Chapter 2: Risk, Return, and Portfolio Theory
Every financial decision involves a trade-off between risk and return. Understanding how to measure risk, calculate returns, and construct portfolios that optimize this trade-off is fundamental to financial management. This chapter covers return calculation, risk measurement, diversification, and basic portfolio theory.
2.1 Return
Return is the gain or loss on an investment over a period, expressed as a percentage of the initial investment.
Types of Return
| Type | Formula | Example |
|---|---|---|
| Holding Period Return | HPR = (Ending Price - Beginning Price + Income) / Beginning Price | Buy NEPSE stock at NPR 500, sell at 600, dividend NPR 20: HPR = (600-500+20)/500 = 24% |
| Expected Return | E(R) = Σ [Pi × Ri] (probability-weighted average) | Boom (0.3): 25%, Normal (0.5): 15%, Recession (0.2): -5% → E(R) = 13% |
| Average Return | Arithmetic Mean = Σ Ri / n | Returns over 3 years: 10%, 15%, 5% → Average = 10% |
2.2 Risk
Risk is the possibility that actual returns will differ from expected returns. In finance, risk is measured by the variability (dispersion) of returns.
Risk Measures
| Measure | Formula | Interpretation |
|---|---|---|
| Variance (σ²) | σ² = Σ Pi × (Ri - E(R))² | Average squared deviation from expected return |
| Standard Deviation (σ) | σ = √Variance | Most common risk measure; same units as return |
| Coefficient of Variation (CV) | CV = σ / E(R) | Risk per unit of return; useful for comparing investments |
Worked Example
| State | Probability | Return | P×R | R-E(R) | [R-E(R)]² | P×[R-E(R)]² |
|---|---|---|---|---|---|---|
| Boom | 0.3 | 25% | 7.5 | 12 | 144 | 43.2 |
| Normal | 0.5 | 15% | 7.5 | 2 | 4 | 2.0 |
| Recession | 0.2 | -5% | -1.0 | -18 | 324 | 64.8 |
E(R) = 7.5 + 7.5 - 1.0 = 14% (corrected: 13%... let me recalculate: 0.3×25 + 0.5×15 + 0.2×(-5) = 7.5+7.5-1 = 14%)
σ² = 43.2 + 2.0 + 64.8 = 110
σ = √110 = 10.49%
CV = 10.49/14 = 0.75
2.3 Types of Risk
| Type | Also Called | Source | Diversifiable? |
|---|---|---|---|
| Systematic Risk | Market risk, non-diversifiable | Economy-wide: inflation, interest rates, political instability | No — cannot be eliminated |
| Unsystematic Risk | Specific risk, diversifiable | Firm-specific: management, labor, competition | Yes — eliminated through diversification |
Total Risk = Systematic Risk + Unsystematic Risk
Beta (β) measures systematic risk. β = 1 means same risk as market. β > 1 means more volatile. β < 1 means less volatile.
2.4 Portfolio Theory
A portfolio is a combination of investments. Portfolio theory (Markowitz) shows that diversification can reduce risk without proportionally reducing return.
Two-Asset Portfolio
Portfolio Return: E(Rp) = w1×E(R1) + w2×E(R2)
Portfolio Risk: σp² = w1²σ1² + w2²σ2² + 2w1w2σ1σ2ρ12
Where ρ12 = correlation coefficient between assets. When ρ < 1, diversification reduces risk.
Diversification Effect
| Correlation (ρ) | Diversification Benefit |
|---|---|
| ρ = +1 | No diversification benefit (risks perfectly correlated) |
| 0 < ρ < +1 | Some risk reduction (most real-world case) |
| ρ = 0 | Significant risk reduction |
| ρ = -1 | Maximum risk reduction (can eliminate all risk) |
2.5 Capital Asset Pricing Model (CAPM) — Detailed
CAPM: E(Ri) = Rf + βi(Rm - Rf)
The CAPM establishes a linear relationship between expected return and systematic risk (beta). It is one of the most important models in finance.
Security Market Line (SML)
| Component | Meaning | Nepal Value |
|---|---|---|
| Rf (Risk-free rate) | Return on government securities (zero default risk) | Nepal govt bond ~8-10% |
| Rm (Market return) | Expected return on market portfolio (NEPSE index) | Historical NEPSE average ~15-18% |
| (Rm - Rf) = Market Risk Premium | Extra return for bearing market risk | ~7-8% for Nepal |
| β (Beta) | Sensitivity of stock to market movements | Varies by company/sector |
CAPM Calculation for NEPSE Stocks
Given: Rf = 8%, Rm = 16%, Market Risk Premium = 8%
| Stock | Beta | Required Return | Interpretation |
|---|---|---|---|
| Nabil Bank | 1.1 | 8 + 1.1(8) = 16.8% | Slightly above market risk |
| Nepal Telecom | 0.6 | 8 + 0.6(8) = 12.8% | Defensive stock (below market risk) |
| Chilime Hydropower | 0.9 | 8 + 0.9(8) = 15.2% | Close to market risk |
| Nepal Life Insurance | 0.7 | 8 + 0.7(8) = 13.6% | Below market risk |
| Startup Company X | 2.0 | 8 + 2.0(8) = 24% | Very high risk, very high required return |
Investment Decision Using CAPM:
If Nabil Bank stock is expected to earn 20% next year and CAPM required return is 16.8%:
Expected return (20%) > Required return (16.8%) → Stock is UNDERVALUED → BUY
If expected return were 14% < 16.8% → Stock is OVERVALUED → SELL
2.6 Beta Calculation and Interpretation
Beta Formula: β = Covariance(Ri, Rm) / Variance(Rm) = ρ(i,m) × σi / σm
| Beta Value | Meaning | Stock Behaviour | Example |
|---|---|---|---|
| β = 0 | No correlation with market | Unaffected by market movements | Government bonds (risk-free) |
| 0 < β < 1 | Less volatile than market | Moves with market but less sharply | Insurance, utilities — defensive stocks |
| β = 1 | Same volatility as market | Moves exactly with market | Diversified index fund |
| β > 1 | More volatile than market | Amplifies market movements | Banking, real estate — cyclical stocks |
| β < 0 | Moves opposite to market | Hedge against market decline | Gold (sometimes), certain derivatives |
2.7 Portfolio Return and Risk — Two-Asset Complete Example
Stock A: E(R) = 18%, σ = 25% | Stock B: E(R) = 12%, σ = 15% | Correlation ρ = 0.3
Portfolio: 60% in A, 40% in B
Portfolio Return:
E(Rp) = 0.6(18) + 0.4(12) = 10.8 + 4.8 = 15.6%
Portfolio Risk:
σp² = (0.6)²(25)² + (0.4)²(15)² + 2(0.6)(0.4)(25)(15)(0.3)
= 0.36(625) + 0.16(225) + 2(0.6)(0.4)(25)(15)(0.3)
= 225 + 36 + 54 = 315
σp = √315 = 17.75%
Diversification Benefit:
Weighted average risk (no diversification) = 0.6(25) + 0.4(15) = 21%
Actual portfolio risk = 17.75%
Risk reduction = 21% - 17.75% = 3.25 percentage points (due to imperfect correlation)
What if ρ = -0.5?
σp² = 225 + 36 + 2(0.6)(0.4)(25)(15)(-0.5) = 225 + 36 - 90 = 171
σp = √171 = 13.08% (much lower — negative correlation provides stronger diversification)
2.8 NEPSE Investment Analysis
| Risk Type | NEPSE Source | How to Manage |
|---|---|---|
| Market/Systematic | NRB policy changes, political instability, Indian market correlation, global crises | Cannot diversify away; adjust portfolio beta; use CAPM for pricing |
| Company/Unsystematic | Bad management decisions, loan defaults, accounting fraud, operational failures | Diversify across 15-20 stocks in different sectors |
| Liquidity Risk | Many NEPSE stocks have low trading volume | Focus on actively traded stocks; avoid illiquid micro-caps |
| Regulatory Risk | NRB directive changes, SEBON rule changes | Monitor regulatory environment; diversify across sectors |
Practice Questions
Short Answer:
1. Define risk and return. How are they related?
2. Differentiate systematic and unsystematic risk.
3. What is beta? How is it interpreted?
4. Explain the concept of diversification.
5. What is the coefficient of variation and when is it useful?
Long Answer:
6. Stock A: Boom(0.25)=30%, Normal(0.50)=18%, Recession(0.25)=-10%. Stock B: Boom=10%, Normal=12%, Recession=8%. Calculate E(R), σ, and CV for both. Which is riskier? (15 marks)
7. Explain Markowitz portfolio theory. How does diversification reduce risk? (15 marks)
8. Portfolio: 60% in Stock X (E(R)=15%, σ=20%) and 40% in Stock Y (E(R)=10%, σ=12%), ρ=0.3. Calculate portfolio return and risk. (15 marks)
9. Compare systematic and unsystematic risk. How does each affect investment decisions in NEPSE? (15 marks)
10. "Don't put all your eggs in one basket." Discuss this principle using portfolio theory. (15 marks)
Exam Tips: ✓ Expected return and standard deviation calculations ALWAYS asked ✓ Show complete probability tables in calculations ✓ Know portfolio return and risk formulas ✓ Systematic vs unsystematic risk is common theory question ✓ CV is useful when comparing investments with different returns