Traders and investors often use the notional exposure of their portfolio constituents in relation to their total capital as a measure of risk, and they build their portfolio weights around this idea. If you are including diverse instruments in your portfolio with very different volatilities, the former might not be the best approach, because you will not be able to allocate the same risk to equal weights.
We can start with an example to be able to go easily to the point: one dollar invested in a US two-year treasury note future will have a very different profit and loss profile, than a dollar invested in S&P 500 futures, right? Well let’s observe the prices (t , p(t) ) of both instruments in the following chart:
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We use continuous price curves here for the futures using the Panama method looking at open interest for the roll (topic for another post). |
Now, calculate the returns from the prices, defined as:
And plot its corresponding chart:
As you can see, the charts above already look ridiculous. Why? Because the instruments are of a very different nature (and shouldn’t compare them like that), i.e., VERY different volatility.
To better manage the above case, we propose here the concept of volatility scaling (also known as well as volatility weighting or volatility normalizing).
We can start by revising a simple and intuitive way of vola scaling, popularized by the Turtle traders, which would serve chart traders and technical analysts to scale their risk allocation. This approach is based on the ATR (average true range), which is a popular price-based indicator. Think of the ATR as a volatility indicator for the instrument. The method is the following:
Assume you made up your mind, are ready to make a trade, and are willing to risk a certain percentage of your capital. For example, 10 basis points of an account value of $1M, which would be 0.001 * $1M = $1000 (so you would like to risk $1000 in this trade). We are still talking about futures, so you need to take into account the size of the contract with the point value. If the contract is in a different currency than your account, also the FX rate. Then,
What about that n? That’s an important part of the equation. It’s the number of ATRs that you will tolerate, that the market moves against you; it’s the price level where you will set a stop loss. So, instead of picking an arbitrary price level, you will set a level that depends on a multiple n of the volatility of your investment asset. Since assets have VERY different volatilities, it’s a better choice to decide price levels depending on volatilities than just picking one. In other words: if you don’t take volatility into account, your stop might be triggered just by usual volatility and you might have been right in the general direction of the market.
Now, for portfolio traders and many other applications, we can continue to generalize the scaling. Let’s pick a standard statistical way to calculate volatility (there are many choices, this is one of them) over a chosen fixed length daily time window {t,…,T}:
The above formula is the unbiased annualised moving standard deviation of the daily returns (1). Where,
is the moving mean of daily returns (1) over the same window.
It’s up to the modeler to choose the time window. Long windows provide more stable estimates, whereas short windows will react quicker.
Now, we can create the main result of this piece: the scaled returns! Analogous to the turtle trader result, if you wish that your investment vehicle to have a certain volatility (or a target volatility v), use the following function to convert your returns:
Observe now our two transformed, very different assets in a constant investment chart of cumulative returns with an annualized target volatility of 12%:
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| To know more about constant investment charts, go to this post |
Interestingly, notice that when running the same risk (meaning the same target volatility) with an equal allocation to the above instruments, the US two-year outperforms the S&P 500 in this time period.
If you use this methodology, you will allocate risk and not cash, and you will be giving all your trade ideas the same chance (risk) to work!
