Theme 40. Finance and the formation of an investment portfolio

The correlation coefficient that indicates the strength of the relationship between two variables can be found using the following formula:

Where:
r_xy – the correlation coefficient of the linear relationship between the variables x and y

x_i – the values of the x-variable in a sample
x̅ – the mean of the values of the x-variable
y_i – the values of the y-variable in a sample
ȳ – the mean of the values of the y-variable

We can use “КОРРЕЛ” formula in Microsoft Excel.

Table with results:

Combinations	Correlation
A and B	0,86
A and Index	0,99
B and Index	0,88

3. Capital asset pricing model (CAPM)

The Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk and expected return for assets, particularly stocks. CAPM is widely used throughout finance for pricing risky securities and generating expected returns for assets given the risk of those assets and cost of capital.

The formula for calculating the expected return of an asset given its risk is as follows:

Investors expect to be compensated for risk and the time value of money. The risk-free rate in the CAPM formula accounts for the time value of money. The other components of the CAPM formula account for the investor taking on additional risk.

The beta of a potential investment is a measure of how much risk the investment will add to a portfolio that looks like the market. If a stock is riskier than the market, it will have a beta greater than one. If a stock has a beta of less than one, the formula assumes it will reduce the risk of a portfolio.
A stock’s beta is then multiplied by the market risk premium, which is the return expected from the market above the risk-free rate. The risk-free rate is then added to the product of the stock’s beta and the market risk premium. The result should give an investor the required return or discount rate they can use to find the value of an asset.
The goal of the CAPM formula is to evaluate whether a stock is fairly valued when its risk and the time value of money are compared to its expected return.
William Sharpe, an economist and Nobel laureate devised CAPM for his 1970 book “Portfolio Theory and Capital Markets.” He notes that an individual investment contains two kinds of risk:
Systematic Risk: In other words, market risk that portfolio diversification can’t reduce. Interest rates, recessions, and wars are examples of systematic risks.
Unsystematic Risk: This “specific risk” relates to a specific company or industry. Strikes, mismanagement or shortage of a necessary component in the manufacturing process all qualify as unsystematic risk.
Unsystematic risk, or specific risk is what modern portfolio theory targets when it suggests diversification of a portfolio. However, diversification doesn’t address systematic risk. CAPM exists for measuring systematic risk.
For example, imagine an investor is contemplating a stock worth $100 per share today that pays a 3% annual dividend. The stock has a beta compared to the market of 1.3, which means it is riskier than a market portfolio. Also, assume that the risk-free rate is 3% and this investor expects the market to rise in value by 8% per year.
The expected return of the stock based on the CAPM formula is 9.5%:

The expected return of the CAPM formula is used to discount the expected dividends and capital appreciation of the stock over the expected holding period. If the discounted value of those future cash flows is equal to $100 then the CAPM formula indicates the stock is fairly valued relative to risk.
The capital asset pricing model is important in the world of financial modeling for a few key reasons. Firstly, by helping investors calculate the expected return on an investment, it helps determine how appropriate a particular investment may be. Investors can use the CAPM for gauging their portfolio’s health and rebalancing, if necessary.
Secondly, it’s a relatively simple formula that’s fairly easy to use. Additionally, the CAPM is an important tool for investors when it comes to accessing both risk and reward. It’s also one of the few formulas that accounts for systematic risk.
That said, CAPM’s critics say it makes unrealistic assumptions. For instance, beta doesn’t acknowledge that price swings in either direction don’t hold equal risk. And, using a particular period for risk assessment ignores that risk and returns don’t distribute evenly over time.
The CAPM also presupposes a constant risk-free rate, which isn’t always the case. A 1% bump in treasury bond interest rates would significantly affect that investment. Meanwhile, using a stock index like the S&P 500 only suggests a theoretical value. That index could perform differently over time.

Alpha is a measure of the performance of an investment as compared to a suitable benchmark index, such as the S&P 500. An alpha of one (the baseline value is zero) shows that the return on the investment during a specified time frame outperformed the overall market average by 1%. A negative alpha number reflects an investment that is underperforming as compared to the market average.

Alpha is one of five standard performance ratios that are commonly used to evaluate individual stocks or an investment portfolio. Alpha is usually a single number (e.g., 1 or 4) representing a percentage that reflects how an investment performed relative to a benchmark index.
A positive alpha of 5 (+5) means that the portfolio’s return exceeded the benchmark index’s performance by 5%. An alpha of negative 5 (-5) indicates that the portfolio underperformed the benchmark index by 5%. An alpha of zero means that the investment earned a return that matched the overall market return, as reflected by the selected benchmark index.
The alpha of a portfolio is the excess return it produces compared to a benchmark index. Investors in mutual funds or ETFs often look for a fund with a high alpha in hopes of getting a superior return on investment (ROI).
The alpha ratio is often used along with the beta coefficient, which is a measure of the volatility of an investment. The two ratios are both used in the Capital Assets Pricing Model (CAPM) to analyse a portfolio of investments and assess its theoretical performance.
Origin of Alpha. The concept of alpha originated from the introduction of weighted index funds, which attempt to replicate the performance of the entire market and assign an equivalent weight to each area of investment. The development as an investing strategy created a new standard of performance.
Basically, investors began to require portfolio managers of actively traded funds to produce returns that exceeded what investors could expect to make by investing in a passive index fund. Alpha was created as a metric to compare active investments with index investing.
The CAPM is used to calculate the amount of return that investors need to realize to compensate for a particular level of risk. It subtracts the risk-free rate from the expected rate and weighs it with a factor – beta – to get the risk premium. It then adds the risk premium to the risk-free rate of return to get the rate of return an investor expects as compensation for the risk. The CAPM formula is expressed as follows: