Chapter 4: Common Stocks. Chapter 5: Forwards and Futures

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1 Part B Valuation Chapter 3: Fixed Income Securities Chapter 4: Common Stocks Chapter 5: Forwards and Futures Chapter 6: Options Lecture Notes

2 Introduction Part B Valuation We have learned that: Business decisions often reduce to valuation of assets/cfs Two elements are important in valuing a CF: time and risk Value of CFs is determined in financial markets From the market, we can learn How time affects value --- time value of money How risk affects value --- risk premium In particular, Two special cash flows: annuity and perpetuity (with and without growth) In this part of the course, we study the market valuation of Bonds Stocks Forwards and futures Options Lecture Notes 2

3 Finance Theory I Haoxiang Zhu MIT Sloan School of Management Lecture 3: Fixed-Income Securities Lecture Notes 3

4 Key concepts Lecture 3: Fixed income securities Overview of fixed-income i markets Term structure of interest rates Spot rates and zero-coupon bonds Yield to maturity and coupon bonds Interest rate risk: Duration and Convexity Forward interest rates Inflation risk Default risk Theories of interest rates Readings: Brealey, Myers and Allen, Chapters 3, 23, 24 Lecture Notes 4

5 Fixed-income income securities Lecture 3: Fixed income securities Fixed-income securities are financial claims with promised cash flows of fixed amount paid at fixed dates. There are exceptions to this definition (e.g. floating rate notes). Fixed-income securities: 1. Treasury/sovereign securities: U.S. Treasury securities (bills, notes, bonds) Bunds, JGBs, U.K. Gilts 2. Federal agency securities: Securities issued by federal agencies (FHLB, FNMA ) 3. Corporate securities: Commercial paper (CP) Corporate bonds 4. Municipal securities (Munies) 5. Mortgage-backed securities (MBS) Derivatives: Interest rate swaps, credit default swaps, etc. Lecture Notes 5

6 Overview of FI markets Lecture 3: Fixed income securities US Debt Market Outstanding ($B, 2012 Q3) Money Market, 2,442.4 Asset Backed, 1, Municipal, 3,719.4 Agency, 2,358.4 Municipal Treasury, 10,716.1 Treasury Corporate, 8,583.8 Mortgage Source: SIFMA Mortgage, 8,205.2 Corporate Agency Money Market Asset Backed Lecture Notes 6

7 Overview of FI markets Lecture 3: Fixed income securities Outstanding U.S. Bond Market Debt, Billions USD (Source: SIFMA) Mortgage Corporate Federal Agency Municipal 7 Treasury 1,6 Related 2,6 Debt Securities 5,6 Money Markets 3 Asset-Backed 4,6 Total , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,914.4 Lecture Notes 7

8 Overview of FI markets Lecture 3: Fixed income securities Issuance in the U.S. Bond Markets, USD Billions (Source: SIFMA) Corporate Federal Agency Year Municipal Treasury 1 Mortgage-Related 2 Debt 3 Securities Asset-Backed Total , , , , , , , , , , , , , , , , (4) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Interest bearing marketable coupon public debt. 2 Includes GNMA, FNMA, and FHLMC mortgage-backed securities and CMOs and private-label MBS/CMOs. 3 Includes all non-convertible debt, MTNs and Yankee bonds, but excludes CDs and federal agency debt. 4 Beginning with 2004, Sallie Mae has been excluded due to privatization. Lecture Notes 8

9 Organization of FI markets Lecture 3: Fixed income securities Issuers: 1. Governments 2. Corporations 3. Commercial Banks 4. States 5. Municipalities 6. SPVs 7. Foreign Institutions Intermediaries: 1. Primary Dealers 2. Other Dealers 3. Investment Banks 4. Credit-rating Agencies 5. Credit Enhancers Investors: 1. Governments 2. Pension Funds 3. Insurance Companies 4. Commercial Banks 5. Mutual Funds 6. Hedge Funds 7. Foreign Institutions 8. Individuals Lecture Notes 9

