Interest Rate Determination Nominal Rate = Real Rate +Expected Inflation.

Interest Rate Determination

Nominal Rate = Real Rate + Expected Inflation

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.

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

TABLE 1

Corporate Bond Market

U.S. Treasury Bond Market

P Pi i

P Pi iSc St

950 5 950 5

P Pi iSc St

DcDcDt

950 5 950 5

P Pi iSc St

DcDcDt

Pre-bailout

N = 1I% = APV = -1068PMT = 100FV = 1000

Post-bailout

N = 1I% = APV = -1023PMT = 100FV = 1000

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 = 1I% = 2.996PV = -1068PMT = 100FV = 1000

Post-bailout

N = 1I% = 7.527PV = -1023PMT = 100FV = 1000

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.

• 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.

Risk Structure Illiquidity risk premium

P Pi i

P Pi iSc St

950 5 950 5

P Pi iSc St

DcDcDt

950 5 950 5

P Pi iSc St

DcDcDt

Treasury

N = 1I% = APV = -1058PMT = 80FV = 1000

Corporate

N = 1I% = APV = -1001PMT = 80FV = 1000

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 = 1I% = 2.079PV = -1058PMT = 80FV = 1000

Corporate

N = 1I% = 7.892PV = -1001PMT = 80FV = 1000

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.

• 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 ifo federal income taxes are lowered or there is talk of doing so

o politicians seriously consider ending the exemption

o the exemption is repealed.

Risk Structure Tax exemption risk discount

Municipal Bond Market

PP iiScSt

950 5950 5

PP iiScSt

DcDcDt

950 5950 5

PP iiScSt

DcDcDt

Tax-free municipal

N = 1I% = 3.5PV = APMT = 80FV = 1000

Corporate

N = 1I% = 3.5PV = APMT = 40FV = 1000

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 = 1I% = 3.5PV = -1043.48PMT = 80FV = 1000

Corporate

N = 1I% = 3.5PV = -1004.83PMT = 40FV = 1000

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

Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics, 1941–1970; Federal Reserve; www.federalreserve.gov/releases/h15/data.htm.

Risk Structure

+ Default Risk Premium+ Illiquidity Risk Premium– Tax exemption discount

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.

Facts that the theory must explain:

1. Interest rates on bonds of different maturities move together over time

Term Structure

Figure 4—Interest rate movements on Treasuries with different maturities

Sources: Federal Reserve; www.federalreserve.gov/releases/h15/data.htm.

Term Structure

2. 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

3. Yield curves almost always slope upward

Term Structure

31February 4, 2005

Term Structure

Figure 7 Yield Curves for U.S. Government Bonds

Term Structure

Figure 6

Term Structure

2. 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

3. Yield curves almost always slope upward

Term Structure

Three Theories that explain these facts

1. Segmented markets theory explains fact three but not the first two

2. Expectations theory explains the first two facts but not the third

3. Liquidity premium theory combines the two theories to explain all three facts

Term Structurematurity 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 o do not prefer bonds of one maturity over anothero do not hold any quantity of a bond if its expected return is less than that of

another bond with a different maturity o consider bonds with different maturities to be perfect substitute

1 2 ( 1)...e e et t t t n

i i i ii

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.

i 1 2 3 4 5e e e e e

t t t t t ti i i i i i

1 2 3 4 5e e e e e

Graph the maturity adjusted yields over maturity

1 2 3 4 5 61.00

1 2 3 4 5 6

maturity premium

for a 1-year bond

1 2 3 4 5 6

maturity premium

for a 2-year bond

0.325%

1 2 3 4 5 6

maturity premium

for a 3-year bond

1 2 3 4 5 6

maturity premium

for a 4-year bond

0.7675%

1 2 3 4 5 6

maturity premium

for a 5-year bond

1 2 3 4 5 6

maturity premium

for a 6-year bond

Term StructureExpectations Theory

1 2 3 4 5 6

Yield Curve

Term Structureliquidity 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

Suppose the liquidity premium is linear in maturity:

lnt = 0.08n

Term Structureliquidity premium

1 2 ( 1)...e e et t t t

n tt nni i i

Term StructureExpectations Theory

1 2 3 4 5 6

Yield Curve

Term StructureLiquidity Premium Theory

1 2 3 4 5 61.00

1 2 3 4 5 6

1 2 ( 1)...e e et t t t

n tt nni i i

Yield Curve

+ Default Risk Premiums+ Illiquidity Risk Premium– Tax exemption risk discount+ Maturity Premium+ Liquidity Premium

Interest Rate Determination Nominal Rate = Real Rate +Expected Inflation.

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