BOND ANALYSIS AND VALUATION

Transcription

1 BOND ANALYSIS AND VALUATION CEFA 003/004 LECTURE NOTES Mats Hansson Svenska handelshögskolan Institutionen för finansiell ekonomi och ekonomisk statistik

2 TU1.UT TUFIXED TU.UT TUBOND TU3.UT TUDAY i Contents INCOME SECURITIES - AN INTRODUCTIONUT 1 TU1.1.UT TUWhat s so special about fixed income securities?ut 1 TU1..UT TUThe risks of investing in debt and why everybody always talks about the yieldut TU1.3.UT TUThe money and bond marketsut 3 TU1.4.UT TUMarket sizeut 4 TU1.5.UT TUFactors affecting the level of the nominal returnut 5 TU1.5.1.UT TUThe real returnut 5 TU1.5..UT TUThe inflation rateut 6 TU1.5.3.UT TUThe risk premiumut 7 AND INTEREST RATE MATHEMATICSUT 8 TU.1.UT TUThe frequency of compoundingut 8 TU.1.1.UT TUEffective money market yieldsut 10 TU..UT TUBuilding blocks: zeros and forwardsut 11 TU..1.UT TUZero-coupon bondsut 11 TU...UT TUForward ratesut 13 TU.3.UT TUZero-coupon pricing of coupon bondsut 15 TU.3.1.UT TUThe coupon rateut 15 TU.3..UT TUThe present value of a coupon bondut 15 TU.3.3.UT TUYield to maturity for a coupon bondut 17 TU.3.4.UT TUThe par yieldut 0 TU.4.UT TUThe yield: common misconceptionsut 1 TU.4.1.UT TUThe yield is not the returnut 1 TU.4..UT TUYields are not additiveut TU.5.UT TUFrom coupon bonds to zeros: bootstrappingut 3 TU.6.UT TUSpot and forward rates with semi-annual compoundingut 5 COUNTS AND ACCRUED INTERESTUT 7 TU3.1.UT TUDay count basisut 7 TU3..UT TUThe money marketut 8 TU3.3.UT TUZero-coupon bonds: annual compoundingut 9 TU3.4.UT TUZero-coupon bonds: semi-annual compoundingut 30 TU3.5.UT TUCoupon bondsut 30 TU3.5.1.UT TUDirty prices and clean pricesut 31 TU3.5..UT TUBehavior of dirty and clean prices over time: convergence towards parut 3

3 TU4.UT TUMEASURING TU5.UT TU6.UT TUAPPLICATIONS TU7.UT TUPRICING ii INTEREST RATE RISK: DURATION AND CONVEXITYUT 34 TU4.1.UT TUThe yield-price relationship for bondsut 34 TU4..UT TUDurationUT 35 TU4..1.UT TUMacaulay durationut 36 TU4.3.UT TUModified duration and PVBPUT 37 TU4.3.1.UT TUThe duration of a bond through timeut 38 TU4.3..UT TUKey rate durationut 39 TU4.4.UT TUConvexityUT 40 TU4.4.1.UT TUDuration matching and the value of convexityut 4 TU4.5.UT TUBond portfolio duration and convexityut 44 TU4.6.UT TUButterfly trades: A critical assessment of yield, convexity and durationut 45 TU4.6.1.UT TUWeighting a butterflyut 45 TU4.6..UT TUA critical assessment of yield, convexity, and durationut 47 TUAPPLICATIONS OF BOND MATHEMATICS I: FRA:S AND BOND FUTURESUT 49 TU5.1.UT TUForward Rate AgreementsUT 49 TU5..UT TUBond futuresut 51 TU5..1.UT TUFutures pricing: The general approachut 51 TU5...UT TURepo transactions in the bond cash and futures marketsut 5 TU5..3.UT TUCoupon paymentsut 53 TU5..4.UT TUNotional bonds and delivery optionsut 54 TU5..5.UT TUFutures pricing using quoted prices and accrued interestut 55 OF BOND MATHEMATICS II: SWAP CONTRACTSUT 58 TU6.1.UT TUInterest rate swapsut 58 TU6.1.1.UT TUThe swap rateut 59 TU6.1..UT TUThe swap rate and FRA-ratesUT 60 TU6.1.3.UT TUInterest rate swap valuationut 61 TU6..UT TUCurrency swapsut 6 TU6..1.UT TUCurrency swap rates and currency forward ratesut 63 TU6...UT TUCurrency swap valuationut 64 CREDIT RISKUT 65 TU7.1.UT TUCredit ratingsut 65 TU7..UT TUThe traditional approach to pricing credit riskut 66 TU7.3.UT TUsing option theory to price credit riskut 67 TU7.4.UT TUDefault probabilities, rating transitions, recovery rates and how to use them to estimate bond returnsut 69 TU7.5.UT TUSelected empirical results on spreadsut 7

