Closed-form Duration of Mortgages?

[mascolinha]

Anyone knows a CLOSED-FORM for calculating the duration of a mortgage? (or an amortizing loan)…I really need a closed form (I know how to perform it by calculating each component and weight-averaging it by the time interval..)
Thanks!

 

[S2000magician]

There isn’t a simple closed-form formula (other than the one you’ve dismissed: summing th PV if each cash flow times the time-to-receipt is a closed-form). The reason is that the PVs form a geometric sequence, while the times form an arithmetic sequence; there’s no simple formula for the sum of geometirc sequence terms multiplied by corresponding arithmetic sequence terms.
Sorry.

 

[MathMan]

Actually, there is a closed-form for this. It can be derived directly by using the closed form for the pv of an annuity equation and the definition of Duration.
Duration = - dP/dr * (1+r)/P
P = D / r * (( (1+r)^n -1 ) / (1+r)^n )
dP/dr = D * ( (1+r) - (1+r)^(n+1) + n*r ) / ( r^2 * (1+r)^(n+1) )
Duration = - D * ( (1+r) - (1+r)^(n+1) + n*r ) / ( r^2 * (1+r)^(n+1) ) * (1+r)/P
It can also be derived generally using a specific form of a combined arithmetic and geometric series:
Note that:
Summation(i = 1 to n) i * x^i
is just equal to dS/dx * x where:
S = Summation(i = 1 to n) x^i
a simple geometric series which he know has a closed form and will obviously remained closed
when deferentiated (if it is differentiable) and multiplied by a number.

 

[S2000magician]

MathMan wrote:Actually, there is a closed-form for this. It can be derived directly by using the closed form for the pv of an annuity equation and the definition of Duration.
Duration = - dP/dr * (1+r)/P
P = D / r * (( (1+r)^n -1 ) / (1+r)^n )
dP/dr = D * ( (1+r) - (1+r)^(n+1) + n*r ) / ( r^2 * (1+r)^(n+1) )
Duration = - D * ( (1+r) - (1+r)^(n+1) + n*r ) / ( r^2 * (1+r)^(n+1) ) * (1+r)/P
It can also be derived generally using a specific form of a combined arithmetic and geometric series:
Note that:
Summation(i = 1 to n) i * x^i
is just equal to dS/dx * x where:
S = Summation(i = 1 to n) x^i
a simple geometric series which he know has a closed form and will obviously remained closed
when deferentiated (if it is differentiable) and multiplied by a number.

I was too hasty in saying that there wasn’t a formula. (The truth is that I should have grabbed a piece of paper and worked it out myself; it’s not that difficult.)
Thanks!