BF422 — Investments

Chapter 6: Efficient Diversification

Comprehensive Overview — Textbook + Course Content

Bodie, Kane & Marcus • Essentials of Investments

Arrow keys or swipe to navigate • N = notes • T = theme • Space = next

Overview

Chapter Road Map

Diversification & Portfolio Risk — Market vs. unique risk, power of diversification
Scenario Analysis & Risk Measurement — Stock/bond fund example, variance, covariance
Two-Asset Portfolio Math — Return & variance formulas, correlation effects
Investment Opportunity Set & Efficient Frontier — Mean-variance criterion, Markowitz optimization
Risk-Free Asset & Optimal Portfolio — CAL, CML, tangency portfolio, complete portfolio

Key insight: Diversification is the only "free lunch" in finance — reduce risk without sacrificing return.

I

Diversification & Portfolio Risk

Why not all risk is created equal

Diversification Fundamentals Bodie Ch. 6, Slide 1

Diversification and Portfolio Risk

Market Risk

Also called: systematic, non-diversifiable risk
Risk factors common to whole economy
Interest rates, GDP, inflation, geopolitics
Cannot be eliminated by diversification

Unique Risk

Also called: firm-specific, non-systematic, diversifiable risk
Risk specific to an individual company
CEO departure, lawsuits, product failures
Can be eliminated by diversification

Total Risk = Market Risk + Unique Risk

σ2_total = σ2_market + σ2_firm

Diversification Fundamentals Bodie Ch. 6, Slide 2

Risk as Function of Number of Stocks

As you add more stocks, unique risk shrinks → only market risk remains

Diversification Fundamentals Bodie Fig. 7.2

Figure 7.2: Risk versus Diversification

Right axis: risk compared to a one-stock portfolio. Most benefit from first 20–30 stocks.

Diversification Fundamentals Bodie Ch. 6, Slide 3

Quick Poll

How many securities do you need to make a diversified portfolio?

Answer: ~20–30 stocks

Going from 1 → 20 stocks reduces portfolio SD from ~49% to ~21%
Beyond 30 stocks, additional diversification benefit is minimal
You cannot eliminate market risk no matter how many stocks you add

Application Bodie Ch. 6, Slide 7

Diversification for Your 401(k)

Employee Contribution

5% of $35K salary (pre-tax)

$1,750

Employer Match

50% on up to 6% of salary

$875

Total Annual

Don't miss your piece of the pie!

$2,625

Target-date funds = Separation Theorem in practice:

One fund holds the "optimal risky portfolio" (diversified stocks + bonds)
Automatically adjusts allocation as you age (more bonds near retirement)
Warning: Never concentrate your 401(k) in employer stock (remember Enron)

II

Scenario Analysis & Risk Measurement

Building intuition with a stock/bond fund example

Scenario Analysis Bodie Ch. 6, Slide 5

Risky Portfolio: Stock Fund & Bond Fund

Scenario	Probability	Stock Fund Return	Prob × Return	Bond Fund Return	Prob × Return
Severe recession	.05	−37	−1.9	−9	−0.45
Mild recession	.25	−11	−2.8	15	3.8
Normal growth	.40	14	5.6	8	3.2
Boom	.30	30	9.0	−5	−1.5
E(r) = Mean Return:			10.0		5.0

Key observation: When stocks do poorly (recession), bonds often do well — and vice versa. This inverse relationship is the foundation of diversification.

Scenario Analysis Bodie Ch. 6, Slide 5

Calculating Individual Fund Risk

Stock Fund

Scenario	Dev	Squared	Prob × Sq
Severe rec.	−47	2,209	110.45
Mild rec.	−21	441	110.25
Normal	4	16	6.40
Boom	20	400	120.00
Variance			347.10

SD = √347.10 = 18.63%

Bond Fund

Scenario	Dev	Squared	Prob × Sq
Severe rec.	−14	196	9.80
Mild rec.	10	100	25.00
Normal	3	9	3.60
Boom	−10	100	30.00
Variance			68.40

SD = √68.40 = 8.27%

Scenario Analysis Bodie Ch. 6, Slide 6

Portfolio Return & Risk: 40/60 Stock-Bond

Portfolio invested 40% in stock fund and 60% in bond fund:

Scenario	Prob.	r_P	Prob × r_P	Dev from E(r)	Sq. Dev	Prob × Sq.
Severe recession	.05	−20.2	−1.01	−27.2	739.84	36.99
Mild recession	.25	4.6	1.15	−2.4	5.76	1.44
Normal growth	.40	10.4	4.16	3.4	11.56	4.62
Boom	.30	9.0	2.70	2.0	4.00	1.20
E(r_P)			7.00	Variance		44.26

SD_P = √44.26 = 6.65%

Weighted-average SD would be 0.4(18.63) + 0.6(8.27) = 12.41% — but actual portfolio SD is only 6.65%! Diversification cuts risk nearly in half.

