The yield curve is a graphical representation of the bond yield at different maturities. A normal yield curve will slope upwards, with higher yields for longer bonds. An inverted yield curve will slope downwards, with higher yields for shorter bonds. This implies that the central bank is raising interest rates in response to an overheating economy, and it usually precedes recession. But what I’m interested in is the convexity or concavity of a normal yield curve.

A normal yield curve is generally expected to be concave:

Short bonds and T-bills usually have the lowest rates. 2-year, 3-year and 5-year bonds have relatively higher yields. Then the yield curve flattens from 10-year to long bonds, as they approach the investors’ required long-term return. Given that we are looking at government bonds, the guarantees are assumed to be dependable.

Recently, the yield curve has been convex:


You’ll notice that rates are very low. Also notice the lack of “hump” on the hill. What gives the yield curve this shape? There are three factors that affect the shape of the yield curve: future rate expectations, term premiums and rate volatility.

Future rate expectations can be calculated from the difference in the yields at different maturities. If the 3-month T-bill yields 0.46% (as currently) and the 6-month T-bill yields 0.55%, that implies that investors expect the 3-month yield to rise to 0.64% in three months because once their T-bill matures, they have to reinvest it at 0.64% to get a total yield over the six months of 0.55%. Currently, the 2-year bond also implies a rate expectation of 0.64% from six months to 2 years. The implied rate expectations from 2 years to 3 years is 0.71%, from 3 years to 5 years is 0.90%, from 5 years to 7 years is 1.70% and from 7 years to 10 years is 2.12%. From 10 years to the long bond (less than 30 years) is 2.31%. We can see from these calculations that investors not only expect rates to remain low, but to remain low for an extended period of time  (at least five years).

There is a term premium that compensates investors for locking their money up for long periods rather than constantly rolling it over. Because of the risk premium, investors will usually require that long terms reward them with higher returns. That is currently the case, and it doesn’t really explain convexity or concavity, only the normal shape of the yield curve. What we can assume, though, is that a concave yield curve is somehow related to reduced (rather than increased) risk premiums in the middle of the curve (5-10 years). Another way of saying this is that a 5-year bond is currently almost as attractive as a 2-year bond, since they have very similar yields (0.75% compared to 0.62%). This may related to the first point, an expectation of low future rates, or it may relate to the next point, the idea that economic outcomes are unpredictable or even unstable, so safety is valuable over the medium term.

Implied volatility seems to have the most influence over the convexity or concavity of the yield curve. “When volatility is high, the curve is likely to be upward-sloping; when volatility is low, the shape of the yield curve is determined mostly by the expectations component, since risk premia are virtually absent.” ( The way I understand this is that when short-term rates are volatile, the yield curve will be more concave (with a hump) because the uncertainty of future rates will cause investors to demand lower prices (higher yields & returns) for longer bonds. (This uncertainty tends to resolve over very long periods, like 10-30 years.) Currently, with low volatility of short-term rates, investors hardly demand any premium for longer bonds. But it may also be that investors bid up the prices (and down the yields) of bonds with maturities of less than 10 years when they feel very uncertain about the economic prospects over the coming 10 years. This would be a “flight to safety,” where investors prefer to own a government-guaranteed investment rather than owning stocks and investing in uncertain future corporate profits.

I would read a convex yield curve as “economic uncertainty” and the related “persistent low interest rates.” Whereas an inverted yield curve warns of economic overheating and an impending recession, a convex yield curve appears to warn of low corporate profits. In that case, bonds are expensive, stocks are expensive and it’s not an easy time to be an investor right now.

Yield curve shape: convex and concave

2 thoughts on “Yield curve shape: convex and concave

  • June 20, 2017 at 3:56 pm

    This definition is backwards compared with mathematical and economic standards of concave and convex curves.

    • June 25, 2017 at 9:16 am

      Thank you for pointing that out. I will admit that I had a lot of trouble finding information about concavity and convexity as it relates to yield curves. Convexity is, however, a common notion relative to bond yields, which should have given me the information I needed. I have updated my post based on your feedback.


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