Fixed Income Management 2024/2025 PDF

MSc Banking and Finance Fixed Income Management 2024/2025 Building Competence. Crossing Borders. Peter Schwendner [email protected] Short Introduction - Peter Schwendner 1991 - 1995 Diplom in Physics, University of Goettingen 1995 - 1998 Ph.D., Max-Planc...

MSc Banking and Finance Fixed Income Management 2024/2025 Building Competence. Crossing Borders. Peter Schwendner [email protected] Short Introduction - Peter Schwendner 1991 - 1995 Diplom in Physics, University of Goettingen 1995 - 1998 Ph.D., Max-Planck-Institut Goettingen 1998 - 2009 Sal. Oppenheim jr. & Cie., Frankfurt: Head of Quantitative Research at Trading Department Developing Equity Derivatives Analytics and Products CFA and FRM certifications 2009 - 2013 Fortinbras Asset Management, Eschborn: Partner, COO Developing Fixed Income Overlay Strategies Since 2013 ZHAW / Institute for Wealth and Asset Management 2 Fixed Income Management Lecture Introduction: Current market situation US Treasury Bonds / valuation / discounting / arbitrage Bond prices, spot and forward rates Yield to maturity; Generalizations and Conventions Term structure Interest rate risk management; Bond futures Swaps, corporate bonds, credit risk Practical exercises using Excel to support your intuition FinanceLab: interactive Rotman Interactive Trader (RIT) cases to better understand dynamics of instruments and risk as markets move 3  Recommended readings Tuckman, Bruce & Serrat, Angel – Fixed Income Securities, 2011 => Trading-oriented, focus on US Treasuries => Most examples from the lecture correspond to this book Martelllini, Lionel; Priaulet Philippe; Priaulet, Stephane – Fixed-Income Securities, 2003 => More in a «model world» compared to Tuckman Petitt, Barbara; Pinto, Jerald; Pirie, Wendy – Fixed Income Analysis, 2015  CFA Literature Advanced pricing blog: https://blog.deriscope.com/index.php/en/ 4 Some WSJ and FT headlines from bond markets Some WSJ and FT headlines from bond markets Trends at the U.S., German, UK, Swiss and Japanese sovereign bond market: yield of 10Y bonds Only few traders have experience with rising interest rates. 7 Trends at the U.S., German, UK, Swiss and Japanese sovereign bond market: yield of 10Y bonds Only few traders have experience with rising interest rates. 8 Trends at the U.S., German, UK, Swiss and Japanese sovereign bond market: performance of futures on 10y bonds From 1980-2020, falling interest rates caused capital gains for bonds in addition to coupon income. In 2022, interest rates were rising and bond markets were falling. 9 Debt is a substantial part of the global capital stock Source: Gadzinski/Schuller, 2016 10 Government debt is primarily owned by domestic investors 11 US is most important bond market, followed by Euro area Source: Gadzinski/Schuller, 2016 12 Case Study: 10Y European sovereign bonds Greek 2015 elections announced Financial crisis becomes Euro area debt crisis. Yield volatilities Italian budget spike up, yield 2018 levels diverge. Convergence before EUR introduction Covid Year 13 Correlations of 10Y European Sovereign Bonds, daily data 14 Correlations of 10Y European Sovereign Bonds, daily data Ref.: Schwendner, Peter; Schüle, Martin; Ott, Thomas; Hillebrand, Martin: European government bond dynamics and stability policies : taming contagion risks. Journal of Network Theory in Finance. 1(4), 1-25 (2015). 15 Correlation regimes from a clustering of yearly correlation matrices “Cityblock” distance matrix between yearly correlation matrices of daily data Average correlations within the k=4 regimes Regime time series as a result from k-means clustering of the distance matrix Ref.: Papenbrock, J., Schwendner, P.: Handling risk-on/risk-off dynamics with correlation regimes and correlation networks. Financ Mark Portf Manag 29, 125–147 (2015). 