Bond Return Calculation Methodology

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1 Bond Return Calculation Methodology Morningstar Methodology Paper June 30, Morningstar, Inc. All rights reserved. The information in this document is the property of Morningstar, Inc. Reproduction or transcription by any means, in whole or in part, without the prior written consent of Morningstar, Inc., is prohibited.

2 Content Introduction 3 Coupon Payment and Accrued Interest 5 Coupon Dates 5 Coupon Payment Amount and Accrued Interest 6 Actual/Actual Day-Count Convention 7 Actual/365 Day-Count Convention 8 Actual/360 Day-Count Convention 9 30/360 Day-Count Convention 10 Bond Total Return 12 Daily Total Return 12 Daily Total Return Index 13 Morningstar Bond Return Calculation Methodology June 30,

3 Introduction A bond is a debt instrument. The borrowing entity promises to pay a specified sum of money at specific future dates. The promised payment consists of two components: interest and principal. For example, the borrower, or issuer, has an obligation to pay interest at a fixed rate twice a year and repay the principal upon the date of maturity. The principal, the amount that the borrower must repay to the lender, is also called the face value or par value. For a bond, the interest payment is called a coupon. The coupon rate is the interest rate expressed in percentage of principal. Since bond issuers do not distribute coupon payments on a daily basis, a bond buyer must pay the seller for the amount of interest earned by the latter since the previous coupon date; this is called accrued interest. Upon transaction, the buyer pays the seller the agreed-upon price, aka the "clean" price, for the bond plus accrued interest. The clean price plus accrued interest is called the "dirty" price. Bonds do not transact every day, so market price is not always observable. Therefore, for the purpose of calculating the daily returns of bonds, Morningstar uses the evaluated price, which is the theoretical fair price of a bond obtained from third-party providers. This methodology document addresses the return calculation of fixed-rate and zero-coupon bonds. Fixed-rate bonds pay periodic coupons that are constant over the bond's life. Zerocoupon bonds do not pay periodic interest and are issued at a discount to par value. The methodology does not currently address other types of bonds such as floating-rate notes, bonds that need principal factors such as mortgage-backed, commercial mortgage-backed, and asset-backed bonds, Treasury Inflation-Protected Securities, and so on. Morningstar Bond Return Calculation Methodology June 30,

4 Introduction (continued) In order to calculate a bond's return, the following information about the bond is required or could be used as a substitute for required information: Coupon payment frequency: expressed as the number of coupon payments in a year, 1=annually, 2=semiannually, 4=quarterly, 12=monthly, and so on. EOM: end-of-month. This indicates whether all coupon payment dates fall on the last day of the month. For example, if the first coupon for a semiannual bond is June 30, the next coupon is paid on Dec. 31 if the bond is EOM and Dec. 30 if it is not. Coupon rate: annual coupon rate of the bond, expressed as a percentage of par value. Day-count convention: the bond's day-count convention for coupon payments and accrued interest calculation. Please refer to the Day-Count Convention sections of this document for details. First coupon date: date of the first coupon payment for the bond. Maturity date: date the bond issuer must return the principal. Clean price: price of the bond without accrued interest. For the purpose of calculating daily return of bonds, Morningstar uses the evaluated price, which is the theoretical fair price of a bond obtained from third-party providers. Morningstar does not calculate a return stream for a bond under the following conditions: Coupon payment frequency, coupon rate, or day-count convention is not available, and the bond is not a zero-coupon bond. The first coupon date and the maturity date are both unavailable, as we would be unable to determine when coupon payments ought to occur. Morningstar stops further updates to a stream of returns under the following conditions: The bond is missing more than five consecutive days of clean price. If the clean price recommences at a future date, the return series for this bond restarts from that day onward. Default, issuer files for bankruptcy protection, delisting, or indefinite suspension. If the issuer emerges from default or bankruptcy at a future date, the return series for this bond restarts from that day forward. Morningstar Bond Return Calculation Methodology June 30,

