Adjust the bank discounted rate to make it comparable n P P r BEY 365

Adjust the bank discounted rate to make it comparable

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Adjust the bank discounted rate to make it comparable n P P r BEY 365 000 , 10 × = % 28 . 8 90 365 800 , 9 800 , 9 000 , 10 = × = BEY r BD BEY r P r = × × 365 360 000 , 10 Example: same as before BDR versus BEY
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38 Spot Zero-Coupon (or Discount) Rate ! Spot Zero-Coupon (or Discount) Rate is the annualized rate on a pure discount bond ! where B(0,t) is the market price at date 0 of a bond paying off $1 at date t ! General pricing formula ( ) t B R t t , 0 ) 1 ( 1 , 0 = + ( ) = = = + = T t t T t t t t t B F R F P 1 1 , 0 0 , 0 ) 1 ( Bond Par Yield ! Recall that a par bond is a bond with a coupon identical to its yield to maturity ! The bond's price is therefore equal to its principal ! Then we define the par yield c(n) so that a n-year maturity fixed bond paying annually a coupon rate of c(n) with a $100 face value quotes at par ! Typically, the par yield curve is used to determine the coupon level of a bond issued at par = = + + = + + + = n i i i n n n n n i i i R R n c R R n c 1 , 0 , 0 , 0 1 , 0 ) 1 ( 1 ) 1 ( 1 1 ) ( ) 1 ( 100 ) 1 ( ) ( 100 100
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39 Zero-Coupon Bonds ! Zero-Coupon Bond ! Does not make coupon payments ! Always sells at a discount (a price lower than face value), so they are also called pure discount bonds ! For example, ! In USA, Treasury Bills are U.S. government zero-coupon bonds with a maturity of up to one year. ! In France BTF : (bon du Trésor à taux fixe) are France government zero-coupon bonds (prepaid interest) with a maturity of up to one year. Zero-Coupon Bonds ! Suppose that a one-year, risk-free, zero-coupon bond with a $100,000 face value has an initial price of $96,618.36. The cash flows would be: ! Although the bond pays no “interest,” your compensation is the difference between the initial price and the face value.
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40 Zero-Coupon Bonds ! Yield to Maturity ! The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond. ! Price of a Zero-Coupon bond (1 ) = + n n FV P YTM Zero-Coupon Bonds ! Yield to Maturity ! For the one-year zero coupon bond: ! Thus, the YTM is 3.5%. 1 100,000 96,618.36 (1 ) = + YTM 1 100,000 1 1.035 96,618.36 + = = YTM
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41 Zero-Coupon Bonds ! Yield to Maturity ! Yield to Maturity of an n -Year Zero-Coupon Bond 1 1 = n n FV YTM P Zero-Coupon Bonds Example ! Problem ! Suppose that the following zero-coupon bonds are selling at the prices shown below per $100 face value. Determine the corresponding yield to maturity for each bond. Maturity 1 year 2 years 3 years 4 years Price $98.04 $95.18 $91.51 $87.14
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42 Zero-Coupon Bonds Example ! Solution: 1/2 1/3 1/4 YTM (100 / 98.04) 1 0.02 2% YTM (100 / 95.18) 1 0.025 2.5% YTM (100 / 91.51) 1 0.03 3% YTM (100 / 87.14) 1 0.035 3.5% = = = = = = = = = = = Zero-Coupon Bonds ! Risk-Free Interest Rates ! A default-free zero-coupon bond that matures on date n provides a risk-free return over the same period. ! Thus, the Law of One Price guarantees that the risk-free interest rate equals the yield to maturity on such a bond. ! Risk-Free Interest Rate with Maturity n = n n r YTM
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43 Zero-Coupon Bonds !
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