An Illustrated Guide to the Yield Curve

AT SIXTY TRILLION DOLLARS, the market value of US fixed-income securities is twice as large as the market capitalization of all US corporations and thrice the size of the American economy as measured by GDP. These debt securities consist of some $12tn in corporate bonds, $15tn in mortgages, another $15tn in interest-rate swaps; all of it anchored on the $16tn market for US Treasuries. When one talks about “the” yield curve one means the term structure of US Treasuries.

The yield curve does not actually exist. It is estimated from market prices of on-the-run Treasury securities using something called a quasi-cubic hermite spline function. (Don’t ask.) You don’t want to know how the sausage is made. What you want to learn above all is how to read the damn thing.


Figure 1. US yield curve as of 15 February 2018. (Source: NYFed.)

Figure 1 displays the yield curve as of 15 February 2018. Here’s how you should read this graph. Think of the bonds as paying $1 in n years time (called the maturity or term). What the graph tells you is that the prevailing market prices of bonds are such that buying one today and holding it to maturity to get that $1 yields the displayed return via the simple time-value-of-money formula:

bond price formula

The relationship between the price and yield of a zero-coupon bond.

Table 1 displays the prices of these hypothetical bonds whose yields are displayed in Figure 1.

Bond prices and yields
Maturity 1 2 3 4 5 6 7 8 9 10
Yield 2.00 2.22 2.41 2.55 2.65 2.74 2.80 2.86 2.90 2.94
Price $0.98 $0.96 $0.93 $0.90 $0.88 $0.85 $0.82 $0.80 $0.77 $0.75

These hypothetical, plain vanilla bonds that promise to pay $1 exactly n years in the future are called zero-coupon bonds. They are the building blocks of all bonds; any bond whatsoever can be mathematically represented as a portfolio of zero-coupon bonds. In other words, all the information contained in the messy markets for default-remote bonds can be read off the yield curve for zero-coupon bonds. And that is what is displayed in Figure 1.

The expectations theory of the yield curve says that the yield curve reflects the expected path of the short rate. More precisely, that the yield at time t on a zero-coupon bond of maturity n equals the market expectation at time t of short rate prevailing at time t+n.

expectations theory


The expectations theory is wrong. Bond prices are dramatically more volatile than that implied by the theory and they routinely move in opposite direction that that implied by the expected path of policy rates. The basic reason why the theory fails is that default-remote bonds, even the obligations of a global military hegemon, are not in fact risk-free. The most obvious risk is that inflation may erode the real value of the bond. But beyond inflation risk, a bond holder faces duration risk: The risk that interest rates may rise faster than expected thereby reducing the market value of the bond in her portfolio. Only bonds of very short maturity are truly risk-free which is why the risk-free rate is approximated by the 3-month Treasury bill. Figure 2 displays the time-variation in the yield curve and the expected path of the Fed’s policy rate since 2010. Note how the former is much more volatile than the latter and has utterly distinct dynamics.


Figure 2. The yield curve and the expected path of short-term rates.  (Source: NYFed.)

Bond yields have two basic components. They reflect on the one hand the expected path of short-term rates and on the other the compensation for the risk they pose to the bondholder’s balance sheet. The latter component is called the term risk premium. In order to extract the expected path and risk premium from bond yield, we have to use a term structure model. We use estimates from the gold standard of term structure models, the ACM model developed by economists at the NY Fed.

Figure 3 displays the evolution of the term structure of the term risk premium, ie the difference between the two “sheets” displayed in Figure 2.


Figure 3. Term structure of the term risk premium. (Source: NYFed.)

Comparing Figure 2 and Figure 3, observe that the bulk of the variation in the yield curve since 2010 has been driven by variation in the term risk premium. Figure 4 displays the two components of the yield on the 5-year note.


Figure 4. Components of the yield on the five-year note. (Source: NYFed.)