10 Main Features of Bonds Issuer: US Treasury/Government States, municipalities, and agencies Corporations Foreign governments (sovereign bonds) 2. Term (number of years to maturity): Short (less than 1 yr) T bills, CD s, Commercial papers Long (more than 1yr) T bonds bonds, corporate bonds 3. Price vs. par value = face value par bond discount bond (price < face value) premium bond (price > face value) Lecture Notes 10

11 Main Features of Bonds Coupon Coupon rate: total annual interest payment per dollar face value Coupon frequency (usually semi-annual) Fixed or variable (floaters and inverse floaters) Nominal or inflation-indexed (TIIS / TIPS) Possibly no coupons (zero-coupon bond) 5. Currency Yankee bonds, Samurai bonds Eurobonds Dim sum bonds 6. Credit risk Risk free Defaultable Lecture Notes 11

12 Main Features of Bonds Seniority and security Senior, subordinated, junior Secured by properties and equipment, other assets of the issuer, income-stream, etc Sinking fund provisions i (sinkers) 8. Covenants Restrictions on additional issues, dividends, and other corporate actions. 9. Option provisions Callability: After a certain period, issuer has the right to pay back the loan before it matures. Putability: After a certain period, bondholder has the right to demand payment of the loan before maturity. Convertibility: After a certain period, bondholder had the right to exchange the bond for stocks of the issuer. Lecture Notes 12

13 Cash flow of FI securities Lecture 3: Fixed income securities Cash flow: 1. Maturity 2. Principal 3. Coupon Example. A 3-year bond with principal of $1,000 and annual coupon payment of 5% has the following cash flow: , t = time Lecture Notes 13

14 Valuation of FI securities Lecture 3: Fixed income securities Valuation: 1. Time value Interest rates 2. Risks: Inflation Credit Timing (callability, prepayment) Liquidity Currency For now, consider riskless debt only Lecture Notes 14

15 Term structure of interest rates Lecture 3: Fixed income securities Our objective here is to value riskless cash flows Given the rich set of fixed-income securities traded in the market, their prices provide the information needed to value riskless cash flows at hand In the market, this information on the time value of money is given in many different forms: 1. Spot interest rates 2. Yields 3. Forward interest rates Convention: All interest rates are quoted as annualized. Lecture Notes 15

16 Term structure of interest rates Lecture 3: Fixed income securities Spot interest rate r t is the current (annualized) interest rate for maturity t, applied from NOW to t. r t is for payments only on date t r t can be different for each different date t Example. Spot interest rates on : 08 01: Maturity (year) 1/4 1/ (longest) Interest Rate (%) The set of spot interest rates for different maturities {r 1,r 2,...,r t,...} gives the term structure of interest rates, which refers to the relation between spot rates and their maturities. Spot rates can be calculatedlated from the prices of zero-coupon bonds. Lecture Notes 16

17 Zero-coupon bonds Lecture 3: Fixed income securities A zero-coupon bond (also called discount bond) with maturity T is a bond that pays $1 only after time T. (e.g. T=3 month or 2 year) The most common zero-coupon bonds are Treasury bills (T-bills) and Treasury STRIPS (Separate Trading of Registered Interest and Principal Securities). Example. Treasury bill interest rates (in %) DATE 4 WEEKS 13 WEEKS 26 WEEKS 52 WEEKS 2/13/ /31/ N/A For T-bills (t<1), PV = 1/(1+r*t) t), where r is the discount rate. Example. Prices of STRIPS Maturity (year) 1/4 1/ Price For the 5-year STRIPS, we have = r (1 + r 5 ) 5 5 = 1=4.64% (0.797) 1/5 Lecture Notes 17

