Portfolio Math

Hey students! 📊 Ready to dive into the fascinating world of portfolio mathematics? This lesson will teach you how to calculate expected returns, understand risk through variance and covariance, and discover why diversification is often called the "only free lunch" in investing. By the end of this lesson, you'll understand the mathematical foundations that help investors build better portfolios and make smarter investment decisions. Let's unlock the secrets behind successful portfolio management! 🚀

Understanding Expected Returns

Expected return is the profit or loss you anticipate from an investment over a specific period. Think of it like predicting your grade on a test based on your past performance – you're using historical data to estimate future outcomes! 📈

For a single asset, we calculate expected return using this formula:

$$E(R) = \sum_{i=1}^{n} P_i \times R_i$$

Where $E(R)$ is the expected return, $P_i$ is the probability of scenario $i$, and $R_i$ is the return in scenario $i$.

Let's say you're considering investing in Apple stock. Looking at historical data, you might find that in good economic years (40% probability), Apple returns 15%. In average years (50% probability), it returns 8%. In poor years (10% probability), it loses 5%. Your expected return would be:

$$E(R) = (0.40 \times 0.15) + (0.50 \times 0.08) + (0.10 \times -0.05) = 0.06 + 0.04 - 0.005 = 9.5\%$$

For a portfolio containing multiple assets, the expected return is simply the weighted average of individual asset returns:

$$E(R_p) = \sum_{i=1}^{n} w_i \times E(R_i)$$

Where $w_i$ is the weight (percentage) of asset $i$ in your portfolio. If you put 60% in Apple (expected return 9.5%) and 40% in Microsoft (expected return 12%), your portfolio's expected return is:

$$E(R_p) = (0.60 \times 0.095) + (0.40 \times 0.12) = 0.057 + 0.048 = 10.5\%$$

This is beautifully simple – portfolio returns are just weighted averages! 🎯

Measuring Risk Through Variance and Standard Deviation

While expected returns tell us what we might gain, variance tells us how uncertain that gain is. Variance measures how much actual returns tend to deviate from the expected return – it's our mathematical measure of risk! 📊

The variance formula for a single asset is:

$$\sigma^2 = \sum_{i=1}^{n} P_i \times [R_i - E(R)]^2$$

Using our Apple example, if the expected return is 9.5%:

• Good year variance contribution: $0.40 \times (0.15 - 0.095)^2 = 0.40 \times 0.003025 = 0.00121$
• Average year: $0.50 \times (0.08 - 0.095)^2 = 0.50 \times 0.000225 = 0.0001125$
• Poor year: $0.10 \times (-0.05 - 0.095)^2 = 0.10 \times 0.021025 = 0.0021025$

Total variance = $0.00121 + 0.0001125 + 0.0021025 = 0.0034225$

The standard deviation is simply $\sigma = \sqrt{0.0034225} = 5.85\%$

This means Apple's returns typically vary by about 5.85% from the expected 9.5% return. Higher standard deviation means higher risk! ⚡

The Magic of Covariance and Correlation

Here's where portfolio math gets really interesting! Covariance measures how two assets move together. When Apple goes up, does Microsoft usually go up too (positive covariance) or down (negative covariance)? 🤔

Covariance formula:

$$Cov(R_i, R_j) = \sum_{k=1}^{n} P_k \times [R_{i,k} - E(R_i)] \times [R_{j,k} - E(R_j)]$$

Correlation is just standardized covariance, ranging from -1 to +1:

$$\rho_{i,j} = \frac{Cov(R_i, R_j)}{\sigma_i \times \sigma_j}$$

Real-world example: During the 2008 financial crisis, most stocks had correlations near +0.9, meaning they all fell together. However, gold had negative correlation with stocks, making it a valuable diversifier! ✨

Portfolio variance (the really important formula!) is:

$$\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{i,j}$$

Where $\sigma_{i,j}$ is the covariance between assets $i$ and $j$ (when $i=j$, this is just the variance of asset $i$).

