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# Figure 1.1: a sequence of MRI images. Note the lesion (white blob) in the top of the brain

To do that you look at historic gains of each of these assets. Assume that you find that a s s e t i h i s t o r i c a l l y g a i n o n a v e r a g e p i , t h a t i s , a t t h e e n d o f t h e p e r i o d i t w a s w o r t h p i x i . T h u s , t h e t o t a l p o r t f o l i o w i l l w o r t h p > x . U n f o r t u n a t e l y , l i f e a r e n o t t h a t e a s y . T h e p r o b is that this gain was only on average. Each gain has also a standard deviation. As you may know from your own experience, the highest earning stocks are often the most risky one. We assume that A is the covariance matrix associated with the assets. Then, the risk can be written as x>Ax. There are a few problems which are associated with finding an optimal portfolio. Here we consider minimizing the risk while making some money. This can be written as l e m

x>Ax

(1.6a)

p>x ηT

(1.6b)

x>e = T

(1.6c)

x0

(1.6d)

min s.t

Where η > 1 determine what is the minimal earning we can live with reducing the risk as much as possible. This is a classical quadratic programming problem that can be solved for the optimal investment strategy.

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