Interactive Comprehensive Overview
Dr. Yulin Li
School of Business · Monmouth University
Bodie, Kane & Marcus · Essentials of Investments
Arrow keys or swipe to navigate · N = notes · T = theme
Market vs. unique risk, power of diversification, 20-30 stock rule
Stock/bond fund example, variance, covariance, correlation
Return & variance formulas, correlation effects, matrix form
Opportunity set, mean-variance criterion, Markowitz optimization
CAL, CML, tangency portfolio, separation theorem, complete portfolio
Core insight: Diversification is the only "free lunch" in finance — reduce risk without sacrificing expected return.
Why not all risk is created equal
Stocks: 0
Portfolio SD: —
Risk Reduction: —
Key insight: ~80% of diversification benefit comes from the first 20 stocks. Beyond 30, you're just asymptotically approaching the market risk floor (~20%).
How many securities do you need to make a diversified portfolio?
Answer: ~20–30 stocks
Analogy: Think of insurance — insuring one house is risky, but insuring 1,000 houses in different cities means individual fires cancel out. But you can't diversify against nationwide floods (market risk).
5% of $35K salary
$1,750
50% on up to 6%
$875
Free money!
$2,625
Target-date funds = the Separation Theorem in practice:
Never concentrate your 401(k) in employer stock. Remember Enron: employees lost both their jobs and their retirement savings simultaneously.
Building intuition with a stock/bond fund example
| Scenario | Prob | Stock | Bond | 40/60 |
|---|---|---|---|---|
| Severe Rec. | .05 | −37 | −9 | −20.2 |
| Mild Rec. | .25 | −11 | 15 | 4.6 |
| Normal | .40 | 14 | 8 | 10.4 |
| Boom | .30 | 30 | −5 | 9.0 |
| E(r) | 10.0 | 5.0 | 7.0 | |
| σ | 18.6 | 8.3 | 6.7 | |
Key observation: When stocks do poorly, bonds often do well — and vice versa. The 40/60 portfolio has σ = 6.7%, far below either fund alone!
Using the same stock fund (E(r)=10%, σ=18.63%) and bond fund (E(r)=5%, σ=8.27%, ρ=−0.49), now invest 60% stocks / 40% bonds:
E(rP) = wS·E(rS) + wB·E(rB)
σ²P = wS²σS² + wB²σB² + 2wSwBρσSσB
E(rP) = 0.60(10) + 0.40(5) = 8.0%
σ²P = (0.6)²(347.1) + (0.4)²(68.4) + 2(0.6)(0.4)(−74.8)
= 124.96 + 10.94 + (−35.90) = 100.0
σP = √100.0 = 10.0%
Note: more stock → higher return (8% vs 7%) but also higher risk (10% vs 6.65%). The trade-off is real!
The formulas that make diversification work
Key: Expected return is linear — no interaction effects. The "free lunch" of diversification shows up only in risk, not return.
Diversification lives in the cross-term! When Cov < 0, the green bar subtracts from portfolio variance.
| wE | E(r) | σP |
|---|
Four portfolios are plotted below. Using the mean-variance criterion, identify which are efficient and which are dominated.
| Portfolio | E(r) | σ | Status |
|---|---|---|---|
| A | 9% | 12% | |
| B | 7% | 14% | |
| C | 11% | 15% | |
| D | 9% | 16% |
A (9%, 12%): Efficient — no other portfolio has ≥9% return with ≤12% risk
B (7%, 14%): Dominated by A — A has higher return AND lower risk
C (11%, 15%): Efficient — highest return, only slightly more risk than A
D (9%, 16%): Dominated by A — same return but much more risk
+0.85
AAPL & MSFT
Same industry, similar drivers — weak diversification
−0.30
Stocks & Gold
Flight to safety — good diversifier
≈ 0
Uncorrelated
No linear relationship — moderate benefit
| SPY | TLT | GLD | |
|---|---|---|---|
| SPY | 1.00 | −0.40 | 0.05 |
| TLT | −0.40 | 1.00 | 0.30 |
| GLD | 0.05 | 0.30 | 1.00 |
Rule of thumb: For diversification, seek assets with ρ ≤ 0.3. Negative correlations are even better — but they're rare among equities.
For N assets, portfolio variance generalizes to:
Where Σ is the N × N covariance matrix:
| N assets | Covariances needed |
|---|---|
| 2 | 1 |
| 10 | 45 |
| 50 | 1,225 |
| 500 | 124,750 |
At scale, covariances dominate: In an equal-weighted portfolio, individual variances shrink to zero as N → ∞, but average covariance persists. This is why market risk cannot be diversified away.
