Week 4 assignment
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Key Concepts and Skills
After studying this chapter, you should be able to:
Calculate expected returns.
Explain the impact of diversification.
Define the systematic risk principle.
Discuss the security market line and the risk-return trade-off.
11-2
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2
Chapter Outline
11.1 Expected Returns and Variances
11.2 Portfolios
11.3 Announcements, Surprises, and Expected Returns
11.4 Risk: Systematic and Unsystematic
11.5 Diversification and Portfolio Risk
11.6 Systematic Risk and Beta
11.7 The Security Market Line
11.8 The SML and the Cost of Capital: A Preview
11-3
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3
Expected Returns
Expected returns are based on the probabilities of possible outcomes.
Where:
pi = the probability of state “i” occurring
Ri = the expected return on an asset in state i
Return to Quick Quiz
11-4
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4
Example: Expected Returns (1 of 2)
11-5
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5
Example: Expected Returns (2 of 2)
11-6
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6
Variance & Standard Deviation (1 of 2)
Variance and standard deviation measure the volatility of returns.
Variance = Weighted average of squared deviations
Standard Deviation = Square root of variance
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11-7
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Variance & Standard Deviation (2 of 2)
11-8
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8
Portfolios
Portfolio = collection of assets
An asset’s risk and return impact how the stock affects the risk and return of the portfolio.
The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets.
11-9
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9
Portfolio Expected Returns
The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio.
Weights (wj) = % of portfolio invested in each asset
Return to Quick Quiz
11-10
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10
Example: Portfolio Weights
11-11
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11
Expected Portfolio Return Alternative Method
Steps:
Calculate expected portfolio return in each state.
Apply the probabilities of each state to the expected return of the portfolio in that state.
Sum the result of Step 2.
Return to Slide 11-15
11-12
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12
Portfolio Risk Variance & Standard Deviation
Portfolio standard deviation is NOT a weighted average of the standard deviation of the component securities’ risk.
If it were, there would be no benefit to diversification.
11-13
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13
Portfolio Variance
Compute portfolio return for each state: RP,i = w1R1,i + w2R2,i + … + wmRm,i
Compute the overall expected portfolio return using the same formula as for an individual asset.
Compute the portfolio variance and standard deviation using the same formulas as for an individual asset.
Return to Quick Quiz
11-14
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14
Portfolio Risk
Calculate Expected Portfolio Return in each state of the economy and overall (Slide 11-12).
Compute the deviation (DEV) of expected portfolio return in each state from total expected portfolio return.
Square the deviations (DEV^2) found in Step 2.
Multiply the squared deviations from Step 3 times the probability of each state occurring (x p(i)).
The sum of the results from Step 4 = Portfolio Variance.
11-15
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15
Announcements, News, and Efficient Markets
Announcements and news contain both expected and surprise components.
The surprise component affects stock prices.
Efficient markets result from investors trading on unexpected news.
The easier it is to trade on surprises, the more efficient markets should be.
Efficient markets involve random price changes because we cannot predict surprises.
11-16
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16
Systematic Risk
Factors that affect a large number of assets
“Non-diversifiable risk”
“Market risk”
Examples: changes in GDP, inflation, interest rates, etc.
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11-17
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Unsystematic Risk
= Diversifiable risk
Risk factors that affect a limited number of assets
Risk that can be eliminated by combining assets into portfolios
“Unique risk”
“Asset-specific risk”
Examples: labor strikes, part shortages, etc.
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11-18
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18
Returns
Total Return = Expected return + Unexpected return
R = E(R) + U
Unexpected return (U) = Systematic portion (m) + Unsystematic portion (ε)
Total Return = Expected return E(R) + Systematic portion m
+ Unsystematic portion ε
= E(R) + m + ε
11-19
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19
The Principle of Diversification
Diversification can substantially reduce risk without an equivalent reduction in expected returns.
Reduces the variability of returns
Caused by the offset of worse-than-expected returns from one asset by better-than-expected returns from another
Minimum level of risk that cannot be diversified away = systematic portion
11-20
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20
Standard Deviations of Annual Portfolio Returns Table 11.7
11-21
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Portfolio Conclusions
As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio.
sp falls very slowly after about 40 stocks are included
The lower limit for sp ≈ 20% = sM
Forming well-diversified portfolios can eliminate about half the risk of owning a single stock.
