he crisis of 2008 brought tail risk hedging into the calculus of investing, and turned risk management into an active rather than passive activity. Since then, the question has been raised: Is it better to dynamically hedge tail risks by rebalancing portfolio exposures or by explicitly purchasing options?
Rather than making an exclusive choice we believe that rebalancing, options purchase and diversification should all be considered on the same footing. The pricing of assets, both relative (among each other) and absolute (with reference to the portfolio’s risk and return posture), should be the primary driver in the end. For the same amount of explicit or implicit premium, investors should choose the strategy that is most likely to provide their desired outcome.
Given the recent increase in risky asset prices, investors are asking the same question again today: Is it better to dynamically de-risk if markets begin to fall to lock in gains, or is it better to purchase explicit tail hedges? While the first alternative incurs no upfront premium, it is subject to market gyrations. The second alternative locks the investor into an upfront premium, but it reduces the need for dynamic rebalancing.
While the choice ultimately depends on the price of the hedges, there is another important perspective which supports the explicit purchase of options, rather than dynamic hedging. This has to do with our tendency, as humans, to be time-inconsistent, that is, our behavior changes as the situation changes, usually with the passage of time. This makes dynamic rebalancing prone to behavioral biases. For instance, in the heat of the battle (as the markets were collapsing in 2008), we discovered that the most rational of investors first held their risky assets. Some even doubled down. But as markets fell further, investors dumped these risky assets as they essentially got stopped out, in traders’ terms. This was in stark contrast to their plans to rebalance to a policy portfolio (60% equities, 40% bonds) more or less continuously.
Before starting an investing program, we usually tell ourselves that if losses start to accrue, at some point we will exit our position and wait on the sidelines. If we can follow this plan, our return distribution will be positively skewed, that is, we have potentially unlimited gains, while we have a strategy designed to limit losses. But in practice investors generally are not able to follow through with the plan, and end up with a negatively skewed return distribution.
Those of us who have been to a casino know this behavior only too well. Gamblers at the table on average stay too long when they are losing (until they cannot lose any more) and get out too early when they are winning. Casinos, of course, encourage gamblers to stay longer at the table – by providing all sorts of free perks. (When I go to Vegas I take the free perks but usually don’t spend a dime or a second at the tables – I have no skill!) This time-inconsistent behavior can be explained (see Nicholas Barberis, “A Model of Casino Gambling,” 2009) by the perception of probabilities changing as the gamble progresses. A gambler enters the casino with the intent of leaving when his losses exceed a particular threshold, and even in a coin flip game with perfectly predictable odds shows the tendency to stay too long when losing and exit too soon when winning. To understand why this happens, it’s important to understand that people generally have the tendency to overweight low-probability events and underweight high-probability events. In other words, objective probabilities are “re-weighted,” depending on the state of the world. Intuitively, as winnings accrue, the investor underweights the 50/50 probability of winning in the next round, and exits too early.
Now, consider the impact of this behavior on downside risk management. Suppose an investor enters the market with the intent of cutting his losses and exiting as soon as total losses reach a predefined level, but doesn’t have a specific control in place to ensure that happens. And suppose he is investing 10-dollar “chips” so he will “lose” or “win” in 10-dollar increments. He is also “gambling” in a market with 50/50 odds. Now, assume that he finds himself in the state where he has accumulated four losses, so his total losses are 40 dollars, and he attaches a “value” to this loss of v(-40) where is a loss-averse “value” function, which typically shows a form displayed in Figure 1. Now his choice is to either exit or play another round. If he wins the next round, his losses decrease to -30, whereas if he loses again his losses increase to -50. If the probability of losing is ½, then his expected value from “gambling” again is v(-50)w (½) + v(-30)(1-w(½)). Here is the subjective probability mapping function that takes objective probabilities and maps them to subjective probabilities. As shown in Figure 2, the moderate probabilities are underweighted and tail probabilities are overweighted, so, since w(½) < ½, and given the convexity of the value function in the region of losses, he decides it’s better to gamble another round.
The horizontal axis represents gain or loss from current value and the vertical axis represents the value attached to the gain or loss. The value function shows concave behavior for winnings and convex behavior for losses. In particular, this investor becomes more risk-seeking when faced with losses.
The horizontal axis represents actual objective probabilities and the vertical axis represents subjective probabilities. The straight line is the undistorted probabilities where objective and subjective probabilities are identical. The curved lines represent probability weighting where tail probabilities are overweighted and moderate probabilities are underweighted.
When evaluating the value functions for loss-averse investors who overweight tail probabilities, we find that the expected value for continuing to “gamble” is higher even though the pre-set loss threshold is breached. The final outcome is this: The investor, without a pre-commitment device in place, fully rationally changes his plan in the heat of the battle. Even though he enters the market with the intent of exiting when his losses hit a particular threshold, when faced with that state, he decides to gamble. The intuition is simple: Before the first gamble begins, the probability of a string of losses is objectively evaluated to be low in absolute terms, but still subjectively evaluated to be higher than the objective probabilities. This leads to a loss-containment plan as a pre-condition to enter the gamble. However, once faced with actual losses, the investor underweights the probability of another loss and chooses to stay at the table. The tendency to take more risk when losing combined with the probability weighting results in the investor abandoning his original plan.
To overcome this very natural bias to be inconsistent in time as the game evolves, one can put a concrete pre-commitment device in place. In the casino, not using your ATM card (or leaving it at home) is one such device. When investing, one way to do this is to invest with systematic rules in place that cannot be changed. The other way is to spend a small amount of premium, if valuations are attractive, to outsource the exit strategy to the options markets. At pricing levels of low option premia the purchase of options to prevent time-inconsistent behavior seems like a judicious decision. To commit to such a plan that creates and maintains positive skewness in portfolio construction makes good investment sense.