# Implied forward rates puzzle

Here's an interesting cocktail puzzle related to the term structure of interest rates.

One of the primary competing theories for explaining the term structure of rates is the Rational Exepctations Hypothesis (REH).

Now generally we test a theory by examining the empirical data and seeing which theory explains the data parsimoniously. For example, if the REH is correct then the test is that implied forward rates are an unbiased predictor (regardless of the quality of the prediction).

My claim is that you can show the REH has a bias on a priori grounds and therefore can be rejected. Is the argument correct or is there a flaw?

Premises:

A. REH asserts that implied forward rates are unbiased predictors of future spot rates.

B. Implied forward rates are always above spot rates when the term structure is upward sloping, and similarly implied forwards are always below spot rates when the term structure is inverted. (This is true by the mathematics of calculating implied forward rates although you can see it by conceptually by considering that implied forward rates can be "locked-in".). A depiction of forward rates and the spot curve is depicted here:

C. By premises #1 and #2, REH will never predict term structure flattening.

D. By #3, REH expectations are biased since the probability distribution of implied rates assigns zero chance to term structure inversion (although we know empirically there are non-zero chances).

Simply put, REH biases future term structure changes upwards when term structure is upward sloping, and biases future term structure changes downwards when the term structure is inverted.

Conclusion: REH is a biased predictor of future rates and therefore the theory is flawed. In particular, the REH biases future predictions of rates upwards when the term structure is upward sloping. Note: This is not to say that implied forward rates cannot predict an inversion with upward term structure. Indeed the chart above shows precisely this case.

Postscript: The Salomon Brothers research team sets up a cross-sectional regression experiment to identify which hypothesis holds up. Turns out they find that the bond risk premium hypothesis does out-performs the REH hypothesis.

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What is premise #3? I only see 2. –  Tal Fishman Jul 30 '12 at 19:33
I'm not sure I understand how you reach conclusion 2 (as per Tal Fishman's comment it seems like premise 3 was left off). My understanding of REH is that if $r_{t}$ is the spot rate in time $t$ and $f_{t,t+k}$ is the forward rate from $t$ to $t+k$, then $E\left(r_{t+k}\right)=f_{t,t+k}$. However, that implies nothing about the variance of the forward rates. I see no reason why some scenarios generated wouldn't have downward sloping term structures. –  John Jul 30 '12 at 20:17
@TalFishman -- looks like a bug. When I edit the post I can see the bullets and the numbers show correctly, however the rendering does not. –  Quant Guy Jul 30 '12 at 21:04
@QuantGuy Numbered lists in Markdown assume that (1) any numbering indicates a list, and (2) the numbered paragraphs are consecutive. So when you interrupted the list with a picture, the Markdown generator assumed you have two separate lists. And since any numbering indicates a list, Markdown restarted the count at 1, despite the explicit 3 in your editor code. Oh, silly Markdown. –  chrisaycock Jul 31 '12 at 1:21
@chrisaycock - Thanks Chris. I switched to letters. –  Quant Guy Jul 31 '12 at 3:22

There are two flaws in the argument. The simpler one is that expectations give information about probability distributions (premise D). I think this is what John was referring to in his comment. The fact that the expectation of a forward rate in period X is Y% tells us nothing about the implied probability distribution in that period, and certainly doesn't preclude the possibility that the rate could be 0% or 100% with non-zero probability. Y% is merely the expectation of that distribution. For a more concrete example, consider a little normal distribution around each forward, representing its probability distribution. The forward rate tells us the mean (center) of that distribution, but tells us nothing about how far it stretches in either direction.

The second flaw is that premise (C) does not follow from (A) and (B). Premise (C) states that REH will never predict term structure flattening, but, as you knowledge in your conclusion, the picture provided to illustrate (B) shows a term structure that will not only flatten but invert.

To demonstrate why this is so, note that while it is true that an monotonically increasing term structure requires forwards that are everywhere higher than spot rate, that does not mean that the structure does not flatten. A counterexample is trivial to construct: 5% forward rate in the first period, and 10% forward rates thereafter. The resulting spot curve starts at 5% and monotonically increases toward 10%, a level it never reaches (as of day 0). But every successive day will yield a flatter curve, ceteris paribus, reaching a limit of a 10% flat curve in the absence of new information that impacts expectations.

In other words: if the forward rates flatten, so will the spot.

For a more complex counterexample, refer again to the image in your post, which shows a monotonically increasing spot curve with forwards that imply a future inversion (an act which necessitates an intermediate flattening).

Finally, don't forget that there are many possible term structures with non-monotonic slopes. In such structures, the forward curve will be sometimes higher and sometimes lower than spot. Therefore, (B) is incomplete, as it only considers two cases, and (C) can not be absolute.

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