FIXED INCOME. FINANCE TRAINER International Fixed Income / Page 1 of 48

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1 FIXED INCOME. The Bond Market Different Criteria for Bonds Common Bonds and Their Abbreviations and Conventions (Excurus)... 7 Interest payments Associations, Abbreviations and Terms in the Capital Markets Quotation of Bonds Pricing of Bonds Influencing Factors Calculation of Bond Prices Formula for "Bullet" Bonds with Annual Coupons General Formula for Pricing Examples for Calculation Assumptions for the Traditional Bond Formulae Pricing with the Zero Curve Calculation of Price Sensitivities/ The Concept of Duration The Simple Duration (Macaulay Duration) Modified Duration Influencing Factors on the Modified Duration The Estimation Failure of the Duration the Convexity Coupon, Yield to Maturity, Par Yield and Zero Coupon Ratings FINANCE TRAINER International Fixed Income / Page of 48

2 . The Bond Market In the bond market bonds/debentures with medium- and long-term maturities are traded. A bond/debenture is a negotiable debt. The issuer obliges himself to pay the owner a specific interest rate for an agreed period of time and to repay the principal on a specified settlement day (or on several settlement days). From the issuer s point of view bonds are debt capital, and the owner of the bond is the creditor. This is also the main difference to shares which securitise an investment in a company. Here, from the issuing company s point of view, the shares are equity capital. Therefore shares do not have any specific term and the interest (dividend) depends on the economical success. Overview financial markets: Financial Markets (market for fundraising) Dept Capital Equity Capital Money Market Short-term (< 2 years) Capital Markets Long-term (> 2 years) Stock Market Unlimited terms Securitised:: CD, CP T-Bill Bill of Exchange Unsecuritised: short-term bank loan Securitised: bonds Unsecuritised: Long-term bank loan Note: Due to the many different criteria, there exist also other classifications. Capital Markets is often also a synonym for the bond market of the financial markets in common. This script focuses on the bond market and also uses the names capital market resp. fixed income. For the issuer, bonds are a very important instrument for fundraising. Due to the securitisation and the tradability the issuer can find a number of potential investors, like private investors, banks, companies or investment fonds. The investors have the advantage that they can choose from a variety of investment opportunities, depending on their investment horizon and their risk appetite. FINANCE TRAINER International Fixed Income / Page 2 of 48

3 . Different Criteria for Bonds Bonds can be divided along many different criteria: I. Issuer II. Primary market III. Interest payment I. Issuer Bonds can be categorised in terms of their issuer into Government bonds The most important issuers are states which use bonds for financing their national debts. Because of the big volumina this market is very liquid and is therefore very important for the whole bond market. Due to the good ratings, government bonds are also regarded as benchmark for non-risk investments. Bank bonds Banks use the bond market for refinancing their loans to customers. Corporate bonds The number of corporate issuances has been strongly increasing in recent years. For corporates, bonds are an alternative to the common bank loan. II. Primary market Domestic bonds Bonds that are issued by the government, banks, or corporations in their own, domestic market, are called domestic bonds. Issuers have to follow the legal regulations. For example, in the US market every public bond issuance has to be registered at the SEC (US Securities and Exchange Commission) what can lead to significantly high costs. To avoid these costs so-called private placements are conducted, i.e. the bonds are sold to only one single investor or a small number of investors. Here only larger financial institutions come into question which reduces the bonds tradability. Foreign bonds Bonds that are issued in the domestic market by a foreign institution; e.g. a bond that is issued by a US bank in Germany. Like domestic bonds also foreign bonds underly the specific local conditions. The US have by far the largest foreign bond market. Foreign bonds often carry typical names. A bond, issued by a foreign (e.g. British) issuer in the FINANCE TRAINER International Fixed Income / Page 3 of 48

4 US is called Yankee. Further names are Bulldog (UK), Matador (Spain), Alpine (Switzerland) or Samurai (Japan). Eurobonds or international bonds Eurobonds are issued in the international market outside the issuer s home country. Conditions under British law are common, but also US law and others are applied. Some years ago every single issuance had its own prospectus. Today standardised bonds are often issued within issuance programs (e.g. debt issuance program, medium term note program, commercial paper program). For these programs a sample documentation, a so-called sample prospectus, is written which shows the description of the issuer. The individual documentation for individual bonds belonging to such a program is reduced to additional writings. Thus the time for preparation and costs can be reduced significantly. Due to the increasing deregulation of domestic markets the differentiation between domestic, foreign and eurobonds is becoming more and more obsolete. III. Interest payments There exist fixed-rate interest bonds on the one hand and floating-rate interest bonds (floating-rate notes, FRNs) on the other hand. With fixed-rate interest bonds, the interest rate (in per cent of the nominal value) is fixed at the time of issue and does not change during the bond s term. With floating-rate interest bonds, the interest rate is defined as a premium or discount to a specific reference/benchmark rate (e.g. LIBOR, EURIBOR). Due to possible changes of the reference rate, the interest payments may vary during the bond's term. The fixing of the interest rate for the next period is done at pre-defined dates. Another criterion to distinct bonds is the frequency of the interest payments. The following types of bonds can be found in the market: With zero-bonds, no interest payments take place during the bond's term. The whole interest payment (including compound interest) is done at maturity. With annual interest payments, a payment has to be made annually. With non-annual interest payments, payments take place either quarterly or six-monthly. FINANCE TRAINER International Fixed Income / Page 4 of 48

