Brosius - Why does the negative binomial result in more link-ratio credibility

[Five in Two]

From the paper:

The negative binomial distribution has more variance than the Poisson distribution with the same mean; as a result, we have less confidence in our prior estimate of expected losses. Given a value of z that is larger than predicted, we are thus relatively more willing to increase our estimated ultimate claim count than we were when Y was Poisson; this implies a larger b.

I assume the Z is larger if the link-ratio is getting more credit. However, which portion of Z is moving (the VHM or the EVPV or both)?

[CHACAL7781]

L(X): estimate of ultimate lossses = Z * (X/d) + (1-Z) * E(Y)

(X/d): Estimate of Ult losses based on the link ratio method
E(Y): Estimate of Ult losses based on the budgeted loss method

Z = VHM / (VHM + EPV)

Under the Poisson (Variance = Mean) case VHM = EVPV, so Z = 0.5. This gives equal weight to both methods Link-ratio and Budgeted loss). This is the B-F method.

Under the Neg Bin Case (Variance > Mean), so Z > 0.5. This will give more weight to the Link-ratio method (again since Z > 0.5) whichi will result in more Link-ratio credibility)

Hope this helps.

[rfa3232]

BF: z=d
your answer doesn't seem right, altho nicely written

[src4sd]

From the paper:

The negative binomial distribution has more variance than the Poisson distribution with the same mean; as a result, we have less confidence in our prior estimate of expected losses. Given a value of z that is larger than predicted, we are thus relatively more willing to increase our estimated ultimate claim count than we were when Y was Poisson; this implies a larger b.

I assume the Z is larger if the link-ratio is getting more credit. However, which portion of Z is moving (the VHM or the EVPV or both)?

VHM is the variance of the ultimate claims distribution (either Poisson or Negative Binomial). EVPV is the variance in the reporting process (which is the same binomial process, regardless of which claims distribution you are assuming). When you switch from Poisson to Negative Binomial in ultimate claims but keep the same reporting process, VHM goes up and EVPV stays the same, so Z goes up, and more weight is given to the link ratio method.

When I forget the details, I like to think about extreme cases. Suppose EVPV = 0. This would mean no variance in reporting, e.g. exactly half (or some other fixed fraction) of the claims will be reported. Then Z=1 and you use the link ratio method. Conversely, if VHM=0, that means there's no variance in the ultimate claims, they will always equal expected losses. Then Z=0 and you use the budgeted loss method.

[2M]

L(X): estimate of ultimate lossses = Z * (X/d) + (1-Z) * E(Y)

(X/d): Estimate of Ult losses based on the link ratio method
E(Y): Estimate of Ult losses based on the budgeted loss method

Z = VHM / (VHM + EPV)

Under the Poisson (Variance = Mean) case VHM = EVPV, so Z = 0.5. This gives equal weight to both methods Link-ratio and Budgeted loss). This is the B-F method.

Under the Neg Bin Case (Variance > Mean), so Z > 0.5. This will give more weight to the Link-ratio method (again since Z > 0.5) whichi will result in more Link-ratio credibility)

Hope this helps.

Take out the reference to the B-F method and this looks OK to me

[src4sd]

L(X): estimate of ultimate lossses = Z * (X/d) + (1-Z) * E(Y)

(X/d): Estimate of Ult losses based on the link ratio method
E(Y): Estimate of Ult losses based on the budgeted loss method

Z = VHM / (VHM + EPV)

Under the Poisson (Variance = Mean) case VHM = EVPV, so Z = 0.5. This gives equal weight to both methods Link-ratio and Budgeted loss). This is the B-F method.

Under the Neg Bin Case (Variance > Mean), so Z > 0.5. This will give more weight to the Link-ratio method (again since Z > 0.5) whichi will result in more Link-ratio credibility)

Hope this helps.

This is not correct. The author is confusing EVPV and true expected value. And, Z=1/2 does not imply the BF method. The BF method corresponds to Z = 1/LDF. The LDF can and does cover a wide range. It can be 1.5, 2, 3 or higher for immature ages and 1.05, 1.01, or smaller for ages near ultimate.

BF DOES correspond to Poisson. Neither BF nor Poisson necessarily correspond to Z=1/2.

[rfa3232]

Take out the reference to the B-F method and this looks OK to me

better get studying

[2M]

Under the Poisson (Variance = Mean) case VHM = EVPV, so Z = 0.5..

OK :duh: I read this too fast

[src4sd]

You were taken in by how well written it was, I think!

[rfa3232]

lol indeed

[Archen]

From the paper:

The negative binomial distribution has more variance than the Poisson distribution with the same mean; as a result, we have less confidence in our prior estimate of expected losses. Given a value of z that is larger than predicted, we are thus relatively more willing to increase our estimated ultimate claim count than we were when Y was Poisson; this implies a larger b.

I assume the Z is larger if the link-ratio is getting more credit. However, which portion of Z is moving (the VHM or the EVPV or both)?

First of all, these concepts are critical.