We can see that, until 2014, most of the variation in the yield on the 5-year note was driven by variation in the risk premium. The big shock in 2013 was, of course, the taper tantrum induced by Bernanke’s announcement that the Fed would slow down its bond buying. Since 2014, the expected rate has gone up but the risk premium has fallen. Note that the term risk premium can, has episodically been, and at present is, negative. The collapse of risk premiums is not confined to the market for US Treasuries. Risk premiums (ie, expected returns in excess of the risk-free rate) get bid away across asset classes by investors’ increasingly desperate search for yield. Put another way, we are in an asset price boom—asset prices are too high. For high quality collateral like US Treasuries, this translates into negative risk premiums.

The expected path of the policy rate reflects the market’s expectation of the Fed’s reaction function on the one hand and the trajectory of core macroeconomic variables on the other. A steeper path implies that the market expects the Fed to tighten faster, say in order to counter inflation. So what has been going on over the past few weeks? Figure 5 zooms in on the components of the 5-year yield in 2018.


Figure 5. Recent movements in the components of the 5-year yield.

Both expected rates and the risk premium have gone up since the year began. Medium term rate expectations (5-year) went up 13 basis points—a basis point is hundredth of a percentage point—while the term risk premium went up 26 basis points. Rate expectations fell dramatically with the return of market volatility in early February. But that has since been priced out. Figure 6 displays the term structure of rate expectations. We see that rate expectations rose uniformly, fell together, and are now priced back in.


Figure 6. Term structure of rate expectations.

What all this means is that the market expects the Fed to hike faster, perhaps because it expects inflation to surprise on the upside, thereby forcing the Fed to hike faster than hitherto expected. Since there is no reason to believe that inflation has returned, this should get priced out soon enough.



Bonus chart: The expected path of policy rates.




Here is Why the Dollar is Weak

The so-called ‘Trump reflation trade’ started unraveling before Christmas Day, 2016. As expectations of inflation eroded and the expected path of the Fed’s policy rate became shallower, the dollar began to weaken. By the summer, all concerned agreed that the whole reflation trade had been priced out and then some. The dollar rebounded. But then…it started falling all over again.


Figure 1. Euro/dollar spot rate. (Source: Federal Reserve.)

This appeared to be a great mystery and generated considerable talk about whether the US Treasury Secretary was talking down the dollar. The FT‘s well-respected market commentator, John Authers, noted,

Has the US dollar stopped making sense? US rates are rising, and a long-run bull market in Treasury bonds seems to be over. This is not happening elsewhere, so the differential between US and European yields has risen to its highest since the euro came into being in 1999. That should mean a rising US dollar and falling US stocks. But US stocks are shooting for the moon and the dollar is tumbling — down 13 per cent from last year’s high on a trade-weighted basis. One retort is that the US has just passed a big tax cut. Of course that raises earnings this year — so buy stocks and sell Treasuries. But it should also be a reason to buy dollars. And stocks that benefit most from the tax cut are doing no better than anyone else this year. This renders the weak dollar the more mystifying.

Matt Klein, one of the sharpest knives in the FT‘s drawer, offered that there was no mystery, it had to do with growth expectations in Europe/RoW rising relative to the United States. This would be the standard (nonfinancial) macro explanation of dollar weakness. As we shall see, he is not entirely wrong. Although the mechanism is not as straightforward as he implies.

In a followup, Authers later shared a comment from a trader that pointed to a massive carry trade underway whereby you borrow dollars to fund euro forwards. According to this trader, there was supermassive 100 basis point (ie, 1 percent) carry in the trade. This, he/she alleged, was the cause of dollar weakness.

I realized that there is a very easy way to check this. Such a carry would only exist if fwd rates deep in the curve were much higher on the continent than in America. Gavyn Davies had already noted the empirical case for this. We can do much, much better than suggestive visual evidence. Indeed, we will see how this can be nailed down mathematically.

We appeal to what’s called uncovered interest rate parity, which says that the home interest rate equals the foreign interest rate plus the expected rate of depreciation of the home currency. It imposes a consistency condition on the euro/dollar spot exchange rate on the one hand and the yield curves prevailing in America and on the continent on the other. Matt is right about changes in expectations about relative growth rates in the United States and the eurozone. What this means is that the US yield curve has become flatter than the German yield curve and that has opened up the carry that Authers’ trader gushed about.


Figure 2. Term spreads in the United States (30yr minus 2yr) and Germany (15-30yr minus 2-5yr).

The proof is a straightforward calculation of uncovered interest rate parity. Figure 3 displays the expected future exchange rate implied by the yield curves displayed in Figure 2 via the UIP equation as well as the spot rate.


Figure 3. Uncovered interest rate parity for the euro/dollar exchange rate.

The interpretation is straightforward. Change is relative growth expectations between the United States and Europe led to relative movements in the yield curves which opened up a huge carry trade opportunity. And that massive carry trade put downward pressure on the spot rate for the dollar. Here’s a graph of the carry implied by the yield curves together with the spot exchange rate.


Figure 4. Carry implied by the yield curves on the continent and in America.

An interesting question that arises then is whether the carry trade consumed so much dealer balance sheet capacity that it precipitated the risk-off that began last week and culminated on Vol Monday. Perhaps that is why US bank stocks (but not European) did so well on Monday despite the volatility shock. If that is the case, we would’ve nailed two birds with one stone.


Capital Formation in US Firms

Brenner suggested on these pages that US capital formation has slowed drastically. Is that true? In trying to answer that question, I found the god of modern economics. More precisely, I figured a way to nail down the correct metric for capital formation and that allowed me to measure residual productivity growth; the great unexplained of modern economics. The reason why economics does not have a theory of innovation is that it falls in the social realm; in the specific sense of Matthew Crawford’s ‘ecologies of attention’. I figure that total factor productivity is a function of the vitality of situated communities of knowledge. People engaging together with the machine face incentives endogenous to the situated community centered on the machine. Firms become more productive when—as situated communities of knowledge—they learn better ways of solving their problems and increasing productivity.

Firms increase productivity in two ways: (1) deploying capital in productive assets (machinery and so on) and (2) figuring out better ways of working said assets. In order to estimate the contribution of (2) we have to nail down (1). For (2) is the residual; the portion of variation in productivity unexplained by (1). The best way to nail down (1) is to consider the net (of depreciation) stock of corporate fixed assets since firms can let capital stock erode by not replacing or repairing equipment and structures et cetera. That is, corporate investment spending may not be enough given the variation in depreciation. Instead, what we really need a handle on is capital accumulation. Figure 1 presents the raw series for growth in the net stock of fixed assets of US firms. It suggests a secular decline in US capital formation since 1966.


Figure 1. Growth in net capital stock. (Source: BEA)

The problem is that much of the variation observed in Figure 1 is an artifact of demographics. Figure 2 shows the scatter plot and series for growth in net stock of corporate fixed assets and growth in prime age population for the United States.

We must control for demographics. There are two ways to factor out the contribution of demographics. (1) Linearly project the variation of one on the other and use the OLS residuals. (2) Look instead at net capital stock per prime age adult. Happily, both yield very much the same dynamical behavior and predict labor productivity equally well. We use (2) because it admits a straightforward interpretation as capital intensity—capital stock per prime age adult. Figure 3 displays our metric for capital formation. We don’t control for real output growth or capacity utilization because of issues concerning endogeniety: Sure, capitalists are investing less because growth is slow but growth is also slow because capitalists are investing less. But assuming that demographics is exogenous is a useful fiction.


Figure 3. Growth in US capital intensity. (Source: BEA, author’s calculations.)

You can think of the graph displayed in Figure 3 in two ways: as detrended capital formation, or more precisely, as growth in capital intensity defined as the natural log of the ratio of the net stock of corporate fixed assets (chained quantity index) to prime age population. We can see that there were investment booms in the nineties, the fifties and the sixties. The two booms in the late sixties and the late nineties stand out. This observation is strengthened by the dynamical behavior of the stock of equipment and the average age of the equipment. Figure 4 shows the growth of the net stock of equipment per prime age adult. We see that average age falls in investment spurts and rises otherwise.