4 1 1. Fixed income securities - An introduction Debt instruments or fixed income securities are financial instruments that commit the issuer to a series of fixed payments (for example a series of coupons and principal). Examples are treasury bonds and bills, corporate bonds and loans, certificates of deposit, and interest rate and currency swaps. The issuer of these securities promise a certain cash flow at certain specified times in the future, hence the definition fixed income. Also, fixed income securities typically have a finite maturity What s so special about fixed income securities? A fixed income security follows the same basic principles of valuation as for e.g. stocks: future cash flows are discounted to present time. Fixed income securities, however, have a number of special features that make a separate treatment of these instruments warranted: 1. Fixed cash flows. Most bonds pay fixed interest (altough floating rate notes are also common), which is also paid on specified dates. Thus, cash flows are known both with respect to size and maturity, except in the case of default.. Finite and known maturity. Except for some rare cases (perpetuities), fixed income securities have a limited maturity, which is known in advance. Item 1) and ) makes it possible to construct special risk measures for bonds, like duration and convexity. 3. Only downside with respect to promised cash flow. The cash flow received from a straight bond can never exceed its promised coupons and face value. 4. A wide variety of instruments are available. A wide range of maturities (1 day to 30 years or more), cash flow structures (zero-coupon bonds, coupon bonds, annuities etc.), issuers (corporations, governments, municipalities etc.), and derivatives (swaps, bond and money market futures, forwards, options etc.). Since a bond or a loan is a legal contract between borrower and lender, the payoff and risk structure (payment schedule, covenants etc.) of the bond/loan is determined in this contract, and hence there is no limit to where product development can go in the debt markets. 5. Lower risks, lower returns. The financial risk associated with fixed income is lower than with equities. This means lower expected and, on average, lower realised returns. This in turn calls for more precision in the pricing process, since if returns are low, every basis point counts. Since upside potential is low, paying too much (mispricing) usually means that the investor s return is ruined for good, while on the stock markets one can always hope for a more substantial increase in value. 6. The term structure of interest rates is used for valuation. When valuing stocks, a single discount rate is used to discount all cash flows. In doing this we assume that all cash flows are equally risky and that the time value of money is the same for all maturities. The wide diversity of instruments available on fixed income markets makes a more precise valuation of debt instruments possible. In the ideal case, we can find information on the interest rates for many different maturities, making it possible to value each cash flow of a bond using a separate interest rate that reflects the risk of that maturity. 7. Arbitrage. The variety of instruments with very low credit risk (interbank market) or in practice no credit risk (Treasury markets) makes arbitrage and arbitrage pricing possible and links prices of instruments to each other.

5 8. Volatility is a function of time. As we will later see, the volatility of a fixed income security depends on 1) changes in the interest rate level, and ) the duration of the bond. Thus, even if interest rate volatility is constant, the volatility of the bond will decrease with time. 9. Changes in the interest rate level is the most important source of risk. The nature of debt instruments implies that the valuation process is to a large extent concerned with the time value of money. Time value of money is closely related to the level of interest rates, and hence debt instruments could also be labeled interest sensitive assets. 10. Institutional details. The practice of expressing prices as interest rates (yields), or clean prices, different day count conventions and compounding frequencies etc. 1.. The risks of investing in debt and why everybody always talks about the yield the most common type of interest rate payment is a fixed coupon since the cash flows are fixed, there is less upside than in stocks, and all the changes in price will come from the change in the discount rate, or the yield hence, much of the fixed income investment analysis is centered around the yield and the yield spread SPREAD (Risk premium) EXCESS RETURN Transaction cost Covenants YIELD BENCHMARK YIELD (Default risk free yield) Credit risk Seniority Real interest rate Interest rate risk Inflation risk many of the risks contributing to total yield are difficult to measure and price excess return is the expected excess return from investing in corporates over Treasuries after taking into account all the risks incorporated in the spread that can materialise excess return must be must be positive in the long run, otherwise a risk-averse investor will not invest in corporates

6 3 up to date there exists no pricing model (a la CAPM) flexible enough to incorporate all the terms and covenants in the debt contract affecting return and risk, yet general enough to lend itself to practical use 1.3. The money and bond markets Fixed income securities can be classified in as many sectors as one likes, but the most common categories are by maturity: 1. Short term securities. Maturity up to 1 year.. Long term securities. Maturity of more than 1 year. This classification comes from the similarity of the pricng technique within one category, short term debt usually pays no interest (zero-coupon bonds, discount bonds), so there is only one cash flow at maturity, and simple interest is used when discounting. Long term bonds usually pay interest, and compound interest is used for discounting. One could also classify securities into: 1. Default risk free securities. These securities are issued by governments (Treasuries) of developed countries, and are considered to be in practice free of default risk.. Securities with default risk. Corporate bonds and for example emerging market sovereign debt are not free of default risk. securities with a maturity of maximum 1 year are frequently referred to as money market instruments typical money market instruments include: 1. short term deposits (nontradable). certificates of deposit (interbank market) 3. commercial paper (corporate sector) 4. Treasury bills (government sector) typical bond market instruments include: 1. Government bonds (also called Treasury or Sovereign bonds). Corporate bonds 3. Mortgage bonds (securitised mortgage debt) The government sector is usually more liquid than the corporate sector due to larger markets, and in many countries the bulk of corporate borrowing is still mainly routed through bank loans, altough this has been changing in Europe since the Euro. Loans of large corporations are usually syndicated loans, a group of banks divides the loan between themselves for diversification. Most bonds and loans are "bullets", where interest is paid annually or semiannually, and the principal is paid back at maturity. In many countries, a substantial part of the government bonds are so called benchmark bonds (or serial bonds), for example the Finnish benchmarks are:

7 4 Bond Maturity CPN Issue price Current price Yield% Amount EUR (m) % Serial bond 003/I Serial bond 001/I Serial bond /II Serial bond /II 9 Serial bond 000/I Serial bond /II Market size many bonds are listed at an exchange, but trading is (so far) mostly OTC global markets by country of issuer: By sector: 1) Government and agency, ) Corporate and financial institutions 0000 USD billions USA Germany France Japan Finland Government Corporate Total risk management and the desire to explore cost-effective borrowing through swaps has lead to enromous global fixed income derivatives markets:

8 5 By sector: 1) Foreign exchange, ) Interest rate, 3) Equity linked, and market (OTC or Exchanges) USD billions OTC FX OTC Interest OTC Equity Ex. FX Ex. Interest Ex. Equity Forwards Swaps Options Total Factors affecting the level of the nominal return Why are interest rates not equal for all time-periods, and why is the term structure of interest rates usually (but not always) upward sloping? To answer these questions we need to have a look at what factors affect required returns, and why these factors need not be equally large for all time periods. The return for any asset can be decomposed into three factors: 1. the required real return. expected inflation over the investment horizon 3. a risk premium the real return and expected inflation affect the returns on all assets in an economy the magnitude of the risk premium is asset specific The real return first suppose there is no inflation and the investment is risk-free: the return consists solely of the investors perception of time value of money, or real return thus, the real return says how much the investor wants his purchasing power to increase when investing investing is delaying consumption to the future, for doing this the investor requires a compensation if this compensation is equally large for each time period (e.g. each year), the yield curve will be flat (compensation proportional to time) in the simplest setting, the level of the real return depends on money supply and demand:

9 6 1. the supply depends on investors willingness to postpone consumption. the demand depends on opportunities for productive investment for example, suppose that investment opportunities improve and firms are willing to invest more at any interest rate level then, interest rates must rise to induce investors to save more investors require a higher compensation to postpone a larger amount of their consumption The inflation rate if there is inflation in the economy, investors will require a premium over the real rate equal to the expected inflation rate for the investment horizon if inflation is constant, this will result in a still flat, but higher yield curve thus, still ignoring risk premiums, the required nominal interest rate (rbnb) on a riskless security depends on: 1) the real required return (rbrb). and ) the expected inflation [E(i)] is approximately: r r + E() i n r UExample:U An investor is investing for 1 year, and wants the purchasing power of his money to increase with 3% over the next year. This is his required real return, a compensation for postponing his consumption 1 year. Also, the investor expects the inflation to be % from today to 1-year ahead. Thus, his nominal required return is r n r r + E( i) % inflation and real returns need not be constant over time if investors expect inflation and/or real rates to increase, longer rates should be higher if investors expect inflation and/or real rates to decline, longer rates should be lower but: even if the level of inflation is not expected to rise, the level of future inflation is still uncertain, investors may require a premium for longer rates due to inflation risk your real return is uncertain require risk premium for risky real return if, in addition, the level of real returns is uncertain further risk of real return risk thus we may have premiums for both inflation risk, and real return risk investors require a premium for investing for long maturities implies an upward sloping yield curve (despite small declines inflation/real rates) downward sloping curves only if future inflation/real rates substantially lower flat curves occur only if future yields expected to decline

10 The risk premium a security that has no default risk is considered riskless with respect to default risk for example government securities are considered free of default risk (or at least have the lowest possible level of default risk) a risk premium must be added on top of required real returns and expected inflation for issuers that have default risk but even in the treasury markets we typically observe that interest rates (for example yields) increase with maturity interest rate risk (typically measured by duration) increases with maturity

11 8. Bond and interest rate mathematics in this chapter we assume there is no credit risk, such that money can be moved back and forth in time without caring about the riskiness of future payments nobody has ever claimed that bond calculations are fun or interesting, but given the size of the market and the amount of money potentially lost because one simply didn t know, one cant t ignore the subject let s have a look at a some numbers we can use to describe a coupon bond: Face value: Present value: Maturity: 3.00 years Coupon rate: 10.00% (annual) Yield to maturity: % Par yield: % Zero-coupon rate (3 yrs.): 4.55% We can make some observations: first, we note that the PV is higher than the face value. Second, we note that the coupon rate (10%) is higher than the yield to maturity (what is a yield to maturity anyway?). Third, we note that the zero-coupon rate (what s that again?) is higher than the yield. Fourth, we have something called par yield (what?) which is different from all previous interest rates (coupon, zero, yield). Fifth, we note that we are confronted with four different interest rates: the coupon rate, the yield, par yield, and the zero-coupon rate. The final blow is that none of these is the return of the bond, despite its fixed income features. All these relations are not an accident: the price and face value are related through coupon rates, yields, and zero yields. This example should make it clear that the expression interest rate can mean a variety of things. In the following chapter(s) we will explore these concepts in detail..1. The frequency of compounding stock returns and standard deviations are usually expressed as percent per year a stock return in the US has the same interpretation as a stock return in Finland interest rates, however, come in many varieties and are usually not directly comparable each market and currency has its own agreed upon rules of how to convert a discount rate into a price or the other way around: 1. when should one use simple interest, and when compound interest?. if compound interest is used, what is the frequency of compounding? 3. how should one count days to arrive at a fraction of a year (one unit of time in finance is 1 year)? rules about how to discount and to define fractions of a year are important in fixed income markets because prices can be given both as discount factors (yields) or prices these are not as important on stock markets, since there is anyway great uncertainty about the timing and size of cash flows, and since prices are never given as yields

