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Analytical portfolio

Dynamics of Portfolio Wealth, generated by Portfolio Strategy with Constant Weights


Assume that discounted prices of assets move according to the analytical model with Mu vector , vector of continuously compounded dividend yields , covariance matrix and risk-free rate . For
simplicity assume that the vector is constant through time. Then dynamics of discounted portfolio wealth , corresponding to portfolio vector , satisfies the
following equation:


where
is a
Wiener process,




Note.
Readers who are not familiar with stochastic differential equations in continuous time may interpret
as very short time range, as a
simple return on portfolio over  and as a normally
distributed random variable with zero mean and variance , independent of other variables ,
.


Definition. Quantity is called Portfolio Mu, or Portfolio Expected Instantaneous Simple Rate of Return.
Definition. Quantity is called Portfolio Dividend Yield.
Definition. Quantity is called Portfolio Excess Mu.
Definition. Quantity is called Portfolio Volatility.
Definition. Quantity is called
Portfolio Expected Excess Growth Rate.
The latter definition reflects the easily derived equality .


Decomposition of Portfolio Variance


The direct consequence of the portfolio volatility definition is the following decomposition of portfolio variance :

where . Obviously,
Element denotes proportion of portfolio variance, contributed by -th asset.
In other words, vector with components, called Contributions to Portfolio Risk, determines alternative representation of portfolio structure, measured in units of portfolio variance rather than wealth, as in case of vector .













 
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