# Market price of risk of different maturities

T. Bjork Arbitrage Theory in Continuous Time Proposition 23.1 "Assume that the bond market is free of arbitrage. Then there exists a process $$\lambda$$ such that the relation $$\frac{\alpha_T(t)-r(t)}{\sigma_T(t)} = \lambda(t)$$ holds for all $$t$$ and for every choice of maturity time $$T$$"

Is there any empirical evidence of this? Could we actually check with the market data that such relation really holds?

Moreover, could we use this relation to check for absance of arbitrage in the actual bonds market?

• Actually there is empirical evidence of the opposite, i.e. the existence of a Term Premium. But this is not evidence of arbitrage, just that a more complicated risk model than assumed here is needed. And the simpler theory is still useful in many ways. Oct 23, 2020 at 15:37
• This is a one uncertainty/factor version of the no-arbitrage condition. Isn't this essentially saying that the Sharpe ratio is constant irrespective of maturity and not that there is no term-premium?
– fes
Oct 24, 2020 at 7:00

What @noob2 said:

Actually there is empirical evidence of the opposite, i.e. the existence of a Term Premium. But this is not evidence of arbitrage, just that a more complicated risk model than assumed here is needed. And the simpler theory is still useful in many ways

I feel it's helpful to unpack this a little. Let's say you are buying a 10 year Treasury/Bund etc. You know what equivalent 1,2...9 year govvies are yielding, so you can work out what the arb-free 1 year yields for years 1,2...9 will be.

Is this an estimate of what Bills then will yield? Yes, it is... But it may not be an unbiased estimate. Which is the Term Premium (TP) point. Is the 1y rate in 9 year's time a biased or unbiased estimate of the actual/likely/expected 1y rate then?

And it's seems to be a structurally biased over-estimate of actual interest rates then... because if I could get X just holding cash for the next T years, why would I ever accept X locking in for those T years rather than just holding cash? I want (X + TP) to buy the 10 year security in preference to the 12m one, and rolling it.

Measuring TP is notoriously difficult; but it exists, and is generally assumed to be non-negative. That's as good as it gets. But you simply cannot assume from (say) 6 and 7 year, that you know what 1 year interest rates in 6 years time will actually be. You can know the forward price of this (the "6y1y" in the jargon); but you cannot know the term premium paying you to receive interest thus (or in any other equivalent forward), compared to genuine market expectations of future interest rates.

This is one of the central challenges of bond fund management; and there is NO theoretical workaround... As noob2 says, there is no arb here. There is an immeasurable risk premium versus genuine expectations. So the genuine expectations cannot be measured, if you can't measure the TP, which you can't. Stuck with a circular logic, best judgement is the ONLY guide.