# How to use convolution of 2 loss distributions?

My project is to set a distribution loss of PNL in CHF into a distribution loss in USD. To do this I will need to have a distribution loss of the spot rate CHF/USD. I have simulated this distribution.

So have 2 distributions:

• A: One is the simulated spot rate of USD/CHF

• B: The other is the loss distribution of the PNL of my fund in CHF

I need to find out a way to multiply these 2 distributions in order have, as final result, the distribution loss of the PNL in USD. Since multiplication is not possible I chose convolution between both distributions.

In R I used the function "convolve. However, If I run

convolve(A,B,"open")

the result does not at all seems right to me, because the total number of entries is the sum of the entries of both distributions. Moreover, the results shows a Gaussian curve with only negative entries.

Could you please tell if I should apply convolution in the case? If no which method should I use to multiply both distributions?

• Why are you convolving those two things? What does that achieve here? – Glen_b Jun 8 '17 at 5:59
• I need to chage the currency of the PNL distribution, using the spot rate distribution. It is a simple multiplication, for distributions I will have to use convolution – mister_nugget Jun 8 '17 at 8:09
• Back up. I understand perfectly well that changing currencies involves multiplication by the exchange rate. But you completely jumped over the part where you explain how that has anything to do with convolution. So I ask again: Why are you convolving those two things? What does that achieve here? .... don't just say "for distributions I will have to use convolution" ... how on earth does convolution achieve what you need to do? You can't just throw a random calculation at a problem, so what is the result you're using that establishes that convolution does what's needed here? – Glen_b Jun 8 '17 at 9:17
• I cannot multiply both definitions points by point, I need somehow a way to smear the values of the exchange rate distribution along the PNL distribtion. It looks like convolution can do this but if you have a better idea , I am open to hear it – mister_nugget Jun 8 '17 at 9:32
• It sounds like you're just guessing, without any clear idea of what you're actually doing. You're actually trying to compute the distribution of the product of the two random variables are you not? If you don't understand how to achieve that, that's what you should ask about. How are you going to deal with the fact that both things change over time? – Glen_b Jun 8 '17 at 9:43

Assuming we're talking about a single time point (or that the simulation model is not dependent on time), let $S$ be the random variable representing the spot US dollar/Swiss franc exchange rate and $L$ be the random variable representing the loss in Swiss francs. Then (ignoring transaction costs) the US dollar loss is $U=SL$.
So if you simulate many values from the joint distribution of $(S,L)$ (i.e. $[(s_1,l_1), (s_2,l_2), ..., (s_n,l_n)]^\top$ then you get simulations from the distribution of of $U=SL$ by calculating the products $u_i = s_i l_i$.