Interest Rate as the Sum of Real Risk-free Rate and Risk Premiums
Interest is a reward a borrower pays for using an asset, usually capital, belonging to a lender. It is compensation for the loss or value depreciation occasioned by the use of the asset. We could also describe it as the…
Interest Rates as Required Rates of Return, Discount Rates, and Opportunity Costs
The time value of money is a concept that states that cash received today is more valuable than cash received in the future. If a person agrees to receive payment in the future, he foregoes the option of earning interest…
Methods of Solving Counting Problems
Counting problems involve determining the exact number of ways two or more operations or events can be performed simultaneously. For instance, we might be interested in the number of ways to choose 7 chartered analysts comprising 3 women and 4…
Updating Probability Using Bayes’ Formula
Bayes’ formula is used to calculate an updated or posterior probability given a set of prior probabilities for a given event. It is a theorem named after the Reverend T Bayes and is used widely in Bayesian methods of statistical…
Covariance of Portfolio Returns Given a Joint Probability Distribution
Covariance between variables can be calculated in two ways. One method is the historical sample covariance between two random variables \(X_i\) and \(Y_i\). It is based on a sample of past data of size \(n\) and is given by: $$\text{Cov}_{X_i,Y_i}=\frac{\sum_{i=1}^{n}{(X_i -\bar{X})(Y_i…
Probability Tree and its Application to Investment Problems
A tree diagram is a visual representation of all possible future outcomes and the associated probabilities of a random variable. Tree diagrams are particularly useful when we have several possible outcomes. They facilitate the recording of all the possibilities in…
Conditional Expectation in Investments
In the context of investments, conditional expectation refers to the expected value of an investment, given a certain set of real-world events that are relevant to that particular investment. This means that in their calculation and prediction of the expected…
Expected Value, Variance, and Standard Deviation of Random Variables
Expected Value The expected value of a random variable is the average of the possible outcomes of that variable, taking the probability weights into account. Therefore: $$ E\left( X \right) =\sum _{ i=1 }^{ n }{ { X }_{ i…
Calculating Unconditional Probability Using Total Probability
We can use the total probability rule to determine the unconditional probability of an event in terms of conditional probabilities in certain scenarios.
Dependent and Independent Events
Two or more events are independent if the occurrence of one event does not influence the occurrence of the other event(s). Let us put this in annotations:
