Chapter 43: Risk, Return, Diversification, and Portfolio Choice. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Beta sets the hurdle rate

How much return must a project offer, given how strongly it moves with the market?

CAPM prices only the market risk an asset adds to a diversified portfolio. The required return rises along a straight line in beta, and the slope is the market premium.

$Equation, written in LaTeX: E[R_i]=R_f+\beta_i(E[R_M]-R_f).$

$Equation, written in LaTeX: E[R]=2.5\%+1.3(6\%)=10.3\%.$

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R_f = 2.5% is the risk-free return, E[R_M] the expected market return and beta the project's market exposure. The book's expansion has beta 1.3, and 0.8 after the redesign.

Predict first. If a prototype failure risk that is independent of the market is added, does the project's dot move?

Your prediction No, beta and the hurdle rate stay the same Yes, the hurdle rate rises

Project beta

Expected market return

Figure: Beta sets the hurdle rate. Security market line from 2.5% at beta 0 through the market at beta 1 and 8.5%. The project at beta 1.3 requires 10.3%. — Project beta: 1.3, Expected market return: 8.5%
Constructed example: the chapter's hypothetical expansion (risk-free 2.5%, market 8.5%, betas 1.3 and 0.8); betas 0.5 and 1.8 and market returns of 6.5% and 10.5% are added.

Market premium: 6.0%
Required return: 10.3%
Benchmark at beta 0.8: 7.3%
Change versus beta 0.8: +3.0 points

The market premium is 8.5 - 2.5 = 6.0%. CAPM gives 2.5 + 1.3 x 6.0 = 2.5 + 7.80 = 10.30%. Against the beta 0.8 benchmark of 7.30% the hurdle is 3.00 points higher.

Worked steps

Premium = 8.5 - 2.5 = 6.0
E[R] = 2.5 + 1.3 x 6.0 = 2.5 + 7.80 = 10.30%
Benchmark = 2.5 + 0.8 x 6.0 = 7.30%
Change = 10.30 - 7.30 = +3.00 points

Use the idea

Use a project beta, not the company beta, when the project's cyclicality differs, and show how the hurdle moves with beta and the premium.

Where the conclusion applies

Frictionless trading, common beliefs, a risk-free asset and a market proxy. Several priced factors, a stale beta or a poor market proxy would change the benchmark.

Check your understanding: With beta 0.8 and a market return of 10.5%, what is the hurdle rate?

2.5 + 0.8 x (10.5 - 2.5) = 2.5 + 6.4 = 8.9%.

Chapter 43 source: section "Capital asset pricing model".

Demonstration 2 of 4

One look or two: myopic loss aversion

Does the same investment feel worse when it is evaluated every period?

Aggregating lets a gain cancel a loss before the loss weight applies, so fewer outcomes are coded as losses. With lambda = 1 the framing does not matter.

Equation, written in LaTeX: 0.5(15)+0.5[2(-10)]=-2.5.

Equation, written in LaTeX: 0.25(30)+0.50(5)+0.25[2(-20)]=0.

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Each period gains 15 points or loses 10 with probability one half. Losses are weighted by lambda (2 in the book). Two independent periods, with compounding ignored.

Predict first. At lambda = 1.5, is the aggregated value positive?

Your prediction Yes, it is 2.5 No, it is zero No, it is negative

Loss weight lambda

Evaluation

Figure: One look or two: myopic loss aversion. Bars of experienced value for the per period outcomes with loss weight 2.0; the value per look is -2.50. — Loss weight lambda: 2, Evaluation: Separate (two looks)
Constructed example: the chapter's hypothetical position (gain 15, loss 10, lambda 2); lambda values 1, 1.5 and 2.5 are added.

Value per look: -2.50
Two separate looks: -5.00
Two periods aggregated: 0.00
Framing that feels better: aggregated

Separate: 0.5(15) + 0.5[2.0(-10)] = 7.5 - 10.00 = -2.50 per look, -5.00 over two looks. Aggregated: 0.25(30) + 0.50(5) + 0.25[2.0(-20)] = 7.5 + 2.5 - 10.00 = 0.00. The aggregated framing feels better by 5.00 points. The payoffs are identical; only the evaluation boundary changed.

Worked steps

Per look = 7.5 - 0.5 x 2.0 x 10 = -2.50
Two looks = 2 x -2.50 = -5.00
Aggregated = 7.5 + 2.5 - 0.25 x 2.0 x 20 = 0.00

Use the idea

Evaluate a long-horizon portfolio at the horizon that matters, unless you truly must sell after one period.

Where the conclusion applies

A piecewise linear value function, independent periods and no compounding. A real liquidity need after one period makes the separate view the right one.

Check your understanding: At lambda = 2.5, what are the separate and aggregated values?

Separate: 7.5 - 0.5 x 25 = -5 per look, -10 over two. Aggregated: 7.5 + 2.5 - 0.25 x 50 = -2.5.

Chapter 43 source: section "Myopic loss aversion".

