Equity Risk Premium - the expected return on inventory exceeding the risk-free rate - is the basic amount of all asset valuations, both for theoretical and practical motives. This is a main degree of aggregate risk aversion and significant indicator of the cost of capital for businesses, individual savings determinations and budget strategies for governments. Lately, the capital risk premium (ERP) has also became a dominant as a most important indicator of economic development, a potential explanation of unemployment recovery and an indicator of financial stability

## ERP definition

Theoretically, ERP is the return that investors need to become indifferent on the boundaries between having a risky market portfolio and a risk-free bond. Because compensation depends on future stock results, the stock price risk premium includes future returns on inventories that are not directly recorded. Ultimately, every ERP model is a representation of investor expectations. The most challenging activity in estimating ERP is fact that it is not clear what return on the market and a risk-free rate really represents in the real world. Practically speaking, the most accepted measures of total market returns are reached from wide-ranging stock market indices like S&P 500. Eventually, each of ERP models is a model of investor expectations. These principles do not include all listed shares and some elements of luxuriousness, for instance - real estate, private equity markets and private equity. Although we limit ourselves to all traded shares, decision makers have many choices, such as using values or indices of equal weight, and excluding penny or rarely listed shares. A similar problem occurs with the risk-free rate. Although we almost always use Treasury yields as a quantity of risk-free rates, they are not completely risk free because nominal Treasuries are lay open to inflation and liquidity risks, even though we assumed that there was no chance of total default (Busseti E. p 1-5 2010).

## We can define ERP mathrmatically

$𝑅_t= 𝐸_t[𝑅_t+_k]+error_t+_k$

• 𝑅t+k make returns among t
• t+k and 𝐸𝑡[𝑅t+k] are expected, which is predicted from t to t + k at time t
• the error t+k formula is random, which is achieved at t and t + k

According to real expectations error t+k (mean 0 and perpendicular to 𝐸𝑡[𝑅t+k]) This is a general model and it fits many models, so we cannot assume rational expectations at this stage.

The horizon model in time t is interpreted as$ERP_t(𝑘) = 𝐸_t[𝑅_t+_k]-R^f_t+_k$,

• 𝑅t+k is risk free rate (from t to t+k)

This definition is very important and reveals some important conditions for ERP.

Initially, ERP excludes future return prospects and is stochastic because the result determined by new data which arrival is arbitrary and unknown. Also, it has a sort of investment horizon k that helps determine expected returns for a month, year, three years or more from today. Last but not least, all speculation is accurate because error t+k is stochastic. The capital risk premium is usually less than the real rate of return (Duarte F., Rosa C. p.2-4 2015).

## References

Author: Karolina Kaproń