Chapter 6

Report
Interest Rate Determination
Nominal Rate (i)
=
+
+
+
–
+
+
Real Rate (r)
Expected Inflation (p e)
Default Risk Premium (d)
Illiquidity Risk Premium (l)
Tax exemption discount (t)
Maturity Premium (int – it)
Liquidity Premium (lnt)
Interest Rate Determination
Nominal Rate (i)
Risk structure
=
+
+
+
–
Real Rate (r)
Expected Inflation (p e)
Default Risk Premium (d)
Illiquidity Risk Premium (l)
Tax exemption discount (t)
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
Risk Structure
Default risk premium
TABLE 1
Risk Structure
Default risk premium
P
i
P
i
Sc
St
950
5
950
5
Dt
Dc
Q
Corporate Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Default risk premium
P
i
P
i
Sc
St
950
5
925
6
950
5
Dt
Dc
Dc
Q
Corporate Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Default risk premium
P
i
P
i
Sc
St
950
5
925
6
975
4
950
5
Dt
Dt
Dc
Dc
Q
Corporate Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Default risk premium
P
i
P
i
Sc
St
4
975
2
925
6
Dt
Dt
Dc
Dc
Q
Corporate Bond
Market
Q
U.S. Treasury Bond
Market
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
Post-bailout
N=1
I% = A
PV = -1068
PMT = 100
FV = 1000
N=1
I% = A
PV = -1023
PMT = 100
FV = 1000
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
Post-bailout
N=1
I% = 2.996
PV = -1068
PMT = 100
FV = 1000
N=1
I% = 7.527
PV = -1023
PMT = 100
FV = 1000
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.
Risk Structure
Illiquidity risk premium
P
i
P
i
Sc
St
950
5
950
5
Dt
Dc
Q
Corporate Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Illiquidity risk premium
P
i
P
i
Sc
St
950
5
925
6
950
5
Dt
Dc
Dc
Q
Corporate Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Illiquidity risk premium
P
i
P
i
Sc
St
950
5
925
6
975
4
950
5
Dt
Dt
Dc
Dc
Q
Corporate Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Illiquidity risk premium
P
i
P
i
Sc
St
4
975
2
925
6
Dt
Dt
Dc
Dc
Q
Corporate Bond
Market
Q
U.S. Treasury Bond
Market
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. Why is the price of Treasuries higher than
corporate bonds with the same attributes? If the price of treasuries
is $1058 and the price of a similar corporate bond with the same
bond rating is $1001, compute the yields on the two bonds.
Treasury
Corporate
N=1
I% = A
PV = -1058
PMT = 80
FV = 1000
N=1
I% = A
PV = 1001
PMT = 80
FV = 1000
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. Why is the price of Treasuries higher than
corporate bonds with the same attributes? If the price of treasuries
is $1058 and the price of a similar corporate bond with the same
bond rating is $1001, compute the yields on the two bonds.
Treasury
Corporate
N=1
I% = 2.079
PV = -1058
PMT = 80
FV = 1000
N=1
I% = 7.892
PV = 1001
PMT = 80
FV = 1000
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
o 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
P
i
P
St
i
Sc
950
5
950
5
Dm
Dt
Q
Municipal Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Tax exemption risk discount
P
i
P
i
St
Sc
950
5
950
5
925
6
Dm
Dt
Dt
Q
Municipal Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Tax exemption risk discount
P
i
P
i
St
Sc
975
4
950
5
Dm
950
5
925
6
Dt
Dc
Dc
Q
Municipal Bond
Market
Q
U.S. Treasury Bond
Market
Risk Structure
Tax exemption risk discount
P
i
P
i
St
Sc
4
975
-2
925
Dt
6
Dt
Dt
Dt
Q
Municipal Bond
Market
Q
U.S. Treasury Bond
Market
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 of 50%.
Tax-free municipal
Corporate
N=1
I% = 3.5
PV = A
PMT = 80
FV = 1000
N=1
I% = 3.5
PV = A
PMT = 40
FV = 1000
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 of 50%.
Tax-free municipal
Corporate
N=1
I% = 3.5
PV = -1043.48
PMT = 80
FV = 1000
N=1
I% = 3.5
PV = -1004.83
PMT = 40
FV = 1000
Risk Structure
Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics,
1941–1970; Federal Reserve; www.federalreserve.gov/releases/h15/data.htm.
Figure 1—Long-Term Bond Yields, 1919–2011
Interest Rate Determination
Nominal Rate (i)
=
+
+
+
–
+
+
Real Rate (r)
Expected Inflation (p e)
Default Risk Premium (d)
Illiquidity Risk Premium (l)
Tax exemption discount (t)
Maturity Premium (int – it)
Liquidity Premium (lnt)
Interest Rate Determination
Nominal Rate (i)
Risk structure
Term structure
=
+
+
+
–
+
+
Real Rate (r)
Expected Inflation (p e)
Default Risk Premium (d)
Illiquidity Risk Premium (l)
Tax exemption discount (t)
Maturity Premium (int – it)
Liquidity Premium (lnt)
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.
Term Structure
Facts that the theory must explain:
1.
Interest rates on bonds of different maturities move together over time
Term Structure
Sources: Federal Reserve; www.federalreserve.gov/releases/h15/data.htm.
Figure 4—Interest rate movements on Treasuries with different maturities
Term Structure
Facts that the theory must explain:
1.
Interest rates on bonds of different maturities move together over time
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
February 4, 2005
33
Term Structure
Figure 7 Yield Curves for U.S. Government Bonds
Term Structure
Figure 6
Term Structure
Facts that the theory must explain:
1.
Interest rates on bonds of different maturities move together over time
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
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 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
e
e
e
int 
it  it 1  it  2  ...  it  ( n 1)
n
– 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 another
o 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
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.
e
e
e
e
e
i

