
PeakTrader.com Economics, Portfolio Optimization, and Technical Analysis

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arthur Guest

Posted: Fri Sep 10, 2004 4:01 pm Post subject: Modern Portfolio Theory 


Modern Portfolio Theory assumes risk is simply a constraint or static. However, I believe, risk is dynamic. Imagine an XY space (I'll add the equations and graphs when they're complete). X is the horizontal axis representing risk and Y is the vertical axis representing return. The point where the X axis and Y axis intersect is the origin representing zero, i.e. zero return and zero risk.
The origin can represent 100% cash in a portfolio (i.e. where return equals zero equals risk; or p = 0 = r). Starting at the origin, taking small amounts of risk can yield greater amounts of return; or the marginal unit of risk is less than the marginal unit of return, i.e. increasing returns to scale. The Optimization point is where the marginal unit of risk equals the marginal unit of return, i.e. constant returns to scale (the derivative of a constant is zero). When marginal risk exceeds marginal return, then there is decreasing returns to scale.
Both the risk and return curves start at the origin. The risk curve is convex, i.e. increases at an increasing rate. The return curve is concave, i.e. increases at an increasing rate and then increases at a decreasing rate. The return and risk curves roughly form the shape of an American football at a 45 degree angle in the XY space. At the opposite end of the origin, the return and risk curves meet again. This is the point where cash equals zero. So, this model is not only an optimization model but also a contemporous model.
Therefore, an optimal portfolio must be balanced where marginal return equals marginal risk (i.e. the elasticities, or slopes, of both the return and risk curves are equal inverses). This is the point where return exceeds risk by the greatest difference. I've noticed the optimization point for an option portfolio is closer to the origin than a stock portfolio. So, less risk is needed to optimize an option portfolio. Also, unless adjustments are made to avoid decreasing returns to scale, e.g. when the size of the portfolio changes, then it will become increasingly easier for the portfolio to fall to zero (may add more info soon). 

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