Sunday, December 17, 2023

Bond risk premiums -- certainty found and lost again

This is a second post from a set of comments I gave at the NBER Asset Pricing conference in early November at Stanford.  Conference agenda hereMy full slides here. First post here, on new-Keynesian models

I  commented on "Downward Nominal Rigidities and Bond Premia" by Fran├žois Gourio  and Phuong Ngo. The paper was about bond premiums. Commenting made me realize that I thought I understood the issue, and now I realize I don't at all. Understanding term premiums still seems a fruitful area of research after all these years.  

I thought I understood risk premiums

The term premium question is, do you earn more money on average holding long term bonds or short-term bonds? Related, is the yield curve on average upward or downward sloping? Should an investor hold long or short term bonds? 

1. In the beginning there was the mean variance frontier and the CAPM. 

Long term bonds have an almost stock-like standard deviation (around 10%, 16% for stocks) with a mean return barely above that of cash or short term bonds. They look like yucky investments. 

(They're not, or not just based on this observation. Bonds are around 40% of the market. Good final exam question: Given the above picture, should a mean-variance investor get out of bonds? Is the market price and quantity irrational? Hint: Individual stocks are also inside the frontier.) 

More precisely, short-term bonds or the "risk free rate" are the best investment for risk-averse investors. Long term bonds are at best part of the risky portfolio. Less risk averse investors hold some of them for slightly better return and diversification. 

That leads to the standard presupposition that long-term bonds have higher returns, and the yield curve slopes up, to compensate for their extra risk. That isn't quite right -- average return depends on betas. Long term bonds have higher returns, if their extra risk covaries with stock risk. They could be "negative beta" securities, but that is unlikely. Higher interest rates lower stock prices too.  
2. Campbell and Viceira neatly reversed this conclusion in a beautiful AER paper. What is the risk-free investment for a long-horizon investor, one who wants a steady stream of consumption? Answer: an indexed perpetuity. A bond that pays (say) $100 per year, indexed for inflation, forever. This is obvious once you look at the payoffs, but not at all obvious from standard portfolio theory. There long-term bonds look like volatile investments whose returns miraculously correlated with innovations to state variables for investment opportunities. In English, long term bonds can have big mark to market losses. But when price goes down yield goes up, you make it back in the long run. Yes, they are risk free for long term investors. Portfolios for long-term investors is a long riff on this theme. 

Now, your presupposition is that long term bonds should have the lowest yields, being safest, and short-term bonds should have a higher mean return to compensate for extra risk. 

But we're talking about nominal bonds, not indexed bonds. The risk-free proposition holds if real interest rates vary, but inflation does not.In that case, short bonds have roll-over risk for long term investors, and long bonds have steady payouts. If inflation varies but real rates are constant, then short-term bonds have less risk for long term investors. 

That suggests an interesting view: Until 1980, inflation was pretty variable, and we should see upward sloping term structure and risk premium. After 1980, or at least after 1990, inflation was stable and real interest rates varied. The risk premium should turn around. 

3. That too is simplistic, because of course I'm looking again at variance not beta. Now, inflation reliably falls in recessions (see graph). Interest rates also fall in recessions, so bond prices rise. That means bonds are great negative-beta investments. Bonds overall should have very low returns. And this pattern has become much stronger since the 1980s, so bond returns should have gone down. 

They did. In all the arguments about "savings glut," "low r*" and so on, I never see this basic mechanism mentioned. Bonds are great negative-beta securities to hold in a recession or financial crisis. 

And, that holds especially for government bonds. Look at 2008, and remember that prices move inversely to yields.  Holding 10 year government bonds would have been much better than holding BAA bonds! That saving grace in a severe financial crisis, when the marginal utility of cash was high, might well account for some of the otherwise much higher yield of BAA bonds. 

But today we're looking at the term premium, long bonds vs short bonds, not the overall value of bonds. Now, short bond yields go down a lot more than long term yields. But price is 1/(1+y)^10, and the short bonds mature and roll over. It's not obvious from the graph which of long or short bonds has a better return after inflation going through the financial crisis. But that is easy enough to settle. 

But I didn't 

Reading Gourio and Ngo made me realize this cozy view was a bit lazy.  I was looking at covariance of return with one-period marginal utility, forgetting the whole long-horizon investor business that brought me here in the first place. The main lesson of Campbell and Vieira's work is that it is nuts to do one-period mean and alpha vs beta analysis of bond returns. More precisely, if you do that you must include "state variables for investment opportunities." When bond prices go down bond yields go up. You will make it all back. That matters. 

Yet here I was thinking about one-period bond returns and how they covary with instantaneous marginal utility. What matters for the long-horizon investor is how a bad outcome covaries with remaining lifetime consumption, remaining lifetime utility. Returns that fall in a recession shouldn't matter much at all if we know the recession will end. 

There is, of course, one special case in which consumption today is a sufficient statistic for lifetime utility -- the time-separable power utility case. To use that, though, you really have to look at nondurable consumption, not other measures of stress. And, of course, I'm assuming that long-term investors drive the market. 

