User talk:7e7/deleted content

Durvexity [der-VEKS-ih-tee] is the duration of the interest-rate yield convexity curve that effectively measures the sensitivity of the price of a fixed income investment to the rate of change of the yield.

History
Durvexity first became an issue following the dislocation of the financial markets pursuant to the announcement of the bankruptcy filing of Lehman Brothers on September 15, 2008. Mathematically, it is the duration of the convexity curve, or the rate of change in the rate of the change of the bond yield/price relationship. This calculation will result in the coefficient of convexity, which gives a more intuitive measure for pricing of interest rate volatility. When volatility was low, durvexity remained near unity. The most pronounced example of recent durvexity increases are seen in the three-month LIBOR vs. one-month LIBOR basis swap. Historically, these two rates were correlated. However, beginning late in the third quarter of 2008, the markets began to demonstrate a preference for short-term financing due to increased liquidity concerns. The resulting disjunction has resulted in what many have deemed inefficient pricing on even plain-vanilla interest rate swap agreements and a subsequent spike in durvexity.

Algebraic definition
If duration (D) is defined, using the definition of modified duration, as:


 * $$D = - \frac{1}{P} \frac{dP}{dy}. $$    (1)

where P is price and y is yield.

Without stating references, being infinitisimal, a definition of yield would be something like

$$y = \int_{t_0}^{\infty} f(t) {dt}. $$

which complicates matters significantly.

Further, if convexity (C) is defined as:


 * $$C = \frac{1}{P} \frac{d^2\left(P(r)\right)}{dr^2}, $$    (2)

where P is again the price and r is the interest rate.

Rearranging the convexity equation to be stated in terms of price renders:


 * $$P = \frac{1}{C} \frac{d^2 P(r)}{dr^2}. $$   (3)

Therefore, the duration of the convexity curve can be expressing as by substituting Equation 3 into Equation 1:


 * $$\text{DURVEXITY} = - \frac{C}{ \frac{d^2 P(r) }{dr^2}} \cdot \frac{dP}{dy}. $$

Graphical explanation
As seen in the graph below, bonds have historically demonstrated a price/yield relationship akin to the convex curve 1. The duration calculated at point a would result in the first derivative depicted by curve 3. For a given increase in yield, the duration curve would predict a new price, point b. After applying the appropriate convexity adjustment, the market participant would expect a new price on the bond, point c.



However, in reality, the market dislocation has resulted in a much more convex yield curve, #2. To arrive at the real price of the bond, it is necessary to apply another, durvexity adjustment. The durvexity adjustment will result in the actual market price of the bond, d.