BOND ANALYSIS AND VALUATION

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

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

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

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. 1.1. 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.

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

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:

4 Bond Maturity CPN Issue price Current price Yield% Amount EUR (m) 9.10.03 % 10.11.0 3 10.11.0 3 Serial bond 003/I 4.7.006.750 99.714 99.040 3.03 6 500 Serial bond 001/I 4.7.007 5.000 100.01 105.040 3.495 6 31 Serial bond 4.7.008 3.000 99.55 96.830 3.754 5 999 003/II Serial bond 5.4.00 5.000 99.500 105.170 3.9 5 753 1998/II 9 Serial bond 000/I 3..01 5.750 99.110 109.350 4.7 5 673 1 Serial bond 4.7.013 5.375 99.666 106.910 4.473 6 000 00/II 36 156 1.4. 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 15000 10000 5000 0 USA Germany France Japan Finland Government 9697 867 791 5316 77 Corporate 8805 190 887 1606 41 Total 1850 3057 1678 69 118 risk management and the desire to explore cost-effective borrowing through swaps has lead to enromous global fixed income derivatives markets:

5 By sector: 1) Foreign exchange, ) Interest rate, 3) Equity linked, and market (OTC or Exchanges) 100000 USD billions 80000 60000 40000 0000 0 OTC FX OTC Interest OTC Equity Ex. FX Ex. Interest Ex. Equity Forwards 1073 879 364 7 13444 4 Swaps 4509 79161 0 0 0 0 Options 338 13746 1944 33 04 307 Total 18470 101699 308 105 35468 79 1.5. 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 1.5.1. 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:

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 1.5.. 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) 0.03 + 0.0 0.05 5% 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

7 1.5.3. 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

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: 1000.00 Present value: 1150.6 Maturity: 3.00 years Coupon rate: 10.00% (annual) Yield to maturity: 4.5188% Par yield: 4.5344% 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

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 1 000 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. 035* 0. 5) 1008. 75 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 1 000 today, how much cash do you have after years? t [ ] [ ] FV PV * 1+ r 1000EUR * 1+ 0. 045 108680. 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):

10 r 1/ N [( 1+ ) ] 1 ( 1+ 0.0868) r Actual [ ] 1/ 1 0. 045 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 +. * 1 1000 * 1 0 045 1088. 3 4 an interest of 4.5/4 1.065% 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 0045. * FV PV * e 1000EUR * e 1088. 7EUR 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 + 1 004. ( 360 / 30) 360/ 30 [[ ] ] + 1. 00 1 1 0. 003333 1 0. 04074 4. 074%

11 The 60-day effective annual yield is Yield EFF + 1 004. ( 360 / 60) 360/ 60 [[ ] ] + 600. 1 1 0. 006666 1 0. 040673 4. 0673% 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 % 1 4.00 4.30 3 4.55 4 4.75 5 4.90 then, we can graph the term structure of these spot rates:

1 6.00 5.00 Spot rate (%) 4.00 3.00.00 1.00 0.00 0 1 3 4 5 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 104.55 back after 3-years. Since the spot rate is expressed as an annually compunded rate, your investment yields: 3 FV 100*( 1+ 0. 0455) 114. 8 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 ( 1+ 0. 0455) 3 1 08750. 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:

rbb (from 13 1.00 0.95 Discount factor 0.90 0.85 0.80 0.75 0 1 3 4 5 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 (1 + 0.0455) 3 875.00 or equivalently: PV CF * Df 1000 * 0.8750 875.00... 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:

rb1b rbb and rb3b rb4b rb5b 14 1. 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

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? (1 + 0.0430) f 1 1 1.046009 1 0.046009 4.60% (1 + 0.040) 3 (1 + 0.0455) f 3 1 1.050518 1 0.050518 5.05% (1 + 0.0430) we can extend the information in the table: Maturity Spot rate Actual return Discount factor Forward 1-year Years % (p.a.) % 1 4.00 4.00 0.9615 4.0000 4.30 8.78 0.919 4.6009 3 4.55 14.8 0.8750 5.0518 4 4.75 0.40 0.8306 5.353 5 4.90 7.0 0.7873 5.50.3. 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?

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 1 + + 1 (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 1 000 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 ) 3 3 50 (1 + 0.04) 50 + (1 + 0.0430) 1050 + (1 + 0.0455) 3 48.07 + 45.96 + 918.79 101.83 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.

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..3.3. 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) 1050 + (1 + y) 3 101.83 the yield can be found by trial-and-error (for example using a solver-function) and is y 0.04539 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 50/101.83 4.9367%... which is a more or less meaningless measure try to value the bond using the yield: 50 PV (1 + 0.04539) 50 1050 + + 47.83 + 45.77 + 919.4 101.83 3 (1 + 0.04539) (1 + 0.04539) 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

18 Stripped bond: Coupon 1 Coupon Principal Coupon 3 0 1 3 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 (1 + 0.04) 100 + (1 + 0.043) 1100 + (1 + 0.0455) 3 1150.6 the yield for this bond is 4.5188% 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 4.5500% 875.00 3 year bullet, 5% coupon 4.539% 101.83 3 year bullet, 10% coupon 4.5188% 1150.6 3 year annuity, 5% coupon 4.3680% 994.47 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 1 000 000 Bond type: Annuity

19 Coupon: 5.00% Annuity: EUR 367 08.56 annual The annuity is calculated as: t 3 r(1 + r) 0.05(1 + 0.05) Annuity PV * 1000000* 36708.56 t 3 (1 + r) 1 (1 + 0.05) 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 36708.56 (1 + 0.05993) 1 36708.56 + (1 + 0.05993) 36708.56 + (1 + 0.05993) 3 98770.06 At this price, the annuity naturally carries a yield of 5.933% II) Using term-structure of zero-coupon rates: PV 36708.56 36708.56 36708.56 + + 3 (1 + 0.04) (1 + 0.05) (1 + 0.06) 1 994469.54 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

0 1 50 000 4.35% 367 08.56 33.33% 50 000 4.35% 367 08.56 33.33% 3 1 050 000 91.30% 367 08.56 33.33% Total 1 150 000 100.00% 1 101 65.68 100.00%.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: 0.9615, 0.919, and 0.8750. Then: Yield Par (1 0.8759) (0.9615 + 0.919 + 0.8750) 0.045344 which is 4.5344%. Thus, if we would issue a 3 year bond with 4.5344% 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.) % 1 4.00 4.00 0.9615 4.0000 4.0000 4.30 8.78 0.919 4.6009 4.937 3 4.55 14.8 0.8750 5.0518 4.5344 4 4.75 0.40 0.8306 5.353 4.738 5 4.90 7.0 0.7873 5.50 4.8639

1 and draw some graphs of the different interest rate curves: 6.00 5.50 Rate (%) 5.00 4.50 4.00 Zeros Forwards Par yields 3.50 3.00 0 1 3 4 5 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 101.83, 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*(1 + 0.04539)^ 56.6357 PndP Coupon: 50*(1 + 0.04539) 5.665

U3UPUrdUPU Coupon + Face value 1050.00 Total future value 1156.90 The price of the bond was PV 101.83, so the horizon return is: 1156.90 r 101.83 1/ 3 1 0.04539 4.539% 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% 961.54 0.4887 4.0000%

3 5 year bullet 5.00% 1005.95 0.5113 4.8631% Portfolio 1967.49 1.0000 First, let s try to calculate the portfolio yield by treating yields as returns: r P N i 1 w r i i 0.4887 * 0.04 + 0.5113* 0.048631 0.044413 or 4.4413% (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: 1000 50 50 1050 + + + + (1 + y) (1 + y) (1 + y) (1 + y) 5 1967.49 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 0.0471349 4.7135% Note the difference (over 7 basis points!) between the true yield y 4.7135%, and y 4.4413% 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.

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% 1000.00 961.54 year bullet years 4.50% 1000.00 1003.88 3 year bullet 3 years 5.00% 1000.00 101.83 The 1 year zero rate is easy: 1000 961.54 1 0.04 Turning to the year zero rate we use the year coupon bond and we now know that: 45 1045 + (1 + 0.04) (1 + r ) 1003.88 Clearly, there is only one solution for the year zero rate (rbb) that satisfies the equation. After some calculation we find that 0.043 4.30% does the trick. We continue in the same fashion with the 3 year coupon bond: 50 50 1050 + + (1 + 0.04) (1 + 0.043) (1 + r ) and find that 3 3 101.83 0.0455 4.55% (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:

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: 104.50 1 0.045 100 and 4.70 104.70 + (1 + 0.045) (1 + r ) 100 and rbb 4.7047%, 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.) 0.5 7.45 1.0 7.68 1.5 7.69.0 7.75 to convert these rates to discount factors:

6 Df t 1 rt t ( 1+ ) * For example for the 1 year, semiannual rate of 7.68%: Df t 1 1 rt t 1+ 1+ 0. ( ) ( 0768 ) * 1 * 0. 974 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+ ( ) 3 3 1 0. 0771 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.) % 0.5 7.45 3.750 0.9641 7.4500 1.0 7.68 7.875 0.974 7.9103 1.5 7.69 11.984 0.8930 7.7100.0 7.75 16.444 0.8589 7.9301

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 15.35 0% 5.96% 41.1 US Treasury 15.35 0% 5.96% 40.60 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:

8 t 31 365 0.084931 thus, if we invest 100 units of currency: FV 31 100 * 1 + 0.04 * 100.3397 365 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 (30/360 0.083333). 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 1 000 000, and the quotes are: Maturity Bid Ask 30 days Quote (%) 3.01%.96%

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: 1000000 P CD + 30 1 0. 0301* 360 1000000 997497. 94 100508. The ask (.96%) again corresponds to a price of EUR 997 539.40. That is: Maturity Bid Ask 30 days Quote (%) 3.01%.96% Price (EUR) EUR 997 497.94 EUR 997 539.40 3.3. 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 4.7.015 is quoted on 1.3.000: Maturity Bid Ask 4.7.015 Quote (yield) 5.96% 5.91% Price (% of face value) 41.1 41.41 The maturity of the zero is 15.35 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 100 000, the bid price in euros is: EUR100000 PV EUR4111.70 15.35 (1 + 0.0596) and the price quote is (K): K 4111.70 100000 0.411 41.1%

rbtb 30 3.4. Zero-coupon bonds: semi-annual compounding suppose that the same market quotes (yield and price quote) are observed on the US Treasury STRIPS markets is the price in currency the same? no, since US Treasury yields are semi-annual (assuming day count basis is the same): USD100000 PV USD40596.9 15.35* 0.0596 1 + and the price quote K 40.60% if there should have been differences in the day count basis between the markets, this difference would have shown up in the calculation of maturity, and the fraction of a year (0.35) might have been different 3.5. Coupon bonds bonds are usually quoted in per cent of their face value, e.g. 110.87% or just 110.87 the yield to maturity (YTM) of a bond is another way to express the price recall that the present value of a bond is simply: PV BOND n CFt ( 1 + r ) t 1 t t where CFBtB t cash flow (coupon and/or face value repayment) at time t spot rate for maturity t time in years Once the yield to maturity is known, one can of course to cut some corners use the yield: PV BOND n CFt ( 1 + y ) t 1 t in most of the forthcoming examples, we will use the yield to demonstrate the calculations let s start our example of dirty and clean prices by pricing a bond:

31 UExample:U Suppose todays date is 17.1.1997, and a government bond that matures 15.3.004 and pays a CPN of 9.50%. Assume a yield of 5.55% and, a face value of EUR 1 000 000. What is the PV of the bond on 17.1.1997? There are 57 days between 17.1 and 15.3 which is 0.156 years under the Actual/Actual basis so: PV 95000 95000 1095000 + + +.156 1.156 (1 + 0.0555) (1 + 0.0555) (1 + 0.0555) 0 7.156 1307995.13 note that: 1. the value of the first coupon can, depending on the market conventions, be calculated using simple or, as in this example, compound interest. there may not be exactly 1 year between the coupon payment dates if payments occur on weekends or holidays, this has not been taken into account here the effect on value of 1. and. is of course minimal, but it will be there the bond will, however, nor be quoted as 1 307 995.13 on the market... 3.5.1. Dirty prices and clean prices the present value is called the dirty price, full price, or invoice price what are clean prices? the price that dealers quote is the clean price, not the PV (dirty price): Clean price Dirty price Accrued coupon interest since last coupon payment The accrued interest (AI) is: v AI * r * CPN N 365 where v number of days since last coupon payment rbcpnb coupon rate N face value of the bond 365 day count basis for the bond, here assumed to be 365 The clean price (K) of a bond is: K PVBOND AI N

3 UExample:U Using the bond from the previous example we know that: Purchase date: 17.1.1997 Next coupon payment date: 15.3.1997 Days to next coupon date: 57 Days of accrued interest (v): 308 PV (dirty price): 1 307 995.13 EUR Face value: 1 000 000.00 EUR (Note again that here, we abstract from taking into account delivery days applied on the market (usually T +1...+3), that is, the bond and the money does not move today, and you actually trade 1-3 day forwards). The days of accrued interest v 365 57 308 using actual/actual day count basis. We can now compute: 308 AI * 0.095*1000000EUR 80164. 38EUR 365 and hence K 1307995.13 80164.38 1.783 1.783% 1000000 Remember that the even if the quote is 1.783%, you still pay the dirty price 1 307 995.13 for the bond (that s why the dirty price is also called the invoice price). 3.5.. Behavior of dirty and clean prices over time: convergence towards par the present value (dirty price) will vary according to the number of days to the next coupon payment: immediately after a coupon payment, the PV will fall, and then rise again as the next coupon approaches in time the clean price, on the other hand behaves more smoothly UExample:U Consider the same bond that matures 15.3.004 and pays a CPN of 9.50%. Assume a (constant) yield of 5.55% and, for ease of exposition, a face value of only EUR 100. Starting from the CPN date 15.3.1996, we calculate the PV for each month. Note how the PV falls on the CPN date 15.3.1997, just to start rising again. (This example still uses the 30/360 day count basis).

33 Date Days to next CPN PV (Dirty Price) Price (Clean Price) 15.3.1996 360 14.97 14.97 15.4.1996 330 15.535 14.744 15.5.1996 300 16.10 14.518 15.6.1996 70 16.671 14.96 15.7.1996 40 17.4 14.075 15.8.1996 10 17.816 13.858 15.9.1996 180 18.393 13.643 15.10.1996 150 18.97 13.430 15.11.1996 10 19.554 13.0 15.1.1996 90 130.138 13.013 15.1.1997 60 130.75 1.805 15..1997 30 131.315 1.607 15.3.1997 360 1.407 1.407 15.4.1997 330 1.960 1.168 15.5.1997 300 13.514 11.931 15.6.1997 70 14.07 11.697 the clean price (K) of the bond approaches 100 (par) when maturity decreases the dirty price (PV) of the bond approaches 100 + last coupon when maturity decreases UExample:U Consider a bond that matures 18.4.006. Assume the yield remains at 6.00%. Date Years Clean price Dirty price 17.1.1997 9.58 108.64 114.060 17.1.1998 8.58 107.913 113.331 17.1.1999 7.58 107.140 11.358 17.1.000 6.58 106.31 111.739 17.1.001 5.58 105.453 110.870 17.1.00 4.58 104.533 109.950 17.1.003 3.58 103.557 108.974 17.1.004.58 10.53 107.940 17.1.005 1.58 101.47 106.844 17.1.006 0.58 100.65 105.68 a discount bond would have started below par and approached par from below

34 4. Measuring interest rate risk: duration and convexity Since cash flows for bonds are usually fixed, a price change can come from two sources: 1. The passage of time (convergence towards par). This is of course totally predictable, and hence not a risk.. A change in the yield. This can be due to a change in the benchmark yield, and/or change in the yield spread. The yield-price relationship is inverse, and we would like to have a measure of how sensitive the bond price is to yield changes. A good approximation for bond price changes due to yield is the duration, a measure for interest rate risk. For large yield changes convexity can be added to improve the performance of the duration. A more important use of convexity is that it measures the sensitivity of duration to yield changes. Similar risk measures are used in the options markets are the delta and gamma. 4.1. The yield-price relationship for bonds we again discuss interest rate changes in terms of yield changes: this is more convenient as the yield is the mostly used interest rate measure for coupon bonds when yields increase, bond prices decrese when yields decrease, bond prices increase for small yield changes, the percentage price change is roughly the same whether the required yield increases or decreases for large yield changes, the percentage price increase is larger than a price decrease UExample:U On 17.1.1997, the (dirty) price of the RoF006 Government bond is 114.060, and the yield is 6.00%. Consider the impact of an one percent yield increase/decrease: Yield (%) Yield change (% units) Dirty Price EUR Price change (%) 5.00-1.00 11.73 +6.73 6.00 0.00 114.060 0.00 7.00 +1.00 107.033-6.16

35 Price and Yield for RoF001 and RoF006 Government bonds 180.00 160.00 Dirty Price 140.00 10.00 100.00 RoF001 RoF006 80.00 60.00 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 Yield 4.. Duration different ways to calculate duration: 1. Macaulay duration (DBMACB): i) The present value of time-weighted cashflows, divided by the present value of the bond. Uses yield to calculate present values. ii) The discounted, average payback time. iii) Balancing point in time between interest rate and price risk.. Fisher-Weil duration (DBFWB): i) The same as DBMACB, but uses zero rates. 3. Modified duration (DBMODB): i) Macaulay duration/(1 + y) ii) The % price increase (decrease) in the bond price if the yield decreases (increases) by a unit of 1%. 4. Key rate duration (DBKRB): i) Calculates the price response separately for a 1% change in each zero-coupon rate used to calculate the PV of the bond. Gives a better picture of which parts of the term-structure is responsible for how much of total interest rate risk. The key rate durations can then be summed up to give an overall interest rate risk measure, comparable with modified duration.

36 4..1. Macaulay duration Macaulay duration (DBMACB) using the yield: 1 CF1* t1 CF * t CFn * t Duration( DMAC ) * t t... PV 1 + + + BOND ( 1+ y) ( 1+ y) ( 1+ y) n tn n t t1 CF t * t ( 1+ y) PV i ti BOND UExample:U Calculate the Macaulay duration for the bond: Maturity:.0 years Face value: EUR 1 000 000.00 PV: EUR 1 141 635.55 Coupon rate: 11.0% annual Yield: 3.54% Duration( D MAC 1 * 1141635.55 1 110000*1.00 ) * 1141635.55 1.00 (1 + 0.0354) [ 10639.13 + 07079.83] 1. 9069 1110000*.00 +.00 (1 + 0.0354) since the yield is used to discount, the Macaulay duration assumes that the term structure is flat and that yield shifts are parallell the duration of a zero coupon bond equals its maturity for example, N 100, y 3.54%, PV 93.8 Duration ( D MAC ) 1 * 93.8 holding everything else equal: 100*.00.00 (1 + 0.0354) 1 9.8 [ ]. 00 * 186.56 1. increasing coupon rates decreases duration. increasing yield decreases duration 3. increasing maturity increases duration (in most cases) Macaulay duration for different maturities (T), yields, and coupon rates (annual): Yield 5% Yield 10% Yield 0% Coupon 0% 5% 10% 0% 5% 10% 0% 5% 10% T 5 5.00 4.55 4.5 5.00 4.49 4.17 5.00 4.36 3.99 10 10.00 8.11 7.7 10.00 7.66 6.76 10.00 6.65 5.7 0 0.00 13.09 11.48 0.00 10.74 9.36 0.00 6.87 6.0 50 50.00 19.17 17.76 50.00 11.4 10.91 50.00 6.01 6.00 100 100.00 0.84 0.54 100.00 11.01 11.00 100.00 6.00 6.00