18 Zero-coupon bonds Lecture 3: Fixed income securities Prices of zero-coupon bonds provide information about spot interest rates 1 B 1 = (1 + r 1 ) r 1 B 2 = 1 (1 + r2 ) 2 r 2 B 3 =. B T = 1 (1 + r 3 ) 3 r 3 1 (1 + r T ) T r T Let B t denote the current (time 0) price of a zero-coupon bond that matures at year t, where t =1 or t>1. Then B t = 1 or r (1 + r t ) t t = 1 Convention for t<1: PV = 1/(1+r * t). B 1/t t 1 Lecture Notes 18

19 Coupon bonds Lecture 3: Fixed income securities A coupon bond pays a stream of regular coupon payments and a principal at maturity. A coupon bond is a portfolio of zero coupon bonds. A coupon bond is also an annuity plus one zero coupon bond. Example. A 3-year bond of $1,000 par and 5% annual coupon , time For now, suppose that coupons are annual. Lecture Notes 19

20 Coupon bonds Lecture 3: Fixed income securities = , time 50 (50 1-year STRIPS) time 50 (50 2-year STRIPS) time ( year STRIPS) Lecture Notes time

21 Coupon bonds Lecture 3: Fixed income securities Suppose that the discount bond prices are as follows t B t What should the price of the coupon bond be? Price = (50)(0.952) + (50)(0.898) + (1050)(0.863) = What if someone quotes you a different price? The price of a coupon bond is given by XTX B = (Ct B t )+(P B T ) = t=1 C 1 1+r C T 1 (1+r T 1 ) T 1 + C T +P (1+r T ) T where P denotes the principal and C s denote the coupons. Lecture Notes 21

22 Yield-to-maturity (YTM) Lecture 3: Fixed income securities The Yield-to-maturity (or yield) of a bond, denoted by y, is defined by TX C t B = (1+y) + P t (1+y) T t=1 1 The yield is an average discount rate, applied to ALL cash flows. Given its maturity, the principle and the coupon rate, the price of a bond is decreasing in the YTM. Example. Current 1- and 2-year spot interest rates are 5% and 6%, respectively. The price of a 2-year Treasury coupon bond with a par value of $100 and a coupon rate of 6% is 6 B = ( ) 06) 2 = Its YTM is %: = ( ) ) 2 Lecture Notes 22

23 Yield-to-maturity (YTM) What s the yield-to-maturity of a bond that pays semi-annual coupon? A bond has maturity of T>1 years, semi-annual coupon of C per year, principal P, and current price B. Its yield-to-maturity y is defined by B 2T t1 C/2 P t (1 y/2) (1 y/2) 2T We are doing PV calculation under semi-annual compounding. US U.S. Treasury bonds pay semi-annual coupons. Note: In /411 we do not go into conventions of days-counting. Lecture Notes 23

24 Term structure of interest rates Lecture 3: Fixed income securities Treasury yields (in %) on 2/13/2013 and12/31/ /13/ /31/ M 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 20Y 30Y See also bonds/government bonds/us/ Lecture Notes 24

25 Term structure of interest rates Lecture 3: Fixed income securities History of U U.S. S Yield Curve (1/1/1962 (1/1/1962 9/1/2010) 9/1/2010) US TREASURY CONSTANT MATURITIES 1 MTH 16 US TREASURY CONSTANT MATURITIES 3 MTH 14 US TREASURY CONSTANT MATURITIES 6 MTH 12 US TREASURY CONSTANT MATURITIES 1 YR 10 US TREASURY CONSTANT MATURITIES 2 YR 8 US TREASURY CONSTANT MATURITIES 3 YR US TREASURY CONSTANT MATURITIES 5 YR 6 US TREASURY CONSTANT MATURITIES 7 YR 4 US TREASURY CONSTANT MATURITIES 10 YR Lecture Notes 1/2/2010 1/2/2008 1/2/2006 1/2/2004 1/2/2002 1/2/2000 1/2/1998 1/2/1996 1/2/1994 1/2/1992 1/2/1990 1/2/1988 1/2/1986 1/2/1984 1/2/1982 1/2/1980 1/2/1978 1/2/1976 1/2/1974 1/2/1972 1/2/1970 1/2/1968 US TREASURY CONSTANT MATURITIES 30 YR 1/2/ /2/1964 US TREASURY CONSTANT MATURITIES 20 YR 1/2/

