A Commerzbank paper proposes a practical way to estimate term premia across interest rate swap markets. The method adjusts conventional yield curves for median error curves, i.e. for recent tendencies of implied future yields to overpredict spot yields. The adjustment produces “neutral curves” or presumed unbiased predictors of future yields. The neutral curves can then be used to back out term premia.

*James, Jessica, Michael Leister and Christoph Rieger (2016), “De-mystifying the term premium”, Commerzbank Interest Rate Strategy, 25 April 2016.
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*There is no free link for the paper at present. For full text please contact Commerzbank AG.*

*This post ties relates to the subject of fundamental value estimation (view summary page here).*

*The below are excerpts from the paper. Headings and cursive text have been added.*

### Understanding the term premium

“It has been observed for many years that interest rates are higher for longer term instruments. Besides expectations regarding future short term rates, this __reflects the additional risk to the lender compared to rolling over short term instruments and, in liquid markets, the higher price sensitivity of longer-dated bonds or swaps__ [*which implies higher mark-to-market risk for a portfolio*]. The difference between genuine short rate expectations and the observable yield is called the term premium.”

“We find that short dated forward rates imply both positive and negative interest rate moves, and that these directional indications have considerable predictive power. Longer dated forwards, however, past one or at the most two years, have little predictive power as they simply almost always imply a rise in rates. This explains that for the entire available historical data set, on average the forward rates have implied positive moves, and on average the actual moves have been negative, as rates have overall fallen over the past two decades. Thus, __as the term premium comes into effect in the longer forwards, it progressively degrades the ability of the forwards to predict__.”

### Estimating the term premium

“While simple in theory, the term premium is not directly observable and more difficult to measure than one might expect.”

“__We have derived a method of calculating the term premium by using the ‘forward errors’ from the swap curve__ i.e., by understanding how moves predicted by forward rates differ from actual realised moves, as an average over time. This allows us to easily discover the term premium for any currency and tenor; previously only USD and EUR have been studied in depth.”

“Forward rates can be interpreted as the market ‘prediction’ of what will occur…if there were no term premium…As it is, the yield curve embodies at least two drivers; a view on the future, and the term premium… __We would like to find the adjustment which makes the curve today an unbiased predictor of the curve in the future__…It…allows us to generate a ‘neutral’ forward curve which lies in the centre of the actual rate evolution…We use this ‘neutral’ forward curve to derive…a term premium.”

“We consider the median of the quantity of the ‘forecast’ move of the yield curve minus the actual move…For each time forward, for a given calculation period, __if we take this median curve away from the forecast curve, we will have the future curve about which 50% of actual rates lie above, and 50% below__.”

“The calculation will be different for every time forward……Below we graph…the median over all data of the predicted move (forward rate – spot rate) minus the actual move (spot rate at maturity – spot rate at inception). __In all tenors and short times forward the predicted move is a little more than the actual move overall. This is very consistent with a term premium effect__. But for longer times forward, we see a very large effect for short tenors (predicted move much larger than actual move) which is not as great for longer tenors.”

“The final calculation for the term premium…consists of the following steps:

__Create a time series of median predicted moves minus actual moves for each swap tenor__. This uses a scrolling window approach with a predefined lookback period, so each item in the series will incorporate an exogenously defined number of data points, like one month or one quarter. The move window is one year. This forward error series is taken to be a set of 1 year forward rates. We want to use it to derive a starting curve.__Use linear interpolation to create fill in the gaps__. This created forward curve will have missing tenors, like 6 years or 11 years. We use simple linear interpolation to fill them in__Back out the discount factors from this set of 1 year forward rates__. Unfortunately we need to make an assumption at either the long end or the short end of the curve to do this, as we have one fewer rate than we need to get a unique solution. We assume that the one year rate is equal to the 1 year forward rate, which gives an approximate value.- Use the discount factors to
__derive the term premium curve__.”

### Learning from the term premium

“Below we graph the 10y term premium for USD, EUR and GBP over the last 12 months. Though highly correlated, we see the EUR-USD divergence in late 2015 as opinions firmed up over the likelihood of a rate hike in the US which would not be matched in the Eurozone. As the likelihood of hikes in any country rapidly receded in 2016, so the term premiums headed south.”

“JPY is always an interesting currency to consider when it comes to all things related to a zero yield environment. The moribund but positive term premium takes a sudden dive in February 2016 as negative rates are introduced, and looks likely to head into negative territory itself.”

### Annex: Other methods for estimating the term premium

“We have certainly derived an empirical data based method of finding the term premium which is consistent with several far more complex models, and is pleasingly intuitive…The more important approaches [*that were used in the past have been*]:

- A vector-autoregression (VAR) approach has been used to forecast interest rates. The idea in this case is that the variation which is not captured by the forecast, on average, is the term premium
- One could also use a macroeconomic model to do the forecast. An example of this is the Rudebusch and Wu 2003 model (1).
- Cochrane and Piazzesi derive a purely empirical forecast model to forecast long term US treasury excess return, with the unexplained part once more being assigned to the term premium.
- Kim and Wright fit a 3-factor affine term structure model to US treasury yields since 1990. A variation of the model also incorporates inflation data. The term premium is defined as the difference between the yield and the expected short term interest rate.
- Adrian, Crump and Moench use a linear regression approach to fit the evolution of the yield curve, using multiple tenors simultaneously, and their term premium is that portion of the curve shifts unexplained by the forward rates. This is an elegant but complex paper.
- Hordahl and Tristani derive a very different model using inflation forecasts and bond yields, maintaining an affine form for curves, but without utilising forward rates as forecasts.
- Survey-based measures. One could in theory survey market participants about their interest rate forecasts and calculate the term premium as difference between the consensus forecast and the market forwards. Yet such surveys are infrequent and may still differ from the market’s genuine rate expectations.”