37 some refinements to the 3 rules of thumb: 1. The duration of a zero-coupon bond equals it s maturity. For coupon bonds, duration approaches D 1 + 1/y when maturity increases.. Duration for coupon bonds with coupon rate higher or equal to the yield will rise with maturity and reach a maximum of D 1 + 1/y. 3. Duration for bonds with coupon rate lower than the yield will first rise with maturity and reach it s maximum and then decline with maturity to D 1 + 1/y. 4.3. Modified duration and PVBP the approximate percent price change for a 1.00% yield change (100 bps) is given by the modified duration: Duration( D ) MOD DMAC ( 1+ y) The percent price change (dp/p) for any yield change dy: dp P D * dy MOD and hence we have the EUR price change (dpv) for any yield change dy: dpv DMOD * PV * dy note the - sign before the expressions, indicating the inverse relationship between yield change and price change A commonly used measure is the Price Value of a Basis Point (PVBP): DMOD PVBP * PV 10000 and says how much the value of the bond changes in EUR when the yield changes with 1 bps (0.01%). UExample:U Calculate the modified duration and the PVBP for the same bond: DBMODB 1.9069/(1 + 0.0354) 1.8417

38 DMOD PVBP * PV 10000 1.8417 *1141635.55FIM 10000 10.6EUR UExampleU: If the yield increases by 1.00% to 4.54%, how much will the price decrease? dp/p -1.8417*0.01-0.018417-1.8417% dp -1.8417*1 141 635.55*0.01-1 06.00 EUR The new price should then be: 1 141 635.55 1 06.00 1 10 609.55 EUR But recalculating the price using the yield 4.54, we find that the true price should be EUR 1 10 905.4, and the error is EUR 95.87 the error occurs because duration assumes a linear price-yield relationship when it in fact is convex 4.3.1. The duration of a bond through time between coupon payments duration decreases one-by-one with time if yields stay unchanged at a coupon payment duration jumps up and increases in the long run duration decreases more slowly than time does this matter in the practical sense? duration increases after a coupon the bond becomes more sensitive to yield changes after a coupon payment but this is only in % terms, the absolute (EUR) value change is unaffected UExample:U The RoF006 bond pays a 7.5% coupon 18.4.1997. Consider what happens to duration and volatility due to a 0.10% yield increase immediately before and after the coupon payment: Duration % change due to EUR change due Date duration to duration 17.4.1997 6.5703-0.6138-7 173.59 18.4.1997 7.0063-0.6610-7 171.7 The % change in value of the bond changes, but the change in EUR value changes only because of the slight duration change from 17.4.1997 to 18.4.1997. does the duration jump mean that we are suddenly exposed to immunization risk? the duration of the investor s portfolio now consists of the bond and 7 500 EUR in cash (cash has zero duration) but the investor still has to decide: 1. how to reinvest the cash. how much of the original bond (portfolio) to hold

rbtb Df 39 implications for an investment with a duration target: 1. should not invest cash in the same bond (duration will exceed target). if the cash is invested in a security with any significant duration, a part of the bond (portfolio) must be sold and invest proceeds in shorter duration assets 4.3.. Key rate duration duration can only measure parallell yield changes, and does not tell us where on the yield curve the largest risks are located for a single bullet bond, the largest risk of course comes from the last payment of face value + last coupon, since it is: 1. the largest payment. located furthest out in time and hence through compounding most affected by a yield change for a bond portfolio or a more complex product than a bullet one can perform a more detailed analysis of interest rate risk along the whole yield curve using key rates we use the following bond as an example: N 100 Maturity 5 years Coupon rate 5% (annual) We compute the PV, Fisher-Weil duration, and Key rate durations assuming the term structure below: CF PV Fisher-Weil Key rate (CF*t)*Df duration 1 4.00% 0.961583 5.00 4.8077 4.8077 0.0460 4.30% 0.91945 5.00 4.596 9.195 0.0876 3 4.55% 0.875040 5.00 4.375 13.156 0.148 4 4.75% 0.830585 5.00 4.159 16.6117 0.1577 5 4.90% 0.78768 105.00 8.6631 413.3157 3.9193 100.595 4.5435 4.3354 For example, the 5 year key rate duration for a +/- 1% zero rate change is calculated: D A PV PV * PV * dr + 0 t where B PVB+B the price of the bond if the zero-coupon rate for maturity t decreases with 1% PVḆ the price of the bond if the zero-coupon rate for maturity t increases with 1% and we have

40 B PVB0B 100.595 (rb5b 4.90%) PVḆ 96.7653 (rb5b 5.90%) PVB+B 104.6505 (rb5b 3.90%) D A PV+ PV * PV0 * dr5 104.6505 96.7653 3.9193 *100.595*0.01 by common sense, calculate the average absolute price change for a +/- 1% in rates: dpv, if rb5b dpv, if rb5b 5.90% (96.7653 100.595)/100.595-3.81% 3.90% (104.6505 100.595)/100.595 +4.03% on average (3.81%+4.03)/ 3.9193% which is the price responsiveness if the 5-year zero rate changes 1% and all other rates remain unchanged. 4.4. Convexity for yield increases, duration overestimates the price decrease for yield decrases, duration underestimates the price increases The convexity for a bond is: n t + t CFt Convexity ( 1 )* 1 t + * t 1 ( 1+ y) PV BOND the first term in brackets is sometimes called dollar-convexity in isolation, this measure means nothing, and does not have a straightforward interpretation like duration or modified duration convexity has two uses: 1. Convexity measures and can be used to correct the price response-error caused by duration. But: convexity is still an approximation for this correction, since the shape of the price-yield cannot be fully described with the two measures modified duration and convexity.. Convexity measures the change in duration due to yield changes. The percent price change due to convexity is:

41 dp P 1 *( Convexity)*( dy) The EUR price change due to convexity is: 1 dp *( Dollar convexity)*( dy) UExample:U Calculate the convexity of the bond used in previous examples. The coupon is 11%, and the yield is 3.54%. 1.00 * (1 + 1.00) *110000.00 * (1 +.00) *1110000 1 Convexity + * 3.00 4.00 (1 0.0354) (1 0.0354) 1141635.55 + + 5.495 UExample:U If the yield increases by 1.00% to 4.54%, how much of the price decrease is due to convexity? dp/p 0.5*5.495*(0.01)^ 0.000648 0.0648% dp 0.5*5 993 039*(0.01)^ 99.65 EUR We can now get a better estimate of the price decrease: Duration + Convexity Estimate of total price change -1 06.00 + 99.65-0 76.35 which is only EUR 3.78 less the actual decrease of -0 730.13 note that convexity is always a positive number, correcting the price upward sometimes convexity is expressed as: Convexity/00 5.495/00 0.06% which is the % correction directly, and as such has a meaningful interpretation

4 some examples of convexities: Yield 5% Yield 10% Yield 0% Coupon 0% 5% 10% 0% 5% 10% 0% 5% 10% T 5 7.1 3.94 1.83 4.79 1.45 19.37 0.83 17.35 15.36 10 99.77 75.00 64.0 91.91 63.40 5.79 76.39 44.11 35.13 0 380.95 11.33 171.95 347.11 146.10 116. 91.67 6.30 50.64 50 31.9 564.13 484.41 107.4 06.74 190.55 1770.8 50.5 50.14 100 9161.0 764.95 73.90 8347.1 00.44 199.85 7013.8 50.00 50.00 one could argue that convexity has value since the price of a highly convex bond 1. increases more when yields fall. decreases less when yields rise one might argue that investors should be willing to pay more for a convex bond, particularly when yields are volatile 4.4.1. Duration matching and the value of convexity if interest rates fall, two opposite effects: 1. your bond portfolio will increase in value. the coupons will be reinvested at a lower rate if interest rates rise, two opposite effects: 1. your bond portfolio will decrease in value. the coupons will be reinvested at a higher rate at an investment horizon equal to the bond s duration these two effects will (approximately) offset duration is the balance point between market risk and reinvestment risk if the investors horizon is h: 1. if h D dw/dy 0 no risk. if h < D dw/dy < 0 market risk 3. if h > D dw/dy > 0 reinvestment risk note that all formulas assume a single yield y a flat yield curve is assumed convexity is beneficial for an investor who buys a bond with duration equal to his horizon, and the yield curve shifts once to some arbitrary level in case of a sudden, large yield change the investor is more protected than with a bond with lesser convexity in the case of a duration matched to the horizon, and a convex bond, a yield increase will yield a higher end-of-period wealth, since the value of the reinvested coupons exceed the loss of the lower redemption value

43 a yield decrease will yield a higher end-of-period wealth, since the higher redemption value will more than compensate for the lower reinvested coupon values UExample:U An investor buys a bond with Macaulay duration of 3.3841, and assume for simplicity that the investor s horizon equals this duration. Thus, the investor is immunized, and he is going to sell the bond 0.4714 years before maturity, since the maturity is 3.8556 years. Suppose the yield is initially 5.15%, and now consider the effect of a sudden, immediate yield change to 4.15%, and 6.15%, respectively: Yield 4.15% 5.15% 6.15% FV of coupons 319 413.7 34 07.3 39 046.01 Redemption value of bond 1 079 113.49 1 074 6.91 1 069 479.73 Total FV 1 398 57.1 1 398 470.14 1 398 56.74 Yearly return 5.1513 5.1500 5.151 note that a flat term structure is assumed (all coupons reinvested at the yield) the change in the reinvestment values of the coupons just about offset the change in the redemption value of the bond the more convex the bond, the larger would the return changes have been convexity is nearly irrelevant if yields change smoothly and continuously rather than in large, sudden jumps if yields change smoothly in small increments, the investor has the opportunity to adjust for duration changes continuously thus, no large yield changes like in the previous example will occur convexity is harmful if there is uncertainty about how the yield curve will change the effect of changes in the shape of the yield curve on the duration of a bond will be greater, the greater the convexity this is beacuse convexity implies larger dispersion of cash flows thus, convexity implies that there is more uncertainty about how the duration of the bond will behave when the yield curve changes immunize risk to meet future obligations, for e.g. pension funds protect invested money from interest rate shifts what if zero-coupon bonds with same maturity do not exist? base-case solution: invest in a bond or bond portfolio with equal (Macaulay) duration and present value as the future obligation or stream of obligations e.g. a payment in 5 years could be matched by a bond portfolio with DBMACB 5.0 protects only against parallell yield curve shifts problems with immunization: 1. duration drift over time not equal to time (duration decreases more slowly than time). yield changes will change duration 3. Macaulay-duration based: immunizised only if yield curve is flat and changes are parallell rebalancing needed: transaction costs versus immunization target