Scenario Analysis Bodie Ch. 6, Slide 6

Covariance Calculation from Scenarios

Scenario	Prob.	Stock Dev	Bond Dev	Product of Dev	Prob × Product
Severe recession	.05	−47	−14	658	32.9
Mild recession	.25	−21	10	−210	−52.5
Normal growth	.40	4	3	12	4.8
Boom	.30	20	−10	−200	−60.0
Covariance = SUM					−74.8

ρ = Cov / (σ_S × σ_B) = −74.8 / (18.63 × 8.27) = −0.49

The negative correlation (ρ = −0.49) means stocks and bonds tend to move in opposite directions — this is why the 40/60 portfolio has such low risk.

III

Two-Asset Portfolio Math

The formulas that make diversification work

Two-Asset Portfolios Bodie Ch. 6, Slide 8

Asset Allocation with Two Risky Assets

How to lower the risk of the portfolio?
The low risk of the portfolio is due to the inverse relationship between performances of stock and bond funds
The question is how investors measure the extent to which two assets move together

Answer: We need two statistical measures:

Covariance — direction and magnitude of co-movement
Correlation — standardized covariance, bounded [−1, +1]

Two-Asset Portfolios Bodie Ch. 6, Slide 9

Portfolio Return Formula

Consider a portfolio made up of two risky assets (equity and debt):

r_P = w_D r_D + w_E r_E

w_D = proportion invested in the bond (debt) fund
w_E = proportion invested in the stock (equity) fund
w_D + w_E = 1 (budget constraint)

E(r_P) = w_D E(r_D) + w_E E(r_E)

Key point: Expected return is a linear function of weights — no interaction effects, no "free lunch" on the return side.

Two-Asset Portfolios Bodie Ch. 6, Slide 10

Portfolio Variance: THE Key Formula

σ_P² = w_D² σ_D² + w_E² σ_E² + 2 w_D w_E Cov(r_D, r_E)

Bond variance

w_D² σ_D²

Equity variance

w_E² σ_E²

Cross-term

2 w_D w_E Cov(r_D, r_E)

Diversification lives in the cross-term! When Cov < 0, the cross-term reduces portfolio variance below the weighted sum of individual variances.

Two-Asset Portfolios Bodie Ch. 6, Slide 11

Portfolios of Two Risky Assets: Covariance

Cov(r_D, r_E) = ρ_DE σ_D σ_E

ρ_DE = Correlation coefficient of returns
σ_D = Standard deviation of bond returns
σ_E = Standard deviation of equity returns

Range of values for ρ_DE ∈ [−1, +1]:

If ρ = +1.0 → perfectly positively correlated (no diversification benefit)
If ρ = 0 → linearly uncorrelated (moderate benefit)
If ρ = −1.0 → perfectly negatively correlated (maximum benefit, zero-risk possible)

Cov(r_S, r_B) = Σ p(i) [r_S(i) − E(r_S)] [r_B(i) − E(r_B)]

Two-Asset Portfolios Bodie Ch. 6, Slide 12

Why is Correlation Important?

Work through each case using σ_P² = w_D²σ_D² + w_E²σ_E² + 2w_Dw_Eρσ_Dσ_E

ρ_DE	Portfolio Variance Simplifies To	Diversification
+1	σ_P = w_Dσ_D + w_Eσ_E (weighted average)	None
0	σ_P² = w_D²σ_D² + w_E²σ_E²	Moderate
−1	σ_P = \|w_Dσ_D − w_Eσ_E\| (can = 0!)	Maximum

Which gives the lowest portfolio variance? ρ = −1. You can achieve zero risk with the right weights!

