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Unformatted text preview: Economics 106V Investments : Lecture Note-7 Daisuke Miyakawa UCLA Department of Economics July 11, 2008 7th lecture (note: this lecture is one-hour) covers the following items for the further discussion of bond pricing: (1) The concept of yield curve, (2) forward rate, (3) expectation hypothesis, and (6) some other theories for the term structure of interest rate. After discussing those items, we will have a midterm exam. 1 Bond Pricing-2 1.1 Yield Curve Recall the following expression for the price of zero-coupon bonds introduced in the previous lecture: P 1 = FV (1+ y 1 ) T ; P 2 = FV (1+ y 2 ) T & 1 ; & & & ; P t = FV (1+ y t ) T & ( t & 1) ; P t +1 = FV (1+ y t +1 ) T & t ; & & & ; P T = FV (1+ y T ) HPR t = P t +1 & P t P t = P t +1 P t 1 where T : Maturity of the zero-coupon bond P t : The price for a bond at the beginning of period- t y t : YTM at the beginning of period- t Here, P 1 is the price of a zero-coupon bond which matures in T years from now and y 1 is its YTM over the next T years, both of which are computed at the beginning of year-1. Also, P t +1 is the price of a zero-coupon bond which matures in ( T t ) years from the point (i.e., the term to maturity) and y t +1 is its YTM over the next ( T t ) years, both of which are computed at the beginning of year- ( t + 1) . These represent the evolutions of the price and yield (or YTM) of one speci&c zero-coupon debt, which is illustrated in Figure-1. Year Year-1 Year-2 Year-3 Year-T Year-(T-1) P 1 P 2 P 3 P T-1 P T YTM=y 1 YTM=y 2 YTM=y 3 YTM=y T-1 YTM=y T Figure-1: The Evolution of the Price One Annual Debt 1 How can we summarize the prices and yields of debts with dierent maturities at a speci&c year? Here, our interest is summarized in Figure-2 Year Year-1 Year-2 Year-3 Year-T Year-(T-1) P T,1 P T-1,1 P T-2,1 P 2,1 P 1,1 YTM=y T,1 YTM=y T-1,1 YTM=y T-2,1 YTM=y 2,1 YTM=y 1,1 Figure-2: The Illustration of Yield Curve as of Year-1 The notation in Figure-2 is the following: P &; 1 : The year-1 price of zero-coupon bond which matures in & years from now y &; 1 : The year-1 yield of zero-coupon bond over the next & years As you may guess from the discussion on the last lecture, there are the following relationship between those two objects: P T; 1 = FV (1+ y T; 1 ) T ; P T & 1 ; 1 = FV (1+ y T & 1 ; 1 ) T & 1 ; & & & ; P 2 ; 1 = FV (1+ y 2 ; 1 ) 2 ; P 1 ; 1 = FV 1+ y 1 ; 1 The plot of those yields for a &xed timing (i.e., t = 1 on this example) is called as a yield curve. And the structure of the yield curve is called as a term structure of interest rate. Figure-3 illustrates various types of yield curves....
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- Summer '08