# 3 – M4 L4 03 Setting Up The Problem Risk V4

Okay, now where does the risk model come in? Well, we discussed previously that if we put the quantity representing the mean in the objective, one option is to place a constraint on the portfolio variance. We already know how to calculate the portfolio covariance matrix using a risk model. All we do to get the portfolio variance is the same thing we did back when we were calculating the portfolio variance before. We write the portfolio variance as the weight vector transpose times the covariance matrix times the weight vector. That’s great, so let’s limit portfolio risk. We’ll say it has to be below some value. We’re already most of the way there. You might have one very reasonable question at this point which is, how do we know what value to set as our limit on portfolio variance? Well, this value represents the tolerable variance of your portfolio returns per time period over which you run the optimization. As a portfolio manager, this is usually given to you, but where does it come from? Well, it represents a business decision. Remember that the investment portfolio we’re designing represents a product, which is sold to investors. When investors are shopping around, they typically compare products with different risk return characteristics. What numbers are reasonable? The general stale volatility can be observed from the market itself, for example, from an index like the S&P 500. The annualized standard deviation of the S&P 500 over 100 years has been 12 percent or so. So, people benchmark their risk against that. If you look at indices that measure hedge fund industry performance, you see volatility of around four to five percent. So, if I’m running a hedge fund, there’s business risk associated with setting my product’s portfolio risk to something dramatically different from those numbers. Over time, there’s been evolutionary convergence to the levels of risk, that makes sense given the market for these products. If you work in a really big firm, and you’re managing a small portfolio, you will likely receive this number as a mandate, such as you can’t lose more than 10 percent. So, you can calibrate your risk so that you are comfortable with where you are currently in relation to that maximum loss threshold.