Duration

The duration of a financial instrument atfixed rate, like a obligation, is the average lifespan of its financial flows balanced by their brought up to date value. All things being equal besides, plus the duration is high, plus the risk is large.

Use

It is about a tool making it possible schematically to compare several instruments or obligations withrate fixes between them, whatever were their conditions of emission. It is primarily a statistical patrimonial measurement , which provides to the managers funds or to the administrative of passive credit/ a size which they will compare with the intermediate duration of a mandate of management, or at one intermediate duration of use of the funds.

It is used before very immunizing wallets, like simple but effective substitute:

• is of a perfect backing, financial flow by financial flow, with obligations zero-coupon, often difficult to realize;
• is of a reliable mathematical modeling of the evolution over one long period of the curve of the rates of interest.

Like, by definition, the duration is lower than the simple average lifespan (i.e. balanced only by flows of refunding of the capital, not brought up to date) of the obligation, its pleasing employment systematically to cover a liability by a obligation of longer duration .

Indeed, if the duration of the liability is M , an obligation of duration NR will have necessarily an average lifespan N , except if it is about a Obligation zero-coupon, in which case N=M . Thus, in the event of fall of the short rates during the life of the obligation, the profit carried out on this one will be in fact higher than the loss incurred on the liability. Nearly 25 years of quasi-uninterrupted fall of interest rates considerably increased the prestige of the duration near the managers of funds. It was quite less in the years 1970, and it is probable that it would drop again in the event of a return of inflation… In short, the immunization in duration of a wallet is perfect only if it is carried out with instruments zero-coupon. The use of obligations or traditional swaps, necessarily longer, creates certainly new a Risque of rate, weaker, but considerable.

The duration is sometimes presented peremptorily like " duration that an obligation puts to refund its price of achat". That is entirely true only in the case of instruments zero-coupon. For all the other obligations, this definition is to be taken with a large salt pinch, because it omits that it is about a median value…

Confusions to be avoided

The duration gives on the other hand a rather approximate measurement of the instantaneous impact of a variation of the Interest rate on the price of this obligation. Admittedly, more the duration is large, more the impact on the title will be it. Nevertheless, this measurement is too vague to be used on the financial markets.

In addition, it takes account of the form of the curve of the rates, neither of its deformations, nor of its dynamics.

Modified duration

The term of " modified duration" in the Anglo-Saxon literature indicates the following formula:

$D^*\; =\; \backslash \; frac\; \{D\}\; \{1\; +\; R\}$

Modified duration measurement sensitivity of a product to the variations of interest rate. While the duration measures this sensitivity into absolute, modified duration measurement expressed as a percentage.

Mathematical formulation

The duration $D\; \backslash ,\; \backslash !$ of an obligation concerning flows $F\_i\; \backslash ,\; \backslash !$ at the time of the $n\; \backslash ,\; \backslash !$ remaining periods, is given by the following formula, where $t\; (I)\; \backslash ,\; \backslash !$ is the time interval, expressed in years, separating the date from actualization of the date of flow $F\_i\; \backslash ,\; \backslash !$:

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

with $r\; \backslash ,\; \backslash !$ the annual percentage rate of the obligation such as the price observed $P\; \backslash ,\; \backslash !$ of the obligation corresponds to the value brought up to date of this one. It is the solution of the equation:

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

It is noticed (cf above, Confusions to avoid ) that the measurement of the risk of instantaneous rate, $\backslash \; frac\; \{dP\}\; \{Dr.\}$ is certainly expressed according to the duration

$\backslash \; frac\; \{dP\}\; \{Dr.\}\; =\; -\; \backslash \; frac\; \{P.D\}\; \{1+r\}\; =\; -\; \backslash \; P.D^*$

but of it is quite different.

In other words, the duration is the elasticity (with the sign near) of the price of the obligation at the annual percentage rate:

$D\; =\; -\; \backslash \; frac\; \{\backslash \; frac\; \{dP\}\; \{P\}\}\; \{\backslash \; frac\; \{Dr.\}\; \{1\; +\; R\}\}$

$\backslash \; frac\; \{dP\}\; \{P\}\; =\; -\; D^*.dr$

See too

• evaluation of obligation
• Actualization
• Curve of rate
• Rate zero-coupon
• Risk of rate
• sensitivity of an obligation
• convexity

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