5 Coupon Payment and Accrued Interest Coupon, the interest payment of a bond, is a significant source of a bond's return. This section describes the calculation of the coupon payment and its related measure, accrued interest. The section is not applicable to a zero-coupon bond as there is no period coupon payment. Coupon Dates The following information is needed to determine the dates on which coupons are paid: first coupon date, coupon payment frequency, and whether the coupon payment dates fall on the last day of the month. The first coupon date indicates when the first coupon payment occurs, and subsequent coupon payments are made on the same day of the month in regular intervals based on the coupon payment frequency. Because each month may have a different number of days, when the first coupon date falls on the month-end of a smaller month, it is important to know whether coupon payment dates always fall on the last day of the month for the bond in question. For example, if the first coupon for a semiannual bond is June 30, the next coupon is paid on Dec. 31 for an end-of-month, or EOM, bond, and it is on Dec. 30 for a bond that is not EOM. Similarly, if the first coupon for a quarterly EOM bond is Feb. 28, the subsequent coupon payments are on May 31, August 31, Nov. 30, and Feb. 29 for a leap year. Morningstar does not make adjustments when coupon dates fall on weekends or national holidays, or for any other reason that may make the timing of the receipt of coupon payments differ from the entitlement date of the coupon. When the first coupon date is not available, Morningstar uses the maturity date and works backward using the coupon payment frequency to generate a series of coupon payment dates. As stated in the Introduction, in the case that both the first coupon date and the maturity date are unavailable for a bond that is not a zero-coupon bond, Morningstar does not calculate a return stream. Morningstar Bond Return Calculation Methodology June 30,

6 Coupon Payment and Accrued Interest (continued) Coupon Payment Amount and Accrued Interest At each coupon payment date, one would intuitively expect the coupon payments to be the annual coupon rate divided by the number of periodic coupons in the year, but this is not always the case. Similarly, it is not always true that adding all coupon payments for the year results in the annual coupon rate. This is because there are different day-count conventions, each with a unique way of handling the fact that the number of days between coupon payments varies, and so does the number of days in a year depending on whether it is a leap year. Accrued interest is also an important part of a bond's return. Since bond issuers do not distribute coupon payments on a daily basis, a bond buyer must pay the seller for the amount of interest earned by the latter since the previous coupon date; this is accrued interest. On a daily basis, the market value of a bond, also known as the dirty price, is the clean price plus accrued interest. While the clean price is obtained directly from third-party providers as the evaluated or theoretical price of the bond, accrued interest and coupon payment must be calculated. The following formula is applicable to both coupon payment and accrued interest calculation since accrued interest is essentially prorated coupon payment: [1] Interest t = Coupon Rate Day-Count Factor t, where Interest t = Coupon payment or accrued interest on valuation day t, expressed as percentage of par value Coupon Rate = Annual coupon rate of a bond, expressed as a percentage of par value Day-Count Factor t = Day-count factor on valuation day t Note: Accrued interest is zero on a coupon date. Morningstar Bond Return Calculation Methodology June 30,

7 Coupon Payment and Accrued Interest (continued) The day-count factor is an essential step in the calculation of coupon payment and accrued interest for a bond. It is based on the bond's day-count convention and coupon payment frequency. The day-count convention determines the effective number of days between two dates in question, regardless of how many actual calendar days set them apart. In the case of coupon payment calculation, it is the effective number of days between the date of the coupon payment that is being calculated and the date of the previous coupon payment. For accrued interest calculation, it is the effective number of days between the previous coupon date and the valuation date. There is no central authority defining day-count conventions, and variations exist within the same general convention. The following sections address in detail Morningstar's methodology for the most commonly used day-count conventions: 1. Actual/actual 2. Actual/ Actual/ /360 Actual/Actual Day-Count Convention The actual/actual day-count convention counts the actual number of days in the interest valuation period as well as the actual number of days of the year in question, taking into account that there are 366 days in a leap year. Under this method, all coupon payments are always for the same amount regardless of the length in the coupon period, and each coupon payment is prorated equally among the days in the coupon period for the purpose of daily accrual calculation. For example, for a bond that pays a 4% coupon rate semiannually, one coupon period may have 181 days and the next coupon period 184 days, but the bondholder is paid the same amount on each coupon date, which is 2% in this example. The coupon payment for the first coupon period is equally prorated among its 181 days for daily accrual, and the coupon payment for the second coupon period is equally prorated among its 184 days. Morningstar Bond Return Calculation Methodology June 30,

