Portfolio Theory: Risk, Return & Diversification

TAKEAWAYS:

• Diversification reduces risk by combining uncorrelated assets, lowering portfolio volatility without sacrificing returns.
• Portfolio weights determine the contribution of each asset to expected return and risk (use $w_i=\frac{{\text{Investment in Asset}}_i}{\text{Total Portfolio Value}}$).
• Correlation matters: Negative correlations (e.g., stocks vs. bonds) amplify risk reduction; positive correlations (e.g., two tech stocks) limit it.
• Efficient frontier shows the optimal risk-return tradeoff: portfolios on this curve offer the highest return for a given risk level.
• Capital Market Line (CML) extends the efficient frontier to include risk-free assets (e.g., Treasury bills), showing the dominant portfolio for any risk tolerance.
• Real-world tie: Nepali investors use portfolio theory to balance NEPSE stocks (high risk) with fixed deposits (low risk) to match their risk appetite.

---

---

1. Core Concepts: Risk and Return in Isolation

Before combining assets, understand how individual assets contribute to risk and return.

Key Definitions

• Expected Return ($E(R_p)$): The weighted average of all possible returns, calculated as: $$E(R_p)=\sum _{i=1}^n{w}_{i}E(R_i)$$ where $w_i$ = weight of asset $i$ in the portfolio, $E(R_i)$ = expected return of asset $i$.

• Risk (Variance/Standard Deviation): Measures how returns fluctuate. For a single asset, risk is its standard deviation (${\sigma }_i$). For a portfolio, risk depends on:

  • Individual asset risks (${\sigma }_i$).
  • Covariance between assets ($Co{v}_{i,j}$), which depends on correlation (${\rho }_{i,j}$).

Why Correlation is Critical

Correlation ($\rho$) measures how two assets move together:

• $\rho =+1$: Perfect positive correlation (e.g., NEPSE index vs. a single bank stock).
• $\rho =-1$: Perfect negative correlation (e.g., gold vs. the US dollar).
• $\rho =0$: No correlation (e.g., stocks vs. agricultural commodities).

Visual: Correlation Impact on Portfolio Risk

Key Insight: Negative correlations reduce portfolio risk more effectively than positive or zero correlations.

---

2. Portfolio Risk: The Math Behind Diversification

Portfolio Variance Formula

The risk of a two-asset portfolio is: $${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\sigma }_1{\sigma }_2{\rho }_{1,2}$$ For N assets, it generalizes to: $${\sigma }_p^2=\sum _{i=1}^n{w}_i^2{\sigma }_i^2+2\sum _{i=1}^n\sum _{j=i+1}^n{w}_i{w}_j{\sigma }_i{\sigma }_j{\rho }_{i,j}$$

Worked Example: Diversifying with NEPSE Stocks

Scenario: You invest in two NEPSE stocks:

• Stock A (Nabil Bank): $E(R_A)=12\%$, ${\sigma }_A=15\%$
• Stock B (Global IME): $E(R_B)=8\%$, ${\sigma }_B=20\%$
• Correlation (${\rho }_{A,B}$): $-0.4$ (inverse relationship, e.g., banking vs. manufacturing).

Portfolio Weights: 60% in Nabil Bank, 40% in Global IME.

Calculations:

1. Expected Return: $$E(R_p)=0.6\times 12\%+0.4\times 8\%=10.4\%$$
2. Portfolio Variance: $${\sigma }_p^2=(0.6^2\times 0.15^2)+(0.4^2\times 0.20^2)+2\times 0.6\times 0.4\times 0.15\times 0.20\times (-0.4)$$ $${\sigma }_p^2=0.0081+0.0064-0.0036=0.0089⟹{\sigma }_p=\sqrt{0.0089}=9.43\%$$ Interpretation:

• Individual Risks: Nabil Bank (15%), Global IME (20%).
• Portfolio Risk: 9.43% (lower than both individual risks due to negative correlation!).

---

3. The Efficient Frontier: Optimal Risk-Return Tradeoff

The efficient frontier is the set of portfolios offering the highest expected return for a given level of risk (or the lowest risk for a given return).

How to Plot It

1. Combine assets with different weights (e.g., 0% to 100% in Stock A, balancing with Stock B).
2. Calculate $E(R_p)$ and ${\sigma }_p$ for each combination.
3. Plot ${\sigma }_p$ (x-axis) vs. $E(R_p)$ (y-axis). The upper curve is the efficient frontier.

Visual: Efficient Frontier for Two Assets

Key Observations

• Portfolios below the frontier are inefficient (higher risk for the same return).
• The global minimum-variance portfolio (leftmost point) has the lowest possible risk but may not be desirable if returns are too low.

---

4. Adding a Risk-Free Asset: The Capital Market Line (CML)

The CML extends the efficient frontier by including a risk-free asset (e.g., Treasury bills, fixed deposits in Nepal).

CML Formula

$$E(R_p)=R_f+\left(\frac{E(R_m)-R_f}{{\sigma }_m}\right){\sigma }_p$$ where:

• $R_f$ = risk-free rate (e.g., 6% for Nepali fixed deposits).
• $E(R_m)$ = expected return of the market portfolio (e.g., NEPSE index).
• ${\sigma }_m$ = standard deviation of the market portfolio.