For a two-asset portfolio, this simplifies to:

$$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_{1,2}$$

The Power of Diversification

Now for the exciting part – diversification benefits! 🎉 This is why portfolio risk is NOT simply the weighted average of individual asset risks. The magic happens through that covariance term!

Let's see this in action with a real example. Suppose you have:

• Asset A: Expected return = 10%, Standard deviation = 20%
• Asset B: Expected return = 12%, Standard deviation = 25%
• Correlation between A and B = 0.3

If you invest 50% in each asset:

• Portfolio expected return = $0.5 \times 10\% + 0.5 \times 12\% = 11\%$
• Portfolio variance = $(0.5)^2(0.20)^2 + (0.5)^2(0.25)^2 + 2(0.5)(0.5)(0.3)(0.20)(0.25)$
• Portfolio variance = $0.01 + 0.015625 + 0.0075 = 0.033125$
• Portfolio standard deviation = $\sqrt{0.033125} = 18.2\%$

Notice something amazing? The weighted average of individual standard deviations would be $0.5 \times 20\% + 0.5 \times 25\% = 22.5\%$, but your portfolio risk is only 18.2%! You've reduced risk by 4.3 percentage points just through diversification! 🎯

This benefit increases as correlation decreases. If the correlation were -0.5 instead of 0.3, your portfolio standard deviation would drop to just 11.2%! This is why investors love combining assets that don't move together – like stocks and bonds, or domestic and international investments.

The famous saying "don't put all your eggs in one basket" has solid mathematical backing. Studies show that a well-diversified portfolio of 20-30 stocks can eliminate about 90% of company-specific risk! 📚

Real-World Applications and Examples

Professional portfolio managers use these concepts daily. Consider the typical "60/40" portfolio (60% stocks, 40% bonds). Historically, stocks have averaged about 10% annual returns with 16% standard deviation, while bonds have averaged 5% returns with 4% standard deviation. The correlation between stocks and bonds is typically around 0.1 to 0.3.

Using our formulas:

• Expected return = $0.6 \times 10\% + 0.4 \times 5\% = 8\%$
• Portfolio standard deviation ≈ $\sqrt{(0.6)^2(0.16)^2 + (0.4)^2(0.04)^2 + 2(0.6)(0.4)(0.2)(0.16)(0.04)} ≈ 9.8\%$

This gives you 80% of stock market returns with only 61% of stock market risk! That's the power of diversification in action. 💪

Conclusion

Portfolio mathematics reveals the beautiful relationship between risk and return in investing. Expected returns help us estimate future performance, while variance and standard deviation quantify risk. The real magic happens through covariance and correlation, which enable diversification benefits that reduce portfolio risk below the weighted average of individual asset risks. These mathematical principles form the foundation of modern portfolio theory and help investors build more efficient portfolios. Remember students, successful investing isn't about eliminating risk entirely – it's about getting the best possible return for the level of risk you're comfortable taking!

Study Notes

• Expected Portfolio Return: $E(R_p) = \sum_{i=1}^{n} w_i \times E(R_i)$ (weighted average of individual returns)

• Portfolio Variance: $\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{i,j}$ (includes covariance effects)

• Two-Asset Portfolio Variance: $\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_{1,2}$

• Correlation Range: -1 ≤ ρ ≤ +1 (perfect negative to perfect positive correlation)

• Diversification Benefit: Portfolio risk < weighted average of individual risks when correlation < 1

• Covariance Formula: $Cov(R_i, R_j) = \sum_{k=1}^{n} P_k \times [R_{i,k} - E(R_i)] \times [R_{j,k} - E(R_j)]$

• Correlation Formula: $\rho_{i,j} = \frac{Cov(R_i, R_j)}{\sigma_i \times \sigma_j}$

• Risk Reduction: Well-diversified portfolio (20-30 stocks) can eliminate ~90% of company-specific risk

• Optimal Diversification: Combine assets with low or negative correlations for maximum risk reduction

• Standard Deviation: $\sigma = \sqrt{\sigma^2}$ (square root of variance, measures risk in same units as returns)