5 Registered bonds / bearer bonds A bond might be issued as either a registered or a bearer bond. Registered bonds: The current owner of the bond and every change of ownership is recorded in a central register. Coupon payments and redemption are booked on the account of the current owner. Bearer bonds: The current bearer of the bond is entitled to receive the interest payments and redemption. The current owner is not registered. Interest payments are made to the bearer of the interest coupons and the repayment is to be made to the bearer or sender of the bond. Redemption Usually, the issuer of a bond obliges himself to pay off the capital at a pre-set date and at a defined rate (usually at a rate of 00). If the whole bond is paid off at once it is called a "bullet". If the bond is paid off gradually, it is called an amortising bond or a sinking fund. In the contract it is fixed that parts of the bond can be redeemed earlier at fixed dates and at a fixed rate. The redemption plan can define a certain amount or the different redemption parts. You can also only define the amount and determine the different redemption parts by lot. Then you know what amount will be repaid earlier, but not to which investor. In addition to the bonds mentioned above, there are the so-called callable bonds, where the issuer has the right to pay off the bond before maturity at a prior defined rate. Convertible bonds Convertible bonds are unsecured fixed-rate bonds that give the owner the right (but not the obligation) to convert the bonds into shares (under conditions that have been specified in advance). Usually, the interest rate for such convertible bonds is below the interest rate for "normal" bonds, since the owner has the right to convert the bond into shares. FINANCE TRAINER International Fixed Income / Page 5 of 48

6 Day counting in the capital markets Country Euro area USA UK Japan Day Counting ACT/ACT ACT/ACT Gilts: s.a. ACT/ACT 30/360 or ACT/ACT Switzerland 30/360 or 30E/360 FINANCE TRAINER International Fixed Income / Page 6 of 48

7 .2 Common Bonds and Their Abbreviations and Conventions (Excurus) USA T-Bill Short-term US government bond on a discount basis T-Bond US government bond with terms of more than 0 years T-Note US government bond with terms of less than 0 years Germany Bobl Bundesobligation, German government bond Bund Bundesanleihe, German government bond OBL Bundesobligation (Bobl), German government bond REX German bonds with variable interest payments which are bound to a long-term benchmark UK Gilt UK government bond Japan JGB Japanese government bond France BTAN Bon de trésor à taux fixe et intérêt annuel, short-term French government bond BTF Bon de trésor à taux fixe et intérêt précompté, French government bond on a discount basis (~T-Bill) OAT Obligation assimilable du Trésor, French government bond OATi French government bond, linked to inflation TEC French bonds with variable interest payments which are bound to a long-term benchmark Italy BTP Buoni del Tresoro Poliennali, Italian government bonds > year with fixed interest payment CCT Certificati di Credito del Tresoro, Italian government bonds > year with floating interest payment CTZ Certificati di Credito del Tresoro Zero Coupon, Italian government bonds > year without interest coupon Day counting in the capital markets Country Day Counting FINANCE TRAINER International Fixed Income / Page 7 of 48

8 Euro area USA UK Japan ACT/ACT ACT/ACT Gilts: s.a. ACT/ACT 30/360 or ACT/ACT Switzerland 30/360 or 30E/360 Interest payments type Eurobonds interest payment annual government bonds Bund Treasury Note UK / Gilts Japan annual semi-annual semi-annual semi-annual FINANCE TRAINER International Fixed Income / Page 8 of 48

9 .3 Associations, Abbreviations and Terms in the Capital Markets ABS BBAIRS Bullet Callable Bond Cedel CGO CTD DVP Euroclear Flat quotation GNMA Gross-Up ISDA ISMA Conversion Price MBS Pari Passu SEC Step Up Bond Swappable Bond Convertible Bond Asset Backed Securities; a bond which is securitised by certain assets (e.g. loans, credit card outstandings, leasings etc.) British Bankers Association Interest Rate Swaps, standard documentation by the British Bankers Association for interest rate swaps Bond which repays 00% of the nominal at maturity Bond where the issuer has the right to repay earlier A clearing house, founded in Luxemburg, after the merger with Deutsche Bank Cedel had been absorbed in Clearstream Central Gilt Office Cheapest to deliver, the bond which is the cheapest at the moment regarding delivery in futures Delivery versus Payment One of the two largest security clearing houses, based in Brussels Bond quotation at dirty price (incl. accrued interest) Government National Mortgage Association (also called Ginnie Mae ), a state agency in the US which buys mortgages from financial institutions, then securitises these mortgages and sells it to investors as bonds. the obligation of an issuer to offset the withholding tax, if such a tax is introduced during the term of the bond. grossing-up clauses are frequently used in eurobonds International Securities and Derivatives Association International Securities Market Association Defined share price for the option of a convertible bond Mortgage Backed Securities, an ABS kind with mortgages A clause which obliges the issuer to treat all investors the same Securities and Exchange Commission (USA) Bond with a rising coupon The coupon of this bond is higher than the actual interest rate swap rate A bond that gives the owner the right to convert the nominal into shares at a defined price FINANCE TRAINER International Fixed Income / Page 9 of 48

