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# INVESTMENTS FIRE 314

Virginia Commonwealth University

GPA 3.64

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This 10 page Class Notes was uploaded by Dr. Evans Lubowitz on Wednesday October 28, 2015. The Class Notes belongs to FIRE 314 at Virginia Commonwealth University taught by David Upton in Fall. Since its upload, it has received 17 views. For similar materials see /class/230641/fire-314-virginia-commonwealth-university in Finance, Banking And Insurance at Virginia Commonwealth University.

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Date Created: 10/28/15

Fixed Income Securities Fixed Income Securities FIRE 314 Fall 2009 THE GOLDEN RULE OF VALUATION The Golden Rule of valuation for all assets is Equivalent assets should have equivalentprices A simpler version of this is the Law of one Price from economics the same asset should not sell for two different prices But often we must value assets that are dissimilar in many ways but still equivalent so that the dollar values will be different The easy example is a difference in size Consider two annuities one of 10 one of 100 Assume the timing and risk of both annuities is the same How can we apply the Golden Rule in such cases The approach is to nd the rate of return required of annuities of the given level of risk Then equivalent price means the price that will result in both assets having the same rate of return So we could restate the Golden Rule as Equivalent assets should have the same rate of return The equivalent price found by equating the rates of return of equivalent assets is sometime called the intrinsic value or the economically justi ed p1ice For the moment we will assume that the required rate of return is known Once we get a feel for how assets are valued given the required rate we will turn to a discussion of risk and the required rate BOND VALUATION AND YIELD VALUATION IMIVIEDIATELY AFTER PAYMENT DATE From the above discussion of time value the intrinsic value of a bond is the present value of the cash ows at the required rate of return EXAMPLE What is the value of a 20 year 10 annual pay bond if the true annual yield to similar bonds is 106 Using the calculator we can nd the present value of the cash ows at a rate of 106 On an HP12C you would enter 20 N 106 I 100 PMT 1000 FV On a Texas Instruments BA35 calculator you would enter 20 N 106 I 100 PMT 1000 FV CPT PV This gives the value of the bond as 95094 Bonds prices are generally not given in dollar terms but rather as a per cent of face value This bond would typically be quoted as 95094 ACCRUED INTEREST The seller of the bond is probably not a total dummy however and will want hisher share of the next interest payment Thus bond prices are understood to include payment of accrued interest unless I Fixed Income Securities speci cally said to be trading quot atquot You get this back since you get all of the next interest payment The method of computing accrued interest differs between corporate 360 day year of twelve 30 day months and Treasury bonds actual number of days in period and actual number of days in the year but both are based on simple interest Simple interest is computed at a constant daily rate equal to the annual rate divided by the number of days in a year Eg if a 10 corporate bond pays interest semiannually and it has been 45 days since the last interest payment the accrued interest would be 45180x50 Note that this can get complicated for corporates because not all months have 30 days There are some complex rules for determining the number of days 7 so we won t go there There is a function to do this is built into the HP12C calculator but is somewhat complicated to use CURRENT YIELD AND YIELD TO MATURITY There are several measures of bond yield a Current yield is the annual cash ow divided by the price of the bond It is a very limited measure because it ignores both capital gainloss and time value b Yield to maturity ytm is the way that bond yields are usually quoted It is the discount rate that sets the present value of the cash ows equal to the price of the bond YTM has several problems we will get to later Unfortunately since there is no way to directly compute the ytm you must use an approximation Your calculator however can do the estimating for you EXAMPLE What is the true yield ofa 20 year 10 annualpay bond callable in 15 yrs at 105 which is priced at 95094 a The current yield is 100 95094 0105159 105159 This ignores capital gain or loss as well as time value b True correct yield to maturity is easily done by calculator On an HP12C you would enter 20 N 95094 PV 100 PMT 1000 FV on a Texas Instruments BA35 calculator you would enter 20 N 95094 PV 100 PMT 1000 FV CPT i ytm 1060 You may note that the calculator sometimes takes several seconds to compute the ytm this is because it is using an iterative approximation algorithm not a closed solution c Yield to rst call The above yield to maturity assumes the bond is held to maturity This will not be the case if the bond is called Yield to rst call is the yield to maturity assuming that the bond is called at the rst callable time Note that there may be steps in the call provision of the bond so that it is also possible to have a yield to other calls EXAMPLE What is the yield if the above bond is called at the rst callable time This assumes 15 years of