motivated by a discussion on the Webring Forum
Once upon a time, people equated risk with Standard Deviation (or Volatility).
Although they continued to use the word "risk", they were actually calculating something that most people didn't associate with that word.
>Once upon a time?
Yes, but these days people more often refer to "uncertainty" when they use Standard Deviation.
>And that's better, eh?
Yes. Better, but I don't like equating it to "risk".
For example, let's define:
risk = Standard Deviation = Volatility = Uncertainty 
Then, if your annual returns were always 10% larger than mine, our risk would be the same.
>You're talking about the Standard Deviation of returns, right?
Right, so ...
>So how about Webster's definition of "risk"?
Okay, let's consider:
risk = the Probability of a Loss 
Now, suppose when I flip a coin I get $1 if it comes up heads and I lose $1 if it comes up tails.
Then my risk is 50%. That's the probability of a loss on my coin toss.
Now suppose that you flip a coin and win $1 for a heads and lose $1,000 if it comes up tails.
Then your risk is also 50%.
The risk is the same. Does that seem reasonable to you?
>Uh ... no. But what definition of "risk" do you like?
Well, for starters, I think it should have something to do with losses ... and how much you stand to lose.
Since many people would like to calculate "risk" when they're talking about their portfolio lasting for 10 or 20 or 40 years, maybe a
reasonable definition would involve withdrawing money from a portfolio (with a prescribed asset allocation)
which is rebalanced annually.
So, let's do this to generate the umpteenth definition of "risk":
 We assume a portfolio with a certain composition (like 25% each of large cap growth & value + 25% small cap value + 25% gov't long bonds).
 We start with a $1M portfolio and withdraw 5% annually.
 Our withdrawals increase annually by 3%.
 We rebalance yearly to maintain the prescribed allocation.
 We select, at random, annual returns for each asset from historical data from year 1928 to year 2000.
 We apply these returns to our portfolio  and repeat for each of N years.
 We repeat steps 5 and 6 a jillion times ... a la Monte Carlo.
 We calculate the percentage of times that our portfolio failed to survive for N years.
 We define:
Risk = Percentage of Portfolios that failed to Survive for N years 
>Why 1928 to 2000?
I just happen to have those numbers here.
>And why 3% inflation and 5% withdrawal and ...?
It's entertainment. Choose whatever you like. It's the idea that I'm suggesting.
>So what's that "jillion", in the Monte Carlo simulations.
I did 10,000. That's plenty ... for me.
>So where's the spreadsheet?
I modified the one described here, at the bottom of the page.
(The spreadsheet can be found here. It's withdrawalchaos.xls)
>You think that's a better definition?
Well, I like it. It makes me feel warm all over and ...
>I assume you've actually played with that definition, right?
Yes, and I got this
>The risk goes down ... eventually?
I guess one might expect that.
I mean, if you can last for 40 years then you'd probably last for 80 years.
Maybe that's reasonable, wouldn't you say?
>How would I know?
 Figure 1 
Remember when we got this chart, in Figure 2?
>No!
Well, they're the values of (Cumulative Inflation)/(Cumulative Gain) that we got
here.
>What's that CPI?
We assume the Consumer Price Index is "1" in 1950 and we change it annually according to the annual inflation.
Then the maximum rate of withdrawal depends upon the area under the chart in Figure 2.
 Figure 2 
Indeed, if I_{n} and G_{n} are the cumulative inflation and portfolio gains after n years, then
f, the maximum withdrawal rate for N years, is given by:
f = 1 / (I_{1}/G_{1}
+ I_{2}/G_{2} + I_{3}/G_{3} + ... + I_{N}/G_{N}
)

Notice anything interesting about Figure 2?
>No ... I don't even understand it!
I told you. The individual heights of those bars are the values of I_{1}/G_{1}, I_{2}/G_{2},
I_{3}/G_{3} etc. etc.
If we sum the areas under the chart (in green),
we'd get the reciprocal of the maximum rate of withdrawal.
What I mean is ... any larger withdrawal rate and your portfolio wouldn't survive N years.
>Starting in 1950.
Yes, of course .... and for a particular portfolio composition.
What it suggests is that it's the first batch of years that make all the difference and ...
>But it depends upon when you start.
Of course! Figure 3 is for the case when you start in 1928.
I think we're led to the conclusion that the first 1520 years will determine whether your portfolio will survive for umpteen years.
 Figure 3 
>Guaranteed?
Are you kidding?
>But if my portfolio lasts, say 20 years, it'll last 40 years, right?
Wait'll I check
>I'm surprised you don't got a spreadsheet. I mean, you usually have ...
Okay, here's one:
click on picture to download the spreadsheet
You pick four assets (from a list of ten) and how much you want to devote to each, then some stuff like inflation and withdrawal rate
(increasing annually with your selected inflation) ... then you click a button.
Then annual returns for each asset are selected at random and applied to your portfolio (which is rebalanced yearly to maintain your
prescribed allocation) and the percentage of portfolios that survive 20 years, 25 years, etc. etc. is determined.
>Sounds like ol' Monte Carlo to me.
Yes ... and you get to choose the number of Monte Carlo simulations. The picture shows the results for 20,000 MC iterations.
>Where are those annual returns ... for each asset?
They're in the spreadsheet ... somewhere.
>And what does that spreadsheet prove?
Prove? Prove? Nothing, but it suggests that once your portfolio has survived 40 years there's a good chance it'll survive 50 or 60 years.
See how quickly the survival rate drops at first. Then is drops more and more slowly.
>And that's a proof?
That's a suggestion.
>Instead of that Monte Carlo stuff, do you have a real, live example?
Sure. Figure 4 gives the Maximum Rate of Withdrawal for a particular portfolio ... starting in 1928.
The portoflio is rebalanced yearly and ...
>And what about other portfolios?
Just download the spreadsheet above. The chart is there ... somewhere.
Just look for it !
>I see it gives that notorious 4% "Safe Withdrawal Rate".
You noticed that, did you? Did you also notice that it's still very nearly 4% for N = 30 and 40 and 50 years.
>Yes, I noticed that.
Good for you.
 Figure 4 
