Sensitivity (economy)

In economy, the sensitivity (in English: sensitivity ) is the variation of an economic size when another varies in absolute value of a unit, or in relative value of 1%. This notion approaches that of the Élasticité (economy) elasticity, this one systematically measured however by comparison of the relative values.

For example the sensitivity to the oil price of the balance of the foreign trade of a country, or of the bottom line of a company very user of this matter, is the variation of this indicator of management when this price on the spot market varies from a dollar.

In Finance, the sensitivity (in English: modified duration ) is related to the concept of Duration. It is an indicator of the Risque of rate related to an instrument atfixed rate, like a obligation.

Financial definition ( sensitivity with interest rate )

It is about the variation for 1% of rate (100 basic points) of the price of the instrument compared to its Annual percentage rate. Like, for the majority of the instruments, the price increases when the rates decrease, one considers the inverse proportion rather.

With:

• $r\; \backslash ,\; \backslash !$ the annual percentage rate of the instrument,
• $P\; (R)\; \backslash ,\; \backslash !$ the price of the instrument according to its annual percentage rate,
• $P\text{'}\; (R)\; =\; \backslash \; frac\; \{dP\; (R)\}\{Dr.\}$ the Derived from the price of the instrument compared to the annual percentage rate,

the sensitivity $S\; (R)\; \backslash ,\; \backslash !$ is thus:

$S\; (R)\; =\; -\; \backslash \; frac\; \{P\text{'}\; (R)\}\{P\; (R)\}$

Orders of magnitude

Schematically, an obligation having a sensitivity of 5 will see its value lowering approximately 5% if its interest rate increases by 1%, and, conversely, its value to increase approximately 5% if the rates drop by 1%.

Nevertheless, it is advisable to remember that the price $P\; (R)\; \backslash ,\; \backslash !$ does not evolve/move in a linear way according to the annual percentage rate $r\; \backslash ,\; \backslash !$ and that the error grows with the variation of rate which one applies.

Objectives and limits

Just like the Duration, the sensitivity is essentiellerment a patrimonial measurement , used by the administrative of funds . It is indeed about the report/ratio of the measurement first of the Risque of rate, $P\text{'}\; (R)\; \backslash ,\; \backslash !$, with the brought up to date Value of the instrument, it is to say to the whole of the funded capital, and not to its Face value.

The professionals of the markets of interest rate use, them, like measurement first risk of annual percentage rate, only $P\text{'}\; (R)\; \backslash ,\; \backslash !$ only, not corrected.

On the other hand, the average sensitivity of a OPCVM to variable capital constitutes, for the public, an important measure of the risk of rate which it shows, since one subscribes to this type of placement and that it is resold, each time, in gross values.

Properties

The longer one obligation withfixed rate is, the more it is sensitive.

In addition, as it is about an elasticity, the sensitivity has paradoxical properties. Thus, more the nominal Taux of an obligation withfixed rate is raised, more the risk of rate $P\text{'}\; (R)\; \backslash ,\; \backslash !$ that it presents is high. Nevertheless, as it is equipped with an high price expressed as a percentage of the nominal one, its sensitivity will be lower than that of an obligation of same characteristics but of lower nominal rate.

Formulas

The sensitivity is given by the following formula:

$S\; =\; \backslash \; frac\; \{1\}\; \{P\}\; \backslash \; \backslash \; sum\_\; \{i=1\}\; ^n\; \backslash \; frac\; \{T\; (I)\; \backslash \; times\; F\_i\}\; \{\backslash \; left\; (1\; +\; R\; \backslash \; right)\; ^\; \{T\; (I)\; +1\}\}$

with:

$P\; \backslash ,\; \backslash !$ the price of the obligation,

$F\_i\; \backslash ,\; \backslash !$ the flow (coupon and capital) of the period $i\; \backslash ,\; \backslash !$,

$t\; (I)\; \backslash ,\; \backslash !$ is the time interval, expressed in years, separating the date from actualization of the date of flow $F\_i\; \backslash ,\; \backslash !$

$r\; \backslash ,\; \backslash !$ the annual percentage rate of the obligation.

That the sensitivity can be expressed according to the Duration $D\; \backslash ,\; \backslash \; is\; noticed!$:

$S\; =\; \backslash \; frac\; \{D\}\; \{\backslash \; left\; (1\; +\; R\; \backslash \; right)\}$

See too

• Annual percentage rate
• evaluation of obligation
• Convexity (economy)
• Duration
• Curve of rate
• Rate zero-coupon

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