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AECO 480/580(9671/9672) Financial
Econometrics
III. Bond and Term Structure Models
Zhongwen Liang
Department of Economics
University at Albany, SUNY
March 29, 2018
Bond Yields and Prices
- Bonds are a financial instrument that will pay the face value (or par value) to its holder at the time of maturity.
- Some bonds also pay interest periodically referring to as coupon payment.
- Zero-coupon bonds do not pay periodic interest.
- Bond yield is the return an investor will receive by holding a bond to maturity.
- In finance, several types of bond yield are used. The common ones are the current yield and yield to maturity (YTM).
- Current Yield. The current yield denotes the percentage return that the annual coupon payment provides the investor.
- Mathematically, we have
Current yield=
Annual interest paid in dollars
Market price of the bond
100 %.
- For example, if an investor paid $90 for a bond with face value of $100, also known aspar value, and the bond paid a coupon rate of 5% per annum, then the current yield of the bond isct= ( 0. 05 100 )/ 90 100 % = 5. 56 %.
- We use the subscripttto signify that the yield is typically time dependent.
- From the definition, current yield does not include any capital gains or losses of the investment.
- For zero-coupon bonds, the yield is calculated as follows:
Crrent yield=
(
Face value
Purchase price
) 1 /k
1 ,
wherekdenotes time to maturity in years.
- For instance, if an investor purchased a zero-coupon bond with face value $100 for $90 and the bond will mature in 2 years, then the yield isct= ( 100 / 90 )^1 /^2 1 = 5. 41 %.
- Yield to Maturity. The current yield does not consider the time value of money, because it does not consider the present value of the coupon payments the investor will receive in the future.
- Therefore, a more commonly used measurement of bond investment is the YTM.
- The calculation of YTM, however, is more complex.
- Simply put, YTM is the yield obtained by equating the bond price to the present value of all future payments.
- Suppose that the bond holder will receivekpayments between purchase and maturity.
- LetyandPbe the YTM and price of the bond, respectively. Then,
P=
C 1
1 +y
+
C 2
( 1 +y)^2
++
Ck+F
( 1 +y)k
,
whereFdenotes the face value andCiis theith cash flow
of coupon payment.
- Suppose that the coupon rate isper annum, the number of payments is m per year, and the time to maturity isn years.
- In this case, cash flow of coupon payment isF/m, and the number of payments isk=mn.
- The bond price and YTM can be formulated as
P =
F
m
[
1
1 +y/m
+
1
( 1 +y/m)^2
++
1
( 1 +y/m)k
]
+
F
( 1 +y/m)k
=
F
y
[
1
1
( 1 +y/m)k
]
+
F
( 1 +y/m)k
- The table below gives some results between bond price and YTM assuming thatF= $100, coupon rate is 5% per annum payable semiannually, and time to maturity is 3 years.
- From the table, we see that as the YTM increases the bond price decreases. In other words, YTM is inversely proportional to the bond price.
Yield Semiannual Bond
to Maturity (%) Rate (%) Price ($)
6 3.0 97.
7 3.5 94.
8 4.0 92.
9 4.5 89.
10 5.0 87.
- In practice, we observed bond price so that YTM must be calculated.
- The solution is not easy to find in general, but calibration can be used to obtain an accurate approximation.
- As an example, suppose that one paid $94 to purchase the bond shown in the prior table.
- From which, we see that the YTM must be in the interval [7,8]%.
- With trial and error, we have
Yield Semiannual Bond
to Maturity (%) Rate (%) Price ($)
7.1 3.55 94.
7.2 3.6 94.
7.3 3.65 93.
7.25 3.625 94.
7.26 3.63 94.
- Therefore, the YTM is approximately 7.26% per annum for the investor.
- Many financial institutions provide online programs that calculate bond YTM and price.
U.S. Government Bonds
- The U.S. Government issues various bonds to finance its debts.
- These bonds include Treasury bills, Treasury notes, and Treasury bonds.
- A simple description of these bonds is given below.
- Treasury bills (T-Bills) mature in one year or less.
- They do not pay interest prior to maturity and are sold at a discount of the face value (or par value) to create a positive YTM.
- The commonly used maturities are 28 days (1 month), 91 days (3 months), 182 days (6 months), and 364 days ( year).
- The minimum purchase is $100.
