The Risk Structure and Term Structure of Interest Rates Chapter 5

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 the 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 3

Response to an Increase in Default Risk on Corporate Bonds – S/D Application 4

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 6

Tax Status and Bond Prices Coupon Payments on Municipal Bonds are exempt from Federal Tax Payments. For 28% tax bracket: After Tax Yield = (Taxable Yield) x (1- Tax Rate). 3.60% = 5% x (1 - .28) Taxable Equivalent Yield = Tax-Exempt Bond Yield / (1- Tax Rate) http://www.bloomberg.com/markets/rates/index.html

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 What it means AAA Aaa Moody’s What it means AAA Aaa Highest quality and creditworthiness AA Aa Slightly less likely to pay principal + interest 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 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 Lowest rated class, most unlikely to reach investment grade D Already defaulted on payments NR 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 Aaa 0% Aa A Baa1 0.29% Baa2 Baa3 0.98% Ba1 2.67% 0.91% Ba2 1.63% 2.82% 0.66% 0.51% Ba3 3.77% 3.92% 1.72% 0.99% B1 4.38% 8.59% 4.35% 3.63% B2 7.41% 22.09% 6.36% 3.84% 0.50% B3 13.86% 28.93% 4.10% 11.72% 1.93%

Default Risk – Price and YTM Suppose risk-free rate is 4% Have a company called Mainland.com that issues one-year, 4% coupon bond, FV=$100. If risk free, the price of bond is Yield= 104/100-1=

Default Risk Suppose 5% probability Mainland.com goes bankrupt – you get nothing Expected Value of FlimFlam.com bond payment Possibilities Payoff Probability Payoff x Probability Full Payment $104 .95 $98.80 Default $0 .05 Expect to receive $98.80 one-year from now. Discount at risk-free rate = Yield as a percentage of Price = ($104 / $95) -1 = .0947 or 9.47% Default risk premium = 9.47% - 4% = 5.47%.

Default Risk Premium The default risk premium is calculated as the market interest rate minus the risk free interest rate

Default Risk Suppose 10% probability FlimFlam.com goes bankrupt – you get nothing Expected Value of FlimFlam.com bond payment Possibilities Payoff Probability Payoff x Probabilities Full Payment $104 .90 $93.60 Default $0 .10 Expect to receive $93.60 one-year from now. Discount at risk-free rate = 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 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://www.bloomberg.com/markets/rates/index.html

Term Structure of Interest Rates

Term Structure of Interest Rates Web Link: Bloomberg.com http://stockcharts.com/index.html

Term Structure of Interest Rates: Facts to Explain Interest rates (Yields) of different maturities tend to move together Yields on short-term bond are more volatile than yields on long-term bonds Long-term yields tend to be higher than short-term yields. Also want to explain the fact that yield curves can be inverted.

Term structure of interest rates Bonds with identical default risk, liquidity, and tax characteristics may still have different interest rates because the time remaining to maturity is different. Yield curve is a plot of the yield on bonds with differing terms to maturity but the same risk, liquidity and tax considerations. Shape of the yield curve: Usually upward-sloping (long-term i > short-term i ) sometimes ‘inverted’ (long-term i < short-term i ) Flat implies short- and long-term rates are similar.

Movements over Time of Interest Rates on U. S Movements over Time of Interest Rates on U.S. Government Bonds with Different Maturities Sources: Federal Reserve: www.federalreserve.gov/releases/h15/data.htm. 24

Three Theories to Explain the Three Facts Pure Expectations Theory explains the first two facts but not the third Segmented Markets Theory explains fact three but not the first two 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 interest rate on 1-year bond today (t). interest rate on 2-year bond today (t). interest rate on n-year bond today (t). interest rate on 1-year bond, 1-year from today (t+1). Expected interest rate on 1-year bond, 1-year from today (t+1). Expected interest rate on 1-year bond, n-years from today (t+n).

A Note on Averages Arithmetic average =

Expectations Theory: Let the current interest rate (i1t) on one-year bond be 6%. You expect the interest rate on a one-year bond next year ( ) to be 9%. Then the expected return from buying 2 one-year 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 Let i1t = interest rate for this year on a 1-year bond ie1,t+1 = next year's expected interest rate on a 1 year bond i2t = today's interest rate on a 2-year bond If i1t = 8% and ie1,t+1 = 12%, then i2t = (i1t + ie1,t+1)/2 = (8 + 12)/2 = 10%. Suppose a 2-year bond had an interest rate above 10%, say, 11%. A 2-year bond would bring the lender a total return of 22% over the two years while a succession of two 1-year loans would only bring a 20% return. Investors will shift to the 2-year bond market and drive down the interest rate to 10%. 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. Investors will shift to the 2-year bond market and drive down the interest rate to 7.5%.

Term Structure of Interest Rates: Expectations Theory From: We can derive the following arithmetic approximation: Which says the long-term interest rate = average of current and expected future short-term interest rates.

Expectations Theory

Expectations Hypothesis –Arithmetic Average 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% 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%. This is the only interest rate that is known at time t

Expectations Hypothesis 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: (5% + 6% + 7% + 8% + 9%)/5 = 5% Interest rate for one, two, three, four and five-year bonds: 7%, 6.5%, 6%, 5.5% and 5%.

Term Structure of Interest Rates: Expectations Theory

Term Structure Facts and the Expectations Theory Expectations Theory Explains: Interest Rates of different maturities tend to move together - long term interest rates are averages of expected future short-term interest rates. Yields on short-term bond are more volatile than yields on long-term bonds – 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. Investors try to achieve the highest possible rate of return from holding one or more securities, but they do not move from one maturity to another.

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 P1s P1l P2l D2s D1l D1s D2l Quantity of Short-term Bonds 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 The segmented markets hypothesis cannot 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 The 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 Liquidity or Risk Premium (explains fact 3) Pure Expectations Theory: average of expected future short-term rates (explains facts 1&2)

Relationship Between the Liquidity Premium and Expectations Theories (if short term interest rates are expected to remain constant)

Numerical Example Term in years (n) 1 2 3 4 5 5% 6% 7% 8% 9% 0% 0.25% 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% 6.5% Actual n-year bond interest rates, accounting for liquidity preference 5.75% 7.25%

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