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Sampling and Estimation(Level I CFA Exam)
Sampling and Estimation(Level I CFA Exam)| compact notes by Professor James Forjan, PhD, CFA
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Quantitative Methods in FinanceQuantitative methods involve the use of mathematical and statistical
techniques to analyze financial data and make informed decisions. These
methods can be used to calculate various financial metrics, such as
expected returns, risk levels, and portfolio performance.
One key concept in quantitative finance is expected return. This is the
expected amount that an investment will earn, on average, over a given
period of time. It is calculated by taking the sum of the probabilities of all
possible outcomes, multiplied by their respective returns.
For example, let's say you are considering investing in a stock that has the
potential to earn a return of either 10%, 5%, or -2% over the next year.
The probabilities of each outcome occurring are 20%, 60%, and 20%,
respectively. To calculate the expected return of this investment, you
would use the following formula:
Expected Return = (0.2 x 0.10) + (0.6 x 0.05) + (0.2 x -0.02) = 0.02 or 2%
This means that, on average, you can expect this investment to earn a 2%
return over the next year.
Another important concept in quantitative finance is risk. Risk refers to
the possibility that an investment will earn less than its expected return, or
even result in a loss. One common way to measure risk is by calculating
the standard deviation of returns.
Standard deviation is a statistical measure that shows how much variation
there is in a set of data. In finance, it is used to show the variability of
returns for an investment. A high standard deviation indicates that there is
a lot of variability in the returns, and therefore a higher level of risk.
For example, let's say you are considering two investments with the
following expected returns and standard deviations:
Investment A: Expected Return = 7%, Standard Deviation = 5%
Investment B: Expected Return = 5%, Standard Deviation = 2%
Based solely on these numbers, Investment A would seem like the better
choice, as it has a higher expected return. However, we also need to
consider the level of risk associated with each investment. The standard
deviation of returns for Investment A is 5%, which is higher than the
standard deviation for Investment B, which is only 2%. This means that
Investment A is riskier than Investment B.
, To further illustrate this point, let's say we have a quote from a well-
known investor:
"Risk comes from not knowing what you're doing" - Warren Buffet
This quote highlights the importance of understanding the risks
associated with an investment before making a decision. Simply chasing
the highest expected return can lead to disastrous results if the associated
risk is not carefully considered.
Another important concept in quantitative finance is portfolio
optimization. This involves choosing the optimal combination of
investments to maximize returns and minimize risk. There are many
different approaches to portfolio optimization, but one common method
is to use modern portfolio theory (MPT).
MPT is a mathematical framework for constructing portfolios that
balances risk and return. It uses the concepts of expected return and
standard deviation discussed earlier, as well as the correlation between
different assets, to create a portfolio that is diversified and effectively
manages risk.
For example, let's say you have a portfolio with two assets: Stock A and
Stock B. The expected returns and standard deviations for each stock are
as follows:
Stock A: Expected Return = 8%, Standard Deviation = 10%
Stock B: Expected Return = 6%, Standard Deviation = 8%
The correlation between the two stocks is 0.5. This means that their
returns are somewhat positively correlated, meaning they tend to move in
the same direction.
Using MPT, we can calculate the expected return and standard deviation
for the portfolio as a whole, which are as follows:
Portfolio Expected Return = (0.5 x 0.08) + (0.5 x 0.06) = 0.07 or 7%
Portfolio Standard Deviation = sqrt((0.5^2 * 0.1^2) + (0.5^2 *
0.08^2) + (2 * 0.5 * 0.5 * 0.1 * 0.08 * 0.5)) = 0.092 or 9.2%
This means that the expected return for the portfolio is 7%, and the
standard deviation of returns is 9.2%. By combining these two assets in a
portfolio, we have effectively balanced risk and return and created a
diversified portfolio.
, In conclusion, quantitative methods are an essential tool for finance
professionals. They allow for the calculation of key financial metrics, such
as expected returns and risk levels, and can be used to construct well-
diversified portfolios that effectively balance risk and return. Whether you
are an investment professional or simply looking to make informed
decisions about your personal finances, understanding quantitative
methods is essential.
Sampling and Estimation Techniques
Sampling and Estimation Techniques are essential tools for statisticians
and data analysts. In sampling, we take a subset of data from a larger
population to make inferences about the population. In estimation, we
use sample statistics to estimate population parameters.
There are different sampling techniques, and the video explains two of
them: simple random sampling and systematic sampling.
In simple random sampling, every member of the population has an equal
chance of being selected. This technique ensures that the sample is
representative of the population, and reduces bias. To illustrate, imagine
you want to estimate the average income of people in a city. You can list
down all the people in the city (the population) and assign a number to
each one. Then, using a random number generator, you can select a few
numbers corresponding to the sample size.
In systematic sampling, you select every nth member of the population,
where n is determined by the size of the population and the desired
sample size. This technique is useful when the population is too large to
list down, and when the data is arranged in a logical order. For example, if
you want to estimate the average score of students in a school, and there
are 1000 students, you can select every 20th student (1000/50, where 50 is
the desired sample size) starting from a random point.
The video also covers two estimation techniques: point estimation and
interval estimation.
Point estimation is the process of using sample data to estimate a
population parameter. For example, using the sample mean to estimate
the population mean. However, point estimates can be inaccurate, so we
use interval estimation to provide a range of possible values for the
population parameter.
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