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**Interest Rate Determination**

Nominal Rate = Real Rate + Expected Inflation

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**The Risk and Term Structures of Interest Rates**

Risk structure: Bonds with the same maturity (n) have different interest rates because of default risk premium (d) illiquidity risk premium (l) income tax risk discount (t) Term structure: For bonds with identical characteristics, the interest rate (i) increases as maturity (n) increases maturity premium (int – it) liquidity premium (lnt) The yield curve is the relationship between i and n.

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**Risk Structure Default risk premium**

Default risk is the 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 Default risk premium (d) is the spread between the interest rates on bonds with default risk and the interest rates on Treasury bonds, holding l, t, n, lnt, and int – it equal

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**Risk Structure Default risk premium**

TABLE 1

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**Risk Structure Default risk premium**

Sc St 950 5 950 5 Dt Dc Q Q Corporate Bond Market U.S. Treasury Bond Market

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**Risk Structure Default risk premium**

Sc St 950 5 950 5 925 6 Dt Dc Dc Q Q Corporate Bond Market U.S. Treasury Bond Market

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**Risk Structure Default risk premium**

Sc St 4 975 950 5 950 5 925 6 Dt Dt Dc Dc Q Q Corporate Bond Market U.S. Treasury Bond Market

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**Risk Structure Default risk premium**

Sc St 4 975 2 925 6 Dt Dt Dc Dc Q Q Corporate Bond Market U.S. Treasury Bond Market

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**Risk Structure Default risk premium**

You own a $1000, 10% GM bond that matures next year. The Obama Administration abrogated 100 years of bankruptcy law when it stripped primary bond holders of their first claim rights on corporate assets during the GM bailout. Explain why corporate bond prices would be lower in the post bailout era, holding all else equal. If the GM bond sold for $1068 before the bailout but sells for $1023, compute the yields on the bonds before and after the bailout. Pre-bailout N = 1 I% = A PV = -1068 PMT = 100 FV = 1000 Post-bailout N = 1 I% = A PV = -1023 PMT = 100 FV = 1000

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**Risk Structure Default risk premium**

You own a $1000, 10% GM bond that matures next year. The Obama Administration abrogated 100 years of bankruptcy law when it stripped primary bond holders of their first claim rights on corporate assets during the GM bailout. Explain why corporate bond prices would be lower in the post bailout era, holding all else equal. If the GM bond sold for $1068 before the bailout but sells for $1023, compute the yields on the bonds before and after the bailout. Post-bailout N = 1 I% = 7.527 PV = -1023 PMT = 100 FV = 1000

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**Illiquidity risk premium**

Risk Structure Illiquidity risk premium Liquidity is 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 Illiquidity risk premium (l) is the spread between the interest rate on a bond that is illiquid and the interest rate on Treasury bonds, holding d, t, n, lnt, and int – it equal. E.g., assume an investor is looking at buying two corporate bonds that have the same coupon rates and maturities, but only one is traded on a public exchange. The investor is not be willing to pay as much for the non-public bond. The difference in yields the investor is willing to pay for each bond is the liquidity premium.

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**Illiquidity risk premium**

Risk Structure Illiquidity risk premium P i P i Sc St 950 5 950 5 Dt Dc Q Q Corporate Bond Market U.S. Treasury Bond Market

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**Illiquidity risk premium**

Risk Structure Illiquidity risk premium P i P i Sc St 950 5 950 5 925 6 Dt Dc Dc Q Q Corporate Bond Market U.S. Treasury Bond Market

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**Illiquidity risk premium**

Risk Structure Illiquidity risk premium P i P i Sc St 4 975 950 5 950 5 925 6 Dt Dt Dc Dc Q Q Corporate Bond Market U.S. Treasury Bond Market

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**Illiquidity risk premium**

Risk Structure Illiquidity risk premium P i P i Sc St 4 975 2 925 6 Dt Dt Dc Dc Q Q Corporate Bond Market U.S. Treasury Bond Market

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**Illiquidity risk premium**

Risk Structure Illiquidity risk premium You are considering owning two $1000 bonds that mature next year. One is a corporate bond, the other is a Treasury, and both have an 8% coupon rate. Explain why the Treasury is selling for $1058 while the corporate bond is selling for $1001 if both have the same bond rating, and compute the yields on the two bonds. Corporate N = 1 I% = A PV = -1001 PMT = 80 FV = 1000

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**Illiquidity risk premium**

