Risk and Return: Measures, Trade-offs, and Portfolio Theory

Key Concepts and Definitions

Risk and Return: The Core Trade-off

Risk and return are inversely related: higher potential returns come with higher risk. Investors must weigh these trade-offs based on their risk tolerance.

Measures of Risk

1. Standard Deviation (σ): Measures the volatility of returns around the mean. Higher σ = higher risk.
2. Variance: The square of standard deviation, representing dispersion of returns.
3. Beta (β): Measures systematic risk (market risk) relative to the market benchmark (e.g., NEPSE index). β > 1 = more volatile than the market; β < 1 = less volatile.
4. Coefficient of Variation (CV): Normalizes risk by return: $CV=\frac{\sigma }{\text{Expected Return}}$. Useful for comparing investments with different returns.

Measures of Return

1. Arithmetic Mean Return: Simple average of returns over time.
2. Geometric Mean Return: Accounts for compounding; more accurate for long-term analysis.
3. Sharpe Ratio: Measures excess return per unit of risk: $\text{Sharpe Ratio}=\frac{R_p-R_f}{{\sigma }_p}$, where $R_p$ = portfolio return, $R_f$ = risk-free rate, ${\sigma }_p$ = portfolio standard deviation.

In the Real World

1. Ncell’s Investment in 5G: Ncell’s decision to invest in 5G technology involved high risk (technological uncertainty, capital expenditure) but promised higher returns (market share growth, premium pricing). The company likely evaluated risk using beta (comparing its stock volatility to the NEPSE index) and Sharpe ratio to justify the investment to shareholders.

2. Pathao’s Ride-Hailing Pricing: Pathao dynamically adjusts fares during peak hours (e.g., Kathmandu traffic jams). Drivers face higher risk (income variability) but also higher potential returns. Pathao’s algorithm uses standard deviation to model demand volatility and optimize pricing.

3. Nepal Investment Bank’s Loan Portfolio: The bank diversifies loans across sectors (retail, agriculture, SMEs) to reduce unsystematic risk. If one sector (e.g., tourism) underperforms, others (e.g., agriculture) may offset losses. The bank’s portfolio beta is likely closer to 1 (market average) due to diversification.

Worked Example: Risk and Return for a Nepali Retailer

Scenario: Kathmandu Mart, a mid-sized retail shop in Thapathali, invests in two projects:

1. Project A: Expanding to a new branch (high risk, high return).
2. Project B: Upgrading inventory management software (moderate risk, moderate return).

Assume:

• Risk-free rate ($R_f$) = 6% (Nepal Treasury Bill rate).
• Market return ($R_m$) = 12% (NEPSE average).
• Project A: Expected return = 20%, standard deviation (σ) = 15%.
• Project B: Expected return = 14%, σ = 8%.

Step 1: Calculate Sharpe Ratios

Interpretation: Project B offers a better risk-adjusted return (higher Sharpe ratio) despite lower absolute return.

Step 2: Diversification Impact

If Kathmandu Mart invests equally in both projects:

• Portfolio Expected Return = $0.5\times 20\%+0.5\times 14\%=17\%$.
• Portfolio Standard Deviation: Assume correlation (ρ) between A and B is 0.3. $${\sigma }_p=\sqrt{(0.5^2\times 15{\%}^2)+(0.5^2\times 8{\%}^2)+2\times 0.5\times 0.5\times 0.3\times 15\%\times 8\%}=9.1\%.$$
• New Sharpe Ratio = $\frac{17\%-6\%}{9.1\%}=1.21$ (better than either project alone).

Visualizing Risk and Return

1. Risk-Return Trade-off Curve

pie
    title Risk-Return Trade-off for Kathmandu Mart Projects
    "Project A (High Risk, High Return)" : 20
    "Project B (Moderate Risk, Moderate Return)" : 14
    "Portfolio (Diversified)" : 17

2. Beta and Systematic Risk

3. Diversification and Portfolio Risk

flowchart TD
    A["Single Stock (High Risk)"]
    B["Portfolio with 5 Stocks (Lower Risk)"]
    C["Portfolio with 20 Stocks (Market Risk Only)"]
    A --> B --> C

Key Insight: Diversification reduces unsystematic risk (company-specific) but cannot eliminate systematic risk (market-wide).

Risk and Return in Practice: Nepalese Context

Case Study: NEPSE Index Investments

Assume an investor holds:

• 60% in Nepal Bank Limited (β = 1.1, σ = 20%).
• 40% in Global IME Bank (β = 0.8, σ = 15%).

Step 1: Calculate Portfolio Beta $${\beta }_p=(0.6\times 1.1)+(0.4\times 0.8)=0.66+0.32=0.98\approx 1.$$ Interpretation: The portfolio’s risk is nearly identical to the market (NEPSE index).

Step 2: Expected Return Using CAPM Assume $R_f=6\%$, $R_m=12\%$. $$R_p=R_f+{\beta }_p(R_m-R_f)=6\%+0.98\times (12\%-6\%)=11.88\%.$$

Comparison Table: Risk Metrics

Advantages and Disadvantages of Risk Measures

Exam Tip

1. Memorize Formulas: Focus on standard deviation, beta, Sharpe ratio, and CAPM. Derive them from scratch in exams.
2. Real-World Application: Always relate answers to Nepali examples (e.g., Ncell’s beta, Pathao’s dynamic pricing).
3. Diversification: Explain how combining assets reduces unsystematic risk but not systematic risk.
4. Graphs: Sketch risk-return trade-off curves and beta comparisons. Label axes clearly (e.g., "Expected Return (%)" vs. "Risk (σ)").
5. Numerical Problems: Practice calculating portfolio beta and Sharpe ratios. Use the CAPM to find expected returns.

Final Worked Example: NTC’s Investment Decision

Scenario: Nepal Telecommunications Company (NTC) is deciding between two projects:

1. Fiber Optic Expansion: High risk (σ = 25%), expected return = 30%.
2. 4G Network Upgrade: Moderate risk (σ = 10%), expected return = 15%.

Given:

• Risk-free rate ($R_f$) = 5%.
• Market return ($R_m$) = 10%.

Step 1: Calculate Sharpe Ratios

Step 2: Diversification Benefit If NTC invests 50% in each:

• Portfolio Return = $0.5\times 30\%+0.5\times 15\%=22.5\%$.
• Portfolio σ: Assume correlation (ρ) = 0.4. $${\sigma }_p=\sqrt{(0.5^2\times 25{\%}^2)+(0.5^2\times 10{\%}^2)+2\times 0.5\times 0.5\times 0.4\times 25\%\times 10\%}=13.0\%.$$
• New Sharpe Ratio = $\frac{22.5\%-5\%}{13\%}=1.35$ (better than individual projects).

Conclusion: Diversification improves the risk-return profile, aligning with NTC’s goal of stable growth.