Z=VHM/(VHM+EVPV)=1/(1+EVPV/VHM)
VHM=Var[E(X|Y)]
EVPV=E[Var(X|Y)]
X=reported calims
Y=ultimate claims

The switch from Poisson to Negative Binomial is only changing the distribution of Y. The distribution of (X,Y) is just switch from Posson-Binomial to Negative Binomial-Binomial. Notice that the second process Binomial(Y,d) for X|Y distribution is not changed at all.

Now let's see what happens.

VHM=Var[E(X|Y)]=Var[Y*d]=d^2*Var(Y)
EVPV= E[Var(X|Y)]=E[Y*d*(1-d)]=d*(1-d)*E(Y)

The switching makes Var(Y) lager but E(Y) unchanged. So VHM becomes larger, and EVPV is unchanged at all.

Look at Z=1/(1+EVPV/VHM) now, and you'll find Z is greater.

Hope this helps.

[peem]

L(X): estimate of ultimate lossses = Z * (X/d) + (1-Z) * E(Y)

(X/d): Estimate of Ult losses based on the link ratio method
E(Y): Estimate of Ult losses based on the budgeted loss method

Z = VHM / (VHM + EPV)

Under the Poisson (Variance = Mean) case VHM = EVPV, so Z = 0.5. This gives equal weight to both methods Link-ratio and Budgeted loss). This is the B-F method.

Under the Neg Bin Case (Variance > Mean), so Z > 0.5. This will give more weight to the Link-ratio method (again since Z > 0.5) whichi will result in more Link-ratio credibility)

Hope this helps.

Actually, if you use Development Formula 2 for L(x), L(x) = Q(x) for the poisson-binomial case. For the Poisson-Binomial, Z=d.

If you think about it, Y~poisson(\mu) and X|Y~binomial(Y,d). So,

EVPV = E_y[Var_y(X|Y)] = E_y[Yd(1-d)] = dE_y[Y] = d(1-d)\mu.

And,

VHM = Var_y[E_x(X|Y)] = Var_y[Yd] = d^2Var_y[Y] = d^2\mu.

Then,

Z = \frac{1}{1 + \frac{EVPV}{VHM}} = \frac{1}{1 + \frac{d(1-d)\mu}{d^2\mu}} = \frac{1}{1 + \frac{(1-d)}{d}} = \frac{d}{d + (1-d)} = d.

Plugging Z = d into Development formula 2,

L(x) = Z\frac{x}{d} +(1-Z)E(Y) = d\frac{x}{d} +(1-d)E(Y) = x +(1-d)\mu = Q(x)_{poisson-binomial.

I think in general, Poisson-Binomial corresponds to B-F method. R(x) is constant in both cases so our reserve estimates stay the same in both cases and do not depend on x claims reported at year end.

It's clear that whenever Z = d (poisson-binomial or not), then L(x) corresponds to B-F. Simply because then R(x) = constant, as Q(x) = x + constant and R(x) = Q(x) - x = constant.

[rfa3232]

First of all, these concepts are critical.

Z=VHM/(VHM+EVPV)=1/(1+EVPV/VHM)
VHM=Var[E(X|Y)]
EVPV=E[Var(X|Y)]
X=reported calims
Y=ultimate claims

The switch from Poisson to Negative Binomial is only changing the distribution of Y. The distribution of (X,Y) is just switch from Posson-Binomial to Negative Binomial-Binomial. Notice that the second process Binomial(Y,d) for X|Y distribution is not changed at all.

Now let's see what happens.

VHM=Var[E(X|Y)]=Var[Y*d]=d^2*Var(Y)
EVPV= E[Var(X|Y)]=E[Y*d*(1-d)]=d*(1-d)*E(Y)

The switching makes Var(Y) lager but E(Y) unchanged. So VHM becomes larger, and EVPV is unchanged at all.

Look at Z=1/(1+EVPV/VHM) now, and you'll find Z is greater.

Hope this helps.

very nice, insightful

[davidchen]

If the conclusion for the Neg Bin - Bin case is that:

"the negative binomial distribution has more variance than the Poisson distribution with the same mean; as a result, we have less confidence in our prior estimate of expected losses, and given a larger value of x than predicted, we are more willing to increase our estimated ultimate claim count",

anyone know why TIA say it doesn't answer any of Huge White's questions. I would think that it answers Huge White's 3rd question if we are increasing our estimates.

[OldSchool]

If the conclusion for the Neg Bin - Bin case is that:

"the negative binomial distribution has more variance than the Poisson distribution with the same mean; as a result, we have less confidence in our prior estimate of expected losses, and given a larger value of x than predicted, we are more willing to increase our estimated ultimate claim count",

anyone know why TIA say it doesn't answer any of Huge White's questions. I would think that it answers Huge White's 3rd question if we are increasing our estimates.

This interpretation parses Hugh White's third answer literally. His third answer states: "Increase the bulk reserve in proportion to the increase of actual reported over expected reported." [emphasis added]

Brosius explains at the top of page 9 that our estimate does increase, but the increase is not proportional.