Normally we do not impose the consumption-based model. So it remains true, if you are thinking about expected returns in terms of betas on various factors, it is absolutely nuts not to think about long term bonds with factors such as yields that are state variables for future investment opportunities. 

Gouio and Ngo use a consumption-based model, but with Epstein Zin utility. (Grumble grumble, habits are better for capturing time-varying risk premeia.) The power utility proposition that today's consumption is a sufficient statistic for information about the future also falls apart with Epstein Zin utility.  A lot of the point of Epstein Zin based asset pricing is that expected returns line up with consumption betas, but also and often predominantly with betas on information variables that indicate future consumption. 

Here, my comment is not critical, but just interpretive. If we want to understand how their or any model of the bond risk premium works, we cannot think as I did above simply in terms of returns and current consumption. We have to think in terms of returns and information variables about future consumption, a set of state-variable betas. Or, following back to Campbell and Viceira's beautiful insight, we should think about returns as increases in the whole stream of consumption. We should think about portfolio theory in terms of streams of payoffs and streams of consumption, not one-period correlations and state variables. 

What's the answer? Why do Gourio and Ngo find a shifting term premium? Well, I finally know the question, but not really the intuition of the answer. 

You can see how my attempt to find intuition for bond premiums follows advances in theory, from mean-variance portfolios and CAPM, to ICAPM with time-varying investment opportunities, which bonds have in spades, to a long-term payoff view of asset pricing, to time-varying multi factor models, to the consequences of Epstein Zin utility. 

But contemporary finance is now exploring a wild new west: "institutional finance" in which leveraged intermediaries are the crucial agents and the rest of us pretty passive; segmented markets, safe asset "shortages" "noise traders" and pure supply and demand curves for individual securities, neither connected across assets by familiar portfolio maximization nor connected over time by standard market efficiency arguments. With this model of markets in mind, obviously, who should (or can!) buy long term bonds, and how we understand term premiums, will be vastly different. 

So, I go from a very settled view with just a little clarification needed -- long vs short term bond recession betas -- to seeing that the basic story of term premiums really is still out there waiting to be found. 


  1. "Good final exam question: Given the above picture, should a mean-variance investor get out of bonds? Is the market price and quantity irrational? Hint: Individual stocks are also inside the frontier.)" An investor using Sharpe Ratio's: 6 month treasury 5.41/10 = .541. 10 Yr treasury 3.933/10 = .3933. For SPY, expected return - long term rate/ standard deviation is (10.757- 3.933)/ 16 = .4245. The best trade is the 6 month treasury until the inverted yield curve reverses. Goes from backwardation.
    to contango. A portfolio of short term paper and the SPY Index could be constructed per the investor's risk aversion coefficient. If the market is efficient, I don't think it's irrational.

  2. I'm quite happy to say that it is most gratifying to know that John Cochrane and I think alike on how to invest long term. I followed his thoughts here when retiring 12 years ago, and despite all of the many shifts in markets have managed to have a growing nest egg which is even larger than when I retired, and an income that has stood the test of time. Vic H.

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  4. What level of economics, or amount of schooling would I need to understand what you write. I've been following for about 6 mos, hoping some it would rub off but it is so far outside my level of knowlegde. I find it very interesting that there is this much academia surrounding something that to be is very basic -bonds, bond yields, etc. But this is just out there. Wish I could understand it. Can you reccomned a starting point, a basic textbook or something that I can begin on?

    1. Most of my posts are for general readers. Some like this one are aimed at research academics, graduate students, or finance professionals. For them, if I stop to define marginal utility of consumption, they fall asleep. For normal mortals, if I don't, you fall asleep. Such is life. I'd say buy my textbook "Asset Pricing," but that also requires some background.

    2. Asset Pricing is for grad students, that's a lot of required background. The biggest barrier is really that even the background is essentially at graduate level. I would say "basic" dynamic macro is what you need but we're talking about the sort that talks about consumption Euler equations, so undergraduate macro won't really cut it. Maybe advanced undergraduate...not sure.

    3. Appreciate the feedback, I'm thinking now I'm even more in over
      my head than I originally thought, but I have been always fairly adept mathematically so that part isn't quite as difficult as the terminology and concepts. Interesting though as a biologist (retired), population dynamics has some similarities in the math.

    4. A basic advanced undergraduate level textbook, such as Williamson, Stephen D., Macroeconomics, 6th Edition, New York: Pearson Education, 2016, [Identifiers: LCCN 2016042605; ISBN-13: 978-0-13-447211-9; ISBN-10: 0-13-447211-X] will provide a basic grounding in the subject of most, if not all, of John's blogspot articles in macroeconomics.