Risk Structure Illiquidity risk premium You are considering owning two $1000 bonds that mature next year. One is a corporate bond, the other is a Treasury, and both have an 8% coupon rate. Explain why the Treasury is selling for $1058 while the corporate bond is selling for $1001 if both have the same bond rating, and compute the yields on the two bonds. Treasury N = 1 I% = 2.079 PV = -1058 PMT = 80 FV = 1000 Corporate N = 1 I% = 7.892 PV = -1001 PMT = 80 FV = 1000

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**Tax exemption risk discount**

Risk Structure Tax exemption risk discount Income tax considerations Interest payments on municipal bonds are exempt from federal income taxes. Tax exemption risk discount (t) is the spread between the interest rate on a tax exempt municipal bond and the interest rate on Treasury bonds, holding d, l, n, lnt, and int – it equal. The discount shrinks if federal income taxes are lowered or there is talk of doing so politicians seriously consider ending the exemption the exemption is repealed.

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**Tax exemption risk discount**

Risk Structure Tax exemption risk discount P i P i St Sc 950 5 950 5 Dt Dc Q Q Municipal Bond Market U.S. Treasury Bond Market

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**Tax exemption risk discount**

Risk Structure Tax exemption risk discount P i P i St Sc 950 5 950 5 925 6 Dt Dc Dc Q Q Municipal Bond Market U.S. Treasury Bond Market

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**Tax exemption risk discount**

Risk Structure Tax exemption risk discount P i P i St Sc 4 975 950 5 950 5 925 6 Dt Dt Dc Dc Q Q Municipal Bond Market U.S. Treasury Bond Market

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**Tax exemption risk discount**

Risk Structure Tax exemption risk discount P i P i St Sc 4 975 -2 925 6 Dt Dt Dc Dc Q Q Municipal Bond Market U.S. Treasury Bond Market

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**Tax exemption risk discount**

Risk Structure Tax exemption risk discount You are considering owning two $1000 bonds that mature next year. One is a corporate bond, the other is a tax-free municipal, and both have an 8% coupon rate. If the bonds have a current yield of 3.5%, and you intend to hold them for their final year, compute the price you would be willing to pay assuming a federal income tax rate is 50%. Corporate N = 1 I% = 3.5 PV = A PMT = 40 FV = 1000

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**Tax exemption risk discount**

Risk Structure Tax exemption risk discount You are considering owning two $1000 bonds that mature next year. One is a corporate bond, the other is a tax-free municipal, and both have an 8% coupon rate. If the bonds have a current yield of 3.5%, and you intend to hold them for their final year, compute the price you would be willing to pay assuming a federal income tax rate is 50%. Tax-free municipal N = 1 I% = 3.5 PV = PMT = 80 FV = 1000 Corporate N = 1 I% = 3.5 PV = PMT = 40 FV = 1000

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**Figure 1—Long-Term Bond Yields, 1919–2011**

Risk Structure Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics, 1941–1970; Federal Reserve; Figure 1—Long-Term Bond Yields, 1919–2011

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**Interest Rate Determination**

Nominal Rate = Real Rate + Expected Inflation + Default Risk Premium + Illiquidity Risk Premium – Tax exemption discount

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**Term Structure Time to maturity affects interest rates because**

Time increases exposure to risk, causing investors to demand higher yields on securities with longer maturities. The term structure of interest rates refers to difference in the yields on instruments that are identical except for term to maturity. Term structure is represented graphically by a yield curve. Yield curves consider only the relationship between maturity or term of a security and its yield at a moment in time, otrs.

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**Facts that the theory must explain:**

Term Structure Facts that the theory must explain: Interest rates on bonds of different maturities move together over time

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Term Structure Sources: Federal Reserve; Figure 4—Interest rate movements on Treasuries with different maturities

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**Facts that the theory must explain:**

Term Structure Facts that the theory must explain: Interest rates on bonds of different maturities move together over time When short-term interest rates are low, yield curves are more likely to have an upward slope; when short-term rates are high, yield curves are more likely to slope downward and be inverted Yield curves almost always slope upward

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Term Structure February 4, 2005

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**Figure 7 Yield Curves for U.S. Government Bonds**

Term Structure Figure 7 Yield Curves for U.S. Government Bonds

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**Term Structure Figure 6 Upward sloping**

Interest rates on securities with longer maturities exceed interest rates on shorter term securities. Downward sloping Interest rates on securities with longer maturities are less than interest rates on shorter term securities. Horizontal Interest rates on long and short term securities have approximately the same interest rates.