11-22
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Portfolio Diversification Figure 11.1
11-23
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Total Risk = Stand-Alone Risk
Total risk = Systematic risk + Unsystematic risk
The standard deviation of returns is a measure of total risk.
For well-diversified portfolios, unsystematic risk is very small.
Total risk for a diversified portfolio is essentially equivalent to the systematic risk.
11-24
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24
Systematic Risk Principle
There is a reward for bearing risk.
There is no reward for bearing risk unnecessarily.
The expected return (market required return) on an asset depends only on that asset’s systematic or market risk.
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11-25
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Market Risk for Individual Securities
The contribution of a security to the overall riskiness of a portfolio
Relevant for stocks held in well-diversified portfolios
Measured by a stock’s beta coefficient, βj
Measures the stock’s volatility relative to the market
11-26
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34
Interpretation of Beta
If β = 1.0, stock has average risk
If β > 1.0, stock is riskier than average
If β < 1.0, stock is less risky than average
Most stocks have betas in the range of 0.5 to 1.5.
Beta of the market = 1.0
Beta of a T-Bill = 0
11-27
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39
Beta Coefficients for Selected Companies Table 11.8
11-28
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Click on this link to access Yahoo finance.
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Example: Work the Web
Many sites provide betas for companies.
Yahoo! Finance provides beta, plus a lot of other information under its profile link.
Click on this link to go to Yahoo! Finance.
Enter a ticker symbol and get a basic quote.
Click on key statistics.
Beta is reported under stock price history.
11-29
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Portfolio Beta
βp = Weighted average of the Betas of the assets in the portfolio
Weights (wj)= % of portfolio invested in asset j
11-30
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47
Quick Quiz: Total vs. Systematic Risk
Consider the following information:
Standard Deviation Beta
Security C 20% 1.25
Security K 30% 0.95
Which security has more total risk?
Which security has more systematic risk?
Which security should have the higher expected return?
11-31
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Beta and the Risk Premium
Risk premium = E(R ) – Rf
The higher the beta, the greater the risk premium should be
Can we define the relationship between the risk premium and beta so that we can estimate the expected return?
YES!
11-32
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32
SML and Equilibrium
11-33
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Reward-to-Risk Ratio
Reward-to-Risk Ratio:
= Slope of line on graph
In equilibrium, ratio should be the same for all assets
When E(R) is plotted against β for all assets, the result should be a straight line
11-34
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34
Market Equilibrium
In equilibrium, all assets and portfolios must have the same reward-to-risk ratio
Each ratio must equal the reward-to-risk ratio for the market
11-35
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35
Security Market Line
The security market line (SML) is the representation of market equilibrium.
The slope of the SML = reward-to-risk ratio:
(E(RM) – Rf) / βM
Slope = E(RM) – Rf = market risk premium
Since β of the market is always 1.0
11-36
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36
The SML and Required Return
The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM).
Rf = Risk-free rate (T-Bill or T-Bond)
RM = Market return ≈ S&P 500
RPM = Market risk premium = E(RM) – Rf
E(Rj) = “Required Return of Asset j”
11-37
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43
Capital Asset Pricing Model
The capital asset pricing model (CAPM) defines the relationship between risk and return.
E(RA) = Rf + (E(RM) – Rf)βA
If an asset’s systematic risk (β) is known, CAPM can be used to determine its expected return.
11-38
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38
SML Example
11-39
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39
Factors Affecting Required Return
Rf measures the pure time value of money
RPM = (E(RM)-Rf) measures the reward for bearing systematic risk
βj measures the amount of systematic risk
11-40
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40
Quick Quiz (1 of 2)
How do you compute the expected return and standard deviation:
For an individual asset? (Slide 11-4 and Slide 11-7)
For a portfolio? (Slide 11-10 and Slide 11-14)
What is the difference between systematic and unsystematic risk? (Slide 11-18 and Slide 11-19)
What type of risk is relevant for determining the expected return? (Slide 11-25)
11-41
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41
Quick Quiz (2 of 2)
4. Consider an asset with a beta of 1.2, a risk-free rate of 5%, and a market return of 13%.
What is the reward-to-risk ratio in equilibrium?
