Chapter 6: The Risk and Term Structure of Interest Rates In previous chapter we analyzed the determination of "the interest rate" as if there were only 1. YTM's, though, differ according to risk and maturity, so in fact at any given time there are many interest rates. Risk Structure of Interest Rates Question: - Why do bonds with the same maturities have different YTM's? Answer: - They vary along at least three other dimensions Default Risk The perceived chance that the issuer will default (i.e. fail to live up to repayment contract) Note: U.S. Treasury bonds considered to have zero default risk (a.k.a. default-free bonds) b/c Treasury can always increase taxes or print money Definition: "risk premium" -- the additional interest people must earn in order to be willing to hold a risky bond...the higher the default risk, the greater the risk premium Think of supply and demand analysis from last chapter (flow of funds framework for bond pricing)
Note: Bonds are rated for their default risk Bond rating agencies: Moody s, Standard & Poor s, others
Risk Structure of Interest Rates (cont'd.) Liquidity The more widely traded, the more liquid. Treasure bonds are most widely traded, ergo most liquid Corporate bonds may face "thinner" trading markets, therefore less liquid (Note: "illiquidity" is a different kind of risk, but risk nonetheless) The more liquid, the greater the demand (ceteris paribus), and hence the higher the price (and the lower the YTM) Definition: liquidity premium -- the difference in YTM on a bond relative to a Treasury bond of equal maturity, due to the difference in liquidity. Note: Treasury bonds are the benchmark both for low default risk and for high liquidity. Thus it isn't easy to distinguish the risk premium from the liquidity premium. Usually when people say the "risk premium" they mean a combination of the two: "the risk and liquidity premium" Income Tax Considerations Recall that "Munis" are tax-free (i.e. generate no income tax liability) If you re in the 40% tax bracket, and the yield is 10%, then your take-home yield is just 6%. So: risk and liquidity held constant, a person in the 40% tax bracket would be just as happy with a municipal bond paying 6% as with a non-municipal bond paying 10%
Yields on Different Instruments: Note: liquidity AND risk AND tax differences between these Most liquid: U.S. Treasuries Least risky: U.S. Treasuries Least taxed: Munis Note: 1. rates tend to move together, though the spreads between different interest rates are not constant 2. Munis have generally had the lowest nominal yields (higher after tax) 3. Surge in yields on all instruments after inflationary experience of the 70s 4. Connection between Muni spread and changes in income tax rates?
Treasury Yields 1960-2006 Note how the spreads expand and contract (and occasionally go negative)
The flight to quality in times of crisis
Term Structure of Interest Rates Aside from differences in default risk, liquidity, and tax treatment, bonds differ along another dimension: term to maturity The "Yield Curve" Definition: A graph illustrating YTM for bonds which are comparable along all other dimensions except term to maturity.
Possible shapes of the yield curve Upward sloping (normally the case) Flat Downward sloping, or "inverted" (unusual, but important) Three Stylized Facts about the yield curve, which a good theory of interest rates must be able to explain 1. Rates tend to move together over time, even for bonds of different maturities 2. The higher the short-term rate, the more likely the yield curve is inverted 3. Yield curves almost always slope up Three Theories which attempt an explanation: 1. Expectations Theory 2. Segmented Markets Theory 3. Liquidity Premium Theory
Expectations Theory "The interest rate on a long-term bond will be the average of the shortterm interest rates that people expect to occur over the life of the long-term bond" For example, If money needed after two years, you could either a) buy a 2-year bond b) buy a 1-year bond, and then after 1 year, buy another 1-year bond If those two approaches are perfect substitutes, then the yields must equate Formally, let i t = time t (today) yield on a 1-year bond i 2t = time t yield on a 2-year bond i e t+1 = expected yield on a 1-year bond starting in t+1 (i.e. next year) The expected return from investing $1 by strategy (a) above: = (1+ i 2t )*(1+ i 2t ) - 1 = 1 + 2*i 2t + (i 2t ) 2-1 = 2*i 2t + (i 2t ) 2 2*i 2t since (i 2t ) 2 0
Meanwhile, the return from investing $1 by strategy (b) above: = (1+ i t )*(1+i e t+1) - 1 = 1 + i t + i e t+1 + i t * i e t+1-1 = i t + i e t+1 + i t * i e t+1 i t + i e t+1 since i t * i e t+1 0 If these two bonds are perfect substitutes then their yields must equate (else one or the other won't be held), i.e. 2*i 2t = i t + i e t+1 or i 2t = (i t + i e t+1)/2 In other words: the two-period (annualized) YTM must be equal to the average of the expected 1-period YTMs In the general, n-period case, the arbitrage condition would be i nt = (i t + i e t+1 + i e t+2 +... + i e t+(n-1))/n Which says simply: "the interest rate on an n-period bond must be equal to the average of the expected 1-period rates over the relevant time horizon"
Problem: Calculate the yield curve for 1- to 5-year bonds if a) the Expectations Theory is correct, and b) the expected trajectory of 1- year interest rates is the following Solution: i t =5% i e t+1=7% i e t+2=8% i e t+3=7.5% i e t+4=7% 2-year bond: i 2t = (5%+7%)/2 = 6% 3-year bond: i 3t = (5%+7%+8%)/3 = 6.67% 4-year bond: i 4t = (5%+7%+8%+7.5%)/4 = 6.875% 5-year bond: i 5t = (5%+7%+8%+7.5%+7%) = 6.9% (Graph) Insight: As long as "interest rates are expected to increase" the yield curve will be upward sloping More exactly: as long as the marginal (expected) interest rate is above the average (expected) interest rate, the yield curve will be upward sloping
So, which of the stylized facts can this Expectations Theory model explain? 1. Rates move together Yes. b/c a change in short-rate gets averaged into longer rates 2. Yield curve slopes up when short rates are low and down when short-rates are high Yes. b/c mean reversion 3. Yield curves almost always slope upwards No. b/c short-rates are equally likely to be above mean as below. Hence expected rates should be falling as often as rising, and yield curve therefore downward sloping as often as upward sloping.
Segmented Markets Theory Holds that "bonds of different maturities are not substitutes" (or perhaps that they are such imperfect substitutes that their relative yields matter little to investors) For example, if investors want to completely avoid interest rate risk, they must hold a bond to maturity. Hence those who need their money in 10 years have no use for a 5-year or a 30- year bond, etc. Assume there are more people with need for their money in the shortterm than in the long term. Demand for the short bond will be higher than demand for the long bond The price of short bond will be higher The yield on the short bond will be lower So: This theory can explain stylized fact 3 ("yield curves almost always slope up") But it fails to explain stylized fact 1 ("rates move together") { or does it...?) And it certainly fails to explain stylized fact 2 ("upward sloping when short rate is low, downward sloping when short rate is high")
Liquidity Premium Theory Expectations Theory assumes perfect substitutability between maturities Segmented Markets Theory assumes no substitutability between maturities Liquidity Premium Theory assumes imperfect substitutability (but substitutability nonetheless) LPT "The interest rate on a long-term bond will equal an average of shortterm expected rates, plus a liquidity premium ('term premium'), which is a positive function of time to maturity" The term premium captures the following: higher interest rate risk embedded in longer term bonds investor preference for shorter terms Thus, according to LPT: i nt = (i t + i e t+1 + i e t+2 +... + i e t+(n-1))/n + l nt where l nt term premium on an n-year bond at time t (and l nt is increasing in n)
(graph l nt )What can we explain with this theory? Stylized fact Explained? 1. rates move together 2. inverted yield curve at high short rates 3. y-curve almost always slopes up