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# Derivatives10Lec3 - Derivatives Click to edit Master...

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Click to edit Master subtitle style 11.1 Derivatives Interest Rates

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21.2 Where we are l Last Session: Fundamentals of Forward and Futures Contracts (Chapters 2-3, OFOD) l This Session: Introduction to Interest Rates and the Value of Future Cash (Chapter 4, OFOD) l Next Session: Swaps Chap 7.
31.3 Plan for This Session l Review some items left over from last time l Interest Rates l The Yield Curve & Zero Curve l Present Value & Future Value l Compounding frequency; Continuous Compounding l Spot Rate & Discount Function l Yield measures

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Yield Curve (Term Structure of Interest Rates) Basics What is the Yield Curve? Interest rates on financial instruments vary because of default risk, liquidity risk, call provisions, etc. Holding all the above constant, it also appears rates vary because of maturity. The relationship between interest rates and maturity, all else fixed, is called the term structure of interest rates or the yield curve. Where do we find the yield curve? Typical yield curve.

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61.6 Zero Rate & Discount Function l A zero rate (or spot rate ), for term T is today’s ( t0 ) rate of interest earned on an investment that provides a payoff only at time t0+T l Lets denote this as R(t0,T) l More formally using our earlier notation; for a cash flow, CF(t0+T) , to be received on t0+T , its present value, PV(t0,T) = CF(t0+T) e-R(t 0 ,T)xT = CF(t0+T) x d(t0,T) where d(t0,T) is the discount function l It is important to recognize that today’s discount function, like the spot rate, is a function of term , T . l A cash flow, CF(t0+T) , happens on a date, t0+T .
71.7 Example – Spot Rate & Discount Maturity (Years) Spot Rate (%) Discount PV of 6% 2-year Bond (SA pay) 0.5 5.0 0.975310 2.925930 1.0 5.8 0.943650 2.830950 1.5 6.4 0.908464 2.725392 2.0 6.8 0.872843 89.902829 TOTAL = 98.385101

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81.8 Bond Pricing l To calculate the cash price of a bond, discount each cash flow at the appropriate zero rate l In our example, the theoretical price of a 2-year bond providing a 6% coupon semiannually is 3 3 3 103 98 39 0 05 0 5 0 058 1 0 0 064 1 5 0 068 2 0 e e e e - × - × - × - × + + + = . . . . . . . . .
91.9 Bond Yield l The bond yield is the single discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond l Suppose that the market price of the bond in our example equals its theoretical price of 98.39 l The bond yield (continuously compounded) is given by solving (numerically) to get y =0.0676 or 6.76%. 3 3 3 103 98 39 0 5 1 0 1 5 2 0 e e e e y y y y - × - × - × - × + + + = . . . . .

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101.10 Example – Bond Yield Maturity (Years) Yield (%) Discount PV of 6% 2-year Bond (SA pay) 0.5 6.76 0.966765 2.900295 1.0 6.76 0.934634 2.803903 1.5 6.76 0.903572 2.710715 2.0 6.76 0.873541 89.974742 TOTAL = 98.389564
111.11 Par Yield or Par Coupon l The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. l In our example we solve g) compoundin s.a. (with get to 87 6 100 2 100 2 2 2 0 . 2 068 . 0 5 . 1 064 . 0 0 . 1 058 . 0 5 . 0 05 . 0 . c= e c e c e c e c = + + + + × - × - × - × -

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121.12 Example – Par Coupon Maturity (Years) Spot Rate (%) Discount PV of 6.87% 2- year Bond 0.5 5.0 0.975310 3.350190 1.0 5.8 0.943650 3.241438 1.5 6.4 0.908464 3.120574 2.0 6.8 0.872843 90.282516 TOTAL = 99.994718
131.13 Par Yield continued l In general if m is the number of coupon payments per year, P is the present value of \$1 received at maturity and A

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Derivatives10Lec3 - Derivatives Click to edit Master...

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