# Term premium 10 year yields

There's a lot of headlines about term premium. If I understand this correctly, term premium compensates investors for holding a longer maturity bond vs rolling over short-term bonds.

So as an example, if a 1 year bond is 2% and let's say it stays at 2% for 10 years, I can continue to roll this over each year for 10 years. The 10 year yield would be the average of the 1-year yields so it would be 2%.

However and correct me if I'm wrong but the market's 10-year yield would be closer to 2.5% because of 50 basis point of term premium (I just made this number up)

However, term premium is negative in this market due to the effects of QE, demographics, demand for fixed income which would pull the 10-year yield to let's say 2.25%.

Is this the correct way to look at it? Is there term premium for 2-year, 5-year?

It's a topic of intense interest to me, so it'll likely be a bit more than you asked for =)

# Decomposing the yield curve

Simply put, a default-free interest rate can be decomposed as follows:

$$\text{default-risk bond yield} = \text{rate expectations} + \text{bond risk premium} + \text{convexity bias}$$

I provided some comments in this post and will omit the details here. But you're pretty much correct – if market expectation is that short-term interest rate will be 2% over the next ten years, and ex-ante average term premium is 50 bp, then the ex-ante 10-year spot yield (ignoring convexity) should be roughly 2.5%.

Term premium, aka bond risk premium (BRP) or maturity risk premium, compensates investors for taking on duration risk. The return of longer-term bonds prior to maturity is not known in advance – you'll sell them with capital gains or losses, depending on how yields have moved. Accordingly, risk-averse investors demand a premium as compensation for this uncertainty.

Based on the decomposition above, if market expectation of future interest rates is completely flat and convexity bias is ignored, the yield curve should still be upward sloping (theoretically speaking), assuming term premium is positive.

# How do we know term premium is nonzero?

There are many ways to see this empirically. The first chart below plots the average yield curves since 1945 till the end of 2016. I've chosen this sample because yield levels are roughly the same at the start and end of the sample, so it's less biased. You can see that on average, the yield curve has been upward sloping. It is difficult to imagine that market participants consistently expected interest rates to rise over such a long period that covers both a secular bear market and a secular bull market for bonds. A more plausible explanation is that there are risk premia priced into bonds.

Alternatively, we can look at the realized returns of bonds. The chart below shows the returns of US Treasuries across maturity buckets. If there's no risk premium, then all bonds should have the same returns. Instead, duration extension (moving out the curve) generally provides higher returns. (This is not true for the longest duration bonds, but we'd have to go into convexity bias and other technical factors associated with super long bonds, which detracts from the current discussion).

Notably, the conclusion remains true on a risk-adjusted basis; that is, intermediate bonds have higher Sharpe ratios than short-term securities.

Other empirical studies of similar nature include looking at ex-ante yield levels and subsequently realized average short-term interest rates. As an example, the post referenced above provides an example of comparing 5-year forward short rate against subsequently realized short rate. You can also look at 5-year spot yields relative to average bill rates over the next five years. They all paint the same picture.

Academic literature documenting bond risk premium is vast. I recommend the work of Antii Ilmanen and John Cochrane as a starting point.

# How do we estimate term premium on an ex-ante basis?

The charts above are based on realized yield curves and realized returns of bonds. The consensus now is that not only does term premium exist, it's time-varying. The holy grail of bond investing, of course, is to ascertain term premium on an ex-ante basis (which is equivalent to forecasting the excess return of bonds). This requires us to figure out the exact amount of "pure" market expectation priced into the curve.

This is incredibly difficult to do! The most popular models nowadays include those pioneered by Kim-Wright and ACM. They are usually term structure models that compute yields in the risk neutral measure and the physical measure, with the difference attributed to risk premium.

Because the official Kim-Wright estimates have a shorter history, I include my own extension below for a longer-term perspective:

# Is term premium really negative?

Both the Kim-Wright and ACM estimates point to negative term premium at the time of this writing. Extensive literature has attributed these to LSAP conducted around the world along with some secular forces.

Because this is more opinionated, I will not delve into the details, other than to voice that I do not personally believe that term premium was negative in the past few years. Not only would one have missed out on a lot of excess returns over the past few years, I think there are dangerous policy implications stemming from the belief of negative bond risk premium. As said, this is opinionated and beyond the scope of quantitative finance, so I'll stop here...

# What drives bond risk premium?

Lots of things, ranging from inflation risk premium / inflation uncertainties (historically a huge drivers, but less so today), real growth uncertainties, safe haven status of bonds, supply/demand dynamics, etc. A short discussion really does not do this topic justice. I recommend chapter 9 of Antii Ilmanen's Expected Returns.

# Does term premium exist at the front end of the yield curve?

Yes, although it's negligible nowadays for understandable reasons. If you go back enough years, the first step in understanding Fed funds futures pricing was to subtract out some estimates of term premium.

The positive risk premium in the front end of the yield curve is also known as "forward/spot premium" or "forward rate bias" and is extensively studied by systematic investors.

# Does term premium matter for us?

It depends on how you trade. If you're buying and holding to maturity, it doesn't matter that much – you'll get the total yield; whether you get it as rate expectations or as risk premium is not that important. However, if your trades are really about harvesting excess returns (as opposed to total return), then it's essential. (As said, I consider forecasting bond risk premium to be the holy grail in bond trading.) Outside of trading, term premium is also important in conducting and interpreting monetary policies.

