Equity risk is "the financial risk involved in holding equity in a particular investment." Equity risk often refers to equity in companies through the purchase of stocks, and does not commonly refer to the risk in paying into real estate or building equity in properties.
The measure of risk used in the equity markets is typically the standard deviation of a security's price over a number of periods. The standard deviation will delineate the normal fluctuations one can expect in that particular security above and below the mean, or average. However, since most investors would not consider fluctuations above the average return as "risk", some economists prefer other means of measuring it.
Equity risk premium is defined as "excess return that an individual stock or the overall stock market provides over a risk-free rate." This excess compensates investors for taking on the relatively higher risk of the equity market. The size of the premium can vary as the risk in the stock, or just the stock market in general, increases. For example, higher risks have a higher premium. The concept of this is to entice investors to take on riskier investments. A key component in this is the risk-free rate, which is quoted as "the rate on longer-term government bonds." These are considered risk free because there is a low chance that the government will default on its loans. However, the investment in stocks isn't guaranteed, because businesses often suffer downturns or go out of business.
To calculate the equity-risk premium, subtract the risk free rate from the return of a stock over a period of time. For example, if the return on a stock is 17% and the risk-free rate over the same period of time is 9%, then the equity-risk premium would be 8% for the stock over that period of time.