Markowitz mean-variance optimization
Mean-Variance Criterion: A dominates B if E(rA) ≥ E(rB) and σA ≤ σB. Move northwest for the best portfolios!
The downward-sloping portion (below MVP) is inefficient — for every point there, a point directly above offers the same risk with higher return.
| wE | E(r) | ρ=−1 | ρ=0 | ρ=.30 | ρ=1 |
|---|---|---|---|---|---|
| 0% | 5.0 | 8.3 | 8.3 | 8.3 | 8.3 |
| 20% | 6.0 | 4.3 | 7.0 | 7.6 | 10.3 |
| 40% | 7.0 | 0.4 | 7.8 | 8.8 | 12.4 |
| 60% | 8.0 | 4.0 | 10.1 | 11.2 | 14.5 |
| 80% | 9.0 | 8.4 | 13.2 | 14.2 | 16.5 |
| 100% | 10.0 | 18.6 | 18.6 | 18.6 | 18.6 |
Lower ρ → more curvature → more diversification. At ρ = −1 the frontier touches the y-axis (zero risk). At ρ = +1 it's a straight line (no benefit).
Harry Markowitz showed portfolio selection should be based on mean and variance. Nobel Prize 1990.
Key insight: MVP depends only on risk parameters — not expected returns. It's purely defensive.
Given: σD=12%, σE=20%, ρ=0.10, E(rD)=6%, E(rE)=12%, rf=3%
If tangency has E(r)=10.2%, σ=14.8%
1. Cov = 0.10(12)(20) = 24. wD* = (400−24)/(144+400−48) = 376/496 = 0.7581
2. σ² = (.758)²(144)+(.242)²(400)+2(.758)(.242)(24) = 82.73+23.43+8.81 = 114.97 → σ = 10.72%
3. S = (10.2−3)/14.8 = 7.2/14.8 = 0.486
From curves to lines — the Capital Allocation Line
The steepest CAL wins! The tangent point that maximizes the Sharpe ratio is the optimal risky portfolio P.
The weight in bonds that maximizes the Sharpe ratio:
E(rD)=5%, E(rE)=10%, rf=5%, σD=8.27%, σE=18.63%, ρDE=−0.49
→ E(rP) = 0.52(5) + 0.48(10) = 7.39%, σP = 5.62%, S = 0.42
y > 100% means borrowing at rf to invest more than your wealth in the risky portfolio (leverage).
Find optimal risky portfolio P (tangency). Same for ALL investors.
Choose position on CML (mix of P and rf). Depends on risk aversion.
| Investor | y | E(rC) | σC |
|---|---|---|---|
| Conservative | 30% | 5.7% | 1.7% |
| Moderate | 70% | 6.7% | 3.9% |
| Aggressive | 100% | 7.4% | 5.6% |
| Leveraged | 150% | 8.6% | 8.4% |
Leverage caveat: In practice, borrowing rates exceed rf, so the CML kinks downward beyond y=100%. The Sharpe ratio is not preserved when borrowing costs are higher.
| A | y* | E(rC) | σC | Type |
|---|---|---|---|---|
| 2 | 149% | 8.6% | 8.4% | Very aggressive |
| 4 | 75% | 6.8% | 4.2% | Moderate |
| 6 | 50% | 6.2% | 2.8% | Conservative |
| 8 | 37% | 5.9% | 2.1% | Very conserv. |
The risk-return trade-off is real: Higher average returns come with higher volatility. The equity risk premium (~8.9%) is the reward for bearing market risk.
Sources: Bodie, Kane & Marcus, Essentials of Investments Ch. 6 · BF422 Web App Modules 1–6
Adding stocks to a portfolio primarily reduces:
Maximum diversification benefit occurs when ρ equals:
The Sharpe ratio measures:
Given: E(rS)=12%, σS=20%, E(rB)=6%, σB=10%, ρ=−0.20, weights wS=50%, wB=50%
1. E(rP) = 0.5(12) + 0.5(6) = 9.0%
2. Cov = ρ·σS·σB = (−0.20)(20)(10) = −40
3. σ² = (.5)²(400) + (.5)²(100) + 2(.5)(.5)(−40) = 100 + 25 − 20 = 105
4. σP = √105 = 10.25%
Given: E(rD)=6%, E(rE)=12%, rf=3%, σD=10%, σE=20%, ρDE=0.30
Cov = ρ · σD · σE
Tangency weight in bonds
Tangency weight in equity
1. Cov = 0.30 × 10 × 20 = 60
2. Num = (6−3)(400) − (12−3)(60) = 1200 − 540 = 660
Den = (6−3)(400) + (12−3)(100) − (3+9)(60) = 1200 + 900 − 720 = 1380
wD = 660/1380 = 47.8%
3. wE = 1 − 0.478 = 52.2%