Risk & Return – Comprehensive Exam Notes

Key Ideas & Learning Objectives

• Understand why investors demand compensation for bearing risk and how this shapes the “opportunity cost of capital.”
• Learn to measure return (mean) and risk (variance/standard deviation) for single assets and portfolios.
• Distinguish between diversifiable (unique) risk and non-diversifiable (market) risk.
• Recognize how diversification and correlation work together to lower the total risk of a portfolio.
• Build intuition through historical data, numerical examples, and practice problems that connect theory to real‐world investing.

Time Value of Money, Risk & Opportunity Cost

• Present value of any cash-flow stream reflects both the time value of money and risk: V0 = \frac{CF1}{1+r} + \frac{CF2}{(1+r)^2} + \frac{CF3}{(1+r)^3}+\dots
  • The discount rate r represents the expected return available on comparable, equally risky investments (i.e., the cost of capital).
• Opportunity cost: money tied up in one project forgoes other investments; riskier projects must offer higher expected returns to compensate.

Historical Evidence: How Much Risk? How Much Return?

• One-Year U.S. Returns (1926-2006, Morningstar)
  • LT Corporate Bonds: Best =43\%, Worst =-8\%, Gap =51\%.
  • Large-Cap Stocks: Best =54\%, Worst =-43\%, Gap =97\%.
  • Small-Cap Stocks: Best =143\%, Worst =-58\%, Gap =201\%.
    → Bigger dispersion for stocks (especially small caps) signals higher risk.
• Long-Run Growth of \$1{,}000 (compounded)
  • Small-Cap Stocks: 12.7\% mean → \$1{,}818\;/\;3{,}306\;/\;10{,}926 after 5/10/20 yrs.
  • Large-Cap Stocks: 10.4\% mean → \$1{,}640\;/\;2{,}690\;/\;7{,}234.
  • LT Bonds: 5.9\% mean → \$1{,}332\;/\;1{,}774\;/\;3{,}147.
    → Higher mean return pairs with higher volatility across decades.

Quantifying Return & Risk for a Single Asset

• Expected (mean) return: E(R)=\sum{i=1}^n pi Ri where pi is state probability.
• Variance: \sigma^2 = \sum{i=1}^n pi \big(R_i-E(R)\big)^2.
• Standard deviation: \sigma = \sqrt{\sigma^2} (same units as return).
  • Squaring emphasizes large deviations, capturing risk that investors dislike.

Worked Example – Motorboat vs Ramen (Three economic states)

• Motorboat
  E(RM)=0.2(-5)+0.5(6)+0.3(20)=8\% \sigmaM^2=0.2(!-13)^2+0.5(!-2)^2+0.3(12)^2=79 → \sigma_M\approx8.9\%.
• Ramen
  E(RR)=0.2(30)+0.5(4)+0.3(-15)=3.5\% \sigmaR^2=243.25 → \sigma_R\approx15.6\%.
• Insight: Higher expected return (Motorboat) need not come with higher stand-alone volatility (here Ramen is riskier but offers lower mean). Portfolio context resolves seeming contradiction.

Portfolios: Basic Definitions

• Portfolio = collection of assets with weights w_j summing to 1.
• Portfolio expected return (simple): E(RP)=\sum{j} wj E(Rj).
  • More robust: compute return in each state, then take expectation → automatically captures correlations.
• Portfolio variance uses the same formula but applied to portfolio state returns.

Portfolio Example 1 – Equal Weights in Assets A & B

• Portfolio R in Boom: 0.5(30)+0.5(-5)=12.5\%.
• Portfolio R in Bust: 0.5(-10)+0.5(25)=7.5\%.
• E(R_P)=0.4(12.5)+0.6(7.5)=9.5\%.
• \sigmaP^2=0.4(12.5-9.5)^2+0.6(7.5-9.5)^2=6 → \sigmaP\approx2.45\%.
  → Dramatic volatility reduction relative to the individual securities (30 % & 25 % each).

Portfolio Example 2 – 65 % Motorboat, 35 % Ramen

• E(R_P)=6.425\%.
• \sigmaP^2\approx1.296 → \sigmaP\approx1.14\% (far below either stock’s \sigma).

Diversification: Where Did the Risk Go?

• Unique (idiosyncratic) risk: firm-specific events (strikes, input prices). Can be diversified away.
• Market (systematic) risk: economy-wide shocks (recession, wars, inflation). Remains after diversification.
• Total Risk =\text{Unique}+\text{Market}.
• Empirical rule of thumb: 90 % of achievable diversification benefits realized with 12–20 reasonably uncorrelated stocks (Reilly & Brown).

Illustration – Ten Individual Std. Devs vs Portfolio

The Role of Correlation (\rho)

• -1\le\rho{ab}\le1 measures co-movement of returns Ra, R_b.
  • \rho=1: perfect positive – no risk reduction from mixing.
  • \rho=0: independent – partial risk reduction.
  • \rho=-1: perfect negative – potential to eliminate all variability.

Two-Stock 50/50 Example (Boom vs Recession, p = 0.5)

Principle of Diversification – Three Key Take-aways

1. Spreading wealth across assets/industries lowers total variability without proportionally lowering expected return.
2. For a well-diversified portfolio, unique risk \to0; only market risk matters and must be priced.
3. The risk contribution of any security equals how its returns covary with the existing holdings (captured later by beta, measured in CAPM).

Part I Summary

• Investors are risk-averse → higher risk demands higher expected return.
• Total risk (standard deviation) decomposes into market + unique parts.
• Diversification erases unique risk; market risk remains and is the focus of asset-pricing models.
• Coming attractions:
  • Measuring market risk ⇒ Beta.
  • Pricing market risk ⇒ Capital Asset Pricing Model (CAPM).

Practice Problem – Independent Two-Stock Gamble

• Setup: Two stocks priced \$10 now. Each goes to \$8 (−20 %) or \$14 (+40 %) with p=0.5, independently.

Case 1 – Concentrated (\$20 in one stock)

• Return outcomes: -20\% or +40\%.
• E(R)=0.5(-0.2)+0.5(0.4)=0.1\;\text{(10 %)}.
  \sigma^2=0.09 → \sigma=0.30\;\text{(30 %)}.

Case 2 – Diversified (\$10 in each stock)

• Portfolio return is the average of two independent draws.
  

• Advisers have fiduciary duty to diversify client portfolios; failure exposes clients to uncompensated unique risk.
• Education matters: naïve investors often equate “lots of stocks” with diversification; remind them correlation, not mere count, matters.
• Historical data underscores equity premium but also highlight behavioral pitfalls (myopic loss aversion during down years).

Connecting to Future Topics

• Beta (systematic risk measure) will refine portfolio analysis by quantifying each asset’s contribution to overall market volatility.
• CAPM will translate beta into required return: E(Ri)=Rf + \betai\big(E(Rm)-R_f\big)$$, linking micro risk assessment to macro asset pricing.