10 2. Quotation of Bonds Face value Every bond has a fixed face value. This face value serves as the basis for the interest payment as the basis to calculate the redemption value which can differ from the face value. If the current price of the bond is above (below) 00, the bond is said to be quoted at a premium (discount). Quotation Bond prices are usually quoted in per cent of the face value. For example, the bond's price of 0.50 for EUR government bonds corresponds to the price of EUR 0.50 per cent of the nominal value. In the eurobond market, prices are usually quoted in decimals (e.g. 0.50), while in the US and in UK they are often quoted with fractions (e.g. 0½ or 06/32). In the bond market, market makers quote both bid and offer on request: the bid price is the price they are willing to pay for bonds while the offer price is the price at which they are willing to sell the bonds. Accrued interest The owner of the bond receives the full amount of interest at coupon dates, even though he might not have possessed the bond during the whole period of interest. Therefore, when a bond is sold or bough, accrued interest has to be taken into account. Since comparing bond prices, that include accrued interest, is a complicated business, bonds are usually quoted without accrued interest. Nonetheless, the buyer of a bond has to pay the accrued interest to the seller when he purchases the bond. The price without accrued interest is called "clean" price. The price including accrued interest is called "dirty" price. FINANCE TRAINER International Fixed Income / Page 0 of 48

11 To calculate the accrued interest, one computes the interest on the face value for the elapsed days (taking into account the respective method of calculation). According to the ISMA conventions the calculation of the accrued interest days includes the start value of the interest period, but not the end value (= trade date of the bond). Dirty Price A USD bond with a 7%-coupon (30/360 annual) and a time to maturity of 3½ years is sold at a price of 0.50 (= clean price). Since the last coupon date, 6 months (= 80 days) have elapsed. 80 Dirty price = Therefore, the buyer of the coupon must pay a price of Start/ end value A EUR-Bund, coupon 6.0% (ACT/ACT annual), had its last coupon payment on 2 th January 200. On 8 th April Bank A sells the bond to Bank B. The accrued interest days based on ACT are calculated as follows: Period Days January 2th 3st 20 February 28 March 3 April 7th 6 Total 95 The accrued interest, that Bank B has to pay, are EUR ( 6 ). 365 FINANCE TRAINER International Fixed Income / Page of 48

12 3. Pricing of Bonds 3. Influencing Factors The price of a fixed-rate interest bond corresponds to the price at which market participants are willing to buy or sell the bond. When speaking of the price of a bond, one usually refers to its market price. This market price is influenced by the following factors:. the time to maturity of the bond 2. the actual market yield of bonds with the same time to maturity 3. the fixed interest rate of the bond 4. the linked credit risk (credit quality of the issuer) 5. the liquidity of the bond's secondary market We will concentrate on the first three of these factors (time to maturity, yield, and coupon). We use government bonds for demonstration purposes, because here the influence of both the premium for credit risks and a possible liquidity premium are reduced to a minimum. 3.2 Calculation of Bond Prices How can the fair bond price be determined? The answer is simple. If you buy a bond you know that you will receive regular coupon payments during the bond s term and the nominal at maturity. Therefore, today, you are willing to pay the present value of these future cashflows. Thus the fair bond price is the present value of all future cash-flows. Following example: Assuming, a 5-year government bond which is issued today at a price of 00 has a coupon of 6%. At a price of 00 a coupon of 6% equals an interest rate of 6% for the nominal amount. Therefore, the market yield for 5-year government bonds is 6%. There is another government bond in the market with the same maturity which pays a coupon of 7%. Which price are you willing to pay for this 7%-bond? Answer: you are willing to pay as much as the interest on the nominal amount corresponds to the market yield of 6%. We look at every single cash-flow and check how much we can FINANCE TRAINER International Fixed Income / Page 2 of 48

13 pay for it in order to have a return of 6%. The purchase of nominal EUR 00 m of the above 7%-bond has the following cash-flows: CF CF2 CF3 CF4 CF5 Cash-flow present value sum: 04.2 How many EUR can you pay today for a cash-flow of EUR 7 in one year to get a return of 6%? Answer: the discounted amount, i.e. 7/(+) = EUR Analogically EUR 7 in 2 years are worth today 7/(+)2 = EUR 6,22998 EUR, etc. If today, we pay the sum of the present values of all cash-flows, we will receive a return on the investment which equals the market yield Formula for "Bullet" Bonds with Annual Coupons The common formula used in the pricing of a bullet bond is as follows: C C C Nom PV... or r r 2 r N r N PV C N n n r r N 00 PV = present value = fair bond price C = coupon, in decimals (6% = ) r = current market yield, in decimals n = ongoing year N = total number of years Nom = nominal = 00 FINANCE TRAINER International Fixed Income / Page 3 of 48

14 Calculate the price for a EUR government bond with a 7%-coupon, 5 years to maturity, and annual interest payments (the latest payment just took place). The current interest rate (yield) for government bonds with 5 years to maturity is 6%. 5 PV 0.07 n 5 n = 04.2 The fair price of this bond is We demonstrate the calculation of the sum 5 n n in the following table: year discount factor Sum: FINANCE TRAINER International Fixed Income / Page 4 of 48