26 15% 10% 5% 0% 5% 10% Interest rate risk Lecture 3: Fixed income securities As interest rates change (stochastically) over time, bond prices also change. The value of a bond is subject to interest rate risk. Price (in log) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` r yield (%) Average Returns Rt on US Treasuries (maturities >10 years), 1972/1 2009/ /12 ` ` ` ` ` Lecture Notes 26

27 Measures of interest rate risk Lecture 3: Fixed income securities Duration and Modified Duration Again, we start with a bond with annual coupon and maturity T years. Recall XTX Ct B = ) + P t t=1 (1+y)t (1+y) T What happens to the bond value if the yield goes up by 1 bp? Modified Duration (MD) gives the interest rate risk. T 1 db 1 C t PT MD Bdy B t1 (1 y) (1 y) t1 T1 Macaulay duration (D) is the PV-weighted average time of cash flows. T T PV ( CF 1 Ct PT t ) D t t t 1 B B t1 (1 y) (1 y) D MD 1 y Lecture Notes 27 T

28 Measures of interest rate risk Lecture 3: Fixed income securities Duration of bonds with semi-annual coupons. Coupon C per year, maturity T years. B 2 T C/2 P t t 1 (1 y/ 2) (1 y/ 2) Modified Duration (MD) MD Bdy B t1 (1 y/ 2) (1 y/ 2) Macaulay duration (D) 2T 2T 1 db 1 C /2 t /2 P 2 T /2 D B y y 2 1 T C/2 t/2 P2 T /2 t 2T t1 (1 / 2) (1 / 2) D MD 1 y /2 t1 2T1 Lecture Notes 28

29 Duration Lecture 3: Fixed income securities Use Macaulay Duration to get the intuition. Use Modified Duration to calculate interest rate risk. If yield goes up by 1 bps, the bond value decreases by (MD) bps. The Macaulay duration of a bond portfolio is the weighted average of the durations of the constituents What is the Macaulay duration of a zero-coupon bond? A coupon bond s Macaulay duration is shorter than maturity. Fixing ing the yield, what happens to the Macaulay duration of a coupon bond if the coupon rate increases? Intuition? (It falls.) Fixing the coupon, what happens to the Macaulay duration of the bond if the YTM increases? Intuition? (It falls.) A perpetuity s Macaulay duration is (1+y)/y; its modified duration is 1/y. Lecture Notes 29

30 Measures of interest rate risk Lecture 3: Fixed income securities Example. Consider a4-year T-note with face value $100 and 7% semiannual coupon, selling at $103.51, yielding 6%. Duration: D=0.5*(1*3.28%+2*3.19%+ +8*78.93%)=3.57 Modified duration: MD = D/(1+0.03) = 3.46 If the annual yield moves up by 1 bp, the bond price decreases roughly by 3.46 bps. Lecture Notes 30

31 How good is the approximation? Lecture 3: Fixed income securities Same example: 4-year T-note, face value $100, 7% coupon, 6% yield 120 Actual Bond Price % 1.3% 1.6% 1.9% 2.2% 2.5% 2.8% 3.1% 3.4% 3.7% 4.0% 4.3% 4.6% 4.9% MD gives an accurate estimate of price change if yield change is small, but an inaccurate one if yield change is large. 5.2% 5.5% 5.8% 6.1% 6.4% 6.7% 7.0% 7.3% 7.6% 7.9% 8.2% 8.5% 8.8% 9.1% 9.4% 9.7% 10.0% Lecture Notes 31

