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 priceimplies 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 ofinterest that the bond will pay The coupon rate does not normally changeduring the life of the bond, instead the price ofthe bond changes as the coupon rate becomesmore or less attractive relative to other interestrates The coupon rate determines the dollar amount ofthe annual interest payment:
The Current Yield The current yield is a measure of the currentincome from owning the bond It is calculated as:
The Yield to Maturity The yield to maturity is the average annual rateof return that a bondholder will earn under thefollowing 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’sinternal rate of return (IRR)
The Modified Yield to Maturity The assumptions behind the calculation of the YTM areoften not met in practice This is particularly true of the reinvestment assumption To more accurately calculate the yield, we can changethe assumed reinvestment rate to the actual rate at whichwe expect to reinvest The resulting yield measure is referred to as the modifiedYTM, and is the same as the MIRR for the bond
The Yield to Call Most corporate bonds, and many older governmentbonds, have provisions which allow them to be called ifinterest rates should drop during the life of the bond Normally, if a bond is called, the bondholder is paid apremium over the face value (known as the callpremium) 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 thebond’s returns The realized yield is simply the average annualrate of return that was actually earned on theinvestment If you know the future selling price,reinvestment rate, and the holding period, youcan calculate an ex-ante realized yield which canbe used in place of the YTM (this might becalled the expected yield)
Bond Valuation in Practice The preceding examples ignore a couple ofimportant details that are important in the realworld: Those equations only work on a payment date. Inreality, most bonds are purchased in between couponpayment dates. Therefore, the purchaser must paythe seller the accrued interest on the bond in additionto the quoted price. Various types of bonds use different assumptionsregarding the number of days in a month and year.
Valuing Bonds Between Coupon Dates Imagine that we are halfway between coupon dates. Weknow how to value the bond as of the previous (or nexteven) coupon date, but what about accrued interest? Accrued interest is assumed to be earned equallythroughout the period, so that if we bought the bondtoday, we’d have to pay the seller one-half of theperiod’s interest. Bonds are generally quoted “flat,” that is, without theaccrued interest. So, the total price you’ll pay is thequoted price plus the accrued interest (unless the bond isin 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 priceof 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 thenext coupon date, and then discount that valuefor the fraction of the period remaining.
Day Count Conventions Historically, there are several different assumptions that have beenmade regarding the number of days in a month and year. Not allfixed-income markets use the same convention: 30/360 – 30 days in a month, 360 days in a year. This is used in thecorporate, 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 countconvention 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 yielddifferential among treasury issues of differingmaturities Thus, the yield curve can be useful indetermining the required rates of return for loansof varying maturity
Today’s Actual Yield CurveMaturity 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 20021.00%2.00%3.00%4.00%5.00%6.00%90DAY180DAYYEAR2YR3YR4YR5YR7YR10YR15YR20YR30YRTerm to MaturityYieldData Source: http://www.ratecurve.com/yc2.html
Explanations of the Term Structure There are three popular explanations of the termstructure of interest rates (i.e., why the yieldcurve is shaped the way it is): The expectations hypothesis The liquidity preference hypothesis The market segmentation hypothesis (preferredhabitats) Note that there is probably some truth in each ofthese hypotheses, but the expectationshypothesis is probably the most accepted
The Expectations Hypothesis The expectations hypothesis says that long-terminterest rates are geometric means of the shorter-term interest rates For example, a ten-year rate can be considered tobe the average of two consecutive five-year rates(the current five-year rate, and the five-year ratefive years hence) Therefore, the current ten-year rate must be:( ) ( ) ( )10 5555510 111 RRR t+++=+
The Liquidity Preference Hypothesis The liquidity preference hypothesis contends thatinvestors require a premium for the increased volatilityof 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 ifthey 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 habitathypothesis because it contends that interest ratesare determined by supply and demand and thatdifferent investors have preferred maturitiesfrom which they do no stray There is not much support for this hypothesis
Bond Price Volatility Bond prices change as any of the variableschange: Prices vary inversely with yields The longer the term to maturity, the larger the changein price for a given change in yield The lower the coupon, the larger the percentagechange in price for a given change in yield Price changes are greater (in absolute value) whenrates fall than when rates rise
Measuring Term to Maturity It is difficult to compare bonds with differentmaturities and different coupons, since bondprice changes are related in opposite ways tothese variables Macaulay developed a way to measure theaverage term to maturity that also takes thecoupon rate into account This measure is known as duration, and is abetter indicator of volatility than term to maturityalone
Duration Duration is calculated as: So, Macaulay’s duration is a weighted average ofthe time to receive the present value of the cashflows The weights are the present values of the bond’scash flows as a proportion of the bond price
Notes About Duration Duration is less than term to maturity, except forzero coupon bonds where duration and maturityare equal Higher coupons lead to lower durations Longer terms to maturity usually lead to longerdurations Higher yields lead to lower durations As a practical matter, duration is generally nolonger than about 20 years even for perpetuities
Modified Duration A measure of the volatility of bond prices is themodified duration (higher DMod = highervolatility) Modified duration is equal to Macaulay’sduration divided by 1 + per period YTM Note that this is the first partial derivative of thebond valuation equation wrt the yield
Convexity Convexity is a measure of the curvature of theprice/yield relationship Note that this is the second partial derivative ofthe bond valuation equation wrt the yieldYieldD =Slope ofTangentLineModConvexity