Figure 4. Equipment growth and ave age of equipment. (Source: BEA.)

Capital intensity and the average age of equipment are good predictors of labor productivity growth. Together they predict two-fifths of the variation in labor productivity growth. Figure 5 displays the scatter plot.


Figure 5. Capital formation and age of equipment predict labor productivity. (Source: BEA, author’s calculations.)

Figure 6 displays the 5-year moving average of labor productivity growth orthogonal to lagged growth in capital formation and average age of equipment. That is, we project labor productivity growth on lagged capital formation and change in average age of equipment, and report the OLS residuals. The origin of the Y axis has been moved to 1 for ease of interpretation. (Excel and area graphs; don’t ask.) Labor productivity orthogonal to capital formation and technological shocks (captured by age of equipment) is residual productivity attributable to gains in knowhow. The portion of growth not explained by capital formation (including the technology embedded in new equipment/machinery). It is thus a finer measure than TFP.


Figure 6. An alternate measure of total factor productivity: US labor productivity growth orthogonal to capital formation and age of equipment.

There have been two spurts in underlying productivity by our metric. A big one in the early sixties and a medium-scale one in the late-90s and early-2000s. The Sixties’ Boom stands out prominently. What explains the productivity miracle of the 1960s? That’s the big explanandum thrown up by the present study.

To gather our findings together: Looking at investment ignores depreciation. Looking at growth in net capital stock suggests a secular decline since the mid-1960s. But that is in fact an artifact of demographic shocks. Looking at the ratio of net capital stock to prime age population controls for these demographic shocks. We find two prominent investment booms in the sixties and nineties. Perhaps not coincidentally, these two periods are also times of significantly positive multi-year pure productivity shocks.


Figure 7. The balance between the three main blocs of US corporations.

The sixties were a period of unabashed hegemony of the Chandlerian firms. Manufacturing corporations accounted for more than half of all corporate profit in 1967-1969. See Figure 7. We must ask a more precise question than what’s special about the sixties. We must ask, What was going on in these Chandlerian firms that was specific to the sixties? or that was premised on conditions that prevailed only in the sixties? Did it have something to do with the passage of the reins of power to the ‘organized intelligence’ running these great corporations, pace Galbraith? seen as ‘ecologies of attention’, pace Crawford? In other words, could it be that corporate freedom from ‘shareholder value’ empowered the engineers running these corporations and allowed them to solve problems faster, thereby increasing effective knowhow? Or was it the flowering of the ‘corporate-liberal synthesis’? or maybe even the full flowering of ‘Fordism’ and ‘the Treaty of Detroit’? What the hell was going on in the sixties?


Cross-Border Banking Flows as a Metric for the Global Financial Cycle

Perhaps Nomi Prins did not choose the title of her piece “The next financial crisis will be worse than the last.” But the idea that there is going to be another financial crisis in the center of the world economy in the near term even vaguely comparable in virulence to the GFC has, as we shall see, no basis in reality. The reason is straightforward. Financial crises are denouements of credit booms, not asset price booms—all credit booms are attended by asset price booms but not the other way around—and while there is certainly an asset price boom in global markets, there is no credit boom at the center of the world economy. The chances of a banking crisis are even more remote than credit indicators suggest because of the extraordinary surveillance of the balance sheets of global banks since the GFC. Even if all legal-regulatory innovations over the past decade—especially the tighter limits on capital ratios—are bull, the sheer fact of enhanced regulatory and independent balance sheet surveillance means that banks find it much more difficult to hide risks on and off their balance sheets.

We now have a good handle on the mechanics of financial booms and crises. Financial booms are banking expansions. Banks are special because, yes, they create money by lending. But do not let the Bank of England distract you. The issue is that the excess elasticity of bank balance sheets mechanically generates lending booms since bank assets are the liabilities of non-banks. As a rule, credit booms emerge from the mutually-reinforcing interaction of property prices and bank lending. As collateral values go up, more can be lent against the same property; in turn, greater lending pushes up property prices further. Credit booms show up in credit gap measures such as credit-to-GDP ratios. We also understand how credit booms end. The stock of outstanding debt lags behind credit gaps. Once the debt burden, which is a function of the stock of outstanding debt not credit growth, becomes intolerable, credit defaults puncture the boom and precipitate a financial crisis. That’s why the best predictors of financial crises are credit gaps and debt ratios.

