Implementation¶

General Idea¶

You have finally done all the methods to get your indicated rates for each of the classes in the form of rate relativities and a base rate. But due to some regulatory constraint you are forced to alter your results.

For example, if you found the rate relativities as such:

Territory	Premium	Current Relativity	Indicated Relativity
1	$195,000	0.85	0.75
2	$475,000	1.00	1.00
3	$330,000	1.30	1.20
Total	$1,000,000

Know that changing the relativity doesn't change the total premium collected. It just changes the distribution of the $1,000,000 between the territories.¹
Thus, when we change the relativities, the base rate also HAS TO CHANGE!

How do we measure the change in the base rate? We do it by finding the Off-Balance Factor. (OBF for short) given by "Exposure Weighted Average Rating Factor" (EWARF, I read it as "e-warph"): $$ OBF=\dfrac{\text{Current EWARF}}{\text{Proposed EWARF}} $$

What is EWARF? Think of it as the effects of all the rating factors distilled into one value. Thus, you can just take the total exposures and multiply them with the EWARF to see how much premiums you are collecting in total². So, in our case here:

\[ \text{Exposures}\times \text{Base Rate (\$)} \times \text{Current EAWRF} = \text{Total Premium} = \$ 1,000,000 \]

When a relativity change happens, then the exposures don't change, neither does the base rate. The only thing that changes is the $\text{EAWRF}$ and so we find the $OBF$ and apply it to the $\text{Base Rate}$ in order to get revenue neutral changes.³ So, with our new rates we have:

\[ \text{Exposures}\times OBF\cdot\text{Base Rate (\$)} \times \text{Indicated EAWRF} = \text{Total Premium} = \$ 1,000,000 \]

OBF Adjustment note¶

If we do not apply the OBF then the Rel change factor then the following would be the projected premiums:

Territory	Current Prem	Projected Prem
1	$195,000.00	$172,058.82
2	$475,000.00	$475,000.00
3	$330,000.00	$304,615.38
	$1,000,000.00	$951,674.21

You can clearly see that we don't have a revenue neutral change in the relativities since the total premiums have decreased.

To Summarize

OBF offsets the effect of the changing relativities

Implementation Scenario¶

Let's say that the company has decided to change the rates (i.e. relativities and base rate) in such a way that the total revenue increases by 20\%. Subject to the constraint:

We cannot increase the premiums collected (e.g. $330,000 currently for Terr 3) to more than 25%, and cap it beyond that point, which may lead to a shortfall in the desired premium.⁴

So, we want the total premium to increase to $1,000,000 \times 1.20 = 1,200,000$.

Territory	Rel chg factor	Total Change
1	0.882	11.26%
2	1.000	26.09%	> 25\%
3	0.923	16.39%
Total	0.952	20.00%

$\text{Rel Chg Factor} =\dfrac{\text{Proposed Rels}}{\text{Current Rels}}$
Total Rel Change Factor = $\dfrac{\sum\limits_{t=1}^3 \text{Premium}_{t}\times \text{Rel Chg Factor}_{t}}{\text{Total Premiums}}$
- The exposure weighted average of the relativity change factors ($\text{EWARF}$)
- This gives the change factor of the total premiums. So, if we had only changed the relativities (and not the base rate, adjusted by the $OBF$) we would have a total premium of $1000k \times0.952= 952k$
$\text{Total change} =\text{Rel Chg Factor}\times OBF \times 120\%$ (for each territory)
- This total change reflects: the effect of the relativity change (Rel Chg Factor), the base rate adjusted for a revenue neutral change (OBF) and the required total change (120\%)

The "Adjustment" part #1¶

We note in the table above that, the relativity of Territory 2 has overshot that cap (26.09\% > 25%) so we need to ensure that the total premium for Territory 2 doesn't increase to more than $475k \times 1.25 = 539.75k$.[^5] In order to do so we will have to reduce the Territory 2 rate. But note the following things that happen:

Question

You might think, the previous and the current base rate relativities are the same (i.e., $1$) and we are increasing all rates by $20\%$ then how will the Territory 2 relativity increase by more than $25\%$ (it should just be $20\%$, right?). Well, this increase happens solely due to the $OBF$ we multiplied with the base rate to create a revenue neutral change for the relativities. So, the $26.09\% = 1.20 \times OBF$

On reducing Territory 2 rate, we will experience a shortfall in total required premium of $1200k$.
But, Territory 2 happens to be the base rate. So we are in effect reducing the base rate.
If we reduce the base rate, premiums for all the other territories will also reduce.
So, to ensure we have a total premium increase of $20\%$ (and not less), we will have to increase the other two relativities (Territory 1 & 3) by some amount that compensates for:
- Capping of Base Rate
- Territory 1 & 3 premiums reduced due to $\text{Base Rate}$ reduction

So, how should we change the Indicated relativities Ter 1 & 3 relativities? We can just inflate them by a factor $X$, which can then be used to formulate the following equation:

\[ 475(1.25) + 195\left(\dfrac{0.75X}{0.85}\right)(1.25)+ 330\left(\dfrac{1.2X}{1.3}\right)(1.25) = 1200 \]

Here, we multiply everything by $1.25$, the increase in the base rate (applies to all)
We also multiply by the $\dfrac{\text{Indicated Rel}\times X}{\text{Current Rel}}$ as that is the final change in relativity.
We are not using the $OBF$ anywhere here, since it was needed to adjust for a revenue neutral change owing to the changing relativities. But now, we have fixed the base rate at $475(1.25)$ and so it doesn't appear in the picture⁵

Then, solving for $X$, we get $1.017$.

Thus the new relativities should be:

Territory	Final rel	How?
1	0.7631	$0.75 \times 1.017$
2	1.0000
3	1.2210	$1.20 \times 1.017$

With the $\text{Final Base Rate =}1.25 \times \text{Current Base Rate}$

For example, if the indicated relativities were (1,1,1) then the premium charged to each of the territories in aggregate would be 333,333.33 each... so that it sums up to 1,000,000. ↩
For example, say our base rate is $100 and total exposures (summed across all territories) is 8000. Note that these values are not given in the table (I am making them up for the example) but they do inherently exist even if not directly stated to us, in the premium values given. Then my EWARF is going to be $\dfrac{1000000}{100\times 8000}=1.25$ which is like the net effect of all the relativities (0.85,1.00,1.30). ↩
Meaning, without changing the total premium ↩
So, say if territory 3's indicated total premium increases to more than $330k\times 1.25= 412.5k$ , (say $420k$) we will have to cap the total premium to $412.5k$. In which case, we will have a shortfall in premium of $420k - 412.5k = 7.5k$ which has to be compensated by increasing the premiums in other territories relativities. ↩
However, if say some other territory had crossed the cap limit (and not the base rate), the base rate would then be $\text{Current Base}\times OBF \times 1.20$ (where 1.20 is the factor increase in total premiums.) ↩

Territory	Final rel	How?
1	0.7631	\(0.75 \times 1.017\)
2	1.0000
3	1.2210	\(1.20 \times 1.017\)