What is the expected return on the asset?
E(R) = 5% + (13% - 5%) × 1.2 = 14.6%
11-42
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END
Chapter 11
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43
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State (i)p(i)E(Ra)E(Rb)
Recession0.25-20%30%
Neutral0.5015%15%
Boom0.2535%-10%
1.00
Stock AStock B
E(R)
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Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| 1.00 | 11.3% | 12.5% |
Sheet2
| Expected Return | |||
| E(R) | |||
| State (i) | p(i) | Stock L | Stock U |
| Recession | 0.5 | -20% | 30% |
| Boom | 0.5 | 70% | 10% |
| 1.0 | 25% | 20% |
Sheet3
State (i)p(i)E(Ra)p(i) x E(Ra)E(Rb)p(i) x E(Rb)
Recession0.25-20%-5.0%30%7.5%
Neutral0.5015%7.5%15%7.5%
Boom0.2535%8.8%-10%-2.5%
E(R)11.25%12.50%
Stock AStock B
E(R)
Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| E(R) | 11.25% | 12.5% |
Pf VAR
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | |||||
| 30% | 17% | 22% | 20% | 12% | Dev | Dev^2 | x p(i) | |||
| State (i) | p(i) | E( R ) | ||||||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% | -13% | 0.01663 | 0.00416 |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% | 3% | 0.00101 | 0.00050 |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% | 7% | 0.00428 | 0.00107 |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% | VAR(Pf) | 0.0057325858 | |
| Std(Pf) | 7.6% |
Pf E(R)
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | ||
| w(j) | 30% | 17% | 22% | 20% | 12% | 100% | |
| State (i) | p(i) | Expected Return | |||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% |
Pf Wgts
| Portfolio Weights | |||||
| Dollars | % of Pf | w(j) x | |||
| Asset | Invested | w(j) | E( Rj ) | E( Rj ) | |
| A | $15,000 | 30% | 12.5% | 3.735% | |
| B | $8,600 | 17% | 9.5% | 1.627% | |
| C | $11,000 | 22% | 10.0% | 2.191% | |
| D | $9,800 | 20% | 7.5% | 1.464% | |
| E | $5,800 | 12% | 8.5% | 0.982% | |
| $50,200 | 100% | 10.000% |
VAR
| Variance & Standard Deviation | |||||||||||
| Stock A | E(R) | ||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | Stock A | Stock B | |||||
| Recession | 0.25 | -20% | 10% | 0.0244141 | State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) | |
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | 35% | 6% | 0.0141016 | Recession | 0.25 | -20% | -5.0% | 30% | 7.5% | |
| 1.00 | Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% | |||||
| Expected Return | 11.3% | Boom | 0.25 | 35% | 8.8% | -10% | -2.5% | ||||
| Variance | 0.049765625 | E(R) | 1.00 | 25% | 11.3% | 20% | 12.5% | ||||
| Standard Deviation | 22.3% | ||||||||||
| Stock B | |||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |||||||
| Recession | 0.25 | 30% | 3% | 0.0076563 | |||||||
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | -10% | 5% | 0.0126563 | |||||||
| 1.00 | |||||||||||
| Expected Return | 12.5% | ||||||||||
| Variance | 0.0316 | ||||||||||
| Standard Deviation | 17.8% |
Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| E(R) | 11.25% | 12.50% |
Sheet2
| Variance & Standard Deviation | |||||||||||
| Stock A | E(R) | ||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | Stock A | Stock B | |||||
| Recession | 0.25 | -20% | 4% | 0.01 | State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) | |
| Neutral | 0.50 | 15% | 2% | 0.01125 | |||||||
| Boom | 0.25 | 35% | 12% | 0.030625 | Recession | 0.25 | -20% | -5.0% | 30% | 7.5% | |
| 1.00 | Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% | |||||
| Expected Return | 0.0% | Boom | 0.25 | 35% | 8.8% | -10% | -2.5% | ||||