DBPB wbib DBiB 44 Contingent immunization less-than-perfect immunization for a higher return set a band within the duration is allowed to fluctuate within these bands, active portfolio management to enhance return if duration hits the bounds, back to strict immunization policy Multiperiod immunization a stream of obligations immunize every payment separately Dedicated portfolio cash flow matching no duration requirements no rebalancing 4.5. Bond portfolio duration and convexity calculating duration, modified duration and convexity for a bond portfolio is easy if one assumes that all bonds in the portfolio has the same yield: D P N i i 1 w D i where duration or modified duration of the portfolio weight of bond i in the portfolio duration or modified duration for bond i portfolio convexity is calculated in the same way UExample:U Assume the following bond portfolio: Bond Coupon Price Weight Yield DBMACB DBMODB 1 y. zero 0.00% 961.54 0.4887 4.0000% 1.0000 0.9615 5 y. bullet 5.00% 1005.95 0.5113 4.8631% 4.5475 4.3366 Portfolio 1967.49 1.0000 then: D MAC N i i 1 w D i 0.4887 *1.0000 + 0.5113* 4.5475.8138

and 45 and D MOD N i i 1 w D i 0.4887 * 0.9615 + 0.5113* 4.3366.687 4.6. Butterfly trades: A critical assessment of yield, convexity and duration Butterfly trades are return-enhancement devices that consist of selling an intermediate maturity issue, and a simultaneous purchase of one shorter and one longer maturity issue. A butterfly is deemed beneficial if the average yield of the purchased bonds exceed the yield of the sold bond, while holding duration constant. 4.6.1. Weighting a butterfly assume three bonds indexed as i 1,, and 3, and that we try to obtain a higher return by selling bond and buying bonds 1 + 3 to satisfy the cost constraint the butterfly requires that Q * P Q * P + Q * P 1 1 3 3 where Q P face value of bond (in millions of units of currency) PV of the bond in % of face value the risk constraint on a butterfly requires that Q * PVBP Q * PVBP + Q * PVBP 1 1 1 1 where PVBP price value of a basis point (in EUR per 1 000 000 face value) the equations can be reduced to solve for the values of QB1B QBB Q 1 P * PVBP3 P3 * PVBP * Q P1* PVBP3 P3 * PVBP1 and Q 3 P1* PVBP P * PVBP1 * Q P1* PVBP3 P3 * PVBP1

46 UExample:U (This example is based on actual market conditions on the Finnish Treasury market in November 1997). Consider the following bonds: Bond 1 Bond Bond 3 Coupon 10.00% 9.50% 7.5% P (%) 117.991 16.39 113.360 Yield 5.15% 5.6% 5.81% PVBP (EUR) 379.74 590.50 69.89 Modified duration 3.184 4.6743 6.113 Macaulay duration 3.3841 4.9370 6.4674 Convexity 14.44 30.131 50.03 Note that we express P in percent. The PVBP is however expressed based on EUR 1 000 000 face value. This scaling is a matter of convenience and does not affect the calculations. Q 3 and 16. 39 * 69. 89 113. 360* 590. 50 1000000 53017EUR 117 991 69 89 113 360 379 74 *. *.. *. Q 3 117. 991* 590. 50 16. 39 * 379. 74 1000000 560655EUR 117 991 69 89 113 360 379 74 *. *.. *. Thus you should buy Bond 1 to a face value of 53 017 EUR and Bond 3 to a face value of 560 655 EUR The values satisfy the cost constraint since (53.017*1.17991 + 56.0655*1.13360) 6.773 + 63.5558 16.39 which equals the price of the bond to be sold (Bond ). This also shows that it is straightforward to weight the butterfly as PV in EUR, since the face value Q can be multiplied with the % price P. The values satisfy the risk constraint since (0.530*379.74 + 0.5606*69.89) 0.081 + 388.4719 590.50 which is equal to the PVBP of Bond, and equivalent to using modified durations and PV:s: DMOD(1) DMOD(3) * PV ( EUR) 1 + * PV ( EUR) 3 10000 10000 3.184 6.113 * 67738 + *635551 0.03 + 388.47 590.50 10000 10000 or even simpler, weighting modified durations with the PV:s: (0.4969*3.184)+(0.5031*6.113)4.6743

47 where 0.4969 and 0.5031 are the weights in the butterfly based on the PV:s. The PV of the portfolio is 16.39, and the weights are 6.773/16.39 0.4969 and 63.5558/16.39 0.5031 4.6.. A critical assessment of yield, convexity, and duration calculating yield on the portfolio using the value-weighted average 5.480% the conventional estimate of the portfolio yield would indicate a 13.80 bp loss in yield versus Bond the yield on the portfolio using the duration-weighted approximation: ( 67738* 3. 184 * 0. 0515) + ( 635551* 6113. * 0. 0581) y P 0. 05584 5584%. ( 67738* 3. 184) + ( 635551* 6113. ) thus, in this case, the butterfly produces a yield loss of 3.58 bp the convexity of the portfolio is a weighted average of the bonds or: (0.4969*14.44) + (0.5031*50.03) 3.39 thus, the butterfly gives away 3.58 bp of yield, but increases convexity with.11 the change in the value of a bond is dp DMOD * PV * dy since the trade was structured as a butterfly dpbb dpb1+3b duration-based measures assume parallell shifts in yield curves if this is not the case, a butterfly not a perfect substitute for one single bond the standard deviation in % of a bond is: s D * σ MOD PVBP * σ * 10000 PV the standard deviation of the EUR-value of a bond is S PV * D * σ PVBP * σ * 10000 MOD where σ the standard deviation of the yield change the EUR-standard deviation of a two-bond portfolio is (using duration): P 1 MOD1 1 MOD 1 MOD1 MOD 1, 1 S PV * D * σ + PV * D * σ + * PV * PV * D * D * Corr * σ * σ

48 the correlation of dpbb with dpb1+3b denoted CorrBdPB is Corr dp ( PV1* PV * DMOD1* DMOD * Corr1, * σ 1* σ ) + ( PV3* PV * DMOD3* DMOD * Corr3, * σ 3* σ ) S * S 13, CorrBdPB < 1 unless the yield shifts are parallell UExample:U Suppose the yield change correlation matrix, yield change standard deviations, and EUR standard deviations for bonds 1,, and 3 on: Bond 1 Bond Bond 3 S (EUR) annual 8 460.15 46 963.19 56 311.57 σ (dy) annual 0.007495 0.007953 0.00817 s (%) annual.411 3.7175 4.9675 Correlation matrix (dy) Bond 1 1.0000 0.896 0.859 Bond 0.896 1.0000 0.9467 Bond 3 0.859 0.9467 1.0000 (Yield changes are daily closing quoted bid yields from.1.1997-7.11.1997) E.g. for Bond : sb B 4.6743*0.007953 0.0371 3.71% and S 3.71% * 1 63 90 EUR 46 963.19 EUR The EUR standard deviation for the portfolio Bond 1 + Bond 3: SB1,3B 45 18.05 EUR which is slightly lower than the risk of Bond. The correlation of the change in value of the portfolio and the single bond is: CorrBdPB 0.9606 note that 1. the correlation between value changes is not perfect. due to a yield correlation of 0.859 between Bond 1 and 3, the actual risk of the portfolio is slightly lower although the risk constraint was satisfied 3. the difference in actual risk is a consequence of yield changes not being parallell, and different maturities having unequal standard deviation, which are not captured by using duration based measures only 4. the non-parallell behaviour was captured using yield change correlations, and standard deviation estimates whose measurement is an empirical issue (frequency, length of time-period) 5. the butterfly is not a perfect substitute for a single bond

rb1b rbb tb1b 49 5. Applications of bond mathematics I: FRA:s and bond futures We will consider three major fixed income derivative instruments available on most markets: Forward Rate Agreements (FRA:s), swaps, and bond futures. FRA and swap pricing are straightforward applications of the term structure of spot rates in the interbank market. Bond futures pricing relies on arbitrage between cash and futures markets. 5.1. Forward Rate Agreements FRA:s or Short Interest Rate Futures are used to hedge future short term borrowing or lending in the money market, or to speculate on future interest rates the future interest rate (FRA or future rate) is fixed today underlying instrument: usually a 1- or 3-month deposit or borrowing in a certain reference rate, e.g. 3-month LIBOR or EURIBOR or other money market rate expiration usually on the third Wednesday in March, June, September, and December the buyer makes a future borrowing in the reference rate (buys money) the seller makes a future deposit in the reference rate (sells money) cash settlement how is the rate for an FRA determined? recall from the relationship between spot rates and forward rates that: ( 1+ f ) 1 ( 1+ r ) ( 1+ r ) 1 ( 1+ r )*( 1+ f ) ( 1+ r ) 1 1 using money market notation: [ 1+ r1 *( t1 / 360) ]*[ 1+ f1 *( t1 / 360) ] [ 1+ r *( t / 360) ] where fb1b stands for the FRA-rate, or the forward rate directly: f 1 [ 1+ r *( t / 360) ] [ 1+ r1*( t1 / 360) ] 1 *( 360 / t1 ) where fb1b tb1b the FRA rate interest rate from today to FRA deposit start date interest rate from today to FRA deposit maturity date the FRA deposit lenght in days days from today to FRA deposit start date

tbb tb1b rbrb + deposit depo: 50 days from today to FRA deposit maturity date UExample:U Suppose that on January 17, the -month (actual: 61 day) money market rate is rb1b.97%, the 5-month (actual: 15 days) money market rate is rbb 3.03%, and thus 61 and tbb 15. What is the correct FRA rate for a 3-month (actual: 91 days) that starts 61 days from today? The setting is: Date: January 17 March 19 June 18 Action: FRA made FRA Deposit Days: tb1b 61 tb1b 91 tbb 15 [ +. *( / )] [ 1+ 0. 097 *( 61 / 360) ] 1 0 0303 15 360 f 1 1 *( 360 / 91) 0. 030548 this relation must hold in order to prevent arbitrage otherwise, if e.g. the tb1b + the FRA yields more than the tbb and invest in rb1b fb1b FRA:s are cash settled up front: The gain or loss for the buyer of an FRA: borrow at rbb Gain / Loss t ( rr rfra )* * N 360 t + rr * 1 360 The gain or loss for the seller of a FRA: Gain / Loss t ( rfra rr )* * N 360 t + rr * 1 360 where rbfrab N t the FRA contract interest rate the reference rate, (e.g. EURIBOR) at maturity or closing nominal amount of contracts maturity of the FRA deposit to be made