15 Market yield and bond price If the market yield rises, the future cash-flow will be discounted at a higher interest rate which leads to a lower present value (and vice versa). Thus we have the following connection: Interest rates Price Interest rates Price In other words: If a bond pays a coupon which is lower than the current market yield, an investor will only be willing to pay a price which is lower than 00 in order to compensate for the difference between the low coupon payments and the actual issuances. Is the coupon higher than the current yield the bond price will be above General Formula for Pricing Since the amount of capital is not necessarily paid back at maturity and the interest payments are not necessarily paid annually, a number of specialised formulae for pricing exist. For this reason, we present a general formula that takes into account different types of interest payments as well as different types of repayment arrangements. First, we determine the cash-flows that result from a nominal amount of 00. Then, these cash-flows are discounted by the current yield. P = N r n n CF n P = price r = current yield, in decimals n = ongoing period N = total number of periods CF n = cash-flow (at a nominal of 00) at time n Since pricing does not necessarily happen at coupon dates (result is a "broken" period at the beginning), therefore we have to generalise the formula again: FINANCE TRAINER International Fixed Income / Page 5 of 48

16 Clean Price (Moosmüller) P c = Pc d AI B CF n CF d r B N CFn d r r B n2 n = clean price = days to the first cash-flow = accrued interest = day base (360/ 365/ ACT) = cash-flow (at a nominal of 00) at time n AI Note: With "broken" periods there are different ways of calculation. In Germany, the "Moosmüller-method" is usually employed, i.e. the first discounting is calculated linearly (+r*d/b); internationally, the "ISMA-method" is commonly used, which discounts cashflows exponentially, i.e. (+r*d/b) is replaced by (+r)d/b. The ISMA-method always leads to a higher result than the Moosmüller-method. FINANCE TRAINER International Fixed Income / Page 6 of 48

17 3.2.3 Examples for Calculation bullet bond at the coupon date bond: German government bond (Bund) coupon: 7.00% fixed time to maturity: 5 years interest payment: annual redemption: bullet current yield: 6.00% Assumption: The latest interest payment has just been made Year Cash-flow Discounting Present value (2x3) Sum: The calculated bond price is As the latest interest payment has just been made, here, clean price = dirty price. FINANCE TRAINER International Fixed Income / Page 7 of 48

18 bullet bond between coupon dates Calculation with Moosmüller-method bond: German government bond (Bund) coupon: 7.00% fixed time to maturity: 4 years, 270 days interest payment: annual redemption: bullet current yield: 6.00% Year Cash- Discounting Present value flow (2x3) 270 days year, 270 days years, 270 days years, 270 days years, 270 days Sum (dirty price): Accrued interest: Price (clean price): with Moosmüller-method The sum of all discounted cash-flows is the dirty price. This amount has to be paid when buying the bond, i.e. clean price plus accrued interest. The clean price is quoted which is the result of the dirty price minus accrued interest. FINANCE TRAINER International Fixed Income / Page 8 of 48

19 Continuation: same bond, calculation with ISMA-method bond: German government bond (Bund) coupon: 7.00% fixed time to maturity: 4 years, 270 days interest payment: annual redemption: bullet current yield: 6.00% Year Cash- Discounting Present value flow (2x3) 270 days , year, 270 days x 0, years, 270 days x 2 0, years, 270 days x 3 0, years, 270 days x 4 0, Sum (dirty price): Accrued interest: Price (clean price): with ISMA-method With the ISMA-method the clean price is Compared to the Moosmüller-calculation ( ) this price is by higher. FINANCE TRAINER International Fixed Income / Page 9 of 48

20 bullet bond with semi-annual coupon coupon: 7.00% fixed time to maturity: 3 years interest payment: semi-annually redemption: bullet current yield: 6.00% (= 3.0% per period) Assumption: The latest interest payment just took place. For coupon periods less than one year the number of interest periods increases. For the price calculation the particular cash-flow is discounted for the number of interest periods by the interest rate divided by the number of the yearly interest periods Year Cash-flow Discounting Present value (2x3) , , ,9547 0, , , Sum: Note: one could also convert the semi-annual rate into an annual rate ( = 6.09%) and then discount with the number of years, i.e. 0.5; ;.5; etc. FINANCE TRAINER International Fixed Income / Page 20 of 48

21 Bond with partial redemption before maturity coupon: 7.00% fixed time to maturity: 5 years interest payment: annual redemption: 50% after 3 years, rest: at end of term current yield: 6.00% Assumption: The latest interest payment just took place. At first, the cash-flows have to be determined. In the years, 2 and 3 the coupon is paid for the whole nominal amount. In year 3 there is an additional partial redemption of 50. In the years 4 and 5 the coupon is only paid for 50 and is therefore only Year Cash-flow Discounting Present value (2x3) Sum: FINANCE TRAINER International Fixed Income / Page 2 of 48

