Risk & Return: Portfolio Theory & CAPM

TAKEAWAYS:

• Risk cannot be eliminated but can be reduced by diversification (spreading investments across assets).
• Expected return is calculated using probabilities of outcomes, not just historical averages.
• Portfolio theory shows that combining assets reduces unsystematic risk (company-specific) while retaining systematic risk (market-wide).
• CAPM links risk (beta) to required return: E(Ri) = Rf + βi(E(Rm) – Rf), where β measures market sensitivity.
• Beta (β) >1 means higher market risk; β<1 means lower; β=1 matches the market.
• Efficient frontier plots portfolios with the highest return for a given risk level.

1. Introduction to Risk and Return

Risk is the uncertainty about future returns on an investment. Investors demand higher returns for taking on more risk. This relationship is central to financial decision-making.

Types of Risk

Risk can be classified into two broad categories:

mindmap
  root((Risk Classification))
    Systematic Risk
      - Market-wide (e.g., recession, inflation)
      - Cannot be diversified away
    Unsystematic Risk
      - Firm-specific (e.g., management failure, lawsuits)
      - Can be reduced by diversification
    Note: Add a **diversification arrow** from Unsystematic Risk to a portfolio icon (⚪) with label 'Diversification'

2. Expected Return and Variance

Investors care about expected return (average return) and variance (spread of returns). Higher variance means higher risk.

Calculating Expected Return

Given a probability distribution of returns, the expected return (E(R)) is:

$$E(R)=\sum _{i=1}^n{P}_i\times R_i$$

Example: Suppose Stock A has:

• 30% chance of 25% return
• 40% chance of 15% return
• 30% chance of -5% return

Then: $$E(R_A)=(0.30\times 25\%)+(0.40\times 15\%)+(0.30\times -5\%)=12.5\%$$

3. Portfolio Theory: Diversification

Diversification reduces risk by combining assets with low or negative correlation. The key insight: unsystematic risk can be eliminated, but systematic risk remains.

Portfolio Risk vs. Individual Risk

4. The Efficient Frontier

The efficient frontier is the set of portfolios offering the highest expected return for a given level of risk. Investors should only hold portfolios on this frontier.

Why?

• Portfolios below the frontier are dominated (lower return, same risk).
• Portfolios above are unattainable with given assets.

5. Capital Asset Pricing Model (CAPM)

CAPM explains the relationship between risk and required return. It states:

$$E(R_i)=R_f+{\beta }_i(E(R_m)-R_f)$$

Where:

• E(Ri) = Expected return on asset i
• Rf = Risk-free rate (e.g., government bond yield)
• βi = Beta of asset i (sensitivity to market)
• E(Rm) = Expected market return

Interpreting Beta (β)

• β > 1: Asset is more volatile than the market (e.g., tech stocks).
• β = 1: Asset moves with the market (e.g., S&P 500).
• β < 1: Asset is less volatile (e.g., utilities).

Example (Nepal Context): Suppose:

• Risk-free rate (Rf) = 5%
• Market return (E(Rm)) = 12%
• A stock has β = 1.5

Then: $$E(R)=5\%+1.5\times (12\%-5\%)=14.5\%$$

6. Real-World Applications

In the Real World

1. NEPSE (Nepal Stock Exchange):

   • Investors use CAPM to price stocks like Ncell or NTC. If a stock’s beta is 1.3, its required return is higher than the market average.
   • Diversification: A pension fund holding Ncell, NTC, and Daraz reduces risk compared to holding just one.

2. Khalti’s Payment Processing:

   • Khalti’s revenue depends on market conditions (systematic risk) and fraud risks (unsystematic risk). By offering multiple payment methods (e.g., UPI, cards), Khalti diversifies unsystematic risk.

3. Pathao’s Ride-Sharing:

   • Pathao’s earnings fluctuate with weather (systematic) and driver availability (unsystematic). A diversified fleet (motorcycles, cars) reduces the latter.

7. Worked Example: Portfolio Return and Risk

Scenario: You hold two stocks:

• Stock X: 60% weight, E(R) = 15%, σ = 20%
• Stock Y: 40% weight, E(R) = 10%, σ = 15%
• Correlation (ρ) between X and Y = 0.3

Step 1: Calculate Portfolio Expected Return $$E(R_p)=(0.60\times 15\%)+(0.40\times 10\%)=13.0\%$$

Step 2: Calculate Portfolio Variance (σ²) $${\sigma }_p^2=w_X^2{\sigma }_X^2+w_Y^2{\sigma }_Y^2+2w_X{w}_Y\rho {\sigma }_X{\sigma }_Y$$ $$=(0.6^2\times 0.2^2)+(0.4^2\times 0.15^2)+2\times 0.6\times 0.4\times 0.3\times 0.2\times 0.15$$ $$=0.0144+0.0036+0.0027=0.0207$$ $${\sigma }_p=\sqrt{0.0207}=14.4\%$$

Result:

• Expected return: 13.0%
• Portfolio risk (σ): 14.4% (lower than holding X alone).

8. Advantages and Limitations of CAPM

9. Exam Tip

• CAPM questions often ask for:
  1. Calculating expected return given beta and market data.
  2. Comparing portfolios on the efficient frontier.
  3. Explaining why diversification reduces risk.
• Portfolio theory questions may involve:
  • Calculating portfolio variance/covariance.
  • Drawing the efficient frontier.
• Key formulas to memorize:
  • Expected return: $$E(R)=\sum P_i{R}_i$$
  • CAPM: $$E(R_i)=R_f+{\beta }_i(E(R_m)-R_f)$$
  • Portfolio variance (with correlation).

Common Pitfalls:

• Forgetting to square standard deviations in variance calculations.
• Misinterpreting beta (e.g., thinking β=2 means double the return, not double the risk).
• Assuming all risks are diversifiable (systematic risk remains).

Final Note: CAPM and portfolio theory are foundational for Nepal’s financial markets (NEPSE, banks) and global investing (Google, WhatsApp). Master these, and you’ll ace both theoretical and numerical questions!