16  Key observable: Italian spread 10Y bond spread Italy – Germany as observable of European convergence or divergence: small spreads imply convergence in funding cost between European “core” and “periphery” and are interpreted to reflect also political convergence. Treasury-Bond- Valuation, Discounting, Arbitrage 18  Largest homogeneous group of bonds: US Treasury Bonds  Elements of a «Termsheet» («bond indenture»)  Coupon rate  # payments per year  Time to maturity/expiry  Face value / notional / nominal  Example on 15.8.1998  a US Treasury-Bond with coupon 51/4%, expiry Aug/15/2003 and a nominal issued amount of $10,000 …  … pays a semiannual coupon of $262.5 ($10,000 times 5.25%/2)  … every six months until Aug/15/2003, as well as the redemption of the $10,000 face value at maturity. 19 «Rolling out» Cashflows of a Term sheet  Example on 15.8.1998  a US Treasury-Bond with coupon 51/4%, expiry Aug/15/2003 and a nominal issued amount of $10,000 …  … pays a semiannual coupon of $262.5 ($10,000 times 5.25%/2)  … every six months until Aug/15/2003, as well as the redemption of the $10,000 face value at maturity.  The notional value is the reference amount to calculate the coupon and redemption cash flows. 20 Quoting market prices Example for treasury bond quotes at 15.2.2001 Quoted in % of notional “ ”  Coupon Expiry Price 12 ticks = 12/32  7.875% Aug/15/01 101-123/4 Half “ ” tick  14.250% Febr/15/02 108-31+  6.375% Aug/15/02 102-5  6.250% Febr/15/03 102-18 1/8  5.250% Aug/15/03 100-27 Quotation is in 1/32 21 Computing discount factors First step: transform quote into decimal price 101+(12 + ¾)/32= 101.3984375 original Bonds Coupon Expiry Quoted Price Price as decimal 7.875 15.08.2001 101-12 3/4 101.3984375 14.250 15.02.2002 108-31+ 108.984375 6.375 15.08.2002 102-5 102.15625 6.250 15.02.2003 102-18 1/8 102.5664063 5.250 15.08.2003 100-27 100.84375 This decimal price is the market value in percent of the nominal value. Example: Buy 10.000$ nominal of the first bond Market value: 10.000$ * 101.3984375% = 10139.84375 $ “today”: 15.2.2001 22 Computing discount factors Second step: define time grid and roll out cash flows original Bonds Roll out cashflows Coupon Expiry Quoted Price Price as decimal 15.08.2001 15.02.2002 15.08.2002 15.02.2003 15.08.2003 7.875 15.08.2001 101-12 3/4 101.3984375 103.9375 14.250 15.02.2002 108-31+ 108.984375 7.125 107.125 6.375 15.08.2002 102-5 102.15625 3.1875 3.1875 103.1875 6.250 15.02.2003 102-18 1/8 102.5664063 3.125 3.125 3.125 103.125 5.250 15.08.2003 100-27 100.84375 2.625 2.625 2.625 2.625 102.625 23 Computing discount factors Third step: compute first discount factor, then iteratively the others DF(T=0.5) = PV1(T=0) / Cashflow1(T=0.5) original Bonds Roll out cashflows Results Coupon Expiry Quoted Price Price as decimal 15.08.2001 15.02.2002 15.08.2002 15.02.2003 15.08.2003 T DF 7.875 15.08.2001 101-12 3/4 101.3984375 103.9375 0.50 0.975571 14.250 15.02.2002 108-31+ 108.984375 7.125 107.125 1.00 0.952471 6.375 15.08.2002 102-5 102.15625 3.1875 3.1875 103.1875 1.50 0.930448 6.250 15.02.2003 102-18 1/8 102.5664063 3.125 3.125 3.125 103.125 2.00 0.907962 5.250 15.08.2003 100-27 100.84375 2.625 2.625 2.625 2.625 102.625 2.50 0.886303 DF(T=0.5) = Price1 / Cashflow1(T=0.5) = = 101.3984375 / 103.9375 = 0.975571 24 Computing discount factors Third step: compute first discount factor, then iteratively the others DF(T=0.5) = PV1(T=0) / Cashflow1(T=0.5) original Bonds Roll out cashflows Results Coupon Expiry Quoted Price Price as decimal 15.08.2001 15.02.2002 15.08.2002 15.02.2003 15.08.2003 T DF 7.875 15.08.2001 101-12 3/4 101.3984375 103.9375 0.50 0.975571 14.250 15.02.2002 108-31+ 108.984375 7.125 107.125 1.00 0.952471 6.375 15.08.2002 102-5 102.15625 3.1875 3.1875 103.1875 1.50 0.930448 6.250 15.02.2003 102-18 1/8 102.5664063 3.125 3.125 3.125 103.125 2.00 0.907962 5.250 15.08.2003 100-27 100.84375 2.625 2.625 2.625 2.625 102.625 2.50 0.886303 Discount factors DF(T=0.5) = Price1 / Cashflow1(T=0.5) = 1 = 101.3984375 / 103.9375 = 0.975571 0.95 DF(T=1) = (108.984375– DF(T=0.5) * 7.125) / 107.125 = 0.9 = 0.952471 0.85 0.8 0.50 1.00 1.50 2.00 2.50 Maturity [years] 25 Apply discount factors: Valuation of other bonds 26 Apply discount factors: Valuation of other bonds Idea: can we use the «relatively cheap» 10 ¾% bond and the original bonds to set up a replication portfolio to realize the difference of the market prices from their valuation? 