Note-8 - Bond Portfolio Management

# Note-8 - Bond Portfolio Management - Economics 106V...

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Economics 106V Investments : Lecture Note-8 Daisuke Miyakawa UCLA Department of Economics July 11, 2008 8th lecture covers the following items: (1) The interest rate risk on the price change of a bond, (2) dura- tion, (3) convexity, (4) the bond management and the term structure of interest rate, and (5) immunization . After discussing those items, we solve some exercise problems for the second half part of lecture-6 to lecture-8. 1 Interest Rate Risk on Price Change 1.1 Basic Idea On the last lecture, we studied how to price a coupon bond. By using correct yields for each maturity, we can compute a fair price of a bond as follows. unknown z}|{ P 1 = c 1 1+ y 1 ; 1 + c 2 (1+ y 2 ; 1 ) 2 + ° ° ° + c T ° 1 (1+ y T ° 1 ; 1 ) T ° 1 + c T + F V (1+ y T; 1 ) T Note that the price of the bond is the unknown on this step. Once we obtain the price of the bond, we can obtain the YTM of this bond in the following way: P 1 = c 1 1 + y | {z } u n k n ow n + c 2 (1 + y ) | {z } u n k n ow n 2 + ° ° ° + c T ° 1 (1 + y ) | {z } u n k n ow n T ° 1 + c T + F V (1 + y ) | {z } u n k n ow n T Note that the YTM is the unknown on this step. Suppose that all the yields for each maturity equal to an "interest rate" r , which is equivalent to a completely °at yield curve. This situation may be same as the one where we have no method to compute the term structure of the interest rate precisely 1 . In either case, all we can know about the connection between the price and the single "interest rate" is P 1 = c 1 1+ r + c 2 (1+ r ) 2 + ° ° ° + c T ° 1 (1+ r ) T ° 1 + c T + F V (1+ r ) T where r : (i) The constant yield or (ii) YTM implied by CF and price Suppose that investors are interested in how the bond price varies in future. Apparently, the "interest rate" r is one important factor a/ecting the price variation. 1 These two situations may look very di/erent. However, the following discussion can make sense only when one of those situations is the case. 1

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It is apparent from the expression that (i) P 1 increases (decreases) as 1 + r decreases (increases). If the investors want to sell the bond before the maturity, this price variation needs to be considered as a main risk associated with the bond investment. On the next sections, we will study how to quantify this risk and how to use such an information for the bond portfolio management. We need to understand that the following analysis is an extremely simpli±ed theory since we assume that either (i) yield curve is °at or (ii) the implied 1 + r is all we can obtain as an interest rate. 1.2 Some Propositions for Interest Rate Risk on Price Change There are several features on the interest rate risk on the bond price. 1. Bond prices and yields are inversely related. 2. An increase in a bond²s yield to maturity results in a smaller price change than a decrease in yield of equal magnitude. 3. Prices of long-term bond tend to be more sensitive to interest rate changes than prices of short-term bonds.
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