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# BUSN380 Week 4 TCO 5 Bond Valuation Notes - Lecture Supplement

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Date Created: 11/06/15

WEEK 4 LECTURE SUPPLEMENT Bond valuation represents a straightforward application of the present value principles presented by Lecture 1 With respect to these present value principles generally speaking there are two major bond categories zerocoupon bonds and coupon bonds The price present value of a zerocoupon bond is determined by discounting the bond s terminal payout by the bond s yield to maturity For example the current price of a zerocoupon bond with a face value of 1000 a yield to maturity of 10 and a fiveyear maturity is 62092 1000ll5 Zerocoupon bonds are illustrative of why ceteris paribus that is holding all other factors constant bonds that have longer maturities have greater price risk in relation to otherwise comparable bonds with shorter maturities This result can be generalized to coupon bonds as well Suppose a zerocoupon bond has a face value of 1000 a yield to maturity of 10 and a six year maturity The price is 56447 1000ll6 Now suppose the yield to maturity on both the fiveyear and sixyear zerocoupon bonds rises to 11 The two bonds respectively will be priced at 59345 10001115 53464 10001116 On a percentage basis the price of the fiveyear bond has changed by 62092 5934562092 004424 On a percentage basis the price of the sixyear bond has changed by 56447 5346456447 005285 Thus the sixyear bond has greater price risk Now let us assume we wish to determine the price of a 10 annual coupon bond with a face value of 1000 a yield to maturity of 10 and a fiveyear maturity The price is determined as follows 100ll 100ll2 100ll3 100ll4 100ll5 l000ll5 100l 11395l l000ll5 l00000 If the above bond pays coupons 9 annually the price is 90ll 90112 90ll5 l000ll5 90l 11395l l000ll5 96209 If the above bond pays coupons 9 semiannually the price is 45105 451052 451053 4510510 100010510 451105391005 100010510 96139 Note Regarding coupon paying bonds a coupon interest rate equal to the yield to maturity will result in a bond that displays a price equal to the bond s face value It must therefore be the case that when a bond displays a price equal to the face value the coupon interest rate must equal the yield to maturity Accordingly a bond displaying a coupon interest rate greater than the yield to maturity must exhibit a price greater than the face value and vice versa Often bond valuation involves finding the bond s yield to maturity Algebraically such valuation problems involve solution through trialanderror and subsequent interpolation Here s an example Suppose we wish to find the annual yield to maturity for a semiannual coupon paying bond that carries an 8 coupon rate has a 10 year maturity a face value of 1000 and a price of 1100 1100 8021 1 i2 20i2 10001 i220 Solve for i by trialanderror Since the price is greater than the face value i lt 4 semiannually We know that at an annual yield to maturity of 8 semiannual yield to maturity of 4 the price equals 1000 Try 4 annually 401 1 04220042 10001 04220 132703 Interpolate 08 100000 x04 1100 10001327 03 1000 x 0122313 110000 08 0122313 0678 z the annual yieldtomaturity 04 132703 By substituting the value of 0678 into the following equation we see that we have arrived at a reasonable approximation of the annual yieldtomaturity 401 1 06782 2 06782 10001 0678220 108757 Through repeated interpolation we could if we wished arrive at the precise yield to maturity that corresponds to a price of 1100 Alternatively enter the value of 1100 in cell A1 of an Excel spreadsheet Enter the value of 40 in cells A2 A20 Enter the value of 1040 in cell A21 In cell A22 type the following formula and hit the Enter key on your computer keyboard IRRA1A21 The semiannual yield to maturity will be re ected as 33085 Multiplication by two 2 provides the annual yield to maturity 233085 006617 Let us now examine several basic approaches used to value price common and preferred stocks Unlike bonds which convey creditor status and have maturity dates common stock conveys ownership status and has no maturity date Similar to bonds however the value of common stock fundamentally is a function of this security s expected future payouts discounted to the present by the applicable riskadjusted discount rate In the case of dividendpaying stocks the expected future payouts involved are represented by the expected future dividend payouts Suppose a stock is expected to pay a perpetual of dividend of 2 annually Further suppose a 10 risk adjusted discount rate Using results derived in Lecture 1 we see that the fundamental value of this stock is given by 20 2ll 2l12 2ll3 2ll 2l Now let us amend the above treatment by assuming a 5 annual growth rate in the dividend payout once the first future 2 dividend has been paid Again using results derived in Lecture 1 the fundamental value is given by 2ll391 2l05ll392 2l052ll393 2l05 391l139 2 l 05 40 Note that in the case of a common stock whose dividend payout is expected to grow at a constant positive rate the formulaic representation is given by P0 D1rs g where P0 represents the current value price D1 represents the dividend one period hence rs represents the riskadjusted discount rate and g represents the growth rate of the dividend Rearranging these terms we obtain the following equivalent relationship rs BlPO g Since preferred shares a hybrid security including a number of features found in common stocks and bonds typically conferring to the owner a fixed zerogrowth payout stream the valuation of these securities is identical to the treatment accorded a common stock that is expected to pay a fixed dividend in perpetuity The CAPM Capital Asset Pricing Model is a widelyused practitioner and academic approach for obtaining a stock s required expected rate of return This expected rate of return is synonymous with the riskadjusted discount rate discussed above The CAPM formally links risk with expected return and is represented as follows HR If ERm 105139 Where Rj represents the equilibrium return of the security Rf represents the riskfree rate ERm represents the expected return of the market Bj represents the beta of the security In the case of an individual asset use of the CAPM presumes that the asset of concern is included or will be included in an efficiently diversified portfolio When this is the case the asset s total risk is reduced to beta systematic risk The unsystematic firmspecific risk portion of the asset s total risk is diversified away Systematic risk refers to market risk how an individual stock s return varies in relation to changes in the market return Formally speaking beta equals the ratio of the covariance between an individual asset s return and the return of the market divided by the variance of the market return Bj Sim62m Interestingly beta also corresponds to the slope of the regression line discussed in the supplement accompanying the course termproject assignment Unsystematic risk refers to risk factors specific to a given company Enron s fraudulent activities are an example of firmspecific risk

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