Use of conditional expectation in investment applications

Expected Value of random variable X is calculated as:

E(X) = ΣP(x_i)x_i

Variance of a random variable
The expected value of squared deviations from the random variable’s expected value. Variance measures the volatility from an average and thus is a measure` of risk.

Var (X) = σ²(X)= E[X – E(X)]² = ΣP(x_i)(x_i – E(X)²]

Where:
P(x_i) is the probability of occuring the return of x_i

Standard deviation is the square root of the variance. Used as a measure of risk shows dispersion of possible outcomes around expected level of outcomes.

σ(x) = (Variance)^1/2

Use all relevant information in making forecast of investment returns and risk. When new information becomes available, need to refine the conditiononal expectation.

E(XlS) = ΣP(x_ilS)
Where:
S – new information available

Bayes’ Formula
Bayes formula adjusts a probability to handle the addition of new information, i.e. to update a given set of prior probability for a given event in response to the arrival of new information. Bayes' Formula, is given as below:

P =(probability of new information given event / unconditional probability of new information) x prior probability of event

That is,
Posterior P(A) = Prior P(A) x [P(BlA) / P(B)]