| Variance | 0.051875 | E(R) | 1.00 | 25% | 11.3% | 20% | 12.5% | ||||
| Standard Deviation | 23% | ||||||||||
| Stock B | |||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |||||||
| Recession | 0.25 | 30% | 9% | 0.0225 | |||||||
| Neutral | 0.50 | 15% | 2% | 0.01125 | |||||||
| Boom | 0.25 | -10% | 1% | 0.0025 | |||||||
| 1.00 | |||||||||||
| Expected Return | 0.0% | ||||||||||
| Variance | 0.0363 | ||||||||||
| Standard Deviation | 19% |
Sheet3
å
=
-
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2
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R
(
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R
(
p
2
σ
Variance & Standard Deviation
State (i) p(i) E(R) DEV^2 x p(i) Recession 0.25 -20% 0.097656 0.0244141
Neutral 0.50 15% 0.001406 0.0007031 Boom 0.25 35% 0.056406 0.0141016
1.00 11.25%
0.03921875 19.8%
State (i) p(i) E(R) DEV^2 x p(i) Recession 0.25 30% 0.030625 0.0076563
Neutral 0.50 15% 0.000625 0.0003125 Boom 0.25 -10% 0.050625 0.0126563
1.00 12.50%
0.0206 14.4%
Standard Deviation
Stock A
Expected Return Variance
Standard Deviation
Stock B
Expected Return Variance
Variance & Standard Deviation
State (i)p(i)E(R)DEV^2 x p(i)
Recession 0.25 -20%0.097656 0.0244141
Neutral0.50 15% 0.001406 0.0007031
Boom 0.25 35% 0.056406 0.0141016
1.00
11.25%
0.03921875
19.8%
State (i)p(i)E(R)DEV^2 x p(i)
Recession 0.25 30% 0.030625 0.0076563
Neutral0.50 15% 0.000625 0.0003125
Boom 0.25 -10%0.050625 0.0126563
1.00
12.50%
0.0206
14.4%
Standard Deviation
Stock A
Expected Return
Variance
Standard Deviation
Stock B
Expected Return
Variance
Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| E(R) | 1.00 | 11.25% | 12.50% |
VAR
| Variance & Standard Deviation | |||||
| Stock A | |||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |
| Recession | 0.25 | -20% | 0.097656 | 0.0244141 | |
| Neutral | 0.50 | 15% | 0.001406 | 0.0007031 | |
| Boom | 0.25 | 35% | 0.056406 | 0.0141016 | |
| 1.00 | |||||
| Expected Return | 11.25% | ||||
| Variance | 0.03921875 | ||||
| Standard Deviation | 19.8% | ||||
| Stock B | |||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |
| Recession | 0.25 | 30% | 0.030625 | 0.0076563 | |
| Neutral | 0.50 | 15% | 0.000625 | 0.0003125 | |
| Boom | 0.25 | -10% | 0.050625 | 0.0126563 | |
| 1.00 | |||||
| Expected Return | 12.50% | ||||
| Variance | 0.0206 | ||||
| Standard Deviation | 14.4% | ||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| 1.00 | |||||
| E(R) | 11.25% | 12.50% |
Pf VAR
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | |||||
| 30% | 17% | 22% | 20% | 12% | Dev | Dev^2 | x p(i) | |||
| State (i) | p(i) | E( R ) | ||||||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% | -13% | 0.01663 | 0.00416 |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% | 3% | 0.00101 | 0.00050 |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% | 7% | 0.00428 | 0.00107 |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% | VAR(Pf) | 0.0057325858 | |
| Std(Pf) | 7.6% |
Pf E(R)
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | ||
| w(j) | 30% | 17% | 22% | 20% | 12% | 100% | |
| State (i) | p(i) | Expected Return | |||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% |
Pf Wgts
| Portfolio Weights | |||||
| Dollars | % of Pf | w(j) x | |||
| Asset | Invested | w(j) | E( Rj ) | E( Rj ) | |
| A | $15,000 | 30% | 12.5% | 3.735% | |
| B | $8,600 | 17% | 9.5% | 1.627% | |
| C | $11,000 | 22% | 10.0% | 2.191% | |
| D | $9,800 | 20% | 7.5% | 1.464% | |
| E | $5,800 | 12% | 8.5% | 0.982% | |
| $50,200 | 100% | 10.000% |
VAR
| Variance & Standard Deviation | |||||
