SmartFolio contains the following optimization criteria:
CRRA Utility Function Maximization
Constant Relative Risk Aversion (CRRA) utility functions Class of utility functions which is exhausted by Logarithmic and Power utility functions. These utility functions are characterized by risk bearing proportional to wealth. form the most common class of utility functions used in econometric models. Under the assumptions of the analytical model The financial market model which assumes that all assets collectively follow a random walk in continuous time. This means that distribution of logarithmic returns is normal, returns are serially uncorrelated, and variance of returns grows linearly with a time interval under consideration., maximization of CRRA utility is equivalent to maximization of a weighted sum of expected portfolio return measured with portfolio Mu and portfolio variance. The latter expression can be considered as an extension of the single-period Markovitz criterion to its multi-period counterpart.
A CRRA utility is unambiguously specified by the parameter called relative risk aversion CRRA utility function parameter which determines degree of investor's attitude to risk. For risk aversed investors this parameter takes positive values. The more is the corresponding value, the more is utility function curvature, which leads to more conservative portfolio strategies. Parameter value equal to 0 corresponds to a risk-neutral investor. An investor who is interested only in his portfolio expected growth rate has relative risk aversion equal to 1. It is widely accepted that for most investors this parameter takes values in the range between 2 and 4. or just aversion. Possible aversion values range from zero to infinity. The more investor's risk aversion is, the less amount of risk the investor is willing to bear. Risk aversion equal to 1 results in logarithmic utility Utility function which belongs to CRRA (Constant Relative Risk Aversion) class of utility functions. Corresponds to Relative risk aversion equal to 1. Maximization of the logarithmic utility is equivalent to maximization of Expected excess growth rate., which is equivalent to maximization of portfolio expected excess growth rate Expected instantaneous rate of return (including Dividend yield) over the risk-free rate.. All other admissible aversion values correspond to a power utility Power utility functions along with Logarithmic utility fully exhaust CRRA (Constant Relative Risk Aversion) class of utility functions..
Note. The most common values of risk aversion for most
investors range from 2 to 4, values close to 4
corresponding to risk preferences of pension funds.
Calculating optimal portfolios for different values of risk aversion, we will obtain a set of efficient portfolios In SmartFolio an efficient portfolio is defined as portfolio which is optimal respective to a CRRA utility function with some value of Relative risk aversion.. Exactly this algorithm is used in the efficient frontier construction procedure
Target Shortfall Probability Minimization
This criterion might be viewed as an alternative representation of the previous one, but it is more suitable for practical usage. The point is that estimation of investor's risk aversion parameter is extremely difficult task. And even more, there are empirical evidences that this parameter changes according to the investor's circumstances.
An alternative approach to expressing investor's attitude to risk is expounded in [Stutzer; 2003]. The main steps of the proposed approach are outlined below:
The investor determines his/her investment horizon Critical date for the investor: when reaching it he/she evaluates success made by the investments. For private persons such date often corresponds to the moment up to which they postpone their consumption..
The investor determines his/her target excess growth rate The difference between growth rate which the investor demands on his investments and the Risk-free rate. All rates are continuously compounded..
The portfolio that minimizes the probability of falling short of the selected investment target is constructed.
In case the corresponding target shortfall probability is unacceptable for the investor (a table of shortfall probabilities corresponding to different investment horizons and target rates is calculated in target shortfall probabilities analysis procedure), then the investor lowers his target, repeating the whole procedure from step 2.
Note. Target Shortfall
Probability and CRRA Utility Function
optimization criteria are equivalent in the following sense: for each
value of target rate there exists such value of risk aversion (and vice
versa) that the corresponding optimal portfolios coincide.
The essence of benchmark tracking is minimization of portfolio deviation from the risk-free asset Other notations are Risk-free asset or Numeraire. It is the asset, in units of which portfolio wealth is measured. As a rule, the investor is nearly indifferent to changes in value of the riskless asset.. Obviously, it makes sense when portfolio consisting of risk-free asset alone is inadmissible. The most common situation where this is the case, is when the investor wishes to "track" some non-tradable or strongly illiquid asset, such as a broad-based market index. To achieve this goal you should perform the following actions:
In the portfolio construction dialog set the riskless asset to an asset you wish to track;
In the optimization constraints dialog check the zero weight in riskless asset checkbox;
Run the benchmark tracking optimization.
If you perform optimization over the analytical portfolio A portfolio which is analyzed using the assumptions of the Analytical model., then the only tracking error measure available is portfolio volatility The most common measure of risk. Defined as annualized standard deviation of returns.. But if, instead, you optimize weights of the historical portfolio As opposed to the Analytical portfolio, which is analyzed based on the assumptions of the Analytical model, historical portfolio is analyzed directly on historical data., then you also have an option of selecting portfolio downside volatility Similar to the notion of Volatility, but accounts for only those price increments which do not exceed the threshold. Semi-volatility is its special case corresponding to the threshold equal to the average increment size. The usage of downside volatility instead of ordinary volatility is justified when the distribution of logarithmic price increments deviates from normal. In the latter case downside volatility better reflects risks of the investor. tracking error measure. To specify the latter you must set a value of minimum acceptance excess rate Threshold value used in the formula for Downside volatility..
More detailed information on this topic is contained in the "Optimization criteria" section of the Theory Help.