# volatility game

**Volatility has been at the forefront of many investors’ minds over the past couple of weeks. But in the midst of the short-term upheavals and reactions, it’s all too easy to lose sight of the longer term impact of volatility. In this blog I take a closer look at volatility as a measure of risk. Whilst it has its drawbacks, it also has some important advantages, particularly for long-term investors.**

Suppose you’re not the gambling type but are forced to repeatedly play a game, much like a long-term investor facing some unrewarded risks. The game involves rolling a fair die and making a gain or loss depending on what number turns up. You can choose between two games:

Payoffs |
||

Result from rolling die |
Game A |
Game B |

1 |
-£10.00 |
-£6.61 |

2 |
£1.00 |
-£3.97 |

3 |
£1.50 |
-£1.32 |

4 |
£2.00 |
£1.32 |

5 |
£2.50 |
£3.97 |

6 |
£3.00 |
£6.61 |

Both games have the same average outcome (zero) but you’d like to reduce risk as much as possible. Which game would you go for? Eyeing them up, Game B has symmetric and evenly spaced payoffs. Game A, on the other hand, offers gains for all rolls other than unlucky snake-eye number “1”, in which case you lose a relatively large amount. We suspect most people would prefer game B, as it avoids this sting in the tail.

But what if you were to play these games repeatedly? What do the profiles of total winnings look like then? The charts below are frequency plots (which show the chances of different outcomes) from playing the games multiple times. The chart on the left shows the results for 10 games of A in a row (in blue) and 10 games of B in a row (in yellow). The chart on the right shows the results for 100 games.

The more and more you play, the closer the two profiles become. In fact, they both tend towards the same bell-shaped distribution*. The perhaps surprising conclusion is that a long-term player should be indifferent between A and B**. The short-term “shape” is irrelevant. All that actually matters is the standard deviation, which describes how wide the distribution is. As you might have guessed, I constructed the standard deviation to be the same for A and B.

It’s interesting to note that I could have made game B slightly more risky, but still symmetric, just by scaling up its payoffs by a fraction. If I did that then many people, not being Vulcans, would still have preferred game B, despite A having a lower standard deviation. These people would be setting themselves up for a wider bell-shaped distribution of outcomes in the long-run, purely to satisfy their short-term feelings.

This might remind you of a well-known behavioural phenomenon: loss aversion. This also leads to people giving up long-term outcomes to satisfy short-term feelings.

**Implications for investors**

It’s fashionable to criticise volatility (the standard deviation of returns) as not allowing for the shape of returns. But the argument above should give us pause for thought. That feature can actually be a *good* thing, because the short-term shape of returns is often a red herring when it comes to long-term outcomes.

It’s true that volatility doesn’t distinguish between upside and downside risk (as critics like to point out) but what this ignores is that most problems are multi-period. Unless you are a genuinely short-term investor with capped upside for some reason, today’s upside is as good as a reduction in tomorrow’s downside.

Understanding this can lead to various changes in strategy, such as:

- Not necessarily shying away from strategies that have a sting in the tail as a diversifying source of return (e.g. writing puts).
- Multi-asset diversification by asset class: this is sometimes criticised because correlations of short-term returns ‘go to 1’ in crises, failing when you need it most. But provided diversification reduces the volatility of overall returns, diversification is still highly beneficial, as this is the measure of risk that matters in the long-run. Watch this space for more on this.
- DB schemes often have lumpy LPI benefits (pension increases with hard caps and floors) that they aim to hedge using a changing mix of fixed and inflation-linked assets. This is impossible to do perfectly, but the primary focus should normally be on reducing volatility, rather than focusing on the short-term shape of the payoffs.

Volatility isn’t a perfect measure of risk – some of the criticisms it faces (which I don’t go into here) are valid. But it also can have important advantages, particularly for investors focusing on their long-term range of outcomes. At times like the present, it’s important to look beyond daily figures and maintain a longer term perspective.

** This follows from a powerful result in statistics called the **Central Limit Theorem**.*

*** although technically this requires that these bets are not too large in the context of overall wealth.*