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Simplified comprehensive information on Bond valuation,
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The Valuation of Bonds
Bond Values
Bond values are discussed in one of two ways: The dollar price The yield to maturity
These two methods are equivalent since a price implies a yield, and vice-versa
Bond Yields
The rate of return on a bond: Coupon rate Current yield Yield to maturity Modified yield to maturity Yield to call Realized Yield
The Coupon Rate
The coupon rate of a bond is the stated rate of interest that the bond will pay
The coupon rate does not normally change during the life of the bond, instead the price of the bond changes as the coupon rate becomes more or less attractive relative to other interest rates
The coupon rate determines the dollar amount of the annual interest payment:
The Current Yield
The current yield is a measure of the current income from owning the bond
It is calculated as:
The Yield to Maturity
The yield to maturity is the average annual rate of return that a bondholder will earn under the following assumptions: The bond is held to maturity The interest payments are reinvested at the YTM
The yield to maturity is the same as the bond’s internal rate of return (IRR)
The Modified Yield to Maturity
The assumptions behind the calculation of the YTM are often not met in practice
This is particularly true of the reinvestment assumption To more accurately calculate the yield, we can change
the assumed reinvestment rate to the actual rate at which we expect to reinvest
The resulting yield measure is referred to as the modified YTM, and is the same as the MIRR for the bond
The Yield to Call
Most corporate bonds, and many older government bonds, have provisions which allow them to be called if interest rates should drop during the life of the bond
Normally, if a bond is called, the bondholder is paid a premium over the face value (known as the call premium)
The YTC is calculated exactly the same as YTM, except: The call premium is added to the face value, and The first call date is used instead of the maturity date
The Realized Yield
The realized yield is an ex-post measure of the bond’s returns
The realized yield is simply the average annual rate of return that was actually earned on the investment
If you know the future selling price, reinvestment rate, and the holding period, you can calculate an ex-ante realized yield which can be used in place of the YTM (this might be called the expected yield)
Bond Valuation in Practice
The preceding examples ignore a couple of important details that are important in the real world: Those equations only work on a payment date. In
reality, most bonds are purchased in between coupon payment dates. Therefore, the purchaser must pay the seller the accrued interest on the bond in addition to the quoted price.
Various types of bonds use different assumptions regarding the number of days in a month and year.
Valuing Bonds Between Coupon Dates
Imagine that we are halfway between coupon dates. We know how to value the bond as of the previous (or next even) coupon date, but what about accrued interest?
Accrued interest is assumed to be earned equally throughout the period, so that if we bought the bond today, we’d have to pay the seller one-half of the period’s interest.
Bonds are generally quoted “flat,” that is, without the accrued interest. So, the total price you’ll pay is the quoted price plus the accrued interest (unless the bond is in default, in which case you do not pay accrued interest, but you will receive the interest if it is ever paid).
Valuing Bonds Between Coupon Dates (cont.)
The procedure for determining the quoted price of the bonds is: Value the bond as of the last payment date. Take that value forward to the current point in time.
This is the total price that you will actually pay. To get the quoted price, subtract the accrued interest.
We can also start by valuing the bond as of the next coupon date, and then discount that value for the fraction of the period remaining.
Day Count Conventions
Historically, there are several different assumptions that have been made regarding the number of days in a month and year. Not all fixed-income markets use the same convention: 30/360 – 30 days in a month, 360 days in a year. This is used in the
corporate, agency, and municipal markets. Actual/Actual – Uses the actual number of days in a month and year.
This convention is used in the U.S. Treasury markets. Two other possible day count conventions are:
Actual/360 Actual/365
Obviously, when valuing bonds between coupon dates the day count convention will affect the amount of accrued interest.
The Term Structure of Interest Rates
Interest rates for bonds vary by term to maturity, among other factors
The yield curve provides describes the yield differential among treasury issues of differing maturities
Thus, the yield curve can be useful in determining the required rates of return for loans of varying maturity
Types of Yield Curves
Today’s Actual Yield Curve
Maturity YLDPRIME 4.75%DISC 1.25%FUNDS 1.75%90 DAY 1.71%180 DAY 1.88%YEAR 2.19%2 YR 3.23%3 YR 3.74%4 YR 4.18%5 YR 4.43%7 YR 4.91%10 YR 5.10%15YR 5.64%20 YR 5.76%30 YR 5.61%
U.S. Treasury Yield Curve24 April 2002
1.00%2.00%3.00%4.00%5.00%6.00%
90 D
AY
180 D
AY
YEAR2 Y
R3 Y
R4 Y
R5 Y
R7 Y
R10
YR
15YR
20 Y
R
30 Y
R
Term to Maturity
Yie
ld
Data Source: http://www.ratecurve.com/yc2.html
Explanations of the Term Structure
There are three popular explanations of the term structure of interest rates (i.e., why the yield curve is shaped the way it is): The expectations hypothesis The liquidity preference hypothesis The market segmentation hypothesis (preferred
habitats) Note that there is probably some truth in each of
these hypotheses, but the expectations hypothesis is probably the most accepted
The Expectations Hypothesis
The expectations hypothesis says that long-term interest rates are geometric means of the shorter-term interest rates
For example, a ten-year rate can be considered to be the average of two consecutive five-year rates (the current five-year rate, and the five-year rate five years hence)
Therefore, the current ten-year rate must be:
10 555
5510 111 RRR t
The Liquidity Preference Hypothesis
The liquidity preference hypothesis contends that investors require a premium for the increased volatility of long-term investments
Thus, it suggests that, all other things being equal, long-term rates should be higher than short-term rates
Note that long-term rates may contain a premium, even if they are lower than short-term rates
There is good evidence that such premiums exist
The Market Segmentation Hypothesis
This theory is also known as the preferred habitat hypothesis because it contends that interest rates are determined by supply and demand and that different investors have preferred maturities from which they do no stray
There is not much support for this hypothesis
Bond Price Volatility
Bond prices change as any of the variables change: Prices vary inversely with yields The longer the term to maturity, the larger the change
in price for a given change in yield The lower the coupon, the larger the percentage
change in price for a given change in yield Price changes are greater (in absolute value) when
rates fall than when rates rise
Measuring Term to Maturity
It is difficult to compare bonds with different maturities and different coupons, since bond price changes are related in opposite ways to these variables
Macaulay developed a way to measure the average term to maturity that also takes the coupon rate into account
This measure is known as duration, and is a better indicator of volatility than term to maturity alone
Duration
Duration is calculated as:
So, Macaulay’s duration is a weighted average of the time to receive the present value of the cash flows
The weights are the present values of the bond’s cash flows as a proportion of the bond price
Notes About Duration
Duration is less than term to maturity, except for zero coupon bonds where duration and maturity are equal
Higher coupons lead to lower durations Longer terms to maturity usually lead to longer
durations Higher yields lead to lower durations As a practical matter, duration is generally no
longer than about 20 years even for perpetuities
Modified Duration
A measure of the volatility of bond prices is the modified duration (higher DMod = higher volatility)
Modified duration is equal to Macaulay’s duration divided by 1 + per period YTM
Note that this is the first partial derivative of the bond valuation equation wrt the yield
Convexity
Convexity is a measure of the curvature of the price/yield relationship
Note that this is the second partial derivative of the bond valuation equation wrt the yield
Yield
D = Slope of Tangent LineMod
Convexity