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Lesson 10

Lesson 10. Relation between two RVs produced in the same experiment. Portfolio construction. We want to spend our money into several securities bought in the market. So we need to study several securities (and therefore RV) together.

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Lesson 10

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  1. Lesson 10 Relation between two RVs produced in the same experiment

  2. Portfolio construction • We want to spend our money into several securities bought in the market. • So we need to study several securities (and therefore RV) together. • If securities were independent (on a market, think of the NYSE, and the dollar) it would be easy to build a portfolio with high return and zero risk.

  3. One RV • Collection of possible values (ai’s) • Set of probabilities • A long series of outcomes • A histogram • The outcomes were produced by replication of an experiment E • In finance, the usual experiment is « wait one year » • Two concepts : the mean and the variance

  4. Two RVs (produced in the same experiment) • All the initial concepts are naturally extended. • But there will also be a new one. • Old ones : • The set of possible values : a collection of pairs (ai, bj)’s • A set of probabilities : each pair has a probability • A long series of outcomes ; we can plot them and construct the extension of a histogram : a scattergram, and we count pairs that fell in each cell of a grid

  5. 2 RVs : the new concept • The new concept is : the variables may or may not be related • Some joint distributions (i.e. the set of probabilities) show independence, and some set of probabilities reveal dependence. • This can also be seen with scattergrams drawn from long series of actual outcomes of pairs. • Concept introduced by Karl Pearson (working with Francis Galton, a cousin of Charles Darwin) in the second half of the XIXth century, while studying the role of genetics in evolution and related topics in biology and agriculture.

  6. Visual interpretation • The best way to « feel » the relationship between two RV is to look at their scattergram (or their joint distribution) • We « fit » an oval shape (with the appropriate technique which we won’t study) through the scattergram • The most important fact is whether there is an angle between the axes of the oval and the x and y axes • If the angle is zero : no relationship between X and Y. • If there is an angle : there is a relationship • The narrowness of the oval is related the strength of the relationship.

  7. Negatively related RVs

  8. Independent RVs

  9. The mathematical concept of covariance • Covariance is the formalisation of the relationship between 2 RVs • Correlation is a slight variation on covariance

  10. Computation of covariance • If we remember how we computed the variance of one RV… • …then we extend this to 2 RVs.

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