# Variance of portfolio future values based on return distribution

I'm trying to estimate the distribution of future portfolio values based on the distribution of a portfolio's returns.

First, to define some variables:

• Rt = simple return for period t
• rt = ln(1+Rt)
• Vt = portfolio value at period t

If we assume rt is a normally distributed random value and an initial investment of $1, ln(Vt) = r1 + r2 + ... + rt. Since ln(Vt) is a sum of normally distributed random variables, ln(Vt) is also normal. Its mean is t * mean(rt), and its variance is t * var(rt). MY PROBLEM: I need to get the distribution of portfolio values based on a starting value that is not 1 dollar and cash inflows/outflows each period thereafter. I can get the mean by adding ln(starting_value). However, I am not sure how to modify the variance from the scenario above to adjust for the fact that my scale is no longer based on a starting value of$1. If I just take the cumulative sum of the variance of the returns like I did in the scenario above, the portfolio's variance each period is far too small.

Any help would be greatly appreciated. I can provide a spreadsheet with a fairly simple example calculation if that helps clarify what I'm trying to do.

Read RiskMetrics Technical Manual here. It describes value-at-risk (VaR) computation. That's what you're trying to do. Particularly, you're implementing so called "historical VaR" with a parametric distribution fit.

• The essence of the method is described on page 8. Look at bullet #1: $V_1=V_0e^{r}$, where $V_1,V_0$ are future and current values of a portfolio, and $r$ is the return.
• Look at bullet #4: when $r<<1$ you can apporoximate $V_1\approx V_0+V_0r$.
• Now, you can compute the variance of the future portfolio value: $Var[V_1]=V_0^2Var[r]$.
• Where to get $r$ from? There are many different approaches. You seem to be trying the simplest one: assume that the variance is constant and estimate it from historical series.

So the rest is a technicality. Get the series of returns from the past: $r_t=\ln(V_t/V_{t-1})$, and estimate its variance and other parameters. Note, as I wrote earlier: the log here is simply to compute the continuously compounded return. See Eq. 4.3 on page 46.

There's a lot of interesting stuff in the document. For instance, the returns in the distance past are probably less relevant to the tomorrow's return than the returns in the recent periods. So, they use EWMA to account for that.

• Hmm. I'm a little lost because I don't think I'll have time to go through that VaR section of the document. I have some portfolio management notes that cover the topic as well, but I never took the course and just need to quickly figure out the calculation for Var(ln(Vtomorrow)). Any idea if that's in the document you referenced? – dvanderb Dec 4 '14 at 21:36
• This is the doc which introduced VaR in the first place. So, yes, this stuff will be there, and it's written for applied folks, i.e. light on theory and easy to read. Also, you don't estimate $Var[\ln(V_t)]$, but $Var[\ln(r_t)]$ - returns, not log values. The diff of a log is a continuous return. – Aksakal Dec 4 '14 at 21:39
• The problem for me is that I haven't done any portfolio management (I'm trying to figure this out for a web application and am learning as I go). I'm not sure why I would be able to estimate E[ln(Vt)] but not the variance. I guess I'll just need to keep reading and figure this out. Thanks for the help pointing me in the right direction. – dvanderb Dec 4 '14 at 21:52
• @dvanderb, I updated the answer with more streamlines exposition – Aksakal Dec 5 '14 at 2:22