Risk and risk premium

Risk is uncertainty. When you make an investment, there is always a risk associated with it whether it is a bond, stock, fund, etc. Risk arises from uncertainty. We don't know market direction next year. As a matter of fact, we don't even know how market will drop/raise our stock price tomorrow. Risk can be separated into 2 different types. One is an internal risk. This is also known as a business specific risk, where uncertainty comes from the company and its operation. The other one is a market risk. This is an external risk that you will have to bear.

Scenario Analysis and  Probability Distributions
In order to examine risk, we can look at the risk/uncertainty using probability. Here 3 important variables that you need to know before moving on the analysis.
1) Mean: most likely value
2) Variance or standard deviation
3) Skewness

This picture demonstrates famous concept known as "normal distribution." The normal distribution is an absolutely continuous probability distribution whose cumulants of all orders above two are zero. Note that norm. distribution has mean equals to median.
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. skewness value can be positive or negative, or even undefined. Qualitatively, a negative skew indicates that the tail on the left side of probability density function is longer  than the right side and the bulk of the values (including the median) lie to the right of the mean. A positive skew indicates that the tail on the right side is longer than the left side and the bulk of the values lie to the left of the mean.

Measuring Mean: Scenario or Subjective Returns

p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
This is a formula to calculate expected return (subjective/scenario return), which is equals to multiplication of each state's probability (in decimal) and return. After you get each value, sum them up to get final E(r) value.

Numerical Example: Subjective or Scenario Distributions

State    Prob. of State    rin State
    1            .1        -.05
    2            .2        .05
    3            .4        .15
    4            .2        .25
    5            .1        .35

    E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)
    E(r) = .15 or 15%


Measuring Variance or Dispersion of Returns
Variance is used as one of several descriptors of a distribution. It describes how far values lie from the mean.

Var = sum of probability (in decimal) multiplied by outcome of individual return minus average. And Standard deviation is simply a square root of variance.

Using the same example above,

Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095 or 10.95%