Question about semi-sophisticated statistics and financial modeling -
lets say you are forming a porfolio comprised of N mutual funds - each
fund has a mean rate of return and a standard deviation of returns.
How do you determine the probability that the portfolio might produce
the a particular rate of return over P periods? (Assuming you know the
weight of each fund in the portfolio?) I'm trying to create an
invesmtment plan in Excel, so I'd like to avoid using something like
iterations (Monte Carlo sim) - is there a a formula for this?

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Clarification of Question by mjstuehler-ga on 24 Mar 2006 06:42 PST

A clarification: here are the inputs:

1. the number of funds
2. the proportion of each fund in the portfolio
3. the mean return of each fund for each period
4. the standard deviation of returns for each fund for each period
5. the number of periods

Based on these, I'd like to determine what the mean return and
standard deviation of returns is for the portfolio after N periods

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Request for Question Clarification by hedgie-ga on 24 Mar 2006 23:15 PST

Well, mjstuehler

       You would have to make some additional assumptions, 
 such as 

 the funds are statisticaly independent,
 (or provide additional inputs, such as cross-corelations)
 ...
  With few such mods, question would be:
 
I have N stochastic processes (time series) P.i, 
characterised by parameters  p.i.1   p.i.2 ...
  
  What would be such parameters of a combined process
formed from these with weight w.i ?

  Something like that could be answered without need for 
 Monte-Carlo simulation.

 for a simple process (simple model of a time series).

Is that what you want?

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Clarification of Question by mjstuehler-ga on 26 Mar 2006 20:09 PST

Hedgie,

Thanks for your response, and my apologies for my delayed clarification.

I would be satisfied with an answer that assumed that the funds were
statistically independent, although a solution that allowed me to
incorporate a correlation matrix would really be appreciated - a
significant improvement.

(Because I don't know what's possible, I don't know how much I can ask for!!)

Please don't hesitate to let me know if there's any additional
clarification or information I can provide for you.

mjstuehler

            I will divide the answer into three sections

A) simplest case : independent stationary variables
B) few words on time-series models
C) stationary correlated variables


A) We will employ the simplest statistical model:

Value describing each component (a fund) is a 
random variable with normal distribution (described by mean and sigma).

Then, the theorem which provides the answer is here:

     if X and Y are independent random variables that are normally
distributed, then X + Y is also normally distributed

You can skip the proof (which is here)
http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
and general background on normal (= Gaussian or 'bell-like curve)
distribution (which is here)
http://en.wikipedia.org/wiki/Normal_distribution

and just use the result: means are additive, and variances (not
sigmas) are additive

meaning:  for N mutual funds (components) (lets say N=2) we have

fund 1 has mean.1  and sigma.1   (average and standard deviation)
fund 2 has mean.2  and sigma.2   (average and standard deviation)

for a particular property (such as price, or dividend, current market
value, or ROI ...)

Let's pick an additive property (such as current market value = V)

Than, when we compose a portfolio,  property of the total is

  V.t = w.1 * V.1  + w.2 * V.2

  (market value of the portfolio IS Sum of the Values of the components,
    That is not so for all properties true, e.g. not for ROI)

-------------- mwaning of weights w
Here V1 can be 100 shares of Goog , V2 1000 shares of GERN ... 
if weight w.1 = 2   and w.2=5, then the composite (portfolio) would have

   200 shares of Goog and 5000 shares of Gern , OK?
-----------------------------------

So, if V1 and V2 are Gaussian with parameters shown above,
  then V.t will be Gaussian with parameters

 mean.t= w.1 * mean.1 + w.2 * mean.2 

and std. deviation sigma.t defined by

 sigma.t ^2 = (w.1 * sigma.1) ^2 + (w.2 * sigma.2) ^2 

where ^ means power:   3^2 = 9,     2^3 =8, ...

B) When you say (in your clarification)

 "standard deviation of returns is for the portfolio after N periods .."

  You are using symbol N in a second meaning.
 
Let's talk about N funds, which we will keep for M time-periods  (e.g.
 weeks, or months..)

 i = 1,  2, ...  N      and time  l=1,2, .... M  

The simplest assumption is that the values   V.i.l (value of dund i at time l)
and independent from each other and of their past.

In such a case, Values (V.1.1 , V.1.2, V.1.3  ) for find i=1 are just
samples taken from the same
distribution  (N(mean.1, sigma.1), and so on for other i's.

That assumes  the fundamentals did not change over time and proces have no memory.

That is a gross oversimplification. The commonly accepted model  a
'random walk ' (aka Markov chain)
in which prices do not have long term memory, but the next period
price (V(t+1)  depends on previous price V(t).

The 'Nobel prize- winning' application based on this model (applied to
options) is described here
http://en.wikipedia.org/wiki/Black-Scholes_formula

Books are available with details and generalizations, but the formulas
are too complex for a spread sheet.
Search Term: Black_sholes
Popular expose of the model:  ( a classic by Burton Malkiel )
http://www.amazon.com/gp/product/0393315290/ref=nosim/102-7446460-2049751?n=283155
http://www.defaultrisk.com/bk_g_arwdws.htm
simpe summary
http://www.ithadtobeyou.net/carpe/archives/000135.html

C) We will stay with memory-less model and just remove the assumption
of independence:

 We have N time-series  V.i  (i =1 , 2  ... N) which we treat is M
samples from  N distributions.

 We construct composite time series V.t as before

 We calculate  'expectation value'  of square  M2=  <V.t * V.t> and get

M2= sum over i  Sum over j    < V.i * V.j > 

We need to understand relation of M2, variance and sigma, defined here:
 http://mathworld.wolfram.com/StandardDeviation.html

and  correlation coefficients  r.i.j  defined here 
  http://mathworld.wolfram.com/CorrelationCoefficient.html
and in (too many) details, here
http://www.mega.nu/ampp/rummel/uc.htm
and a bit more general way - as a Cross Correlation here
http://astronomy.swin.edu.au/~pbourke/other/correlate/
   
For our purposes:
M2 = second moment
 Variance is M2 - mean ^2 
 sigma is variance^.5    (square root)

Inserting cross-correlations s.i.j  and sigmas.i  into the above expression,
we obtain formula for means of the sigma.t in terms of sigmas and r's.
mean.t is same as in the case A).

If all cross-correlations s.i.j (for i != j) are zero, then this
formula is reduced to the
previous (independent variables) case. (Here the symbol !=  means 'not equal' ). 

I did try to make it as simple as possible, but there is always a
space for improvement:
Please do ask for clarification (RFC) if what I wrote is nor clear. 
After all is clear, rating is appreciated.

Hedgie

There is a major flaw in the proposed logic. Fund movements are not
independent. Most US funds have a very high correlation to the S&P
500. Thus, the Standard deviation of the portfolio will be much higher
than if the components were independent.

http://www.fasttrack.net provides a good tool box for this kind of modeling.