Two-Asset Portfolios Web App: Module 2

Correlation in Practice

+0.85

AAPL & MSFT

Same industry, similar drivers — weak diversification

−0.30

Stocks & Gold

Flight to safety — good diversifier

0.00

Uncorrelated

No linear relationship — moderate benefit

Correlation Matrix Example

	SPY	TLT	GLD
SPY	1.00	−0.40	0.05
TLT	−0.40	1.00	0.30
GLD	0.05	0.30	1.00

Multi-Asset Extension Web App: Module 2

Matrix Form: N-Asset Portfolio

For N assets, portfolio variance generalizes to:

σ_P² = w′ Σ w

Where Σ is the N × N covariance matrix:

Diagonal: individual variances (σ_i²)
Off-diagonal: pairwise covariances (Cov_ij)
Symmetric: Cov_ij = Cov_ji

Curse of Dimensionality

N	Covariances
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.

IV

Investment Opportunity Set & Efficient Frontier

Markowitz mean-variance optimization

Efficient Frontier Bodie Ch. 6, Slide 13

Asset Allocation with Two Risky Assets

Investment Opportunity Set

All available portfolios with different combinations of stocks and bonds
100% stock, 0% bond
90% stock, 10% bond
80% stock, 20% bond
… down to 0% stock, 100% bond

Mean-Variance Criterion

Maximize return and minimize risk!
If E(r_A) ≥ E(r_B) and σ_A ≤ σ_B → A dominates B

"Northwest for the win!"

Higher return (up), Lower risk (left)

Efficient Frontier Bodie Ch. 6, Slide 14

Investment Opportunity Set

Downward-sloping portion is inefficient — dominated by portfolios directly above
Upward-sloping portion is the efficient frontier
Portfolio Z: same SD as stocks but lower return — dominated

Efficient Frontier Bodie Ch. 6, Slide 15

Portfolio SD for Different Correlations

w_D	w_E	E(r_P)	ρ = −1	ρ = 0	ρ = 0.30	ρ = 1
0.00	1.00	13.00	20.00	20.00	20.00	20.00
0.20	0.80	12.00	13.60	16.18	16.88	18.40
0.40	0.60	11.00	7.20	12.92	14.20	16.80
0.60	0.40	10.00	0.80	10.76	12.26	15.20
0.80	0.20	9.00	5.60	10.40	11.45	13.60
1.00	0.00	8.00	12.00	12.00	12.00	12.00

Minimum Variance Portfolio

	ρ = −1	ρ = 0	ρ = 0.30
w_D	0.6250	0.7353	0.8200
E(r_P)	9.8750	9.3235	8.9000
σ_P	0.0000	10.2899	11.4473

Efficient Frontier Bodie Ch. 6, Slide 16

E(r) as Function of SD: Multiple Correlations

Lower ρ → more curvature → more diversification benefit. At ρ = −1, the frontier touches the y-axis (zero risk).

Efficient Frontier Web App: Module 3

Markowitz Mean-Variance Optimization

The 1952 Revolution

Harry Markowitz showed that portfolio selection should be based on mean and variance of portfolio returns, not just individual security analysis.

Nobel Prize in Economics (1990).

The Optimization Problem

Minimize: σ_P² = w′Σw
Subject to:
w′μ = target return
w′1 = 1 (budget)

Efficient Frontier: The set of portfolios that offer the highest expected return for each level of risk. Everything below the frontier is suboptimal.

Efficient Frontier Web App: Modules 2–4

Minimum Variance Portfolio (MVP)

The weight in asset D that minimizes portfolio variance:

w_D* = (σ_E² − Cov(r_D,r_E)) / (σ_D² + σ_E² − 2 Cov(r_D,r_E))

Key Insight

MVP depends only on risk parameters (σ, Cov) — not on expected returns. It is a purely defensive allocation.

Edge Cases

ρ = +1: w_D* = σ_E/(σ_D+σ_E)
ρ = −1: w_D* = σ_E/(σ_D+σ_E) → σ_P = 0

Short-Selling & Constraints

Unconstrained: weights can be negative (short-selling) — frontier extends further. Long-only constraint (0 ≤ w ≤ 1): truncates the frontier at the asset endpoints.

