Conversion formulas 17 l notation r c continuously

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Conversion Formulas 17 l Notation: R c : continuously compounded rate l Equivalent interest rate , R m , compounded m times per year: l Conversion formulas: ( ) 1 1 ln / = + = m R m m c c e m R m R m R e R c = (1 + R m m ) m Fin330
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Conversion Formulas: Examples 18 l What rate of interest with continuous compounding is equivalent to 15% per annum with monthly compounding? Answer: l A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. What is the equivalent quarterly compounded rate? Answer: Fin330
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Conversion Formulas: Examples 19 l What rate of interest with continuous compounding is equivalent to 15% per annum with monthly compounding? e x =(1+15%/12) 12 => x=12 ln(1+15%/12)=14.91% l A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. What is the Equiv. quarterly compounded rate? e 12% =(1+x/4) 4 => x=4(e 12%/4 -1)=12.18% Fin330
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Overview 1. Types of rates 2. Measuring interest rates 3. Bond pricing : 1. Zero coupon bond and Zero Rate 2. Bond’s Yield 3. Term structure of interest rates Fin330 20
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21 Zero coupon bond and Zero Rate l Zero-coupon bonds : bonds with no payment before maturity. l When a zero coupon bond matures, the investor receives its face value (= par value ). l A t -year zero rate (= spot rate ) is the rate of interest earned on a t-year zero coupon bond. Fin330
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22 Bonds in general l In general, bonds do pay coupons. l Example: l A bond has a face value of $1,000, a coupon rate of 5% and maturity of 2 years. l The flow of funds for the investor is as follows: l t =0: - $Price of bond l t =1: + $50 l t =2: + $1,050. l How to price bonds which pay coupons? Fin330
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Bond pricing Fin330 23 l A zero rate with maturity t defines the discount rate for a cash flow occurring at time t . l We use the spot rates to determine the price. l In the example above, the bond can be viewed as a portfolio of zero coupon bonds with one- and two-year maturities.
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Bond pricing: Example Fin330 24 l Assume the following Treasury zero rates (continuous compounded): l A 2-year Treasury bond with a face value of $100 provides coupons at the rate of 6% per annum semiannually. l What is the theoretical price of this bond? Maturity (years) Zero rates (%) (cont. compounding) 0.5 5.0 1.0 5.8 1.5 6.4 2.0 6.8
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Bond pricing: Example Fin330 25 l Recall that the PV of investment with continuous compounding is l The flow of funds for an investor: l Theoretical price of the bond is $.
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