## Thursday, December 8, 2011

### 8 Steps to Monte-Carlo Insight

I attended a day-long symposium several years back where a portion of the agenda was devoted to risk management. The primary speaker’s company had a policy of hedging 100% of their exposure to a raw material, and they executed this by locking in prices one year in advance.
How different is this to making spot purchases, only one year in advance? Is this really hedging? It seems like putting all the “year-ahead eggs in one basket” would produce similar volatility. Wouldn’t a dollar-cost average or roll-in strategy be better?
The great thing about today’s age is that we can answer these questions using the wonders of technology!

Can We Test It? Yes We Can!
In “Blogful of Thanks”, I mentioned Lee Crumbaugh and his focus on scenario analysis. “What is the weather going to be like today?”. “Hmm, that makes me think of scenario analysis” “What do you think of Sears threatening to move out of Illinois because the state is not going to provide tax breaks?” “Hmmm, my mind immediately jumps to scenario analysis
I am really no different than Lee, except we need to replace “scenario analysis” with “Monte-Carlo analysis”. Let me tell you, a lot of things make me want to do a Monte-Carlo analysis!
If we think about it, the two are really not that different. A Monte-Carlo analysis is merely several thousand scenarios played out in rapid succession.
So let’s do a Monte-Carlo analysis!

Step 1 – Define the Model
For many interest rate, currency, and commodity products, prices tend to “mean revert”. This term is “statistical-ese” for the fact that when prices are too high, they tend to move lower, and when prices are too low, they tend to move higher. In other words, they are more likely to (but not guaranteed to) move towards the mean.
This can be described by the following simple mean reversion equation:

The alpha term (α) is the mean-reversion parameter. Mu (µ)is the long-run mean. Sigma (σ) is the standard deviation, and ϵ is the error term ( a random variable from a defined distribution).
Step 2 – Populate the Parameters
One of the features of many interest rate, currency, and commodity markets is that volatility decreases as we “move out the forward curve”. To see this, look at US corn at two different time periods:

The price for the “near” months is widely different, yet for the “out” months is pretty much the same. For some interest rate markets, the near month volatility can be twice that of the out months.
Often, determining the parameters of the model will involve statistical analysis of historic data (this can be a blog post onto itself, so I will not go into it here).
For the sake of this analysis, we are going to populate our corn price curves assuming that the long-run mean is \$5.00.
We will set the alpha (i.e. the speed of mean reversion) parameter at .02 for the near months, and increase it gradually in .005 increments. This has the effect of making the out months’ center on the mean more quickly than the near months, consistent with the graph we have just seen.
We will set the standard deviation at .2 and gradually decrease it, again to capture the higher price movements of the near months vs. the out months phenomenon.
We will “draw” (i.e. simulate) our error term from a normal distribution with a 0 mean and 1 standard deviation.
Finally, since each month’s price for corn will to some extent reflect underlying fundamentals that affect every month’s price, we use a correlation matrix and establish correlations through our price curve using a Cholesky decomposition of the correlation matrix. Correlations for this model were .98 for neighbor months and decreasing by .02 for each month further.
These then are the assumptions through the 12 month forward curve:

Step 3 – Identify the Comparisons
In this step, we decide what we are going to model (now that we have a model!).
In this analysis, we are going to look at three different alternatives:
·            No Hedge - 12 units purchased at the prompt month price
·            Year-Ahead Hedge – 12 units purchased at the 12-month forward price
·            Roll-In Hedge – one unit purchased each month for 12 months (i.e. one 12-month purchased 12 months ahead, one 11-month purchased 11 months ahead, etc.)
Step 4 – Push the Button!
Run the Monte Carlo! In this analysis we use 10,000 runs (determining how many runs to use is also a blog post in itself). I ran this in Excel and downloaded the results into R.

Step 5 – Assess the Results
The results of this analysis are as follows:

The lowest average price was the “Year-Ahead Hedge”, followed by the “Roll-In Hedge”. Spot was the highest average price, as well as the highest standard deviation. Graphically (brown is "No-Hedge", red is "Roll-In", and orange is "Year-Ahead"):

Based on this, the symposium presenter was correct in using a “Year-Ahead Hedge” strategy, if (and that’s a big if) prices move as modeled.

Step 6 – Interpret the Results
This result disappointed me because I thought the “Roll-In Hedge” strategy would be better, since it averages in different prices as opposed to the an “all eggs in one basket” strategy. So does it make sense that the opposite would be the case? Why?
The reason I propose is that the “Year-Ahead Hedge”, because it transacts at the highest point of the curve in terms of mean-reversion, and the lowest point of the curve in terms of volatility, is able to overcome the averaging benefit of the “Roll-In Hedge”.

Step 7 – Question the Results
If this is indeed the reason, we should be able to “question the data” or “question the model” to refute or confirm our conclusions.
If the rate of mean reversion and volatility made a difference, we should be able to eliminate the hedge effectiveness by making both of these factors constant across all 12 forward month periods.
This has the following results:

As we can see, there is not a difference in any of the hedges.
If rate of mean reversion and volatility make a difference, then we should also be able to put them in a different place and by doing so generate different results.
To do this, we put the highest mean reversion rate and the lowest volatility at month 6, and then increased gradually on both sides of the forward curve (i.e. towards month 1 and month 12).  If we looked at a graph of these we would see a big “U”.
Under this scenario, the “Roll-In Hedge” does better than either alternative:

The “Roll-In Hedge” does better under this because it is the only hedge scenario which takes advantage of the low volatility / high mean reverting period 6 (and to a slightly lesser extent periods 7 and 8, 6 and 9, etc.). The other two strategies are at the high end of volatility and low end of mean reversion.
These results enhance our initial conclusion, and allow us to generalize, “it is better to concentrate hedging activity in low volatility months and/or higher mean-reverting time periods”.

Step 8 – Identify Further Areas of Research
From Step 7 we have decided that concentrating hedge instruments in low volatility / high mean reverting periods produces the most optimal results.
One area of interest might be to conduct a break even analysis of the parameters. This would address questions such as “By how much does volatility need to be increased at the tail end for hedge improvement to significantly disappear?” or “What mean-reversion level accomplishes a similar effect?”
Another set of questions might surround hedging volumes – “how different are the results of purchases are seasonal rather than constant throughout the year?” or “What level of base load capacity do we need vs. contingent capacity to make this strategy likely to be effective?”

Key Takeaways
A Monte-Carlo analysis of major risks can provide valuable insights into your risk management strategies and processes. In addition, it can provide indicators as to the true value drivers of your organization’s risk and allows you to make adjustments in real-time.
A Monte-Carlo analysis can also provide questions that either allows you to feel more confident in your results, or point to areas where further research is warranted.

Questions
·         Have you performed a Monte-Carlo analysis in the past month?
·         What insights were learned through this analysis?
·         What strategies were undertaken?
·         What critical value drivers are you paying attention to as a result?
·         What are the unanswered questions to pursue in the next round of studies?