Pure Expectations Theory

Can someone talk me through the logic here:

An analyst collects the following info regarding spot rates of interest:

1yr rate = 4%, 2yr rate = 5%, 3yr 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 is closest to:

ANSWER is 9.04%

Pure expectations says that tomorrow’s spot rate is today’s forward rate (starting tomorrow); i.e., if you want a future spot rate for some time t in the future, look at today’s forward rate starting at time t.

So the expected 2-year spot rate two years from today is today’s 2-year forward rate starting two years from now: 2f2. You get that from today’s 2-year spot rate and today’s 4-year spot rate:

(1 + 2f2)² = (1 + s4)^4 / (1 + s2)²

(1 + 2f2)² = (1.07)^4 / (1.05)²

(1 + 2f2)² = 1.188931

1 + 2f2 = 1.090381

2f2 = 0.090381 = 9.0381%

Thank you! Still a tad shaky but I think I have it. Is this a must memorize formula?

You’ll see it in the second Fixed Income study session as well. I’d encourage you to understand it.

If you understand the difference between the spot curve and the forward curve, then pure expectations is easy to explain: you’re just moving “today” along the forward curve (to the right), or, alternatively, moving the forward curve itself to the left.