# Do portfolio mean and portfolio variance have probability distributions?

• If $$X$$ is a $$T\times N$$ matrix of multivariate asset returns,
• and $$w$$ is some optimal portfolio weight vector,

then the portfolio return series is $$r_p = X w \in\mathbb{R}^{T}$$. This return series can then be used to form the portfolio's return distribution $$f(r_p)$$. and the portfolio mean and portfolio variance of this portfolio distribution would therefore be estimated as scalars.

Instead of scalars, can an empirical distribution of the portfolio mean and portfolio variance be constructed somehow?

• Yes in statistics the distribution of statistic like mean is called “sampling distribution”. – fesman Oct 3 '20 at 7:12
• Portfolio mean at a concrete time point is a concrete number. You may assume this number is the same for all time points or perhaps varying over time. In the first case, what do you mean by empirical distribution of a concrete number? The same holds for variance. In the second ase, what do you mean by empirical distribution of a concrete vector of numbers (or matrix of numbers for variance)? – Richard Hardy Oct 3 '20 at 18:19

Yes, they can/do. But you have to drink the proverbial Kool-Aid(or taking the blue pill is probably the more relevant metaphor these days ;-), and approach this as a Bayesian inference problem.

So instead of mu, you have a normally distributed probability distribution of mu, depending on mu-of-mu and variance-of-mu. And the same for variance (mu-of-var, and var-of-var). These four parameters, call them theta, determine the distribution of your two mu and sigma parameters.

So we have p(mu|theta) and p(variance|theta) as normally distributed. We can use Bayes to work out posterior p(theta|outcomes), being proportional to p(outcomes|theta) * prior p(theta). Since the Bayesian conjugate for a normal distribution is another normal distribution, we don't have to calculate every possible level; and the output posterior will be in the same form as the input prior. In effect, the initial prior ceases to matter very much, once you feed the model with outcomes!

Given this posterior p(theta|outcomes), forecasting p(outcome|theta,new data) becomes trivial, and gives you a data-based distribution around your new data, rather than a hard-and-fast point estimate.

• Bayesian inference provides a nice thought framework, but I wouldn't know how to begin implementing Bayesian calculations for the problem at hand. if a guideline procedure or algorithm could be provided that executes all the conditional probability statements above, it would be helpful since I'm looking for data-driven results, not only a theoretical compass. what is a "Bayesian conjugate"? – develarist Oct 9 '20 at 8:14

Given a set of returns, say 500 days, and a fixed portfolio construction, you can derive the 500 daily portfolio valuation changes.

You can easily measure the mean and variance of these valuation changes. Since this is a sample you are interested in the confidence of your estimators (i.e. the mean and variance).

One method that is often used is a resampling procedure called bootstrap sampling. Perform 1000 simulations by selecting 500 datapoinst from the original 500 datapoints (WITH replacement). Each of the 1000 simulations will yield a different mean and different variance. You can estimate the variance of the means and the variance of the variances from this data.

You can read a little more about pros and cons of bootstrap sampling over at wikipedia.

### A coin toss

Personally I think bootstrap sampling is an underappreciated/underused area of statistics. Let me give highlight a simple example.

Suppose you toss a coin 20 times and receive the result:

1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 1


What does that tell us? It tells us the mean is 0.55 and the variance is 0.26. But could my mean be wrong and how wrong could it be?

In this scenario we know the true probability of a coin toss is 50% and the distribution is binomial. But consider plotting the real and parametric distributions of outcomes compared with 200 bootstrap samples of 20 datapoints with replacement:

I think that's a powerful result from such a simple technique.

• Re: "by selecting 500 datapoinst from the original 500 datapoints (WITH replacement)". Shouldn't the bootstrapped sub-samples be of shorter length than the real dataset? – develarist Oct 8 '20 at 12:38
• Also, is your first paragraph suggesting a rolling windows approach for the 500 days of valuation changes, whereas the remaining paragraphs deal with something separate: bootstrap? – develarist Oct 8 '20 at 12:39
• The key is that you subsample with replacement, then sampling 500 from 500 is not an issue, obviously if you do without then you get the same 500 datapoints in each sim! There may be a theory on optimal subsampling number of items but I tend to use the same number of points as the original dataset. I dont make any comment about a rolling window, You assume a TxN matrix of returns, I simply assert T is 500, and your $r_p$ has length 500. – Attack68 Oct 8 '20 at 12:44
• alright, was just clarifying. i was expecting a bootstrap answer, but as you know, the bootstrap is an artificial impression of what the portfolio moments' distributions could be – develarist Oct 8 '20 at 12:46
• in the new image, what are the blue, orange and grey densities? – develarist Oct 9 '20 at 6:46

In the limit, the distribution of the mean of samples taken from an independent identically distribution is always normally distributed according to the Central limit theorem.
So the empirical distribution for the portfolio mean should be normally distributed according to the Central limit theorem. But is the question then, how to calculate the mean and variance for that normal distribution?

• edit with how this answer applies to the portfolio question – develarist Oct 3 '20 at 6:05
• No i am not asking how to calculate the mean and variance. There's an empirical distribution for the portfolio return series, but tell me, how would you calculate the empirical distribution of the portfolio mean – develarist Oct 3 '20 at 9:45