32 How good is the approximation? Lecture 3: Fixed income securities Another example: 30-year zero-coupon bond, y=5% % 1.2% 1.4% 1.6% 1.8% 2.0% 2.2% 2.4% 2.6% 2.8% 3.0% 3.2% 3.4% 3.6% 3.8% 4.0% 4.2% 4.4% 4.6% 4.8% 5.0% 5.2% 5.4% 5.6% 5.8% 6.0% 6.2% 6.4% 6.6% 6.8% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.0% 9.2% 9.4% 9.6% 9.8% 10.0% Actual With duration Lecture Notes 32

33 Convexity Lecture 3: Fixed income securities Bond price is not a linear function of the yield. For large yield changes, the effect of curvature (i.e., nonlinearity) becomes important. Convexity, CX, measures the curvature of the bond price as a function of the yield: CX 1 B d 2 dy B 2 1 B (1 1 y) 2 T t1 t( t 1) CF (1 t y) t MD and Convexity give a better measurement of interest rate risk. 2 db 1 d B 2 2 B ( y) ( y)... dy 2 dy B 1 MD ( y ) CX ( y ) B 2 Lecture Notes 33 2

34 How good is the approximation? Lecture 3: Fixed income securities % 1.2% 1.4% 1.6% 1.8% 2.0% 2.2% 2.4% 2.6% 2.8% 3.0% 3.2% 3.4% 3.6% 3.8% 4.0% 4.2% 4.4% 4.6% 4.8% 5.0% 5.2% 5.4% 5.6% 5.8% 6.0% 6.2% 6.4% 6.6% 6.8% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.0% 9.2% 9.4% 9.6% 9.8% 10.0% Actual With duration With Duration + Convexity Zero coupon bond, y5%t30 y=5%, T=30. Actual and approximated prices (using duration, or duration and convexity adjustments). Lecture Notes 34

35 Forward interest rates Lecture 3: Fixed income securities So far, we have focused on spot interest rates: rates for a transaction between today, time 0, and a future date, time t. Now, we study forward interest rates: rates for a transaction between two future dates, for instance, t 1 and t 2 For a forward transaction to borrow money in the future: Terms of the transaction are agreed on today, t = 0 Loan is received on a future date t 1 Repayment of the loan occurs on date t 2 Future spot rates can be different from current corresponding forward rates Lecture Notes 35

36 Forward interest rates Lecture 3: Fixed income securities Example. As the CFO of a U.S. multinational, you expect to repatriate $10M from a foreign subsidiary in 1 year, which will be used to pay dividends 1 year later. Not knowing the interest rates in 1 year, you would like to lock into a lending rate one year from now for a period of one year. What should you do? The current interest t rates are time to maturity t (years) 1 2 spot interest rate r t Lecture Notes 36

37 Forward interest rates Lecture 3: Fixed income securities Strategy: Borrow $9.524M now for one year at 5% Invest the proceeds $9.524M for two years at 7% Outcome (in million dollars): Year yr borrowing yr lending Repatriation Net The locked-in 1-year lending rate 1 year from now is 9.04% Lecture Notes 37

38 Forward interest rates Lecture 3: Fixed income securities The (one-year) forward interest rate between time t-1 and t is or (1 + r t ) t =(1+r t 1 ) t 1 (1 + f t ) ) f t = B t 1 1= (1+r t) t B t (1 + r t 1 ) 1 t 1 6 Spot and forward rates r 4 - f 5 - r 3 f r 2 - f 3 - r 1 f 2 r 1 f r 1 = f Lecture Notes 38 - year

39 Forward interest rates Lecture 3: Fixed income securities Example. Suppose that discount bond prices are as follows: t B t r t A customer wants a forward contract to borrow $20M three years from now for one year. Can you (a bank) quote a rate? f 4 = 8.51% Lecture Notes 39

40 Forward interest rates Lecture 3: Fixed income securities What should you do today to get rid of the risks in the cash flows? 1. Buy 20,000,000 of 3 year discount bonds, costing (20,, 000,, 000)(0.8278) ) = $16,, 556,, Finance this by selling 4 year discount bonds of amount 16,556,000/ = 21,701, This creates a liability in year 4 in the amount $21,701,403 Cash flows from this strategy (in million dollars): Year Purchase of 3-year bonds Sale of 4-year bonds Total The future rate is given by: 21, 701, , 000, 000 1=8.51% Lecture Notes 40

