duration of inflation linked P&C reserves

[Gareth]

The text http://www.casact.org/library/finanalysis/89val117.pdf says that:

"The cash flows from General Liability losses, however, are inflation
sensitive. If liability losses are sensitive to inflation through the
settlement date (with no lag between inflation and its effects on losses)
then the reserve is equivalent to an asset with a duration of zero years."

I don't quite see why this is true. As a simple example, consider a inflation linked payment in n years time.

The duration of the liability is:

n x amount x (1+inf)^n x (1+ i)^-n / ( amount x (1+inf)^n x (1+ i)^-n )
= n, not zero

Obviously, inflation and interest will vary in reality, so I guess you could break it down into:

interest = real interest rate + inflation risk premium

but the inflation risk premium could be small, large and correlated to real interest rates depending on economic conditions. So it seems very unclear how you would calculate duration allowing for this.

Can anyone clarify what I am missing here?

[Third Eye]

You have the wrong definition of duration. You should think of duration as measuring the derivative of the economic value of the liability with respect to the interest rate. If the liability and the interest rate move exactly together, then there is no change in the discounted cost of the liability when interest rates change, so the duration is 0. In practice, inflation of liabilities and total return on investments are not closely correlated over the short term, but the theory is good. (I am emphatically not an expert on this, but I hope this helps.)

[Gareth]

That's not the definition of financial duration. In terms of derivatives, duration is normally defined as: -(dV/dr) / V

Also, if the liabilities are inflation linked, then its true that in general inflation and interest rates are correlated, so the assets should more or less match the liability increases... but that's not the same as saying zero duration.

If duration is zero then the liabilities will not change when interest rates change. This is a fallacy, since the NPV of the liabilities will change according to the effect of the real interest rate.

It's also worth noting that you can define real duration using the real interest rate, and then match using inflation linked bonds in the same way as you would for nominal liabilities with nominal bonds.

I am of the opinion this zero duration statement is an error that has propagated through a number of papers over the years.

[PAC]

I think issue boils down to SF's working definitions:
- [intuitive duration] = lag in economic/financial recognition of interest rate change (my working defn)
- [financial duration] = - (dV/dr) / V (well established bond result)

Using the 'intuitive' definition SF says...
=> [intuitive duration of equities]
= [intuitive duration of (most) property losses]
= [intuitive duration of inflation sensitive/uncertain timing liabilities] = zero
i.e. immediate recognition at the margin, while

=> [intuitive duration of bonds]
= [intuitive duration of nominal value/fixed timing liabilities] > 0
i.e. delayed financial recognition when disposing of bonds to make loss payments.

SF notes that the 'intuitive' and 'financial' definitions of duration:
- differ for inflation-sensitive/uncertain timing instruments
- are consistent for nominal/fixed timing instruments

[PAC]

A little more discussion on this seems warranted given the following statement...

Originally Posted by Gareth

I am of the opinion this zero duration statement is an error that has propagated through a number of papers over the years.

[Gareth]

PAC: Where in the paper did you find this intuitive duration definition?

As far as I can see from the paper, they refer to Macaulay duration, in particular, if you are trying to do duration convexity matching then you can only work with the financial definition.

There's lot of discussion in investment literature on the duration of inflation linked bonds, but generally little conclusions. It's also interesting to look at real duration and real convexity (i.e. change in liability w.r.t. real interest rate instead of nominal rate).

I've seen the zero duration of inflation linked P&C liabilities mentioned in a couple of papers, I'll dig them out later on.

[PAC]

Try thinking of the equity example this way:
- the current tangible value of an equity is some amount that represents, assuming no future growth opportunities i.e. no reinvestment, no new businesses etc, the PV of expected dividend payments.
- changes in expected inflation rate => changes in the expected dividend amounts, the opposite of the vanilla bond case.
- if the linkage between expected inflation and expected dividends is "perfect" then the current tangible value remains unchanged and duration = 0

An inflation sensitive insurance liability - with some (notional) current tangible value and future payments that adjust based on expected inflation - has analogous duration.

[Gareth]

But if you consider duration to represent sensitivity to changes in interest rates, I'm still not convinced this is right.

Value of equity = d / (i - g) where d is dividend rate, i interest rate and g is growth rate

So, if g = inflation and i = nominal interest rate then i-g = real interest rate.

Since the real interest rate does change continually as a stochastic process, I would argue that equities have a non-zero duration. The same argument applies to inflation linked insurance liabilities.

Another paper that takes this zero duration view is "Interest Rate Risk of Property-Liability Insurers" by Jin Park (see page 32) - see http://www.aria.org/meetings/2006pap...20Insurers.pdf

[PAC]

There you have it...

Your model (DDM) assumes that the PV equity is derived from some fixed dividend amount, inflated/discounted in perpetuity.

What I said is that the future dividend stream is derived from the current tangible/intrinsic equity value. This is not the DDM, but that's the point; DDM for equity durations doesn't work well.

I'm not saying that you will realize perfect inflation sensitivity - and a zero duration - in practice; studies I've seen peg equity durations between, say, 2 and 10 years which, while greater than zero, are also much lower than the DDM-based durations.

[Gareth]

Thanks, I guess what you are really saying is we can come up with a potentially better model of the equity value following a fundamental analysis approach - and under this approach you arrive at a lower duration of equities.

Now returning to the liability side. I personally would measure the economic value by projecting the expected future payments taking into consideration the term structure of claims inflation and the current yield curve for calculating the NPV.

It's probably fair to say that this is a sensible definition of economic value and under this framework you can calculate financial duration in the classical sense. This will lead to a duration very different to zero.

At the end of the day, my view is that the measure of duration should be appropriate for implementing duration convexity matching. I'm just not convinced that the alternative definition makes sense from this perspective.