Monte Carlo simulation

Post on: 16 Март, 2015 No Comment

>But there’s a spreadsheet, right?

Yes, you can use a spreadsheet which looks like this:

if’n that’s what you like to see. Actually, the spreadsheet has several sample portfolios and a red graph which gives the deterministic portfolio evolution.

the Math

We note that a collection of returns < r > is log-Normal if r = EXP(x) where the collection < x > is Normal.
If the prescribed Mean and Standard Deviation of the log-Normal distribution are m and v respectively *. the associated Normal Mean (M) and Standard Deviation (V) are obtained via

M = Log(1 + m) — V 2 / 2

V = SQRT(Log(1 + (v / (1 + m)) 2 )) see the Magic Formula here . * Sorry. Here v stands for volatility (= standard deviation), not variance :^(

>Sure! Make it confusing!

The above transformation is done for each portfolio component, yielding M_s. V_s for stocks and M_b. V_b for bonds.

If there were no correlation between the portfolio components, Normally distributed gains for each component could be generated according to the following prescription:

Pick a number x between 0 and 1.

Determine where F() = x and F is the Normal, cumulative probability function

(whose values lie between 0 and 1, eh?) and looks like this:

Starting at the mean M, go either left or right a distance V

where V is the Standard Deviation and is that random multiplier.

You’re now at some Normal gain, M + V, so convert to a log-Normal Gain:

Gain = EXP ( M + V ) — 1

where (remember?) is a number generated according to the scheme described above and

x is a uniformly distributed random number (or probability ), between 0 and 1

Now generate an annual gain for a two-component portfolio according to:

Gain = [ EXP ( M_s + V_s ) — 1 ] P + [ EXP ( M_b + V_b ) — 1 ] (1 — P )

where P is the stock fraction.

However, taking into consideration the prescribed Correlation( C ) between components, an annual portfolio Gain is generated according to: Gain = [ EXP ( M_s + w V_s ) — 1 ] P + [ EXP ( M_b + V_b ) — 1 ] (1 — P )

where and are numbers generated according to the scheme described above and (magic!)

w = SQRT(1 — C 2 ) + C

Note: If C = 0, w = . the two gains are independent, eh?

If C = 1, w = . both gains are above (or below) their respective means, together

hence are positively correlated (meaning stock and bond gains move up or down together).

If C = -1, w = — . when one gain is above (or below) its means, the other is below (or above)

hence they’re negatively correlated (meaning stock and bond gains move in opposite directions).

(See correlation stuff .)

>So what’s a good value to pick?

Pay attention.

The portfolio is increased by this Gain, a Withdrawal is made

then increased according to the prescribed inflation Factor(f): Portfolio(n+1) = (1+Gain) Portfolio(n) — Withdrawal(n)

It’s interesting to see the difference between final portfolio survival rates and the Standard Deviation of the portfolio components.

Of course, we might also set both Standard Deviations to ZERO

And you may want to see the effect of using a Normal (rather than log-Normal) distribution of annual returns.

And you might want to peer 30 years into the future using the Mean & Standard Deviation

from the 1950s or maybe the 1960s or. like so:

And what about the probability that your portfolio will last for 5 years or 10 years or 15. like so:

And what about the variability of your portfolio, how far it might deviate from the deterministic component (the red curve)

as the years progress to 30 years. like this sample set of possible portfolio evolutions:

Or, stare raptly at this sample chart (below) which shows the growth of a $1.00 Deterministic Portfolio and the graphs if one adds or subtracts the Standard Deviation (over a thousand Monte Carlo simulations):

>Wow! Just think if you felt satisfied with a Deterministic Gain! How often did the portfolio survive, after thirty years?

Anyway, you can play with the spreadsheet and extract inofrmation to see the effect of high inflation, high withdrawal rates, high investment returns. and high hopes. However, there’s a spreadsheet better suited for Saving for Retirement. described here .

And you might want to enter a negative withdrawal rate meaning that you’re adding to your portfolio annually. so you can watch it grow (before you retire and start withdrawing)

Then, of course, you may not like the idea of using a fixed Mean and Standard Deviation and some fictional Distribution of Returns (like Normal or Log-normal) but prefer to select your stock returns, at random, from the actual S&P 500 returns (from Jan, 1926 to May, 2001 which I got from Richard’s Page ). You’ll see some significant difference in results, like mebbe:

>If you want to scare somebuddy about a 5% withdrawal rate, pick a Normal distribution, eh?

You got it. About 80% of the actual monthly S&P 500 returns lie within one Standard Deviation from the Mean. unlike, for example, a Normal distribution where it’s 68% and that means.

Oh. one other thingy. If, in the spreadsheet, you pick not N or L or S but T. then the spreadsheet selects random T SE 300 returns and Canadian long Bond returns.