22 3.2.4 Assumptions for the Traditional Bond Formulae When calculating the bond price with the traditional pricing formula (Moosmüller or ISMA) we have discounted all cash-flows with the current market yield for comparable bonds. However, this method is not perfect. On closer examination one can see that two assumptions are made: Flat yield curve All future cash-flows are discounted with the same interest rate, the market yield for the bond s maturity. However, we know that in practice in most cases different interest rates for different maturities are paid. Thus, for the calculation with the traditional bond formulae a flat yield curve is assumed. Re-investment of the interest returns at the same interest rate When calculating the present value the question is: how much are we willing to pay today for a future cash-flow? We then discount the future cash-flow with the interest rate for a comparable alternative investment. The future value of this alternative investment should then be the same as the future cash-flow. However, when calculating the present value with the formula /(+r)n, you are assuming that during the term of the bond the interest payments can always be invested at the same rate. In practice interest rates are changing which makes it uncertain at which rate interest payments during the term can be re-invested. Even when assuming unchanged rates the re-investment at the same rate would only be possible for a flat yield curve. Thus, in practice, it is not sure if the investment of the present value results in the underlying future value Pricing with the Zero Curve The calculation with the traditional formulae does not give you any exact fair price but only a result which is true, assuming a flat yield curve and a re-investment of the coupon payments at the same rate (unrealistic scenario). Theoretically, one should use a calculation based on the so-called zero curve that is already state-of-the-art in the swap market, but has not yet been fully accepted in the bond market. With the zero bond method, interest payments during the term of the bond are eliminated so FINANCE TRAINER International Fixed Income / Page 22 of 48

23 you do not have to make any assumptions concerning their re-investment. When calculating the bond price with the zero-curve, each cash-flow is treated like a cash-flow from a zero bond. This way, every single cash-flow is discounted with the particular zero rate for the particular period. Thus no flat yield curve is assumed anymore but the actual interest rates for the different periods are used. The calculation of zero rates - bootstrapping Where do the zero-rates come from? Talking about interest rates, they are normally the rates for coupon instruments (interest rate swaps, government bonds, etc.). One also calls them yields, par yields or yield to maturity. It is important to understand that the zero curve is derived from one of these other curves, i.e. the zero rates can be calculated from the interest rates of coupon instruments. This transaction is called bootstrapping. The zero curve concept is therefore a mathematical method which gives us an exact result and does not make any assumptions regarding the re-investment of the interest payments during the term. How can these zero rates be calculated? First, the interest payments during the term are eliminated, or more precisely, they are hedged at the actual market rates. After this there are only two cash-flows left: the underlying amount in the beginning and the back payment at maturity. Thus we have a present value and a future value for which an interest rate can easily be computed. Unfortunately, the effective calculation is a bit more complex and is therefore a typical case for a spread sheet analysis. However, it is important to understand the underlying principle which should be demonstrated in the following example. From this yield curve the zero rates have to be derived. Period yield year 4.00% 2 years 4.50% 3 years 5.00% 4 years 5.50% 5 years 6.00% FINANCE TRAINER International Fixed Income / Page 23 of 48

24 -year zero rate: With a term of year, there is no interest payment during the term (annual payments assumed). Therefore, by definition the one-year zero rate (Z) is the same as the one-year yield; i.e. here 4.00%. 2-year zero rate: At first, we look at the cash-flow which is produced when investing in a 2-year bond at a market rate of 4.50%: CF0 CF CF2 Cash-flow (bond) At time 0 we pay the price for the bond, the cash-flow is 00. After one year we receive a coupon payment over 4.50 and after 2 years another coupon payment plus 00 redemption. In the next step we eliminate the coupon payment in one year over 4.50 (CF). The question is: how much can we borrow today in order to pay back exactly 4.50 in one year? Answer: 4.50/(+0.04) = After the hedging we have the following new cash-flow: CF0 CF CF2 Cash-flow (bond) ,50 +04,50 Cash-flow (hedge) + 4,327-4,50 [4,50 / (+0,04)] Total: -95, ,50 The zero rate can now be calculated from the present value (CF0) and the future value (CF2): CFn Z n 2 n CF % Zn = zero rate at time n n = ongoing year CF n = cash-flow at time n FINANCE TRAINER International Fixed Income / Page 24 of 48

25 Thus for the 2 nd year we have a zero rate at 4.53%. This would be the rate for the period of 2 years after hedging all cash-flows during the term. At the moment we have the following rates: Period Yield Zero rate year 4.00% % 2 years 4.50% 4.53% 3 years 5.00%??? 4 years 5.50% 5 years 6.00% What is the zero rate for 3 years? 3-year zero rate: We use the same principle and, first, look at the cash-flows of a 3-year bond with the current market yield (5%): CF0 CF CF2 CF3 Cash-flow (bond) Now we have to eliminate resp. hedge 2 cash-flows. Again the question is: how much can we borrow today in order to pay back exactly 5.00 after year and year 2? Therefore we discount the cash-flows with the particular zero rates: CF0 CF CF2 CF3 Cash-flow (bond) Cash-flow (hedge CF) [5 / (+0.04)] Cash-flow (hedge CF2) [5 / ( )2] Total CF CF Z % The 3-year zero rate is 5.034%. FINANCE TRAINER International Fixed Income / Page 25 of 48