27 Apply discount factors: replication portfolio Idea: can we use the «relatively cheap» 10 ¾% bond and the original bonds to set up a replication portfolio to realize the difference of the market prices from their valuation? Trade: buy this 10 ¾% Bond, short positions in the original bonds: How can we calculate the necessary («face amounts») of these original bonds? 28 Apply discount factors: replication portfolio Solution: «Cash-Flow-Matching»: determine nominal values of the original bonds so that their cash flows mirror the cash flows of the «cheap» 10 ¾ bond. 29 Apply discount factors: replication portfolio  Fi = nominal of bond i in the replication portfolio    61   F1  0 + F2  0 + F3  0 + F4  100 + 4 % = −105.375  F = −102.182   2  4       63  61  F1  0 + F2  0 + F3  100 + 8  % + F4  4 % = −5.375  F = −2.114   2  2 3       14 1 4  6 38 6 14   F1  0 + F2  100 +  % + F3  % + F4  % = −5.375  F2 = −1.974   2   2 2     7 78  14 1 4 6 38 6 14   F1  100 +  % + F2  % + F3  % +F4  % = −5.375  F1 = −1.899   2   2 2 2  Solution: «Cash-Flow-Matching»: determine nominal values of the original bonds so that their cash flows mirror the cash flows of the «cheap» 10 ¾ bond. 30 Implications for bond valuation  The proceedings of selling the replication portfolio equal the theoretical value of the «cheap» 103/4 bond.  How can we valuate bonds?  1) Compute discount factors from a set of reference bonds and compute present value of the new bond.  2) Alternative: use reference bonds to replicate the cash flows of the new bond. The value of the replication portfolio is then the value of the new bond. 31 Arbitrage gains  If we buy the «cheap» 103/4 bond and sell the replication portfolio, wie can theoretially earn 0.103% today without risk.  If we invest $500 Mio: Arbitrage gain of $513,512.  Really?  Is this possible with liquid instruments like treasury bonds?  Usually, the market of liquid instruments would move until the arbitrage opportunity disappears.  Why is there a theoretical arbitrage opportunity? 32 Exercises (Tuckman Chapter 1) 33 Exercises (Tuckman Chapter 1) 34 Bond prices, spot rates and forward rates 30 35 Spot Zero-Coupon-Rates  Spot rates: refer to an interest rate period that begins immediately (in contrast to a forward)  Spot Zero-Coupon-Rates are the annualized rates of a simple zerobond 1 t = B(0, t ) (1+ R 0,t )  B(0,t) is market value of a bond that pays $1 at time t. This is the definition of a discount factor. General valuation formula: T T P0 =  =  CFt  B(0,t ) CFt t t=1 (1+ R 0,t ) t=1 36 Semiannual Compounding  Relevant as convention for US Treasuries, as those pay seminannual coupons. Valuation formula gets more complex: 2T 2T  =  CFt  B(0,t ) CFt P0 = t t =1  R0,t  t =1 1 +   2   The semiannually compounded future value w for the present value x in T years at an annual rate r becomes:  2T  r w = x1+   2  Let’s look at a 2-year zero coupon bond trading at $92 with a notional of $100. The 2-year zero-coupon-rate R(0,2) matches the equation: 100 92 = 4  R0,2 = 4.21%  R0,2  1+   2  37 Time basis and compounding frequency  Time basis → rates are usually states on a yearly basis (p.a.)  But: «compounding» frequencies can be different: Examples:  Invest $100 at 6% for two years; yearly rate with semiannual compounding  100 × (1+ 6%/2) after 6 months  100 × (1+ 6%/2)2 after 1 year  100 × (1+ 6%/2)3 after 1.5 years  100 × (1+ 6%/2)4 after 2 years  Invest $100 at 6% for one year with monthly compounding:  100 × (1+ 6%/12) after 1 month  100 × (1+ 6%/12)2 after 2 months  …. 