51 UExample:U On January 17PthP, a firm decides to hedge a EUR 5 000 000 future deposit to be made on March 19PthP. The depo matures on June 18PthP, or in 91 days. The firm sells 5 March FRAs. The bid for FRAs is 3.0548. On March 19PthP, the 3-month EURIBOR is at.96. Gain / Loss 91 (. 0 0306 0. 096)* * 5000000 360 119817... + 1189 7 91. * 100748 1 0 096 360 The gain from the contract is EUR 1189.7. On March 19PthP the buyer thus deposits EUR 5 001 189.7 for 91 days at.96%: 5 001 189.7*[1 + 0.096*(91/360)] 5 038 609.8 To check that this really is the payoff at the FRA contract rate of 3.0548: [5 038 145.1/5 000 000)-1]*(360/91) 0.0306 3.0548% 5.. Bond futures underlying instrument is a physical asset a, typically a coupon paying treasury bond most markets have expiration dates in March, June, September, and December settlement can be either 1) delivery or ) cash the buyer of a future buys the underlying bond on the expiration day the seller of a future sells the underlying bond on the expiration day need to forecast future prices? no the forces of arbitrage again determines the correct futures price 5..1. Futures pricing: The general approach cash and carry pricing: you have two alternative strategies: 1. buy the bond today at the cost PV and finance the purchase by borrowing until future expires. buy future at the futures price F, and pay F when future expires implies that: 1. with both strategies 1 and you own the bond at the future time t. no net cash outlay today 3. cost of both strategies are known today 4. cost of both strategies must equal to prevent arbitrage

5 that is, the futures price F must be: t F S* + r 1 * 360 where where 1. the left-hand side is cost of strategy at time t, and. the right-hand side is the cost of strategy 1 at time t F S r t the futures price the spot price (the price of the bond today PV) the money market rate (interest rate at which the purchase of the bond can be financed) time until future expires (in days) UExample:U The PV of a Government bond is 1 100. The future expires in 60 days, and the 60 day money market rate is 3.50%. What is the futures price F? F 1100 + 1 0 035 60 *. * 360 1106. 4 If the futures price is higher, say, F 1 110 you could: 1. Buy the bond today, and finance it at 3.5% at the cost: 1 106.4. Make a futures contract today to sell the bond at F 1 110 3. After 60 days you deliver the bond to the buyer of the future and get F 1100 in cash and pay off your borrowing 1 106.4 4. Your riskless arbitrage profit is thus 3.58 5... Repo transactions in the bond cash and futures markets a bond can be purchased by financing the purchase in the sale and repurchase (repo) market (if such a market exists) in a repo the financial asset is is lent ( repo d out ) to a second party who in turn as a collateral lends an equal amount of cash to the owner of the asset (in practice the collateral is usually slightly higher than the bond s PV) when the repo is terminated, the second party returns the bond to the owner, and the owner returns the cash loan (collateral) plus interest to the second party the interest rate paid on the cash loan is called the repo rate For simplicity, we assume that:

53 1. collateral for shorting is 100% of the bond s value today. there is only one rate of interest, 3.5% Then, if the futures price is lower, say F 1 105 you could: 1. Short the bond in the repo market (borrow the bond). Pay collateral 1 100, which earns 3.5% interest. Finance the collateral by selling bond at market price S 1 100 3. Make a futures contract to buy the bond back at F 1 105 4. After 60 days you get the bond from the seller of the futures contract, and deliver to the bond lender in 1). 5. You pay F 1 105 for this bond, but get 1 106.4 from interest paying collateral. 6. Your riskless arbitrage profit is 1.4 5..3. Coupon payments suppose a coupon is paid before the future expires then the price of the (reinvested) coupon must be deducted from the futures price F UExample:U Consider the same bond as before, but now the bond pays a 5% (50 EUR) coupon after 30 days. The coupon can be reinvested at the 30 day money market rate (rbrb), which is assumed to be 3.00%. What is the futures price F? t t R F S + r CPN rr + * 1 * * 1 * 360 360 60 F + + 30 1100* 1 0. 035* 50 * 1 0. 03* 1106. 4 5015. 1056. 30 360 360 Why? If you want to hold the bond after months, you can: 1. Buy the cash bond now and finance it at 3.5%. The cost is EUR 1106.4, and the EUR 50 coupon is reinvested at 3.00%: EUR 50.15 which can be deducted from your total costs from buying the bond: EUR 1106.4 - EUR 50.14 EUR 1056.30 total after months.. Buy the future. It's now clear that the futures price F must equal the cost in 1., since F is also paid after months in exchange for the bond, and both strategies lead to holding the bond after months. similar arbitrage positions like demonstrated earlier can be created for a bond that pays a coupon to exploit mispriced futures contracts one has to remember that when shorting a bond, the coupon payments belong to the owner of the bond, not the arbitrageur holding the bond short

54 5..4. Notional bonds and delivery options a common practice on futures exchanges is to use a notional bond as the underlying security a notional bond does not exist (!) the notional bond is defined only subject to 1) face value, ) coupon rate, and 3) a maturity range for example, for LIFFE s Long Gilt Future, the notional bond is simply defined as 100 000 nominal value notional Gilt with 7% coupon several existing treasury bonds can be chosen by the short futures holder to deliver the use of a notional bond instead of a real one, and the use of many deliverable bonds are used so that no market participant(s) can corner the market in the underlying security since the deliverable bonds will have different present values the price paid by the future s buyer at delivery must be adjusted using a conversion factor the conversion factor is determined by the future s exchange and calculated by pricing the deliverable bonds using the yield stated in the futures contract UExample:U A bond futures contract has the (very simplified) following contract specifications: Five-Year Bund Future Unit of trading: Delivery months: Contract standard: Deliverable bonds: EUR 100 000 nominal value notional Bund with 6% coupon March, June, September, and December Delivery may be made of any Bunds on the List of Deliverable Bunds Any Bund with the following characteristics: redemption in a single installment not earlier than 4 years, and not later than 6 years having no terms permitting early redemption bearing interest at a single fixed rate Further, to more clearly demonstrate the process, we assume that all bonds, when delivered, have no accrued interest. Suppose that the List of Deliverable Bunds contains 3 bonds: Bond: Coupon Face value PV calculated PV using Conversion at delivery date contract yield factor (6%) at delivery date 4-year 5% annual EUR 1 000 1 009.90 965.35 0.9653 5-year 6% annual EUR 1 000 1 049.69 1 000.00 1.0000 5-year 7% annual EUR 1 000 1 093.4 1 04.1 1.041 The PV is as usual found by discounting the bond with the term structure of zeros. To find the conversion factor, the bond is discounted with the yield stated in the futures contract. This value is then divided with the face value to find the conversion factor.

55 Since the future s contract is for EUR 100 000 nominal value, and all deliverable bonds have a nominal value of EUR 1 000, this means that 100 bonds must be delivered, no matter which bond the seller decides to deliver. The conversion factor then decides how much the buyer must pay for the 100 bonds to be delivered, such that: Invoice price Contract size * Futures settlement price * Conversion factor + accrued interest Suppose that the futures contract settles at: F 105%, and that the seller decides to deliver the 5-year, 7% bond with conversion factor 1.041, then: Invoice price EUR 100 000 * 1.05 * 1.041 + 0 EUR 109 40.50 Thus, the seller delivers 100 of the 5-year 7% bonds, and the buyer pays EUR 109 40.50 how is the future s price determined when there is no underlying security? usually, one of the deliverable bonds will, despite the use of conversion factors be cheaper to deliver than the other bonds (since the conversion factor is fixed) this bond is called the cheapest to deliver bond (CTD) the CTD-bond can be found by calculating the return from the following exercise for each deliverable bond: 1. Buy the bond.. Make a futures contract to sell the bond. 3. Deliver the bond at the future s settlement date. 4. The return is called the implied repo rate, and the bond with the highest implied repo is the cheapest to deliver bond. because a rational investor will deliver the CTD, the futures will be derived from the CTD-bond the CTD bond may of course change over the life of the futures contract 5..5. Futures pricing using quoted prices and accrued interest the previous examples showed how futures contracts are generally priced in practice, bond futures prices are quoted as clean prices, and we have to take accrued interest into account when calculating the clean futures price cash and carry pricing: you have two alternative strategies: 1. buy the bond today at K + AI and finance [K + AI] by borrowing until future expires. buy future at F + AI today, and pay F + AI when future expires alternatively you could think that you have the money now and can earn interest on [K + AI] by investing this amount in the money market

rbtb 56 implies that: 1. with both strategies 1 and you own the bond at time t (when future expires). cost of both strategies are known today 3. cost of both strategies must equal The price of a bond future can be solved from: [K + (rbcpnb * v/365 * N)]*[1 + (rbtb * t/360)] [F + (rbcpnb * (v+t)/365 *N)] where F t v the future price days to maturity for the future days since last coupon payment money market rate from now to maturity of the future more compactly [K + AIBcB]*[1 + (rbtb * t/360)] [F + AIBtB] where AIBcB AIBtB accrued interest on the contract date accrued interest on the maturity date the left-hand side of the equation: cost of K + AIBcB financed in the money market to maturity date of future the right-hand side of the equation: cost of F + AIBtB on maturity date of future there is no need to forecast future bond prices to price a future rearranging F [K + AIBcB]*[1 + (rbtb * t/360)] - AIBtB UExample:U On January 17PthP, a Treasury bond with 10% CPN bond trades at K EUR 1 18 000. The -month EURIBOR is at.96%. Assume there are 1 days since the last coupon, and 61 days to the maturity of the future. What is the March 19PthP futures price? F [1 18 000 + (0.10*1/365*1 000 000)]*[1 + (0.096*61/360)] - [0.10*(1 + 61)/365*1 000 000] F [1 18 000 + 33 44.66]*[1.000495] - [50 136.99] 1 07 564.6

