Lecture_03 - 3 The Term Structure of Interest Rates This...

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1 3 The Term Structure of Interest Rates This chapter presents the preliminaries of the model. A The Economy We consider a frictionless, competitive, and discrete trading economy.
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2 By frictionless we mean: no transaction costs, no bid/ask spreads, no restrictions on trade (legal or otherwise) such as margin requirements or short sale restrictions, and no taxes. The frictionless markets assumption can be justified on two grounds. First, large institutional traders approximate frictionless markets since their transaction costs are minimal. If these traders determine prices, then this model approximates actual pricing and hedging well. The second argument is that frictionless markets is a necessary prelude to understanding friction-filled markets.
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3 The markets are assumed to be competitive. This implies that the market for any financial security is perfectly (infinitely) liquid. This is an idealization more nearly satisfied by large volume trading on organized exchanges than it is in the over-the-counter markets. Last, we consider a discrete trading economy with trading dates { 0 , 1 , 2 , . .., τ } . This assumption is a reasonable approximation if is large and the time interval between trading periods is small.
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4 B The Traded Securities Traded in this economy are zero-coupon bonds of all maturities { 0 , . .., τ } and a money market account . The price of a zero-coupon bond at time t that pays a sure dollar at time T t is denoted P(t,T) . All zero-coupon bonds are assumed to be default free and have strictly positive prices. The money market account represents an investment portfolio in the shortest term zero- coupon bond. It is initialized at time 0 with a dollar investment, and its time t value is denoted B(t) ( so, B( 0 ) = 1).
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5 C Interest Rates Markets quote bond prices using interest rates. This section defines the most important of these: yields, forward rates, and spot rates. As a convention in this book, all rates will be denoted as one plus a percentage (these are sometimes called dollar returns).
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6 The yield at time t on a T -maturity zero-coupon bond, denoted y(t,T) , is defined by: ) /( 1 ) , ( 1 ) , ( t T T t P T t y with y(t,T)>0 . (3.1) It is often called the holding period return. Alternatively written, [] ) ( ) , ( 1 ) , ( t T T t y T t P = . (3.2) The yield is the internal rate of return on the zero- coupon bond.
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7 The time t forward rate for the period [T,T+ 1 ] , denoted f(t,T) , is defined by ) 1 , ( ) , ( ) , ( + T t P T t P T t f . (3.3) The forward rate can be understood from two perspectives. First, looking at expression (3.3), the only difference between P(t,T) and P(t,T+1) is that P(t,T+1) earns interest for one more time period, the period [T,T+1]. Taking the ratio as in expression (3.3) isolates the implicit rate earned on the longer maturity bond over this last time period.
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8 The second interpretation of the forward rate is that it corresponds to the rate that one can contract at time t for a riskless loan over the time period [T,T+ 1 ] .
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Lecture_03 - 3 The Term Structure of Interest Rates This...

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