Metropolitan banking is international. As the day progresses, the trading book of global banks passes from Hong Kong to London to New York. The transatlantic circuit is especially important. The mid-2000s financial boom was driven in large part by a transatlantic, European banking glut. In other words, there is good reason to believe that cross-border banking flows are a especially good barometer of the global financial cycle. I therefore decided to analyze the JEDH database on cross-border banking and debt flows.

I’ll probably have much more to report later. But here’s the basic picture. Figure 1 displays three variables; all standardized to have mean 0 and variance 1. “CoreFPC” is the first principal component of the cross-border flows of Japan, Germany, France, Italy, Netherlands, Norway, Sweden, Denmark, and Finland. Roughly speaking, it captures the common variation in the series. “G2” is the sum of the cross-border flows of the United Kingdom and the United States. “China” is the sum of the cross-border flows of China, Hong Kong, and Macau.


Figure 1. Cross-border banking flows.

A few observations are in order. The tight coupling of Anglo-Saxon finance (“G2”) and the rest of the core (“coreFPC”) is manifest; thus allowing us to interpret either as providing a fair metric for the global financial cycle. We will use the former because (a) it is a tighter, more parsimonious definition; (b) it is applicable in more general settings in the sense that we can extend many macrofinancial variables back to the 19th century without changing our center countries. Hélène Rey’s notion of the global financial cycle—as the covariation of risk premia embedded in global asset prices—is less relevant to macrofinancial stability than our metric. Although banking expansions can be read off of asset prices, it is a noisier metric precisely because not all asset price booms are attended by real financial booms that end in tears. Cross-border banking flows provide a finer measure of banking gluts than the compression of risk premia because all lending booms are attended by cross-border banking flows and vice-versa.

This metric confirms what credit gaps and debt service ratios can tell us about the buildup of financial imbalances in the center of the world economy. Interestingly, by this metric the Chinese cycle seems to have turned. This was not clear when we looked the credit gap (Figure 2).


Figure 2. China’s macrofinancial vitals.

Finally, as a sanity check we look at the bond market. The slope of the yield curve is the best predictor of US recessions out there. When the yield curve inverts it heralds a recession in the near term. There is good reason for this. The inversion of the yield curve destroys banks’ net interest margins; forward-looking measures of banks’ net worth fall; banks respond by shedding assets; finally, the attendant fall in bank lending pushes the macroeconomy into recession—this is Adrian and Shin‘s risk-taking channel of monetary policy. The term spread has indeed compressed, but the yield curve is still upward-sloping.


Figure 3. The term spread.

Because the current asset price boom is unattended by a credit boom in the center of the world economy, the possibility of a financial crisis comparable to the GFC is remote. And despite the “age” of the expansion, a normal recession does not yet seem to be on the cards either.





The President and the Stock Market

Authers’ Note today sounded downright exasperated with the President’s tweets. Trump claimed credit for the fastest 1,000 point gain in the Dow’s history:


That’s too cute by half. Each 1000 percent gain gets mathematically smaller as the Dow goes up; eg, the move from 10,000 to 11,000 is a 10 percent move whereas that from 24,000 to 25,000 is just 4 percent. Properly compared, the performance of the Dow under Trump lies between equivalent periods in Obama’s two terms.

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But the Dow is a stupid aggregate of stocks for the simple reason that it is price-weighted. “Over the past year,” Auther notes, “two great industrial giants have been the alpha and omega of the Dow: Boeing has doubled, gaining $85bn in market cap, while GE has collapsed by more than 40 per cent, shedding a staggering $118bn. But due to the ridiculous way in which the Dow is calculated, Boeing accounts for a rise in the Dow of 1,064 points, while GE accounts for a fall of only 84 points.” That’s because Boeing has 600 million shares outstanding valued at $309 each for total market cap of $184 billion; while GE has 8.7 billion shares worth just $18.5 each for a total of $161 billion. The differential sensitivity of the index is an artifact of the practically irrelevant question of how many shares the firms have outstanding.

