Normal Distribution, Probability after multiple periods

Water Cooler Investing
#1

If I have a normal distributon, Expected return of: 10%, vol of 5% and a one year holding period I can easily look at a table and find out what is the probability of having a return of 3% or less for example.

How can I do the same for a longer holding period, say I have a ten year holding period and I want to see what is the probability that portfolio value will be less than X in ten years. I am currently doing it using simulation and it is very slow from long periods.

Is there a mathmatical solution?

#2

Yes. Assuming your 5% volatility above is the 1y annualized standard deviation, the 10y (non annualized) standard deviation is 5% x 10^0.5. Your expected return is presumably 110%^10. The other calculations are the same as before.

#3

i actually have an excel spreadsheet that can do all the thigns you’re looking for using 10k simulations/run.

so say 100k portfolio. 10% return. 5%std. 10 year holding period.

95% chance it’ll be greater than ~$200k as an ending value or 100k profit.

I ran it like 5x and it ranged from 195k to 205k for the 95% level

#4

ohai wouldnt a simulation still be better as the true risk is path dependent and that doesnt really get captured well when you do a multi period analysis like that?

#5

^yes.

https://sites.google.com/site/cfalevel3examprep/managing-portfolio-risk/ss14/reading-26/26e

the answer using monte carlo vs ohai formula deviate greatly.

5 percentile using ohai formula is -1.65 as z score:

1.1^10 + .05*(10^.5)*(1.65) =

2.59 -.261 = 2.329

or 233k as ending value or 133k profit

#6

The terminal value at year 10 is not path dependent and so, there is no value to be gained from using a MC simulation over a close form solution. I do not know what the simulation is doing relative to the formula, or maybe I am interchanging portfolio and returns standard deviation or something. However, any the answer from the similation should approximate the equivalent close form solution.

#7

i think the reason they deviate is cuz std as well as the number of periods affects path dependency.

lower std use formula.

lower periods use formula.

mc simulation is better with higher std and multiple periods.

#8

Thanks, simple enough concept that was there in the CFA curriculum somewhere. Thanks for the refresh.

#9

I have made something for myself, would be nice if you can share yours and I can share my output.

https://1drv.ms/x/s!AgEdDvW0qb2MujRD4T-Yrq38I2ja

Mine is still in the infant stage, but I plan to have many tabs for different purposes.

What I am building is tools for retirement planning. I will use the suggested formula for simple cases, but some are path dependent and will still require simulation.

Once thing I need to model is inflation, and how that will correlate to the rest of the assets being modeled. Any idea guys? Can I really assume inflation is normally distributed… That is giving a lot more negative inflation years than there has been historically…Or is it better to pull inflation numbers from historical inflation and pick from those randomly without assuming correlation to stock market for example.

#10

thanks mate, no matter how long I stay away from the site whenever I come back you jump on my questions :wink:

#11

This article is showing that returns are not normally distributed after multiple periods

https://web.stanford.edu/~wfsharpe/mia/rr/mia_rr3.htm

if that is the case how can te approach suggested by ohai still work ?

Thanks,

#12

the discussion of whether even single period returns are normally distributed is pretty well documented, for brevity of a model people assume they are because its much more difficult to not make that assumption