| Stock A | |||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |
| Recession | 0.25 | -20% | 0.097656 | 0.0244141 | |
| Neutral | 0.50 | 15% | 0.001406 | 0.0007031 | |
| Boom | 0.25 | 35% | 0.056406 | 0.0141016 | |
| 1.00 | |||||
| Expected Return | 11.25% | ||||
| Variance | 0.03921875 | ||||
| Standard Deviation | 19.8% | ||||
| Stock B | |||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |
| Recession | 0.25 | 30% | 0.030625 | 0.0076563 | |
| Neutral | 0.50 | 15% | 0.000625 | 0.0003125 | |
| Boom | 0.25 | -10% | 0.050625 | 0.0126563 | |
| 1.00 | |||||
| Expected Return | 12.50% | ||||
| Variance | 0.0206 | ||||
| Standard Deviation | 14.4% | ||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| 1.00 | |||||
| E(R) | 11.25% | 12.50% |
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AssetInvested w(j)E( Rj )E( Rj )
A $15,000 30%12.5% 3.735%
B$8,600 17%9.5%1.627%
C$11,000 22%10.0% 2.191%
D$9,800 20%7.5%1.464%
E$5,800 12%8.5%0.982%
$50,200 100% 10.000%
Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| E(R) | 1.00 | 11.3% | 12.5% |
VAR
| Variance & Standard Deviation | |||||||||||
| Stock A | E(R) | ||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | Stock A | Stock B | |||||
| Recession | 0.25 | -20% | 10% | 0.0244141 | State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) | |
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | 35% | 6% | 0.0141016 | Recession | 0.25 | -20% | -5.0% | 30% | 7.5% | |
| 1.00 | Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% | |||||
| Expected Return | 11.3% | Boom | 0.25 | 35% | 8.8% | -10% | -2.5% | ||||
| Variance | 0.049765625 | E(R) | 1.00 | 25% | 11.3% | 20% | 12.5% | ||||
| Standard Deviation | 22.3% | ||||||||||
| Stock B | |||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |||||||
| Recession | 0.25 | 30% | 3% | 0.0076563 | |||||||
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | -10% | 5% | 0.0126563 | |||||||
| 1.00 | |||||||||||
| Expected Return | 12.5% | ||||||||||
| Variance | 0.0316 | ||||||||||
| Standard Deviation | 17.8% |
Pf Wgts
| Portfolio Weights | |||||
| Dollars | % of Pf | w(j) x | |||
| Asset | Invested | w(j) | E( Rj ) | E( Rj ) | |
| A | $15,000 | 30% | 12.5% | 3.735% | |
| B | $8,600 | 17% | 9.5% | 1.627% | |
| C | $11,000 | 22% | 10.0% | 2.191% | |
| D | $9,800 | 20% | 7.5% | 1.464% | |
| E | $5,800 | 12% | 8.5% | 0.982% | |
| $50,200 | 100% | 10.000% |
Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| E(R) | 1.00 | 11.3% | 12.5% |
VAR
| Variance & Standard Deviation | |||||||||||
| Stock A | E(R) | ||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | Stock A | Stock B | |||||
| Recession | 0.25 | -20% | 10% | 0.0244141 | State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) | |
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | 35% | 6% | 0.0141016 | Recession | 0.25 | -20% | -5.0% | 30% | 7.5% | |
| 1.00 | Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% | |||||
| Expected Return | 11.3% | Boom | 0.25 | 35% | 8.8% | -10% | -2.5% | ||||
| Variance | 0.049765625 | E(R) | 1.00 | 25% | 11.3% | 20% | 12.5% | ||||
| Standard Deviation | 22.3% | ||||||||||
| Stock B | |||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |||||||
| Recession | 0.25 | 30% | 3% | 0.0076563 | |||||||
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | -10% | 5% | 0.0126563 | |||||||
| 1.00 | |||||||||||
| Expected Return | 12.5% | ||||||||||
| Variance | 0.0316 | ||||||||||
| Standard Deviation | 17.8% |
Pf Wgts
| Portfolio Weights | |||||
| Dollars | % of Pf | w(j) x | |||
| Asset | Invested | w(j) | E( Rj ) | E( Rj ) | |
