Why Treasury Bond Yields Can Invert Across Different Coupons and Maturities

# Convexity in long end treasury bonds


# Convexity in long end treasury bonds












According to published research, yield curves plotted against duration typically exhibit steepening forces as investors demand higher return for longer yields, but that often in the very long-end, the convexity premium will weigh more on yields, causing the yield spread to tip over and be inverted and that if the market expects higher volatility this 'inversion' can increase.

Currently, teasuries that mature in 25 years, i.e. 2050-2051, have a higher duration and have higher convexity than the bonds that mature in 30 years, 2055. Yet the 25y v 30y curve is inverted. If the research is correct, shouldn't the 25y maturities have lower yields given they have the convexity benefit and this is more valuable in the long end? I don't understand why 25y v 30y treasuries is inverted and what drives changes in this spread over time. thanks

## Answer by Chris Taylor (score 5)

https://quant.stackexchange.com/a/84119

This is just a consequence of the mathematical definition of yields. It's important to remember that the bond yield is simply the constant discount rate that prices the current bond. You cannot interpret it as an expected return, and it's not directly comparable between bonds that have different economics (e.g. different time to maturity, different coupon or different issuer).

For example, consider the following par rate and forward rate curves. These are simply derived from a discount factor curve, i.e. a curve that prices a zero coupon bond at each future date, but I've presented them as par and forward rate curves for ease of interpretation. Note that the forward rate curve is approximately flat between 25 and 30 years, and the par rate curve is upward sloping.

We can price two bonds from this curve:

- A bond with 25 years to maturity and a 1% coupon (i.e. issued as a 30 year bond five years ago, when rates were very low).

- A bond with 30 years to maturity and a 4.5% coupon (i.e. issued recently, when rates are higher)

The 25 year bond has a price of \$45.63, which is low because the majority of the bond's value comes from the repayment of the principal, which is discounted at a high rate. The 30 year recently-issued bond has a price of \$97.01 because the regular coupon payments offer a lot of near-term income that is missing from the 25 year bond.

Given the prices, we can calculate yields for these bonds. Remember that the yield is just the constant discount rate that gives the current price. The yield of the 25 year bond is 4.70% and the yield of the 30 year bond is 4.66%. The 25 year bond has a higher yield, even though the par rate at 25 years is lower than the par rate at 30 years! That is, the "yield curve" appears inverted, even though we know that the forward rate curve and par rate curve are not inverted. This is simply an artifact of the definition of yield, and a good reminder that you cannot directly compare the yield of bonds with different economics.

Similarly we can look at the duration and convexity of the two bonds:

- The 25 year bond has a duration of 19.88 years and convexity of 463.96

- The 30 year bond has a duration of 16.19 years and a convexity of 378.40

The reason for the higher duration and convexity of the short-term bond is that it has a low coupon - most of the value is concentrated at the 25 year tenor, whereas the 30 year bond with its higher coupon has the value more evenly spread over the lifetime of the bond.

Another way of looking at it is that the 25 year bond has a lower price, and since the duration and convexity measure relative price movements in response to yields, a similar-sized absolute price change for the two bonds results in a higher relative price change for the lower priced bond. Indeed, if we computed dollar DV01 and dollar gamma instead of duration and convexity (this would measure sensitivity in terms of dollars, not relative price change) then we would find that the 30 year bond has a higher dollar DV01 and dollar gamma.