      For narrow topics, such as new-Keynesian dynamic stochastic general equilibrium models that pepper John's blogspot articles on central banking (interest rates, fiscal theory of the price level, etc.), see, for example, Costa, Celso Jose, Jr., Understanding DSGE models--Theory and Applications, 2016, Wilmington, DE: Vernon Press [Identifiers: LCCN 2015952652; ISBN-13: 978-1-62273-133-6] provides clear explanations of the basis of the DSGE models and the development of linearized approximations of the non-linear governing equations that solve the economic planner's optimization problem. Worked examples included in the text give step-by-step guidance to the principal macroeconomic models covered in contemporary theory at the basic first-year graduate course level.

      For a text on advanced finance econometrics, a classic textbook that may be recommended is Campbell, J. Y., A. W. Lo, and A. C. MacKinlay, The Econometrics of Financial Markets, 1997, Princeton, NJ: Princeton Univ. Press [ISBN-10: 0-691-04301-9] is a comprehensive treatment of the subject of econometrics as applied to financial securities, asset returns, asset pricing models, option pricing, market microstructure, bond yield curve and bond yield curve premia (the subject of this blogspot article) and more.

      No single textbook or collection of textbooks will do justice to the depth and breadth of the discipline. For the layman, such as this writer, the best that can be hoped for is a basic understanding of the mechanics and an appreciation for the difficulties practitioners face in coming to grips with a social science in which the object of examination is non-stationary, highly intelligent, strongly motivated, and infinitely adaptable to constraints and incentives—i.e., human beings.

      The textbooks named above are but a sampling of the range of reference books and articles that could provide useful background information to the blogspot articles published here. John has named his own publication, Asset Pricing, but in addition to that reference book, his website contains his articles and teaching materials which are accessible, and, for the most part, open-access. Exploration of his website and other websites featured on this website revels an unusually rich collection of information for those who are so inclined.

    5. Eduard Challe, Macroeconomic Fluctuations and Policies is also a good elementary intro. For my taste Asset Pricing itself and Campbell's new book are better than CLM. Asset Pricing doesn't quite cover the econometrics, as stated above you're supposed to already know what GMM and MLE are, but the explanations of how it applies and what you're actually doing are superior.

    6. Also, if you are comfortable with the math around population dynamics then maybe Euler equations are no bid deal. Sargent's 1987 book Dynamac Macro Theory is not for the faint of heart but chapters 1 and 3 basically contain most of what you need to get started on Asset Pricing, along with a basic econometrics book.

  5. It always struck me that the basic consumption stochastic discount rate model looks like Ho Lee. Now yield curve modelers learned early on, that it doesn't do a good job of explaining YC dynamics because long rates are pinned and the volatility term structure is downward sloping. They quickly learned you need to introduce mean reversion and a stochastic mean reversion level. 2 factors, one represented by changes in the short rate the other changes in something like a perpetual annuity. Each has an MPR, which can vary through time.

    Couple of other comments, I think when you look back to gold standard YCs we're generally inverted? Also if there is a shortage of inflation linked perpetuities (there is a big one), what is the second best inflation hedge for the longer term investor? I think cash is still a better option than nominal bonds? So do you get downward sloping real curves and upwards sloping nominals?

  6. > But we're talking about nominal bonds, not indexed bonds. The risk-free proposition holds if real interest rates vary, but inflation does not.In that case, short bonds have roll-over risk for long term investors, and long bonds have steady payouts. If inflation varies but real rates are constant, then short-term bonds have less risk for long term investors.

    I have just looked at the term structure for TIPS set at auction during
    December 2023. Not only, it is much more flatter than nominal yields, but there have been number of days where the whole term structure was almost fully downward sloping, say on December 8th, except for 30-year yields.

  7. The following paper from the Richmond Federal Reserve, Research Div., may be of particular interest to your readers:
    (1) Wolman, Alexander L., "Bond Price Premiums", Economic Quarterly, Fall 2006, Richmond, VA: Federal Reserve Bank of Richmond.

    From the paper's Abstract: "Consumption-based bond pricing theory implies that if investors are risk-averse then interest rates deviate from the Fisher relationship and the expectations hypothesis, and the deviations are described by forward premiums and inflation-risk premiums."

    Based on the theory developed in A. L. Wolman's paper, the expectations hypothesis holds if the investor's marginal utility of consumption is independent of the bond price and/or the rate of interest. For a risk neutral investor, one whose utility of consumption is equal to consumption, i.e., u(c(t)) = c(t), u'(c(t)) = 1, a constant, the expectations hypothesis is true -- the covariance of return and marginal utility of consumption is zero, for all time t t ∈ (0,∞] -- a necessary (and, sufficient) condition for the EH to be applicable.

    Wolman's paper "[ ] provide[s] a detailed introduction to the theory, using the two Fisherian interest rate decompositions and the corresponding premiums as the organizing framework." [1]

  8. BTW, on the subject of Asset Pricing, a long while ago you put on your website a draft of a new chapter to be added to the second edition. Yet no second edition has ever been published, what's that about?

  9. This is a fascinating subject (or puzzle?). But for a long time now I cannot understand why the focus is generally on nominal and not real rates (inflation indexed bonds)


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