What AA did you use?
kramer wrote:Before I retired, I built my own monte carlo engine in Matlab, a mathematical modeling tool.
I verified the correctness of the code partly by generally duplicating Bernstein's numbers in the retirement calculator from hell:
http://www.efficientfrontier.com/ef/101/hell101.htm
Also, he suggests a Standard Deviation of 12% in there for stock returns and explains why (explaining why it is not even higher, that is). And a Standard Deviation of 5% for intermediate bond instruments.
I assumed a relatively low stock SD of 12%. This is worth a comment here. When one runs Monte Carlo simulations of stock returns using the historical 15%-20% SD, one comes up with higher long-period (>20 years) variability than is actually observed. The reason is, over the long haul stock returns have a tendency to mean-revert. To correct for this, I’ve lowered the SD a bit.
Raybo wrote:
Attempting to simulate a low interest rate, low return, reasonably high inflation environment, I set the distributions to
Inflation mean: 3.5% variance 1.5% (note can't go below -2%)
Investment mean: 5% variance 10% (nominal returns)
Interest mean: 2% variance 1%
The results from 10,000 runs of this model show a 10% chance of running out of money in 30 years. However, in all of those "out of money" years, stock returns averaged 2.6% or less, inflation averaged no less than 3.5%, and interest rates were never higher than 2.2%.
If I raise the investment mean to 7%, the percentage of 30 year failures falls to 1.3% and in all the failures, the average investment return was 3.5% or less.
grayfox wrote:What I'm saying is that when people report that, over the long term, stocks have returned about 10% per year, they are referring to the annualized return, which is the geometric mean. Same thing when Jeremy Siegel says that stocks had 7% real return per year over the long term. They are reporting the annualized return, i.e. geometric mean.
But because of volatility drag, the annualized return is less than the average annual return. On average by that formula: Volatility Drag = .5 * sigma^2. [I think volatility drag is a random variable, so the number is not exact.] In other words, the underlying probability distribution doesn't have a mean of 10%. The underlying distribution might have mean return mu = 12% and standard deviation = 20%
Now a lot of people are forecasting lower returns for stocks going forward, say only 4.5% real return instead of the historical 7%. But again, that forecast is for the annualized return. The underlying distribution they are forecasting might have a mean mu = 6.5% and sd = 20%. But because of volatility drag, you will own get about 4.5% annualized return.
So if you were going generate random returns from a normal distribution, and you were forecasting 4.5% real annualized return, you might use mu=6.5% and sd=20%. After you run it, you can calculate average return and annualized return. If you do enough runs, the average return should be close to 6.5%, but the annualized return would be less, probably averaging closer to 4.5%
Raybo wrote:Just so I understand what greyfox and rodc are saying. It isn't that I need to change how my simulation works, it is that the mean and standard deviation I've chosen might not be accurate to how the market numbers actually work. In fact, the numbers I am using might be too low for what I am trying to simulate.
Is this correct?
ret.avg ret.ann vol.drag
Mean 6.546 4.512 2.03454
SD 3.894 3.845 0.04874
Histogram of Volatility Drag
for 1000 Runs
Midpoint Count
0.75 3
1.25 144
1.75 383
2.25 279
2.75 149
3.25 34
3.75 4
4.25 2
4.75 2
ret.avg ret.ann vol.drag
Mean 5.016 4.393 0.62320
SD 1.831 1.785 0.04535
Midpoint Count
[1,] 0.25 2
[2,] 0.35 39
[3,] 0.45 152
[4,] 0.55 256
[5,] 0.65 279
[6,] 0.75 173
[7,] 0.85 71
[8,] 0.95 21
[9,] 1.05 7
ret.avg ret.ann vol.drag
Mean 5.724 5.082 0.64240
SD 1.836 1.787 0.04976
Histogram of Volatility Drag
Midpoint Count
[1,] 0.35 21
[2,] 0.45 121
[3,] 0.55 238
[4,] 0.65 315
[5,] 0.75 193
[6,] 0.85 74
[7,] 0.95 27
[8,] 1.05 10
[9,] 1.15 1
Along with probability of success, another important output is the size of the shortfall for those simulation iterations where the plan failed (ran out of money). A plan with a reasonably high probability of success that usually fails in the early years may be less robust than a plan with a lower probability of success that usually fails near the end of the plan. The Average Spending Shortfall shows the average percent of total planned retirement spending that couldn’t be funded in those simulation iterations that failed. For example, consider a retirement plan with level spending planned for 40 years. Further, assume that when the retirement plan fails, on average it fails in simulation year 30. Such a plan would have an average spending shortfall of 25%. This is because on average 1/4 of the retirement plan’s spending wouldn’t get funded in those iterations that failed.
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