Lecture 2 with solutions.pdf

# Lecture 2 with solutions.pdf - Lecture 2 Outline Term...

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Lecture 2: Outline – Term structure of interest rates – Risk free rates and zero coupon bonds – Bootstrapping the yield curve – Coupon bond YTMs – Discount factors Woman accidentally dumps mother’s mattress filled with \$1M

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The term structure of interest rates In reality, there are different risk free rates for different maturities The term structure (yield curve) is the relationship between interest rates and maturity Y axis is the interest rate X axis is the maturity for which the rate applies
The Yield Curve

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Shape of the Yield Curve Upward sloping: spot yields increase with the passage of time Reflects investor view that interest rates will rise over time Interest rates increase at a decreasing rate Rate of increase is a reflection of the volatility of that rise over the term (at a point in the yield curve)
Term structure of interest rates (2) We will let r 0,t denote the t-year risk free rate at t = 0. Also called the t-year spot rate. The PV of a risk free cash flow at time t is obtained by discounting the cash flow back t years with the t-year spot rate, r 0,t t , 0 t ) r (1 C PV t + =

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Term structure and zero coupon bonds One can also think about the yield curve as plotting the relationship between the maturity of zero coupon bonds (x-axis) and their yields to maturity (y-axis) • Why? Price of zero coupon bond is given by either thus t t ) YTM (1 F P + = t t 0, ) r (1 F P + = t r , 0 t YTM =
Term structure: Example The 1-year risk free rate, r 0,1 , is 3% The price of a 2-year zero coupon bond with face value \$10,000 is \$9,000 What is the price of a 2-year, 6% coupon bond, with face value \$1,000? Step 1: Find the 2-year risk free rate r 0,2 , which satisfies the following equation 9,000 = 10,000/(1+ r 0,2 ) 2 à r 0,2 = 0.054 or 5.4% Now we compute the price of the 6% coupon bond by discounting the bond’s cash flows at the maturity matched discount rates: P 0 = 60/(1+r 0,1 ) + 1060/(1+r 0,2 ) 2 = 60/1.03 + 1060/1.054 2 = 1,012.25

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Example: Bootstrapping the yield curve Given the following information about coupon bonds (each bond has \$1,000 face value) What are r 0,1 , r 0,2 , r 0,3 ? (i.e., what are the YTM on 1, 2, and 3 year zero coupon bonds)? What is the price of a 3-year, 10% coupon bond with face value \$1000? Maturity 1 2 3 Coupon rate 4% 5% 3% Price 1000 940 880
Bootstrapping the yield curve: Solution For these bootstrapping questions, the key is to start with the shortest term bond The 1-year coupon bond makes a payment only at t = 1, which means you can back out the 1-year risk free rate as follows Price of bond at time 0 = PV = 1,000 = (40 + 1,000)/(1+r 0,1 ) à r 0,1 = 4% (indeed when bond is priced at par YTM = coupon rate) The price of the 5%, 2-yr bond can be expressed as 940 = 50/(1+r 0,1 ) + 1,050/(1+r 0,2 ) 2 = 50/(1+0.04) + 1,050/(1+r 0,2 ) 2 Solve this equation with one unknown à r 0,2 = 8.50%

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Bootstrapping the yield curve: Solution (2) Finally, the price of the 3%, 3-yr bond can be expressed as 880 = 30/(1+r 0,1 ) + 30/(1+r 0,2 ) 2 + 1030/(1+r 0,3 ) 3 = 30/(1+0.04) + 30/(1+0.085) 2 + 1030/(1+r 0,3 ) 3 Solve this equation with one unknown à r 0,3 = 7.65% Now that we have r 0,1 , r 0,2 , r 0,3 we can find the price of any 1, 2, or 3-year coupon bond. For example, for the 3-year, 10% coupon Bond: P = 100/(1+r 0,1 ) + 100/(1+r 0,2 ) 2 + 1100/(1+r 0,3 ) 3 = 1062.86
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