*Will Dison and David Elliott.*

Financial market prices provide information about market participants’ Bank Rate expectations. But central expectations can be measured in different ways. Mean expectations, derived from forward interest rates, represent the average of the range of possible outcomes, weighted by their perceived probabilities. On the other hand, modal expectations, which can be estimated from interest rate options, represent the perceived single most likely outcome. Currently, these market-implied mean and modal expectations for the path of Bank Rate over the coming few years differ starkly, with the mode lying well below the mean. In this post we argue that this divergence primarily reflects the proximity of the effective lower bound to nominal interest rates.

**Chart 1** shows recent estimates of market-implied mean and modal expectations for the path of Bank Rate over the next two years. Mean expectations are for Bank Rate to rise gradually to around 1.1% in two years’ time. On the other hand, modal expectations, estimated from interest rate options, are for Bank Rate still to be close to 0.5% at that horizon.

**Chart 1. Market-implied mean and modal expectations for Bank Rate(a)(b)**

###### (a) The blue line is the forward 3-month OIS rate. The pink line is the mode of the option-implied distribution for 3-month Libor, less the spread between the forward 3-month Libor rate and the forward 3-month OIS rate.

###### (b) Estimated from data as of 17 November 2015.

These two paths are strikingly different. The mean path suggests that markets expect Bank Rate increases to be slow, potentially reflecting persistent headwinds to economic growth. But while mean expectations are for gradual increases in Bank Rate over this horizon, estimated modal expectations see no rate rises at all, with Bank Rate remaining at its current level for some time to come.

The modal expectations in this chart are estimated with some uncertainty, so you should not place too much weight on the precise numbers. (In particular, the kinks in the estimated modal path are likely to reflect estimation uncertainty.) Furthermore, the prices of the financial market instruments on which all these estimates are based will contain risk premia, which compensate investors for uncertainty about future interest rate outcomes. The presence of risk premia means that the estimates of both the mean and modal expectations will differ from market participants’ actual expectations for Bank Rate.

But the large disparity between these market-implied mean and modal paths, and between the different economic outlooks these could imply, makes it important to understand this divergence. On the one hand, the low level of modal expectations could be consistent with markets pricing in expectations of ‘secular stagnation’, in which long-term trends lead to a persistent shortfall in demand. An alternative hypothesis, which we argue for in this post, is that the low level of the modal path is driven by the proximity of the effective lower bound to nominal interest rates.

**How can we estimate market-implied mean and modal paths for Bank Rate?**

Forward interest rates – that is, the rates at which investors can agree today to borrow or lend over some period beginning in the future – provide a measure of market participants’ expectations of future interest rates. The Bank estimates forward rates based on derivative contracts called overnight index swaps (OIS). OIS are contracts involving payments based on the average overnight interest rate that prevails over their lifetime. For sterling contracts, the relevant overnight interest rate is the sterling overnight index average (SONIA). Since SONIA is typically close to Bank Rate, forward OIS rates are a common measure of the expected future path of Bank Rate.

But as the future path of Bank Rate is uncertain, there is a probability distribution for the level of Bank Rate at each future date. Forward OIS rates represent the mean of this distribution. And options prices can be used to produce an estimate of the rest of the distribution that is consistent with market prices.

Options on OIS are not widely traded. But the Bank estimates market-implied distributions for the future level of Libor (the London interbank offered rate) using options on Libor futures. Libor futures are based on the interest rates at which banks judge they can borrow unsecured from other banks over a period of three months, so will typically include more compensation for credit risk than SONIA (which is an overnight rate). In order to estimate a distribution for the future level of Bank Rate, we shift the option-implied Libor distribution down by the forward Libor-OIS spread, to remove the credit risk component. This gives an estimate of the market-implied distribution for the future level of Bank Rate, as shown in **Chart 2**.

**Chart 2. Estimated market-implied distribution for Bank Rate in 1 year’s time(a)(b)**

###### (a) Option-implied distribution for 3-month Libor in 1 year’s time, less the 3-month Libor-OIS spread, 1 year forward. The blue line is the forward 3-month OIS rate. The pink line is the mode of the option-implied distribution for 3-month Libor, less the spread between the forward 3-month Libor rate and the forward 3-month OIS rate.

