Unit 8 answers the most fundamental question in investing: how are risk and return related? The answer shapes every investment decision — from where BBYM families put their savings to how The Swanson Initiative trust fund should be allocated. The central insight: not all risk is rewarded. Only systematic risk (risk that cannot be diversified away) earns a return premium. The CAPM model quantifies exactly how much return any investment should earn given its systematic risk — measured by beta. This unit connects directly to personal portfolio construction and community wealth building.
Part 1 — Core Topics Explained
Every major concept tested on the Unit 8 assessment
📋 Learning Objectives
- Calculate expected return using a probability-weighted distribution
- Calculate standard deviation as a measure of investment risk
- Distinguish systematic (market) risk from unsystematic (company-specific) risk
- Explain how diversification eliminates unsystematic risk — and what it cannot eliminate
- Explain the role of correlation in portfolio risk reduction
- Define beta and interpret beta values for specific stocks
- Use the CAPM formula to calculate required return on any stock
- Interpret the Security Market Line (SML) and identify over/undervalued stocks
- Assess personal risk tolerance and connect it to appropriate asset allocation for BBYM community members
1. Expected Return — Probability-Weighted Outcomes
The expected return is the weighted average of all possible returns, where each return is weighted by its probability of occurring. It is the single-number summary of what an investment is expected to earn.
Expected Return
E(R) = Σ [Pᵢ × Rᵢ]
Pᵢ = probability of scenario i | Rᵢ = return in scenario i | Probabilities must sum to 1.0 (100%)
Worked Example — BBYM Community Investment Fund:
| Economic Scenario | Probability | Return | P × R |
|---|---|---|---|
| Strong Growth | 25% | +30% | 7.5% |
| Normal | 50% | +12% | 6.0% |
| Recession | 25% | −10% | −2.5% |
| E(R) | 100% | 11.0% |
2. Standard Deviation — Measuring Risk as Variability
Standard deviation (σ) measures how much actual returns deviate from the expected return. A higher standard deviation means more variability — more risk. It is the most common quantitative measure of investment risk.
Standard Deviation (Risk)
σ = √[ Σ Pᵢ × (Rᵢ − E(R))² ]
For the example above: σ = √[0.25(30−11)² + 0.50(12−11)² + 0.25(−10−11)²]
= √[0.25(361) + 0.50(1) + 0.25(441)] = √[90.25 + 0.5 + 110.25] = √201 = 14.18%
= √[0.25(361) + 0.50(1) + 0.25(441)] = √[90.25 + 0.5 + 110.25] = √201 = 14.18%
How to Interpret Standard Deviation:
For a normally distributed investment with E(R) = 11% and σ = 14.18%:
About 68% of annual returns will fall between −3.18% and +25.18% (±1σ)
About 95% of annual returns will fall between −17.36% and +39.36% (±2σ)
The S&P 500 historically averages ~10.5% annual return with ~15.6% standard deviation. This means in roughly 1 out of every 3 years, returns are outside the −5.1% to +26.1% range. A 30%+ annual drop (like 2008’s −38%) is roughly a 2σ event — rare but expected to occur occasionally.
For a normally distributed investment with E(R) = 11% and σ = 14.18%:
About 68% of annual returns will fall between −3.18% and +25.18% (±1σ)
About 95% of annual returns will fall between −17.36% and +39.36% (±2σ)
The S&P 500 historically averages ~10.5% annual return with ~15.6% standard deviation. This means in roughly 1 out of every 3 years, returns are outside the −5.1% to +26.1% range. A 30%+ annual drop (like 2008’s −38%) is roughly a 2σ event — rare but expected to occur occasionally.
3. Systematic vs. Unsystematic Risk — The Most Critical Distinction
Not all risk is the same. Finance distinguishes two fundamentally different types, and only one of them earns compensation from the market:
| Risk Type | Also Called | What Causes It | Can Be Diversified Away? | Does Market Compensate? |
|---|---|---|---|---|
| Systematic Risk | Market risk, non-diversifiable risk, beta risk | Economy-wide forces: recessions, inflation shocks, Fed rate changes, wars, pandemics — factors affecting all stocks simultaneously | No — adding more stocks cannot eliminate risk that affects everything at once | Yes — investors demand and receive a risk premium for bearing this risk |
| Unsystematic Risk | Company-specific risk, diversifiable risk, idiosyncratic risk | Firm-specific events: CEO scandal, product recall, factory fire, lawsuit, competitor disruption — affects one company while others are unaffected | Yes — can be eliminated by holding a diversified portfolio of many uncorrelated assets | No — market does not compensate because rational investors can eliminate this risk for free through diversification |
The Critical Implication — Why Diversification Is Non-Negotiable:
If you hold only one stock and it crashes 80% due to a CEO scandal, you bear the full loss. This is unsystematic risk — and the market pays you nothing extra for taking it, because it was avoidable. A diversified investor holding 50 stocks experiences almost no impact from one company's scandal because it is offset by the other 49.
This is why CAPM measures only systematic (beta) risk — the only risk that rational investors are exposed to and therefore the only risk the market prices into expected returns. Holding a concentrated portfolio is taking on uncompensated risk.
If you hold only one stock and it crashes 80% due to a CEO scandal, you bear the full loss. This is unsystematic risk — and the market pays you nothing extra for taking it, because it was avoidable. A diversified investor holding 50 stocks experiences almost no impact from one company's scandal because it is offset by the other 49.
This is why CAPM measures only systematic (beta) risk — the only risk that rational investors are exposed to and therefore the only risk the market prices into expected returns. Holding a concentrated portfolio is taking on uncompensated risk.