| A | $15,000 | 30% | 12.5% | 3.735% | |
| B | $8,600 | 17% | 9.5% | 1.627% | |
| C | $11,000 | 22% | 10.0% | 2.191% | |
| D | $9,800 | 20% | 7.5% | 1.464% | |
| E | $5,800 | 12% | 8.5% | 0.982% | |
| $50,200 | 100% | 10.000% |
Pf VAR
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | |||||
| 30% | 17% | 22% | 20% | 12% | Dev | Dev^2 | x p(i) | |||
| State (i) | p(i) | E( R ) | ||||||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% | -13% | 0.01663 | 0.00416 |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% | 3% | 0.00101 | 0.00050 |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% | 7% | 0.00428 | 0.00107 |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% | VAR(Pf) | 0.0057325858 | |
| Std(Pf) | 7.6% |
Pf E(R)
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | ||
| w(j) | 30% | 17% | 22% | 20% | 12% | 100% | |
| State (i) | p(i) | Expected Return | |||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% |
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Stock VStock W Stock XStock YStock ZPortfolio
w(j)30%17%22%20%12%100%
State (i) p(i)
Recession 0.25-20.0% 18.0% 5.0%-8.0% 4.0% -3%
Neutral 0.5017.5%15.0%10.0%11.0% 9.0% 13%
Boom 0.2535.0%-10.0% 15.0%16.0%12.0% 17%
E(R) 1.0012.5% 9.5%10.0% 7.5%8.5% 10%
Expected Return
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Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| E(R) | 1.00 | 11.3% | 12.5% |
VAR
| Variance & Standard Deviation | |||||||||||
| Stock A | E(R) | ||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | Stock A | Stock B | |||||
| Recession | 0.25 | -20% | 10% | 0.0244141 | State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) | |
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | 35% | 6% | 0.0141016 | Recession | 0.25 | -20% | -5.0% | 30% | 7.5% | |
| 1.00 | Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% | |||||
| Expected Return | 11.3% | Boom | 0.25 | 35% | 8.8% | -10% | -2.5% | ||||
| Variance | 0.049765625 | E(R) | 1.00 | 25% | 11.3% | 20% | 12.5% | ||||
| Standard Deviation | 22.3% | ||||||||||
| Stock B | |||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |||||||
| Recession | 0.25 | 30% | 3% | 0.0076563 | |||||||
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | -10% | 5% | 0.0126563 | |||||||
| 1.00 | |||||||||||
| Expected Return | 12.5% | ||||||||||
| Variance | 0.0316 | ||||||||||
| Standard Deviation | 17.8% |
Pf Wgts
| Portfolio Weights | |||||
| Dollars | % of Pf | w(j) x | |||
| Asset | Invested | w(j) | E( Rj ) | E( Rj ) | |
| A | $15,000 | 30% | 12.5% | 3.735% | |
| B | $8,600 | 17% | 9.5% | 1.627% | |
| C | $11,000 | 22% | 10.0% | 2.191% | |
| D | $9,800 | 20% | 7.5% | 1.464% | |
| E | $5,800 | 12% | 8.5% | 0.982% | |
| $50,200 | 100% | 10.000% |
Pf E(R)
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | ||
| w(j) | 30% | 17% | 22% | 20% | 12% | 100% | |
| State (i) | p(i) | Expected Return | |||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% |
Pf VAR
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | |||||
| 30% | 17% | 22% | 20% | 12% | Dev | Dev^2 | x p(i) | |||
| State (i) | p(i) | E( R ) | ||||||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% | -13% | 0.01663 | 0.00416 |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% | 3% | 0.00101 | 0.00050 |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% | 7% | 0.00428 | 0.00107 |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% | VAR(Pf) | 0.0057325858 | |
| Std(Pf) | 7.6% |
Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| E(R) | 1.00 | 11.3% | 12.5% |
VAR