V

Risk-Free Asset & Optimal Portfolio

From curves to lines — the Capital Allocation Line

Risk-Free & CAL Bodie Ch. 6, Slide 17

The Optimal Risky Portfolio with a Risk-Free Asset

Slope of CAL is Sharpe Ratio of Risky Portfolio:

S_P = [E(r_P) − r_f] / σ_P

Optimal Risky Portfolio

Best combination of risky and safe assets to form portfolio
Maximize Sharpe ratio
Consider various CALs constructed from the risk-free rate and different risky portfolios
The steepest CAL = the best risk-return trade-off

Key question: Which point on the efficient frontier, combined with r_f, gives the steepest line?

Risk-Free & CAL Bodie Ch. 6, Slide 18

Two Feasible CALs

Portfolio A

E(r_A) = 8.9%, σ_A = 11.45%

Portfolio B

E(r_B) = 9.5%, σ_B = 11.70%

Steeper CAL → Better!

Risk-Free & CAL Bodie Ch. 6, Slide 19

Optimal CAL(P) — The Tangency Portfolio

Optimal Risky Portfolio P

E(r_P)	11%
σ_P	14.2%
Sharpe Ratio	0.42

S_P = (11% − 5%) / 14.2% = 0.42

CAL: max the slope

Curve: mean-variance optimization

The tangent point is where both goals meet.

Risk-Free & CAL Bodie Ch. 6, Slide 20

Calculating the Tangency Portfolio

Fund managers call it Portfolio O (optimal). For two risky assets:

w_D = { [E(r_D)−r_f]σ_E² − [E(r_E)−r_f]σ_Dσ_Eρ_DE } / { [E(r_D)−r_f]σ_E² + [E(r_E)−r_f]σ_D² − [E(r_D)−r_f+E(r_E)−r_f]σ_Dσ_Eρ_DE }

w_E = 1 − w_D

D means bond fund, E means stock fund, f means risk-free asset (T-bill)
Every fund manager should use portfolio O as the risky asset to set up the complete portfolio (risky + risk-free)

Risk-Free & CAL Web App: Module 6

Risk-Free Asset: Properties

Definition

Variance = 0, return known with certainty

Proxy: U.S. Treasury Bills (T-Bills)

Key Property

Cov(r_f, r_i) = 0 for all risky assets

Zero covariance simplifies portfolio math dramatically

Why Adding r_f Changes Everything

σ_P = w_risky × σ_risky → Linear! (no squared terms, no cross-terms)

Allocation	% Risk-Free	% Risky	E(r_P)	σ_P
Conservative	70%	30%	6.8%	4.3%
Moderate	30%	70%	9.2%	9.9%
Aggressive	0%	100%	11.0%	14.2%
Leveraged	−20%	120%	12.2%	17.0%

Risk-Free & CAL Web App: Module 6

The Capital Market Line (CML)

The optimal CAL that is tangent to the efficient frontier defines the CML:

E(r_C) = r_f + [(E(r_M) − r_f) / σ_M] × σ_C

The Market Portfolio (M)

Theoretically: all risky assets, market-cap weighted
Practical proxies: S&P 500, MSCI World
The tangency portfolio everyone should hold

CML Dominance

CML lies above the efficient frontier everywhere
Except at the tangency point M (where they touch)
No risky-only portfolio can beat the CML

Risk-Free & CAL Web App: Module 6

The Separation Theorem

Two independent decisions:

1. Investment Decision

Find the optimal risky portfolio (tangency point M)

Same for ALL investors regardless of risk preferences

2. Financing Decision

Choose where on the CML to sit (mix of M and r_f)

Depends on individual risk aversion

Conservative investor: 70% r_f + 30% M (lend)
Moderate investor: 30% r_f + 70% M
Aggressive investor: Borrow at r_f, invest >100% in M (leverage)

Risk-Free & CAL Web App: Module 6

Leverage on the CML

Borrowing at r_f to invest more than 100% in the market portfolio:

Example: 150% Market Allocation

w_M = 1.50, w_f = −0.50 (borrowing 50% of wealth)

E(r_C) = −0.50(5%) + 1.50(11%) = 14.0%
σ_C = 1.50 × 14.2% = 21.3%

Higher expected return, but proportionally higher risk. The Sharpe ratio stays the same (0.42) — leverage doesn't improve risk-adjusted returns.

Caveat: In practice, borrowing rates exceed r_f, so the leveraged portion of the CML kinks downward.