i

i

i

i

i
i nt  t t 1 t  2 t 3 t  4 t 5
n
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.
e
e
e
e
e
i

i

i

i

i

i
i nt1t  t t 1 t  2 t 3 t  4 t 5
n
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.
e
e
e
e
e
i

i

i

i

i

i
i 2t  t t 1 t  2 t 3 t  4 t 5
n
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.
e
e
e
e
e
i

i

i

i

i

i
i 3t  t t 1 t  2 t 3 t  4 t 5
n
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.
e
e
e
e
e
i

i

i

i

i

i
i 4t  t t 1 t  2 t 3 t  4 t 5
n
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.
e
e
e
e
e
i

i

i

i

i

i
i 5t  t t 1 t  2 t 3 t  4 t 5
n
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.
e
e
e
e
e
i

i

i

i

i

i
i 6t  t t 1 t  2 t 3 t  4 t 5
n
Term Structure
maturity premium
Graph the maturity adjusted yields over maturity
2.20
2.00
i
1.80
1.60
1.40
1.20
1.00
1
2
3
4
n
5
6
Term Structure
maturity premium
Graph the maturity adjusted yields over maturity
2.20
2.00
i
1.80
1.60
1.40
maturity premium
for a 1-year bond
0%
1.20
1.00
1
2
3
4
n
5
6
Term Structure
maturity premium
Graph the maturity adjusted yields over maturity
2.20
2.00
i
1.80
1.60
maturity premium
for a 2-year bond
0.325%
1.40
1.20
1.00
1
2
3
4
n
5
6
Term Structure
maturity premium
Graph the maturity adjusted yields over maturity
2.20
2.00
i
1.80
maturity premium
for a 3-year bond
0.57%
1.60
1.40
1.20
1.00
1
2
3
4
n
5
6
Term Structure
maturity premium
Graph the maturity adjusted yields over maturity
2.20
2.00
i
1.80
maturity premium
for a 4-year bond
0.7675%
1.60
1.40
1.20
1.00
1
2
3
4
n
5
6
Term Structure
maturity premium
Graph the maturity adjusted yields over maturity
2.20
2.00
i
maturity premium
for a 5-year bond
0.93%
1.80
1.60
1.40
1.20
1.00
1
2
3
4
n
5
6
Term Structure
maturity premium
Graph the maturity adjusted yields over maturity
2.20
2.00
i
maturity premium
for a 6-year bond
1.06%
1.80
1.60
1.40
1.20
1.00
1
2
3
4
n
5
6
Term Structure
Expectations Theory
Yield Curve
2.20
2.00
i
1.80
1.60
1.40
1.20
1.00
1
2
3
4
n
5
6
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
Term Structure
liquidity premium
Suppose the liquidity premium is linear in maturity:
lnt = 0.08n
Term Structure
Expectations Theory
Yield Curve
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
1
int 
2
3
4
5
it  ite1  ite 2  ...  ite ( n 1)
n
6
 lnt
Term Structure
Liquidity Premium Theory
Yield Curve
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
1
int 
2
3
4
5
it  ite1  ite 2  ...  ite ( n 1)
n
6
 lnt
Interest Rate Determination
Nominal Rate (i)
Risk structure
Term structure
=
+
+
+
–
+
+
Real Rate (r)
Expected Inflation (p e)
Default Risk Premium (d)
Illiquidity Risk Premium (l)
Tax exemption discount (t)
Maturity Premium (int – it)
Liquidity Premium (lnt)

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