Modified convexity: Difference between revisions
imported>Administrator (CSV import) |
imported>Doug Williamson (Layout.) |
||
| (9 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
(MC). Broadly speaking, modified convexity measures the curvature of an instrument’s price function, as yields change - from a given starting point - by a small amount. | (MC). | ||
Broadly speaking, modified convexity measures the curvature of an instrument’s or a portfolio's price function, as yields change - from a given starting point - by a small amount. | |||
More strictly, it is the rate of change of modified duration with respect to yield - at the given starting yield. | More strictly, it is the rate of change of modified duration with respect to yield - at the given starting yield. | ||
Modified convexity can be calculated from Convexity as follows: | Modified convexity can be calculated from Convexity as follows: | ||
'''Modified Convexity = C<sub>MOD</sub> = Convexity / (1+r)<sup>2</sup>''' | '''Modified Convexity = C<sub>MOD</sub> = Convexity / (1+r)<sup>2</sup>''' | ||
Where: | |||
r = periodic yield = [[Nominal annual rate]] / compounding frequency per year | |||
The estimation of price change for a given small change in yield can then be calculated as follows: | The estimation of price change for a given small change in yield can then be calculated as follows: | ||
Price change estimation using Modified Duration (MD) only: | Price change estimation using Modified Duration (MD) only: | ||
= - Price x MD x Change in yield | |||
Price change estimation using Modified Convexity (C<sub>MOD</sub>): | Price change estimation using Modified Convexity (C<sub>MOD</sub>): | ||
Because the value v yield relationship is a curve and not a straight line (values do not change linearly as yields change) the estimate of change in value using only modified duration will generally underestimate the new value (because the curve lies above its tangent). Therefore the modified convexity adjustment is always positive - it always adds to the estimate of the new price whether yields increase or decrease. | = - (Price x MD x (Change in yield) ) + ½ x (Price x C<sub>MOD</sub> x (Change in yield)<sup>2</sup> ) | ||
Because the value v yield relationship is a curve and not a straight line (values do not change linearly as yields change) the estimate of change in value using only modified duration will generally underestimate the new value (because the curve lies above its tangent). | |||
Therefore the modified convexity adjustment is always positive - it always adds to the estimate of the new price whether yields increase or decrease. | |||
It is also possible to estimate the MD and the C<sub>MOD</sub> from given observations of Price and Yield, by rearranging them to solve for MD and C<sub>MOD</sub> - effectively running the price change estimation formulae in the other direction. | It is also possible to estimate the MD and the C<sub>MOD</sub> from given observations of Price and Yield, by rearranging them to solve for MD and C<sub>MOD</sub> - effectively running the price change estimation formulae in the other direction. | ||
== See also == | == See also == | ||
| Line 23: | Line 41: | ||
* [[Matching]] | * [[Matching]] | ||
* [[Modified duration]] | * [[Modified duration]] | ||
* [[Nominal annual rate]] | |||
* [[Portfolio]] | |||
[[Category:Manage_risks]] | |||
[[Category:Risk_frameworks]] | |||
Latest revision as of 21:43, 16 November 2016
(MC).
Broadly speaking, modified convexity measures the curvature of an instrument’s or a portfolio's price function, as yields change - from a given starting point - by a small amount.
More strictly, it is the rate of change of modified duration with respect to yield - at the given starting yield.
Modified convexity can be calculated from Convexity as follows:
Modified Convexity = CMOD = Convexity / (1+r)2
Where:
r = periodic yield = Nominal annual rate / compounding frequency per year
The estimation of price change for a given small change in yield can then be calculated as follows:
Price change estimation using Modified Duration (MD) only:
= - Price x MD x Change in yield
Price change estimation using Modified Convexity (CMOD):
= - (Price x MD x (Change in yield) ) + ½ x (Price x CMOD x (Change in yield)2 )
Because the value v yield relationship is a curve and not a straight line (values do not change linearly as yields change) the estimate of change in value using only modified duration will generally underestimate the new value (because the curve lies above its tangent).
Therefore the modified convexity adjustment is always positive - it always adds to the estimate of the new price whether yields increase or decrease.
It is also possible to estimate the MD and the CMOD from given observations of Price and Yield, by rearranging them to solve for MD and CMOD - effectively running the price change estimation formulae in the other direction.