100 × (1+ 6%/12)12 = $106.1678 after 1 year  Equivalent to a 6.1678% effective rate with yearly compounding. 38 Transform rates belonging to different compounding frequencies In general Amount x invested at annual rate r , but n times compounded during one year for T years Gets the end value r x(1+ )nT n Effective annual rate (paid out once a year) ra is defined as solution of the equation r x(1+ ) nT = x(1+ ra )T n n r a =  1+ r  −1  n 39 Continuous compounding If the rate r is compounded continuously, n tends to infinity and we get: Effective annual rate ra:  The effective annual rate of a continuously compounded rate of 6% ist e6% –1 = 6.1837%  ra>r; to get the same end value, a continously compounded rate needs a smaller rate than a larger compounding frequency. 40 Annualized spot zero rates that belong to semiannual compounding derived from discount factors 41 Annualized US spot zero rates for longer terms at 15.2.2001 42 Forward rates = Rate for a forward loan (a contract to lend money later and to pay it back even more later)  Let’s compare two alternatives for a 2-year- horizon with yearly compounding:  Alternative A: buy 2-year-zerobond  Alternative B: buy 1-year-zerobond and roll out cashflows for a further year  What yield from year 1 to year 2 will make you indifferent between the two choices?  F0,1,1= the rate that can be guaranteed now for a loan starting in 1 year and repayable 2 years from now. (1+ R0,2 ) 2 1+ F0,1,1 = (1+ R0,1 ) 43 Forward rates – general formula The forward zero-coupon rate between the years x and y given annual compounding is Example 1  Let the spot zero rate with annual compounding be 4% p.a. and the 18-month rate be 4.5% p.a.  Compute the 6-month forward rate starting in 1 year. (1+ R0,18m )3/ 2 = (1+R0,1 )(1+ F0,12m,6m )1/2 (1.045)3 / 2 = (1.04)(1+F0,12m,6m )1/ 2 ; F0,12m,6m = 5.5072% «from T=12M for 6 months» 45 Example 2  Same numbers, but with semiannual compounding: 3 2  R0,18m   R0,1   F0,12m,6m  1+  = 1+  1+   2   2   2  F0,12m,6m = 5.5037% 46 6-month forwards as of 15.2.2001, computed from spot rates with semiannual compounding 5.050% 5.000% 4.950% 4.900% 4.850% 4.800% 4.750% 4.700% 4.650% 4.600% 4.550% 0.5 1 1.5 2 2.5 Spot rate f_T-1,T 47 Spot and Forward curve of the US Treasury Market as-of 15. February 2001 48  Spot and Forward curve of the US Treasury Market as-of 4. November 2016 (Bloomberg: FWCV) 12M Forward curve of 4.11.2016 6M Forward curve of 4.11.2016 Spot curve of 4.11.2016 49  Spot and Forward curve of the US Treasury-Markt as-of 4. November 2015 12M Forward curve of 4.11.2015 Actual spot curve of 11.2016 Spot curve of 4.11.2015 50 Which bond maturity should an investor pick? (I)  When should we pick short-term or long-term bonds?  Investor A invests $10,000 in rolling 6-month zerobonds from today until in 2 ½ years.  Investor B invests $10,000 in a single bond with a coupon of 51/4 and expiry 15.8.2003 at a price of 100-27, and reinvests paid coupons in rolling 6-month zerobonds.  Which end value do the two investors achieve, if both decide their investment strategy at 15.2.2001? 51 Which bond maturity should an investor pick? (II)  Assumption: the forwards as computed at 15.2.2001 become the later spot rates.  End value for investor A as of 15.8.2003:  5.008%  4.851%  4.734%  4.953%  4.888%  $10,0001+ 1+ 1+ 1+ 1+  = $11,282.83  2  2  2  2  2   For investor B:  Pays $10,000 and buys $9,916.33 nominal value of this bond  Each coupon payment = $9,916.33×5.25%/2=$260.3 52 Which bond maturity should an investor pick? (III) For Investor B:  Coupon I gets reinvested at forwards until 15 August 2003:  4.851%  4.734%  4.953%  4.888%  $260.301+ 1+ 1+ 1+  = $286.52  2  2  2  2   Coupon II gets reinvested at forwards until 15. August 2003:  4.734%  4.953%  4.888%  $260.301+ 1+ 1+  = $279.74  2  2  2   End value of Coupon III =$273.27; Coupon IV=$266.67; Coupon V=$260.30  Redemption of nominal at 15. August 2003=$9,916.33  Total = $11,282.83 53 What are conditions for investor A to make the better deal?  