Cumulative Distribution Function
Cumulative Distribution function is the sum of the probabilities of all outcomes up to and including a particular outcome.
X

p(x)

1

0.10

2

0.20

3

0.30

4

0.40

Discrete Uniform Random Variable
It is that for which all the outcomes have the same probabilities. If X = (1, 2, 3, 4, 5), then p(1) = p(2) = p(3) = p(4) = p(5). The cumulative distribution function for a discrete uniform random variable is n*p(x). The probability for a range of outcomes is p(x)*k, where k is the number of possible outcomes in the range.
A Binomial random variable
It is defined as the no. Of successes in a n trials where the probability of success , p , of each trial is constant. The binomial random distribution is defined as;
Tracking error
Tracking error is defined as the difference between the portfolio returns and the benchmark returns. Normally in investing you have a portfolio that tracks a benchmark. Eg: BSE SENSEX is a benchmark index of 30 stocks. If you have a portfolio that contains stocks included in the SENSEX, your portfolio will track the SENSEX. If you have 1 of each stock in the SENSEX in your portfolio, if the index rises by 12% your portfolio will also give you 12% returns. Similarly if the index tanks by 5%, your portfolio will also have 5% losses. This is called passive management. Sometimes in an attempt to earn better returns than the benchmark portfolio managers follow different strategies in terms of stock selection and positions taken. This is called active management. Active portfolio management will have portfolio returns that differ from the benchmark either on the upside or the downside. This difference in returns is called tracking error, i.e. by how much does the portfolio returns differ from the returns of the benchmark it tracks.
Normal Distribution:
Univariate normal distribution
Univariate normal distributions are the distributions of the outcomes of a single random variable.
Multivariate normal distribution
Multivariate normal distribution are distributions of the relationship between two or more variables. Eg. The relationship between the returns of two assets. For a multi variate distribution it is necessary that there be a relationship between the two variables. That is there should be some correlation between the two variables, positive or negative. For a multi variate distribution with two variables there would be two means, two variances and the pair wise correlations will be given as, 0.5*n*(n1).
Standard normal distribution
It is a normal distribution that has been standardised so that it has a mean of 0 and a standard deviation of 1. Standard normal distribution is given by the z value which is the no. Of standard deviations an observation is from the population mean. It is given as ;
Roy’s Safety First
It measure is given by;
It sets a minimum required return for a given level of risk. The Roy’s safetyfirst criterion allows portfolios to be compared based on the probability that their returns will fall below this minimum desired threshold. The optimal portfolio will be the one that minimizes the probability that the portfolio’s return will fall below a threshold level. Safety first ratio calculates the portfolio returns are how many standard deviations away from the threshold. It calculates the standard deviations below the mean and therefore is a negative figure. The larger the SFR the better because it means the lower the probability of the returns falling below the threshold and losing money or earning negative returns.
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