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Monte Carlo simulation 183
Table 7.2 – Estimating probabilities from the simulation results
Net cash flow
($)
Number of simulations
resulting in this
net cash flow
Probability
estimate based
on simulation
Calculated
probability
−20 000 4 4/10 = 0.4 0.165
−10 000 1 1/10 = 0.1 0.220
0 2 2/10 = 0.2 0.300
10 000 2 2/10 = 0.2 0.180
20 000 1 1/10 = 0.1 0.135
Table 7.3 – The effect of the number of simulations on the reliability of the
probability estimates
Probability estimates based on:
Net cash flow Calculated
($) 50 1000 5000 simulations probability
−20 000 0.14 0.164 0.165 0.165
−10 000 0.18 0.227 0.216 0.220
0 0.42 0.303 0.299 0.300
10 000 0.12 0.168 0.184 0.180
20 000 0.14 0.138 0.136 0.135
of this net cash flow occurring is 4/10. Note that the table also shows
the probability distribution which we would have derived if we had
used a probability tree to calculate the probabilities. The discrepancies
between the two distributions show that the result based on only ten
simulations gives a poor estimate of the real distribution. However, as
more simulations are carried out we can expect this estimate to improve.
This is shown in Table 7.3, which compares the ‘real’ distribution with
estimates based on 50, 1000 and 5000 simulations which were carried
out on a microcomputer.
How many simulations are needed to give an acceptable level of
reliability? This question can be answered by using relatively complex
iterative statistical methods, but a simpler approach is to start off with
a run of about 250 simulations and then increase the length of the
runs until there is virtually no change in the estimates produced by the
simulation. It is unlikely that runs of more than 1000 simulations will
be required.