Diversification

Risk refers to the probability of losing money on an investment.
Return, on the other hand, is the gain or loss generated by an investment. They are often measured as a percentage. There’s a positive relationship between risk and return: the higher the risk, the higher the potential return. Investors will only accept higher risk if there’s a chance for greater reward.
This is why low-risk investments like government bonds tend to have lower returns compared to high-risk assets like stocks. So, the risk-return tradeoff is a key principle in investment to earn more, we often need to accept more risk.

Historical stock returns often follow a normal distribution. It is fully defined by two key statistics: expected return and standard deviation.
The expected return is the average of observed past returns. This indicates what investors can reasonably expect to earn.
Standard deviation, a risk indicator, shows us how much returns typically vary from its average.
For instance, Investment A with more spread-out returns has higher risk than Investment B.
Standard deviation is the square root of variance, where variance is the sum of squared differences from the mean, divided by (n − 1).

In a two-stock portfolio, risk depends not only on individual variances and weights, but also on their correlation.
Considering a portfolio with 60% of investment in Southwest and 40% in Amazon. If the correlation is –1, risk drops sharply to just 7.0%.
In reality, at ρ = 0.38, risk still falls to 24.9%. Therefore, diversification could reduce the portfolio risk below its weighted average risk due to the effect of correlations.

The curved line shows how expected return and standard deviation change with different two-stock portfolios.
While higher returns usually mean higher risk, diversification improves the risk-return trade-off.
Expected return is a weighted average, but portfolio risk also reflects how assets move together.
By combining assets that are not perfectly correlated, investors can reduce total risk without lowering returns.
Especially, with perfect negative correlation and optimal weights, risk can be fully eliminated, although it may change the expected return.
In conlusion, the goal of portfolio construction is clear: to earn more per unit of risk taken.

Portfolio risk consists of two components: specific risk and systematic risk.
Specific risk—also known as unsystematic, idiosyncratic, or diversifiable risk—affects only individual companies or specific industries.
Investors can reduce this risk by holding a diversified portfolio of unrelated assets.
In contrast, systematic risk impacts the entire market and all businesses within it.
Events like economic downturns, interest rate changes, or geopolitical shocks affect most assets simultaneously.
Because this type of risk facilitates the market as a whole, it cannot be diversified away, so it is also called undiversifiable risk.

To calculate the variance of a portfolio with N stocks, we use a matrix of variance and covariance terms.
Portfolio variance equals the double summation over i and j from 1 to N, representing all variance and covariance terms in the portfolio.
With equal investment in each stock (x = 1/N), the portfolio variance becomes:
Portfolio variance = (1/N) × average variance + (1 – 1/N) × average covariance.
As N increases, the portfolio variance approaches the average covariance.
This explains how diversification reduces specific risk.
After about 20–30 stocks, the benefits remain stable, with only systematic risk remaining.

The Capital Market Line (CML) represents the best risk-return combinations by mixing the risk-free asset with the market portfolio.
To find it, firstly, we should identify the efficient frontier using only risky assets. Then, by drawing a straight line from the risk-free rate that is tangent to the efficient frontier, we get the CML.
The point of tangency is the market portfolio, which is the optimal risky portfolio with the highest Sharpe ratio.
Now, all efficient portfolios lie on the CML, and investors choose their position along the line by lending or borrowing, depending on their risk preference.