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The Risk Structure and Term Structure of Interest Rates Chapter 6 Risk Structure of Interest Rates • Bonds with the same maturity have different interest rates due to: • Default risk • Liquidity • Tax considerations Risk Structure of Interest Rates • Default risk: probability that the issuer of a bond is unable or unwilling to make interest payments or pay off the face value • U.S. Treasury bonds are considered default free (government can raise taxes). • Risk premium: the spread between the interest rates on bonds with default risk and the interest rates on (same maturity) Treasury bonds Response to an Increase in Default Risk on Corporate Bonds – Supply/Demand Application Russian Default Risk Structure of Interest Rates • Liquidity: the relative ease with which an asset can be converted into cash • Cost of selling a bond • Number of buyers/sellers in a bond market • Income tax considerations • Interest payments on municipal bonds are exempt from federal income taxes. Interest Rates on Municipal and Treasury Bonds Taxes and Bond Prices • Coupon payments on municipal bonds are exempt from federal Income taxes • For 28% tax bracket: • After tax yield = (taxable yield) x (1 – tax rate) 3.60% = 5% x (1 – 0.28) tax exempt yield • Tax equivalent yield = 1 - tax rate http://www.bloomberg.com/markets/ratesbonds/government-bonds/us/ Risk Structure of Long-Term Bonds in the United States Bond (credit) Ratings and Risk Bond Ratings • Moody’s and Standard and Poor’s Ratings Groups • Investment Grade • Non-Investment – Speculative Grade • Highly Speculative Bond (credit) ratings S&P Moody’s What it means AAA Aaa Highest quality and creditworthiness AA Aa Slightly less likely to pay principal + interest A A Strong capacity to make payments, upper medium grade BBB Baa Medium grade, adequate capacity to make payments BB Ba Moderate ability to pay, speculative element, vulnerable B B Not desirable investment, long term payment doubtful CCC Caa Poor standing, known vulnerabilities, doubtful payment CC Ca Highly speculative, high default likelihood, known reasons C C Lowest rated class, most unlikely to reach investment grade D NR Already defaulted on payments No public rating has been requested + or - & 1,2,3 Within-class refinement of AA to CCC ratings Credit rating & historic default frequencies Moody’s Rating 1985 1990 1995 2000 2006 2008 2009 2010 Aaa 0% 0% 0% 0% 0% 0% 0% 0% Aa 0% 0% 0% 0% 0% 0% 0% 0% A 0% 0% 0% 0% 0% 1.201% 0% 0.36% Baa1 0% 0% 0% 0.29% 0% 0.271% 1.144% 0% Baa2 0% 0% 0% 0% 0% 0.794% 0.74% 0% Baa3 0% 0% 0% 0.98% 0% 0.321% 0.70% 0% Ba1 0% 2.67% 0% 0.91% 0% 0% 2.27% 0% Ba2 1.63% 2.82% 0% 0.66% 0.51% 0% Ba3 3.77% 3.92% 1.72% 0.99% 0% 2.715% 4.01% 0% B1 4.38% 8.59% 4.35% 3.63% 0.66% 1.783% 4.10% 0.85% B2 7.41% 22.09% 6.36% 3.84% 0.50% 0.825% 8.68% 0% B3 13.86% 28.93% 4.10% 11.72% 1.93% 3.198% 8.52% 0.56% 0.60% 0% Default Risk – Price and YTM • Suppose risk-free rate is 4% • Suppose there is a company called FlimFlam that issues one-year, 4% coupon bond, FV=$100. • If risk free, the price of the FlimFlam bond is $4 $100 $104 P $100 1.04 1.04 Default Risk Suppose 5% probability FlimFlam goes bankrupt – you get nothing Expected Value of FlimFlam bond payment Possibilities Payoff Probability Payoff x Probability Full Payment $104 $0 .95 .05 $98.80 $0 Default •Expect to receive $98.80 one-year from now. •Discount at risk-free rate = $98 .80 1.04 •P = $95 $95 Default Risk Premium • We can calculate the probability of repayment from the interest rates. • Let 1+k be the return on a one-year corporate debt and 1+ i be the return on a one-year default risk-free treasury. 