Instructor: Jay Coughenour

Transcription

1 Instructor: Jay Coughenour 1

2 Education: Morristown High School, Morristown, NJ University of Pittsburgh (BA) Northeastern University (MBA) Arizona State University (PhD) Work Experience: Professor at UD since 1999 Short gigs at Securities & Exchange Commission, Krent Paffett & Associates, + various consulting Currently on Christiana Care Board of Trustees (Investments) Personal Married with 2 teenage daughters (both gone bonkers!) Distance runner (5Ks marathon)

3 The objective of this course is to analyze the pricing and use of derivative securities. Derivatives are securities whose payoff depends on the value or level of an underlying asset or measurement. Derivative types: Forwards, futures, swaps, and options will be emphasized, with some coverage of exotic ( path dependent ) options and structured notes. Underlying assets or measurements: We will focus on various commodity markets. However equity, interest rate, and currency contracts will receive light coverage, as will credit, structured and weather derivatives. 6-3

4 FINC314 (and all it s pre- requisites) FINC311, MATH201/202, MATH221 (or equivalent) This course is for well- motivated students with strong math, stat, finance, and memory skills! Students will apply: algebra, binomial and normal probability theory, statistics through linear regression, and cash- flow valuation techniques. In addition, we will enhance our understanding of commodity pricing with some geography and chemistry! 6-4

5 All About Derivatives (2 nd ed.), by Michael Durbin, McGraw Hill, 2011, ISBN: All About Commodities, by Tom Taulli, McGraw Hill, 2011, ISBN: The two paperbacks can be found in most Barnes & Noble stores and other bookstores. Much of our reading list will be found on the internet. A calculator will be necessary for each test. A financial calculator is best. At a minimum you need a calculator with y x, e x, and ln keys. 6-5

6 3 tests account for 70% 3 commodity review sets account for 30% Test schedule (same date review sets are due) October 1: Test 1 (20%); forwards and futures, metals markets. October 29: Test 2 (20%), swaps and options, energy markets December 3: Test 3 (30%), credit, structured, weather derivatives; agricultural and livestock markets 6-6

7 Problem sets (with solutions) will be extended periodically throughout the semester. Probably each Wednesday, or after we complete a subject area. First one today! 6-7

8 You will pretty much find everything, including these notes and all forthcoming notes at this webpage: derivatives- and- risk- management/ We will also be using lots of other internet sites for information such as WSJ, CME, etc. 6-8

9 Jay Coughenour August 27, 2014 FINC

10 Transforming between types is important for understanding forward rates and much more! 6-10

11 r 0 (t 1,t 2 ) is the interest rate from time t 1 to t 2 prevailing at time t 0 on a zero coupon bond r 0 (0,1) is the current one-year rate r 0 (0,2) is the current two-year rate r 0 (1,2) is the current one-year forward rate from t=1 to t=2. P 0 (t 1,t 2 ) is the price of a zero coupon bond purchased at time t 1 and maturing at time t 2 with par value = $

12 r 0 (0,1) t=0 t=1 t=2 r 0 (0,2) r 0 (1,2) Note that all of the above rates are available for contract today 6-12

13 One year zero-coupon bond: P(0,1)= Pay $ today to receive $1 at t= = 1/(1 + r(0,1)) 1 r(0,1)=0.06=6% Two year zero-coupon bond: P(0,2)= Pay $ today to receive $1 at t= =1/(1+r(0,2)) 2, which implies r(0,2)=0.065=6.5% Note that both 6% and 6.5% are YTMs on these bonds! Note that.9434 and.8817 are often called PVIFs (present value interest factors) 6-13

14 Suppose current one-year rate r(0,1) and two-year rate r(0,2) Current forward rate from year 1 to year 2, r 0 (1,2), must satisfy [1 + r 0 (0,1)][1 + r 0 (1,2)] = [1+ r 0 (0,2)] 2 If not, there will be arbitrage opportunities in forward rate market 6-14

