h./fax: +7 (812) , ext kv@cbons.info Elena Skurikhina, Hea of rojects h./fax: (812) , ext sea@cbons.info Cbons.ru 19