# Modelling limitations and understanding of long term goverment bonds

Been trying to understand the yield curve for a while now. This is what I collected so far,

There is a relation between short rates and long rates that goes via the forward rate, and so by the expectation hypotheses one could infer that the long rate is given via the forward rates+term premium. Is this correct?

Given that this is true, could one think of the long rate being a result of iterating the forwards rates+term premiums, each carrying an expectations part as well as a term premium giving in an iterative manner the expectations part and term premium on the long bond?

The only thing we can model is the expectations part via say HJM while the term premium containing uncertainties such as tradeoff to other investments and inflation is impossible to model?

Also what kind of uncertainties does the forward rate contain? that term premium dosnt?

• Term Premium is not really "impossible to model", but it is generally estimated empirically (with considerable uncertainty) eg newyorkfed.org/research/data_indicators/term_premia.html For some purposes (such as valuing i.r. derivatives) Term Premium is left out (set to zero). Are you interested in derivatives pricing or in real world interest rates ? – noob2 May 30 '20 at 13:04
• @noob2 real world rates, today is my day off, and so it is macro time :) – user123124 May 30 '20 at 13:05
• Yes, framework is right: LT yield = cumulative expected short rates plus TP. The problem is the lack of an available expected path for short that excludes TP, from which you could then derive TP. Split a 10 year into a (known) 5y and a (derived) 5y5y, the latter is the expected 5y yield in 5 year's time plus TP. So you have to be willing to make assumptions about the former to try to model the split. The model only being as good as its assumptions, predicated on something other than market prices! (Or some equivalent assumption about the term structure consistency of term premia). Not trivial! – demully May 31 '20 at 15:58
• @demully thanks, and the forward rate is by defintionen the expectation $E_{t}$ on page 4 here ?riksbank.se/globalassets/media/rapporter/staff-memo/svenska/… – user123124 Jun 1 '20 at 18:31
• Yes, exactly. Riksbank are saying that the hypothesis that forward rates (as derived from the yield curve, to prevent arb) reflect JUST expectations (about growth, inflation, policy rates) is weak. There is “something else”, ie what we just term TP. Measuring either individually, both being relevant to asset allocation, is far more difficult than it sounds. – demully Jun 1 '20 at 21:56

OK, your framework on this is right. Long-term yields embed "expectations" about future short rates. That is: what do I think I'll get if I sat for T years with cash in the bank for swaps, or the same rolling Govvie Bills with respect to Bonds?

Plus a cherry-on-top called "term premium". One can rationalise the existence of this in lots of intuitive ways, consistent for pretty much any preferred way of thinking about markets. Economists will tell you it's a risk premium for future inflation uncertainty. Others will tell you it's a simple time-value-of-money liquidity preference. And risk people will observe that if a 5 year bond and a 10 year bond just represent cumulative short-rates for the next 5 and 10 years respectively, then the convexity produces very different risk/volatility profiles for the same returns. I'm getting the same returns for twice the vol in years 1-5; and I have no incentive to lock in uncertainty about years 6-10 today (because I could just buy and roll 5s, 2s, 3m...) So I need a risk premium to incentivise me to go out and play along the curve - rather than just sit and roll cash/bills.

This term premium (TP) is one of the most important numbers in finance. But sadly, the great numbers can't be measured. I'm thinking of the "equity risk premium" here. TP is directly analogous: it's a bond risk premium (versus cash/bills).

So the problem arises when I start measuring forward rates, which are simply arbitrage-free derivatives of the yield curve. Give me a complete yield curve, and I'll quote you a 2y3m, 5y3m and 10y3m forward rate, ie 3m rates in 2/5/10 years time. But because the 2/5/10 year bonds contain Expectations plus TP, so too will the forwards.

As a bond investor, there are two things I want to know. What are the market's expectations for growth, inflation, policy rates? And what carry do I receive for the convexity risk of a longer-dated bond? The forward rate doesn't tell me either of these things directly; but both combined, in a fashion where there is no definitive way to separate them out. That's the catch.

So, yes, one then has to start making (untestable ex-ante) assumptions about either expectations or term premia to pin one of them down, to allow one to then try to measure the other. Cue the inevitable slew of cliches about models and assumptions ;-)

Models like HJM allow you to compute say a 5y5y forward in such a way that its behaviour will be consistent with the behaviour of the surrounding 5y4y, 5y6y, 4y6y, 6y4y etc forwards. Which is powerful stuff, and very useful in preventing arbitrage in the forwards (on top of the forwards preventing arbitrage between the underlying 9 and 11 year bonds etc.).

But they cannot solve the killer problem - the disaggregation of expectations and term premium. If I have a 5y rate of say 2% and a 10y rate of 2.25%, then my 5y5y forward rate for years 6-10 is 2.5% (and loose change). But I don't know how much of this marginal 50bps is the market telling me about growth, inflation and policy rates then; and how much is carry for convexity. And this matters.

Consider two scenarios, extreme by design: - Unseen expectations are for no growth and inflation then, with central banks running negative policy rates to lean against the liquidity trap, and QE morphing into Japan-style Yield Curve Control. Call it 0bp nominal expectations and 250bps TP. I think I'm a buyer.

• Unseen expectations are for rampant inflation, as fiscal authorities man the pumps to squeeze out weak growth, running deficits that take 100% debt:GDP towards Japan-style 200%. Call it 1% growth plus 5% inflation = 600bps nominal plus -350bps term premium = 250bps 5y5y. No thanks, do you know a good precious metals broker ;-)

Both of the above are obviously over-the-top, for effect. But hopefully the point is clear enough. There is a Nobel Prize almost guaranteed to whoever can solve this problem.

But if the parallel in the equity market holds, there are reasons there to believe that the problem is insoluble. There's another Nobel Prize out there for whoever can definitively solve and measure the "equity risk premium". Indeed: even more ink has been spilt on this one than on term premia. The problem there is that there is no such thing as "a risk premium" in the first place. That is to say there are four of them - that are related, but different; and mean different things. Anyone's model might be very good at measuring one of these flavours without error. But the danger then becomes it becomes very misleading applying this flavour to the other flavours.

Imagine you could come up with a flawless model for TP/expectations, that gave you very good forecasts for both. What exactly is that telling us?

Is it 1) a historical realised premium? IE policy rates were 2.3% in aggregate versus 2.8% in yields equals 50bps TP received. The same as stocks did 10% and bonds 5% so ERP was 5%. This is measurable; but only after the event.

2) a historic ex-ante premium? Investors have historically needed an incremental Xbp TP on 5s and Y on 10s to justify taking the incremental duration risk. This is obviously more useful; but does this figure still apply today?

3) a forward ex-ante required premium? How much do investors today think they need or should expect to receive to justify switching from 2s into 5s, or 5s into 10s (from bonds into stocks)? What is "fair compensation" for risk?

4) a forward ex-ante realised premium? How much will I actually get buying today, irrespective of history, current valuations, fair value, and economic or portfolio theory?

So this problem is like the hydra - lop one head off, and more emerge. See URL below for a more thorough exposition. It's a credit investor talking about stocks; but the parallel exactly holds for credit and duration risk.

https://www.oaktreecapital.com/docs/default-source/memos/2013-03-13-the-outlook-for-equities.pdf?sfvrsn=2

sorry to be the bearer of bad news ;-(

• Thanks for clearing things out, I wasnt able to find a reference that did this, I suppose all the references I found on this problem is on such an advanced level that they assume that you know this "basic" stuff. – user123124 Jun 3 '20 at 6:21
• are you implying that the finance profession (like law and medicine) mystifies with jargon, that is never plainly explained??? For shame!!! ;-) – demully Jun 3 '20 at 8:04
• maybe even like priests :O, since from my experience there is no constant(over time) objective truth in economics. This however does not make the problems of economics less important or interseting merely adding a dimension of complexity. – user123124 Jun 4 '20 at 4:56