This paper discusses the value-at-risk (VaR) and conditional value-at-risk (CVaR) as risk measures, their properties, and their use in portfolio optimization. VaR is defined as the α-quantile of a random cost variable Y, while CVaR is defined as the solution to an optimization problem involving the expected value of Y given that Y exceeds VaR. It is shown that CVaR equals the conditional expectation of Y given that Y is greater than or equal to VaR. The paper proves that the minimizer in the CVaR optimization problem is VaR, even when the distribution function is not differentiable. It also provides alternative representations of CVaR, including its expression as the expected value of Y given that Y is above VaR, and as an integral over the distribution function. The paper studies the relationship between VaR and CVaR, highlighting their differences and similarities. It shows that CVaR is a coherent risk measure, while VaR is not. The paper concludes that CVaR is a more robust risk measure than VaR, as it provides a more comprehensive assessment of risk. The paper also discusses the use of VaR and CVaR in portfolio optimization, showing that they can be used to construct portfolios that minimize risk. The paper concludes that CVaR is a more useful risk measure for portfolio optimization than VaR.This paper discusses the value-at-risk (VaR) and conditional value-at-risk (CVaR) as risk measures, their properties, and their use in portfolio optimization. VaR is defined as the α-quantile of a random cost variable Y, while CVaR is defined as the solution to an optimization problem involving the expected value of Y given that Y exceeds VaR. It is shown that CVaR equals the conditional expectation of Y given that Y is greater than or equal to VaR. The paper proves that the minimizer in the CVaR optimization problem is VaR, even when the distribution function is not differentiable. It also provides alternative representations of CVaR, including its expression as the expected value of Y given that Y is above VaR, and as an integral over the distribution function. The paper studies the relationship between VaR and CVaR, highlighting their differences and similarities. It shows that CVaR is a coherent risk measure, while VaR is not. The paper concludes that CVaR is a more robust risk measure than VaR, as it provides a more comprehensive assessment of risk. The paper also discusses the use of VaR and CVaR in portfolio optimization, showing that they can be used to construct portfolios that minimize risk. The paper concludes that CVaR is a more useful risk measure for portfolio optimization than VaR.