8 Coupon Payment and Accrued Interest (continued) For the actual/actual day-count convention, the day-count factor on a given coupon or valuation date is [2] Day-Count Factor t = ( 1 / Freq ) ( Num Prev,t / Num Prev,Next ), where Freq = Coupon payment frequency, expressed as number of coupon payments per year Num Prev,t = Number of days between the previous coupon date and the valuation date Num Prev,Next = Number of days between the previous coupon date and the next coupon date Notes: The coupon payment frequency is expressed as the number of coupon payments in a year: 1=annually, 2=semiannually, 4=quarterly, 12=monthly, and so on. When calculating the day-count factor for a coupon payment on a given coupon date, the previous coupon date refers to the date of the preceding coupon, while both the valuation date and the next coupon date are the coupon date in question. For example, when calculating the day-count factor for the semiannual coupon payment on Aug. 15, 2010, the previous coupon date is May 15, 2010, and both the valuation date and the next coupon date are Aug. 15, Intrinsically, this reduces the formula to 1 over coupon frequency, which is why the coupon payments are always the same throughout the bond's life. Actual/365 Day-Count Convention The actual/365 day-count convention counts the actual number of days in the interest valuation period but assumes there are always 365 days in a year, regardless of leap year. Under this method, coupon payments vary from coupon period to coupon period due to the differing length of each period. Furthermore, the sum of coupon amounts exceeds the annual coupon rate in a leap year for having gained an extra day of interest. Morningstar Bond Return Calculation Methodology June 30,

9 Coupon Payment and Accrued Interest (continued) For the actual/365 day-count convention, the day-count factor on a given coupon or valuation date is [3] Day-Count Factor t = Num Prev,t / 365. Note: Similar to the actual/actual day-count convention, when calculating the day-count factor for a coupon payment on a given coupon date, the previous coupon date refers to the date of the preceding coupon, while the valuation date is the coupon date in question. For example, when calculating the day-count factor for the semiannual coupon payment on Aug. 15, 2010, the previous coupon date is May 15, 2010, and the valuation date is Aug. 15, Actual/360 Day-Count Convention The Actual/360 day-count convention counts the actual number of days in the interest valuation period but assumes there are always 360 days in a year, regardless of leap year. Under this method, coupon payments vary from coupon period to coupon period due to the differing length of each period. Furthermore, the sum of coupon amounts always exceeds the annual coupon rate for having gained five or six additional days of interest. For the actual/360 day-count convention, the day-count factor on a given coupon or valuation date is [4] Day-Count Factor t = Num Prev,t / 360. Note: Similar to the actual/365 day-count convention, when calculating the day-count factor for a coupon payment on a given coupon date, the previous coupon date refers to the date of the preceding coupon, while the valuation date is the coupon date in question. Morningstar Bond Return Calculation Methodology June 30,

10 Coupon Payment and Accrued Interest (continued) 30/360 Day-Count Convention The 30/360 day-count convention assumes there are 30 days in a month and 360 days in a year, regardless of leap year. The spirit of this method is for all coupon payments to be the same amount regardless of the length in the coupon period, while holding the daily accrual rate constant. Since the method assumes there are 30 days in a month and the daily accrual rate is constant, the daily accrual is zero for seven days each year. Those seven days are either the 31st or the first calendar day of a month that follows a 31-day month, depending on whether the bond is EOM. Similarly, in a regular year, three days' worth of accrual is assigned to Feb. 28 or March 1, depending on whether the bond is EOM that pays a coupon in February. In a leap year, this is reduced to two days' worth of accrual. For the 30/360 day-count convention, the day-count factor on a given coupon or valuation date is [5] Day-Count Factor t = [(360 ( Y t Y Prev ) + 30 ( M t M Prev ) + ( D t D Prev )]/360, where Y t = Year number that the valuation day t falls in, expressed as a number, such as 2010 for Aug. 15, 2010 Y Prev = Year number that the previous coupon date falls in, expressed as a number M t = Month number that the valuation day t falls in, expressed as a number, such as 8 for Aug. 15, 2010 M Prev = Month number that the previous coupon date falls in, expressed as a number D t = Day number of the valuation day t (exceptions below), expressed in number, such as 15 for Aug. 15, 2010 D Prev = Day number of the previous coupon date (exceptions below), expressed as a number Note: Similar to the actual/360 day-count convention, when calculating the day-count factor for a coupon payment on a given coupon date, the previous coupon date refers to the date of the preceding coupon, while the valuation date is the coupon date in question. Morningstar Bond Return Calculation Methodology June 30,

