In the context of investment, risk may be defined as the statistical distribution that describes the probabilities of various levels of future returns relative to the expected return. The normal distribution is often assumed — but is invariably a simplification.
A key difference from the use of the word risk in the most other contexts is in an investment context the word risk may also cover the possibility that the actual outcome may be better than expected, as well as worse than expected.
The terms upside risk and downside risk are used to describe the positive and negative risk respectively. An analyst who comments on the upside risk to a share price is talking about the chances of the price rising.
When investors talk of managing risk, it is mainly downside risk that is being minimised. Strategies for reducing downside risk involve trade-offs, so will either reduce the expected return (for example, hedging with options) or the upside risk (for example, selling a very volatile share to buy something more stable).