# Portfolio volatility of discontinuous portfolio

I would like to calculate an investor's average portfolio volatility as a measure of risk aversion. My problem is, that the portfolios are not continuous:

• the investor can have an open position for the first five days of the month, then have a week brake and then open further positions
• the investor can start by opening five positions, close two of them on day 3 and hold the other three until the last day of his investment horizon, which is not determined up front

So I cannot realistically calculate the end-day portfolio worth and use the series of this values to calculate day-to-day returns and from this calculate the volatility as a standard deviation of the returns. And as in the second case, the end-day worth from day 3 to 4 will drop by the amount of closed positions, however, this does not yield a loss for the investor by default, which would be the case when I compare end-day worth from day3 to day 4.

Are you familiar with approaches that would facilitate calculating portfolio volatility for such cases?

• To some extent, cash position is also position May 18, 2018 at 19:25
• Wealth does not suddenly disappear, if you sell an asset it is replaced by cash. The wealth of the investor (including cash) is not discontinuous. May 19, 2018 at 18:13
• @AlexC It does not, but in my case I am observing the investors transactions (from a trading platform's perspective) and I do not have information about their current wealth. So I assume that their only portfolio is what they trade on this particular platform and that is my interest.
– abu
May 19, 2018 at 19:55
• there's the daily position thus daily (dollar) pnl by marking all positions to the market. you'd be able to get some volatility metric based on that May 21, 2018 at 17:38

I apologize for being a little bit blunt but it seems to me you don't really understand the terms you are using

For one: it does not make sense to calculate the volatility of the portfolio "as a function of the risk aversion". Risk aversion is used in portfolio optimization as a way to decide upon optimal risk allocation as a function of the portfolio risk. Portfolio risk/volatility is a function of the time series of the portfolio allocations and the assets universe covariance matrix.

Next, to get to the heart of your question: i don't understand neither what you mean by "the portfolio are not continuous" despite your explanations. Indeed portfolio are usually assumed to be self-financing: so at time of rebalancing the portfolio value is continuous because whatever you buy is financed by sales of other assets.

Whether the investor opens positions, or closes positions, or whatever the investor does with her portfolio of course does not change the fact that this portfolio has a well defined fair value at any time. And this is of course fundamental for all sorts of reasons if only compliance one.

Given that portfolio value is continuous it totally makes sense to look at the times series of its mark-to-market, then daily returns and thus eventually variance/volatility.

In other words: for a self-financed portfolio its volatility is totally well defined.

My opinion is ask yourself the underlying goal of calculating portfolio volatility? Then to determine whether any adjustments need to be made.

For example, if it's a portfolio managed by a fund and both 1 & 2 are part of the strategy, then I don't think you need to make any adjustments to calculate portfolio volatility or performance measures such as sharpe ratio. However, if 1 & 2 are due other reasons that have nothing to do with the trading strategies, then volatility calculation might be adjusted. You might exclude the break period to calculate the true volatility of the portfolio.