Demonstration 3 of 4

Home bias and portfolio volatility

How much volatility does a household add by keeping most of its equity at home?

Mixing two imperfectly correlated markets lowers volatility. With equal expected returns, moving toward the capitalization weight cuts risk at no cost in return.

$Equation, written in LaTeX: \sigma_p^2=w^2(0.22)^2+(1-w)^2(0.16)^2 +2w(1-w)(0.30)(0.22)(0.16).$

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w is the domestic weight. Domestic volatility is 22%, foreign 16%, and the book's correlation is 0.30. The country is 10% of world equity; the household holds 75% of 200,000 dollars at home.

Predict first. Does lowering the correlation to 0 widen or narrow the volatility gap between w = 0.75 and w = 0.10?

Your prediction Narrow it Widen it Leave it unchanged

Domestic weight

Correlation

Portfolio variance: 0.032785
Volatility: 18.11%
Volatility at world weight: 15.21%
Gap versus world weight: 2.90 points
Domestic dollars: $150,000

Variance = 0.75^2(0.0484) + 0.25^2(0.0256) + 2(0.75)(0.25)(0.30)(0.0352) = 0.027225 + 0.0016 + 0.00396 = 0.032785. Volatility is 18.11% against 15.21% at the world weight, a gap of 18.11 - 15.21 = 2.90 points with the same expected return.

Worked steps

w^2 x 0.22^2 = 0.75^2 x 0.0484 = 0.027225
(1 - w)^2 x 0.16^2 = 0.25^2 x 0.0256 = 0.0016
2w(1 - w)(rho)(0.22)(0.16) = 0.00396
Variance = 0.032785, volatility = sqrt(0.032785) = 18.11%
Domestic dollars = 0.75 x 200,000 = $150,000

Use the idea

Measure home bias against world weights, then ask whether costs, taxes or domestic liabilities justify the remaining gap.

Where the conclusion applies

Equal expected net returns, fixed volatilities and correlation. Foreign-investment costs or liabilities tied to domestic shares can justify a larger home weight.

Check your understanding: At w = 1.00 and correlation 0.30, what is volatility?

Variance = 1.00^2 x 0.0484 = 0.0484, so volatility is 22%.

Chapter 43 source: section "Home-bias puzzle".

Demonstration 4 of 4

Pooling shrinks risk, correlation brings it back

How much does pooling more stores reduce the risk of the average loss?

Independent losses average out at the rate of one over the square root of the pool size. A common component does not, and sets a floor under the risk.

$Equation, written in LaTeX: 0.04(0.96)(5{,}000)^2=960{,}000,$

$Equation, written in LaTeX: \sqrt{225(960{,}000)}\approx14{,}696.94.$

$Equation, written in LaTeX: 960{,}000(0.10+\frac{0.90}{225})=99{,}840,$

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Each store has a 4% chance of a 5,000-dollar loss. n stores pool their losses, and rho is the common pairwise loss correlation (0 or 0.10 in the book).

Predict first. At correlation 0.10, does going from 225 to 900 stores cut the SD in half?

Your prediction No, it barely moves Yes, it halves

Stores in pool

Pairwise loss correlation

Figure: Pooling shrinks risk, correlation brings it back. Standard deviation of the average loss per store against pool size on a log scale, correlation 0.00. At 225 stores it is 65.32 dollars. — Stores in pool: 225, Pairwise loss correlation: 0
Constructed example: the chapter's hypothetical bookstores (225 stores, 4%, 5,000 dollars, correlation 0 and 0.10); pools of 25, 100 and 900 and correlation 0.05 are added.

Expected loss per store: $200
Individual SD: $979.80
Average-loss SD: $65.32
Floor as the pool grows: $0.00
Illustrative premium: $240

Expected loss is 0.04(5,000) = 200 and individual variance 0.04(0.96)(5,000)^2 = 960,000, an SD of 979.80. Average-loss variance is 960,000(0.00 + 1.00/225) = 4,266.67, so the SD is 65.32 dollars. Independent losses shrink it by sqrt(225) = 15.00. The premium of 240 is 200 of expected claims plus 25 + 15 for administration and capital.

Worked steps

Expected loss = 0.04 x 5,000 = 200
Individual variance = 0.04 x 0.96 x 5,000^2 = 960,000
Average variance = 960,000 x (0.00 + 1.00/225) = 4,266.67
Average SD = sqrt(4,266.67) = 65.32
Premium = 200 + 25 + 15 = 240

Use the idea

Before relying on a large pool, ask what share of its losses come from a shared cause.

Where the conclusion applies

Identical stores, equal pairwise correlation and a single loss size. Catastrophes or contagion make the correlation larger in bad years.

Check your understanding: With 100 independent stores, what is the average-loss SD?

979.80 / sqrt(100) = 97.98 dollars.

Chapter 43 source: section "Risk Pooling, Insurance, and Diversification".

Beta sets the hurdle rate

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

One look or two: myopic loss aversion

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Home bias and portfolio volatility

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Pooling shrinks risk, correlation brings it back

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies