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## Interest Rates

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**Week One**Interest Rates**What four factors affect the level of interest rates?**• Production opportunities • Time preferences for consumption • Risk • Expected inflation**Figure 5.1 U.S. Interest Rates and Inflation**Rates,1960–2009**Figure 5.2 Term Structure of Risk-Free U.S. Interest Rates,**November 2006, 2007, and 2008**“Nominal” vs. “Real” Rates**r = represents any nominal rate r* = represents the “real” risk-free rate of interest. Like a T-bill rate, if there was no inflation. Typically ranges from 1% to 5% per year. rRF = represents the rate of interest on Treasury securities.**Determinants of Interest Rates**r = r* + IP + DRP + LP + MRP r = required return on a debt security r* = real risk-free rate of interest IP = inflation premium DRP = default risk premium LP = liquidity premium MRP = maturity risk premium**Yield Curve and the Term Structure of Interest Rates**• Term structure – relationship between interest rates (or yields) and maturities. • The yield curve is a graph of the term structure. • The October 2008 Treasury yield curve is shown at the right.**Step 1 – Find the average expected inflation rate over**Years 1 to N: Constructing the Yield Curve: Inflation**Constructing the Yield Curve: Inflation**Assume inflation is expected to be 5% next year, 6% the following year, and 8% thereafter. Must earn these IPs to break even vs. inflation; these IPs would permit you to earn r* (before taxes).**Constructing the Yield Curve: Maturity Risk**• Step 2 – Find the appropriate maturity risk premium (MRP). For this example, the following equation will be used to find a security’s appropriate maturity risk premium. MRPt = 0.1% (t – 1)**Constructing the Yield Curve: Maturity Risk**Using the given equation: Notice that since the equation is linear, the maturity risk premium is increasing as the time to maturity increases, as it should be.**Add the IPs and MRPs to r* to Find the Appropriate Nominal**Rates Step 3 – Adding the premiums to r*. rRF, t = r* + IPt + MRPt Assume r* = 3%,**Default Risk Factor**• One attribute of a bond that influences its interest rate is its risk of default, which occurs when the issuer of the bond is unable or unwilling to make interest payments when promised. • U.S. Treasury bonds have usually been considered to have no default risk because the federal government can always increase taxes to pay off its obligations (or just print money). Bonds like these with no default risk are called default-free bonds.**Default Risk Factor (cont.)**• The spread between the interest rates on bonds with default risk and default-free bonds, called the risk premium, indicates how much additional interest people must earn in order to be willing to hold that risky bond. • A bond with default risk will always have a positive risk premium, and an increase in its default risk will raise the risk premium.**Default Risk Factor (cont.)**• Default risk is an important component of the size of the risk premium. • Because of this, bond investors would like to know as much as possible about the default probability of a bond. • One way to do this is to use the measures provided by credit-rating agencies: Moody’s and S&P are examples.**Liquidity Factor**• Another attribute of a bond that influences its interest rate is its liquidity; a liquid asset is one that can be quickly and cheaply converted into cash if the need arises. The more liquid an asset is, the more desirable it is (higher demand), holding everything else constant. • The differences between interest rates on corporate bonds and Treasury bonds (that is, the risk premiums) reflect not only the corporate bond’s default risk but its liquidity too. This is why a risk premium is sometimes called a risk and liquidity premium.**Hypothetical Yield Curve**• An upward sloping yield curve. • Upward slope due to an increase in expected inflation and increasing maturity risk premium. Interest Rate (%) 15 Maturity risk premium 10 Inflation premium 5 Real risk-free rate Years to Maturity 0 1 10 20**Relationship Between Treasury Yield Curve and Yield Curves**for Corporate Issues • Corporate yield curves are higher than that of Treasury securities, though not necessarily parallel to the Treasury curve. • The spread between corporate and Treasury yield curves widens as the corporate bond rating decreases.**BB-Rated**AAA-Rated Illustrating the Relationship Between Corporate and Treasury Yield Curves Interest Rate (%) 15 10 Treasury Yield Curve 6.0% 5.9% 5 5.2% Years to Maturity 0 0 1 5 10 15 20**Term Structure of Interest Rates**Now that we understand risk, liquidity, and taxes, we turn to another important influence on interest rates – maturity. Bonds with different maturities tend to have different required rates, all else equal.**Term Structure Facts to Be Explained**Besides explaining the shape of the yield curve, a good theory must explain why: • Interest rates for different maturities move together. We see this on the next slide.**Term Structure Facts to Be Explained**Besides explaining the shape of the yield curve, a good theory must explain why: • 1) Interest rates for different maturities move together. • 2) Yield curves tend to have steep upward slope when short rates are low and downward slope when short rates are high. • 3) Yield curve is typically upward sloping.