2 Quantitative Methods - Quantitative Methods for CFA
1. Time Value of Money (TVM)
The Time Value of Money (TVM) is a fundamental concept in finance that states that a sum of money is worth more now than the same sum will be at a future date due to its potential earning capacity. This concept is crucial for making informed financial decisions, such as investing, borrowing, and saving.
Key Components of TVM
- Present Value (PV): The current worth of a future sum of money or stream of cash flows given a specified rate of return.
- Future Value (FV): The value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today.
- Interest Rate (r): The rate at which money earns interest over a given period.
- Number of Periods (n): The number of time periods over which the money is invested or borrowed.
Example
Suppose you have $1,000 today and you can invest it at an annual interest rate of 5%. In one year, the Future Value (FV) of your investment can be calculated using the formula:
FV = PV * (1 + r)n
FV = $1,000 * (1 + 0.05)1 = $1,050
This means that $1,000 today is equivalent to $1,050 in one year, given a 5% interest rate.
2. Probability Distributions
Probability Distributions are mathematical functions that describe the likelihood of different possible outcomes in an experiment or survey. They are essential in quantitative methods for understanding and predicting the behavior of random variables, which are variables whose values are subject to variations due to chance.
Types of Probability Distributions
- Discrete Probability Distribution: Describes the probabilities of discrete (individual, separate) outcomes, such as the roll of a die.
- Continuous Probability Distribution: Describes the probabilities of a continuous range of outcomes, such as the height of individuals in a population.
Common Probability Distributions
- Normal Distribution: A continuous distribution that is symmetric and bell-shaped, often used to model natural phenomena.
- Binomial Distribution: A discrete distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success.
Example
Consider a stock that has a 60% chance of increasing in value and a 40% chance of decreasing. If you want to know the probability that the stock will increase in value exactly 3 times out of 5 days, you can use the Binomial Distribution:
P(X = k) = C(n, k) * pk * (1 - p)n-k
Where:
- n = 5 (number of trials)
- k = 3 (number of successes)
- p = 0.6 (probability of success)
P(X = 3) = C(5, 3) * 0.63 * 0.42
P(X = 3) = 10 * 0.216 * 0.16 = 0.3456
This means there is a 34.56% chance that the stock will increase in value exactly 3 times out of 5 days.