DEFINITION
Effective duration or volatility of a series of cash flows can be interpreted as the sensitivity of the cash flows, to changes in interest rates. It can be expressed as the negative value of the first derivative of the present value function of a series of cash flows divided by its present value.
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FORMULA AND DERIVATION
Consider a series of cash flows \(c_t\) at times \(t=1,\ 2,\ 3,\ \dots \) Let their present value at rate \(i\) be \(P\).
\[P=\ \sum^n_{t=1}{c_t{\left(1+i\right)}^{-t}=\ \sum^n_{t=1}{c_tv^t}}\]
Then, the effective duration, ν (Greek letter"nu"), would be
\[\nu \left(i\right)=\ \frac{{-P}^{'}}{P}\]
Where, \(P^{'}\) is the first derivative of the present value function, \(P\).
To arrive at the formula we first need to determine \(P^{'}\), the first derivative of \(P\).
\[P=\ \sum^n_{t=1}{c_t{\left(1+i\right)}^{-t}}\]
\[P^{'}=\ \sum{c_t\left(-t\right){\left(1+i\right)}^{-t-1}}\]
\[\ \ \ \ =\ \left(-1\right)\sum{c_t.t.{\left(1+i\right)}^{-\left(t+1\right)}}\]
Substituting the above value in the formula for \(v\left(i\right)\) , we would get:
\[\nu \left(i\right)=\ \left(\frac{1}{\sum{c_t{\left(1+i\right)}^{-t}}}\right)\times \ \sum{c_t.t.{\left(1+i\right)}^{-\left(t+1\right)}}\]
The effective duration can also be represented in terms of the Macaulay duration or Discounted Mean Term(DMT), as follows:-
\[v\left(i\right)=\tau \ \times \ {\left(1+i\right)}^{-1}\]
Where, τ is the DMT of the cash flows.
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