#### A new paper by the Financial Engineering team shows how simple strategies based on astute risk management can generate additional returns for investors

In 1863, the French broker Jules Regnault introduced the idea that stock prices move randomly. The French mathematician Louis Bachelier pursued this idea in his PhD thesis entitled “The Theory of Speculation” in 1900. This early work resulted in what is today known as the random walk^{[1]} theory of stock prices.

A typical feature of the random walk theory is to assume that the noise in stock prices can be described by Gaussian distributions with a given volatility constant over time. If this were true, risk management would be a relatively boring activity. Investors allocating a certain amount of wealth to a given financial risky asset would not only know how much of that wealth would be at risk, but would also expect risk to be constant over time.

However, the volatility of financial asset prices is not constant. In 1986, Tim Bollerslev introduced the GARCH^{[2]} model, which provides a much better fit with the observed changes in prices of financial assets. This model takes into account the reality that volatility does change over time, but with predictable patterns. It so happens that when the volatility is low it tends to stay low; conversely, when it is high it tends to stay high. These high and low regimes of volatility are also known as volatility clusters.

Yet this departure from the work of Regnault and Bachelier is still not sufficient to describe observed changes in prices of financial assets. Other more sophisticated GARCH models can however incorporate the missing features. For example, the fact that financial asset returns exhibit fat tails, i.e. a higher frequency of large returns, can be modelled by generating noise not from a Gaussian distribution but from a t-Student distribution. The fact that, e.g. for equities, returns tend to be lower in periods of high volatility and higher in periods of low volatility can also be plugged into GARCH models. These more sophisticated GARCH models are effective in predicting the volatility of asset price returns and in modelling those same returns.

In our most recent paper^{[3]} we show that these sophisticated GARCH models do indeed do a good job at describing the returns of various asset classes and the changes in the volatility of those returns over time. The models are particular good for risky asset classes like equities and high yield bonds, and to some extent commodities, which exhibit significant volatility clustering, fat tails and a negative relationship between returns and volatility. They are less relevant for government bonds or investment grade corporate bonds, which show smaller changes in the volatility of returns over time with less clustering.

Our paper also shows how investors can take advantage of this complex behaviour of riskier financial asset returns using, in the end, very simple strategies. Indeed, investors can substantially improve their returns by implementing the simple strategy of selling risky assets when the volatility rises and buying more when the volatility falls. The amount invested in the risky assets should be such that the product of the percentage of wealth invested in risky assets multiplied by the volatility of the risky assets is constant over time. We call this strategy *inter-temporal risk parity* because it aims at keeping the risk constant over time.

This strategy reduces the allocation to risky assets when their volatility is high and stays high and returns are typically low, and increases such an allocation when the volatility is low, stays low and the returns are typically high. For that reason, the strategy generates higher returns than a simpler buy and hold strategy. Also, because fat tails are bigger in high volatility regimes, the strategy also filters these out and reduces the drawdowns in the investor portfolio when compared to a more simple buy and hold strategy.

Financial assets do have much more complex behaviour than the simple random walk approaches proposed in the 19^{th} century. But with advanced risk modelling approaches and simple strategies to keep the risk of their portfolios constant over time, investors can turn that complexity to their advantage and earn higher returns than simply buying and holding risky assets.

[1]* **A Random Walk Down Wall Street*, written by Burton Gordon Malkiel, professer of economics at Princeton, is an influential book on the subject of stock markets which popularized the random walk hypothesis. Malkiel argues that asset prices typically exhibit signs of random walk and that one cannot consistently outperform market averages. The book is frequently cited by those in favor of the efficient market hypothesis.

[2] Tim Bollerslev, "Generalized Autoregressive Conditional Heteroskedasticity", *Journal of Econometrics*, Vol. 31 (1986), pp. 307-327.

[3] Romain Perchet, Raul Leote de Carvalho, Thomas Heckel, Pierre Moulin. “Inter-temporal risk parity: a constant volatility framework for equities and other asset classes”. Working paper (2014) http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2384583