4. Diversification and Correlation
Diversification reduces portfolio risk by combining assets whose returns do not move in perfect lockstep. The key variable is correlation (ρ) — ranging from −1.0 (perfect negative) to +1.0 (perfect positive).
| Correlation (ρ) | What It Means | Diversification Benefit | Example |
|---|---|---|---|
| +1.0 (Perfect Positive) | Assets always move together in the same direction by the same magnitude | None — no risk reduction from combining these assets | Two identical stocks or two funds tracking the same index |
| 0 to +0.5 (Low Positive) | Assets tend to move in the same direction but not perfectly — the typical relationship between diversified stocks | Significant — most of the benefits of diversification are achieved | A tech stock and a consumer staples stock; S&P 500 and most large-cap stocks |
| 0 (Zero Correlation) | Assets move independently — no relationship between their returns | Maximum benefit for two assets — variance of portfolio approaches zero with many such assets | Some commodity returns vs. equity returns in certain periods |
| −1.0 (Perfect Negative) | Assets always move in opposite directions — when one rises, the other falls by the same amount | Maximum possible — can theoretically eliminate all portfolio risk | Perfect hedge positions; some options strategies |
Practical Diversification — How Many Stocks?
Research shows that most unsystematic risk is eliminated by holding approximately 20–30 randomly selected stocks across different industries. Adding the 31st stock provides very little additional diversification benefit. This is why low-cost index funds (which hold hundreds or thousands of stocks) are the most efficient way for individual investors to achieve maximum diversification at minimal cost.
For BBYM community members: a single low-cost S&P 500 index fund provides instant diversification across 500 companies, 11 sectors, and the entire US large-cap market — eliminating virtually all unsystematic risk with one purchase.
Research shows that most unsystematic risk is eliminated by holding approximately 20–30 randomly selected stocks across different industries. Adding the 31st stock provides very little additional diversification benefit. This is why low-cost index funds (which hold hundreds or thousands of stocks) are the most efficient way for individual investors to achieve maximum diversification at minimal cost.
For BBYM community members: a single low-cost S&P 500 index fund provides instant diversification across 500 companies, 11 sectors, and the entire US large-cap market — eliminating virtually all unsystematic risk with one purchase.
5. Beta — Measuring Systematic Risk
Beta (β) measures a stock's sensitivity to market-wide movements — specifically, how much the stock tends to move when the overall market moves 1%. It is the standardized measure of systematic risk used in CAPM.
β = 0
Risk-Free
No market sensitivity. Example: US Treasury Bills. Return is independent of market movements.
β < 1
Defensive
Moves less than the market. Example: utilities, consumer staples (β ≈ 0.4–0.8). Less volatile.
β = 1.0
Market Average
Moves exactly with the market. The S&P 500 index has β = 1.0 by definition.
β > 1
Aggressive
Moves more than the market. Example: tech stocks, growth stocks (β ≈ 1.2–2.0). Higher risk and return.
β < 0
Inverse
Moves opposite to the market. Rare — some gold stocks, certain inverse ETFs. Natural hedge.
What Beta Tells You in Practice:
If the market rises 10% and a stock has β = 1.5, the stock is expected to rise approximately 15%.
If the market falls 10%, the same stock is expected to fall approximately 15%.
Beta amplifies both gains and losses. High-beta stocks are not "better" investments — they are higher-risk investments that, per CAPM, must offer higher expected returns to compensate investors for their greater systematic risk exposure.
If the market rises 10% and a stock has β = 1.5, the stock is expected to rise approximately 15%.
If the market falls 10%, the same stock is expected to fall approximately 15%.
Beta amplifies both gains and losses. High-beta stocks are not "better" investments — they are higher-risk investments that, per CAPM, must offer higher expected returns to compensate investors for their greater systematic risk exposure.
Part 2 — CAPM & the Security Market Line
The Capital Asset Pricing Model — the most widely used tool in finance for pricing risk
The CAPM Formula
The Capital Asset Pricing Model (CAPM) states that the required return on any investment equals the risk-free rate plus a risk premium that is proportional to the investment's beta (systematic risk).
Capital Asset Pricing Model (CAPM)
rᵢ = rᵣᵓ + (RPᵖ) × βᵢ
rᵢ = required return on the stock | rᵣᵓ = risk-free rate (T-Bill rate)
RPᵖ = market risk premium = E(rᵖ) − rᵣᵓ (the extra return market pays above risk-free)
βᵢ = the stock's beta (measure of systematic risk)
RPᵖ = market risk premium = E(rᵖ) − rᵣᵓ (the extra return market pays above risk-free)
βᵢ = the stock's beta (measure of systematic risk)
Assessment Q8 — Worked Out:
Risk-free rate = 4% | Market risk premium = 6% | Beta = 1.5
rᵢ = 4% + 6% × 1.5 = 4% + 9% = 13%
The required return is 13%. A stock with beta 1.5 takes on 50% more systematic risk than the market, so investors demand 4% (risk-free) + 9% (1.5 × 6% market premium) = 13% to hold it. If the stock's expected return is below 13%, it is overvalued; above 13%, it is undervalued.
Risk-free rate = 4% | Market risk premium = 6% | Beta = 1.5
rᵢ = 4% + 6% × 1.5 = 4% + 9% = 13%
The required return is 13%. A stock with beta 1.5 takes on 50% more systematic risk than the market, so investors demand 4% (risk-free) + 9% (1.5 × 6% market premium) = 13% to hold it. If the stock's expected return is below 13%, it is overvalued; above 13%, it is undervalued.
| Stock | Beta (β) | Risk-Free = 4%, MRP = 6% | Required Return via CAPM | Interpretation |
|---|---|---|---|---|
| T-Bill (risk-free) | 0.0 | 4% + 6% × 0.0 | 4.0% | Baseline — no systematic risk |
| Utility stock (defensive) | 0.5 | 4% + 6% × 0.5 | 7.0% | Half the market's risk — half the premium |
| S&P 500 index fund | 1.0 | 4% + 6% × 1.0 | 10.0% | Market average risk and return |
| Assessment Q8 stock | 1.5 | 4% + 6% × 1.5 | 13.0% | 50% more risk than market — 50% more premium |
| High-growth tech stock | 2.0 | 4% + 6% × 2.0 | 16.0% | Double market risk — double the premium |
The Security Market Line (SML)
The Security Market Line is the graphical representation of CAPM — a straight line plotting required return (y-axis) against beta (x-axis). Every fairly priced stock should plot exactly on the SML. Stocks above the line are undervalued (offer more return than required); stocks below are overvalued.
