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Lecture 4 with solutions.pdf

# Lecture 4 with solutions.pdf - Lecture 4 Outline Recap of...

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Lecture 4: Outline Recap of term structure theories and implications Simple examples Bond sensitivity to changes in interest rates Duration and Modified Duration Applications: Curve steepeners Portfolio Duration Par Curve Asset Swaps

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Traditional Expectations Hypothesis Long term rates are determined by current short term rates and expectations of future short term interest rates Forward interest rates are unbiased estimates of future interest rates: f n,m = E[r n,m ] Investors are almost risk neutral with respect to interest rate risk Expected 1-year return on long-term bonds almost equal to 1-year risk free rate Flat yield curve  expected future interest rates = current rates Upward sloping yield curve  interest rates expected to rise Downward sloping yield curve  interest rates expected to fall
Local Expectations Hypothesis Long term rates are determined by current short term rates and expectations of future short term interest rates Investors are risk neutral with respect to interest rate risk Expected 1-year return on long-term bonds equal to 1-year risk free rate Forward interest rates are slightly downward biased estimates of future interest rates: f n,m < E[r n,m ] Following equivalences only approximately true Flat yield curve  expected future interest rates = current rates Upward sloping yield curve  interest rates expected to rise Downward sloping yield curve  interest rates expected to fall

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Example Local Expectations Hypothesis: In order to obtain an expected return of 8% for each of the k- year bonds, with assumed face value of 100, it must be that 8% = [0.5(100/(1.1) k-1 )+ 0.5(100/(1.06) k-1 )]/P 0,k 1 Where 100/(1.1) k-1 and 100/(1.06) k-1 are the price of the bond at t = 1 if rates went up to 10%, or down to 6%, respectively Where P 0,k = 100/(1+r 0,k ) k = Price of bond maturing in k years , at t=0 The results for the r 0,k are plotted on the next slide where the above equation is reversed to solve for r 0,k
Example: Solution (2) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 10 20 30

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Expected 1 year return 0.000% 1.000% 2.000% 3.000% 4.000% 5.000% 6.000% 7.000% 8.000% 9.000% 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 E1,k