As the shape parameter goes to infinity (with the scale parameter also going there), the limit is normal. In general, a gamma distribution with a large shape parameter looks fairly normal. So in simulating the sum of many gammas, it is not surprising that the sum looks like a gamma. The Central Limit Theorem tells us that the sum looks normal and with the gamma also looking normal, the sum also looks gamma.
You can use EXCEL to perform Glenn's suggestion (however, his implementation of his suggestion is more clever and more efficient than doing it in EXCEL). You first have to replace the true gamma distributions with a discrete version with probability at equally spaced points. Apply the Fast Fourier Transfer to each, then multiply. Inverting the Transformation produces the answer.
I have prepared an EXCEL spreadsheet that does this for 6 gammas. A comparison to the normal approximation indicates that 6 is probably not enough. I could not find a way of automating the FFT (it can be called from Tools|Data Analysis, but that puts numbers, not formulas, into the worksheet), which makes it less useful for ongoint work.
Limits and deductibles can be applied directly to the discretized distributions.
The spreadsheet can be downloaded from www.drake.edu/cbpa/acts. You must enter the address with all lower case letters.
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Stuart Klugman, FSA, PhD
Principal Financial Group Professor of Actuarial Science
Drake University
2507 University Avenue
Des Moines, IA 50311 USA
ph: 515-271-4097
e-mail: Stuart.Klugman@drake.edu
Drake Act. Sci. web site: www.drake.edu/cbpa/acts
Visit the CAS Web Site at http://www.casact.org
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