**Three Theories of Term Structure**• Expectations Theory • Pure Expectations Theory explains 1 and 2, but not 3 • Market Segmentation Theory • Market Segmentation Theory explains 3, but not 1 and 2 • Liquidity Premium Theory • Solution: Combine features of both Pure Expectations Theory and Market Segmentation Theory to get Liquidity Premium Theory and explain all facts**Assumptions of the PEH**• Assumes that the maturity risk premium for Treasury securities is zero. • Long-term rates are an average of current and future short-term rates. • If PEH is correct, you can use the yield curve to “back out” expected future interest rates. • Bonds of different maturities are perfect substitutes**Pure Expectations Hypothesis**• The PEH contends that the shape of the yield curve depends on investor’s expectations about future interest rates. • If interest rates are expected to increase, L-T rates will be higher than S-T rates, and vice-versa. Thus, the yield curve can slope up, down, or even bow.**An Example:Observed Treasury Rates and the PEH**If PEH holds, what does the market expect will be the interest rate on one-year securities, one year from now? Three-year securities, two years from now?**One-Year Forward Rate**6.0% x% 0 1 2 6.2% • PEH says that one-year securities will yield 6.4004%, one year from now. • Notice, if an arithmetic average is used, the answer is still very close. Solve: 6.2% = (6.0% + X)/2, and the result will be 6.4%.**One-Year Forward Rate**(1.062)2 = (1.060) (1 + X) 1.12784/1.060 = (1 + X) 6.4004% = X 6.0% x% 0 1 2 6.2% • PEH says that one-year securities will yield 6.4004%, one year from now. • Notice, if an arithmetic average is used, the answer is still very close. Solve: 6.2% = (6.0% + X)/2, and the result will be 6.4%.**Three-Year Security, Two Years from Now**(1.065)5 = (1.062)2 (1 + X)3 1.37009/1.12784 = (1 + X)3 6.7005% = X 6.2% x% 0 1 2 3 4 5 6.5% • PEH says that three-year securities will yield 6.7005%, two years from now.**Three-Year Security, Two Years from Now**6.2% x% 0 1 2 3 4 5 6.5% • PEH says that three-year securities will yield 6.7005%, two years from now.**Expectations Theory**To illustrate what this means, consider two alternative investment strategies for a two-year time horizon. • Buy $1 of one-year bond, and when it matures, buy another one-year bond with your money. • Buy $1 of two-year bond and hold it.**Expectations Theory**The important point of this theory is that if the Expectations Theory is correct, your expected wealth is the same (a the start) for both strategies. Of course, your actual wealth may differ, if rates change unexpectedly after a year.**Expectations Theory**• Expected return from strategy 1 • Since it(iet+1) is also extremely small, expected return is approximately • it + iet+1**Expectations Theory**• Expected return from strategy 2 Since (i2t)2 is extremely small, expected return is approximately 2(i2t)**Expectations Theory**• From implication above expected returns of two strategies are equal • Therefore Solving for i2t (1)**Expectations Theory**• To help see this, here’s a picture that describes the same information:**More generally for n-period bond…**• Don’t let this seem complicated. Equation 2 simply states that the interest rate on a long-term bond equals the average of short rates expected to occur over life of the long-term bond. (2)**More generally for n-period bond…**• Numerical example • One-year interest rate over the next five years are expected to be 5%, 6%, 7%, 8%, and 9% • Interest rate on two-year bond today: • Interest rate for five-year bond today: • Interest rate for one- to five-year bonds today:**More generally for n-period bond…**• Numerical example • One-year interest rate over the next five years are expected to be 5%, 6%, 7%, 8%, and 9% • Interest rate on two-year bond today: (5% + 6%)/2 = 5.5% • Interest rate for five-year bond today: (5% + 6% + 7% + 8% + 9%)/5 = 7% • Interest rate for one- to five-year bonds today: 5%, 5.5%, 6%, 6.5% and 7%**Expectations Theory and Term Structure Facts**• Explains why yield curve has different slopes • When short rates are expected to rise in future, average of future short rates = intis above today's short rate; therefore yield curve is upward sloping. • When short rates expected to stay same in future, average of future short rates same as today's, and yield curve is flat. • Only when short rates expected to fall will yield curve be downward sloping.**Expectations Theory and Term Structure Facts**• Pure expectations theory explains fact 1—that short and long rates move together • Short rate rises are persistent • If it today, iet+1, iet+2 etc. average of future rates int • Therefore: itint(i.e., short and long rates move together)**Expectations Theory and Term Structure Facts**• Explains fact 2—that yield curves tend to have steep slope when short rates are low and downward slope when short rates are high • When short rates are low, they are expected to rise to normal level, and long rate = average of future short rates will be well above today's short rate; yield curve will have steep upward slope. • When short rates are high, they will be expected to fall in future, and long rate will be below current short rate; yield curve will have downward slope.**Expectations Theory and Term Structure Facts**• Doesn't explain fact 3—that yield curve usually has upward slope • Short rates are as likely to fall in future as rise, so average of expected future short rates will not usually be higher than current short rate: therefore, yield curve will not usually slope upward.**Liquidity Premium Theory**• Investors prefer short-term rather than long-term bonds. This implies that investors must be paid positive liquidity premium, int, to hold long term bonds.**Liquidity Premium Theory**• Results in following modification of Expectations Theory, where lnt is the liquidity premium. (3) • We can also see this graphically…**Numerical Example**• One-year interest rate over the next five years: 5%, 6%, 7%, 8%, and 9% • Investors' preferences for holding short-term bonds so liquidity premium for one- to five-year bonds: 0%, 0.25%, 0.5%, 0.75%, and 1.0%**Numerical Example**• Interest rate on the two-year bond: • Interest rate on the five-year bond: • Interest rates on one to five-year bonds: • Comparing with those for the pure expectations theory, liquidity premium theory produces yield curves more steeply upward sloped