57 if a coupons are paid between the the contract date and maturity date, the money market rate for investing the coupon must be deducted from K [K + AIBcB]*[1 + (rbtb * t/360)] [F + AIBtB] + Σ CPNBiB *(1 + rbcpn,tb) F [K + AIBcB]*[1 + (rbtb * t/360)] - AIBtB - Σ CPNBiB *(1 + rbcpn,tb) where rbcpn,tb money market rate from the receipt of the i:th coupon to maturity UExample:U On January 17PthP, a Treasury bond with CPN 7.5% trades at K EUR 1 086 43.6. A coupon is received on 18.4. The 5-month (rbtb) EURIBOR is at 3.04%, and the -month (rbcpn,tb) EURIBOR is at.96%. What is the June 18PthP futures price? 1. Buy bond on Jan 17PthP price paid is [K + AIBcB] 1 140 596.87 EUR, financed at money market rate, and total cost is 1 140 596.87 EUR*[1 + (0.0304*15/365)] 1 155 036.51 EUR which is paid on June 18PthP. 7.5% coupon is paid on April 18PthP invested at money market rate to yield 7 500*[1 + (0.096*61/365)] 7 858.65 EUR on June 18PthP 3. Total cost on June 18PthP is [1 155 036.51-7 858.65] 1 08 177.86 EUR 4. Total cost for future on January 18PthP must equal 1 08 177.86 EUR [K + AIBtB] Using the formula: F [1 086 43.6 + (0.075*69/360*1 000 000)]*[1 + (0.0304*15/365)] - [0.075*60/360*1 000 000] - 7 500*[1 + (0.096*61/365)] F [1 086 43.6 + 54 173.61]*[1.01660] - [1 083.33] - [7 858.65] 1 070 094.53

58 6. Applications of bond mathematics II: Swap contracts a forward contract is a contract to exchange one payment in the future a swap contract is a contract to exchange a stream of payments at a series of specified dates in the future swap contracts are usually used together with bonds or loans, to exchange the cash flows (interest rate payments and principal) to some other interest rate base or currency than the originally specified in the bond an interest rate swap is a contract between two parties to exchange fixed interest rate payments against floating interest rate payments (fixed-to-floating swap), or the other way round a currency swap is a contract between two parties to exchange interest rate payments in one currency against interest rate payments in another currency (the interest rates can be either fixed or floating interest) swap contracts are usually OTC contracts, and are made between banks and large corporations an interest rate swap can be regarded (same PV) as a series of interest rate forward contracts a currency swap can be regarded (same PV) as a series of currency forwards swaps are used for: 1. Hedging purposes. A stream of cash flows can be hedged. For example, floating rate loans can be hedged against rising interest rates by swapping the interest rate payements to fixed rate payements, or a EUR based bond issuer might swap a USD-nominated bond to EUR.. Cost saving. A bond issuer might be able to borrow floating rate debt at better terms (lower rates) than fixed rate debt. Then a package of floating rate debt + swap to fixed can be more cost effective than to borrow directly at a fixed rate. A bond issuer might also find for example the domestic bond market saturated, and that demand for bonds in the issuers home currency is low. Then a better price (lower yield) might be obtained in a foreign currency, and a swap back to home currency might be desired to hedge the currency risk. 6.1. Interest rate swaps with interest swaps, a customer (corporation or bank) can transform floating rate debt to fixed rate or the other way round swaps are usually semi-annual and the floating rate in swap contracts is usually 6 month LIBOR or EURIBOR UExample:U Consider a corporation borrowing a floating rate EUR 100 million for 3-years. Interest rate payments are annual, and the reference rate is 1 month LIBOR flat (the company has a good rating and can borrow at LIBOR without a credit spread). The company fears increasing interest rates, and decides to swap the floating payments to

59 fixed. The current 1-month LIBOR rate is 4.00%, and bank offers a fixed swap rate of 4.3885% against 1 month LIBOR. The company agrees to pay the bank an annual 4.3885%, and receives LIBOR from the bank. The cash flows are (EUR millions): LIBOR debt Swap floating to fixed TOTAL Maturity LIBOR Receive LIBOR Pay fixed Debt + swap 1-4.0000 +4.0000-4.3885-4.3885 -LIBOR +LIBOR -4.3885-4.3885 3 -LIBOR +LIBOR -4.3885-4.3885 Note that the payements from the LIBOR debt and the swap LIBOR payments cancel, out and what is left are the fixed rate swap payments. Also note that the first payment of the LIBOR debt is known, since the first payment is set at the reference rate, and reset at the LIBOR rate after each interest rate payment. note that principal (EUR 100 m) is not exchanged the interest rate payments, must however be based on this value, why it is called notional principal 6.1.1. The swap rate how is the fixed swap rate set? the swap rates that swap-dealers quote are usually valid only for well-known counterparties with a good credit rating swap rates are typically slightly higher than the yield for treasuries of the same maturity, but lower than yields on corporate debt swap rates are lower than corporate bond yields since the principal is not exchanged, and hence the counterparty risk is limited to the exchange of interest payments differences in counterparty credit quality can be managed by for example charging an upfront fee, which will be higher for counterparties with lower credit quality, collateral requirements, or other provisions since bond yields depend on the coupon level, the swap rate is defined as a par bond yield (the yield on a bond that trades at par, coupon rate yield) there is a tight link between swap rates and the interest rate futures (money market) markets, since both contracts are priced used the same yield curve, swap dealers often operate on both markets and can use futures for hedging swaps, and since a series of futures can be constructed to replicate a swap the par yield again: 1 Df Par yield T Df where: i 1 T i DfBTB DfBiB the discount factor using zero-coupon rates for the maturity T the discount factors using zero-coupon rates up to maturity T

60 UExample:U Calculate the swap rate if the zero-coupon rates are 4.00% for 1 year, 4.0% for years, and 4.40% for 3 years. Then: Par yield 1 0.8788 (0.9615 + 0.910 + 0.8788) 0.043885 we will now show why this must be the case, since an interest rate swap can be constructed using interest rate forwards 6.1.. The swap rate and FRA-rates the 1-year interest rate forwards fb1b, and fb3b are: (1 + 0.04) f 1 1 0.044004 (1 + 0.40) (1 + 0.044) f 3 1 0.04801 (1 + 0.4) Now, we construct a table where we instead hedge the LIBOR debt with forwards. We buy 1-year FRA:s up to 3 years, which means that settlement is based on (LIBOR-FRA rate). Assume the FRA:s are settled in cash. This means that if the LIBOR exceeds the FRA rate, the difference is paid in cash. Cash flows in EUR millions: LIBOR TOTAL TOTAL Difference debt Maturity LIBOR FRA-rate FRA settlement Debt+FRA Debt+swap Swap FRA 1-4.0000 4.0000-4.0000-4.3885 +0.3885 -LIBOR 4.4004 LIBOR-4.4004-4.4004-4.3885-0.0119 3 -LIBOR 4.801 LIBOR-4.801-4.801-4.3885-0.417 The hedged interest rate payments are clearly not the same as the swap cash flows. How about the present value of the series of FRA:s versus the swap? PV Swap PV FRA 4.3885 4.3885 4.3885 + + 4.197 + 4.0419 + 3.8567 1.1183 3 (1 + 0.04) (1 + 0.04) (1 + 0.044) 4.0000 4.4004 4.801 + + 3.846 + 4.058 + 4.193 1.1183 3 (1 + 0.04) (1 + 0.04) (1 + 0.044) That is, even if the individual cash flows are not equal, the PV of the swap payments equals the PV of the swap. If the principal of EUR 100 million (PV 87.8817) is included in the calculations the PV is of course EUR 100 million.

FBtB FB0B FBtB is and 61 6.1.3. Interest rate swap valuation a swap is priced at par, such that the NPV is zero to both parties after this the value of the swap may change a swap can be valued as either: 1. valued using the same principle as when valuing outstanding forwards: compare the contract rate with the current market rate for the same maturity, and disocunt the difference to present value, the result is a net present value (NPV). valued as a bond, the result is a present value (PV), including the principal the values for outstanding forwards or futures: Long forward: PV ( F 0 ) F F t 0 (1 + r ) t Short forward: where PV ( F 0 ) F 0 F (1 + r ) t t the new forward rate prevailing at time t, for a forward identical to FBTB FBtB the new swap rate for the same maturity T as the original swap to be valued (FB0B) note that since there usually are multiple cash flows left, the equation will be extended to a sum of all the differences FBtB FB0B we also assume that the PV of the floating rate leg is zero at the coupon payment date (not true between coupon payment dates) UExample:U Suppose that after 1 year, the 1 and year zero-coupon rates are 4.10%, and 4.5%. Then the year swap rate is 4.469%, with maturity T equal to the old 3 year swap. What is the market value of the old swap now? We are long in the swap, since we pay the fixed leg, and the bank is short. Then 4.3885% 4.469% and: NPV T Ft F ) (1 + r ) 4.469 4.3885 4.469 4.3885 + (1 + 0.041) (1 + 0.045) 0 ( F0 t i 1 t 0.664 which is in this example is in millions, or roughly EUR -66 400. The negative value for the fixed rate payer is a result of lower swap rates, the fixed rate payer is now paying above the market rate for swaps with equal maturity T. The bank, who is short the swap, has naturally made an equally large gain.

