Chap015The Term Structure of Interest Rates(金融工程-南开大学,王小麓))
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McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-17
Using Spot Rates to Price Coupon Bonds
A coupon bond can be viewed as a series of zero coupon bonds. To find the value each payment is discount at the zero coupon rate. Once the bond value is found, one can solve for the yield. It’s the reason that similar maturity and default risk bonds sell at different yields to maturity.
15-1
Chapter 15
The Term Structure of Interest Rates
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-2
Overview of Term Structure of Interest Rates
15-16
Market Segmentation and Preferred Habitat
Short- and long-term bonds are traded in distinct markets. Trading in the distinct segments determines the various rates. Observed rates are not directly influenced by expectations. Preferred Habitat: - Modification of market segmentation - Investors will switch out of preferred maturity segments if premiums are adequate.
(1 yn ) (1 f n ) (1 yn 1 ) n 1
n
fn = one-year forward rate for period n yn = yield for a security with a maturity of n
(1 yn ) (1 yn1 ) (1 f n )
Investors will demand a premium for the risk associated with long-term bonds.
The yield curve has an upward bias built into the long-term rates because of the risk premium.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-10
Forward Rates for Downward Sloping Yield Curve
1yr Forward Rates
The relationship between yield to maturity and maturity.
Information on expected future short term rates can be implied from yield curve. The yield curve is a graph that displays the relationship between yield and maturity.
15-12
Expectations Theory
Observed long-term rate is a function of today’s short-term rate and expected future short-term rates. Long-term and short-term securities are perfect substitutes. Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-13
Liquidity Premium Theory
Long-term bonds are more risky.
Forward rates contain a liquidity premium and are not equal to expected future short-term rates.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
- Upward bias over expectations
Market Segmentation
- Preferred Habitat
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-5
Pricing of Bonds using Expected Rates
1 PVn (1 r1 ) (1 r2 )...(1 rn )
15-6
Long-Term Rates and Bond Prices using Expected Rates
Time to Maturity Price of Zero* Yield to Maturity 1 $925.93 8.00%
2
3 4
841.75
758.33 683.18
8.995
=
=
0.063336
0.063008
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-11
Theories of Term Structure
Expectations
Liquidity Preference
3yr = 9.660
fn = ?
(1.0993)4 = (1.0966)3 (1+fn) (1.46373) / (1.31870) = (1+fn) fn = .10998 or 11% Note: this is expected rate that was used in the prior example.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-9
Downward Sloping Spot Yield Curve
Zero-Coupon Rates Bond Maturity 12% 1 11.75% 2 11.25% 3 10.00% 4 9.25% 5
n
n 1
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-8
Example of Forward Rates using Table 15.2 Numbers
4 yr = 9.993
15-4
Expected Interest Rates in Coming Years (Table 15.1)
Expected One-Year Rates in Coming Years
Year
0 (today) 1 2 3
Interest Rate
8% 10% 11% 11%
ቤተ መጻሕፍቲ ባይዱ
McGraw-Hill/Irwin
PVn = Present Value of $1 in n periods r1 = One-year rate for period 1
r2 = One-year rate for period 2
rn = One-year rate for period n
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-3
Yield Curves
Yields Upward Sloping Flat
Downward Sloping
Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-14
Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Forward Rates
Liquidity Premium Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Three major theories are proposed to explain the observed yield curve.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
1yr
[(1.1175)2 / 1.12] - 1
=
0.115006
0.102567
2yrs [(1.1125)3 / (1.1175)2] - 1 =
3yrs [(1.1)4 / (1.1125)3] - 1
4yrs [(1.0925)5 / (1.1)4] - 1
McGraw-Hill/Irwin
15-15
Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Forward Rates Liquidity Premium
Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
9.660 9.993
* $1,000 Par value zero
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-7
Forward Rates from Observed Long-Term Rates
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-17
Using Spot Rates to Price Coupon Bonds
A coupon bond can be viewed as a series of zero coupon bonds. To find the value each payment is discount at the zero coupon rate. Once the bond value is found, one can solve for the yield. It’s the reason that similar maturity and default risk bonds sell at different yields to maturity.
15-1
Chapter 15
The Term Structure of Interest Rates
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-2
Overview of Term Structure of Interest Rates
15-16
Market Segmentation and Preferred Habitat
Short- and long-term bonds are traded in distinct markets. Trading in the distinct segments determines the various rates. Observed rates are not directly influenced by expectations. Preferred Habitat: - Modification of market segmentation - Investors will switch out of preferred maturity segments if premiums are adequate.
(1 yn ) (1 f n ) (1 yn 1 ) n 1
n
fn = one-year forward rate for period n yn = yield for a security with a maturity of n
(1 yn ) (1 yn1 ) (1 f n )
Investors will demand a premium for the risk associated with long-term bonds.
The yield curve has an upward bias built into the long-term rates because of the risk premium.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-10
Forward Rates for Downward Sloping Yield Curve
1yr Forward Rates
The relationship between yield to maturity and maturity.
Information on expected future short term rates can be implied from yield curve. The yield curve is a graph that displays the relationship between yield and maturity.
15-12
Expectations Theory
Observed long-term rate is a function of today’s short-term rate and expected future short-term rates. Long-term and short-term securities are perfect substitutes. Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-13
Liquidity Premium Theory
Long-term bonds are more risky.
Forward rates contain a liquidity premium and are not equal to expected future short-term rates.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
- Upward bias over expectations
Market Segmentation
- Preferred Habitat
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-5
Pricing of Bonds using Expected Rates
1 PVn (1 r1 ) (1 r2 )...(1 rn )
15-6
Long-Term Rates and Bond Prices using Expected Rates
Time to Maturity Price of Zero* Yield to Maturity 1 $925.93 8.00%
2
3 4
841.75
758.33 683.18
8.995
=
=
0.063336
0.063008
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-11
Theories of Term Structure
Expectations
Liquidity Preference
3yr = 9.660
fn = ?
(1.0993)4 = (1.0966)3 (1+fn) (1.46373) / (1.31870) = (1+fn) fn = .10998 or 11% Note: this is expected rate that was used in the prior example.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-9
Downward Sloping Spot Yield Curve
Zero-Coupon Rates Bond Maturity 12% 1 11.75% 2 11.25% 3 10.00% 4 9.25% 5
n
n 1
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-8
Example of Forward Rates using Table 15.2 Numbers
4 yr = 9.993
15-4
Expected Interest Rates in Coming Years (Table 15.1)
Expected One-Year Rates in Coming Years
Year
0 (today) 1 2 3
Interest Rate
8% 10% 11% 11%
ቤተ መጻሕፍቲ ባይዱ
McGraw-Hill/Irwin
PVn = Present Value of $1 in n periods r1 = One-year rate for period 1
r2 = One-year rate for period 2
rn = One-year rate for period n
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-3
Yield Curves
Yields Upward Sloping Flat
Downward Sloping
Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-14
Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Forward Rates
Liquidity Premium Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Three major theories are proposed to explain the observed yield curve.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
1yr
[(1.1175)2 / 1.12] - 1
=
0.115006
0.102567
2yrs [(1.1125)3 / (1.1175)2] - 1 =
3yrs [(1.1)4 / (1.1125)3] - 1
4yrs [(1.0925)5 / (1.1)4] - 1
McGraw-Hill/Irwin
15-15
Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Forward Rates Liquidity Premium
Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
9.660 9.993
* $1,000 Par value zero
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-7
Forward Rates from Observed Long-Term Rates