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### Definition of Certainty Equivalence

Suppose, a Bernoulli utility function is u(·), the certainty equivalent of a lottery F(·), denoted c(F, u), is the quantity that fulfill the following equation:

u (c(F, u)) = R ∞ −∞ u(x)dF(x)

An individual will be exactly indifferent between a lottery which place probability one on the certainty equivalent and the lottery F(·).

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### Definition of the risk premium:

When a Bernoulli utility function is u(·) and a lottery F(·), then the risk premium which is denoted by ρ(F, u), is the difference between the mean of F and the certainty equivalent c(F, u) :

ρ(F, u) = R ∞ −∞ xdF(x) − c(F, u),

The risk premium is considered as the maximum amount that an individual is willing to pay to in order to eliminate the risk.

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### Certainty and cash flow

Certainty equivalent is also linked with the cash flow concept in the business organizations.  It is Certainty cash flow which is also considered as the risk free cash flow which is often suggested for the investors or the managers.  It is somewhat equal to the different expected cash flow in the business which is not only high but also risky.

The method or formula for calculating the certainty equivalent cash flow is

Certainty equivalent cash flow = expected cash flow / (1 + risk premium)

In which, risk premium = risk-adjusted rate of return – risk-free rate

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