6 Using the bond valuation approach: T CFt PV Bond) (1 + r ) 4.3885 104.3885 + (1 + 0.041) (1 + 0.045) ( t i 1 t 100.664 the swap can naturally be valued at any time, not only at interest payment dates then the value of the floating rate leg is not zero the first payment should be valued as a money market security 6.. Currency swaps now consider a borrower wishing to swap fixed payments in one currency to fixed payments in another currency using the previous 3 year EUR 100 debt issue as an example, consider that the issuer wants to swap the debt to USD the issuer may for example be a US issuer, finding better borrowing terms in EUR, and wishing to hedge EUR outflows or a EUR-based issuer financing its US operations with the debt, and wanting to use the US-revenue to repay the debt an interest rate swap could be regarded as a sequence of FRA:s a fixed-to-fixed currency swap can be regarded as a sequence of currency forwards the swap rates for both legs are set as par yields as before with interest rate swaps, principals are usually exchanged, and a currency swap then consists of the following steps: 1. Initial exchange of currency principals. The currency spot rate is used to determine the principal values. If for exampel S EUR/USD 0.95, and EUR 100 million is swapped to USD, the notional principal in USD is USD 105.6 million. The interest rate payments will be based on these values. (If both parties agree, no principal is exchanged, since the exchange is done at the current spot rate and carries no risk, and thus does not require hedging).. Exchange of interest rate payments during the life of the swap. 3. Exchange back of principal. This step is taken even if principals were not exchanged in 1., since the exchange back is risky and needs to hedged, why it is included in the contract. NOTE!: In these lecture notes a currency quote S(X/Y) means how many units of currency Y does it take to buy 1 unit of currency X. That is EUR/USD 0.95 means it takes 0.95 EUR to buy 1.00 USD. I know this is frustrating for those used to the conventions used in the foreign currency markets. The reason is that this way of quoting is algebraically correct, and hence more easily lends itself to exercises involving inverting the quote, cross-rates, etc. UExample:U Consider a corporation borrowing a fixed rate EUR 100 million for 3-years. Interest rate payments are annual and the rate is 4.3885%. The company wishes to swap the debt into USD. A bank offers to swap these EUR payments against a rate of 4.7906%

63 in USD. The spot currency rate is EUR/USD 0.95, and the USD principal is USD 105.6 million. The interest rate cash flows in %: EUR debt Swap EUR to USD TOTAL Maturity Receive EUR (%) Pay USD (%) Debt + swap (%) 1-4.3885 +4.3885-4.7906-4.7906-4.3885 +4.3885-4.7906-4.7906 3-4.3885 +4.3885-4.7906-4.7906 The interest rate + principal cash flows in currency (millions): EUR debt Swap EUR to USD TOTAL Maturity Receive EUR Pay USD Debt + swap USD 1-4.3885 +4.3885-5.048-5.048-4.3885 +4.3885-5.048-5.048 3-104.3885 +104.3885-110.3059-110.3059 Note that the payments from the EUR debt and the swap payments cancel out and what is left are the USD swap payments. Note the exchange back of the principals. 6..1. Currency swap rates and currency forward rates note that the swap from EUR to USD is identical to a series of FX forwards the swap is a series of forwards to buy EUR against USD the exchange rates for three years implied by the swap: Year 1: EUR 4.3885/USD 5.048 EUR/USD 0.8703 Year : EUR 4.3885/USD 5.048 EUR/USD 0.8703 Year 3: EUR 104.3885/USD 110.3059 EUR/USD 0.9464 when the current spot is EUR/USD 0.95. Is something wrong? calculate the 1,, and 3 years currency forwards using CIRP, assume the USD zeros are 4.50%, 4.65%, and 4.80% for years 1-3: F S 1+ r 1+ r (1 + 0.040) 0.95* (1 + 0.045) 1 d 0 * 1 f 0.9455 and similarly for the and 3 years forwards: EUR/USD 0.9418, and EUR/USD 0.939. Year 0 1 3 EUR debt +100.0000-4.3885-4.3885-104.3885 Swap to USD: USD payments 5.048 5.048 110.3059 PV of USD payments 105.63 4.856 4.6046 95.8330 FX to USD: Forward rate 0.9500 0.9455 0.9418 0.939 USD payments 4.6417 4.6595 111.1505 PV of USD payments 105.63 4.4418 4.546 96.5667

rbrb rbpb 64 Both with the swap and the FX hedge, the firms EUR receipts are set equal to the cash outflow of the debt, and their PV is then EUR 100 m. Note that the PV of the individual cash flows of the swap versus the FX hedge are not equal, but their sums are equal the value of the swap equals the value of the series of currency forwards. 6... Currency swap valuation the currency swap is priced at par, such that the NPV is zero to both parties we will price the currency swap using the bond approach, since now principals matter beacuse they are in different currencies and subject to currency risk the value of the swap is the difference between the two bonds that constitute the swap: PV CF(Receipts) CF( Payments) T T t S t t i 1 (1 + rr ) i 1 (1 + rp ) t where CF(Receipts) cash flow received in the swap CF(Payments) cash flow paid in the swap zero-coupon discount rate for receipt when swap is valued zero-coupon discount rate for receipt when swap is valued S spot currency rate between the two currencies when swap is valued note that receipts and payments are in different currencies, and interest rates used should be for that particular currency UExample:U Suppose that after 1 year, the 1 and year zero-coupon rates in EUR are 4.10%, and 4.5%, and in USD 4.60%, and 4.75%. The spot rate is EUR/USD 0.93. What is the market value of the EUR-USD swap made at EUR 4.3885%-USD 5.048, with principals EUR 100, and USD 105.63? PV PV 4.3885 104.3885 5.048 110.3059 + 0.93* + (1 + 0.041) (1 + 0.04) (1 + 0.046) (1 + 0.0475) 100.664 97.9754.910 which in this example is in millions, or roughly EUR 91 000 for the USD payer. like an interest rate swap the currency swap can naturally be valued at any time, not only at interest rate payment dates

65 7. Pricing credit risk credit risk is usually priced by determining an appropriate additional credit spread on top of a Treasury yield with same maturity to date there exist no established and widely accepted model for pricing credit risk one can roughly distinguish between two approaches: 1. Traditional models. These are not really pricing models, since this approach resembles much of equity analysis and there is no theoretically or other established way in which to in a consistent manner combine company-specific information (accounting etc.) into a yield, credit spread, or price. One can of course use empirical information as guidelines, that is compare issuers with similar characteristics and use price and spread information on these to get an idea of what spread should be charged.. Models based on option pricing. This approach is strongly based on financial pricing theory. Here, the value of the firm is seen as options, where equity seen as a long call (due to limited liability of shareholders), and bonds as short put options. Hence, bonds can be priced as options. 7.1. Credit ratings a credit rating is an assessment of a borrower s willingness and ability to meet its obligations for timely payments of principal debt and interest the most influential credit rating agencies: Moody s, Standard & Poor s, and Fitch issuers pay credit agencies an annual fee for maintaining a credit rating selling a bond issue without ratings can be difficult or impossible ratings scale and definition, long term-debt: Rating agency Brief definition I: Investment grade - High creditworthiness Moody s S&P Aaa AAA Gilt edge, prime, maximum safety Aa1 AA+ Aa AA Very high grade and quality Aa3 AA- A1 A+ A A Upper medium grade A3 A- Baa1 BBB+ Baa BBB Lower medium grade Baa3 BBB- II. High yield - Low creditworthiness Ba1...Ba3 BB+...BB- Low grade, speculative B1...B3 B+...B- Highly speculative III. Predominantly speculative - Substantial risk or in Default Caa CCC+...CCC- Substantial risk, in poor standing Ca CC May be in default, extremely speculative C C Even more speculative than those above D Default Table adapted from: Fabozzi. Frank J. (1997): Handbook of fixed income securities. p. 6

66 the better the rating, the lower the credit spread the markets roughly divide the bond markets into two sectors by rating: 1. Investment grade. Moody s: Aaa Baa3, S&P: AAA BBB-. High yield, or junk bond. Moody s: Ba1 C, S&P: BB+ D the scale for short term debt (commercial paper): Moody s: P-1, P-, P-3, NP (Not Prime) S & P: A-1, A-, A-3, B, C, D credit ratings lowers the barrier for an investor to invest in a bond, since most bond investors think that ratings quite accurately reflect the credit risk of the issuer, and hence there is less uncertainty for the bond investor when making the investment decision ratings are thus a natural starting point for evaluating credit risk note that ratings are only an assessment of credit risk (repayment risk), not interest rate risk, liquidity risk, call risk or other risks ratings are not a recommendation to buy or sell a bond, a good rating does not make a bond a better investment in terms of risk/return -> markets price credit risk and ratings, and lower ratings come with higher yields the ratings and rating agencies have a very strong role in debt markets, since many investors define their investment strategies according to ratings (for example mutual funds), or are prohibited by regulators to invest in certain rating classes (for example insurers) examples of some Nordic issuers (September 003): Issuer Moody's S&P EUR CPN Maturity Spread Swap S ABB Ba3 BB+ 500 9.500 15-01-08 534 5 UPM-Kymmene Baa1 BBB 50 6.350 01-10-09 95 80 UPM-Kymmene Baa1 BBB 600 6.15 3-01-1 107 87 Stora Enso Baa1 BBB+ 850 6.375 9-06-07 70 59 SCA A3 A- 700 5.375 5-06-07 57 35 Elisa Baa BBB+ 300 6.375 31-06-06 141 111 Sonera Group Baa1 A 300 4.65 16-04-09 7 53 Metso Baa3 BBB 500 6.5 11-1-06 155 131 Ericsson LM B1 BB 000 7.875 31-05-06 537 51 7.. The traditional approach to pricing credit risk the most important factors can be summarised into the 6 C:s of credit risk: 1. Character. Management, strategy, track record as borrower (reputation), debt strategy.. Capital. Simply leverage. 3. Capacity. Cash flow, volatility of earnings, competitive position, industry competition, cash liquidity, company structure 4. Cycle. How sensitive is the company to business cycles, what cycle are we in now.