###### (b) Estimated from data as of 17 November 2015.

This estimated distribution provides us with richer information about market participants’ expectations for Bank Rate than looking at the forward OIS curve alone. In particular, it allows us to see how the mode (the single most likely outcome) differs from the mean (the probability-weighted average outcome). The blue line in **Chart 1** (the forward OIS curve) plots the means of successive distributions like that in **Chart 2** over time, while the pink line plots the modes.

**What is driving the gap between mean and modal expectations?**

Before the financial crisis, interest rates were well above the effective lower bound, and option-implied distributions for future interest rates were broadly symmetric. That is, relative to central expectations, the risks of higher or lower rates were seen as broadly in balance. Over this period, mean and modal expectations for Bank Rate were close (**Chart 3**).

**Chart 3. Gap between estimated mean and modal expectations for Bank Rate in 2 years’ time(a)**

###### (a) 3-month Libor rate, 2 years forward, less the mode of the option-implied distribution for 3-month Libor in 2 years’ time.

But, since the crisis, the low level of short-term interest rates and the proximity of the effective lower bound have induced a positive skew in option-implied interest rate distributions, as can be seen from the shape of the distribution in **Chart 2**. As there is limited scope for large cuts in Bank Rate, the distribution is effectively truncated on the downside. On the other hand, the distribution is unbounded on the upside, reflecting tail risks of significantly higher rates. This positive skew in the distribution is reflected in the mean lying above the mode. This phenomenon is not unique to the UK either – for much of the period since the crisis the mean has also been above the mode in estimated interest rate distributions for the US.

**Which measure of central tendency should you focus on, the mean or the mode?**

At first sight it might appear most natural to focus on the mode as the preferred measure of central tendency, as this represents the most likely outcome. Indeed, Michael Bauer at the Federal Reserve Bank of San Francisco has argued that “to avoid distortion from the asymmetrical distribution caused by the zero lower bound, it is preferable […] to consider the mode of the future short rate, because it reflects the most likely path of policy”. And given that the MPC’s fan charts for growth and inflation are centred around the mode, it might appear consistent to condition these forecasts on a modal path for Bank Rate.

But there are good arguments for focussing on the mean instead. The mean is directly observable from the forward OIS curve, while the mode can only be estimated using a model, so there is more uncertainty around the estimate of the mode. And when the dispersion of the distribution is high – meaning that the distribution is relatively flat – the uncertainty around the estimate of the mode will be particularly high. In these circumstances small changes in the shape of the distribution can potentially lead to large movements in the mode. In part because of the greater difficulty in estimating the mode, market participants tend to focus primarily on forward OIS rates (the mean of the distribution) as the most commonly used measure of Bank Rate expectations.

Even if it was possible to estimate modal expectations with great certainty, there are still good reasons for focusing on the mean. When using a measure of central tendency as a forecast, the best choice of measure will depend on your attitude to forecast errors. If all that matters is whether the forecast is right or wrong, and all forecast errors are weighted equally, then the mode is the best measure. A gambler placing a bet on a horse to win outright would be in this situation: picking the correct winner is all that matters, while betting on the horse that finishes second is no better than betting on the horse that finishes last. But if large forecast errors matter more than small ones, and if very large errors matter a lot more, then focusing on the mean is preferred.

**Conclusion**

The divergence between market-implied mean and modal expectations for the future level of Bank Rate is stark. While the low level of modal expectations, as estimated from interest rate options, could reflect pessimism about the economic outlook, it is more likely that the mean-mode gap primarily reflects the proximity of the effective lower bound to nominal interest rates. And while it is important to take into account the whole distribution of Bank Rate expectations, if a single measure of central tendency is needed we believe it is usually most appropriate to focus on the mean, as this is estimated with more certainty and incorporates more information from the rest of the distribution.

**Will Dison works in the Bank’s Macro Financial Analysis Division and David Elliott works in the Bank’s Sterling Markets Division.**

**If you want to get in touch, please email us at bankunderground@bankofengland.co.uk**

**Bank Underground is a blog for Bank of England staff to share views that challenge – or support – prevailing policy orthodoxies. The views expressed here are those of the authors, and are not necessarily those of the Bank of England, or its policy committees.**