# Yearly Aggregated Loss Distribution (operational risk)

Firstly I should mention I am quite unfamiliar with the subject (operational risk). And I am also beginner in risk management. It is also worth to mention that this task is on academic level .

I intend to create something called yearly Aggregated Loss Distribution (also known as Loss Distribution Approach or LDA). It should represent probability of achieving certain amount of loss in entire year that is created by combining frequency of losses and severity of losses distributions.

My data are daily entries which defines the number of losses (frequency) and amount (severity) and covers 7 years, so I can easily transform them to monthly/quarterly/yearly. I am concerned about my methodology, however it seems to me silly and quite obvious since I am beginner in the subject I wish to make sure I do not make any logical mistake. So in steps my procedure looks like this:

1. Transformation from daily to yearly
2. Fit discrete distribution to loss frequency of yearly data
3. Fit continuous distribution to loss severity of yearly data
4. Use fitted distribution and their parameters for simple Monte Carlo simulation where I generate randomly number of losses and severity of each loss.
5. I repeat the process till I have XXXX number of records, then I add each loss (each has assigned own random severity) for each record
6. I make histogram out of summarized frequency and severity which is mine Aggregated Loss Distribution for a year (since distributions where fitted to yearly data).

If my description seems unclear here is the short video that basically shows exact same thing I am doing:

So simplifying my question: To achieve same thing like in this video should I use distributions fitted for yearly data? Intuitionally I would say yes, but I need to be sure.

• Given that links can easily be broken, I'd advise removing the link and explaining the contents of the attached video. – Ed P Jan 14 '17 at 16:03
• Is there anything I haven't discussed in my answer that needs further exploration? – Ed P Jan 15 '17 at 5:28
• Actually it's fine to keep the link, but still describe anything relevant that it contains (that you haven't already) on the assumption that the link may disappear at any time – Glen_b -Reinstate Monica Jan 15 '17 at 10:17
• Since this appears to be academic work, please see our help center (the section under homework there, but the guidelines there apply to a broader class of questions than just homework). – Glen_b -Reinstate Monica Jan 15 '17 at 10:26
• You know that LDA fell out of favor, right? In fact it failed in a spectacular fashion. Here's why oprisklive.prod.incisive.pro.pugpig.com – Aksakal Mar 25 '18 at 2:49

Your approach is as silly as everyone else's. Look at what the FRB is doing to forecast losses for CCAR Banks, their approach is described in: Dodd-Frank Act Stress Test 2016: Supervisory Stress Test Methodology and Results, June 2016. Read “Operational Risk Model Enhancement” section in Box 1 and “Losses Related to Operational-Risk Events” Section in Appendix B.

They using forecast combination approach where they average outputs of historical simulation and panel regression on macroeconomic variables.

The historical simulation is essentially a bootstrapping variation of LDA. It assumes that the losses are compound random variables, such as Poisson compound. They don't call it LDA anymore, because the term is out of favor in US banking supervision. The main difference with LDA is that instead of modeling the severity they bootstrap it from actual losses in the event database. In other words each loss in the compound Poisson is a random draw from the actual event losses observed historically.

The regression part is a simple panel regression on a bunch of variables such as firm characteristics and macroeconomy. You can get a flavor of the model from the FRB paper: U.S. Banking Sector Operational Losses and the Macroeconomic Environment

We have seen this question ( or one like this ) before . It involves using daily data to compute aggregated forecasts yielding the probability of making a goal. Look at Predict number of users for a discussion of how Proctor & Gamble phrased the question. You might also look at http://www.autobox.com/cms/index.php/blog for a discussion of how to actually use daily data to form a useful model. I have been one of the developers of AUTOBOX which might be useful to you in showing you an approach. This of course requires the user having daily data for every day . If you have any possible supporting variables they can also be considered for inclusion. In specific it might be interesting to predict/model the amount of losses as it relates to the # of losses.

Since you are new to this, I think it's best to walk through an example. Let's consider the case of a single risk $Z$ (i.e. a certain type of operational risk).