41 Inflation risk Lecture 3: Fixed income securities Most bonds give nominal payoffs. In the presence of inflation risk, real payoffs are risky even when nominal payoffs are safe. Example. Suppose that inflation next year is uncertain ex ante, with equally possible rate of 10%, 8% and 6%. The real interest rate is 2%. The 1-year nominal interest rate will be (roughly) 10%. Consider the return from investing in a 1-year Treasury security: Year 0 value Inflation rate (%) Year 1 nom. payoff Year 1 real payoff Lecture Notes 41

42 Default risk Non-government bonds carry default risk Lecture 3: Fixed income securities A default occurs when a debt issuer fails to make a promised payment (interest or principal). Credit ratings by rating agencies (e.g., Moody's and S&P) provide indications of the likelihood of default by each issuer. Credit Risk Moody's S&P Fitch Investment Grade Highest Quality Aaa AAA AAA High Quality (Very Strong) Aa AA AA Upper Medium Grade (Strong) A A A Medium Grade Baa BBB BBB High Yield Somewhat Speculative Ba BB BB Speculative B B B Highly Speculative Caa CCC CCC Most Speculative Ca CC CC Imminent Default C C C Default C D D Lecture Notes 42

43 Default risk Source: Federal Reserve Lecture Notes 43

44 Default risk Lecture 3: Fixed income securities Example. Suppose all bonds have par value $1,000 and 10-year Treasury strip is selling at $463.19, yielding 8% 10-year zero-coupon bond issued by XYZ Inc. is selling at $ Expected payoff from XYZ's 10-year zero-coupon bond is $ The XYZ bond: Promised YTM = ³ Expected YTM = 1/10 1=12% ³ /10 1 = 9% Default Premium = Promised YTM Expected YTM = 12% 9% = 3% Risk Premium = Expected YTM Default-free YTM = 9% 8% = 1% Promised YTM: the yield if default does not occur Expected YTM: the probability-weighted average of all possible yields Default premium: the difference between promised yield and expected yield Bond risk premium: the difference between the expected yield on a risky bond and the yield on a risk-free bond of similar maturity and coupon rate Lecture Notes 44

45 Default risk Lecture 3: Fixed income securities Yield-to-maturity for a risky bond 12% - Promised YTM Yield spread Default premium Risk premium 9% - Expected YTM 8% - Default-free YTM Default-free rate Lecture Notes 45

46 Default risk Source: Moody s, 2009 Lecture Notes 46

47 Hypothesis on interest rates Lecture 3: Fixed income securities What determines the term structure of interest rates? 1. Expected future spot rates 2. Risk of long bonds Hypotheses of interest rates: Expectations Hypothesis Liquidity Preference Market Segmentation ti Lecture Notes 47

48 Hypothesis on interest rates Lecture 3: Fixed income securities Expectations Hypothesis: Forward rates predict future spot rates f t =E[r 1 (t)] The slope of the term structure reflects the market's expectations of future short-term interest rates Liquidity Preference: Investors regard long bonds as riskier than short bonds f t =E[r 1 (t)] + LiquidityPremium Long bonds on average receive higher returns than short bonds Forward rates on average ``over-predict'' future short-term rates Term structure reflects a) expectations of future interest rates, and b) risk premium demanded by investors in long bonds Market Segmentation: Interest rate in each maturity depends on supply and demand for borrowing/lending at that maturity. Lecture Notes 48

49 Key concepts Lecture 3: Fixed income securities Overview of fixed-income income markets Term structure of interest rates Spot rates and zero-coupon bonds Yield to maturity and coupon bonds Interest rate risk: Duration and Convexity Forward interest rates Inflation risk Default risk Theories of interest rates Lecture Notes 49

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