1 i p • The probability of repayment is 1 k • the probability of default is 1 – p • The probability of repayment: 1.04 0.95 1.0947 Default Risk Suppose 10% probability FlimFlam goes bankrupt – you get nothing Expected Value of FlimFlam bond payment Possibilities Payoff Probability Payoff x Probabilities Full Payment $104 $0 .90 .10 $93.60 $0 Default •Expect to receive $93.60 one-year from now. •Discount at risk-free rate = $93.60 $90 1.04 •Yield = ($104 / $90) -1 = .1555 or 15.55% •Default risk premium = 15.55% - 4% = 11.55%. Bond Ratings and Risk • Increased risk reduces bond demand. • The resulting shift to the left causes a decline in equilibrium price and an increase in the bond yield. • Bond Yield = U.S. Treasury Yield + Default Risk Premium • Risk spread or default risk premium = Bond Yield - U.S. Treasury Yield Information Content of Interest Rates: Risk Structure • When the economy starts to slow, this puts a strain on private firms. • A slower economy means a higher default probability • Risk Spreads increase. Information Content of Interest Rates: Risk Structure Risk spread = Baa Corporate minus 10-year Treasury Term Structure of Interest Rates Definition of the Term Structure: The relationship among bonds with the same risk, liquidity and tax characteristics but different maturities is called the term structure of interest rates. Yield Curve: A plot of the term structure, with the yield to maturity on the vertical axis and the time to maturity on the horizontal axis. http://finance.yahoo.com/bonds/composite _bond_rates?desktop_view_default=true Term Structure of Interest Rates Term Structure of Interest Rates http://stockcharts.com/index.html Term Structure of Interest Rates: Facts to Explain 1. Interest rates (Yields) on different maturities tend to move together 2. Yields on short-term bond are more volatile than yields on long-term bonds 3. Long-term yields tend to be higher than short-term yields. • Also want to explain the fact that yield curves can be inverted. Movements over Time of Interest Rates on U.S. Government Bonds with Different Maturities Sources: Federal Reserve: www.federalreserve.gov/releases/h15/data.htm. Three Theories to Explain the Three Facts 1. Pure Expectations Theory explains the first two facts but not the third 2. Segmented Markets Theory explains fact three but not the first two 3. Liquidity Premium Theory combines the two theories to explain all three facts Pure Expectations Theory • The interest rate on a long-term bond will equal an average of the short-term interest rates that people expect to occur over the life of the long-term bond • Key Assumption: Buyers of bonds do not prefer bonds of one maturity over another. • Bonds of different maturities are considered to be perfect substitutes Expectations Theory Notation i1t interest rate on 1-year bond today (t). i2t interest rate on 2-year bond today (t). int interest rate on n-year bond today (t). i1t 1 interest rate on 1-year bond, 1-year from today (t+1). i1et 1 Expected interest rate on 1-year bond, 1-year from today (t+1). i1et n Expected interest rate on 1-year bond, n-years from today (t+n). A Note on Averages • Geometric average of i1t and ((1 i1t )(1 i1t 1 )) 1/ 2 • Arithmetic average = i1t 1 = 1 i1t i1t 1 2 Expectations Theory: • Let the current interest rate on one-year bond (i1t) be 6%. • You expect the interest rate on a one-year bond next year ( i1e t 1 ) to be 9%. • Then the expected return from buying 2 oneyear bonds averages (6% + 9%)/2 = 7.5%. • Under the Expectations Theory the current interest rate on a two-year (i2t) bond must be 7.5% for you to be willing to purchase that bond. • Why? Example: 2 