5-1 Three Theories of Term Structure 1.Expectations Theory –Pure Expectations Theory explains 1 and 2, but not 3 2.Market Segmentation Theory –Market Segmentation Theory explains 3, but not 1 and 2 3.Liquidity Premium Theory –Solution: Combine features of both Pure Expectations Theory and Market Segmentation Theory to get Liquidity Premium Theory and explain all facts

5-2 Expectations Theory - Notation We use zero-coupon bonds for this analysis to avoid the need to make reinvestment assumptions Zero coupon rates are “spot” rates We will designate spot rates as S 1, S 2, S 3 etc. where S 1 = a one-year bond We will designate forward rates (rates for periods that start in the future, not today) as 1 f 1, 1 f 2, 2 f 1 etc. 2 f 1 = the two-year forward rate starting one year from now 1 f 2 = the one-year forward rate starting two years from now

5-3 Expectations Theory Expected return from strategy 1 (invest in one year bond and then reinvest in a one-year bond one year from today) (1+S 1 )(1+ 1 f 1 ) = Ending Balance (End Balance/Beginning Balance)-1 = HPR (1+ HPR)^.5 -1 = S 2 = expected return

5-4 Expectations Theory Expected return from strategy 1 (1+S 1 )(1+ 1 f 1 ) = Ending Balance (End Balance/Beginning Balance)-1 = HPR (1+ HPR)^.5 -1 = S 2 = expected return (annualized) S1S11f11f1 S2S S2S2

5-5 Expectations Theory Expected return from strategy 2 (Invest in two- year bond today) (1+S 2 )(1+ S 2 ) = Ending Balance (End Balance/Beginning Balance)-1 = HPR (1+ HPR)^.5-1 = S2 = expected return (annualized) S2S2 S2S2 0 12

5-6 Expectations Theory From implication above expected returns of two strategies are equal Therefore Solving for 1 f 1 (1) (1+S 2 )(1+ S 2 )-1 = (1+S 1 )(1+ 1 f 1 )-1 1 f 1 = ((1+S 2 )(1+ S 2 ))/(1+S 1 ) – 1 1 f 1 = (End Balance/Beg Balance)-1 Note that 1 f 1 is a HPR. We do not need to annualize It because it is exactly a one-year period

5-7 Annualizing HPR Remember that if we did need to annualize, we would use our same formula as always: (1+HPR)^ (# of HPs in one year) – 1

5-8 Expectations Theory Example 1 Assume S 1 = 5% and S 2 = 7%, solve for 1 f 1 : 1 f 1 = ((1+S 2 )(1+ S 2 ))/(1+S 1 ) – 1 1 f 1 = (End Balance/Beginning Balance)-1 1 f 1 = ((1+.07)(1+.07))/(1+.05) – 1 1 f 1 = / = =.0904 S 1 = 5% 1 f 1 = ? S 2 = 7% 0 12

5-9 Expectations Theory Example 2 Assume S 1 = 5% S 3 = 10% 1 f 1 = 9.04%, solve for 1 f 2 : 1 f 2 = ((1+S 3 )(1+ S 3 )(1+S 3 )/(1+S 1 )(1+ 1 f 1 ) – 1 1 f 2 = (End Balance/Beginning Balance)-1 1 f 2 = ((1+.10)(1+.10)(1+.10))/(1+.05)(1.0904) – 1 1 f 2 = / = =.1625 S 1 = 5% 1 f 1 = 9.04 S 3 = 10% f 2 = ? S 3 = 10%

5-10 Expectations Theory Example 3 Assume S 1 = 4% S 3 = 6%, solve for 2 f 1 : 2 f 1 hpr = ((1+S 3 )(1+ S 3 )(1+S 3 )/(1+S 1 ) – 1 2 f 1 hpr = (End Balance/Beginning Balance)-1 2 f 1 hpr = ((1+.06)(1+.06)(1+.06))/(1+.04) – 1 2 f 1 hpr = / = = f 1 = ( )^.5 -1 = annualized S 1 = 4% 2 f 1 = ? S 3 = 6% f 1 = ? S 3 = 6%

5-11 Expectations Theory Example 4 Assume S 1 = 4% 1 f 1 = 4.5% 1 f 2 = 5.2% solve for S 3 S 3 hpr = ((1+S 1 )(1+ 1 f 1 )(1+ 1 f 2 )) – 1 S 3 hpr = ((1.04)(1.045)(1.052)-1 = ( ) -1 = S 3 = ( )^ (1/3) – 1 = annualized S 1 = 4% 1 f 1 = 4.5% S 3 = ? f 2 = 5.2% S 3 = ?