coupon payments and a maturity value of 1050 You would enter 15 N 95094 PV 100 PMT 1050 FV CPT i ytm 10823 Note that the yield to rst call is higher than the yield to maturity Does this mean that you would hope the bond gets called NO for two reasons a The rm will not recall the bonds unless they are worth more than the call price otherwise they would buy them in the market The call price is an upper bound on the price of the bond b You will have to reinvest the money But if the bond value is high that means that yields are low 7 so you will not be able to reinvest at the same rate you are presently making on the bond 2 Fixed Income Securities BOND EQUIVALENT YIELD AND TRUE CORRECT ANNUAL YIELD The above calculations were for an annual pay bond In the US however most corporate bonds pay interest semiannually sometimes quarterly sometimes monthly This changes the calculations somewhat and it also affects the way yields are quoted in practice EXAMPLE Suppose that our twentyyear 10 bond paid interest semiannually and had a quoted yield of 106 Its value would be slightly different To calculate the present value of the payments note that we now have 40 payments instead of 20 payments but each payment is 50 instead of 100 and the discount rate is halfthe yearly rate 11 40 i 53 PMT 50 FV 1000 CPT PV gt 95057 But this seems incorrect 7 the semiannual pay bond gives you your payments sooner 7 ie you get half of each annual payment at siX months so shouldn t it be worth more than an annual pay bond The problem lies in the way yield is quoted in practice 7 it is reported on a bond equivalent yield BEY basis The ytm calculation for a semiannual pay bond will yield a semiannual ytm which must be annualized The annualized ytm quoted in the US nancial press is called the bond equivalent yield also referred to as the simple annual interest rate or the annual percentage rate APR basis It is computed by simply multiplying the semiannual ytm by 2 Bond equivalent Yield 2 X Semiannual ytm For the 20 year 20 semiannual pay bond priced at 95057 for a semiannual yield of 53 this would be Bond equivalent Yield 2 X 530 106 Interest computed without compounding is also referred to as simple interest This calculation is wrong 7 it ignores the compounding of the payment over the second siX months The correct annualized ytm is actually True correct Annual ytm 1 semiannual ytm2 10 For the 20 year 20 semiannual pay bond priced at 95057 for a semiannual yield of53 this would be True correct Annual ytm 1 00532 10 11088 10 01088 1088 The reason the value of the semiannual bond is less than the value of the annual bond then is because we are actually discounting it at an annual rate of 1088 not 106 Further bond equivalent yield is used even for an annual pay bonds 7 the semiannual rate is correctly computed from the true annual rate but then incorrectly doubled Incidentally the 20 year 10 annual pay bond callable in 15 yrs at 105 and priced at 95094 has a true yield of 106 but would have a quoted yield of 0106 2 7 l X 100 X 2 1033 In other words for an annual bond quote you compute the correct semiannual yield 7 and then incorrectly annualize it by multiplying by two Yes this is incorrect 7 but it is consistent with the quoted yield for semiannual bonds Fixed Income Securities Unfortunately quoting ytm on a BEY basis is the established norm You will just have to get used to it But there are several reasons Why use of bond equivalent yield is not a big deal a the rst thing you do in calculations is divide by two undoing the error b Kozma Prutkov c its easier to use Whenever you are given a bond yield quote it will be an annual rate on a bond equivalent basis unless specifically stated otherwise REINVESTMENT RATE RISK The yield to maturity is an internal rate of return The internal rate of return implicitly assumes that all interim cash ows can be reinvested at the internal rate of return If this implicit assumption is incorrect the realized yield sometimes horizon yield will be different than the computed yield to maturity This is easily demonstrated a If the underlying assumption of reinvestment at the ytm is correct at maturity the investor will have the compounded value of the coupon payments FVA plus the maturity value For the semiannual pay bond FVA of coupon paymentsquot 650099 Maturity value 100000 Total 750099 Total realized return semiannual 75009995057 0 1053 gt 53 N 40 i 53 PMT 50 PV 0 CPT FV b If on the other hand interest rates drop immediately after purchase and the actual reinvestment rate is only 10 at maturity the investor will have FVA of coupon payments 603999 Maturity value 100000 Total 703999 Total realized return semiannual 70399995057 40 1051332 gt 513 N 40 i 500 PMT 50 PV 0 CPT FV Thus if the rate at which cash ows are reinvested is not the ytm the realized return will be different than the ytm The possibility that this will occur is called quotreinvestment rate riskquot note that yield to maturity is the yield under three conditions 1 You hold the bond to maturity 2 All payments are made as speci ed 3 All payments are reinvested at the yield to maturity Fixed Income Securities HPR vs HPY Most folks think of return in terms of the percent return Ending Beginning return X 100 Beginning This is also called the Holding Period Yield HPY Another measure of return called Holding Period Return HPR is often used Holding Period