Business•Chapter 3•4 min read

Risk and Return — Portfolio Theory and the Capital Asset Pricing Model

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OIYO EditorialContributor
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1. The Risk-Return Relationship

Risk: The possibility of unexpected variability in returns.
Risk-return trade-off: The higher the risk, the higher the return investors require.

Expected Return=Risk−Free Rate+Risk Premium\text{Expected Return} = \text{Risk}-\text{Free Rate} + \text{Risk Premium}

2. Measuring Returns

(1) Expected Return

E(R)=∑Pi×RiE(R) = \sum P_i \times R_i
  • P_i: Probability of each scenario
  • R_i: Return in each scenario

Example:

  • Boom (30%): +20% return
  • Normal (50%): +8% return
  • Recession (20%): −5% return
E(R)=0.3×20%+0.5×8%+0.2×(−5%)=6%+4%+(−1%)=9%\begin{aligned} E(R) &= 0.3 \times 20\% + 0.5 \times 8\% + 0.2 \times (−5\%) \\ &= 6\% + 4\% + (−1\%) = 9\% \end{aligned}

(2) Variance and Standard Deviation

  • Variance = Σ P_i × [R_i − E(R)]^2
  • Standard Deviation = √Variance

Using the example above

  • Var = 0.3×(20−9)^2 + 0.5×(8−9)^2 + 0.2×(−5−9)^2 = 0.3×121 + 0.5×1 + 0.2×196 = 36.3 + 0.5 + 39.2 = 76
  • Std Dev = √76 ≈ 8.72%

3. Portfolio Theory

(1) Portfolio Expected Return

E(Rp)=∑wi×E(Ri)wi:Weight of assetiin the portfolio\begin{aligned} E(R_p) &= \sum w_i \times E(R_i) \\ &w_i: \text{Weight of asset} i \text{in the portfolio} \end{aligned}

(2) Portfolio Variance

Two-asset portfolio:

Var(Rp)=wA2×σA2+wB2×σB2+2×wA×wB×σABσAB=Cov(A,B)=ρAB×σA×σBρAB:Correlation coefficient(−1≤ρ≤+1)\begin{aligned} \text{Var}(R_p) &= w_A^2 \times \sigma_A^2 + w_B^2 \times \sigma_B^2 + 2 \times w_A \times w_B \times \sigma_AB \\ \sigma_AB &= \text{Cov}(A, B) = \rho_AB \times \sigma_A \times \sigma_B \\ &\rho_AB: \text{Correlation coefficient} (−1 \leq \rho \leq +1) \end{aligned}

Diversification Effect:

ρ=+1:No diversification benefit(risk unchanged)ρ=−1:Perfect diversification possible(risk fully eliminated)0<ρ<+1:Partial diversification(the realistic case)\begin{aligned} \rho &= +1: \text{No diversification benefit} (\text{risk unchanged}) \\ \rho &= −1: \text{Perfect diversification possible} (\text{risk fully eliminated}) \\ &0 < \rho < +1: \text{Partial diversification} (\text{the realistic case}) \end{aligned}

4. Systematic vs. Unsystematic Risk

Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk (Market Risk)

  • Cannot be eliminated through diversification
  • Driven by economy-wide factors: business cycles, interest rates, inflation
  • Measured by Beta (β)

Unsystematic Risk (Idiosyncratic Risk)

  • Can be eliminated through diversification
  • Affects only a specific company or industry
  • Disappears in a well-diversified portfolio

5. CAPM (Capital Asset Pricing Model)

CAPM: A model that determines the required return on an asset based on its systematic risk (beta).

E(Ri)=Rf+βi×[E(Rm)−Rf]E(R_i) = R_f + \beta_i \times [E(R_m) − R_f]
  • R_f: Risk-free rate (e.g., 10-year US Treasury yield)
  • β_i: Beta of asset i (measure of systematic risk)
  • E(R_m): Expected return of the market portfolio (e.g., S&P 500)
  • [E(R_m) − R_f]: Market risk premium

Interpreting Beta (β):

  • β = 1: Same risk as the market (moves with the market)
  • β > 1: Higher risk than the market (aggressive stock)
  • β < 1: Lower risk than the market (defensive stock)
  • β = 0: Risk-free asset
  • β < 0: Moves opposite to the market (hedge asset)

CAPM Application:

Example:Rf=3%,Market risk premium=6%,β=1.5Required return=3%+1.5×6%=3%+9%=12%\begin{aligned} &\text{Example}: \\ R_f &= 3\%, \text{Market risk premium} = 6\%, \beta = 1.5 \\ \text{Required return} &= 3\% + 1.5 \times 6\% = 3\% + 9\% = 12\% \end{aligned}

6. The Security Market Line (SML)

SML: A straight line that shows the relationship between beta and expected return in the CAPM framework.

SML:

y−axis=Expected returnx−axis=Beta\begin{aligned} y-\text{axis} &= \text{Expected return} \\ x-\text{axis} &= \text{Beta} \end{aligned} Slope=Market risk premium=E(Rm)−Rfy−intercept=Rf\begin{aligned} \text{Slope} &= \text{Market risk premium} = E(R_m) − R_f \\ y-\text{intercept} &= R_f \end{aligned}
  • Asset above SML: Undervalued (actual return > required return)
  • Asset below SML: Overvalued (actual return < required return)

7. Key Concept Cards

CAPM Formula ★★★★★ : E(R) = R_f + β × (R_m − R_f). Risk-free rate + Beta × market excess return. Memory tip: CAPM = Risk-free + β × (Market − Risk-free)

Beta Interpretation ★★★★★ : β = 1 same as market, β > 1 aggressive, β < 1 defensive. Measure of systematic risk. Memory tip: β > 1 aggressive, β < 1 defensive

Diversification and Correlation ★★★★☆ : Diversification benefit is greatest when correlation is closest to −1. No benefit when correlation = +1. Memory tip: Lower correlation → greater diversification benefit

8. Practice Quiz

Q. R_f = 2%, market return = 10%, β = 0.8. What is the CAPM required return?

E(R) = 2% + 0.8 × (10% − 2%) = 2% + 6.4% = 8.4%.

Q. Two assets have a correlation of −0.5. What is the benefit of combining them in a portfolio?

Negative correlation → large diversification benefit. The portfolio’s standard deviation will be less than the weighted average of the individual assets’ standard deviations.

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