Exercise 1: COMBINED THEORY: LP varies with maturity according to LP = 2 + 0.5*ln(t) where t=years to maturity. The expected rates of inflation for years [1,2,3,4,5,6,7] are [3.0,4.0,3.5,3.5,3.5,3.5,3.5]. What is the required rate from a 7-year T-note according to the combined theory?

• note: treasuries do not have default risk
• the computed value of LP for a maturity of 7 years is 2+0.5*ln(7) = 2.973
• invest $1 today and compound the rate for each year. at the end of 7 years a dollar will be worth FV dollars
• FV = (1.0297)^7(1.03)(1.04)(1.035)^5 = 1.5615
• the geometric average annual rate is k = 1.5615^(1/7) - 1
• = 1.5615^0.14286 - 1 = 0.0657
• the required rate from this t-bill is 6.57%

• pure expectation theory = long term rates are predictors of future short term rates
• method: set up alternate investment plans with short term and long term instruments and equate the yields
• let the one-year rate next year be x
• $1 invested today in 2-year treasury will be worth (1.06)^2=1.1236
• $1 invested and re-invested in 1-year treasuries will be worth 1.056*(1+x)
• equate and solve: 1.056*(1+x) = 1.1236
• 1+x = 1.064, x = 6.4%

• let the 2-year rate 4 years from now be x
• invest $1 in 4-year tbill to get: 1.07^4 = 1.3107 (4 years from now)
• reinvest this in 2-year bills to get: 1.3107*(1+x)^2 (6 years from now)
• alternately, invest $1 in 6 year bills to get 1.075^6 = 1.5433 (6 years from now)
• pure expectation: the 6-year amounts should be the same: so equate and solve
• 1.3107*(1+x)^2 = 1.5433
• (1+x)^2 = 1.5433/1.3107 = 1.17738
• 1+x = square root of 1.17738 = 1.085
• the 2-year rate 4 years from now will be 8.5%

• let the 1-year rate 1 years from now be x
• invest $1 in 2-year bills to get: 1.045^2 = 1.092
• invest $1 in 1-year bills and re-invest to get: (1.03)*(1+x)
• equate and solve: (1.03)*(1+x) = 1.092
• 1+x = 1.0602, x=0.602, the 1-year rate 1 year from now will be 6.02%
• let the expected inflation rate in years [1,2] be [x1,x2]
• k = (1+LP)*(1+inflation) since no default risk
• year 1: k = 1.01*(1+x1) = 1.03, 1+x1=1.03/1.01 = 1.0198. expected inflation in year 1 is 1.98%
• year 2: k = 1.01*(1+x1) = 1.0602, 1+x1=1.0602/1.01 = 1.0497. expected inflation in year 1 is 4.97%

• let the LP for 2-year bonds be x2 and that for 5-year bonds be x5
• 2-year bond: (1+k)^2 = (1+x2)^2 * (1.08)*(1.05) = (1+x2)^2 * 1.134
• 1+k = (1+x2)*1.0649 = 1.10
• solve for x2: x2 = 1.033 - 1 = 3.3%: LP is 3.3% for maturity = 2 years
• 5-year bonds: (1+k)^5 = (1+x5)^5 * (1.08)*(1.05)*(1.04)^3 = (1+x5)^5 * 1.2756
• 1+k = 1.10 = (1+x5)(1.2756)^(1/5) = (1+x5)*1.0499
• solve for x5: 1+x5 = 1.1/1.0499 = 1.0477: LP is 4.77% for maturity = 5 years

• (a) 1+k = (1.03)(1.04)(1.1) = 1.1783, k=17.83%
• (b) 1+k = (1.03)(1.06)(1.1) = 1.2010, k=20.01%
• (c) 1+k = (1.03)(1.04)(1.08) = 1.1569, k=15.69%