# Portfolio Optimization
## Utility functions approach vs. mean-variance approach
Assume that the analytical model holds in the market with assets, excess Mu vector and the covariance matrix . By we denote the vector of constant portfolio weights.
**Mean-Variance Approach** can be formulated in several equivalent ways.
- Minimization of portfolio
**volatility** subject to the lower constraint on portfolio **excess Mu** (over all admissible portfolios ):
subject to
- Maximization of portfolio
**Excess Mu** subject to the upper constraint on portfolio **volatility** (over all admissible portfolios ):
subject to , where is a strictly positive constant.
- Maximization of the following expression (over all admissible portfolios ):
, where is a strictly positive constant.
**Utility Function Approach** consists in maximizing the expected value of utility function from portfolio terminal wealth over all admissible portfolios .
**Key Result**
*Under the assumptions of the analytical model the maximization of expected CRRA utility with the relative risk aversion coefficient from terminal wealth and the maximization of (both maximizations are taken over all admissible portfolio vectors ) result in the same optimal portfolio .*
Important! *The above statement allows all the results obtained in ***single-period framework** to be easily extended to **multi-period** one with *CRRA Utility Functions**.*
**Note. *** is also known as the ***Risk-Adjusted Expected Excess Rate of Return**, corresponding to the *relative risk aversion coefficient** .*
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