Introduction to Fixed Income Securities PDF

Introduction to Fixed Income Securities What is a bond and its features? Bond is a debt instrument issued by a company, the holder of which gets periodical interest either annually/semi- annually/quarterly/monthly and gets back principal along with interest accrued on redemptio...

Introduction to Fixed Income Securities What is a bond and its features? Bond is a debt instrument issued by a company, the holder of which gets periodical interest either annually/semi- annually/quarterly/monthly and gets back principal along with interest accrued on redemption date at the end of maturity period. Bonds provide the borrower with external funds to finance investments, or, in the case of government bonds, to finance expenditure. Salient features of bonds:- 1. Face Value is the amount on which the issuer pays interest, and generally has to be repaid at the end of the term. Some structured bonds can have a redemption amount different from the face value and can be linked to performance of particular assets such as a stock or commodity index, foreign exchange rate or a fund. This can result in an investor receiving less or more than his original investment at maturity. The normal face value of the bonds is Rs 1 lakh, likely to be reduced to 10,000. 2. Coupon or Interest rate, is the rate that the issuer pays to the holder of the bond. Usually, this rate is fixed throughout the life of the bond, in case of a fixed rate bond. This rate can also vary with a standard benchmark, such as T-Bill rate, Repo rate, MIBOR, MCLR etc. Generally, coupon is paid either annually or semi-annually, also on certain bonds paid either quarterly or monthly. 3. Maturity or redemption date, is the date in the future on which the investor's principal will be repaid. As long as all due payments have been made, the issuer has no further obligations to the bond holders after the maturity date. There have been some issuance like AT 1 with no maturity date (perpetual) but has a call option embedded.  Trends in Indian Corporate Debt Market For Indian companies, the bond market was the preferred route for fundraising in FY24. Total amount raised through privately placed corporate bonds touched a record high peak of 10.02 lakh crores in FY 2023-24. In a span of 5 years, amount raised through corporate bonds increased from 6.55 lakh crores in FY 2018-19 to 10.02 lakh crores for the first-time last year witnessing a CAGR of approx.~ 9%. Also, a significant increase of approx.~17% from FY 22-23 to FY 23-24 is setting up a platform for outstanding size of corporate bond market to more than double by fiscal 2030. As per CRISIL press release on corporate bond market, outstanding size of bond market may clock an increase from ~Rs 43 lakh crore as of last fiscal to Rs 100-120 lakh crore by fiscal 2030. Growth of Indian corporate bond market in last 10 years is set out in chart below: Corporate Bond Market Issuances (last 10 yrs) 1,200,000 1,002,432 1,000,000 800,000 600,000 477,082 400,000 200,000 - FY 15 FY 16 FY 17 FY 18 FY 19 FY 20 FY 21 FY 22 FY 23 FY 24 Note: The data above has been taken from Prime database. As per SEBI data, Listed debt private placements for FY 23-24 is 7.37 lakh crores. Bond Market in India and US Indian Bond Market - ($ 2.3 trillion) US Bond Market - $ 50 trillion 1.3% 2% 0.02% 5% 21% 24% 47% Corporate Bonds Dated G-Secs SDLs T-Bills CPs CDs Muni Bonds 5 Source: SEBI,RBI,NSDL,SIFMA Corporate Bonds vs Equity Shares Particulars Equity Share Corporate Bonds Investor relation with the Part owner of the company Lender to the company company May enjoy voting rights and have special voting Do not enjoy voting rights rights also Investor’s earning on Returns in form of dividend and capital Interest paid on bonds and capital appreciation on investment appreciation bonds (owing to favorable interest rates) Risk on Investment Equity shares are considered to be the riskiest Less risky than equity shares but riskier than G-sec. form of investment. However, corporate bonds also have their own set of risks (explained in slide 9). How liquid is the If equity shares are listed, then it is easily If listed, then can be easily traded. However, corporate investment? tradeable. bonds are not as liquid as equity shares. If unlisted/illiquid shares, then difficult to liquidate If unlisted, investor will find it difficult to liquidate and might be costlier to sell. before maturity date. What happens if company Investor will get face value of the investment back Investor will get preference in repayment over goes into liquidation? only after all other stakeholders are paid off. equity/preference shareholders. However, only after statutory dues like taxes etc. are paid off. Cash Flow on bonds Investor 3 On purchase of bonds Over tenure of bond, Redemption amount= Investment amount= payment of Periodical Principal + Interest No of bonds * Price of interest= Face value of accrued bonds bonds * Coupon rate 2 On redemption of bonds 1 Issuer Mode of Investment in Bonds Private Placement Primary market Public issue Listed bonds Request for Quote (RFQ) Mode of investment in bonds Secondary market Over the counter (OTC) Unlisted bonds  OVERVIEW OF PUBLIC ISSUE PROCESS Kick Off Meeting Listing & Trading Due Diligence, Basis of allotment X + 4 Weeks Pre- Issue other regulatory X + 7 Weeks approvals and Post Issue Drafting Finalization of Technical Rejection Filing of DP/DSP with SE & SEBI Issue Closes Receipt of In principle Approval X + 5 Weeks Issue Opens from Stock Exchange Marketing Determination of ROC Filing of Management Pre-Marketing Price band P/SP/TP Road shows Key Intermediaries Key Activities Due Diligence of the Issuer Lead Managers End to end coordination of the process till Listing of Securities Assists in due diligence Legal Counsel Provides legal guidance for the Issue Co-ordination with the Issuer and Bankers regarding collections, reconciliation, refunds etc Registrar Post issue co-ordination, collation & reconciliation of data Assist in process for Listing and Trading Debenture Trustee Responsible for maintenance of adequate security Takes care of interest of investors even after listing of securities Assist the Company and LM on formulation and execution of the Media and PR Strategy Advertising/ PR Agency Organizing the Road Shows Ensure adequate coverage of IPO & positive news flow Collection of funds raised in the IPO Bankers to the Issue Issue provisional and final certificates to aid in the post issue process Issue of refund cheques Bulk printing of the Prospectus Printers Printing of application forms Ensure timely dispatch & distribution of stationery Credit Rating Agency Provide credit rating for the Issue Private Placement - Parties Involved For Rating the Securities Credit Rating Agency Act as Trustee to ensure security of Filing of Security related documents interest of Bond Holders Registrar of Debenture Company Trustee Process the application and credit in Providing Platform for bidding and trading of securities NSE/BSE Issuer R&T Agent DEMAT account Collecting NSDL/CDSL For ISIN Creation Banker For DEMAT Credit Collect the funds on behalf of Issuer Arrangers Act as advisor Assist in mobilizing funds Primary/Secondary Market & Public Issue/Private Placement Particulars Primary Market Secondary Market A public issue of bonds is when bonds are Meaning Bonds offered by the company Trading (Buy/Sell) of the offered to public in general and any other directly through private bonds subsequently after investor at large through issue of prospectus. placement/public issue is listing is referred to as referred to as primary market secondary market A private placement of bonds is generally Purchase of Directly from the company From the subsequent holder made to certain number of persons which is bonds who wishes to sell generally institutional investors and are not offered to retail investors. Purchase Generally, at face value. But an Generally, at a price higher or Price issuer may also issue at a lower than face value, As per Cos. Act, 2013 an offer for private premium or a discount depending on prevailing placement shall not be made to more than 50 market conditions persons and not more than 200 persons in Mode of Private placement or Public NSE/BSE Request for quote aggregate in a FY, excluding QIBs and buying issue of bonds platform (RFQ) or Over the ESOPs counter trade (OTC) Basic Features and Key Terminologies 13  BASIC FEATURES of a BOND multinational organization Creditworthiness sovereign (national) Issuer government investment-grade bonds non-sovereign (local) non-investment-grade government bonds quasi-government entity company 14 BASIC FEATURES & TERMINOLOGIES Pricing Face Value / Principal Issue Price Market Price Redemption Value Interest Zero Coupon / Coupon bearing Bonds Coupon Rate (Fixed/Floating) Coupon Frequency Repayment types Single Repayment Amortization Embedded Options Plain Vanilla bonds Call / Put Conversion 15 Bond’s cash flows The most common payment structure by far is that of a plain vanilla bond, as 1,080 depicted below. Coupons + Redemption 80 80 80 80 80 80 80 80 80 Annual coupons 1,000 Purchase price 16 Key Participants Issuers Govt. & Govt. Bodies/Authorities Banks/FIs Corporates Investors Institutional Investors Banks, FIs., MFs, Insurance Companies, PFs, Pension Funds, FPIs, etc. Corporates Individual Investors Intermediaries Merchant Banks / Primary Dealers Stock Exchanges Debenture Trustees Credit Rating Agencies Brokers / Market makers 17 FIS Issuers In India Government Government Securities (G-Secs) Treasury bills (T-bills) Bonds issued by State Govt.s & Uts (SDLs) Government Authorities Bonds issued by Govt. Controlled Institutions and PSUs Bonds issued by Local Bodies and Municipalities (Municipal Bonds) Banks / Financial Institutions Bonds/NCDs CPs/CDs Corporates Bonds/Debentures (NCDs) Preference Shares (NCRPs) 18 19 Role of Regulators Reserve Bank India (RBI) manages the borrowing of the Central and State Governments including the Union Territories. The RBI acts as the regulator for the Money market and the G-Sec market. The RBI also governs instruments issued by Commercial Banks and other Institutions regulated by it. The Securities and Exchange Board of India (SEBI) is the regulator for the corporate bond market including instruments issued by Commercial Banks and PSUs, provided such issuances by the above regulated entities are of more than ONE year of maturity. The role of SEBI is paramount when the funds are raised through public issuance. Role of Monetary Policy in Debt Markets Key Challenge - illiquidity Concentrated Market (Only Institutional Investors) Buy & Hold Investors Fragmented Liquidity (Large number of ISINs) Investment Prudential Norms Recent Initiatives by SEBI Amendment in Municipal Bonds Regulations Introduction of Framework for Green Bonds Electronic Book Platform (EBP) Consolidation and Reissuance of Bonds Market Making Integrated Trade Repository Timely disclosure of delays/defaults Streamlining continuous disclosures Thank you 24 Types of Fixed Income Securities Markets India market composition Types of Fixed Income Securities Markets Money Market Government Securities Market Corporate Debt Market  Money Market Money market instruments are short-term financing instruments aiming to increase the financial liquidity of businesses. What are Money Market Instruments? The main characteristic of money market instruments is that they can be easily converted to cash, thereby preserving an investor's cash requirements. The money market and its instruments are usually traded over the counter and, therefore, cannot be done by standalone individual investors themselves. It has to be done through certified brokers or a money market mutual fund. Types of Money Market Instruments Certificate of Deposit Commercial Paper Treasury Bills Repurchase Agreements Banker's Acceptance What is Call Money Market? The call money market is the most liquid of all short-term money market segments, with a maturity period of one day. The tenure of call money loan ranges from one day to fourteen days after the disbursement of the amount is made by the lending institution. In the call money market, any amount can be lent or borrowed at a market-determined interest rate that is acceptable to both the borrower and the lender. Key Rates : https://dbie.rbi.org.in/DBIE/dbie.rbi?site=statistics Government Securities Market A Government Security (G-Sec) is a tradeable instrument issued by the Central Government or the State Governments. It acknowledges the Government’s debt obligation. Such securities are short term (usually called treasury bills, with original maturities of less than one year) or long term (usually called Government bonds or dated securities with original maturity of one year or more). In India, the Central Government issues both, treasury bills and bonds or dated securities while the State Governments issue only bonds or dated securities, which are called the State Development Loans (SDLs). G-Secs carry practically no risk of default and, hence, are called risk-free gilt-edged instruments.  Treasury Bills (T-bills) Treasury bills or T-bills, which are money market instruments, are short term debt instruments issued by the Government of India and are presently issued in three tenors, namely, 91 day, 182 day and 364 day. Treasury bills are zero coupon securities and pay no interest. Instead, they are issued at a discount and redeemed at the face value at maturity. For example, a 91 day Treasury bill of ₹100/- (face value) may be issued at say ₹ 98.20, that is, at a discount of say, ₹1.80 and would be redeemed at the face value of ₹100/-. Cash Management Bills (CMBs) In 2010, Government of India, in consultation with RBI introduced a new short-term instrument, known as Cash Management Bills (CMBs), to meet the temporary mismatches in the cash flow of the Government of India. The CMBs have the generic character of T-bills but are issued for maturities less than 91 days. Dated G-Secs Dated G-Secs are securities which carry a fixed or floating coupon (interest rate) which is paid on the face value, on half-yearly basis. Generally, the tenor of dated securities ranges from 5 years to 40 years.  How are the G-Secs issued? G-Secs are issued through auctions conducted by RBI. Auctions are conducted on the electronic platform called the E-Kuber, the Core Banking Solution (CBS) platform of RBI. Commercial banks, scheduled UCBs, Primary Dealers, insurance companies and provident funds, who maintain funds account (current account) and securities accounts (Subsidiary General Ledger (SGL) account) with RBI, are members of this electronic platform. All members of E-Kuber can place their bids in the auction through this electronic platform. The results of the auction are published by RBI at stipulated time (For Treasury bills at 1:30 PM and for GoI dated securities at 2:00 PM or at half hourly intervals thereafter in case of delay). The RBI, in consultation with the Government of India, issues an indicative half-yearly auction calendar which contains information about the amount of borrowing, the range of the tenor of securities and the period during which auctions will be held. The Reserve Bank of India conducts auctions usually every Wednesday to issue T-bills of 91day, 182 day and 364 day tenors. NDS OM SECONDARY MARKET NDS OM is an anonymous screen-based order matching system for secondary market dealing in government securities. This is an order driven electronic system, where the participants can trade anonymously by placing their orders on the system or accepting the orders already placed by other participants. Anonymity ensures a level playing field for various categories of participants. NDS-OM is owned by RBI. Direct access to the NDS-OM system is currently available only to select financial institutions like Commercial Banks, Primary Dealers, well managed and financially sound UCBs and NBFCs, etc. Other participants can access this system through their Primary members i.e. with whom they maintain Gilt Accounts. The securities and funds are settled on a net basis i.e. Delivery versus Payment System-III (DvP- III). CCIL guarantees settlement of trades on the settlement date by becoming a central counter-party (CCP) to every trade through the process of novation, i.e., it becomes seller to the buyer and buyer to the seller. All outright secondary market transactions in G-Secs are settled on a T+1 basis. Corporate Debt Market Corporate bonds are debt instruments issued by companies to borrow capital for various projects or to manage their day-to-day expenses. When you buy a bond, you lend money to the company in exchange for a return on the loaned amount. There are various categories of corporate bonds in India and they’re classified based on a number of factors such as security, equity characteristics and interest payments among others. Difference between G-Secs and Corp Bond Market Liquidity: G-Sec market is more liquid. On a typical trading day, if there is Rs 40,000 crore of trade in G- Secs, trades in corporate bonds would be Rs 6,000 crore. Nature of participants: nature of participants are similar in that corporate bond market also is populated by institutional investors, but in the corporate bond market, there is a relatively higher preponderance of Mutual Funds and HNIs. OTC: corporate bond trades are OTC, executed verbally and formalized on the NSE / BSE settlement platform. Arguably, G-Sec trades are Exchange-driven, but it is not an Exchange like NSE or BSE. NDS OM is a platform hosted by CCIL and there is no Exchange-standardized contract traded at NDS OM. Non-wholesale participation in the market RBI Retail Direct: This platform is meant for retail. Companies / institutions are not allowed. Instruments available are G-Secs, SDLs and SGBs. Investment vehicles: Mutual Funds for retail to wholesale quantum. AIF / PMS with defined minimum quantum. Bond houses: There are brokers / NBFCs, popularly referred to as bond houses, who offer bonds to non-institutional investors. Online bond platforms: there are online platforms offered by broking houses, where one can purchase bonds from the available list. Exchanges: corporate bonds are mostly listed, at NSE/BSE. One can purchase through a broker. However, traded volumes are limited. Thank You Fixed Income Securities - Investment Risks Risks in FIS 1. Credit Risks 2. Interest Rate Risks 3. Illiquidity Risks 4. Exchange Rate Risks 5. Inflation Risks 6. Reinvestment Risks 7. Optionality Risks (Call and Prepayment Risks) 8. Volatility Risks 9. Event Risks (1) Interest rate risks Price & yield - Inverse relationship (2) Call risk / prepayment risk (3) Reinvestment Risk (4) Credit Risk Credit ratings Rating scales (5) Liquidity risks (6) Exchange risk ▪ Currency Risks in cases where bonds are being settled in currency other than my primary currency ▪ Both Issuer and Investor can be exposed to Exchange Risk ▪ E.g. An FPI investing in Rupee denominated bond issued by an Indian issuer; or an Indian company raising money off shore in Dollar denominated bonds ▪ Forex derivatives can be used to hedge above risks ▪ Masala Bonds vs. ECBs (7) Inflation Risk (8) Volatility Risk ▪Any change in expected volatility of interest rates would change the value of a bond with an imbedded option ▪A bond with an embedded option is typically priced as under: ▪Value of a callable bond = Value of similar plain vanilla bond - Value of the call option ▪Value of a putable bond = Value of similar plain vanilla bond + Value of the put option ▪Any change in expected volatility of interest rates would change the value of the embedded option in the bond, and thus affect the bond value as under: Movement in Expected Int. Rate Effect on Price of Bonds Volatility Callable Bond Putable Bond + - (9) Event Risk Thank you Pricing of Bonds  Concept of “Par Value” “Par" is the face value of a debt instrument which is promised to be paid as principal at the maturity of the debt instrument. Typically, it is ₹100 for a Government bond but ₹10,000 for a corporate bond. This is the amount that an issuer is bound to pay back to the bond investor as per the indenture of the debt issuance. The periodic interest/coupon paid on a debt instrument is on the basis of the face value. The face Value is also known as the redemption value for a plain vanilla bond. The market trades bonds as a percentage of face value. If a trader quotes a bid price of ₹106.35, the trader is willing to buy the security at 106.35% of the face value of the security. Many debt instruments are issued at a discount to the par value. Treasury Bills, Commercial Papers, etc. are always issued at a discount to the par value and the investor receives par value at maturity.  Review of Time Value ❖ Future Value The future value (Pn) of any sum of money invested today is: Pn = P0(1+r)n n = number of periods Pn = future value n periods from now (in dollars) P0 = original principal (in dollars) r = interest rate per period (in decimal form) (1+r)n represents the future value of $1 invested today for n periods at a compounding rate of r  Review of Time Value (continued) ❖Future Value When interest is paid more than one time per year, both the interest rate and the number of periods used to compute the future value must be adjusted as follows: r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years The higher future value when interest is paid semiannually, as opposed to annually, reflects the greater opportunity for reinvesting the interest paid.  Review of Time Value (continued) ❖Future Value of an Ordinary Annuity When the same amount of money is invested periodically, it is referred to as an annuity. When the first investment occurs one period from now, it is referred to as an ordinary annuity. The equation for the future value of an ordinary annuity (Pn) is: (1 + r ) n − 1 Pn = A    r  A = the amount of the annuity (in dollars). r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years  Review of Time Value (continued) ❖ Example of Future Value of an Ordinary Annuity Using Annual Interest: If A = $2,000,000, r = 0.08, and n = 15, then Pn = ? (1 + r ) n − 1 Pn = A    r  (1 + 0.08)15 − 1 Pn = $2, 000, 000    0.08  Pn = $2,000,000 [27.152125] = $54,304,250  Review of Time Value (continued) ❖ Example of Future Value of an Ordinary Annuity Using Semiannual Interest: If A = $2,000,000/2 = $1,000,000, r = 0.08/2 = 0.04, and n = 15(2) = 30, then Pn = ? (1 + r ) n − 1 Pn = A    r  (1 + 0.04 )30 − 1 Pn = $1,000,000    0.04  Pn = $1,000,000 [56.085] = $56,085,000  Review of Time Value (continued) ❖Present Value The present value is the future value process in reverse. We have:  1  Pn =  n (1 + r )  r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years For a given future value at a specified time in the future, the higher the interest rate (or discount rate), the lower the present value. For a given interest rate, the further into the future that the future value will be received, then the lower its present value.  Review of Time Value (continued) ❖Present Value of a Series of Future Values To determine the present value of a series of future values, the present value of each future value must first be computed. Then these present values are added together to obtain the present value of the entire series of future values.  Review of Time Value (continued) ❖Present Value of an Ordinary Annuity When the same amount of money is received (or paid) each period, it is referred to as an annuity. When the first payment is received one period from now, the annuity is called an ordinary annuity. When the first payment is immediate, the annuity is called an annuity due. The present value of an ordinary annuity (PV) is: 1 − 1/ (1 + r ) n  PV = A    r  A = the amount of the annuity (in dollars) r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years  Review of Time Value (continued) ❖ Example of Present Value of an Ordinary Annuity (PV) Using Annual Interest: If A = $100, r = 0.09, and n = 8, then PV = ? 1 − 1/ (1 + r ) n  PV = A    r  1 − 1/ (1 + 0.09 )8  PV = $100    0.09  PV = $100 [5.534811] = $553.48  Review of Time Value (continued) ❖Present Value When Payments Occur More Than Once Per Year If the future value to be received occurs more than once per year, then the present value formula is modified so that 1) the annual interest rate is divided by the frequency per year 2) the number of periods when the future value will be received is adjusted by multiplying the number of years by the frequency per year Thank You Pricing of FIS  Pricing a Bond ❖ Determining the price of any financial instrument requires an estimate of 1) the expected cash flows 2) the appropriate required yield 3) the required yield reflects the yield for financial instruments with comparable risk, or alternative investments ❖ The cash flows for a bond that the issuer cannot retire prior to its stated maturity date consist of 1) periodic coupon interest payments to the maturity date 2) the par (or maturity) value at maturity  Pricing a Bond (continued) In general, the price of a bond (P) can be computed using the following formula: n Ct Mt P = + t = 1 (1 + r ) (1 + r )n t P = price (in dollars) n = number of periods (number of years times 2) t = time period when the payment is to be received C = semiannual coupon payment (in dollars) r = periodic interest rate (required annual yield divided by 2) M = maturity value  Pricing a Bond (continued) Example of Computing the Value of a Bond: ❖ Consider a 20-year 10% coupon bond with a par value of $1,000 and a required yield of 11%. ❖ Given C = 0.1($1,000) / 2 = $50, n = 2(20) = 40 and r = 0.11 / 2 = 0.055, the present value of the coupon payments (P) is: 1 − 1/ (1 + r ) n  P =C   r  1 − 1/ (1 + 0.055 ) 40  P = $50    0.055  P = $50 [16.046131] = $802.31  Pricing a Bond (continued) Example of Computing the Value of a Bond: ❖ The present value of the par or maturity value of $1,000 is:  M   $1,000  (1 + r ) n  =  1 + 0.055 40  = $117.46   ( )  Continuing the computation from the previous slide: The price of the bond (P) = present value coupon payments + present value maturity value = $802.31 + $117.46 = $919.77.  Pricing a Bond (continued) For zero-coupon bonds, the investor realizes interest as the difference between the maturity value and the purchase price. The equation is: Mt P= (1 + r ) n P = price (in dollars) M = maturity value r = periodic interest rate (required annual yield divided by 2) n = number of periods (number of years times 2)  Pricing a Bond (continued) ❖ Zero-Coupon Bond Example Consider the price of a zero-coupon bond (P) that matures 15 years from now, if the maturity value is $1,000 and the required yield is 9.4%. Given M = $1,000, r = 0.094 / 2 = 0.047, and n = 2(15) = 30, what is P ? Mt $ 1, 0 0 0 P = = = $ 252. 12 (1 + r ) n (1 + 0. 0 4 7 ) 30 Thank You Valuation of Fixed Income Securities  Pricing Yield Relationship ❖ Price-Yield Relationship A fundamental property of a bond is that its price changes in the opposite direction from the change in the required yield. The reason is that the price of the bond is the present value of the cash flows. If we graph the price-yield relationship for any option-free bond, we will find that it has the “bowed” shape as shown in Exhibit 2-2 Price-Yield Relationship for a 20-Year 10% Coupon Bond Yield Price ($) Yield Price ($) Yield Price ($) 0.050 1,627.57 0.085 1,143.08 0.120 849.54 0.055 1,541.76 0.090 1,092.01 0.125 817.70 0.060 1,462.30 0.095 1,044.41 0.130 787.82 0.065 1,388.65 0.100 1,000.00 0.135 759.75 0.070 1,320.33 0.105 $958.53 0.140 733.37 0.075 1,256.89 0.110 $919.77 0.145 708.53 0.080 1,197.93 0.115 883.50 0.150 685.14 Shape of Price-Yield Relationship for an Option-Free Bond Maximum Price Price Yield ❖Relationship Between Coupon Rate, Required Yield, and Price When yields in the marketplace rise above the coupon rate at a given point in time, the price of the bond falls so that an investor buying the bond can realize capital appreciation. The appreciation represents a form of interest to a new investor to compensate for a coupon rate that is lower than the required yield. When a bond sells below its par value, it is said to be selling at a discount. A bond whose price is above its par value is said to be selling at a premium. ❖ Relationship Between Bond Price and Time if Interest Rates Are Unchanged For a bond selling at par value, the coupon rate equals the required yield. As the bond moves closer to maturity, the bond continues to sell at par. Its price will remain constant as the bond moves toward the maturity date. The price of a bond will not remain constant for a bond selling at a premium or a discount. Exhibit 2-3 shows the time path of a 20-year 10% coupon bond selling at a discount and the