When formulating investment strategies and assessing risk, traders and investors often make use of two quite different techniques: scenario analysis under specific market scenarios, and Monte Carlo simulation. Scenario analysis is useful for analyzing performance under subjective market views, but it cannot answer probabilistic questions such as: What is the portfolio value-at-risk assuming a bear market occurs? Also, since the results are very sensitive to the precise scenario specification, it cannot give a robust answer to the question: How poorly does a stop-loss strategy perform when the market whipsaws?
Can one answer these questions by combining scenario analysis with Monte Carlo simulation? In its usual form, Monte Carlo simulation draws random paths from a "neutral" probability distribution on the space of paths, ignoring the subjective views of the investor or risk manager. Is it possible to take these views into account? For example, suppose one wants to compute value-at-risk under a bear market assumption. One way would be to carry out a Monte Carlo simulation which drew only paths under which security prices fell by at least 20%. It is only recently that efficient methods have been developed for carrying out this kind of simulation--the so-called Markov chain Monte Carlo algorithms. The results in this article are derived using Gibbs sampling, an algorithm from this family which was initially developed for use in image processing.
The outcomes of option strategies or investment rules, under non-neutral market assumptions, have probability distributions with shapes that are far from obvious. As an example, this article presents a probabilistic analysis of several investment strategies applicable to an individual stock, under a variety of scenarios. The strategies are as follows:
* Invest all funds in the stock, and hold it to the horizon date.
* Invest in a combination of the stock and a put expiring on the horizon date.
* Invest all funds in the stock, with a once-and-for-all stop-loss rule at a specified level.
* Invest using a stop-loss rule, but also buying back at a specified, higher, level.
In each case a horizon of one year is used, with monthly time steps. Stop orders are assumed to be executed at the end of the month, so that considerable slippage is possible. The scenarios were defined as follows: "neutral" means all possible paths are allowed, with probabilities determined by the specified volatility; "was_bear" only includes paths under which the stock price fell by 20% at some stage, though it may or may not have recovered later; and "whipsaw" only includes paths under which the stock price fell by 20% at some stage and recovered to its initial value at some later stage.
Note that each scenario is imprecisely specified: the conditions are satisfied by an infinite number of paths. Gibbs sampling was used to generate one million random paths drawn from the conditional distribution corresponding to each scenario, and thus to estimate the probability distribution of returns.
Figure 1 shows maximum loss estimates, i.e. 95% quantiles of the estimated distribution of investment returns; these may be regarded as the investment manager's analog of "value-at-risk", in this simple single-security case. Under a neutral assumption on market direction, the put and stop loss strategies are about equally effective at limiting losses--the cost of the option roughly offsets the risk of overshooting inherent in the stop loss rule. However, greater losses are possible with the stop/buy-back strategy if the stop is tight. Under a bear market assumption the put is significantly better at limiting losses than the stop or stop/buy-back strategies. Finally, under a whipsaw assumption the simple buy-and-hold strategy has the smallest losses, and the put strategy is superior to the stop loss rules.
Figure 2 shows the full return distributions of four investment strategies under the assumption that a bear market occurred at some stage. The put strategy limits losses to an absolute maximum, unlike the stop-loss strategies, and also has slightly more upside in the cases where the market rallies back. However, the distribution peaks at a more negative return level because of the option cost.
The analysis can easily be extended to more complex instruments such as barrier options. But even for simple instruments and strategies, it shows how return distributions can change qualitatively as well as quantitatively when strategy parameters such as put strikes and stop loss levels are varied--illustrating why it is critical to supplement intuition and single scenario simulation with a more rigorous probabilistic analysis of risk.
This week's Learning Curve was written by Wesley Phoa, Los Angeles-based director of research at Capital Management Sciences.