If we ignore the Dow and look, as all serious investors and analysts do, at the S&P500, even Obama’s 1st year in office comes out ahead of Trump’s.

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In a more longer term perspective, the Trump rally hardly stands out either.

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More generally, Presidents have little influence over the stock market; central banks much more so. But we should not even exaggerate the influence of the latter. Stock valuations are sky-high not just because of monetary accommodation but mostly because the near-term global macroeconomic outlook is especially positive for risk assets. Not only is global growth more robust than it has been in a decade; there is little sign of inflation, which implies that there is little incentive for central banks to take away the punch bowl any time soon. It is this benign macro environment that is compressing market-wide risk premia.

Investors are facing two main risks in the near term. The first is that growth may falter and cash flow expectations may need to be marked down, with attendant corrections in asset valuations. In the extreme, the economy may even plunge into a recession; although that scenario seems unlikely in the near term—the yield curve may be shallower than before but it is still sloping upwards. The second is that inflation might finally show up, forcing the Fed to hike much faster than anticipated, thereby precipitating a risk-off. In other words, markets have been in a sort of Goldilocks Zone. Either a significant deceleration in global growth or a significant acceleration that eliminates global slack can precipitate a sell-off.

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Source: US Treasury.


Does Consumer Confidence Predict Output Growth?

Tip of the hat to Ted Fertik for flagging Mian, Sufi and Khoshkhou (2017). The authors examine the role that partisan bias plays in consumer expectations. One of their findings is that consumer confidence is extraordinarily high among Trump supporters. But that this has not translated so far into higher spending.

I was skeptical of the notion that consumer confidence—as opposed to CEO confidence, investor confidence, and the risk appetite of intermediaries—had discernible and predictable effects on real economic activity. So I examined the data. Turns out that it does. And that it does so through a clear channel: Consumer confidence predicts discretionary consumer spending which in turn is a strong correlate of real GDP (RGDP) growth.

The evidence can be read off Figure 1. Read clockwise from top-left. (1) University of Michigan’s Index of Consumer Sentiment (MCSI) predicts RGDP growth even after controlling for lagged RGDP. (2) MCSI predicts growth in real discretionary consumer spending. (3) MCSI does not predict RGDP growth shocks orthogonal to discretionary consumer spending. That is, the residuals obtained by projecting change in log RGDP onto change in log consumer spending are not correlated with lagged consumer spending. (4) Discretionary consumer spending is contemporaneously correlated with growth in RGDP.


Figure 1. The empirical evidence for the confidence channel. The data is at the quarterly frequency and for the period 1960Q1-2017Q3. Source: Haver Analytics.


Markets Celebrate the GOP’s Trillion Dollar Giveaway to the Rich

So you thought fiscal hawks were serious about the US deficit? Haha. Despite last minute snags the Republicans are expected to push through a trillion dollar tax-cut for their paymasters. One is told by Very Serious People that markets, bond vigilantes in particular, punish fiscal profligacy by demanding higher costs of borrowing. That’s poppycock. Both stocks and bonds are booming for good reason. The tax cut delivers a large, positive wealth shock to investors and the market is repricing to reflect investors’ greater appetite for risk. The bond market refuses to believe that the tax cut—a major fiscal shock when the economy is at full employment—will have much of an effect on economic activity. For the compression of the term spread means that the market expects subdued inflation and a shallow path of policy rates. Were a major pickup in economic activity around the corner, the yield curve would become steeper not shallower. So the euphoria in the stock markets is due not to expectations that animal spirits unleashed by the tax cut and regulatory “reform” would deliver higher growth. Rather it is due to the straightforward transfer of resources from the public sector to corporations and investors. What the market is celebrating in other words, is not the expected growth of the pie but redistribution of the pie in its favor.