same bond selling at a premium as it approaches maturity. (See truncated version of Exhibit 2-3) ✓ The discount bond increases in price as it approaches maturity, assuming that the required yield does not change. ✓ For a premium bond, the opposite occurs. ✓ For both bonds, the price will equal par value at the maturity date. Time Path for the Price of a 20-Year 10% Bond Selling at a Discount and Premium as It Approaches Maturity Price of Discount Bond Price of Premium Bond Year Selling to Yield 12% Selling to Yield 7.8% 20.0 $ 849.54 $1,221.00 16.0 859.16 1,199.14 12.0 874.50 1,169.45 10.0 885.30 1,150.83 8.0 898.94 1,129.13 4.0 937.90 1,074.37 0.0 1,000.00 1,000.00 ❖ Reasons for the Change in the Price of a Bond The price of a bond can change for three reasons: 1) there is a change in the required yield owing to changes in the credit quality of the issuer 2) there is a change in the price of the bond selling at a premium or a discount, without any change in the required yield, simply because the bond is moving toward maturity 3) there is a change in the required yield owing to a change in the yield on comparable bonds (i.e., a change in the yield required by the market)  Complications ❖ The framework for pricing a bond assumes the following: 1) the next coupon payment is exactly six months away 2) the cash flows are known 3) the appropriate required yield can be determined 4) one rate is used to discount all cash flows  Complications (continued) ❖ The next coupon payment is less than six months away When an investor purchases a bond whose next coupon payment is due in less than six months, the accepted method for computing the price of the bond is as follows: n C M P = t −1 + t = 1 (1 + r ) (1 + r ) v (1 + r )v (1 + r )t −1 where v = (days between settlement and next coupon) divided by (days in six-month period)  Complications (continued) ❖ Cash Flows May Not Be Known For most bonds, the cash flows are not known with certainty. This is because an issuer may call a bond before the maturity date. ❖ Determining the Appropriate Required Yield All required yields are benchmarked off yields offered by Treasury securities. From there, we must still decompose the required yield for a bond into its component parts. ❖ One Discount Rate Applicable to All Cash Flows A bond can be viewed as a package of zero-coupon bonds, in which case a unique discount rate should be used to determine the present value of each cash flow.  Pricing Floating-Rate and Inverse-Floating-Rate Securities ❖ The Cash Flow is not known for either a floating-rate or an inverse-floating-rate security; it depends on the reference rate in the future. ❖ Price of a Floater The coupon rate of a floating-rate security (or floater) is equal to a reference rate plus some spread or margin. The price of a floater depends on 1) the spread over the reference rate 2) any restrictions that may be imposed on the resetting of the coupon rate  Pricing Floating-Rate and Inverse-Floating-Rate Securities (continued) ❖ Price of an Inverse-Floater In general, an inverse floater is created from a fixed-rate security. ✓ The security from which the inverse floater is created is called the collateral. ✓ From the collateral two bonds are created: a floater and an inverse floater. (This is depicted in Exhibit 2-4) The price of a floater depends on (i) the spread over the reference rate and (ii) any restrictions that may be imposed on the resetting of the coupon rate. For example, a floater may have a maximum coupon rate called a cap or a minimum coupon rate called a floor. The price of an inverse floater equals the collateral’s price minus the floater’s price.  Exhibit 2-4 Creation of an Inverse Floater Collateral (Fixed-rate bond) Floating-rate Bond Inverse-floating-rate bond (“Floater”) (“Inverse floater”)  Price Quotes Price Quotes ❖ A bond selling at par is quoted as 100, meaning 100% of its par value. ❖ A bond selling at a discount will be selling for less than 100. ❖ A bond selling at a premium will be selling for more than 100.  Accrued Interest ❖ Coupon interest is paid every six months. For mortgage-backed and asset-backed securities, interest is usually paid monthly. If an investor sells a bond between coupon payments, then the amount of interest over this period is called accrued interest. In many countries, the bond buyer must pay the bond seller the accrued interest.  Accrued Interest (continued) ❖The amount that the buyer pays the seller is the agreed-upon price plus accrued interest. This amount is called the dirty price. The price of a bond without accrued interest is called the clean price. The agreed-upon bond price without accrued interest is called the clean price.  Accrued Interest (continued) ❖ A bond in which the buyer must pay the seller accrued interest is said to be trading cum-coupon. ✓ Bonds are always traded cum-coupon. ❖ If the buyer forgoes the next coupon payment, the bond is said to be trading ex-coupon. ✓ One exception is when the issuer has not fulfilled its promise to make the periodic payments. ✓ In this case, the issuer is said to be in default. ✓ In such instances, the bond’s price is sold without accrued interest and is said to be traded flat.  Accrued Interest (continued) ❖ When calculating accrued interest, the following are needed: ✓ the number of days in the accrued interest period (represents the number of days over which the investor has earned interest); ✓ the number of days in the coupon period; and, ✓ the dollar amount of the coupon payment. ❖ Given these values, the accrued interest (AI) assuming semiannual payments is calculated as follows:  Annual coupon   Days in AI period  AI =     2   Days ❖ To illustrate, suppose that there are 50 days in coupon per iod  in the accrued interest period, 183 days in a coupon period, and the annual coupon per $100 of par value is $8. The accrued interest is:  $8  50  AI =    = $4 ( 0.273224 ) = $1.0929 per $100   2 183   Accrued Interest (continued) ❖The trade date (also referred to as the transaction date) is the date on which the transaction is executed (referred to as “T”). ❖The settlement date is the date a transaction is deemed to be completed and the seller must transfer the ownership of the bond to the buyer in exchange for the payment. ✓ The settlement date varies by the type of bond.  Accrued Interest (continued) ❖ In calculating the number of days between two dates, the actual number of days is not always the same as the number of days that should be used in the accrued interest formula. ❖ The number of days used depends on the day count convention for the particular security. ❖ For coupon-bearing Treasury securities, the day count convention is used to determine the actual number of days between two dates. ✓ This is referred to as the “actual/actual” day count convention.  Accrued Interest (continued) ❖ Consider a coupon-bearing Treasury security whose previous coupon payment was March 1 with the next coupon payment being on September 1. ❖ Suppose this Treasury security is purchased with a settlement date of July 17th. ✓ The actual number of days between July 17 (the settlement date) and September 1 (the date of the next coupon payment) is 46 days, which is computed as July 17 to July 31 (14 days) plus August 1 to August 31 (31 days) + September 1 (1 day) = 14 days + 31 days + 1 day = 46 days. ✓ The number of days in the coupon period is the actual number of days between March 1 and September 1, which is 184 days. ✓ The number of days between July 17 to the last coupon payment (March 1) is: (184 days − 46 days) = 138 days.  Accrued Interest (continued) ❖ For coupon-bearing agency, municipal, and corporate bonds, a different day count convention is used. ❖ It is assumed that every month has 30 days, that any 6-month period has 180 days, and that there are 360 days in a year. ❖ This day count convention is referred to as “30/360.” ❖ In other words, the same procedure is used above but 30 days is used for each month. ❖ To illustrate (using the “30/360” convention), the number of days between July 17 (the settlement date) and September 1 (the date of the next coupon payment) is 44 days, which is computed as July 17 to month ending of 30 (13 days) plus August 1 to month ending of 30 (30 days) + September 1 (1 day) = 13 days + 30 days + 1 day = 44 days. ✓ The number of days in the coupon period is the number of days used between March 1 and September 1, which is 180 (instead of 184 days given that March, May, July and August have 31 days that are treated as 30 days). ✓ Thus, the number of days between the last coupon payment (March 1) to July 17 is: (180 days − 44 days) = 136 days. Thank You Analysis of Bonds with Embedded Options  Callable Bonds and Their Investment Characteristics ❖ The presence of a call option results in two disadvantages to the bondholder: 1) callable bonds expose bondholders to reinvestment risk 2) price appreciation potential for a callable bond in a declining interest-rate environment is limited o This phenomenon for a callable bond is referred to as price compression. ❖ If the investor receives sufficient potential compensation in the form of a higher potential yield, an investor would be willing to accept call risk.  Callable Bonds and Their Investment Characteristics (continued) ❖ Traditional Valuation Methodology for Callable Bonds o When a bond is callable, the practice has been to calculate a yield to worst, which is the smallest of the yield to maturity and the yield to call for all possible call dates. o The yield to call (like the yield to maturity) assumes that all cash flows can be reinvested at the computed yield—in this case the yield to call—until the assumed call date. o Moreover, the yield to call assumes that 1) the investor will hold the bond to the assumed call date 2) the issuer will call the bond on that date. o Often, these underlying assumptions about the yield to call are unrealistic because they do not take into account how an investor will reinvest the proceeds if the issue is called.  Callable Bonds and Their Investment Characteristics (continued) ❖ Price-Yield Relationship for a Callable Bond o The price–yield relationship for an option-free bond is convex. o Exhibit 19-4 shows the price–yield relationship for both a noncallable bond and the same bond if it is callable. o The convex curve a–a' is the price–yield relationship for the noncallable (option-free) bond. o The unusual shaped curve denoted by a–b is the price–yield relationship for the callable bond. o The reason for the shape of the price–yield relationship for the callable bond is as follows. When the prevailing market yield for comparable bonds is higher than the coupon interest, it is unlikely that the issuer will call the bond. o If a callable bond is unlikely to be called, it will have the same convex price–yield relationship as a noncallable bond when yields are greater than y*.  Exhibit 19-4 Price-Yield Relationship for a Noncallable and Callable Bond Price a’ Noncallable Bond a’- a b Callable Bond a-b a y* Yield Callable Bonds and Their Investment Characteristics (continued) ❖ Price-Yield Relationship for a Callable Bond o As yields in the market decline, the likelihood that yields will decline further so that the issuer will benefit from calling the bond increases. o The exact yield level at which investors begin to view the issue likely to be called may not be known, but we do know that there is some level, say y*. o At yield levels below y*, the price-yield relationship for the callable bond departs from the price-yield relationship for the noncallable bond. o For a range of yields below y*, there is price compression– that is, there is limited price appreciation as yields decline. o The portion of the callable bond price-yield relationship below y* is said to be negatively convex. Callable Bonds and Their Investment Characteristics (continued) ❖ Price-Yield Relationship for a Callable Bond o Negative convexity means that the price appreciation will be less than the price depreciation for a large change in yield of a given number of basis points. For a bond that is option-free and displays positive convexity, the price appreciation will be greater than the price depreciation for a large change in yield. The price changes resulting from bonds exhibiting positive convexity and negative convexity are shown in Exhibit 19-5 (see Overhead 19-18). o It is important to understand that a bond can still trade above its call price even if it is highly likely to be called. Exhibit 19-5 Price Volatility Implications of Positive and Negative Convexity Absolute Value of Percentage Price Change Change in Interest Rates Positive Convexity Negative Convexity -100 basis points X% Less than Y% +100 basis points Less than X% Y%  Components of a Bond with an Embedded Option ❖ To develop a framework for analyzing a bond with an embedded option, it is necessary to decompose a bond into its component parts. ❖ A callable bond is a bond in which the bondholder has sold the issuer a call option that allows the issuer to repurchase the contractual cash flows of the bond from the time the bond is first callable until the maturity date. ❖ The owner of a callable bond is entering into two separate transactions: 1) buys a noncallable bond from the issuer for which she pays some price 2) sells the issuer a call option for which she receives the option price ❖ A callable bond is equal to the price of the two components parts; that is, callable bond price = noncallable bond price – call option price ❖ The call option price is subtracted from the price of the noncallable bond because when the bondholder sells a call option, she receives the option price. ❖ Graphically, this can be seen in Exhibit 19-6. ❖ The difference between the price of the noncallable bond and the callable bond at any given yield is the price of the embedded call option.  Exhibit 19-6 Decomposition of a Price of a Callable Bond Note: At y** yield level: PNCB = noncallable bond price PCB = callable bond price Price PNCB - PCB = call option price a’ PNCB Noncallable Bond PCB a’- a b Callable Bond a a-b y** y* Yield  Components of a Bond with an Embedded Option (continued) ❖ The logic applied to callable bonds can be similarly applied to putable bonds. ❖ In the case of a putable bond, the bondholder has the right to sell the bond to the issuer at a designated price and time. ❖ A putable bond can be broken into two separate transactions. 1) The investor buys a noncallable bond. 2) The investor buys an option from the issuer that allows the investor to sell the bond to the issuer. ❖ The price of a putable bond is then putable bond price = non-putable bond price + put option price  Valuation Model ❖ The bond valuation process requires that we use the theoretical spot rate to discount cash flows. ❖ This is equivalent to discounting at a series of forward rates. ❖ For an embedded option the valuation process also requires that we take into consideration how interest-rate volatility affects the value of a bond through its effects on the embedded options. ❖ Depending on the structure of the security to be analyzed, three models can be used to account for the valuation effect of embedded options. 1) The first model is for a bond that is not a mortgage-backed security or asset-backed security and which can be exercised at more than one time over its life. 2) The second case is a bond with an embedded option where the option can be exercised only once. 3) The third model is for a mortgage-backed security or certain types of asset-backed securities.  Valuation Model (continued) ❖ Valuation of Option-Free Bonds o The price of an option-free bond is the present value of the cash flows discounted at the spot rates. To illustrate this, we can use the following hypothetical yield curve: Maturity Yield to Market Years M a t u r i t y (%) Value 1 3.50 100 2 4.00 100 3 4.50 100 o We can simplify the illustration by assuming annual-pay bonds. Using the bootstrapping methodology, the spot rates and the one- year forward rates can be obtained. Spot O ne – Y e a r Years R a t e (%) F or w a r d R a t e 1 3.500 3.500 2 4.010 4.523 3 4.541 5.580  Valuation Model (continued) ❖ Valuation of Option-Free Bonds o EXAMPLE. Consider an option-free bond with three years remaining to maturity and a coupon rate of 5.25%. o The price of this bond can be calculated in one of two ways, both producing the same result. 1) The coupon payments can be discounted at the zero-coupon rates: $5.25 $5.25 $100 + $5.25 + + = $102.075 1.035 (1.0401) 2 (1.04541)3 2) The second way is to discount by the one-year forward rates: $5.25 $5.25 $100 + $5.25 + + = $102.075 1.035 (1.035)(1.04523) (1.035)(1.04523)(1.05580 )  Valuation Model (continued) ❖ Introducing Interest-Rate Volatility o When we allow for embedded options, consideration must be given to interest-rate volatility. o This can be done by introducing an interest-rate tree, also referred to as an interest-rate lattice. o This tree is nothing more than a graphical depiction of the one-period forward rates over time based on some assumed interest-rate model and interest-rate volatility.  Valuation Model (continued) ❖ Interest-Rate Model o As explained in the previous chapter, an interest-rate model is a probabilistic description of how interest rates can change over the life of a financial instrument being evaluated. o An interest-rate model does this by making an assumption about the relationship between the level of short-term interest rates and interest-rate volatility (e.g., standard deviation of interest rates). o The interest-rate models commonly used are arbitrage-free models based on how short-term interest rates can evolve (i.e., change) over time. o The interest-rate models based solely on movements in the short-term interest rate are referred to as one-factor models. o More complex models would consider how more than one interest rate changes over time.  Valuation Model (continued) ❖ Interest-Rate Lattice o Exhibit 19-7 shows an example of the most basic type of interest-rate lattice or tree, a binomial interest-rate tree. The corresponding model is referred to as the binomial model. o In this model, it is assumed that interest rates can realize one of two possible rates in the next period. In the valuation model we present in this chapter, we will use the binomial model. o Valuation models that assume that interest rates can take on three possible rates in the next period are called trinomial models. o More complex models exist that assume that more than three possible rates in the next period can be realized.  Exhibit 19-7 Three-Year Binomial Interest-Rate Tree r3HHH NHHH r2HH NHH r1H r3HHL NH NHHL r0 r2HL N NHL r1L r3HLL NL NHLL r2LL NLL r3LLL NLLL Today 1 Year 2 Years 3 Years  Valuation Model (continued) ❖Interest-Rate Lattice o Returning to the binomial interest-rate tree in Exhibit 19-7, each node (bold blue circle) represents a time period that is equal to one year from the node to its left. o Each node is labeled with an N, representing node, and a subscript that indicates the path that one-year forward rates took to get to that node. o H represents the higher of the two forward rates and L the lower of the two forward rates from the preceding year. o For example, node NHH means that to get to that node the following path for one-year rates occurred: The one-year rate realized is the higher of the two rates in the first year and then the higher of the one-year rates in the second year.  