An analyst collects the following information regarding spot rates of interest: 1 yr rate = 4% 2 yr rate = 5% 3 yr rate = 6% 4 yr rate = 7% Utilizing the pure expectations theory of the term structure of interest rates, the expected annualized 2 yr interest rate two years from today will be: A. 7.02% B. 8.03% C. 9.04% D. 18.89% How do you calc this?
1.07^4/1.05^2 = 1.188931 so you’d be tempted to choose D but then it says expected ANNUALIZED not the 2 yr so it’d be sq rt = 1.09 and change… I like my odds on choice C. Mind you, I haven’t done this stuff in years, I might be way off. Looking at the question logically, though, you can almost knock out A and D right away- 7% for a 1 yr rate eyeballing seems low and 18% seems ridiculously high. 1.0803 you get by doing 1.06^3/1.05^2 which I’d think would be a 1 yr rate but not the 2 yr so that only backs up the fact I like choice C. So even if you had no clue how to do this- knock out choices that don’t make sense and you’re at worst a 50/50.
If my understanding is correct, you are asked f2,4, i.e., the interest rate between year 2 and year 4, per annum. So: (1+5%)^2*(1+x)^2=(1+7%)^4 (it should be equivalent to invest at the two-year rate, i.e., 5% per annum for two years, and then at the two-year rate you are looking for for two years, i.e., x, or to invest at the four-year rate, i.e., 7% per annum for four years; otherwise, there would be an arbitrage opportunity) So: x = 9.04%. Answer c.
The way I remember it from econ class, which is less mathematically rigorous than above but gets you pretty close is to say: Since the current 1yr is 4% and the 2yr is 5%, then the period between years 1 and 2 is 6% because it has to average to the current 2yr (5%). Using the same reasoning the period from 2yr to 3yr must be 8% to average the 6% on the 3yr, given the last calculation. And the period from 3 to 4 must be 10% to average 7% on the 4yr, given the previous two. Lastly to get the expected 2 yr, two years from now you just average period 2 to 3 and 3 to 4, which is the average of 8% and 10% or 9%, (c.).
paul_ledin Wrote: ------------------------------------------------------- > The way I remember it from econ class, which is > less mathematically rigorous than above but gets > you pretty close is to say: > > Since the current 1yr is 4% and the 2yr is 5%, > then the period between years 1 and 2 is 6% > because it has to average to the current 2yr > (5%). > > Using the same reasoning the period from 2yr to > 3yr must be 8% to average the 6% on the 3yr, given > the last calculation. > > And the period from 3 to 4 must be 10% to average > 7% on the 4yr, given the previous two. > > Lastly to get the expected 2 yr, two years from > now you just average period 2 to 3 and 3 to 4, > which is the average of 8% and 10% or 9%, (c.). Here’s the “quicker and dirtier” quick and dirty method, that you can do in your head. The 4 yr spot rate pays approximately 28% total over 4 years (forgetting about compounding). The 2 yr rate spot pays 10% total over 2 years. So, the 2 year forward rate must make up the 18% (28-10) difference, and pay about 9% per year. Since the compounding on the 4 yr rate (at 7%) will exceed the compounding on the 2 year rate (at 5%), the actual rate must be a bot above 9%.