Security Market Line — CAPM Visualization
Required Return (%)
rᵣᵓ
β=0
4%
β=0
4%
Utility
β=0.5
7%
β=0.5
7%
Market
β=1.0
10%
β=1.0
10%
Q8 Stock
β=1.5
13%
β=1.5
13%
Tech
β=2.0
16%
β=2.0
16%
Beta (β) →
Stocks plotting above the SML are undervalued (expected return > required). Stocks below are overvalued.
Using the SML to Identify Over/Undervalued Stocks:
Suppose a stock has β = 1.2. CAPM required return = 4% + 6% × 1.2 = 11.2%.
If the stock's expected return (based on analysis of future dividends and growth) = 14%:
→ Expected return ABOVE the SML → Undervalued — buy signal. The stock offers more return than its risk level requires. Investors will buy, price will rise, expected return will fall back to 11.2%.
If expected return = 9%:
→ Expected return BELOW the SML → Overvalued — sell signal. The stock offers less return than its risk demands. Investors will sell, price will fall, expected return will rise back to 11.2%.
Suppose a stock has β = 1.2. CAPM required return = 4% + 6% × 1.2 = 11.2%.
If the stock's expected return (based on analysis of future dividends and growth) = 14%:
→ Expected return ABOVE the SML → Undervalued — buy signal. The stock offers more return than its risk level requires. Investors will buy, price will rise, expected return will fall back to 11.2%.
If expected return = 9%:
→ Expected return BELOW the SML → Overvalued — sell signal. The stock offers less return than its risk demands. Investors will sell, price will fall, expected return will rise back to 11.2%.
Historical Risk-Return by Asset Class
| Asset Class | Avg Annual Return | Std Deviation | Risk Level | Typical Beta |
|---|---|---|---|---|
| US T-Bills (3-month) | ~3.4% | ~3.1% | Very Low | 0.0 |
| US Treasury Bonds | ~5.0% | ~8.0% | Low | 0.0* |
| Corporate Bonds (AAA) | ~5.8% | ~8.5% | Low-Mod | ~0.1 |
| Large-Cap Stocks (S&P 500) | ~10.5% | ~15.6% | Mod-High | 1.0 |
| Small-Cap Stocks | ~12.5% | ~20%+ | High | ~1.2–1.5 |
*Treasury bonds have interest rate risk (duration risk) but essentially zero default/market (beta) risk.
The Risk-Return Tradeoff Is Real — But Only for Systematic Risk:
The data confirms CAPM's prediction: higher systematic risk (higher beta) correlates with higher historical returns. T-Bills have the lowest return and lowest risk. Small-cap stocks have the highest return and highest risk. Investors are rewarded for bearing systematic risk — over the long run.
The caveat: these are long-run averages. In any single year, small-cap stocks might lose 40% while T-Bills earn 5%. Risk tolerance, time horizon, and liquidity needs all determine the appropriate position on this risk-return spectrum for each BBYM family.
The data confirms CAPM's prediction: higher systematic risk (higher beta) correlates with higher historical returns. T-Bills have the lowest return and lowest risk. Small-cap stocks have the highest return and highest risk. Investors are rewarded for bearing systematic risk — over the long run.
The caveat: these are long-run averages. In any single year, small-cap stocks might lose 40% while T-Bills earn 5%. Risk tolerance, time horizon, and liquidity needs all determine the appropriate position on this risk-return spectrum for each BBYM family.
Part 3 — Portfolio Theory & Asset Allocation
Building a diversified portfolio—and connecting theory to BBYM community wealth building
Portfolio Beta — Combining Assets
The beta of a portfolio is the weighted average of the betas of its individual holdings, where weights are the proportions invested in each asset.
Portfolio Beta
βẐ = w₁β₁ + w₂β₂ + … + wₙβₙ
wᵢ = weight (fraction of portfolio) in asset i | βᵢ = beta of asset i | Weights must sum to 1.0
Swanson Initiative Portfolio Example:
This portfolio has β = 0.725 — slightly below market average risk. CAPM required return: 4% + 6% × 0.725 = 8.35%. A moderately conservative allocation appropriate for a community trust fund with long-term but not purely speculative goals.
| Asset | Weight | Beta | w × β |
|---|---|---|---|
| S&P 500 Index Fund | 50% | 1.00 | 0.500 |
| Corporate Bonds | 30% | 0.10 | 0.030 |
| Small-Cap Fund | 15% | 1.30 | 0.195 |
| T-Bills (cash) | 5% | 0.00 | 0.000 |
| Portfolio Beta | 100% | 0.725 |
This portfolio has β = 0.725 — slightly below market average risk. CAPM required return: 4% + 6% × 0.725 = 8.35%. A moderately conservative allocation appropriate for a community trust fund with long-term but not purely speculative goals.
Risk Tolerance and Asset Allocation — BBYM Community Framework
| Risk Profile | Typical Allocation | Expected Return | Worst-Year Scenario | Best For |
|---|---|---|---|---|
| Conservative | 20% stocks / 80% bonds & cash | 4–5% | −5 to −10% | Near-retirement, short horizon (1–3 years), low risk tolerance |
| Moderate | 60% stocks / 40% bonds | 6–7% | −15 to −20% | Mid-career savers, 5–10 year horizon, moderate risk tolerance |
| Aggressive | 90% stocks / 10% bonds | 8–10% | −30 to −40% | Young investors, 15+ year horizon, high risk tolerance and stability |
| Community Trust (Swanson) | 50–60% stocks / 40–50% bonds | 6–8% | −15 to −25% | Perpetual endowment — balance growth with capital preservation for distributions |
BBYM Wealth-Building Principle — Age-Based Allocation:
A widely used rule of thumb: Stock allocation % ≈ 110 minus your age.
Age 20: 110 − 20 = 90% stocks — young investors have time to recover from downturns
Age 40: 110 − 40 = 70% stocks — still growth-oriented but beginning to protect gains
Age 60: 110 − 60 = 50% stocks — balanced approach as retirement approaches
Age 70: 110 − 70 = 40% stocks — capital preservation becomes a priority
This rule is a starting point, not a prescription. Risk tolerance, income stability, and specific financial goals should all modify it. A 25-year-old BBYM entrepreneur with volatile self-employment income might choose 70% stocks rather than 85% for stability.
A widely used rule of thumb: Stock allocation % ≈ 110 minus your age.
Age 20: 110 − 20 = 90% stocks — young investors have time to recover from downturns
Age 40: 110 − 40 = 70% stocks — still growth-oriented but beginning to protect gains
Age 60: 110 − 60 = 50% stocks — balanced approach as retirement approaches
Age 70: 110 − 70 = 40% stocks — capital preservation becomes a priority
This rule is a starting point, not a prescription. Risk tolerance, income stability, and specific financial goals should all modify it. A 25-year-old BBYM entrepreneur with volatile self-employment income might choose 70% stocks rather than 85% for stability.
Why Low-Cost Index Funds Win for Most Investors
The Index Fund Argument — CAPM's Practical Implication:
If CAPM is correct, then in an efficient market, no analyst can consistently identify stocks that plot above the SML (undervalued) before everyone else does. The market quickly prices in any available information.
The practical result: most actively managed mutual funds underperform their benchmark index over 10–15 year periods after fees. An S&P 500 index fund charges 0.03–0.05% annually. The average actively managed fund charges 0.5–1.0%+. That 0.5–1.0% fee difference compounds dramatically over 30–40 years:
$10,000 invested at 10% for 30 years = $174,494
$10,000 at 9.5% (after 0.5% fee) for 30 years = $152,203
Fee cost: $22,291 on a $10,000 initial investment — just from a 0.5% annual fee difference.
For BBYM families building long-term wealth: low-cost index funds in tax-advantaged accounts (Roth IRA, 401k) are the highest-probability path to wealth accumulation.
If CAPM is correct, then in an efficient market, no analyst can consistently identify stocks that plot above the SML (undervalued) before everyone else does. The market quickly prices in any available information.
The practical result: most actively managed mutual funds underperform their benchmark index over 10–15 year periods after fees. An S&P 500 index fund charges 0.03–0.05% annually. The average actively managed fund charges 0.5–1.0%+. That 0.5–1.0% fee difference compounds dramatically over 30–40 years:
$10,000 invested at 10% for 30 years = $174,494
$10,000 at 9.5% (after 0.5% fee) for 30 years = $152,203
Fee cost: $22,291 on a $10,000 initial investment — just from a 0.5% annual fee difference.
For BBYM families building long-term wealth: low-cost index funds in tax-advantaged accounts (Roth IRA, 401k) are the highest-probability path to wealth accumulation.
Part 4 — Key Terms Defined
Master these 16 terms for the Unit 8 assessment
Expected Return E(R)
The probability-weighted average of all possible returns for an investment. Calculated as E(R) = Σ[Pᵢ × Rᵢ]. Represents the single best estimate of what an investment will earn, incorporating all known scenarios and their likelihoods. Not a guaranteed outcome — actual returns will differ.
Standard Deviation (σ)
The square root of the variance — the most common measure of investment risk. Measures the average deviation of actual returns from the expected return. A higher standard deviation means more variability and more risk. σ = √[ΣPᵢ(Rᵢ−E(R))²]. Expressed in the same units as the return (percentage points).
Systematic Risk
Market-wide risk that affects all investments simultaneously and cannot be eliminated through diversification. Caused by economy-wide factors: recessions, inflation shocks, interest rate changes, geopolitical events. Also called market risk or non-diversifiable risk. Measured by beta. The ONLY risk that the market compensates investors for bearing.
Unsystematic Risk
Company-specific or industry-specific risk that affects only individual firms and can be eliminated through diversification. Caused by firm-specific events: CEO departure, product failure, lawsuits, regulatory changes. Also called diversifiable risk or idiosyncratic risk. The market does NOT compensate for this risk because rational investors can eliminate it for free.
Diversification
The strategy of combining multiple investments whose returns are not perfectly correlated, thereby reducing total portfolio risk. Eliminates unsystematic risk but cannot eliminate systematic risk. Most of the benefit is achieved with 20–30 stocks across different industries. The only "free lunch" in finance — reduces risk without necessarily reducing expected return.
Correlation (ρ)
A statistical measure of how two assets' returns move relative to each other, ranging from −1.0 (perfect negative) to +1.0 (perfect positive). The lower the correlation between portfolio assets, the greater the diversification benefit and the more risk is reduced. Combining assets with ρ < 1.0 always reduces portfolio risk below the weighted average of individual risks.
Beta (β)
A measure of a stock's systematic risk relative to the overall market. Beta = 1.0 means the stock moves exactly with the market. Beta > 1.0 means it is more volatile than the market (amplifies both gains and losses). Beta < 1.0 means less volatile than the market. Beta = 0 means no market correlation (like T-Bills). The key input into the CAPM formula.
Capital Asset Pricing Model (CAPM)
A model that defines the relationship between systematic risk (beta) and required return: rᵢ = rᵣᵓ + RPᵖ × βᵢ. States that the required return on any investment equals the risk-free rate plus a risk premium proportional to its beta. The most widely used model in finance for estimating the cost of equity and evaluating investment performance.
Risk-Free Rate (rᵣᵓ)
The return available with zero risk of default and high liquidity — the y-intercept of the Security Market Line. Approximated by the yield on short-term US Treasury Bills. Represents the minimum return any rational investor will accept. The base from which all risk premiums are added in CAPM.
Market Risk Premium (RPᵖ)
The additional return above the risk-free rate that the overall stock market (beta = 1.0) is expected to earn. Equals E(rᵖ) − rᵣᵓ. Historically approximately 5–7% for the US stock market. Represents compensation for bearing the systematic risk of investing in equities rather than Treasury Bills. The slope of the Security Market Line.
Security Market Line (SML)
The graphical representation of CAPM — a straight line with beta on the x-axis and required return on the y-axis. Y-intercept = risk-free rate; slope = market risk premium. All fairly priced securities plot on the SML. Securities above the line are undervalued (expected return > required); below the line are overvalued.
Portfolio Beta
The weighted average of the betas of all assets in a portfolio: βẐ = Σwᵢβᵢ. Because beta is additive with portfolio weights, an investor can precisely control the systematic risk of a portfolio by choosing the right mix of low-beta and high-beta assets. Adding T-Bills (β=0) to a portfolio reduces portfolio beta proportionally.
Efficient Portfolio
A portfolio that offers the highest expected return for a given level of risk, or equivalently, the lowest risk for a given expected return. Efficient portfolios lie on the "efficient frontier" — the set of optimal risk-return combinations. Any portfolio below the efficient frontier is suboptimal — it takes more risk than necessary for its return level.
Risk Tolerance
An investor's psychological and financial ability to endure investment losses without making poor decisions (panic-selling). Determined by time horizon, income stability, financial obligations, emotional temperament, and investment goals. Risk tolerance should drive asset allocation: higher tolerance justifies higher stock allocations with higher expected returns and higher short-term volatility.
Asset Allocation
The strategic decision of how to divide investment capital among broad asset classes (stocks, bonds, cash, real estate). The single most important investment decision for long-term investors — research shows asset allocation determines approximately 90% of portfolio return variation over time. More important than stock selection or market timing.
Index Fund
A low-cost investment fund that passively tracks a market index (such as the S&P 500) rather than actively selecting stocks. Provides instant diversification, eliminates virtually all unsystematic risk, and charges minimal fees (0.03–0.1%). CAPM supports index funds: if markets are efficient, active management cannot consistently add value after fees. The preferred vehicle for most long-term BBYM investors.
Part 5 — Practice Questions
Show all work — these mirror the Unit 8 assessment format exactly
Conceptual Questions
Q1Using CAPM: risk-free rate = 4%, market risk premium = 6%, beta = 1.5. What is the required return? A) 9% B) 10% C) 13% D) 15%. (This is the Unit 8 curriculum assessment question.)▼
Answer: C — 13%
rᵢ = rᵣᵓ + RPᵖ × β = 4% + 6% × 1.5 = 4% + 9% = 13%
The risk premium is 6% × 1.5 = 9% (not simply 6%, which would give 10%). The beta of 1.5 means the stock takes on 50% more systematic risk than the market, so its risk premium is 1.5 times the market premium. Option A (9%) incorrectly adds rᵣᵓ + RPᵖ without beta. Option B (10%) uses beta = 1.0. Option D (15%) uses an incorrect calculation.
rᵢ = rᵣᵓ + RPᵖ × β = 4% + 6% × 1.5 = 4% + 9% = 13%
The risk premium is 6% × 1.5 = 9% (not simply 6%, which would give 10%). The beta of 1.5 means the stock takes on 50% more systematic risk than the market, so its risk premium is 1.5 times the market premium. Option A (9%) incorrectly adds rᵣᵓ + RPᵖ without beta. Option B (10%) uses beta = 1.0. Option D (15%) uses an incorrect calculation.
Q2Explain the difference between systematic and unsystematic risk. Why does the market only compensate investors for systematic risk? Give one real-world example of each type.▼
Systematic risk (market risk) affects all investments simultaneously — it cannot be eliminated by diversification. Caused by economy-wide forces like recessions, Fed rate hikes, or pandemics. Example: The 2020 COVID-19 pandemic — virtually every stock fell simultaneously regardless of company quality, because the entire economy was affected.
Unsystematic risk (company-specific risk) affects only individual firms and can be eliminated by holding a diversified portfolio. Example: Boeing's 737 MAX groundings in 2019 — this hurt Boeing specifically but most other stocks were unaffected. A diversified investor with 50 stocks barely noticed.
Why only systematic risk is compensated: Because unsystematic risk is avoidable. A rational investor can eliminate it for free by diversifying. The market will not pay a premium to investors for bearing a risk they chose to take when they could have diversified it away. It would be like charging extra for insurance against a risk you chose to accept unnecessarily. Only unavoidable risk (systematic) earns a return premium.
Unsystematic risk (company-specific risk) affects only individual firms and can be eliminated by holding a diversified portfolio. Example: Boeing's 737 MAX groundings in 2019 — this hurt Boeing specifically but most other stocks were unaffected. A diversified investor with 50 stocks barely noticed.
Why only systematic risk is compensated: Because unsystematic risk is avoidable. A rational investor can eliminate it for free by diversifying. The market will not pay a premium to investors for bearing a risk they chose to take when they could have diversified it away. It would be like charging extra for insurance against a risk you chose to accept unnecessarily. Only unavoidable risk (systematic) earns a return premium.
Q3A stock has a beta of 1.8. The market drops 15%. How much is this stock expected to drop? A stock has a beta of 0.4. The market rises 20%. How much is this stock expected to rise?▼
Beta 1.8, market drops 15%:
Expected stock change = Beta × Market change = 1.8 × (−15%) = −27%
A high-beta aggressive stock amplifies the market's loss. If the market falls 15%, this stock is expected to fall 27%.
Beta 0.4, market rises 20%:
Expected stock change = 0.4 × 20% = +8%
A low-beta defensive stock participates only partially in market gains. When the market surges 20%, this stock captures only 8% of that upside.
The tradeoff: The low-beta stock protects more during downturns (falls only 6% when market falls 15%) but captures less upside (gains only 8% when market gains 20%). The high-beta stock gains more in bull markets but loses more in bear markets. Neither is inherently better — the right choice depends on time horizon and risk tolerance.
Expected stock change = Beta × Market change = 1.8 × (−15%) = −27%
A high-beta aggressive stock amplifies the market's loss. If the market falls 15%, this stock is expected to fall 27%.
Beta 0.4, market rises 20%:
Expected stock change = 0.4 × 20% = +8%
A low-beta defensive stock participates only partially in market gains. When the market surges 20%, this stock captures only 8% of that upside.
The tradeoff: The low-beta stock protects more during downturns (falls only 6% when market falls 15%) but captures less upside (gains only 8% when market gains 20%). The high-beta stock gains more in bull markets but loses more in bear markets. Neither is inherently better — the right choice depends on time horizon and risk tolerance.
Q4Why does adding more stocks to a portfolio reduce risk, but eventually the risk reduction flattens out and stops? What kind of risk remains no matter how many stocks you add?▼
Adding stocks to a portfolio reduces risk because each new stock's unsystematic risk (company-specific) partially offsets the others — when one company has a bad quarter, the others may be unaffected or even doing well. This averaging effect reduces the portfolio's total variability.
However, the risk reduction flattens out because each additional stock contributes less marginal benefit. The first 10 stocks dramatically reduce risk (each one eliminates a large chunk of concentrated unsystematic risk). By the time you have 25–30 stocks, most unsystematic risk is already gone — the 31st stock adds almost no additional diversification benefit.
What remains no matter how many stocks you hold: Systematic (market) risk. When the 2008 financial crisis hit, virtually every stock in every sector fell simultaneously — a diversified portfolio of 500 stocks still lost 38%. There is no way to diversify away the risk that the entire economy contracts, because you cannot diversify within the economy against the economy itself. This is why the SML has a positive slope — systematic risk is unavoidable and must be compensated.
However, the risk reduction flattens out because each additional stock contributes less marginal benefit. The first 10 stocks dramatically reduce risk (each one eliminates a large chunk of concentrated unsystematic risk). By the time you have 25–30 stocks, most unsystematic risk is already gone — the 31st stock adds almost no additional diversification benefit.
What remains no matter how many stocks you hold: Systematic (market) risk. When the 2008 financial crisis hit, virtually every stock in every sector fell simultaneously — a diversified portfolio of 500 stocks still lost 38%. There is no way to diversify away the risk that the entire economy contracts, because you cannot diversify within the economy against the economy itself. This is why the SML has a positive slope — systematic risk is unavoidable and must be compensated.
Calculation Questions
Q5Calculate the expected return for an investment with three scenarios: Boom (probability 30%, return 40%), Normal (probability 50%, return 12%), Bust (probability 20%, return −15%).▼
E(R) = Σ [Pᵢ × Rᵢ]
= (0.30 × 40%) + (0.50 × 12%) + (0.20 × −15%)
= 12.0% + 6.0% + (−3.0%)
= 15.0%
The expected return is 15.0%. Note that the probabilities sum to 100% (30+50+20=100 ✓). The boom scenario contributes the most to expected return (12%) because it combines a moderate probability (30%) with a high return (40%).
= (0.30 × 40%) + (0.50 × 12%) + (0.20 × −15%)
= 12.0% + 6.0% + (−3.0%)
= 15.0%
The expected return is 15.0%. Note that the probabilities sum to 100% (30+50+20=100 ✓). The boom scenario contributes the most to expected return (12%) because it combines a moderate probability (30%) with a high return (40%).
Q6Using the scenario from Q5 (E(R) = 15%), calculate the standard deviation. Show all steps.▼
σ = √[Σ Pᵢ × (Rᵢ − E(R))²]
Step 1 — Deviations from expected return:
Boom: 40% − 15% = +25% | (25%)² = 625
Normal: 12% − 15% = −3% | (−3%)² = 9
Bust: −15% − 15% = −30% | (−30%)² = 900
Step 2 — Probability-weighted squared deviations (Variance):
= 0.30 × 625 + 0.50 × 9 + 0.20 × 900
= 187.5 + 4.5 + 180.0
= 372.0 (variance in %²)
Step 3 — Standard deviation:
σ = √372.0 = 19.29%
Interpretation: This investment has an expected return of 15% with a standard deviation of 19.29%. In a normal distribution, about 68% of returns will fall between −4.29% and +34.29%.
Step 1 — Deviations from expected return:
Boom: 40% − 15% = +25% | (25%)² = 625
Normal: 12% − 15% = −3% | (−3%)² = 9
Bust: −15% − 15% = −30% | (−30%)² = 900
Step 2 — Probability-weighted squared deviations (Variance):
= 0.30 × 625 + 0.50 × 9 + 0.20 × 900
= 187.5 + 4.5 + 180.0
= 372.0 (variance in %²)
Step 3 — Standard deviation:
σ = √372.0 = 19.29%
Interpretation: This investment has an expected return of 15% with a standard deviation of 19.29%. In a normal distribution, about 68% of returns will fall between −4.29% and +34.29%.
Q7Use CAPM to find the required return for: (a) a stock with β = 0.7, (b) a stock with β = 1.2, (c) a stock with β = 2.0. Assume rᵣᵓ = 3.5%, RPᵖ = 5.5%.▼
rᵢ = rᵣᵓ + RPᵖ × β = 3.5% + 5.5% × β
(a) β = 0.7: r = 3.5% + 5.5% × 0.7 = 3.5% + 3.85% = 7.35%
(b) β = 1.2: r = 3.5% + 5.5% × 1.2 = 3.5% + 6.60% = 10.10%
(c) β = 2.0: r = 3.5% + 5.5% × 2.0 = 3.5% + 11.00% = 14.50%
These results reflect the linear relationship of the SML: each unit of additional beta adds exactly 5.5% (the MRP) to the required return. The β = 2.0 stock must offer nearly double the expected return of the β = 0.7 stock to justify the additional systematic risk.
(a) β = 0.7: r = 3.5% + 5.5% × 0.7 = 3.5% + 3.85% = 7.35%
(b) β = 1.2: r = 3.5% + 5.5% × 1.2 = 3.5% + 6.60% = 10.10%
(c) β = 2.0: r = 3.5% + 5.5% × 2.0 = 3.5% + 11.00% = 14.50%
These results reflect the linear relationship of the SML: each unit of additional beta adds exactly 5.5% (the MRP) to the required return. The β = 2.0 stock must offer nearly double the expected return of the β = 0.7 stock to justify the additional systematic risk.
Q8A stock has β = 1.3. Its CAPM required return is 11.8% (rᵣᵓ = 4%, RPᵖ = 6%). An analyst estimates its expected return at 14%. Is this stock overvalued, undervalued, or fairly priced? What should happen to its price?▼
Verify CAPM required return: 4% + 6% × 1.3 = 4% + 7.8% = 11.8% ✓
Expected return (14%) > Required return (11.8%) → Stock plots above the SML → Undervalued
What should happen: Rational investors recognize that this stock offers more return than its risk level requires. They will buy it, increasing demand and driving up the price. As the price rises, the expected return falls (because the same future cash flows are now divided by a higher price). This process continues until the expected return falls back to the CAPM-required 11.8%, at which point the stock is fairly priced and the buying pressure stops.
Market efficiency note: In an efficient market, this gap closes quickly — sometimes within minutes of information becoming public. This is why consistently identifying undervalued stocks is so difficult: the market processes available information very rapidly.
Expected return (14%) > Required return (11.8%) → Stock plots above the SML → Undervalued
What should happen: Rational investors recognize that this stock offers more return than its risk level requires. They will buy it, increasing demand and driving up the price. As the price rises, the expected return falls (because the same future cash flows are now divided by a higher price). This process continues until the expected return falls back to the CAPM-required 11.8%, at which point the stock is fairly priced and the buying pressure stops.
Market efficiency note: In an efficient market, this gap closes quickly — sometimes within minutes of information becoming public. This is why consistently identifying undervalued stocks is so difficult: the market processes available information very rapidly.
Q9The Swanson Initiative portfolio has: 40% in S&P 500 index (β=1.0), 35% in corporate bonds (β=0.1), 15% in small-cap fund (β=1.4), 10% in T-Bills (β=0). (a) Calculate portfolio beta. (b) Calculate CAPM required return (rᵣᵓ=4%, RPᵖ=6%). (c) Is this appropriate for a community endowment?▼
(a) Portfolio Beta:
βẐ = 0.40(1.0) + 0.35(0.1) + 0.15(1.4) + 0.10(0.0)
= 0.400 + 0.035 + 0.210 + 0.000
= 0.645
(b) CAPM Required Return:
r = 4% + 6% × 0.645 = 4% + 3.87% = 7.87%
(c) Appropriateness assessment:
Yes, this is a reasonable allocation for the Swanson Initiative endowment. The portfolio beta of 0.645 is moderately below market risk — appropriate for a perpetual fund that must:
• Generate consistent returns to fund annual distributions (currently targeting $30,000/year per the perpetuity analysis from Unit 5)
• Preserve capital over very long time horizons
• Avoid catastrophic drawdowns that would compromise the mission
The 7.87% expected return exceeds the 6% return needed to fund $30,000/year from a $500,000 endowment, providing a buffer. The 35% bond allocation reduces volatility significantly. One consideration: if the fund grows, increasing the equity allocation slightly (to 50–55% stocks) would increase expected returns for larger perpetual distributions.
βẐ = 0.40(1.0) + 0.35(0.1) + 0.15(1.4) + 0.10(0.0)
= 0.400 + 0.035 + 0.210 + 0.000
= 0.645
(b) CAPM Required Return:
r = 4% + 6% × 0.645 = 4% + 3.87% = 7.87%
(c) Appropriateness assessment:
Yes, this is a reasonable allocation for the Swanson Initiative endowment. The portfolio beta of 0.645 is moderately below market risk — appropriate for a perpetual fund that must:
• Generate consistent returns to fund annual distributions (currently targeting $30,000/year per the perpetuity analysis from Unit 5)
• Preserve capital over very long time horizons
• Avoid catastrophic drawdowns that would compromise the mission
The 7.87% expected return exceeds the 6% return needed to fund $30,000/year from a $500,000 endowment, providing a buffer. The 35% bond allocation reduces volatility significantly. One consideration: if the fund grows, increasing the equity allocation slightly (to 50–55% stocks) would increase expected returns for larger perpetual distributions.
Q10Two BBYM community members are investing. Keisha, 24, is a teacher with stable income and no dependents. Marcus, 58, is 7 years from retirement with a mortgage. Using CAPM and risk tolerance concepts, recommend an asset allocation for each and explain your reasoning.▼
Keisha (24, stable income, no dependents) — Aggressive Growth Allocation:
Recommended: 85–90% stocks (70% S&P 500 index, 15–20% small-cap or international), 10–15% bonds
Portfolio Beta: ~0.85–0.95
CAPM expected return: ~9–10%
Reasoning: Keisha has 40+ years until traditional retirement — the longest possible time horizon. Time horizon is the most important factor in risk tolerance. She can weather multiple complete market cycles and will benefit enormously from compounding at higher expected returns. If her portfolio drops 30–40% in a recession, she has decades to recover and continues contributing. The marginal cost of being wrong (short-term volatility) is low; the marginal cost of being too conservative (foregone compounding) is enormous at her age.
Marcus (58, 7 years from retirement, mortgage) — Moderate-Conservative Allocation:
Recommended: 50–55% stocks (broad index funds), 35–40% bonds, 10% cash/T-Bills
Portfolio Beta: ~0.50–0.60
CAPM expected return: ~7–8%
Reasoning: Marcus has only 7 years to retirement — a significant market crash could devastate his retirement savings with insufficient time to recover. The mortgage adds financial stress that reduces his ability to tolerate volatility emotionally and financially. He still needs growth (7 years is not zero) but capital preservation is increasingly important. The 50/50 split is a reasonable balance — after retirement, he should shift further toward 40/60 stocks/bonds to fund distributions while preserving capital for a 20–30 year retirement horizon.
Recommended: 85–90% stocks (70% S&P 500 index, 15–20% small-cap or international), 10–15% bonds
Portfolio Beta: ~0.85–0.95
CAPM expected return: ~9–10%
Reasoning: Keisha has 40+ years until traditional retirement — the longest possible time horizon. Time horizon is the most important factor in risk tolerance. She can weather multiple complete market cycles and will benefit enormously from compounding at higher expected returns. If her portfolio drops 30–40% in a recession, she has decades to recover and continues contributing. The marginal cost of being wrong (short-term volatility) is low; the marginal cost of being too conservative (foregone compounding) is enormous at her age.
Marcus (58, 7 years from retirement, mortgage) — Moderate-Conservative Allocation:
Recommended: 50–55% stocks (broad index funds), 35–40% bonds, 10% cash/T-Bills
Portfolio Beta: ~0.50–0.60
CAPM expected return: ~7–8%
Reasoning: Marcus has only 7 years to retirement — a significant market crash could devastate his retirement savings with insufficient time to recover. The mortgage adds financial stress that reduces his ability to tolerate volatility emotionally and financially. He still needs growth (7 years is not zero) but capital preservation is increasingly important. The 50/50 split is a reasonable balance — after retirement, he should shift further toward 40/60 stocks/bonds to fund distributions while preserving capital for a 20–30 year retirement horizon.
Q11A BBYM entrepreneur is tempted to put all her retirement savings into her own business (single investment, high confidence). From a portfolio theory perspective, what is wrong with this strategy even if the business performs well?▼
From a portfolio theory perspective, this strategy has multiple serious problems even if the business succeeds:
1. Concentrated unsystematic risk — uncompensated: Putting all savings in one business maximizes unsystematic risk. The market does not pay a premium for this risk because it is avoidable. The entrepreneur is taking maximum risk for no guaranteed extra return — exactly what CAPM says rational investors should not do.
2. Human capital concentration: The entrepreneur's income is already 100% dependent on the business (employment risk). Investing retirement savings in the same business means both income AND wealth are correlated to the same risk. If the business struggles, she simultaneously loses income AND retirement savings — a double loss when she can least afford it. This is the opposite of diversification.
3. No compensation for idiosyncratic risk: Even in a business that outperforms on average, unsystematic risks (fire, lawsuit, key employee departure, competitor disruption) can destroy value without warning. A diversified investor holds 50 businesses — one disaster is absorbed. A concentrated investor holds one — one disaster is catastrophic.
The BBYM solution: Invest retirement savings in diversified index funds separate from business ownership. Run the business for its operational income and equity upside — but don't conflate business equity with retirement security. Keep them in separate mental and literal accounts.
1. Concentrated unsystematic risk — uncompensated: Putting all savings in one business maximizes unsystematic risk. The market does not pay a premium for this risk because it is avoidable. The entrepreneur is taking maximum risk for no guaranteed extra return — exactly what CAPM says rational investors should not do.
2. Human capital concentration: The entrepreneur's income is already 100% dependent on the business (employment risk). Investing retirement savings in the same business means both income AND wealth are correlated to the same risk. If the business struggles, she simultaneously loses income AND retirement savings — a double loss when she can least afford it. This is the opposite of diversification.
3. No compensation for idiosyncratic risk: Even in a business that outperforms on average, unsystematic risks (fire, lawsuit, key employee departure, competitor disruption) can destroy value without warning. A diversified investor holds 50 businesses — one disaster is absorbed. A concentrated investor holds one — one disaster is catastrophic.
The BBYM solution: Invest retirement savings in diversified index funds separate from business ownership. Run the business for its operational income and equity upside — but don't conflate business equity with retirement security. Keep them in separate mental and literal accounts.
Part 6 — Quick Reference Summary
Read this the night before the assessment
Unit 8 in 5 Essential Sentences
Sentence 1
Expected return is the probability-weighted average of outcomes: E(R) = Σ[Pᵢ×Rᵢ]; standard deviation σ = √[ΣPᵢ(Rᵢ−E(R))²] measures risk as the average deviation from that expected return.
Sentence 2
Systematic risk (market-wide, non-diversifiable) is the ONLY risk the market compensates for; unsystematic risk (company-specific, diversifiable) earns no premium because rational investors can eliminate it for free through diversification.
Sentence 3
Beta measures systematic risk: β < 1 = defensive (less volatile than market), β = 1 = market average, β > 1 = aggressive (more volatile); portfolio beta = weighted average of individual betas.
Sentence 4
CAPM: rᵢ = rᵣᵓ + RPᵖ × β — the Assessment Q8 answer is 4% + 6% × 1.5 = 13%; the Security Market Line plots this relationship, and stocks above the SML are undervalued, below are overvalued.
Sentence 5
For BBYM community members, the practical implication is to hold low-cost diversified index funds in tax-advantaged accounts, with asset allocation (% stocks vs. bonds) driven by time horizon and risk tolerance — not stock-picking.
Must-Know Facts for the Assessment
| Concept / Formula | Answer |
|---|---|
| Expected return formula | E(R) = Σ [Pᵢ × Rᵢ] (probability-weighted sum) |
| Standard deviation formula | σ = √[Σ Pᵢ × (Rᵢ − E(R))²] |
| CAPM formula | rᵢ = rᵣᵓ + RPᵖ × βᵢ |
| Assessment Q8 answer | 4% + 6% × 1.5 = 13% |
| Systematic risk | Market-wide, non-diversifiable, compensated by market; measured by beta |
| Unsystematic risk | Company-specific, diversifiable, NOT compensated; eliminated by holding 20–30+ stocks |
| Beta = 0 | Risk-free (T-Bills) — no market sensitivity |
| Beta = 1.0 | Market average (S&P 500 index) |
| Beta > 1 | More volatile than market — amplifies gains and losses |
| Beta < 1 | Less volatile than market — defensive, cushions downturns |
| Portfolio beta formula | βẐ = Σ wᵢ × βᵢ (weighted average of individual betas) |
| SML: above the line | Undervalued — expected return > required; buy signal |
| SML: below the line | Overvalued — expected return < required; sell signal |
| Correlation for max diversification | Lower (ideally negative) correlation between assets maximizes risk reduction |
| Market risk premium historical | ~5–7% for US equities historically; slope of the SML |