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**Facts that the theory must explain:**

Term Structure Facts that the theory must explain: Interest rates on bonds of different maturities move together over time When short-term interest rates are low, yield curves are more likely to have an upward slope; when short-term rates are high, yield curves are more likely to slope downward and be inverted Yield curves almost always slope upward Three Theories that explain these facts Segmented markets theory explains fact three but not the first two Expectations theory explains the first two facts but not the third Liquidity premium theory combines the two theories to explain all three facts

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**Term Structure maturity premium**

Expectations theory says the yield on a long-term bond equals the average of the short-term interest rates people expect to occur over its life Maturity Premium is the spread between the interest rates on bonds with n years and 1 year to maturity, holding d, l, t, and lnt equal. int – it Buyers of bonds do not prefer bonds of one maturity over another do not hold any quantity of a bond if its expected return is less than that of another bond with a different maturity consider bonds with different maturities to be perfect substitute Interest rates on different maturity bonds move together over time; explained by the expectations term in the equation Yield curves tend to slope upward when short-term rates are low and to be inverted when short-term rates are high; explained by the liquidity premium term in the first case and by a low expected average in the second case Yield curves typically slope upward; explained by a larger liquidity premium as the term to maturity lengthens

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**Term Structure maturity premium**

The table below shows current and expected future one-year interest rates, as well as current interest rates on multiyear bonds. Use the table to calculate the liquidity premium for each multiyear bond.

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**Term Structure maturity premium**

The table below shows current and expected future one-year interest rates, as well as current interest rates on multiyear bonds. Use the table to calculate the liquidity premium for each multiyear bond.

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**Term Structure maturity premium**

The table below shows current and expected future one-year interest rates, as well as current interest rates on multiyear bonds. Use the table to calculate the liquidity premium for each multiyear bond.

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**Term Structure maturity premium**

The table below shows current and expected future one-year interest rates, as well as current interest rates on multiyear bonds. Use the table to calculate the liquidity premium for each multiyear bond.

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**Term Structure maturity premium**

The table below shows current and expected future one-year interest rates, as well as current interest rates on multiyear bonds. Use the table to calculate the liquidity premium for each multiyear bond.

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**Term Structure maturity premium**

The table below shows current and expected future one-year interest rates, as well as current interest rates on multiyear bonds. Use the table to calculate the liquidity premium for each multiyear bond.

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**Term Structure maturity premium**

The table below shows current and expected future one-year interest rates, as well as current interest rates on multiyear bonds. Use the table to calculate the liquidity premium for each multiyear bond.

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**Term Structure maturity premium**

Graph the maturity adjusted yields over maturity i n

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**Term Structure maturity premium**

Graph the maturity adjusted yields over maturity i n maturity premium for a 1-year bond 0%

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**Term Structure maturity premium**

Graph the maturity adjusted yields over maturity i n maturity premium for a 2-year bond 0.325%

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**Term Structure maturity premium**

Graph the maturity adjusted yields over maturity i n maturity premium for a 3-year bond 0.57%

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**Term Structure maturity premium**

Graph the maturity adjusted yields over maturity i n maturity premium for a 4-year bond 0.7675%

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**Term Structure maturity premium**

Graph the maturity adjusted yields over maturity i n maturity premium for a 5-year bond 0.93%

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**Term Structure maturity premium**

Graph the maturity adjusted yields over maturity i n maturity premium for a 6-year bond 1.06%

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**Term Structure Expectations Theory**

Yield Curve i n

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**Term Structure liquidity premium**

The interest rate on a long-term bond will equal an average of short-term interest rates expected to occur over the life of the long-term bond plus a liquidity premium that responds to supply and demand conditions for that bond Bonds of different maturities are partial (not perfect) substitutes Liquidity premium is the spread between the interest rates on bonds with n and one years to maturity, holding d, l, t, and int – it equal lnt Segmented markets theory is the basis for the assumption that Bonds of different maturities are partial (not perfect) substitutes Interest rates on different maturity bonds move together over time; explained by the expectations term in the equation Yield curves tend to slope upward when short-term rates are low and to be inverted when short-term rates are high; explained by the liquidity premium term in the first case and by a low expected average in the second case Yield curves typically slope upward; explained by a larger liquidity premium as the term to maturity lengthens

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**Term Structure liquidity premium**

Suppose the liquidity premium is linear in maturity: lnt = 0.08n

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**Term Structure Expectations Theory**

Yield Curve

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**Liquidity Premium Theory**

Term Structure Liquidity Premium Theory Yield Curve

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**Interest Rate Determination**

Nominal Rate = Real Rate + Expected Inflation + Default Risk Premiums + Illiquidity Risk Premium – Tax exemption risk discount + Maturity Premium + Liquidity Premium

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