11 Morningstar Bond Return Calculation Methodology June 30,

12 Coupon Payment and Accrued Interest (continued) Exceptions: The day number of the previous coupon day, D Prev, is changed to 30 when the previous coupon occurs on the 31st of the month. The day number of the previous coupon day, D Prev, is changed to 30 when both of the following conditions occur: the previous coupon falls on the last day of February, and the bond is EOM. The day number of the valuation day, D t, is changed to 30 when both of the following conditions occur: the valuation day is the 31st of the month, and D Prev = 30 after applying the exceptions outlined in the previous two bullet points. The day number of the valuation day, D t, is changed to 30 when all of the following conditions occur: the valuation day falls on the last day of February, the bond is EOM, and the end of February is one of the periodic coupon dates for this bond. Morningstar Bond Return Calculation Methodology June 30,

13 Bond Total Return Daily Total Return The daily total return of a bond takes into consideration both capital appreciation and income. It is the combination of the change in market value of the bond and coupon payment received on the valuation date, assuming that the latter occurs at the end of the day. The market value of a bond, also known as the dirty price, is the clean price plus accrued interest. The clean price is obtained directly from third-party providers as the evaluated or theoretical price of the bond, and the accrued interest and coupon payment are calculated based on formula [1] above. The formula for the daily total return of a bond is as follows: [6] TR t-1,t = ( P t + AI t + Coupon t ) / ( P t-1 + AI t-1 ) - 1, where TR t-1,t = Total return on day t, which is earned from the end of the previous day to the end of the valuation day P t = Clean price on the valuation day t AI t = Accrued interest on the valuation day t P t-1 = Clean price on the day prior to the valuation day AI t-1 = Accrued interest on the day prior to the valuation day Coupon t = Coupon payment on the valuation day t Notes: Coupon and accrued interest for a zero-coupon bond are zero. Accrued interest is zero on a coupon date. Missing clean price: when the clean price of a bond is missing, the clean price from the most recent past is carried over for up to five days. For example, if a bond is missing prices from Oct. 27 to Oct. 31, the clean price from Oct. 26 is carried over to fill the missing clean price on these five days. Only the clean price is carried over; the dirty price could be different on each of these five days as coupon payments and accrued interest continue being calculated on a daily basis. If the clean price is also missing on Nov. 1, this exceeds the five-day carryover limit and the return stream is stopped as of Oct. 31. If the clean price recommences at a future day, the return series for this bond restarts from that day onward, and historical return prior to the Morningstar Bond Return Calculation Methodology June 30,

14 Bond Total Return (continued) recommencement is erased because the interruption is too large to be considered continuous. Daily Total Return Index To efficiently calculate the return between any two days, it is often helpful to think in terms of index level, in other words, growth in wealth such as growth of $100. Morningstar calls this data stream the daily total return index. The formula for it is as follows: [7] DRI t = DRI t-1 (1+TR t-1,t ), where DRI t = Daily total return index on the valuation day t DRI t-1 = Daily return index on the day prior to the valuation day This concept assumes that total return compounds over time. When compounding, coupon payments are assumed to be reinvested on the coupon payment date at the day's price (the dirty price is the same as the clean price as there is no accrued interest). An alternative view on the concept of compounding is to assume that the coupon is not reinvested but instead returned directly to the investor so that this cash flow exits the portfolio and does not produce a cash drag. In the absence of a cash drag, the portfolio is growing at the total return. Therefore, compounding makes sense no matter whether one believes that coupon payments ought to be reinvested or withdrawn. Morningstar Bond Return Calculation Methodology June 30,

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