15 In general 2 1 [ 1+ r ( t, t )] = t! t [ 1+ r ( 0, t )] 0 2 [ 1+ r ( 0, t )] 0 1 t t 2 1 = P( 0, t1) P( 0, t ) 2 Example 7.1 What are the implied forward rate r 0 (2,3)? (use Table 7.1) P r0 ( 2, 3) ( 0, 2) =! =! P( 0, 3) =

16 Suppose r(0,4)=.085 and r(0,5)=.0825, what is the implied one-year forward rate from year 4 to 5, r(4,5)? [1+r(0,5)] 5 = [1+r(0,4)] 4 [1+r(4,5)] r(4,5)= ( / ) -1 = 7.26% OR P(0,4) = 1/ [1+r(0,4)] 4 = 1/ = P(0,5) = 1/ [1+r(0,5)] 5 = 1/ = r(4,5)= P(0,4)/ P(0,5)-1 =.7216/ = 7.26% 6-16

17 Suppose r(0,4)=.085 and r(0,7)=.0825, what is the implied annual three-year forward rate from year 4 to 7, r(4,7)? [1+r(0,7)] 7 = [1+r(0,4)] 4 [1+r(4,7)] 3 r(4,7)= ( / ) 1/3-1 = 7.92% OR P(0,4) = 1/ [1+r(0,4)] 4 = 1/ = P(0,7) = 1/ [1+r(0,5)] 7 = 1/ = r(4,5)= P(0,4)/ P(0,5)-1 = (.7216/.5741) 1/3-1= 7.92% 6-17

18 Given the zero coupon bond prices, what coupon rate would price the bond at par? Recall, the price of a $1 face value bond is: Discounted value of coupon payments Therefore: n t! t i t i= 1 B ( t, T, c, n) = cp ( t, t ) + P ( t, T) where t i = t + i(t - t)/n (typically initial t=0) c = 1! n " i = 1 P ( t, T) t P ( t, t ) t i Discouted value of $1 principal payment 6-18

19 What is the par coupon rate for the 3 year bond in Table 7.1? c = ( ) / ( ) c = / =

20 Suppose we buy the 3-year bond with face value of $1, with c= , for $1. Suppose we also re-invest all coupon payments (using today s forward rates). This strategy constructs an effective 3- year zero out of a 3-year coupon bond However, to preclude arbitrage, such a strategy had better yield same return as the 3-year zero coupon bond (7%)! 6-20

21 yr 1, we get FV 3 = (1+r(1,3)) 2 FV 3 = ( ) 2 = yr 2, we get FV 3 = (1+r(2,3)) FV 3 = ( ) = yr 3, we get So, at yr 3 we have $ ( = ) Our yield-to-maturity = ( /1) 1/3 1= 7% 6-21

22 We are indifferent between the zero bond market and coupon bond market (combined with forward market). Both markets lock-in yields of 7%. If not indifferent then arbitrage would exist across the two markets 6-22

23 Note that we are also indifferent between investing $1 in the coupon bond and rolling an initial $1 investment through the forward market. (1+.06)(1+.07)(1+.08) = The coupon rate is a constant rate that leaves us indifferent to the rates earned in the forward market. Beautiful! 6-23

24 Recall, PV = FVe -rt For a zero coupon bond paying $1 at time t, this equates to P(0,t) = 1e -rt Therefore, r cc = (1/t)ln(1/P(0,t)) Example: For a 2 year zero, P(0,2)= Therefore r cc = (1/2)ln(1/ )=

25 Used almost exclusively in the pricing of derivatives. Why? Conceptually matches an underlying asset whose price movements are continuous not discrete Manipulating continuous rates is often easier e x e y = e x+y, therefore $1000e r e r e r e r = $1000e r4 (or more generally FV=PVe rt If interest rates vary over time, say r1 for t1, and r2 for t2 then total gross return is e r1t1+r2t2 ln(e x )=x, so if P(0,t) = 1e -rt, then (1/P(0,t)=ert, then taking nat logs we have ln(1/p(0,t)=rt, or r=ln(1/p(0,t)(1/t) 6-25

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