year investment horizon • • • • • Strategy 1: Invest $1,000 for 2-years at 8%: Ending Balance = (1+0.08)2($1,000) = $1,166.40 Strategy 2: Invest $1,000 1-year at 6% and expect 9% one year later: • Ending Balance = (1 +0.06)(1+0.09)($1,000) = $1,155.40 • Come out $11 ahead with Strategy 1. • What happens to S and D? Expectations Theory ( Math) 1. Return from a 2-year bond over 2 years (1 i2t )(1 i2t ) 1 2. Return from a 1-yr bond and then another 1-yr bond (1 i1t )(1 i1te 1 ) - 1 3. If one and two year bonds are perfect substitutes, then: (1 i2t )(1 i2t ) (1 i1t )(1 i ) e 1t 1 Term Structure of Interest Rates: Expectations Theory From: (1 i 2t )(1 i 2t ) (1 i1t )(1 i e 1t 1 ) We can derive the following arithmetic approximation: i1t i i 2t 2 e 1t 1 Which says the long-term interest rate = average of current and expected future short-term interest rates. Here is how we get the approximation: Expected return over the two periods from investing $1 in the two-period bond and holding it for the two periods (1 + i2t )(1 + i2t ) 1 1 2i2t (i2t ) 2 1 2i2t (i2t ) 2 Since (i2t ) 2 is very small the expected return for holding the two-period bond for two periods is 2i2t Here is how we get the approximation: If two one-period bonds are bought with the $1 investment (1 it )(1 ite1 ) 1 1 it ite1 it (ite1 ) 1 it ite1 it (ite1 ) it (ite1 ) is extremely small Simplifying we get it i e t 1 Expectations Theory Both bonds will be held only if the expected returns are equal 2i2t it ite1 it ite1 i2t 2 The two-period rate must equal the average of the two one-period rates For bonds with longer maturities int it ite1 ite 2 ... ite( n 1) n The n-period interest rate equals the average of the one-period interest rates expected to occur over the n-period life of the bond Actual math: No Approximation (1 i 2t )(1 i 2t ) (1 i1t )(1 i e 1t 1 (1 i 2t ) [(1 i1t )(1 i 2 e 1t 1 (1 i 2t ) [(1 i1t )(1 i e 1t 1 i2t [(1 i1t )(1 i e 1t 1 )] 1/2 )] )] 1 1/2 This is a geometric average ) Expectations Hypothesis - Arithmetic Average int i1t i e 1t 1 i e 1t 2 .... i e 1t n 1 n In words: The interest rate on a bond with n years to maturity at time t is the average of the n expected future one-year rates. Numerical example: One-year interest rate over the next five years 5%, 6%, 7%, 8% and 9%: Interest rate on a two-year bond: (5% + 6%)/2 = 5.5% This is the only interest rate that is known at time t Interest rate for a five-year bond: (5% + 6% + 7% + 8% + 9%)/5 = 7% Interest rate for one, two, three, four and five-year bonds are: 5%, 5.5%, 6%, 6.5% and 7%. Expectations Hypothesis int i1t i e 1t 1 i e 1t 2 .... i e 1t n 1 n Another example: One-year interest rate over the next five years 7%, 6%, 5%, 4% and 3%: Interest rate on a two-year bond: (7% + 6%)/2 = 6.5% Interest rate for a five-year bond: (7% + 6% + 5% + 4% + 3%)/5 = 5% Interest rate for one, two, three, four and five-year bonds: 7%, 6.5%, 6%, 5.5% and 5%. Recall the Fisher Equation: i = r + πe • Holding r constant: • If inflation is expected to rise in the future, expected one-year interest rates will rise and the yield curve will slope upward. • If inflation is expected to fall in the future, expected one-year interest rates will fall and the yield curve will slope downward. • If inflation is expected to remain the same in the future, expected one-year interest rates will remain the same and the yield curve will be flat. Term Structure of Interest Rates: Expectations Theory Using the Pure Expectations Theory to Solve for Expected 1-year (forward) Interest rates From the formula for the yield on a 2-year bond: e it ite1 i t 1 2i2t it i2t 2 From the formula for the yield on a 3-year bond: i it ite1 ite 2 i3t 3 e t 2 i 3i3t (it i ) e t 1 In general: e t ( n1) i nint (n 1)i(n1)t e t 2 i 3i3t 2i2t Actual math: No Approximation (1 i2t )(1 i2t ) (1 i1t )(1 i e 1t 1 (1 i e 1t 1 (1 i2t ) ) 1 i1t (1 i2t ) 1 1 i1t 2 e 1t 1 i 2 ) Term Structure Facts and the Expectations Theory Expectations Theory Explains: 1. Interest Rates of different maturities tend to move together - long term interest rates are averages of expected future short-term interest rates. 