Return Beginning Note the HPR l HPY Note also that it is HPR that gets compounded This is a different usage than the text WhatI call the HPY the text calls the HPR and it does not introduce a concept equal to the HPR that I use My de nition of HPY and HPR are the proper and generally accepted de nitions Sometimes HPY is stated as a percent sometimes as a fraction 7 eg 010 or 10 I will not always point out when I am changing from one form to the other 7 you are expected to do this on your own Also please note that the yield of a bond portfolio is not the average or the weighted average of the yields of the individual assets although its close The portfolio yield must be separately calculated from the cash ows of the portfolio EQUIVALENT TAXABLE YIELD Interest on municipal bonds is not subject to federal taxes although they may be subject to state and local taxes Because of this tax exemption the investor gets to keep more of the coupon payments Bond yields are computed on a pretax basis however and the tax exempt status is not re ected in the quoted yields In order to make the yields comparable the municipal yield is changed to the yield a corporate bond would have to pay to result in the same aftertax yield called the equivalent taxable yield Tax exempt Yield Equivalent Taxable Yield l Marginal Tax Rate Three items to note about the equivalent taxable yield 1 While coupon payments on municipals are exempt from federal taxes capital gains are not If the computed municipal yield includes capital gains the equivalent taxable yield is in error 2 Since the marginal tax rate is that of the investor in question the equivalent taxable yield is different for investors in different tax brackets 3 Municipal bonds held by residents of the state of issue are not subject to state tax but if you holds municipal bonds from other states you will be subject to state tax State tax is not included in the 5 Fixed Income Securities above formula It is not unusual to turn the equation around and nd the taX rate that would make the yield on the municipal equal to the yield on a taxable instrument BOND PRICE VOLATILITY There are four rules that are demonstrated below 1 Bond prices move inversely to ytm Compare A to B C to D 2 Longer term bonds m to be more volatile Compare A to C and B to D M Compare E to F At sufficiently high yields medium term bonds may be more volatile than long term bonds 3 For a given absolute change in the ytm the price increase due to a decrease in ytm is larger than the decrease due to an increase in ytm Compare A and C to B and D 4 For a given absolute change in ytm bonds of higher coupon have less volatility Compare G and H to A and B PRICE VOLATILITY CHARACTERISTICS annual pay bondst Coupon Maturity YTM Price A Price A 6 20 yrs 8 196 196 6 3 8 51 51 6 20 6 6 3 6 6 20 4 271 271 6 3 4 56 56 15 20 8 1687 G 345 1698 15 20 6 203Zlt 15 20 4 2495 H 463 2279 6 30 20 30295 6 30 18 33798 E 350 1156 6 20 20 31826 6 20 18 35767 F 3941 1238 MACAULEY DURATION There are two ways to interpret Macauley duration A Macauley duration as the sensitivity of the price of the bond to the interest rate Ie the greater the duration the greater the change in bond price as yield changes Fixed Income Securities For those who talk calculus this is lPdPdy ie the de1ivative rate of change of price with respect to the change in the yield to matuiity For those with an economics background duration is the negative of elasticity of price with respect to yield AP Duration P AHPR HPR The negative is because the relationship is inverse Rewriting the equation E X 100 Duration X 7R X 100 P HPR APDXAHPR Note that given this de nition duration is then the change in price for a 1 change in HPR If the change in HPR is 1 the price change in will just be equal to duration It can be thought of as a measure of volatilitv the higher the duration the higher the volatility Note that the concept of duration can be applied to other cash ows Thus they can be applied to both the assets and the liabilities of a nancial entity Since duration is a measure of change when interest rates change many nancial entities try to keep the duration of their assets close to the duration of their liabilities If they do this the size of assets and liabilities go up and down together as interest rates change which helps avoid bankruptcy B Macauley duration as the Immunization Point Another interpretation of Macauley duration arises from the desire to control risk Consider the sources of1isk in a bond a interest rate sensitivity price volatility b reinvestment rate 1isk c actual or potential change in cash ows such as default call sinking fund obligations subordination collateral dilution There is not much you can do about c except stay safe and buy high quality bonds But consider a and b interest rate increase interest rate decrease Reinvestment effect positive negative price effect negative positive Is there a way to quotbalancequot price risk and the reinvestment rate 1isk The answer is yes 7 Macauley duration is the point in time at which interest rate risk cancels out reinvestment rate 1isk so that you have the same amount of at the duration point no matter what happens to interest rates this is the basis of immunization For instance for an 8 5yr annual pay bond priced to yield 10 which has a 7 Fixed Income Securities Macauley duration of 428 years I If the interest rate remains