• Thanks this is very helpful. This leads to a follow up question. How do Wall street strategist forecast yields? For example, do they forecast the path of short-term rates (Fed Funds) and then apply a term premium to get forecasts for 2-year, 5-year, 10-year yields across the curve? – VanillaCall Mar 11 '18 at 14:14
• That's certainly a good way; classical strats do this a lot (myself included). Others use econometric techniques. More often than not, it's gut feeling... – Helin Mar 11 '18 at 21:03
• Is there an easy way to do this on Bloomberg? Getting the 10y yield assuming it follows the market path of Fed funds....just curious – VanillaCall Mar 13 '18 at 23:49
• @VanillaCall Not 100% sure what you mean. The observed bond yield IS the market's discounting of forward short-term interest rates (Fed funds + some sprd). If you are trying to construct your subjective forwards, I don't think you can do that in BBG, but a simple spreadsheet would do the work. – Helin Mar 14 '18 at 4:36
• +1. this is nice addition to SE.QF – Attack68 Aug 18 '18 at 15:44

Instead of "making numbers up" why not examine the term premia for 1 to 10 years estimated by ACM at the Federal Reserve Bank of NY (link). The latest estimate (for March 8, 2018) of ACMTP10 (the 10 year term premium) was -31.088 basis points.

And don't listen to claims that "there is no term premium" from 15 years ago. That theory (the "Pure Expectations Hypothesis") has been decisively rejected by economists since then. Financial Economics has moved on.

The curve is more complex than saying it is shaped by 'term premium'. The number of different studies and attempts to quantify what term premium is demonstrate the difficulty in truly characterising it and quantifying it.

There are a certain number of other factors that affect term structure of interest rate curves (bonds or swaps):

1) The future expectations of central bank deposit rates (as driven by subsequent inflation expectations). At the short end this is the dominant mechanism of determining interest rates through the transmission mechanism. The further forward you forecast the less certain this effect becomes.

2) Volatility and convexity. Higher volatility usually warrants higher reward, and longer bonds are typically more volatile so command higher yields. Convexity is a positive attribute though and compounded by volatility so this can negate some of those higher yields. (See "Volatility and the Yield Curve" - by Litterman and Weiss)

3) Asset allocation cycles - considering interest rates in isolation is unwise since all financial markets are intimately linked. Personally I suspect that there may well be nonlinear interconnecting forces that no one has yet realised (or at least published).

4) Specific events such as credit worthiness. As an example in 2008/9 UK government long bond yields steepened significantly, at the same time as short bond yields fell. The reason being that the financial crisis had two effects: a) the CB depo rate fell considerably dragging down short yields, and b) the financial system is a major contributor to London's and the UK's economy, with a considerable financial crisis the threat to UK's debt position was enough to cause concerns about the viability of longer maturity bonds.

Anecdotally, my first boss some 15 years ago told me he didn't believe in the term premium and I shouldn't waste much time on it. Rightly or wrongly I followed his advice and it hasn't done me any harm!

• In regards to 3) Asset allocation cycles: can you provide an example (even if contrived or minutiae) of what you mean by nonlinear forces yet to be realized? With your experience, I bet you may have something in mind. – Jared Aug 19 '18 at 23:33
• 1) A good example is regulation changes. The Leverage Ratio in recent years considerably changed the way banks could offer repo to clients at different points of the year essentially impacting on going supply/demand for short rates. This vastly depressed short dated yields and gave rise to increased volatility, in turn this affects the steepness the curve (e.g. German bond yield curve Sep-Dec 2016) – Attack68 Aug 20 '18 at 8:58
• 2) Collateral is hugely important to fin system. When unfunded positions became a huge problem for banks in 07/08 it caused great demand for USD collateral, which drove the xccy basis markets a long way. This allows corporates to issue bonds in other currencies or speculators to do cross-currency asset trades which drive curves in unusual directions dependent on how the xccy basis market has evolved. – Attack68 Aug 20 '18 at 9:02
• 3) Long only bond funds which have some element of active investment can be tweaked to express the fund managers view. If he want more exposure he will shift cash from shorter bonds to longer bonds (this was a common ploy for lots of fixied income funds in the QE regimes of many currencies). Of course this affects curve shape. – Attack68 Aug 20 '18 at 9:03
• These are just some of the more simple observances that have played out in my time but my core belief is that markets are more interconnected that people realise and minor tweaks here and there (regulations or economic conditions or political events) can force changes that impact other markets. And one has to very astute to predict or even observe it (and I am not saying I am - I just I believe that one has to be!) @Jared – Attack68 Aug 20 '18 at 9:07

Term premium is also lower than that in previous cycles due to the Fed's inflated balance sheet. Last I checked, the Fed has ~\$4.0 trillion of Treasury and MBS securities that it has accumulated on its balance sheet. They're running down their balance sheet slowly and still reinvesting at the auctions, however most people think the size of their normalized balance sheet is still larger than pre-crisis due to growth in currency in circulation and other liabilities. Hence, the Fed will also be a source of demand of Treasuries in the future.