Visual: CML vs. Efficient Frontier

Key Insight: The CML is a straight line tangent to the efficient frontier. Any portfolio on the CML is optimal for an investor’s risk tolerance.

---

5. Real-World Applications in Nepal and Globally

## In the Real World

1. eSewa/Khalti Investments:

   • Users diversify by investing in fixed deposits (risk-free) and NEPSE stocks (higher risk) via platforms like eSewa Invest or Khalti Trade.
   • How it works: Negative correlation between fixed deposits (stable) and stocks (volatile) reduces overall portfolio risk.

2. Pathao Driver Income:

   • Drivers face uncertain daily earnings (high risk). To stabilize income:
     • Diversify rides across different routes (e.g., high-demand areas like Thapathali vs. low-demand areas like Bhaktapur).
     • Correlation insight: Rush-hour traffic (high earnings) in Kathmandu may be negatively correlated with off-peak hours (lower earnings), reducing variability.

3. Nepal Investment Bank (NIBL) Portfolio Management:

   • NIBL offers mutual funds that combine:
     • Bonds (low risk, 6-8% return).
     • Equities (high risk, 12-15% return).
     • Correlation: Bonds and equities often have low positive or negative correlation, reducing portfolio volatility.

4. Google’s Alphabet Stock vs. Treasury Bonds:

   • Google (Alphabet) stock: High risk ($\sigma \approx 20\%$), high return ($E(R)\approx 15\%$).
   • US Treasury Bonds: Low risk ($\sigma \approx 2\%$), low return ($E(R)\approx 3\%$).
   • Portfolio effect: A mix of both (e.g., 70% Alphabet, 30% Bonds) can achieve 12% return with 10% risk, far better than holding either alone.

---

6. Numerical Example: Diversifying a Nepali Retailer’s Investments

Scenario: Mr. Sharma runs a Kathmandu retail shop and wants to invest Rs. 500,000. He considers:

1. NEPSE Stocks (e.g., Nabil Bank): $E(R)=14\%$, $\sigma =20\%$.
2. Fixed Deposits (Nepal Bank): $E(R)=7\%$, $\sigma =0\%$ (risk-free).
3. Gold: $E(R)=5\%$, $\sigma =15\%$, ${\rho }_{\text{Stocks,Gold}}=-0.3$.

Goal: Achieve 10% expected return with minimum risk.

Step 1: Calculate Portfolio Weights

Let:

• $w_1$ = weight in Nabil Bank stock.
• $w_2$ = weight in Fixed Deposits.
• $w_3$ = weight in Gold.

Constraints: $$w_1+w_2+w_3=1$$ $$E(R_p)=0.14w_1+0.07w_2+0.05w_3=0.10$$

Assume: $w_2=0.3$ (30% in risk-free deposits). Then: $$0.14w_1+0.05w_3=0.07\text{(since }0.07\times 0.3=0.021\text{)}$$ $$w_1+w_3=0.7$$

Portfolio Variance: $${\sigma }_p^2=w_1^2(0.20{)}^2+w_3^2(0.15{)}^2+2w_1{w}_3(0.20)(0.15)(-0.3)$$ Optimal Weights: Solve numerically (or use Excel Solver) to find:

• $w_1=0.45$ (45% in Nabil Bank).
• $w_3=0.25$ (25% in Gold).

Result:

• Expected Return: $0.14\times 0.45+0.07\times 0.3+0.05\times 0.25=10\%$.
• Portfolio Risk: ${\sigma }_p\approx 8.5\%$ (vs. 20% if all in stocks).

---

7. Advantages and Limitations of Portfolio Theory

---

8. Exam Tip: How to Score Full Marks

1. Memorize Key Formulas:

   • Expected return: $E(R_p)=\sum w_{i}E(R_i)$.
   • Portfolio variance: ${\sigma }_p^2=\sum w_i^2{\sigma }_i^2+2\sum w_i{w}_j{\sigma }_i{\sigma }_j{\rho }_{i,j}$.
   • CML equation: $E(R_p)=R_f+\left(\frac{E(R_m)-R_f}{{\sigma }_m}\right){\sigma }_p$.

2. Practice Numerical Problems:

   • Always show steps for calculating weights, expected returns, and variances.
   • Assume missing data (e.g., if correlation isn’t given, assume $\rho =0$ unless stated).

3. Diagrams Are Worth Marks:

   • Draw the efficient frontier and CML clearly.
   • Label axes: Risk (${\sigma }_p$) on x-axis, Return ($E(R_p)$) on y-axis.

4. Real-World Applications:

   • Link answers to Nepali examples (e.g., NEPSE stocks + fixed deposits).
   • Mention correlation effects (e.g., "Negative correlation reduces risk").

5. Common Pitfalls:

   • Ignoring weights: Always ensure $\sum w_i=1$.
   • Misapplying correlation: Negative $\rho$ reduces risk; positive $\rho$ increases it.
   • Forgetting the risk-free asset: CML requires $R_f$ for optimal portfolios.

---

Final Note: Portfolio theory is not just math—it’s about balancing risk and reward like a Kathmandu shopkeeper balancing inventory costs and sales. Master the formulas, but always think: "How would a real investor use this?"