Valuation Model (continued) ❖Interest-Rate Lattice o Exhibit 19-7 shows the notation for the binomial interest-rate tree in the third year. o We can simplify the notation by letting rt be the lower one- year forward rate t years from now because all the other forward rates t years from now depend on that rate. o Exhibit 19-8 shows the interest-rate tree using this simplified notation. o Before we go on to show how to use this binomial interest- rate tree to value bonds, we first need to focus on 1) what the volatility parameter ( ) in the expression e2 represents 2) how to find the value of the bond at each node  Exhibit 19-8 Three-Year Binomial Interest- Rate Tree with One-Year Forward Rates r 3 e6  NHH r 2 e4  NHH r 1 e2  r 3 e4  NH NHH r0 r 2 e2  N NHL r1 NL r 3 e2  NHLL r2 NLL r3 NLLL Today 1 Year 2 Years 3 Years Lower 1-yr forward rate r1 r2 r3  Valuation Model (continued) ❖Volatility and the Standard Deviation o In the binomial model, it can be shown that the standard deviation of the one-year forward rate is equal to r0. o The standard deviation is a statistical measure of volatility. o For now it is important to see that the process that we assumed generates the binomial interest-rate tree (or equivalently, the forward rates) implies that volatility is measured relative to the current level of rates. EXAMPLE. If  is 10% and the one-year rate (r0) is 4%, what is the standard deviation of the one-year forward rate ? What is if r0 = 12%? r0 = 4% × 10% = 0.4% or 40 basis points r0 = 12% × 10% = 1.2% or 120 basis points  Valuation Model (continued) ❖Determining the Value at a Node o In the binomial model, we find the value of the bond at a node is as illustrated in Exhibit 19-9. o Calculate the bond’s value at the two nodes to the right of the node where we want to obtain the bond’s value. o The cash flow at a node will be either 1) the bond’s value if the short rate is the higher rate plus the coupon payment 2) the bond’s value if the short rate is the lower rate plus the coupon payment. the value is the present value of the expected cash flows o To get the bond’s value at a node we follow the fundamental process for valuation: the appropriate discount rate to use is the one-year forward rate at the node.  Exhibit 19-9 Calculating a Value at a Node Bond’s Value in Higher-Rate State One Year Forward Cash Flow in VH + C Higher-Rate State One-Year Rate at V Node Where Bond’s Value Is Sought r* Cash Flow in VL + C Lower-Rate State Bond’s Value in Lower-Rate State One Year Forward  Valuation Model (continued) ❖ Constructing the Binomial Interest-Rate Tree o To construct the binomial interest-rate tree, we use current on- the-run yields and assume a volatility, σ. o The root rate for the tree, r0, is simply the current one-year rate. o In the first year there are two possible one-year rates, the higher rate and the lower rate. o What we want to find is the two forward rates that will be consistent with the volatility assumption, the process that is assumed to generate the forward rates, and the observed market value of the bond. o There is no simple formula for this. It must be found by an iterative process (i.e., trial and error). o The steps are described and illustrated in Exhibits 19-10 and 19- 11.  Valuation Model (continued) ❖ Constructing the Binomial Interest-Rate Tree → Step 1: Select a value for r1. Recall that r1 is the lower one-year forward rate one year from now. In this first trial we arbitrarily selected a value of 4.5% for r1. → Step 2: Determine the corresponding value for the higher one-year forward rate. This rate is related to the lower one-year forward rate as follows: r1(e2). This value is reported at node NH. → Step 3: Compute the bond’s value one year from now. This value is determined as follows: → 3a. The bond’s value two years from now must be determined. → 3b. Calculate the present value of the bond’s value found in 3a using the higher rate. This value is VH. → 3c. Calculate the present value of the bond’s value found in 3a using the lower rate. This value is VL. → 3d. Add the coupon to VH and VL to get the cash flow at NH and NL, respectively. → 3e. Calculate the present value of the two values using the one-year forward rate using r*, so we can compute: VH + C and VL + C. 1 + r* 1 + r*  Valuation Model (continued) ❖ Constructing the Binomial Interest-Rate Tree → Step 4: Calculate the average present value of the two cash flows in step 3. This is the value at a node is 1 VH + C + VL + C . 2  1 + r* 1 + r*   → Step 5: Compare the value in step 4 with the bond’s market value. If the two values are the same, the r1 used in this trial is the one we seek. This is the one-year forward rate that would be used in the binomial interest-rate tree for the lower rate, and the corresponding rate would be for the higher rate. If, instead, the value found in step 4 is not equal to the market value of the bond, this means that the value r1 in this trial is not the one-period forward rate that is consistent with (1) the volatility assumption of 10%, (2) the process assumed to generate the one-year forward rate, and (3) the observed market value of the bond. In this case the five steps are repeated with a different value for r1. [Note. If we get a value less than $100, then the value for r1 is too large and the five steps must be repeated, trying a lower value for r1.] → In this example, when r1 is 4.5% we get a value of $99.567 in step 4, which is less than the observed market value of $100.Therefore, 4.5% is too large and the five steps must be repeated, trying a lower value for r1.  Valuation Model (continued) ❖ Constructing the Binomial Interest-Rate Tree → After we compute r1, we are still not done. Suppose that we want to “grow” this tree for one more year—that is, we want to determine r2. Now we will use the three-year on-the-run issue to get r2. The same five steps are used in an iterative process to find the one-year forward rate two years from now. But now our objective is as follows: Find the value for r2 that will produce an average present value at node NH equal to the bond value at that node and will also produce an average present value at node NL equal to the bond value at that node. When this value is found, we know that given the forward rate we found for r1, the bond’s value at the root—the value of ultimate interest to us—will be the observed market price. The binomial interest-rate tree constructed is said to be an arbitrage-free tree. It is so named because it fairly prices the on-the-run issues. Exhibit 19-10 Finding the One-Year Forward Rates for Year 1 Using the Two-Year 4% On-the-Run: First Trial V = 100 C = 4.00 V = 98.582 NHH r2,HH = ? C = 4.00 NH r1,H = 5.496% V = 99.567 V = 100 C=0 C = 4.00 N r0 = 3.500% NHL r2,HL = ? V = 99.522 C = 4.00 NL r1,L = 4.500% V = 100 C = 4.00 NLL r2,LL = ? Exhibit 19-11 One-Year Forward Rates for Year 1 Using the Two-Year 4% On-the-Run Issue V = 100 C = 4.00 V = 99.070 NHH r2,HH = ? C = 4.00 NH r1,H = 4.976% V = 100 V = 100 C=0 C = 4.00 N r0 = 3.500% NHL r2,HL = ? V = 99.929 C = 4.00 NL r1,L = 4.074% V = 100 C = 4.00 NLL r2,LL = ?  Valuation Model (continued) ❖ Application to Valuing an Option-Free Bond o To illustrate how to use the binomial interest-rate tree, consider a 5.25% corporate bond that has two years remaining to maturity and is option-free. o Also assume that the issuer’s on-the-run yield curve is the one given earlier, and hence the appropriate binomial interest-rate tree is the one in Exhibit 19-12. o Exhibit 19-13 shows the various values in the discounting process and produces a bond value of $102.075. o It is important to note that this value is identical to the bond value found earlier when we discounted at either the zero-coupon rates or the one-year forward rates. o We should expect to find this result because our bond is option free. This clearly demonstrates that the valuation model is consistent with the standard valuation model for an option-free bond. Exhibit 19-12 One-Year Forward Rates for Year 2 Using the Two-Year 4.5% On-the-Run Issue V = 97.886 C = 4.50 V = 98.074 NHH r2,HH = 6.757% C = 4.50 NH r1,H = 4.976% V = 100 V = 100 C=0 C = 4.50 N r0 = 3.500% NHL r2,HL = 5.532% V = 99.926 C = 4.50 NL r1,L = 4.074% V = 99.972 C = 4.50 NLL r2,LL = 4.530% Exhibit 19-13 Valuing an Option-Free Corporate Bond with Three Years to Maturity and a Coupon Rate of 5.25% V = 97.886 C = 4.50 V = 98.074 NHH r2,HH = 6.757% C = 4.50 NH r1,H = 4.976% V = 100 V = 100 C=0 C = 4.50 N r0 = 3.500% NHL r2,HL = 5.532% V = 99.926 C = 4.50 NL r1,L = 4.074% V = 99.972 C = 4.50 NLL r2,LL = 4.530%  Valuation Model (continued) ❖ Valuing a Callable Corporate Bond o The valuation process for a callable corporate bond proceeds in the same fashion as in the case of an option-free bond but with one exception: When the call option may be exercised by the issuer, the bond value at a node must be changed to reflect the lesser of its value if it is not called (i.e., the value obtained by applying the recursive valuation formula described previously) and the call price. o For example, consider a 5.25% corporate bond with three years remaining to maturity that is callable in one year at $100. Exhibit 19-14 shows the values at each node of the binomial interest-rate tree. Exhibit 19-14 Valuing a Callable Corporate Bond with Three Years to Maturity and a Coupon Rate of 5.25%, and Callable in One Year at 100 V = 98.588 C = 5.25 V = 99.461 NHH r2,HH = 6.757% C = 5.25 NH r1,H = 4.976% V = 101.432 V = 99.732 C=0 C = 5.25 N r0 = 3.500% NHL r2,HL = 5.532% V = 100 (100.001) C = 5.25 NL r1,L = 4.074% V =100 (100.689) C = 5.25 NLL r2,LL = 4.530%  Valuation Model (continued) ❖ Impact of Expected Interest-Rate Volatility on Price o Expected interest-rate volatility is a key input into the valuation of bonds with embedded options. To see the impact on the price of a callable bond, Exhibit 19-15 shows the price of four 5%, 10-year callable bonds with different deferred call structures (six months, two year, five years, and seven years) based on different assumptions about the expected volatility of short-term interest rates. We observe the following from the exhibit: 1) The price of the option-free bond is the same regardless of the interest-rate volatility assumed. This is expected since there is no embedded option that is affected by interest-rate volatility. 2) For any given level of interest-rate volatility, the longer the deferred call, the higher the price. Again, as expected the value of the option- free bond has the highest price. 3) The price of a callable bond moves inversely to the interest-rate volatility assumed.  Exhibit 19-15 Effect of Interest-Rate Volatility and Years to Call on Prices of 5%, 10-Year Callable Bonds 109 107 105 Price (%) 103 101 99 97 12 14 16 18 20 22 24 26 28 30 Volatility of Short-Term Interest Rate (%)  Valuation Model (continued) ❖ Determining the Call Option Value (or Option Cost) o The value of a callable bond is expressed as the difference between the value of a noncallable bond and the value of the call option. o This relationship can also be expressed as follows: value of a call option = value of a noncallable bond – value of a callable bond o But we have just seen how the value of a noncallable bond and the value of a callable bond can be determined. o The difference between the two values is therefore the value of the call option. o In our previous illustration, the value of the noncallable bond is $102.075 and the value of the callable bond is $101.432, so the value of the call option is $0.643.  Valuation Model (continued) ❖ Extension to Other Embedded Options o The bond valuation framework presented here can be used to analyze other embedded options, such as put options, caps and floors on floating-rate notes, and the optional accelerated redemption granted to an issuer in fulfilling its sinking fund requirement. o Exhibit 19-16 (see Overhead 19-50) shows the binomial interest-rate tree with the bond values altered at two nodes. o Because the value of a non-putable bond can be expressed as the value of a putable bond minus the value of a put option on that bond, this means that value of a put option = value of a non-putable bond – value of a putable bond Exhibit 19-16 Valuing a Putable Corporate Bond with Three Years to Maturity and a Coupon Rate of 5.25%, and Putable in One Year at 100 V = 100 (98.588) C = 5.25 V = 100.261 NHH r2,HH = 6.757% C = 5.25 NH r1,H = 4.976% V = 102.523 V = 100 (99.732) C=0 C = 5.25 N r0 = 3.500% NHL r2,HL = 5.532% V = 101.461 C = 5.25 NL r1,L = 4.074% V =100.689 C = 5.25 NLL r2,LL = 4.530%  Valuation Model (continued) ❖ Incorporating Default Risk o The basic binomial model explained above can be extended to incorporate default risk. o The extension involves adjusting the expected cash flows for the probability of a payment default and the expected amount of cash that will be recovered when a default occurs. ❖ Modeling Risk o The user of any valuation model is exposed to modeling risk. o This is the risk that the output of the model is incorrect because the assumptions upon which it is based are incorrect. ❖ Implementation Challenge o To transform the basic interest-rate tree into a practical tool requires refinements. For example, the spacing of the node lines in the tree must be much finer. Although one can introduce time-dependent node spacing, caution is required; it is easy to distort the term structure of volatility. Other practical difficulties include the management of cash flows that fall between two node lines.  Option-Adjusted Spread (continued) ❖ The option-adjusted spread (OAS) was developed as a measure of the yield spread (in basis points) that can be used to convert dollar differences between value and price. ❖ Thus, basically, the OAS is used to reconcile value with market price. ❖ The OAS is a spread over the spot rate curve or benchmark used in the valuation. ❖ The reason that the resulting spread is referred to as option-adjusted is because the cash flows of the security whose value we seek are adjusted to reflect the embedded option. Option-Adjusted Spread (continued) ❖ Translating OAS to Theoretical Value o Although the product of a valuation model is the OAS, the process can be worked in reverse. o For a specified OAS, the valuation model can determine the theoretical value of the security that is consistent with that OAS. o As with the theoretical value, the OAS is affected by the assumed interest-rate volatility. o The higher (lower) the expected interest-rate volatility, the lower (higher) the OAS. ❖ Determining the Option Value in Spread Terms o The option value in spread terms is determined as follows: option value (in basis points) = static spread – OAS  Effective Duration and Convexity ❖ There is a duration measure that is more appropriate for bonds with embedded options that the modified duration measure. ❖ In general, the duration for any bond can be approximated as follows: P_ − P + duration = 2 ( P0 ) ( dy ) P_ = price if yield is decreased by x basis points P+ = price if yield is increased by x basis points P0 = initial price (per $100 of par value) ∆y (or dy) = change in rate used to calculate price (x basis points in decimal form)  Effective Duration and Convexity (continued) ❖ When the approximate duration formula is applied to a bond with an embedded option, the new prices at the higher and lower yield levels should reflect the value from the valuation model. ❖ Duration calculated in this way is called effective duration or option-adjusted duration. ❖ The differences between modified duration and effective duration are summarized in Exhibit 19-17. ❖ The standard convexity measure may be inappropriate for a bond with embedded options because it does not consider the effect of a change in interest rates on the bond’s cash flow.  Exhibit 19-17 Modified Duration Versus Effective Duration Duration Interpretation: Generic description of the sensitivity of a bond’s price (as a percent of initial price) to a parallel shift in the yield curve Modified Duration Effective Duration Duration measure in which it is assumed Duration measure in which recognition that yield changes do not change is given to the fact that yield changes may the expected cash flow change the expected cash flow Thank You Concept of Yield  Understand the Sources of Return Fixed income investors or investors who purchase a bond may benefit from that investment in many ways: – periodic coupon as per the terms in the indenture of the bond; – capital gain if interest rate falls (it may also result in capital loss if interest rate rises), in case the instrument is traded before maturity; – reinvestment of the cash flows received during the life of the bond (akin to interest on interest).  Coupon Income A coupon income is the regular flow of money or return to the investor or lender as promised by the borrower. When we say, for example, 7% GS 2027, we mean the security is a Government Security, paying an Annual Coupon of Rs 7 on a Face value of Rs 100. Typically, the coupons are paid semi-annually and hence, the investor would receive the coupon of Rs 3.50 every half year. Coupon is the promise of the borrower to pay a certain amount of money at regular intervals to the lender during the life of a bond or a note. In earlier years, bonds issued by Governments or central banks on behalf of sovereign Governments used to be bearer bonds and were repaid on physical presentation of the appropriate instrument.  Capital appreciation During the life of investment in a bond, market interest rate changes and the present value of the Bond would also change as the Coupon is fixed. If the investor holds till maturity, her rate of return would remain fixed but if the investor desires to exit the investment anytime during the life of the bond, her investment would either gain in value or it would have reduced in value. There will be capital appreciation during the life of the bond, if the interest rate in the market falls and the original investor sold the bond at such times, though at maturity the return would be only the face value. For example, the issuer issuing the AAA rated bond of 10 years maturity would pay the interest rate prevailing in the market on the date of issue for similar kind of bonds.  Reinvestment income The investor receives periodic interest or coupon on the debt investment. The same is reinvested on assets which would yield further income. For example, when a semi-annual coupon payment is received by our investor from the investment on 7.17% GS 2028, the investor would have the ability to invest that cash flow of Rs 3.585 in another asset on such coupon receiving date. The yield to maturity (YTM) assumes the reinvestment of the future coupons at the same rate (i.e., at YTM). This is because the bond price equation assumes the same yield to discount all future cash flows (i.e., both the coupons and the redemption value).  Traditional Yield Measures Bonds exhibit various characteristics with respect to being securities as below: – Different maturities with the same coupon – Different coupons for the same maturity – Different ratings for the same coupon and maturities – Different redemption values with varied maturity  Coupon rate The simple bond yield, also called coupon rate, is calculated by dividing its interest payment by the face value of the bond. For a bond with face value of ₹100 making coupon payments of ₹10 per year, the coupon rate is 10% (₹10/₹100 = 10%). Current Yield Current yield relates the annual coupon interest to the market price. The formula for the current yield is: current yield = annual dollar coupon interest / price The current yield calculation takes into account only the coupon interest and no other source of return that will affect an investor’s yield. The time value of money is also ignored. Bond 7.17% GS 2028 (8-Jan-2030) is trading at 7.85% or at Rs 96.2290. The current yield of this bond is 7.45% [ = (7.17/96.2290)*100) ] vis-à-vis the coupon of 7.17%. Thank you Concept of Yield  Yield to Maturity (YTM) YTM - An IRR or an interest rate that equates PV of all future CFs of a bond to the current price of the bond ▪ Better measure than Current Yield ▪ Takes Capital gains/losses into account ▪ Considers the Time Value of money Computing the Yield or Internal Rate of Return on any Investment ❑ The yield on any investment is the interest rate that will make the present value of the cash flows from the investment equal to the price (or cost) of the investment. ❑ Mathematically, the yield on any investment, y, is the interest rate that satisfies the equation. P = CF1 + CF2 2 + CF3 3 +... + CFN N 1 + y (1 + y ) (1 + y ) (1 + y ) where P = price of the investment, CF = cash flow in year t =1,2,3,… N, y = yield calculated from this relationship (and also called the internal rate of return), N = number of years.  Computing the Yield or Internal Rate of Return on any Investment (continued) ❑ Absent a financial calculator or computer software, solving for the yield (y) requires a trial-and-error (iterative) procedure. ❑ The objective is to find the yield that will make the present value of the cash flows equal to the price. ❑ Keep in mind that the yield computed is the yield for the period. ▪ That is, if the cash flows are semiannual, the yield is a semiannual yield. ▪ If the cash flows are monthly, the yield is a monthly yield. ❑ To compute the simple annual interest rate, the yield for the period is multiplied by the number of periods in the year.  Computing the Yield or Internal Rate of Return on any Investment (continued) ❑ Special Case: Investment with Only One Future Cash Flow ▪ When the case where there is only one future cash flow, it is not necessary to go through the time-consuming trial-and-error procedure to determine the yield. ▪ We can use the below equation: 1 y=  CFn  n −1  P  where y is the yield, CFn is the cash flow that occurs in n periods, and P is today’s value and n is the number of periods.  Computing the Yield or Internal Rate of Return on any Investment (continued) ❑ Annualizing Yields ▪ To obtain an effective annual yield associated with a periodic interest rate, the following formula is used: effective annual yield = (1 + periodic interest rate)m – 1 where m is the frequency of payments per year. ▪ To illustrate, if interest is paid quarterly and the periodic interest rate is 0.08 / 4 = 0.02, then we have: effective annual yield = (1.02)4 – 1 = 1.0824 – 1 = 0.0824 or 8.24%  Computing the Yield or Internal Rate of Return on any Investment (Continued) ❑ Annualizing Yields ▪ We can also determine the periodic interest rate that will produce a given annual interest rate by solving the effective annual yield equation for the periodic interest rate. ▪ Solving, we find that periodic interest rate = (1 + effective annual yield )1/m – 1 ▪ To illustrate, if the periodic quarterly interest rate that would produce an effective annual yield of 12%, then we have: periodic interest rate = (1.12)1/4 – 1 = 1.0287 – 1 = 0.0287 or 2.87%  Conventional Yield Measures ❑ Bond yield measures commonly quoted by dealers and used by portfolio managers are: 1) Current Yield 2) Yield To Maturity 3) Yield To Call 4) Yield To Put 5) Yield To Worst 6) Cash Flow Yield 7) Yield (Internal Rate of Return) for a Portfolio 8) Yield Spread Measures for Floating-Rate Securities Yield To Maturity ▪ The yield to maturity is the interest rate that will make the present value of the cash flows equal to the price (or initial investment). ▪ For a semiannual pay bond, the yield to maturity is found by first computing the periodic interest rate, y, which satisfies the relationship: P = C + C 2 + C 3 +...+ C n + M n 1 + y (1 + y ) (1 + y ) (1 + y ) (1 + y ) where P = price of the bond, C = semiannual coupon interest (in dollars), M = maturity value (in dollars), and n = number of periods (number of years multiplied by 2). Yield To Maturity (continued) ▪ For a semiannual pay bond, doubling the periodic interest rate or discount rate (y) gives the yield to maturity. ▪ The yield to maturity computed on the basis of this market convention is called the bond-equivalent yield: 1/ n y =  M − 1  P where M = maturity value (in dollars), P = price of the bond, and n = number of periods (number of years multiplied by 2). ▪ The yield-to-maturity calculation takes into account the current coupon income, any capital gain or loss realized by holding the bond to maturity, and the timing of the cash flows. Comparing Current Yield and YTM Bond Selling at: Relationship between Coupon Rate (CR), Current Yield (CY) and YTM Par CR = CY = YTM Discount CR < CY < YTM Premium CR > CY > YTM YTM is a better measure than Current Yield since it takes into consideration the timing of cash flows and the expected capital gains/losses on redemption. Refer Excel Illustration Thank you Measuring Yields  Assumptions of YTM The YTM is the estimated annual rate of return that a bond is expected to earn until reaching maturity, with three notable assumptions: – Assumption 1 → The return assumes the bond investor held onto the debt instrument until the maturity date. – Assumption 2 → All the required interest payments and principal repayment were made on schedule. – Assumption 3 → The coupon payments were reinvested at the same rate as the yield-to-maturity (YTM).  Limitations of YTM ▪ Ignores reinvestment risk (Assumes all coupons can be reinvested at YTM) ▪ Assumes bond will be held till maturity ▪ Assumes flat yield curve  Total Return ▪ Three sources of Bond return: ▪Periodic Coupon income ▪Capital Gain/Loss on sale/redemption of ▪Interest income generated from reinvestment of coupons and sale/redemption proceeds ▪Consideration to future interest rate expectations ▪Expectations regarding reinvestment rates in future ▪Expectations regarding yield/discount rates in future ▪Steps for computation - Excel Illustration  Total Return ❑ The yield to maturity is a promised yield because at the time of purchase an investor is promised a yield, as measured by the yield to maturity, if both of the following conditions are satisfied: 1) the bond is held to maturity 2) all coupon interest payments are reinvested at the yield to maturity ❑ The total return is a measure of yield that incorporates an explicit assumption about the reinvestment rate. ❑ The yield-to-call measure is subject to the same problems as the yield to maturity because it assumes that the: 1) bond will be held until the first call date 2) coupon interest payments will be reinvested at the yield to call  Total Return (continued) ❑ Computing the Total Return for a Bond ▪ The idea underlying total return is simple. 1) The objective is first to compute the total future dollars that will result from investing in a bond assuming a particular reinvestment rate. 2) The total return is then computed as the interest rate that will make the initial investment in the bond grow to the computed total future dollars.  Applications of the Total Return Horizon Analysis ❑ Horizon analysis refers to using total return to assess performance over some investment horizon. ❑ Horizon return refers to when a total return is calculated over an investment horizon. ❑ An often-cited objection to the total return measure is that it requires the portfolio manager to formulate assumptions about reinvestment rates and future yields as well as to think in terms of an investment horizon.  Calculating Yield Changes ❑ The absolute yield change (or absolute rate change) is measured in basis points and is the absolute value of the difference between the two yields as given by absolute yield change = │initial yield – new yield│ × 100. ❑ The percentage change is computed as the natural logarithm of the ratio of the change in yield as shown by percentage change yield = 100 × ln (new yield / initial yield) where ln in the natural logarithm.  Annualizing yields Two approaches for calculating annualized yields ▪Bond-equivalent Yield ▪Effective Annualized Yield Example : A bond with semi annual coupon frequency ▪Ys = Semi-annual yield (computed using YTM model) ▪ Bond-Equivalent Yield ▪ Ya = Ys * 2 ▪ Effective Annualised Yield ▪ Ya = (1 + Ys)2 - 1 Thank you Measuring Yield  Conventional Yield Measures ❑ Bond yield measures commonly quoted by dealers and used by portfolio managers are: 1) Current Yield 2) Yield To Maturity 3) Yield To Call 4) Yield To Put 5) Yield To Worst 6) Cash Flow Yield 7) Yield (Internal Rate of Return) for a Portfolio 8) Yield Spread Measures for Floating-Rate Securities  Conventional Yield Measures (Continued) Yield To Call ▪