Example: the 6-month rate stays at 5.008% until 15.8.2003  Investor A:  5.008%  5 $10,0001+  = $11,316.29  2   Investor B: $260.3  5.008% 5    1+  −1 + $9,916.33 = $11,284.66 5.008%  2    2 General: End value of N coupons C, rate i: 𝐶 [(1 + 𝑖)𝑁 −1] 𝑖 54 What are conditions for investor B to make the better deal?  Example: at the day after the initial investment (16.2.2001), the 6-month rate falls to 4.75% and stays there until 15.8.2003  Investor A:  4.75%  5 $10,0001+  = $11,245.26  2   Investor B: $260.3  4.75% 5   −1 + $9,916.33 = $11,281.14 4.75%  1+  2   2 55 Exercises (Tuckman Chapter 2) 56 Yield To Maturity (YTM); 57 Yield to Maturity (YTM) =if we discount all cash flows using the YTM, we get the present value of the bond.  T: number of years at annual compounding T P=  CFt t t =1 (1+YTM )  YTM reflects the internal rate of return (IRR) of the bond cash flows  YTM can be computed using a solver algorithm (e.g. Excel solver). 58 YTM and present values  Bond with semiannual coupons c, nominal value F and YTM y: 2T Fc/2 F Fc 1 + F P = i+ 2T = 1− i=1 (1+y /2) (1+y /2) y  (1+y/2)  (1+y/2) 2T 2T  c=100y; F=100, P=100 «at par»  c>100y; F=100, P>100 «above par»  c Dividing both sides by (1-z) proves the proposition. Application: 61 Prices for bonds with YTM=5.5% at different coupons and years to maturity 62 YTM as «total return»  Today, we buy a 3-year, 5% bond with YTM=10% for $87.57  This bond pays yearly a $5 coupon and the nominal of $100 at redemption.  Thus, we get: $5 after one year, $5 at the end of the 2nd year, and $105 at the end of the third year.  If we invest the intermediate cash flows at 10%, we receive at the end of the third year:  5×(1.1)2 +5 ×(1.1)+105=$116.55  Our investment generates a total return of y, so that  (1+y)3=116.55/87.57 => y=10%  But: this does not account for reinvestment risk! 63 More precise… If the YTM of a bond is constant across a 6-month period, the realized total return across this period is the YTM. 𝑃0 : today’s PV of a bond with T years to expiry 𝑃1/2 : PV just before the next coupon at unchanged YTM level during the 6-month period until then. c/2 c /2 1+c/2 P0 = + +...+ 1+ y /2 (1+ y/2) 2 (1+ y/2) 2T c c /2 1+ c/ 2 P1 = + +...+ 2 2 (1+y / 2) (1+ y / 2) 2T−1  P1  P1 = (1+y / 2)P0  y = 2 2 −1 2  P0    64 YTM and spot-zero-rates PV formula for a 6 ¼% Treasury Bond with 2 years to expiry, 1 market price 102 − 18 8 above: expressed with YTM, below: with separate spot-zero-rates for each cash flow date 3.125 3.125 3.125 103.125 102+18.125/ 32 = + + + 1+ y /2 (1+ y / 2) (1+ y / 2) (1+ y / 2)4 2 3 3.125 3.125 3.125 103.125 = + + + 1+ R0.1 /2 (1+ R0.2 / 2) (1+ R0.3 / 2) (1+ R0.4 / 2)4 2 3  The YTM of 4.8875% is a weighted average of four discrete spot- zero-rates (5.008%, 4.929%, 4.864% und 4.886%) ’  The YTM is most alike the last spot-zero-rate, as the «center» of the cashflows is near the last payment on a time axis. 65 If the term structure of zero rates…  rises:R0.4>R0.3>R0.2>R0.1  then YTM < R0.4  is flat: R0.4=R0.3=R0.2=R0.1  then YTM= R0.4  falls: R0.4 2.99: «safe zone» several financial ratios as input to 1.81 < Z < 2.99: «grey zone» classify companies into three «zones» Z < 1.81: «distress zone» of increasing default probability. E. Altman, “Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy,” Journal of Finance, September 1968; https://doi.org/10.2307/2978933 140  Rotman credit risk case: combine Merton and Altman Z-Score models Financial statements News Spread derived from current rating Merton model Altman Z-Score model Output: Implied Output: classification credit spread Expected credit spread using weighted probabilities for up- and downgrades Fair value for corporate bond Compare with market value RIT - Case Brief - FI6 - Credit Risk.pdf and make trading decision 141 CDS contracts: hedge credit risk 142