Risk-Free & CAL Web App: Module 6

Utility Function & Complete Portfolio

Utility Function

U = E(r) − ½ A σ²

A = risk aversion coefficient
Higher A → steeper indifference curves
Optimal allocation where indifference curve is tangent to CAL

Optimal Allocation

y* = [E(r_P) − r_f] / (A × σ_P²)

A	y*	Investor
2	1.49	Very aggressive
4	0.74	Moderate
6	0.50	Conservative
8	0.37	Very conservative

All investors hold the same risky portfolio (P) — they differ only in how much they allocate to it (y*). This is the Separation Theorem in action.

Historical Context Web App: Module 1

Historical Asset Class Performance (1926–2023)

Asset Class	Avg. Annual Return	Std. Deviation	Risk Premium
Small-Cap Stocks	16.1%	31.2%	12.8%
Large-Cap Stocks (S&P 500)	12.2%	20.0%	8.9%
Long-Term Corp. Bonds	6.5%	8.5%	3.2%
Long-Term Gov't Bonds	5.9%	9.8%	2.6%
Treasury Bills	3.3%	3.1%	—
Inflation	2.9%	4.0%	—

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.

Summary

Chapter 6: Key Takeaways

Diversification works — unique risk eliminated with 20–30 stocks; market risk remains
Covariance & Correlation drive diversification benefit; ρ < +1 reduces portfolio risk below weighted average
Portfolio Variance Formula: σ_P² = w_D²σ_D² + w_E²σ_E² + 2w_Dw_ECov(r_D,r_E)
Efficient Frontier — "northwest" portfolios dominate; lower ρ = more curvature = more benefit
MVP depends only on risk parameters (not returns)
Adding r_f creates linear CAL; steepest CAL = CML tangent to frontier
Tangency Portfolio = optimal risky portfolio; maximizes Sharpe ratio
Separation Theorem — all investors hold the same risky mix; differ only in leverage
Utility function U = E(r) − ½Aσ² determines optimal allocation y*

Sources: Bodie, Kane & Marcus, Essentials of Investments Ch. 6 • BF422 Web App Modules 1–6

Chapter 6: Efficient Diversification

Chapter Road Map

Diversification & Portfolio Risk

Diversification and Portfolio Risk

Market Risk

Unique Risk

Risk as Function of Number of Stocks

Figure 7.2: Risk versus Diversification

Quick Poll

Diversification for Your 401(k)

Employee Contribution

Employer Match

Total Annual

Scenario Analysis & Risk Measurement

Risky Portfolio: Stock Fund & Bond Fund

Calculating Individual Fund Risk

Stock Fund

Bond Fund

Portfolio Return & Risk: 40/60 Stock-Bond

Covariance Calculation from Scenarios

Two-Asset Portfolio Math

Asset Allocation with Two Risky Assets

Portfolio Return Formula

Portfolio Variance: THE Key Formula

Portfolios of Two Risky Assets: Covariance

Why is Correlation Important?

Correlation in Practice

Correlation Matrix Example

Matrix Form: N-Asset Portfolio

Curse of Dimensionality

Investment Opportunity Set & Efficient Frontier

Asset Allocation with Two Risky Assets

Investment Opportunity Set

Mean-Variance Criterion

Investment Opportunity Set

Portfolio SD for Different Correlations

Minimum Variance Portfolio

E(r) as Function of SD: Multiple Correlations

Markowitz Mean-Variance Optimization

The 1952 Revolution

The Optimization Problem

Minimum Variance Portfolio (MVP)

Key Insight

Edge Cases

Short-Selling & Constraints

Risk-Free Asset & Optimal Portfolio

The Optimal Risky Portfolio with a Risk-Free Asset

Optimal Risky Portfolio

Two Feasible CALs

Optimal CAL(P) — The Tangency Portfolio

Optimal Risky Portfolio P

Calculating the Tangency Portfolio

Risk-Free Asset: Properties

Definition

Key Property

Why Adding rf Changes Everything

The Capital Market Line (CML)

The Market Portfolio (M)

CML Dominance

The Separation Theorem

1. Investment Decision

2. Financing Decision

Leverage on the CML

Example: 150% Market Allocation

Utility Function & Complete Portfolio

Utility Function

Optimal Allocation

Historical Asset Class Performance (1926–2023)

Chapter 6: Key Takeaways

Why Adding r_f Changes Everything