Multi period portfolio risk/return

[hotvette]

TL;DR Summary

How to compute mean and variance of multi period investment

I saw an interesting table in Asset Allocation (Roger Gibson) showing the distribution of portfolio annualized returns for a hypothetical portfolio with mean of 5.8% and standard deviation of 6%. It shows the return percentiles for various holding periods from 1 to 25 years. Can this distribution be determined in closed form for a given mean, standard deviation, and holding period, or must one use monte carlo simulation?

 

[fresh_42]

The first idea which comes to mind is to anchor all data at the starting point. Say we set up the portfolio at $t_0$. Then we can make quantities comparable by comparison to this point in time. The return e.g. would be the difference of prices: $r(t)= p(t)-p(t_0)$ adjusted by things like: where do dividends go, do we consider inflation rates, and do we track and calculate with market capitalization. Different holding periods would then be calculated as $r(t_2)-r(t_1)=p(t_2)-p(t_0)-p(t_1)+p(t_0)=p(t_2)-p(t_1)$.

Similar can be done with expectation values and risk, since they change over time. We get functions $\mu(t)$ and $\sigma(t)$.

What did you mean by given values? They are determined by the overall dependencies defined by the market and calculated for the specific portfolio at time $t$. You cannot choose them, they are benchmarks of the portfolio.

 

[pbuk]

Without seeing the table it is a bit difficult to understand what you mean: is @fresh_42 on the right track or is it more like this:

• We have a portfolio whose return in anyone year is modeled by a normal distribution with mean 5.8% and SD 6%.
• We model the return over an n-year period by n successive trials of this model.
• We want to tabulate quartiles/deciles/confidence limits or whatever for this distribution.

If this is the case then the distribution of the cumulative return is the product of n normal distributions, and whilst it is fairly easy to show that the variance of this distribution is equal to the product of the variances, the distribution is definitely not normal - see Mathworld for a summary of the cases n=2 and n=3.

 

[pbuk]

The fact that the results will not be scale-invariant demonstates the weakness of modelling the annual return with a normal distribution (1 year at 4.04% return should have an identical distribution to 2 x 6 months at 2% return).

Edit: perhaps my assumption that the author was assuming a normal distribution was incorrect; log-normal would be more appropriate and this would have a simple analytic solution for n years.

 

[hotvette]

The page is attached. The portfolio is hypothetical whose returns follow a normal distribution with mean 5.8% and SD of 6%. What I mean by given is that for a specified mean, SD, and number of periods, what is the distribution of returns after the n periods assuming mean and SD are constant for each period. For the below, the author chose mean 5.8% and SD of 6%. We could choose anything.

 

[pbuk]

If the author were using a normal distribution then my post #3 would apply. But that chart and table are clearly NOT normal - the tails are not symmetrical. It looks as though (I haven't checked the maths) the author may be, correctly, using a log-normal distribution in which case my post #4 would apply - the properties of the product of n log-normal distributions are easily determined.

 

[pbuk]

Shame he can't spell annualized though.

 

[fresh_42]

Shame he can't spell annualized though.

With that distribution I'd fire the fonds manager anyway.

 

[BWV]

the distribution is lognomal, as a normal distribution has a non zero probability of a return < -100%

be careful though, the mean and median returns are different - the distribution is skewed right, so mean > median

more specifically,

Mean = $exp(\mu + {\sigma^2/2})$

median= $exp(\mu)$

the kth percentile scales with the square root of the standard deviation, so
z= zscore for kth percentile, t = number of years

the kth percentile (log) return then is

$\mu +z*\sigma/ {\sqrt{t}}$