# Contribution by each factor

Must admit it has been many years since I have done advanced calculations or anything remotely like it, so please bear with me. I have problem that I hope to get some help with. Say I have below dataset:

|   Quarter |   Exposure |   Loss Given   |   Probability of  |   Estimated Loss  |
|           |            |   Default (%)  |   Default (%)     |                   |
|-----------|------------|----------------|-------------------|-------------------|
|       Q1  |   100,000  |          3.0%  |             5.0%  |            150.0  |
|       Q2  |   150,000  |          2.8%  |             5.5%  |            231.0  |


Where Estimated Loss = Exposure x Loss Given Default x Probability of Default.

My problem now is, I would like to be able to explain the difference between Estimated Loss in Q2 and Q1, by explaining how much of the delta is due to the change in each parameter, or how much each parameter contributes of the change - if that makes sense?

Ideally I would like to be able to say something like below:

231 - 150 = 81, of which:
+76 due to increase in Exposure
-10 due to decrease in Loss Given Default
+15 due to increase in Probability of Default


The reason behind showing it like this, is to make it more "understandable" for the reader who is not aware of the mechanics behind the numbers.

My initial thinking was to see what effect it had on Estimated Loss by changing only one parameter at a time; but I will miss some multiplier-effect, since the parameters are not independent.

Also if done this way - the order of how I show the change affects the outcome. Say I first change Exposure and see the result, next up LGD and lastly PD - then, because they are dependent, the effect is larger in PD than in Exposure - and other way around if I change the order. That was when I realised that I should probably have brushed up on my math skills.

1) Would it even be possible/make sense to explain the delta as above, where it can be determined how much each paramter contributes of the delta?

2) If it is possible; I assume that I need to do some sort of Partial Differential equation?

Thanks for you help :-)

The easiest consistent explanation is using multiplication: The Q2 estimated loss of $231$ is $\frac{231}{150}=1.54$ times the Q1 estimated loss of $150$. This is a combination of

• The exposure being $1.5$ times the previous figure
• The loss given default being about $0.933$ times the previous figure
• The probability of default being $1.1$ times the previous figure

and this is consistent with $1.5\times 0.933 \times 1.1 \approx 1.54$ - it does not matter which order you do the multiplication

It is not that far away from saying $50\% -6.7\% + 10\%$ is about $54\%$, except that that is not precise (they actually add up to $53.3\%$ and other examples could be worse) but this would be talking about percentage changes in percentages, which is often confusing

If you looked at the effect on the old estimated loss of $150$ of each change if the others had not changed, you would have got $+75$ (not $+76$), $-10$, $+15$ which add up to $+80$ rather than the actual $+81$, and other examples could be worse

• To get the +76, I took each delta's relative size of the total change (50% / 53.3% = 93.8% of the total change of 81 = 76). I guess based on your answer, that no matter what, it will be some approximation? Which one would you consider using. – ssn Aug 3 '18 at 8:47
• @ssn - I would just give the multiplicative factors $1.5,0.933,1.1$ and stop there. I suspect all the other approaches are as likely to confuse as enlighten – Henry Aug 3 '18 at 9:17