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FIXED INCOME Valuation Approaches historical cost replacement cost market valuation model valuation Main Function of Capital Markets Central role in the effective allocation of capital resources to risky projects Help to time the usage of funds or consumption Transfer of risk money across time Effectively allocate risk RISK single most important characteristic of capital markets risk by market and risk by origin Transformation of Financial Assets Collateralized mortgage obligations pool of mortgages set up legal entity trust that holds the pool as sole asset sell securities backed by mortgages MATURITY RISK and LIQUIDITY can be transformed Derivatives good creation of new securities allow repackaging of risk make markets more efficient bad creation of new complicated securities is immoral allow smart bankers to take advantage of dumb clients treasury bills certificates of deposits loans etc Fixed income securities are closely related to the concepts of debt and loan Two Markets Money Market Bond Market Bond Pricing PV 100 1 rT T Perpetuity 1 r Annuity 1 r 1 1 r T Bond Yields Yield to Maturity rate of return YTM the discount rate that equates the PV of the bond s cash treasury bills and notes municipal and corporate mortgage backed securities etc flows to the bond s price Current Yield yield over the next period c P If YTM coupon rate then bond sells at par If YTM coupon rate then the bond sells at a discount If YTM coupon rate the bond sells at a premium Why investors agree to hold bonds with low coupons b c there might be high capital gains Inverse Relationship between Prices and Yield CONVEX if the line connecting two dots is inside or above Measures of Bond s Return USING PRICES OF ZERO COUPON BONDS WE CAN 1 Realized Compound Return includes proceeds from coupons reinvested at available rate period two parts EXAMPLE 2 Hold Period Return the rate of return over a particular investment the total return on bond held to maturity Construct zero yield curves Calculate any spot rate 1 2 3 Obtain prices of similar bonds with any cash flows 4 a b The future value of cash flows reinvested at various rates The remaining value of the bond at that horizon Abuses of YTM used as a measure of the bond s return used as a tool for choosing between different bonds YTM and Returns Zero Coupon Bonds The YTM of a zero is the spot rate corresponding to the bond s time to maturity HOWEVER the return of investing in the bond and selling it after one year is unknown today YTM and Returns Coupon Bonds The YTM of a coupon is the return of investing in the bond holding it until maturity and reinvesting the coupons at a rate equal to the YTM Treasuries are Inconsistent use treasury strips or zeroes to construct zero yield curve used as the basis for fixed income valuations Discount Factors dT 1 1 rT T PT 100 Price of a coupon bond using zeroes c d1 c d2 100 c dT Synthetically replicate other bonds by a portfolio of zero coupon bonds Bootstrapping procedure of obtaining spot rates from the prices of coupon bonds EXAMPLE Determinants of Yield Curve 1 Expectation Theory claims that the shape of yield curve depends on market participant s expectations of future short term interest rates investors expect to earn the same return over a given period regardless of the total holding periods and bonds maturities Forward Rates Expected Return 1 yn n 1 yn 1 n 1 1 fn Slop of the Yield curve can be used to predict future interest rates if the slope is steep this month then the interest rates are expected to increase in the following month 2 Liquidity Preference Theory forward rates offer a positive liquidity premium over expected spot rates yield curve is determined by the expectations of market participants AND a liquidity premium which is increasing with maturity because the longer term bonds are riskier 1 y1 1 E r2 LP 1 y2 2 Liquidity premium is compensation to investors for risk and issuers of bonds prefer to issue long term bonds and are willing to pay higher yields 3 Macroeconomics and Yield Curve monetary policy control the short term interest rates to influence economy recession Fed decreases interest rates boom Fed increases rates The sensitivity of bond prices to the changes in interest rates can be approximated by the duration of bonds Three factors capture the rich dynamics of yield curve level most important slope and curvature Level Change r parallel shifts in yield curve Performance Profile Value of the investment at r r 1 P r P r r interest rate Downward slowing value of investment goes down Zero coupon bonds are more sensitive to changes in when interest rates go up Convex the value goes down at a decreasing rate Long term bonds are more sensitive to changes in interest rate Duration If duration is large bond investment more sensitive to the changes in interest rate Macaulay Duration w T PVt 1 Price 100 c 1 r 2 1 P r It always decreases when coupon rate increases always decreases with YTM The year around which the bond s discounted cash flows are balanced increases with time to maturity effective maturity of the bond Modified Duration D D 1 y P P D r approximate IMMUNIZATION 3yr bond with coupon rate 4 term structure of interest rates is flat 3 paid 10 283 for 100 units of this bond immunize position using 1yr and 5yr zero coupon bonds with current prices of 97 09 and 86 26 1 Value of Assets A A1 A2 x1 P1 x5 P5 2 Value of Current Assets 10 283 97 09 x1 86 25 x2 3 Macaulay Duration of 3yr bonds Price 4 1 03 4 1 032 104 1 033 102 829 W1 4 1 03 1 102 829 03776 W2 4 1 032 1 102 829 03666 W3 104 1 033 1 102 829 92556 03776 1 03666 2 92556 3 2 8878 4 Modified Duration of 3yr bonds D 2 8878 1 03 2 804 5 Modified Duration of 1 yr 1 1 03 971 5 yr 5 1 03 4 85 6 A x1 P1 D 1 r x5 P5 D 5 r 7 L L D r 10 283 2 804 r 28833 532 r 8 A L x1 P1 D 1 r x5 P5 D 5 r 28 833 532 r 97 09 971 x1 86 26 4 85 x5 28 833 532 94 274x1 418 71x5 28 833 52 9 Solve x5 56 27 x1 55 92 1 yr 2 yr 2 yr Bond A Bond B Bond C 95 24 98 44 103 37 0 coupon 5 coupon 7 coupon 0 100 0 A 0 1 2 0 5 105 B y units 0 1 2 Regression Output R squared HOLDING PERIOD RETURN R2 V a bX V a bX V Portfolio Return X1 X R1 X2 X R2 OR W1 R1 W2 R2 EQUITIES Maturity 4 years coupon rate 4 face value 100 current price 95 hold bond for 2 …

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