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# ecg590i_lecture06 - ECG590I Asset Pricing Lecture 6...

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ECG590I Asset Pricing. Lecture 6: Interest Rates 1 6 Interest Rates Bond: you pay now to receive an amount later (the face value) and possibly coupon payments. 6.1 Zero rates The n -year zero rate (short for zero-coupon rate) is the rate of interest earned on a zero-coupon investment that starts today and lasts n years. Interest rates are usually function of the maturity. John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 6: Interest Rates 2 Formula for T -period asset: P 0 e rT = P T e rT = P T P 0 rT = ln P T P 0 ! r = 1 T ln P T P 0 ! ± 1 T P T ² P 0 P 0 = average percentage change in price over T periods John Seater, North Carolina State University, Fall 2007
ECG590I Asset Pricing. Lecture 6: Interest Rates 3 6.2 Coupon rate The coupon rate is the stated (contracted) rate used to determine periodic payments of income during the life of the bond. Formula for bond price: B = cP 1 + r 1 + cP (1 + r 1 )(1 + r 2 ) + ±±± + cP Q T i =1 (1 + r i ) + P Q T i =1 (1 + r i ) where B = current value of bond c = coupon rate P = principal (face value) r i = 1-period market interest rate in period i (a forward rate , discussed later) T = term to maturity John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 6: Interest Rates 4 6.3 Bond yield The bond yield is the constant discount rate that equates the present Formula: B = cP 1 + r ? + cP (1 + r ? ) 2 + ±±± + cP (1 + r ? ) T + P (1 + r ? ) T where B = cP 1 + r 1 + cP (1 + r 1 )(1 + r 2 ) + ±±± + cP Q T i =1 (1 + r i ) + P Q T i =1 (1 + r i ) John Seater, North Carolina State University, Fall 2007
ECG590I Asset Pricing. Lecture 6: Interest Rates 5 6.4 Par yield its face value. Formula: P = B ? = c ? P 1 + r 1 + c ? P (1 + r 1 )(1 + r 2 ) + ±±± + c ? P Q T i =1 (1 + r i ) + P Q T i =1 (1 + r i ) Canceling P we get 1 = c ? 1 + r 1 + c ? (1 + r 1 )(1 + r 2 ) + ±±± + c ? Q T i =1 (1 + r i ) + 1 Q T i =1 (1 + r i ) John Seater, North Carolina State University, Fall 2007

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ECG590I Asset Pricing. Lecture 6: Interest Rates 6 6.5 Computing treasury zero rates: the bootstrap method We want to know the zero rates implied by the prices of existing bonds. The problem is that bonds usually pay coupons. For example, suppose the current data are bond principal time to maturity semi annual coupon bond price \$ 100 0.25 0 \$ 97.5 \$ 100 0.50 0 \$ 94.9 \$ 100 1.0 0 \$ 90.0 \$ 100 1.5 \$ 4 \$ 96.0 \$ 100 2.0 \$ 6 \$ 101.6 John Seater, North Carolina State University, Fall 2007
ECG590I Asset Pricing. Lecture 6: Interest Rates 7 1. Start with the shortest existing bond and compute its zero rate:

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ecg590i_lecture06 - ECG590I Asset Pricing Lecture 6...

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