26 4-year zero rate: CF0 CF CF2 CF3 CF4 CF (bond 4 years) CF (hedge CF) [5.5 / (+0.04)] CF (hedge CF2) [5.5 / ( )2] CF (hedge CF3) [5.5 / ( )3] Total CF Z CF % The 4-year zero rate is 5.579%. 5-year zero rate: CF0 CF CF2 CF3 CF4 CF5 CF (bond 5 years) CF (hedge CF) [6.0 / (+0.04)] CF (hedge CF2) [6.0 / ( )2] CF (hedge CF3) [6.0 / ( )3] CF (hedge CF4) [6.0 / ( )4] Total CF CF Z % The 5-year zero rate is 6.289%. Result: zero rates derived from the yield curve Period yield Zero-rate year 4.00% % 2 years 4.50% 4.53% 3 years 5.00% 5.034% 4 years 5.50% 5.579% 5 years 6.00% 6.289% FINANCE TRAINER International Fixed Income / Page 26 of 48

27 General formula for zero-calculation Z N N N n r N rn Z n n r N N Z n n = bond yield for period n years in decimals = total period in years = zero rate at time n in decimals = ongoing year Note: Only recursive calculation is possible: to get the 5-year zero-rate you first have to calculate the zeros for the years through 4. Therefore the calculation without a spread sheet analysis is very complex. Summary The zero curve is derived from the yield curve, i.e. from interest rates for couponinstruments (e.g. IRS, government bonds, etc.) The zero curve includes the costs/returns for hedging the future cash-flows at actual market rates. The zero curve does not make any assumptions regarding the form of the curve and the re-investment of coupon payments during the bond term. FINANCE TRAINER International Fixed Income / Page 27 of 48

28 Comparison zero rates vs. yields If you compare the yields to the zero rates in the above example you can see that the zero rates are higher. This is the case whenever you have a normal yield curve. With an inverted yield curve zero rates are lower than the yields, with a flat curve they are about the same. steep inverted flat yield zero yield zero yield zero year 4.00% 4.00% 8.00% 8.00% 6.00% 6.00% 2 years 4.50% 4.5% 7.50% 7.48% 6.00% 6.00% 3 years 5.00% 5.03% 7.00% 6.95% 6.00% 6.00% 4 years 5.50% 5.57% 6.50% 6.42% 6.00% 6.00% 5 years 6.00% 6.3% 6.00% 5.88% 6.00% 6.00% Price of a EUR government bond with the following specifications coupon: 7.00% fixed time to maturity: 5 years interest payment: annual redemption: bullet current yield: 6.00% Assumption: The latest interest payment has just been made. Year Interest rate Zero rate 4.00% 4.00% % 4.5% % 5.03% % 5.57% % 6.3% FINANCE TRAINER International Fixed Income / Page 28 of 48

29 Year Cash- Zero Discounting Present value flow rate (2x3) % % % % % Sum: Taking into account the yield curve for the price calculation, you get a theoretical price difference for this example of approx. 7 basis points (cp. other example, to ). FINANCE TRAINER International Fixed Income / Page 29 of 48

30 4. Calculation of Price Sensitivities/ The Concept of Duration For dealers of short-term positions in bonds (trading book) it is useful to estimate how the bond's price changes when the interest rates change. For this purpose several concepts have been developed. 4. The Simple Duration (Macaulay Duration) The concept of the simple duration was developed by Frederick R. Macaulay in 938. If rates change you have two impacts on a bond position. On the one hand the bond price will change, on the other hand the coupon payments can be re-invested at a different rate. These two effects are always directly opposed. If, for example, rates rise, on the one hand the bond price falls, on the other hand the future coupon payments can be re-invested at a higher interest rate level and vice versa. The Macaulay duration describes the time where these two effects compensate each other. In other words, it is a time dimension in years which indicates the period which is needed for a fixed-rate interest bond in order to exactly compensate the price and compound interest effects which result from a change in rates. This model assumes, however, a singular, parallel yield curve shift. D Macaulay N n N n n x CF r CF r n n n n D Macaulay n N C Fn r = Macaulay Duration = ongoing year = total number of years = cash-flow (at a nominal of 00) at time n = current yield in decimals FINANCE TRAINER International Fixed Income / Page 30 of 48

31 Bond: 3 years Coupon: 5% Market yield: 5% Price: The Macaulay duration is 2.86 years. That means that after 2.86 years the price loss due to a rate rise is compensated by higher compound interest for the coupon payments and vice versa. The following table shows the results at different times with and without interest rate changes: rates at 5% rates at 6% year CF price coupon + com- price + price coupon + com- price + pound interest interest pound interest interest After 2.86 years the price loss of the bond is compensated by higher compound interest. For both scenarios the result is This conclusion, however, is only valid assuming that during the term there are no further interest rate changes. FINANCE TRAINER International Fixed Income / Page 3 of 48

32 4.2 Modified Duration When speaking of duration one usually does not mean the Macaulay duration but the modified duration. It is said to be the best known technique for the determination of price sensitivities. It shows the leverage how the bond price reacts to a market rate movement. Modified Duration MD N n N ncf n CF n n r r n n r MD n N CF n r = modified duration = ongoing year = total number of years = cash-flow (at a nominal of 00) at time n = current yield in decimals If you compare this formula one can realise that this formula is a modification of the Macaulay Duration: MD D Macaulay x r On closer examination of the calculation you can look at the duration as the term-weighted present value of all cash-flows or, vice versa, as the PV-weighted average binding period. FINANCE TRAINER International Fixed Income / Page 32 of 48

33 Calculate the Modified Duration for a 5-year bullet bond. The coupon is 7%, the current market yield is 6% Year Cash- Discounting Present value Present value flow of weighted of cash flow cash-flow (2x3) (x2x3) Sum: MD The duration for this 5-year bond is 4.5. By means of the duration one can now estimate the price change of the bond when market rates change. Assuming the current bond price is If this bond has a MD of 4.5, this means that if rates change by %, the bond price will change by 4.5%, i.e x 4.5% = FINANCE TRAINER International Fixed Income / Page 33 of 48

34 In combination with our rules concerning price changes in the wake of interest rate changes, we receive the following result: if interest rates rise by %, the price of the bond falls by 4.32 percentage points. if interest rates falls by %, the price of the bond rises by 4.32 percentage points. CP MD P CY CP = change of the bond price MD = modified duration P = price of the bond (dirty price) CY = change of the market yield in decimals (% = 0.0) Specifications of a fixed-rate interest bond: current price modified duration 3.50 current yield 6.00% Estimate what will be the bond price if the market yield falls to 5.75% or rises to 6.75%? 5.75%: CP = (+)0.9 If rates fall to 5.75% the price of the bond will rise to 04.9 ( ) %: CP = ( )2.73 If rates rise to 6.75% the price of the bond will fall to 0.27 ( ). FINANCE TRAINER International Fixed Income / Page 34 of 48

35 4.2. Influencing Factors on the Modified Duration The modified duration depends on: Term of the bond: the longer the term of the bond, the stronger the impact of rate changes on the bond price. Coupon: a high coupon means that parts of the bond are paid back earlier through coupon payments. Thus the modified duration is lower, i.e. a shorter average minimum lockup period. Therefore the price sensitivity for high coupon bonds is lower than for zero coupon bonds. Market yield: the cash-flows are discounted with the market yield. The higher the market yield, the lower the present value and thus the MD. Therefore the impact of a rate change by % is weaker on a high interest rate level than on a low one. To sum up, we can state the following rules regarding price sensitivity of fixed-rate interest bonds: High duration at (= high price sensitivity) long term low coupon low market yield Low duration at (= low price sensitivity) short term high coupon high market yield The following table shows the modified duration for a 0-year bond with different coupons and for different market yields: Yield Coupon 0% 5% 0% 0% % % FINANCE TRAINER International Fixed Income / Page 35 of 48

36 All calculations with the modified duration assume a flat yield curve like also for the bond price calculation with the traditional formulae. Also for the determination of the price sensitivity the effect of the yield curve can be considered. Then you call it effective duration. The impact of the yield curve on the duration, however, is not that strong, therefore here we set aside the calculation of the effective duration The Estimation Failure of the Duration the Convexity The concept of the duration assumes a linear connection between the yield and the price change, i.e. you are assuming that the change of the bond price is the same for rates rising by % and rates falling by %. In practice, when rates change, the bond price does not run on a straight line but on a curve. Dirty Price 00 actual price estimated price (MD) Yield When applying the duration you therefore have an estimation failure. The convexity is the measure for this estimation failure resp. the non-linear (contorted) price-yield-curve. The larger the interest rate change, the more the actual price will diverge from the estimated price. Thus the duration gives satisfactory results only for relatively small rate changes. As the yield itself is an input factor for the calculation of the duration, also the modified duration changes when the interest rate level moves. If the interest rate level rises, the duration falls and therefore the actual bond price fall will be smaller than originally estimated. On the contrary, the duration rises when yields fall and therefore the actual bond price rise will be higher than originally estimated. FINANCE TRAINER International Fixed Income / Page 36 of 48

37 Bond: 0 years Coupon: 6% Yield: 5% Price: Modified duration: 7.52% Compare the actual rate to the estimated rate when rates change by +/- percentage point. When rates change by +/- percentage point the estimated price change is 8.0 (07.72 x 7.52%). Yield 4% 5% 6%. Estimated rate Calculated rate Difference The example shows that for the owner of a bond the convexity effects are always positive, i.e. when rates fall the profit is higher and when rates rise the loss is lower than expected. Modified Duration for a bond with partial redemption before maturity time to maturity: 5 years coupon: 7.00% current yield: 6.00% interest payment: annual redemption: 50% after 3 years, rest: end of term FINANCE TRAINER International Fixed Income / Page 37 of 48

38 Year Cash- Discounting Present value of Present value of cash flow weighted flow (2x3) cash-flow (x2x3) Sum: MD FINANCE TRAINER International Fixed Income / Page 38 of 48

39 5. Coupon, Yield to Maturity, Par Yield and Zero Coupon Knowing the interest rate does not tell you all about the real return on this investment. As we already know we need several additional information like the day counting, interest payment frequency, if interest is calculated on the start amount or future value (as for a discount rate), etc.. Even if we are assuming that all these factors are unchanged we have to define more precisely what we mean when speaking about interest rates. Coupon (Nominal Interest Rate) Every interest rate defined in the conditions of a bond is called coupon. It is the basis for the yearly interest payments. These are always calculated on the basis of the nominal amount (00), and are therefore independent from the actual bond price. One is also speaking of the nominal interest rate. The coupon only determines the bond s cash-flows. But it does not give any exact information regarding the return on the invested amount as the bond is usually purchase at a price above or below 00. Yield to maturity The most common concept in order to determine the effective return of a bond is the yield to maturity (YTM) resp. effective interest rate. If you want to determine the YTM you have to reverse the bond price calculation: while for a price calculation you are looking for the bond price at a given market yield, for the YTM calculation the yield as result of a given bond price is determined. The YTM can also be called internal rate of return (IRR) of a cash-flow. If we refer to the YTM as y we can formally describe the dirty price of a bond (= present value of all cash-flows) with the following equation: P dirty CF y CF 2... CF N 2 y y N P dirty y CF n n N = dirty price (= present value of all cash-flows) = yield to maturity (= IRR) = cash-flow at time n = ongoing year = total number of years FINANCE TRAINER International Fixed Income / Page 39 of 48