12 9 simple interest is used when no interest is paid before maturity [ ] FV PV * 1 + ( r * t) and PV FV [ 1+ ( r * t) ] FV PV r t future value present value interest rate expressed as decimals on a per annum basis maturity in years compound interest is used when interest is paid and added to the principal FV PV *(1 + r) and t PV FV t ( 1+ r) the usual rule is that short-term rates (money market) are treated as simple interest rates, while long-term rates are treated as compound interest rates bonds pay interest before maturity, and the opportunity cost can be seen as a longterm deposit that pays interest m times a year and is added to the capital this is standard when analysing returns on any market, for example long-term returns on stock markets always assume dividends are reinvested ( compound interest) all interest rates are expressed as annual rates (unless stated otherwise) UExample:U A bank offers a 3-month deposit rate of 3.50%. If you deposit EUR today, how much cash do you have after 3 months? (For simplicity, assume 3 months is 0.5 years) [ ] [ ] FV PV * 1+ ( r * t) 1000EUR * 1+ ( * 0. 5) EUR note that the actual return earned over 3 months is only 0.875% UExample:U A bank offers a -year deposit rate of 4.5%. The deposit pays interest annually. If you deposit EUR today, how much cash do you have after years? t [ ] [ ] FV PV * 1+ r 1000EUR * note that the actual return earned over a -year period is 8.68% note that the time t, is seldom an integer value (this happens once a year), hence the need to convert a number of days to a fraction of a year by some defined rules converting all rates to annual rates makes comparison easier converting actual returns over N years to a one-year return (using the previous example):

13 10 r 1/ N [( 1+ ) ] 1 ( ) r Actual [ ] 1/ suppose that the compounding frequency (m) is not 1 year but 0.5 years instead (interest is paid quarterly): FV t* m 4 * r PV + EUR EUR m +. * * an interest of 4.5/ % is paid each 0.5 years and compounded 8 times the higher frequency of compounding increases the return to 8.8% if the annual rate remains unchanged since interest can be added to principal more frequently in the limit: continuous compounding when the interval of frequency becomes very small: r* t * FV PV * e 1000EUR * e EUR rates based on continuous compounding are mainly used in theoretical literature, never in practice.1.1. Effective money market yields money market rates are simple yields and cannot be directly compared due to differences in compounding frequency 1 month deposit can be rolled over 1 times during a year, but a month deposit only 6 times conversion to annual effective yields is required: Yield EFF r 1 + SIMPLE (360 / ) t 360 / t 1 UExample:U Both the 30-day and the 60-day simple annual interest rates are 4%. What are the effective annual yields? The 30-day effective annual yield is Yield EFF ( 360 / 30) 360/ 30 [[ ] ] %

14 11 The 60-day effective annual yield is Yield EFF ( 360 / 60) 360/ 60 [[ ] ] % if the simple rates are at level, the 30-day effective yield is higher because it can be rolled over more frequently note also that the roll-over of the 30-day investment is risky, since the second 30-day rate was unknown at the beginning of the 60-day investment period.. Building blocks: zeros and forwards when analyzing fixed income securities, sooner or later one will be confronted with zero coupon rates (also called spot rates) a zero coupon rate is the discount rate (yield) for a zero-coupon bond, that is for a bond paying a single cash flow received at time t zero rates are the building blocks of all fixed income analysis, and as we will see later, using the yield of a coupon bond for valuation can lead to severe mispricing since the zero coupon rate is the only unambiguous interest rate for a particular maturity, everything else needed can be calculated using zero rates: discount factors, coupon bond prices and yields, par yields, forwards, swap rates etc...1. Zero-coupon bonds zero-coupon bonds exist almost exclusively on treasury markets, for example US Treasury or German Bund STRIPS the discounted value of the face value (negative cash flow) is invested and the face value is received at maturity (positive cash flow) the time period (t) may be anything from one day to several years a separate spot rate is required for each period in time (t) to value a cash flow at time t, and hence, we have a term structure of spot rates consider for example the following sequence of spot rates: Maturity Spot rate Years % then, we can graph the term structure of these spot rates:

15 Spot rate (%) Maturity (Years) all interest rates are expressed on an annual basis (p.a.) to be more easily compared because most securities with maturity of over 1 year pay interest, the convention is to express all interest rates for maturities over 1 year as annually or semi-annually compounded rates thus, even if a zero-coupon bond pays no interest, we like to compare it with interest paying securities, and do this by expressing them as annually compounded rates UExample:U The 3-year zero-coupon rate is 4.55%. This does UnotU mean that if you invest 100 today in a 3-year zero-coupon security, you get back after 3-years. Since the spot rate is expressed as an annually compunded rate, your investment yields: 3 FV 100*( ) which corresponds to an actual interest over the 3-year period of 14.8%. But comparing this figure for example with a 3-year deposit that pays 4.5% p.a. is not very meaningful. Hence, the conversion of the spot rate to an annually compounded rate. another way of expressing an interest rate is as a discount factor in the above case the discount factor is: Df 1 3 ( 1+ r) t ( ) the discount factor is the PV of one unit (1) of currency received at time t the discount factor reflects both 1) time and ) the discount rate discount factors are decreasing with maturity and always between 0 and 1:

16 rbb (from Discount factor Maturity (Years) the discount factor is also the price (in % of face value) of a zero-coupon bond for example, the price for a 3-year zero quoted at a yield 4.55% with face value 1000 (promises to pay 1000 after 3 years) is of course: PV 1000 (1 + r) t 1000 ( ) or equivalently: PV CF * Df 1000 * Forward rates a forward interest rate is a rate set today for an investment that starts at a specified time in the future spot rates for different maturities are linked by forward rates e.g. the interest rate for a 1-year investment that starts in the future is called a forward rate e.g. the -year spot rate, rbb period 0 to ), can be expressed using the 1-year spot rate (from 0 to 1) and a forward rate from year 1 to year, which we denote fb1b : rb1 BfB1 B an investor with a -year investment horizon has two choices:

17 rb1b rbb and rb3b rb4b rb5b invest at the -year spot rate rbb. invest at the 1-year spot rate rb1b roll over the deposit with the forward. fb1b since all rates (rb1b, rbb, and fb1b) are known today the two investments can be compared: ( 1+ r ) ( 1+ r )*( 1+ f ) 1 1 this is an important arbitrage statement: the payoff from the two investments are known today, and must be the same to prevent arbitrage if, for example: ( 1+ r ) > ( 1+ r )*( 1+ f ) 1 1 we could borrow at the 1-year rate, roll over the borrowing with the forward and invest at the -year rate for an arbitrage profit (since all rates are known today) note that the time periods need not be 1 year, they could be e.g. 3 months, and show how 3-month forwards are linked to 3- and 6-month money market rates the figure below shows how spot rates are built up of one-period forward rates: fb01b fb1b fb3b fb34b fb45b Period 1 Period Period 3 Period 4 Period 5 the forward rate for period (fb1b) can be found by setting: f 1 ( 1+ r ) ( 1+ r ) 1 1 or, more generally: f nt, ( 1+ r ) t ( 1+ r ) n t n 1 t n maturity for the longer spot rate maturity for the shorter spot rate

18 15 UExampleU: Assume the 1-year spot rate is 4.00%, the -year spot rate is 4.30%, and the 3- year spot rate is 4.55%. What are the one-year forward rates for year and 3? ( ) f % ( ) 3 ( ) f % ( ) we can extend the information in the table: Maturity Spot rate Actual return Discount factor Forward 1-year Years % (p.a.) % Zero-coupon pricing of coupon bonds.3.1. The coupon rate most bonds pay annual or semi-annual fixed interest payments, called coupons coupon is paid on the face value, and is thus for fixed rate bonds a fixed value, since the face value of a bullet bond does not change the interest paid can also be a floating rate, based on some benchmark interest rate (e.g. LIBOR), and is reset at each coupon payment date (FRN floating rate notes) the coupon rate is usually set to reflect the current interest rate level, and rounded to the nearest 5 or 1.5 basis points, and the issue price adjusts to reflect the difference between investor s required yield and the coupon rate.3.. The present value of a coupon bond suppose the (rising) term structure previously used applies, and an investor chooses between two bonds by the same issuer: 1. a 3-year zero-coupon bond with face value 100. a 3-year bond paying 5% annual coupons and face value 100 what return should the investor require from investing in these two bonds? we know that rb3b 4.55% so this seems a reasonable yield for bond 1 should we require the same yield from bond just because the last cash flow occurs at the same time?

19 rbib 16 no, there is no reason to let maturity alone determine the discount rate! these are clearly two different bonds, 4.55% is a yield for one single payment at t 3, and the second bond provides us with a series of payments in t 1, and 3 theoretically, a coupon bond is a collection of zero-coupon bonds, where each payment (coupon or principal) can be seen as a separate zero-coupon bond hence, the price of a coupon bond is the sum of all the individual payments (zeros): PV CF CF CF 1 T CF * Df CF * Df T (1 + r1 ) (1 + r ) (1 + rt ) + CF T * Df T where CFBiB DfBiB cash flow received at time i zero-coupon rate for maturity i discount factor for maturity i For a bullet bond the cash flow is coupon payments until the last cash flow at maturity T (CFBTB) which is the last coupon + face value. The time periods (1...T) are usually fractions of a year (e.g. the first payment could occur after 0.8 years, the second after 1.8 years etc.) from the PV-equation, it would seem very odd to use rb3b 4.55% for all payments instead we use a series of zero coupon rates to value a the bond: UExample:U The value of a 3-year bond that pays a 5% annual coupon on EUR face value assuming the spot rate 4.00% for one year, 4.30% for two years, and 4.55% for three years: PV CF1 (1 + r ) 1 CF + (1 + r ) CF3 + (1 + r ) ( ) 50 + ( ) ( ) The coupon bond sells above par (101.83% of the face value), since the coupon payments exceed the current interest rate level (term structure of zeros). A market for zeros guarantees that the bond must be priced using zero rates: otherwise, a bond could be stripped and the parts (coupon and/or principal) could be sold at a different price in the strips market. Or, one could assemble a bond from strips and sell the package as a coupon-paying bond. This arbitrage/replication approach to bond pricing is of course is directly applicable where a liquid zero-coupon market exists along a coupon-bond market. Nevertheless, zero-coupon pricing captures the whole shape of the termstructure, and correctly prices each part of a specific cash-flow structure.

20 17 This is particularly important when observed market yields correspond to certain cash-flow structures (zeros, deferrals, step-ups, coupon bonds, annuities etc.), and one is to price non-standard cash flow structures Yield to maturity for a coupon bond what is the yield to maturity for a coupon bond priced using zeroes? the yield to maturity for a zero is unambiguous: it is the zero-coupon rate for coupon bonds the yield is the internal rate of return for the bond thus, once the price of the bond is known, we must solve the present value equation with respect to yield (y): PV CF1 (1 + y) CF + (1 + y) CF3 + (1 + y) 3 50 (1 + y) 50 + (1 + y) (1 + y) the yield can be found by trial-and-error (for example using a solver-function) and is y or 4.539% note that since the yield is a kind of weighted average of the zero coupon rates, and the largest cash flow (principal + last coupon) is paid at year 3, the yield is close to the longest zero rate another, less frequently used measure is the current yield: Current yield Coupon/PV for example: Current yield 50/ %... which is a more or less meaningless measure try to value the bond using the yield: 50 PV ( ) ( ) ( ) the values of the individual cash flows have changed a bond can be stripped into zero-coupon instruments: Coupon bond: Coupon 1 Coupon Principal Coupon 3

21 18 Stripped bond: Coupon 1 Coupon Principal Coupon Time the law-of-one-price does not hold, since the stripped coupons and zeros do not have the same price if the bond is valued using the yield now, consider the same spot rates but a bond that pays a 10% coupon: PV 100 ( ) ( ) ( ) the yield for this bond is % or 0.014% lower than for the the 5% coupon bond despite the same spot rates and the same maturity, bonds can have different yields: Bond type Yield Price 3 year bullet, 0% coupon % year bullet, 5% coupon 4.539% year bullet, 10% coupon % year annuity, 5% coupon % a tricky (that is, impossible) question to answer is what is the yield for 3-year bonds : there is no unambiguous answer to that question the only result that prevails is that the only unambiguous interest rate for a certain maturity is the zero-coupon rate a yield for a 3-year coupon bond is clearly not a true 3-year rate since cash flows are distributed over the time span 1-3 years even if the yield has limitations in measuring the interest rate level, it is still a convenient summary measure, and used as such in practice one more example of the pitfalls of pricing with the yield: UExample:U The treasury is offering a new product: Maturity: 3 years Face value: EUR Bond type: Annuity

22 19 Coupon: 5.00% Annuity: EUR annual The annuity is calculated as: t 3 r(1 + r) 0.05( ) Annuity PV * * t 3 (1 + r) 1 ( ) 1 You work for the treasury, and your task is to price the bond to decide which treasury auction bids to accept. You observe some treasury bond yields on the market: Treasury zero-coupon bonds: 1 year: 4.00% years: 5.00% 3 years: 6.00% Treasury 3 year benchmark, 5% coupon bond: 3 years 5.933% You have never even heard about the CEFA-program, and hence, you are unaware of term-structure theory and bond mathematics, and you decide to price the new annuity bond using the 3-year treasury yield for the 5% coupon bond. You find that the price should be (you discount the annuities with the yield): I) Using yield (5.993%) for coupon bond: PV ( ) ( ) ( ) At this price, the annuity naturally carries a yield of 5.933% II) Using term-structure of zero-coupon rates: PV ( ) ( ) ( ) At this price, the annuity carries a yield of 5.965% Now, your pricing adventures of having applied a 5% coupon bond yield to an annuity has three consequences: 1) you mispriced (underpriced) the annuity with about 70 basis points, ) investors would kill to lay their hands on the annuity, 3) you will lose your job or alternatively the Treasury will send you to next year s CEFA program. The following table will highlight the problem of using the yield for a particular cash flow structure when pricing a different cash flow structure: 5% CPN bond Annuity Maturity Cash flow % of total CF Cash flow % of total CF

23 % % % % % % Total % %.3.4. The par yield we now know that the yield for bonds of a certain maturity will depend on the coupon rate (or more generally the cash flow structure, for example the annuity) thus, it is impossible to say what the yield is for a certain maturity a commonly used yield is, however, the par yield a bond whose price equals its face value is said to sell at par (100% of the face value) the yield for such a bond is then the par yield for par bonds: Yield coupon rate PV 100% of face value the par yield for maturity T can easily be calculated using discount factors: Yield Par (1 Df T i 1 Df T i ) which says that you 1) calculate the discount factors for all cash flows upt to T, ) divide 1 minus the discount factor for maturity T with the sum of all discount factors. UExample:U Calculate the par yield for 3-year bonds. From previous tables, we know that the discount factors for years 1 to 3 are: , 0.919, and Then: Yield Par ( ) ( ) which is %. Thus, if we would issue a 3 year bond with % annual coupons, it would trade at par. We can complete the table: Maturity Spot rate Actual Discount Forward Par yield return factor (1-year) Years % (p.a.) %

24 1 and draw some graphs of the different interest rate curves: Rate (%) Zeros Forwards Par yields Maturity (Years).4. The yield: common misconceptions from the previous section, it should have become clear that in some circumstances, the yield to maturity can be a misleading measure for coupon paying bonds however, since the cash flows for most bonds look similar (fixed coupon payments inside a certain range, no negative cash flows), the yield is a convenient summary measure of the bond price relative to its cash flows as an annual per cent rate one should, however, be aware of what a yield is and what it is not:.4.1. The yield is not the return one could easily think that the yield for a bond is the promised return for the bond, since the bond is a fixed income security the yield return only for zeros that are held until maturity, in all other cases this will not hold the yield is a discount rate, not a return! we will analyze a special case when the yield at the time of purchase actually will equal the return: UExample:U Earlier, we priced the 3-year, 5% coupon bond at PV , and calculated the yield, y 4.539%. Suppose we intend to keep the bond until maturity, and want to calculate the return over this investment horizon (3 years). We have a problem of reinvesting the 5% annual coupons, but we assume this can be done at a reinvestment rate that equals the yield. Then the future cash flows are: 1PstP Coupon: 50*( )^ PndP Coupon: 50*( ) 5.665

25 U3UPUrdUPU Coupon + Face value Total future value The price of the bond was PV , so the horizon return is: r / % which equals the yield at the time of purchase. Anybody would, however, agree that this scenario is unrealistic: 1. The bond is seldom held until maturity. The interest rate level at the time of selling the bond is uncertain, and hence the price is uncertain price risk. The coupons can usually not be reinvested at a rate that equals the yield. This would require a flat term-structure. The reality is uncertainty about the future value of the reinvested coupons reinvestment risk the previous exercise is called horizon analysis this is a useful approach in bond investing, since unlike stocks, the life of the bond is finite, and hence invest-and-forget (buy-and-hold) strategies are not applicable the investor might be interested in possible outcomes for the future value of the investment at a certain pont in time (before or at maturity) for example, insurance companies have known liabilities which require funds to be invested such that the liability can be met at that point in time the total, or horizon return from a bond consists of: 1. Coupon interest payments. Income from reinvesting the coupons 3. Capital gain or loss if the bond is sold before maturity it s clear that today s yield cannot capture all these sources of return of course, let s not forget that for example treasury bonds usually have higher yields than treasury bills, and also tend to outperform treasury bills in terms of return the point is that the yield is merely and indication of return, not a promise.4.. Yields are not additive what this means is that yields for bond portfolios can not be calculated like returns for stock portfolios UExampleU: Let s construct a simple bond portfolio that contains one 1-year zero coupon bond, and one 5-year coupon bond that pays 5% annual coupons. We use the term structure from previous examples to price the bonds, and assume that both bonds have a face value of 1000: Bond Coupon Price Weight Yield 1 year zero 0.00% %

26 3 5 year bullet 5.00% % Portfolio First, let s try to calculate the portfolio yield by treating yields as returns: r P N i 1 w r i i * * or % (which is wrong) The only correct way to calculate the yield on a bond portfolio is, however by solving for the yield for the portfolio, given the portfolio price and cash flows: (1 + y) (1 + y) (1 + y) (1 + y) where, in this example, the first cash flow comes from the first bond, and all other cash flows from the second bond. We solve the yield (y), and find that: y % Note the difference (over 7 basis points!) between the true yield y %, and y % calculated earlier. of course, nothing prevents the bond portfolio manager from expressing the yield of his portfolio as a weighted average, but in that case care should be taken to make clear how this figure has been obtained to a very close approximation, the portfolio yield can be calculated using weights, if the equation is adjusted for modified duration.5. From coupon bonds to zeros: bootstrapping suppose you need zeros for pricing, but no zeros with the same credit risk exist zero-coupon or spot rates reflecting a specific level of default risk can be found by: 1. Observing yields in the zero-coupon market (strips). Bootstrapping a yield curve using coupon paying bonds or the swap curve (these two spot rate curves of course have different credit risks) using observable zero-coupon rates for pricing coupon bonds may be problematic: 1. Liquidity. Lower liquidity in the zero market might lead to higher yields.. Taxes. Zeros and coupon bonds might be taxed differently along the whole maturity spectrum, which translates to differences in required yields.

27 rbb rb3b 4 3. The preferred habitat hypothesis. There might be maturity sectors where taxation between zeros and coupon bonds differ, if principal strips are taxed differently than coupon strips. an important application of bootstrapping is to derive theoretical zero-coupon rates from swap rates, which represent par-yields in the interbank market these bootstrapped zero-rates can then be used as a benchmark for pricing nonstandard cash flow structures in the corporate market the need for zero rates is more obvious the more non-standard the bond or valuation need is UExample:U Suppose that a market participant needs zero-coupon rates up to 3-years maturity, but no zero market exist. Instead, he observes the prices for the following bonds: Bond Maturity Coupon rate Face value Price 1 year zero 1 year 0.00% year bullet years 4.50% year bullet 3 years 5.00% The 1 year zero rate is easy: Turning to the year zero rate we use the year coupon bond and we now know that: ( ) (1 + r ) Clearly, there is only one solution for the year zero rate (rbb) that satisfies the equation. After some calculation we find that % does the trick. We continue in the same fashion with the 3 year coupon bond: ( ) ( ) (1 + r ) and find that % (which does not surprise the careful reader, who might have suspected that the coupon bonds in the example were priced using the same term structure as before.) the same technique applies to swap-rates, assuming payments are annual:

28 rbtb 5 Maturity Maturity Swap rate 1 year 1 year 4.50% year years 4.70% 3 year 3 years 4.80% since swaps are priced at par (the swap rate is a par-yield), and a price of 100 can be assumed: and ( ) (1 + r ) 100 and rbb %, and so on for the rest of the swap-curve..6. Spot and forward rates with semi-annual compounding bonds and swaps may pay semi-annual or even quarterly or monthly coupons, and hence there is a need to handle spot rates and forwards for higher compounding frequencies than 1 years in general, the present value formula can be expressed: PV CF rt 1 + m mt where m t the zero coupon rate for maturity t the compounding frequency the maturity of the cash flow in years we will exemplify this using the following semi-annual zero-coupon rates: Maturity Spot rate Years % (p.a.) to convert these rates to discount factors:

29 6 Df t 1 rt t ( 1+ ) * For example for the 1 year, semiannual rate of 7.68%: Df t 1 1 rt t ( ) ( 0768 ) * 1 * the forward rates are now 6-month forwards e.g. the forward from year 1 to year 1.5 (from period to period 3 in half-years): f 3 r ( 1+ ) * r 1+ ( ) note the multiplication by to get the 6-month forward to an annual rate we can produce a similar table like in the case for the 1-year periods: Maturity Spot rate Actual return Discount factor Forward 6-month Years % (p.a.) %

30 7 3. Day counts and accrued interest so far, we have worked in a fairly unrealistic setting: we have analyzed bonds and rates with maturity of integers of one year the reason for abstracting from details is to make the concepts and technique clear we will now take a small step from the classroom towards the cruel world of day count conventions and accrued interest reading stock market quotes is quite clear: you can observe the price at which you buy and the price at which you sell in fixed income markets, things become a mess: prices are frequently quoted as yields, or in percent of face value less accrued interest, which means that just by looking at bond quotes you can never tell what the price is further, a yield quote is not unambiguous across the world: 5% in the US treasury markets does not mean the same thing as 5% in the Finnish or German treasury markets: there are differences in day count conventions and compounding frequency as an example, compare the following zero-coupon bonds: Security Maturity Coupon Quote Price (Years) (yield to maturity) (% of face value) German Bund % 5.96% 41.1 US Treasury % 5.96% What s wrong? Shouldn t the price also be equal? No, since all Bund quotes are based on annual compounding, while US Treasury bond quotes are based on semi-annual compounding they will have different prices if the yields are the same. the lesson is that one should always be aware of what market conventions apply to the interest rate you are analyzing the next section is by no means intended to give a comprehensive treatment of the subject, but merely to introduce the reader to some concepts, and to make the reader aware of the pitfalls that exist 3.1. Day count basis the bond market is like no other place: 31 days can be 30 days, and a year can be more or less than 365 days recall that all interest rates are expressed on an annual basis day count conventions deal with how to compute fractions of a year to give an example of what day count basis means: UExample:U Suppose you are investing in a 1-month short-term deposit. The financial institution promises an interest of 4.00%. This is of course a per annum figure. Since the interest rate is an annual rate, we must know how large a fraction is this particular month of 1 year. Suppose we count the days, we find that there are 31 days in that month. We also know that there are 365 days in a year, so the maturity of the investment in years is:

31 8 t thus, if we invest 100 units of currency: FV * * This day count basis is called Actual/Actual, because all days are counted, in our case 31/365. This need not be the case, since there exist many ways of counting days. The most common day count conventions include: Actual/Actual The denominator is the number of days in the coupon period times the coupon frequency. Actual/365 Like Actual/Actual but always uses 365. Actual/360 Like Actual/365, but always uses 360 (assumes there are only 360 days in a year). (Euro money market) 30/360 Assumes that there are 30 days in each month and 360 days in a year. In our previous example, we would have measured the time period as only 30 days instead of 31, and divided with 360 (30/ ). There are som further variations for this rule. 3.. The money market prices for money market instruments are usually expressed as yields on the Euro market, the day count is Actual/360 the price (PBCDB) of a money market security: P CD where CF t + r 1 * 360 CF r t cash flow at maturity interest rate for period t time to maturity in days UExample:U The maturity of a money market security is 30 days, face value EUR , and the quotes are: Maturity Bid Ask 30 days Quote (%) 3.01%.96%

32 9 What s the price? Note that bid and ask are from the dealer s point of view: the dealer buys at 3.01%, and sells at.96%. Using the bid to find out what the dealer is willing to pay for the security: P CD * The ask (.96%) again corresponds to a price of EUR That is: Maturity Bid Ask 30 days Quote (%) 3.01%.96% Price (EUR) EUR EUR Zero-coupon bonds: annual compounding bond prices are expressed as yields or prices as percent of the face value to calculate the price in currency, one has to know the face value, day count basis, and compounding frequency UExample:U On the Bund STRIPS-market a zero that matures is quoted on : Maturity Bid Ask Quote (yield) 5.96% 5.91% Price (% of face value) The maturity of the zero is years using Actual/Actual day count basis. Further we need to know that on the Bund market, annual compounding is used. If the face value of the zero is EUR , the bid price in euros is: EUR PV EUR ( ) and the price quote is (K): K %