Ch-02 - Fixed Income Securities Fixed and Markets Chapter 2...

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Unformatted text preview: Fixed Income Securities Fixed and Markets Chapter 2 Bond Prices, Spot Rates, and Forward Rates Effective Annual Rates and Compounding Quoted as “5% per annum Quoted compounded semiannually”, or “5% compounded semiannually”. Means paying 2.5% every six months. Means Semiannual Compounding Investing $100 at an annual rate of Investing 5% compounded semiannually for six months generates $100 X (1+0.05/2)=$102.50 Investing $100 at the same rate for Investing one year instead generates $100 X (1+0.05/2) X (1+0.05/2) = $105.0625 In general In Investing x at an annual rate of r Investing compounded semiannually for T years generates r x(1 + ) 2T 2 Semiannually compounded holding period return What is the semiannually compounded What return from investing x for T years and having w at the end? ⎡w 1 ⎤ r = 2 ⎢( ) 2T − 1⎥ ⎣x ⎦ Spot Rates The spot rate is the rate on a spot The loan. It can vary across different maturities. It The t-year spot rate is denoted (t). (t). The Unless otherwise specified, assume Unless semiannual compounding frequency. ⎡ 1 21t ⎤ r = 2 ⎢( ) − 1⎥ ⎣ d (t ) ⎦ d (t ) = 1 ˆ r (t ) 2t (1 + ) 2 Term Structure, downward-sloping or Term inverted, upward-sloping Term structure of spot rates A five-year zero-coupon bond and a five-year ten-year zero-coupon bond are discounted using different rates Coupon bond: each payment must be Coupon discounted at a different rate, according to the term of the payment. From Chapter 1, the 14.25s of Feb. 15, 2002 108+31.5/32=7.125 d(.5) + 107.125 d(1) 108+31.5/32=7.125 Using Using d (t ) = 108+31.5/32= 1 ˆ r (t ) 2t (1 + ) 2 7.125 107.125 + ˆ ˆ r(.5) r(1) 2 (1+ ) (1+ ) 2 2 Forward Rates Forward Forward loan and forward rates A forward loan is an agreement made forward to lend money at some future date. The rate of interest on a forward loan, The specified at the time of the agreement as opposed to the time of the loan, is called a forward rate. Forward rates Forward Define r(t) to be the semiannually Define compounded rate earned on a six month loan ( t - 0.5 ) years forward. r(.5)= (.5)=5.008% r(.5)= ˆ ⎛ r (.5) ⎞ ⎛ r (1) ⎞ ⎛ r (1) ⎞ ⎜1 + ⎟ × ⎜1 + ⎟ = ⎜1 + ⎟ 2⎠⎝ 2⎠ ⎝ 2⎠ ⎝ 2 ˆ ⎛ r (.5) ⎞ ⎛ r (1) ⎞ ⎛ r (1.5) ⎞ ⎛ r (1.5) ⎞ ⎜1 + ⎟ × ⎜1 + ⎟ × ⎜1 + ⎟ = ⎜1 + ⎟ 2⎠⎝ 2⎠⎝ 2⎠⎝ 2⎠ ⎝ ˆ ⎛ r (.5) ⎞ ⎛ r (t ) ⎞ ⎛ r (t ) ⎞ ⎜1 + ⎟ ×⋯ × ⎜1 + ⎟ = ⎜1 + ⎟ 2⎠ 2⎠ ⎝ 2⎠ ⎝ ⎝ 2t Spot sloping upward, forward above spot Spot Spot sloping downward, forward below spot Spot 3 If r(2.5) is above If (2). ˆ ⎛ r (t − .5) ⎞ ⎜1 + ⎟ 2⎠ ⎝ 2 t −1 (2), then (2.5)> (2), ˆ ⎛ r (t ) ⎞ ⎛ r (t ) ⎞ × ⎜1 + ⎟ = ⎜1 + ⎟ 2⎠ ⎝ 2⎠ ⎝ 4 ˆ ˆ ⎛ r (2) ⎞ ⎛ r (2.5) ⎞ ⎛ r (2.5) ⎞ ⎜1 + ⎟ × ⎜1 + ⎟ = ⎜1 + ⎟ 2⎠ ⎝ 2⎠⎝ 2⎠ ⎝ 2t 5 Bond prices can be expressed in terms of forward rates 108+31.5/32=7.125 d(.5) + 107.125 d(1) 108+31.5/32=7.125 108+31.5/32= 108+31.5/32= 7.125 107.125 + ˆ ˆ r(.5) r(1) 2 (1+ ) (1+ ) 2 2 7.125 107.125 + r(.5) ⎞ r(.5) ⎞ ⎛ r(1) ⎞ ⎛ ⎛ ⎜ 1+ ⎟ ⎜1+ ⎟ ⎜ 1+ ⎟ 2⎠ 2 ⎠⎝ 2⎠ ⎝ ⎝ Maturity and Bond Price When are bonds of longer maturity When worth more than bonds of shorter maturity, and when is the reverse true? Consider five imaginary 4.875% Consider coupon bonds with terms from six months to 2.5 years. As of Feb. 15, 2001, which bond would have the greatest price? ...
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