40 If now the YTM for a given bond has to be calculated, both the dirty price and the cash-flows are known. Therefore the equation has only be solved for y. This is done by means of approximation procedures and can be done with programmable calculators or Excel quite easily. A 5-year bond with a coupon of 5% has a price of What is the yield to maturity? Cash-flows: CF0 CF CF2 CF3 CF4 CF The question is: with which interest rate do the cash-flows CF through CF5 have to be discounted in order to receive the present value CF0? Therefore we set up the following equation: y y 2 y 5 This equation can only be solved by approximation (trial and error). In Excel we have used the function goal seek under Extras and get the solution y = 9.00%. If we buy this bond we have a yield of 9.00%. You can verify this result by calculating the bond price for a given market yield of 9.00% with the traditional bond price formula: year YTM CF PV 9% % % % % Sum: FINANCE TRAINER International Fixed Income / Page 40 of 48

41 When using the traditional bond price formulae the calculated values for price and yield show consistent results. For each way the same equation is solved in opposite directions (one time for the price, the other time for the yield). Problems with the YTM As the YTM applies the same method as the traditional bond price formulae, you have the same problem here: a flat yield curve is assumed. Let us go back to the bond price calculation. We have seen that the fair bond price is the present value of all future cash-flows. For the precise calculation of the present values we have discounted every single cash-flow with the particular zero rate. Thus we have considered the yield curve for the precise calculation what led to a number of different interest rates. We have realised that this result had diverged from the price determined by the traditional formula. In order to determine the return on an investment with a consistent interest rate the YTM is used. It is the rate with which all cash-flows have to be discounted to get the actual bond price. If the actual price equals the price determined by the zero curve, the YTM is a kind of nominal-weighted average of the yield curve. As most averages also the YTM has to be regarded critically. The following example is supposed to show the problem, We would like to apply the yield to maturity in order to analyse two investment alternatives. Assuming you have the choice between two 5-year US Treasury bondsbund notes. Bond Price 5% % To find out which of the bonds is more attractive you have to calculate the YTM. For this purpose you have to take the cash-flows and determine the internal rate of return: Bond CF0 CF CF2 CF3 CF4 CF5 YTM 5% % 0% % FINANCE TRAINER International Fixed Income / Page 4 of 48

42 Obviously the 5% bond is more attractive as it has a higher YTM. Therefore it is self-evident that this bond is relatively regarded cheaper than the 0% bond. We want to verify this assumption and calculate the fair bond price with the current zero curve. 5% bond 0% bond year zero CF PV CF PV 5 % % % % % Sum: Comparison: Market prices: Against our first assumption both prices correspond exactly to the fair prices determined by the zero curve. So both bonds are priced fairly, i.e. equally attractive. Obviously the YTM does not tell the whole truth. What can be the reasons? As we have learned for the calculation of the YTM only one interest rate is used for discounting the cash-flows. So a flat yield curve is assumed and it is further assumed that all interest payments can be re-invested at the YTM. In practice you usually have different rates for different periods. Let us have a look at bonds with different coupons: With a high coupon a (partial) redemption on the invested capital is done already during the term by means of higher coupon payments. With a steep interest rate curve these early redemptions would have to be discounted only with a lower short-term interest rate. When calculating the YTM, however, all cash-flows are discounted with an average (higher) rate what means that the result in this case looks worse than it actually is. FINANCE TRAINER International Fixed Income / Page 42 of 48

43 Summary The YTM only gives a consistent result when calculating bond prices with the traditional formula. Due to the simplification (flat interest rate curve and re-investment) bonds with different coupons fairly calculated with the zero curve have different YTM. The YTM is a nominal-weighted average interest rate. Like most averages the YTM does not give all information. An evaluation of an investment only on the basis of the YTM can lead to wrong results. Par yield We often apply yields, e.g. when we examine a yield curve or calculate a bond price with the current market yield. The chapter before, however, has shown that the concept of yield calculation can be a problem because bonds with different coupons have different yields. To solve this problem you usually use the so-called par yield when speaking of e.g. the market yield. The par yield is the yield of a bond which is quoted at (or close) par, i.e. the price is around 00. This way the problem of a wrong YTM calculation for bonds with prices 00 can be avoided. Zero coupon rate The zero rates have been discussed in detail in the chapter pricing with the zero curve. Summary: Zero rates are derived from the yield curve. The advantage of the zero rates is that there is no assumption made regarding the form of the yield curve or the re-investment of interest payments during the term. Zero rates take into account the costs/returns for the hedging of the interest payments during the term. FINANCE TRAINER International Fixed Income / Page 43 of 48