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ExamTortoise
11-22-2006, 01:53 PM
In the new Panning paper on Table 1, column 3 he shows "Duration (D)" of Premium ("7.85"), Losses ("7.62") and Expenses ("6.67").

How does he derive those numbers? I understand where all the other numbers on the table come, just not the duration of the components.

Are we to assume those were derived outside the paper?

JABA
03-01-2007, 03:47 PM
In the new Panning paper on Table 1, column 3 he shows "Duration (D)" of Premium ("7.85"), Losses ("7.62") and Expenses ("6.67").

How does he derive those numbers? I understand where all the other numbers on the table come, just not the duration of the components.

Are we to assume those were derived outside the paper?

This bothered me too. I spent far too much time figuring out where those numbers were coming from, but I finally figured it out.

First, take the formula F = [P - E - L/(1+y)] * [d/(1-d)] and break it up into it's compenents (i.e. multiply d/(1-d) through to each term).

Next, for each term separately take the derivative in terms of y.

Third, for each term divide by the PV of the corresponding U/W item.
for example divide [-dP/dy (P * d/(1-d))] by [P * d/(1-d)].

Finally, plug in all of the variables into these formulas.

Note how the formula in the third step mirrors the formula for the total duration [-dF/dy / F]

Hope this helps.

Exponential Decay
03-19-2007, 01:15 PM
I’m still stumped over this one. Let’s just take dP/dy for example.

Given d = cr/(1+y) and d/(1-d) = cr /(1 + y – cr)

Then, P*d/(1-d) = P* cr/(1 + y – cr)

dP/dy = -P * cr /(1 + y – cr)^2

–(dP/dy) / [P*d/(1-d)] = P * cr * (1 + y – cr)^ -2 * (1 + y – cr)/(P * cr)

= 1/(1 + y - cr)

With y = 5% and cr = 0.90, we have 1/(1 + 0.05 – 0.90) = 1/0.15 = 6.67

But this is not what Panning has for premium duration.

What am I missing?

JABA
03-21-2007, 09:05 PM
dP/dy = -P * cr /(1 + y – cr)^2

This isn't quite right.
Recall that P = [S*(k-y) + L]/(1+y) + E
Because P itself is dependent on y the derivative is more complicated than this.

So really we're taking the derivative of:
{[S*(k-y) + L]/(1+y) + E}* cr/(1 + y – cr) with respect to y.

I had several unsuccessful attempts at taking this derivative before I finally got it. I think you'll find that this will work if you can get through the derivative.

You might want to test this with E and L to see that this works.
dE/dy of E*cr/(1+y-cr) and
dL/dy of L*cr/[(1+y-cr)(1+y)]

Hope this helps.

Exponential Decay
03-22-2007, 01:12 PM
Thanks, JABA. I'll give that a try.