The Loss Distribution Approach can be described as:

$$Z=\sum_{i=1}^{N}X_{i}$$ where $N$ is the number of events (frequency) over one year and $X_{i}$ is the severity of loss $i$. $N$ is modelled as a discrete random variable with probability mass function:

$$\quad\quad\quad p_{k}=\text{Pr}[N=k],\,\,\,k=0,1,2,\ldots$$ $X_{i}$ are iid and modelled with a continuous distribution function $F_{X}(x)$. Now, it is important to note the assumption we make that $N$ and $X_{i}$ are independent for all $i$.

Now, based on your data, you can find suitable distributions to describe the frequency and severity of your losses. The exact method you use to find a suitable distribution will depend on the context, but finding the MLE is usually a good option.

I'll describe an example now. Let's assume we're considering a single (operational) risk $Z$. Let's assume (based on suitable fitting methods) that the distribution of severity of losses are independent and identical and follow:

$$X_{i}\sim \text{LN}(\mu=1,\sigma=2)$$

Similarly, we can say the frequency of losses follows:

$$N\sim \text{Poisson}(\lambda=1)$$

Now, I can only assume (having not watched the linked video) that your goal is to evaluate $E[Z]$, $\text{SD}(Z)$, $\text{VaR}_{q}[Z]$ and $\text{ES}_{q}[Z]$ etc. via Monte Carlo methods. Luckily for us there are closed-form, analytical solutions for the expectation and standard deviation (allowing us to check our simulation results).

To perform the simulations I used MATLAB with $K=10^{6}$ simulations.

%Set vector of number of simulations for loss Z:
K=10^6;

%Set parameters to be used for Lognormal and Poisson random variables:
lambda=1;
mu=1;
sigma=2;

%Initialize annual loss amount vector:
Z_vec=zeros(K,1);

%Iterate for size of annual loss sample:
for k=1:1:K

%Simulate Poisson value:
p_rnd=poissrnd(lambda);

%Initialize loss severity vector, if Poisson>0:
if p_rnd>0

X_vec=zeros(p_rnd,1);

for m=1:1:p_rnd

%Simulate Lognormal value:
X_vec(m,1)=lognrnd(mu,sigma);

end

%Otherwise, set severity vector to zero:
else

X_vec=0;

end

Z_vec(k,1)=sum(X_vec);

end


So the vector Z_vec contains the $10^{6}$ simulations for $Z$. From here it's all very straightforward, calculating the mean, standard deviation and whatever else you are interested in.

From my simulations, I obtained:

\begin{align} E[Z]&=20.1318\\ \text{SD}(Z)&=143.7883 \end{align}

The analytical solutions are (simple compound distribution formulae):

\begin{align} E[Z]&=E[N]E[X]=20.0855\\ \text{SD}(Z)&=\big(E[N]\text{Var}(X)+\text{Var}(N)E[X]^{2}\big)^{1/2}=148.4132 \end{align}

Similar calculations can be made for any sort of risk measure you would like. Keep in mind this method can be generalized to include more risks (i.e. $Z_{i}$, $i=1,2,\ldots$) and include dependencies between the $Z_{i}$.

In terms of your confusion about the time horizon of losses, the time horizon you set is purely up to you. If you want to consider yearly losses, then partition your 7-year period into $j$ yearly periods. For any given yearly period $j$, the observed frequency of losses, $n_{j}$, will be the count of losses. The $n_{j}$ go towards estimating the frequency distribution $N$. Similarly, the severity of those losses in all the yearly periods go towards estimating the severity distribution $X_{i}$.

Hopefully the following diagram illustrates the point well, where in this example there is a 3-year period split into $j=3$ 1-year periods. Each $n_{k}$, $k=\{1,\ldots,j\}$ contributes to estimating $N$ and there are 90 $X_{i}$ observed over the 3-year period which go toward estimating $X$. • Given this appears to be someone's homework (at least they explicitly say that it's academic, and it's very close to a number of homework questions we've had before), our guidelines on homework ask for answerers to give guidance and hints (see the section on homework at the link) rather than what looks like essentially a complete solution (what have you left them to do?). Please keep our guidelines in mind for future answers – Glen_b -Reinstate Monica Jan 15 '17 at 10:22
• Thank you very much for clarification and lot of helpful additional information – Alexandros Jan 20 '17 at 17:51
• @Alexandros Is there anything else that needs further discussion? – Ed P Jan 25 '17 at 10:31