2. Yields on short-term bond are more volatile than yields on long-term bonds – - long term interest rates are averages of expected future short-term interest rates. But Expectations Theory does not explain: 3. Long-term yields tend to be higher than short-term yields. Segmented Market Theory • Bonds of different maturities are not perfect substitutes for each other. Segmented Markets Hypothesis • Assumptions: • Investors have specific preferences about the maturity or term of a security. • Investors do not stray from their preferred maturity. Segmented Markets Hypothesis • The slope of the yield curve is explained by different demand and supply conditions for bonds of different maturities. • If the yield curve slopes up, it does so because the demand for short term bonds is relatively greater than the demand for long term bonds. • Short term bonds have a higher price and a lower yield as a result of the relatively greater demand. So the yield curve slopes upward. Segmented Markets Hypothesis Price Price S S P2s P1l P1s D2s P2l D1l D1s 0 Quantity of Short-term Bonds D2l 0 Quantity of Long-term Bonds Upward Sloping Yield Curve Segmented Markets Hypothesis • The segmented markets hypothesis explains why…. • Yield curves typically slope upward. • On average, investors prefer bonds with shorter maturities that have less interest rate risk. • Therefore, the demand for short term bonds is relatively greater than the demand for long-term bonds Segmented Markets Hypothesis • But, the segmented markets hypothesis does not explain why… • Interest rates on different maturities move together. • The segmented markets hypothesis assumes that short and long markets are completely segmented. Liquidity Premium Theory of the Term Structure of Interest Rates • Yield curve upward slope is explained by the fact that long-term bonds are riskier than short-term bonds. • Bondholders face both inflation risk and interest rate risk. • The longer the term of the bond, the greater both types of risk. • Investors need to be compensated for the greater risk. Term Structure of Interest Rates Liquidity Premium Theory int i1t i e 1t 1 i e 1t 2 .... i n Pure Expectations Theory: average of expected future short-term rates (explains facts 1&2) e 1t n 1 RPn Liquidity or Risk Premium (explains fact 3) Numerical Example Term in years (n) 1 2 3 4 5 One year interest rate expectations 5% 6% 7% 8% 9% Liquidity premium 0% 0.25% 0.5% 0.75% 1.0% Pure expectations predicted n-year bond interest rates 5% 5.5% 6% 6.5% 7% Actual n-year bond interest rates, accounting for liquidity preference 5% 5% 6% 2 5.75% 5% 6% 7% 5 6 7 8 % 4 3 6.5% 7.25% 56789 % 5 8% Relationship Between the Liquidity Premium and Expectations Theories (if short term interest rates are expected to remain constant) Information Content of Interest Rates: Term Structure • When the yield curve slopes down, it is called inverted • An inverted yield curve is a very valuable forecasting tool • It signals an economic downturn Information Content of Interest Rates:10-year T-bond compared to 3-month T- bill Market Predictions of Future Short Rates The actual math is a lot more interesting. Refer to the note on “Term Structure and Forward Interest Rates.”