67 5. Collateral. Collateral, it s market value, priority in default, what possibilities to settle with other investors in default process. 6. Covenants. Callability, putability, restrictions on new debt, asset sales, dividends etc. examples of how to mix these (usually) accounting-related variables into an empirical model include for example Altman s (1968) model or later variations of the same idea some special issues concerning high-yield (HY) issuers: 1. Debt structure. i) HY-issuers on average rely more on bank loans which usually are senior to bonds (Cornish, 1990), partly because HY issuers have limited access to CP-markets, ii) bank debt usually floating rate, short term, can lead liquidity problems if short-term rates rise, iii) shortterm debt leads to frequent refinancing risk, and can lead to asset sales if new debt is difficult to raise.. Enterprise structure. Leads to questions like: which part of the enterprise is responsible for the debt, what is the operative and legal structure, can money and assets be transferred from subsidiary to another, what responsibility does the subsidiaries have? 3. Covenants. Understanding covenants becomes more important, as financial distress is more likely. 4. What do the stock markets tell us? HY-debt have more companyspecific risk than investment grade issuers, what information can we use from the stock markets/stock price about the state of the issuer? 7.3. Using option theory to price credit risk the idea of using option theory in bond pricing started with Robert Merton s article in Journal of Finance (1974), hence, the term Merton-like models Merton s simple model deals with valuing a default risky zero-coupon bond, but the analysis can of course be extended (with some additional effort...) to coupon paying bonds, realising that a coupon bond is a series of zero-coupon bond the idea of regarding a bond as an option is the limited upside of the bond: the value of a zero can never exceed it s par value, but the bond still has downside in terms of default risk the example below shows the present value (PV), and Recovery Rate (RR) of a 3-year zero-coupon bond relative to total company assets (belong to bondholders in case of default) the idea is that if asset value is below the bond s par value (100), and the bond defaults, the Recovery Rate is received and 100 RR is then the credit loss the same analysis can be made for each coupon to arrive at a series of present values and adding them up to total bond value

68 10 Merton's model (1974) Bond: N 100, T 3, CPN 0%, r 5% Value of debt (PV and RR) 100 80 60 40 0 RR PV 0 0 10 0 30 40 50 60 70 80 90 100 110 10 130 140 150 Asset value the difference between the standard Black-Scholes-Merton model for pricing stock options, and Mertons model for valuing risky debt is: BSM put option: P f(s, X, r, s, T) Merton s model: L f(a, B, r, sa, T) A B r sa T asset value nominal value of zero-coupon bond riskfree rate (Treasury) volatility of firm s assets (A) maturity of bond in years The model as presented in Saunders et al. (00) and used for the above chart: rt 1 L Be N( h1 ) + N( h ) d 1 h1 s T ln( d) /( s T ), h 1 s T + ln( d) /( s T ) where levarage is measured as: d Be-rT/A

69 For example, if B 100, A 10, d 0.80, T 3, r 0.05, s 0.1 Bond price (L) 84.63 Spread 0.56% Price of defaultfree bond 86.07 the main difference is that the underlying asset is now the assets of the issuer, and volatility is then the volatility of these assets this simple model suffers from obvious problems: 1. What is the market value of the firm s assets A?. What is the volatility of these assets and how to measure it? 3. Based on European option pricing models, but default can happen at any time. 4. Model allows default only if A 0, in practice must happen before this. extension of this idea include the Black & Cox (1976) model where default can happen before all assets are exhausted (A 0) KMV (now a part of Moody s) have attempted to solve problems 1-4 by: 1. Attempting to model the statistical process of A over time (how does the value of A evolve over time, and what is its volatility).. A is modeled by modeling the stock price (E) behavior, and using leverage as a link between the stock price process and asset value process. The less debt, the closely A follows E. 3. Default if A goes below a specified value (makes this a barrier option). problems still remain: 1. How to in a reliable way model asset value process or default process?. How to reliably estimate recovery rate in case of default? 3. How to define the level of A that triggers default? 4. Does default occur because of low A or other reasons not (yet) in the model such as liquidity crunch? 5. How to model the evolution of leverage since the relation between A and E depends on this. 6. Very difficult to include all covenants and options (callability, putability) in the model. the more realistic the model, the more difficult it becomes to estimate, use and understand it -> significant model risk! the state of the world today: no accepted model 7.4. Default probabilities, rating transitions, recovery rates and how to use them to estimate bond returns what is the default probability (DP), recovery rate (RR), or loss given default (LGD) empirical results on US bond data by Altman & Kishore (1998) on high-yield:

70 Year DR(%) RR(%) Loss AADR 71-97.613 78-97.849 4.9 1.83 85-97 3.745 WADR 71-97 3.311 78-97 3.34.18 Median 71-97 1.5 Max. 91 10.7 36.0 7.16 Min. 81 0.16 1.0 0.15 AADR WADR RR Loss arithmetic average of default rates (DR) par-value weighted average of default rates recovery rate as % of par value (LGD + next coupon) x DR note the rise and variability in the figures some statistics on recovery rates and seniority of bonds: Seniority RR(%) Senior secured 58.7 Senior unsecured 48.9 Senior subordinated 35.0 Subordinated 31.7 Zeros 0.7 investing in bonds with high spreads (yields) does not necessarily lead to high returns: the risks priced into the spread might be realised causing a further widening of the spread and price decline and loss of return rating ans spread changes are an essential part of corporate bond investing 1-year rating transition probabilities (RTP) by Carty & Fons (1994), using Moody s ratings 1970-1994: Rating Rating at end of year (T 1) T 0 Aaa Aa A Baa Ba B C, D Total Aaa 91.90 7.38 0.7 0.00 0.00 0.00 0.00 100 Aa 1.13 91.6 7.09 0.31 0.1 0.00 0.00 100 A 0.10.56 91.0 5.33 0.61 0.0 0.00 100 Baa 0.00 0.1 5.36 87.94 5.46 0.8 0.1 100 conclusions: 1. Downgrades more likely than upgrades (fallen angels more likely than rising stars). Spread differences between rating categories widen with lower ratings, adding to return loss due to price decline.

71 3. Adding up 1 and : Return < initial spread how use historical bond market information to estimate returns? 1. Estimate the bond s RTP for the holding period.. Calculate the expected price change due to rating and spread change. 3. Calculate the expected return, take into account coupon payements and reinvestment. for example, assume a 3-year AA-rated bond that now trades at a 30 bp spread, and you want to estimate the expected return over a 1-year holding period: Horizon Horizon Return Transition Excess rating spread over Treas. probability return Aaa 5 38 1.13% 0.43 Aa 30 30 91.6% 7.38 A 35 1 7.09% 1.49 Baa 60-4 0.31% -0.07 Ba 130-147 0.1% -0.31 8.9 Horizon spread Return over Treasuries Excess return spread now return in excess of Treasury yield assuming that the rating of the bond changes to rating XXX (for example, a downgrade to A increases spread to 35 bp and lowers the price that of the initial spread of 30 bp, only 1 bp is left) Transition probability x Return over Treasuries using these values, the expected return of a AA-bond is not the initial spread of 30 bp, but 8.9 bp some figures based on historical averages for different rating classes over different holding periods (highest returns are bolded): 3-year 5-year 10-year Rating Initial Excess Initial Excess Initial Excess T 0 spread return spread return spread return Aaa 5.0 4. 30.0 8.4 35.0 31.7 Aa 30.0 8.9 34.0 31.4 40.0 30.3 A 35.0 31.1 45.0 37.3 55.0 37.9 Baa 60.0 46.3 70.0 39.9 85.0 1.9 that is for both 3- and 5-year horizons, Baa has the highets returns, but for a 10-year horizon, A is the best rating class note that all such estimates are based on historical averages

7 7.5. Selected empirical results on spreads Bevan & Garzarelli (000): Moody s Baa-index spread 1% increase in GDP growth -> +18 bp 1% increase in financing gap -> +46 bp 1 standard deviation increase on S&P 500 volatility -> +5 bp in the long run, correlation between spreads and benchmark yields positive, in the short run negative Campbell & Taksler (00): AA BBB rated bonds spreads decrease with S&P equity index returns spreads increase with increase in S&P equity volatility stock market volatility more important factor than ratings ratings explain more of spread than accounting information equity volatility more important factor for bonds with longer duration Collin-Dufresne, Goldstein, Martin (001): leverage increases spreads, stock returns decreases spreads S&P 500 returns 7 times more important factor than company specific equity return Treasury yield increases decreases spreads S&P 100 index option implied volatility increases spreads only 5% of spreads can be explained market-wide factors more important than company-specific factors, not consistent with Merton-type models 75% of spreads unexplained, is strong bond/stock market segmentation the reason? liquidity, supply-demand shocks explanations?

i Mind-expanding reading (NOT required for the CEFA-exam): Das, Satyajit (1994): Swap & Derivative financing. Irwin Professional Publishing. Eales, Brian A. (1995): Financial risk management. McGraw-Hill. Fabozzi, Frank J. (000): Bond markets, analysis and strategies. Prentice-Hall International. Galitz, Lawrence C. (1995): Financial engineering: Tools and techniques to manage financial risk. Irwin Professional Publishing. Garbade, Kenneth D. (1996): Fixed income analytics. The MIT Press. de la Grandville, Olivier (001): Bond pricing and portfolio analysis. The MIT Press. Hull, John C. (000): Options, futures, and other derivative securities. Prentice-Hall International. Jorion, Philippe (1996): Value at risk: The new benchmark for controlling market risk. Irwin Professional Publishing. Sundaresan, Suresh M. (1997): Fixed income markets and their derivatives. South- Western. Tuckman, Bruce (1996): Fixed income securities: Tools for today s markets. John Wiley & Sons.