Primer: What Is The Term Premium?

One of the largest complications associated with using rate expectations as a modelling technique to determine bond yields is the question: what is the term risk premium? This article is an introduction to this fairly important topic.

Note that the term risk premium is different than other potential premia in bond yields, such as that needed to compensate for default risk. In this discussion, I assume that we are talking about government bonds that are free of default risk, which I will assert applies to the central government bonds of countries like Canada, the United Kingdom, United States (assuming the Tea Party quiets down), and not benighted countries like those in the euro area.

As a basic example, at the time of writing, the December 2014 30-Day Federal Funds futures contract is trading with an implied yield of 0.27%. The payoff of this contract is the monthly average of the effective fed funds rate. At present, the 20-day moving average of effective fed funds is 0.08%. Two possible interpretations of the December 2014 contract value are:

1. The weighted average of market participants’ expectations for the fed funds rate in December 2014 implies that there will be a rate hike by the Fed by that time.
2. On average, market participants do not expect a rate hike, but there is a 0.19% (19 basis points) risk premium in the December 2014 contract.

If you are a market participant who has the capacity to enter and hold the position to maturity, the profitability of the position is based on the realised future value relative to the entry level, and so you could ignore the risk premium, and go with the first interpretation: do I agree or disagree with the raw implied rate expectation embedded in the instrument? However, if you are subject to short-term mark-to-market worries, or if you have to post prices, you need to take into account what sort of risk premium needs to be embedded in the price.

I will now give a simple example based on an alternative world where the central bank targets the 1-year zero coupon rate (and meets once a year*), and the current zero coupon yield is 5%. (To interpret the numbers, you can buy the one-year coupon bond now for $0.9524, and it will mature and pay you $1 in one year. The two-year coupon bond costs less, at $0.9053, and pays $1 in two years.)

Term	Zero Coupon Rate	Zero Coupon Price	Modified Duration
1 Year	5.00%	$  0.9524	0.952
2 Year	5.10%	$  0.9053	1.903
Memo Item:	1-year rate, 1-year forward		5.20%

For this example, I assume that “everyone” expects that the one-year rate will be 5% in one year, or at least expectations are symmetric around an unchanged rate. But, I assumed that there is a 0.20% risk premium in the forward rate, which pushes the 2-year zero rate 0.10% above the 5% level that raw expectations would imply. (I have supplied these numbers to underline one point: a gentle upward slope in the bond yield curve implies a much steeper slope in the term premium. In this case, the premium is double the slope, and it would be an even higher multiple of a par coupon slope.)

In equation form:

Observed bond yield = (Average Expected Short-Term Rates Over Bond’s Lifetime) + (Term Premium).

Why would there be this premium? There are two explanations:

1. A buy-and-hold investor has to tie up capital for two years rather than one, and so should demand a yield premium for having to do this. Keynes referred to this as a “liquidity premium”, which is unfortunately somewhat confusing, as “liquidity” can often be thought to apply to things like one-the-run versus off-the-run spreads, which are bonds of nearly equivalent time-to-maturity.
2. As the table notes, the 2-year zero coupon bond has roughly twice the modified duration (interest rate sensitivity) than the 1-year bond. We see in market data that longer duration instruments will have more return volatility than shorter duration instruments. (Their yields are slightly more volatile - when rates are far from zero - but the higher price sensitivity to yield changes trumps this.) Modern Portfolio Theory argues that there should be a return premium for more volatile assets.
3. In a situation like the present (and not the data in the table), there is considerable “one-way” risk to interest rates. If the forward rate is just above zero, risks are skewed towards higher rates, since rates cannot go too far below zero, but there is plenty of room for rates to rise. This creates an option-like asymmetric payoff profile, and and so there should be an option premium embedded in forward rates. For this reason, I think JGB yields are near a floor level at the time of writing, even though I do not expect to see any BoJ rate hikes any time soon.
4. If you believe there are supply-and-demand factors determining long-term bond yields, they show up in the risk premium (since expected policy rates should not be affected).

Comparison To The Equity Risk Premium

There is a very large literature around the equity risk premium. This working paper by Pablo Fernández discusses 3 concepts of the equity risk premium:

1. Required risk premium. This matches the concepts discussed above: what premium do you need to add to the expected return to account for the higher returns volatility. It assumes you have a model for spitting out a premium based on volatility, which I, at least, do not have.
2. Historical risk premium. The historical outperformance of the asset over a base asset (for bonds, this would be Treasury Bills).
3. Expected risk premium. What premium is priced into the market right now?

The historical risk premium is straightforward to calculate. It will be more interesting in a century or so, when we have enough modern bond market data to be useful. Before the 1970’s, investors operated in a Gold Standard mindset, and so their bond pricing rules are not useful for fiat currencies. We now have inflation stability, they operated with an assumption of price level stability. And until there was widespread use of digital computers in the 1980’s, forwards were not examined, and bond market analysis was done in a primitive fashion. This caused all manner of relative value opportunities that were later crushed out of the market.

From the data set we have available, the historical risk premium has been massive. This is not surprising, as we have been in a generational bull market in bonds. This muddies discussions of what should be expected, as I will discuss further below.

Turning to the expected term premium, we see that it is unmeasurable. We do not have access to actual investor expectations for future short rates. We can try the following:

• Use a survey measure of expected rates. This begs the question: do investors actually believe the economists who provide the survey measure? Another severe problem with using surveys is that they are updated sporadically, whereas observed yields move in response to each piece of news. This technique may be workable for a low-frequency (e.g., monthly) model, although some care is needed to align the data timing.
• An investor could plug in their own expectations. However, this is just a restatement of what the investor already thinks, and so it provides no new information.
• A model for the term premium could be created. This is a small growth industry in academic and central bank research (affine term structure modelling). I will cover this subject in later articles.

Unfortunately, we cannot calibrate off of historical data. We are at the end of mega bull market in bonds, and historic realised term premia have been “unreasonably” large. (For example, if we applied the historical term premium for 10-year bond yields to the current yield curve, the implied average “expected” rates would probably be negative.) There are two main plausible explanations**: investors and economists have been systematically incorrect in their rate forecasts for two decades; or else term premia were too high. Since the first possibility is rather impolite, I tend to write that historical term premiums were too high, but the term premium should be more sensible going forward.

Since the payoff on financial instruments is based on observed forward rates, and that I feel that term premia should be small for a variety of reasons, I think the best modelling strategy is to assume that expected term premia are small and stable.  I will cover this point in more depth when I discuss affine term structure models, where we see that the modelled term premia are not small and stable.

* The Swiss central bank targets a 3-month rate, and hence normally sets the target every 3 months. A central bank cannot target a policy interest rate which has a maturity longer than its next regular policy meeting.

**Another explanation is that the economy has been hit with a set of random shocks that all coincidentally led to lower yields. Yeah, right.

(c) Brian Romanchuk 2013