- The discount yield of T-Bills is calculated via
Discount yield (%)=
FP
F
360
Days till maturity
100 (%),
whereFandPdenote the face value and purchase price,
respectively.
- The U.S. Treasury Department announces the amounts of offering for 13- and 26- week bills each Thursday for auction on the following Monday and settlement on Thursday.
- Offering amount for 4-week bills are announced on Monday for auction the next day and settlement on Thursday.
- Offering amounts for 52-week bills are announced every fourth Thursday for auction the next Tuesday and settlement on Thursday.
- Treasury notes (T-Notes) mature in 110 years.
- They have a coupon payment every 6 months and face value of $1000.
- These notes are quoted on the secondary market at percentage of face value in thirty-seconds of a point.
- For example, a quote 95 : 08 on a note indicates that it is trading at a discount$( 95 + 8 / 32 ) 1000 / 100 = $ 952 .5.
- The 10-year Treasury note has become the security most frequently quoted when discussing the U.S. government bond market.
- The next figures show, respectively, the time plots of the daily yield and its return of the 10-year T-Notes.
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- Treasury bonds (T-Bonds) have longer maturities, ranging from 20 to 30 years.
- They have a coupon payment every 6 months and are commonly issued with maturities 30 years.
- The 30-year bonds were suspended for a 4-year and 6- month period starting October 31, 2001, but they were reintroduced in February 2006 and are now issued quarterly.
- The term structure of interest rates is the relationship between time to maturity and yield to maturity and the yield curve is a plot of the term structure of yield to maturity against time to maturity at a specific time.
- The next figure presents scatter plots of observed United States zero-coupon bond yield curves for the months of March, May, July and August 1989, for yields yields ranging from 1 to 120 months.
- The yields are computed from the end-of-month price quotes taken from the CRSP government bonds files and is the data used by Diebold and Li (2006).
YIELDS 21
This expression shows that the yield is inversely proportional to the natu- ral logarithm of the price of the bond, where the proportionality constant is 1/. Moreover as the price of the bond ntis always less than $1 then from the properties of logarithms, ntis a negative number and the yield in equa- tion (1.19) will always be positive. Governments issue bonds of differing lengths to maturity. Bonds at the shorter end of the maturity spectrum (maturity less than 12 months) are generally zero-coupon bonds, while the coupon bonds can have a maturity as long as 30 years. Theterm structureof interest rates is the relationship between time to urity and yield to maturity and the ld curveis a plot of the term struc- ture of yield to maturity against time to maturity at a specific time. Figure presents scatter plots of observed United States zero-coupon bond yield curves for the months of March, May, July and August 1989, for yields yields ranging from 1 to 120 months. The yields are computed from the end-of- month price quotes taken from the CRSP government bonds files and is the a used by Diebold and Li (2006).
8.8
9
9.2
9.4
9.6
9.8
Yield (Percent)
(^024) Maturity (Months) 48 72 96 120 March 1989 4.5 5 5.5 6 6.5 Yield (Percent) (^024) Maturity (Months) 48 72 96 120 May 1989 7.4 7.5 7.6 7.7 7.8 7.9 Yield (Percent) (^024) Maturity (Months) 48 72 96 120 July 1989 4.7 4.8 4.9 5 5.1 Yield (Percent) (^024) Maturity (Months) 48 72 96 120 August 1989 Figure 1.4: Observed yield curves for the months of March, May, July and Au- for United States zero coupon bonds. The data are taken from CRSP government bonds files and is the data used by Diebold and Li (2006) The plots of the yield curve in Figure 1.4 reveal a few well-known features.
- At any one time when the yield curve is observed, all the maturities not be represented. This is particularly true at longer maturities where the number of observed yields is much sparser than at the short end of the maturity spectrum.
- The plots of the yield curve in the above figure reveal a few well-known features. 1. At any one time when the yield curve is observed, all the maturities may not be represented. This is particularly true at longer maturities where the number of observed yields is much sparser than at the short end of the maturity spectrum. 2. The yields at longer maturities tend to be less volatile than the yields at the shorter end of the maturity spectrum. 3. Based on the assumption that longer-term financing should carry a risk premium, the expectation may be that the yield curve slopes upward. shows however that the yield curve assumes a variety of shapes, including upward sloping, downward sloping, humped and even inverted humped.
- Modelling bond yields and the term structure are important problems in financial econometrics and various aspects relating to the modelling of bond yields will be addressed.