Forward Rates as Expectations of Future Spot Rates
Summary
The document considers when a current forward rate can be viewed as an unbiased indicator of a future spot rate. It cites the pure expectations hypothesis, under which forward rates predict future spot rates, and mentions weaker empirical support, term-premium adjustments, and research reporting a cointegrating relation between currency-market forward and future spot rates with a unit coefficient.
The discussion also stresses that unbiasedness does not make a forward rate a reliable daily predictor: a changing risk premium may cause forward rates to fluctuate around an underlying value while short-horizon outcomes remain difficult to forecast. A separate answer uses the forward measure associated with a zero-coupon bond maturing at the relevant date, under which a forward rate is expressed as the conditional expectation of the future short rate. The source offers hints rather than a definitive set of conditions, and it gives no detailed derivation or empirical specifications to assess the cited claims.
Key ideas
- The pure expectations hypothesis treats forward rates as unbiased forecasts of future spot rates.
- Term premiums can affect the relationship between forward rates and later spot rates.
- A long-run unbiased relationship does not ensure useful daily prediction.
- Under a maturity-matched forward measure, a forward rate can be represented as an expectation of the future short rate.
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Full text
# Future spot price versus current forward price
# Future spot price versus current forward price
Which are the two conditions necessary to claim that the future spot price will have as many chances to be above or below the current forward price?
## Answer by Simon (score 1, accepted)
https://quant.stackexchange.com/a/12818
I'm not expert on this field so may not able to answer your question precisely, but I can try the best to offer you some hints.
According to the pure expectations hypothesis(PEH), forward rates provide unbiased predictions about future spot rates. Even if the PEH can be rejected, various scholars including Fama has provided evidence for the weaker form of this hypothesis. Stephen has also argued that forward rate adjusted for term-premium can be used as a predictor for future spot rate.
Francisco de Castro and Alfonso Novales has also found a robust cointegration relation between forward and future spot rate in the currency market with a unit coefficient, confirms the unbiasedness of the hypothesis. However, he also finds unbiasedness does not imply that the daily forward price is good predictor for future spot rate, as in fact, the opposite is true. This unpredictability suggests a non-stationary risk premium equal to the forward premium, making the implied forward rates to be constantly fluctuating around the true value. Therefore, if the conintegration relation is true and the conclusion of unpredictability is verified, then there is some support for your claim.
## Answer by wsw (score 0)
https://quant.stackexchange.com/a/19249
Using the $T$-forward measure $Q^T$, where the numeraire is the price of the zero-coupon bond $p(t, T)$ maturing at time $T$, we can see that the forward rate is the expectation of the future short rate $r_T$:
$$ f(t,T) = \mathbb{E}^T \left[ r_T \mid \mathcal{F}_t \right] \, . $$
See chapter 26 of Tomas' book http://www.amazon.com/Arbitrage-Theory-Continuous-Oxford-Finance/dp/019957474X.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.