at 10 in 428 years you will have Payment 1 80 X 110 10936 Payment 2 80 X 1102 28 9942 Payment 3 80 X 110128 9038 Payment 4 80X 110028 8216 38132 Value ofBond 10801100 72 100837 TOTAL 138969 II If the interest rate goes to 102 in 428 years you will have Payment 1 80 X 1102328 11001 Payment 2 80 X 11022 28 9983 Payment 3 80 X 1102128 9059 Payment 4 80X 1102028 8221 38264 Value ofBond 10801102072 100705 TOTAL 138969 Thus ifyou should need 138969 in 428 years ifyou buy the above bond for 92418 you will have 138969 even if interest rates change This implies that you will have a rate of return of 13896992418 4 28 1 010 10 no matter what happens to interest rates This technique of risk avoidance in xed income investing is referred to as quotimmunizationquot ie you have immunized your portfolio from interest rate 1isk MODIFIED DURATION The basic Macauley Duration desc1ibed above is somewhat awkward Macauley duration is the change in price for a 1 change in the HPR 1 yield Most analysts would not think of yield changes in as a change in HPR however Rather they would think in terms terms of basis points A basis point is one hndredth of one percent 7 eg 001 or as a decimal 00001 Ie ifthe yield goes from 10 to 104 this is a 40 basis point change in the yield Because of this Modi ed Duration is often used instead of Macauley Duration where Macauley Duration 1 y 2 Modi ed Duration Price change using Modi ed Duration is computed as AP MODD X Ay Where Ay is the change in yield 7 ie ifyield changes from 10 to 1040 Ay is 040 Note that this indicates Modi ed Duration is the change in price for a 100 basis Quint change in HPY 8 Fixed Income Securities 7 a 100 basis point change in HPY is not the same as a 1 change in HPR Eg if the HPR is 106 a 1 increase would be to 106 X 101 10706 or 106 basis points Given the equivalent HPY of 006 or 6 a 100 basis point increase would move the HPY to 007 or 7 The duration and modi ed duration discussed here are based on quotMacauley Duration quot computed from derivativebased formulas These formulas assume that the yield curve is at and any changes in the yield curve are parallel movements The derivativebased formulas also ignore any effects of embedded options callable putable sinking fund Rather than being computed from the derivative based formulas duration is often calculated based on modeled or observed yield and price changes Depending on the model used these alternative computations can be used to extend duration and convexity to situations violating the Macauley assumptions or to include the effect of embedded options CONVEXITY The duration is based on the derivative and is accurate only for in nitesimally small changes in ytm The de1ivative gives the slope of the relationship at a point but since the relationship is actually curvilinear the slope is different at other points Thus the price change computed from duration over a nite change in yield will be in error The error will be small and negligible for small changes in yield but can increase rapidly for larger changes in yield Geomet1ically this can be visualized as Price Yield A measure called convexity is sometimes used to provide a partial or approximate correction for curvature This concept is beyond the reach of this course COMPUTATIONS Consider a 20yr 10 semiannual pay bond priced at 950 to provide a semiannual yield of 53 106 bond equivalent yield 109 actual yield The Macauley duration of this bond is 8766 years and the modi ed duration is 8766 1053 Ifthe required annual yield increases by 00005 5 basis points to 1065 the estimated per cent price change due to duration is Change in Price Mod Duration x Change in ytm Macauley Duration 8766 Chan ein m lytm g yt 1053 005 04162 Fixed Income Securities OR 83248 x 005 04l62 Note that for a semiannual bond modi ed duration and price change is computed using the semiannual HPR in the denominator Sorry you simply must remember this The yieldp1ice relationship is not linear however so that the duration which is essentially dPdytm is only accurate for in nitesimally small changes in ytm Partial compensation for this error is achieved by using the convexity measure which we do not get into COMPUTING DURATION AND EFFECTIVE DURATION As noted above duration is the de1ivative of price with respect to yield One way of computing Macauley Duration is to take the de1ivative of the pricing equation for a bond This leads to the rather long procedure described in the teXt Besides being cumbersome this approach is only applicable to straight bonds 7 those which have set cash ows ie bonds that are not callable not putable or have no prepayment options An alternative approach to computing duration and one which re ects any uncertainties in the cash ows is called the effective duration Remember that AP MODD X Ay 0 MODD 1 EXAMPLE You are conside1ing a 20 year semiannual pay 10 bond which is selling at 94271 to yield 107 Using the bond valuation equation you would eXpect to observe Yield Price 106 95057 108 93496 However this is a callable bond which is approaching yield levels such that the probability of a call is not negligible Instead of the predicted values you observe Yield Price 106 99056 107 98650 108 98000 It is clear that using Macauley duration would be quite misleading We can compute the Effective modi ed duration by computing AP and Ay

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