Question

Today’s one-year zero rate is 6.50% pa. The unanimous forecast among bond dealers is that, a year from today, the term structure of interest rates will be: 7.00% pa (for 1 year), 9.15% pa (for 2 years) and 10.00% pa (for 3 years).

Assume that the pure expectations theory of the term structure is correct. What is today’s term structure? That is, what are today’s interest rates (pa) for terms of 1, 2, 3 and 4 years? Show your calculations.

Answer 1

Given that,

Today's one year rate r1 = 6.5%

The unanimous forecast among bond dealers is that, a year from today, the term structure of interest rates will be: 7.00% pa (for 1 year), 9.15% pa (for 2 years) and 10.00% pa (for 3 years).

=> f(1,1) = 7%

f(2,1) = 9.15%

f(3,1) = 10%

So, today's term structure can be calculated as below

So, today's 1 year rate r1 = 6.5%

today's 2 year rate r2 = ((1+r1)*(1+f(1,1))^(1/2) - 1 = (1.065*1.07)^(1/2) - 1 = 6.75%

today's 3 year rate r3 = ((((1+r1)*((1+f(2,1))^2))^(1/3)) -1) = (((1.065*(1.0915^2))^(1/3)) - 1) = 8.26%

today's 4 year rate r4 = ((((1+r1)*((1+f(3,1))^3))^(1/4)) -1) = (((1.065*(1.1^3))^(1/4)) - 1) = 9.11%