| Variance & Standard Deviation | |||||||||||
| Stock A | E(R) | ||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | Stock A | Stock B | |||||
| Recession | 0.25 | -20% | 10% | 0.0244141 | State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) | |
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | 35% | 6% | 0.0141016 | Recession | 0.25 | -20% | -5.0% | 30% | 7.5% | |
| 1.00 | Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% | |||||
| Expected Return | 11.3% | Boom | 0.25 | 35% | 8.8% | -10% | -2.5% | ||||
| Variance | 0.049765625 | E(R) | 1.00 | 25% | 11.3% | 20% | 12.5% | ||||
| Standard Deviation | 22.3% | ||||||||||
| Stock B | |||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |||||||
| Recession | 0.25 | 30% | 3% | 0.0076563 | |||||||
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | -10% | 5% | 0.0126563 | |||||||
| 1.00 | |||||||||||
| Expected Return | 12.5% | ||||||||||
| Variance | 0.0316 | ||||||||||
| Standard Deviation | 17.8% |
Pf Wgts
| Portfolio Weights | |||||
| Dollars | % of Pf | w(j) x | |||
| Asset | Invested | w(j) | E( Rj ) | E( Rj ) | |
| A | $15,000 | 30% | 12.5% | 3.735% | |
| B | $8,600 | 17% | 9.5% | 1.627% | |
| C | $11,000 | 22% | 10.0% | 2.191% | |
| D | $9,800 | 20% | 7.5% | 1.464% | |
| E | $5,800 | 12% | 8.5% | 0.982% | |
| $50,200 | 100% | 10.000% |
Pf Var
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | ||
| w(j) | 30% | 17% | 22% | 20% | 12% | 100% | |
| State (i) | p(i) | Expected Return | |||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% |
State (i)p(i)
Recession0.25-3%-13%0.016630.00416
Neutral0.5013%3%0.001010.00050
Boom0.2517%7%0.004280.00107
E(R)1.0010%VAR(Pf)0.0057326
Std(Pf)0.0757138
Std(Pf) as %7.6%
Portfolio
DevDev^2x p(i)
E( R )
VAR
| Variance & Standard Deviation | |||||||||||
| Stock A | E(R) | ||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | Stock A | Stock B | |||||
| Recession | 0.25 | -20% | 10% | 0.0244141 | State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) | |
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | 35% | 6% | 0.0141016 | Recession | 0.25 | -20% | -5.0% | 30% | 7.5% | |
| 1.00 | Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% | |||||
| Expected Return | 11.3% | Boom | 0.25 | 35% | 8.8% | -10% | -2.5% | ||||
| Variance | 0.049765625 | E(R) | 1.00 | 25% | 11.3% | 20% | 12.5% | ||||
| Standard Deviation | 0.2230821 | ||||||||||
| Standard Deviation expressed as a % | 22.3% | ||||||||||
| Stock B | |||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |||||||
| Recession | 0.25 | 30% | 3% | 0.0076563 | |||||||
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | -10% | 5% | 0.0126563 | |||||||
| 1.00 | |||||||||||
| Expected Return | 12.5% | ||||||||||
| Variance | 0.031563 | ||||||||||
| Standard Deviation | 0.177658 | ||||||||||
| Standard Deviation expressed as a % | 17.8% |
Pf VAR
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | |||||
| 30% | 17% | 22% | 20% | 12% | Dev | Dev^2 | x p(i) | |||
| State (i) | p(i) | E( R ) | ||||||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% | -13% | 0.01663 | 0.00416 |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% | 3% | 0.00101 | 0.00050 |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% | 7% | 0.00428 | 0.00107 |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% | VAR(Pf) | 0.0057326 | |
| Std(Pf) | 0.0757138 | |||||||||
| Std(Pf) as % | 7.6% |
Exp Return
| Expected Return | |||||
| E(R) | |||||
| Stock A | Stock B | ||||
| State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) |
| Recession | 0.25 | -20% | -5.0% | 30% | 7.5% |
| Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% |
| Boom | 0.25 | 35% | 8.8% | -10% | -2.5% |
| E(R) | 1.00 | 11.3% | 12.5% |
Pf VAR
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | |||||
| 30% | 17% | 22% | 20% | 12% | Dev | Dev^2 | x p(i) | |||
| State (i) | p(i) | E( R ) | ||||||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% | -13% | 0.01663 | 0.00416 |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% | 3% | 0.00101 | 0.00050 |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% | 7% | 0.00428 | 0.00107 |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% | VAR(Pf) | 0.0057325858 | |
| Std(Pf) | 0.0757138 | |||||||||
| Std(Pf) as % | 7.6% |
Pf E(R)
| Stock V | Stock W | Stock X | Stock Y | Stock Z | Portfolio | ||
| w(j) | 30% | 17% | 22% | 20% | 12% | 100% | |
| State (i) | p(i) | Expected Return | |||||
| Recession | 0.25 | -20.0% | 18.0% | 5.0% | -8.0% | 4.0% | -3% |
| Neutral | 0.50 | 17.5% | 15.0% | 10.0% | 11.0% | 9.0% | 13% |
| Boom | 0.25 | 35.0% | -10.0% | 15.0% | 16.0% | 12.0% | 17% |
| E(R) | 1.00 | 12.5% | 9.5% | 10.0% | 7.5% | 8.5% | 10% |
Pf Wgts
| Portfolio Weights | |||||
| Dollars | % of Pf | w(j) x | |||
| Asset | Invested | w(j) | E( Rj ) | E( Rj ) | |
| A | $15,000 | 30% | 12.5% | 3.735% | |
| B | $8,600 | 17% | 9.5% | 1.627% | |
| C | $11,000 | 22% | 10.0% | 2.191% | |
| D | $9,800 | 20% | 7.5% | 1.464% | |
| E | $5,800 | 12% | 8.5% | 0.982% | |
| $50,200 | 100% | 10.000% |
VAR
| Variance & Standard Deviation | |||||||||||
| Stock A | E(R) | ||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | Stock A | Stock B | |||||
| Recession | 0.25 | -20% | 10% | 0.0244141 | State (i) | p(i) | E(Ra) | p(i) x E(Ra) | E(Rb) | p(i) x E(Rb) | |
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | 35% | 6% | 0.0141016 | Recession | 0.25 | -20% | -5.0% | 30% | 7.5% | |
| 1.00 | Neutral | 0.50 | 15% | 7.5% | 15% | 7.5% | |||||
| Expected Return | 11.3% | Boom | 0.25 | 35% | 8.8% | -10% | -2.5% | ||||
| Variance | 0.049765625 | E(R) | 1.00 | 25% | 11.3% | 20% | 12.5% | ||||
| Standard Deviation | 22.3% | ||||||||||
| Stock B | |||||||||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) | |||||||
| Recession | 0.25 | 30% | 3% | 0.0076563 | |||||||
| Neutral | 0.50 | 15% | 2% | 0.0112500 | |||||||
| Boom | 0.25 | -10% | 5% | 0.0126563 | |||||||
| 1.00 | |||||||||||
| Expected Return | 12.5% | ||||||||||
| Variance | 0.0316 | ||||||||||
| Standard Deviation | 17.8% |
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RETURN
| Expected Return | |||
| E(R) | |||
| State (i) | p(i) | Stock L | Stock U |
| Recession | 0.5 | -20% | 30% |
| Boom | 0.5 | 70% | 10% |
| 1.0 | 25% | 20% |
VARIANCE
| Variance & Standard Deviation | ||||
| Stock L | ||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) |
| Recession | 0.5 | -20% | 20% | 0.10125 |
| Boom | 0.5 | 70% | 20% | 0.10125 |
| 1.0 | ||||
| Expected Return | 25% | |||
| Variance | 0.2025 | |||
| Standard Deviation | 45% | |||
| Stock U | ||||
| State (i) | p(i) | E(R) | DEV^2 | x p(i) |
| Recession | 0.5 | 30% | 1% | 0.005 |
| Boom | 0.5 | 10% | 1% | 0.005 |
| 1.0 | ||||
| Expected Return | 20% | |||
| Variance | 0.0100 | |||
| Standard Deviation | 10% |
Pf E(R)
| Portfolio Return | ||
| Stock | PF % | E(R) |
| L | 50% | 25% |
| U | 50% | 20% |
| Portfolio | 22.50% |
COV
| Covariance | |||||||
| Stock L | Stock U | ||||||
| State (i) | p(i) | E(R) | Dev L | E(R) | Dev U | Dev*Dev | x p(i) |
| Recession | 0.5 | -20% | -45% | 30% | 10% | -4.5% | -0.0225 |
| Boom | 0.5 | 70% | 45% | 10% | -10% | -4.5% | -0.0225 |
| 1.0 | |||||||
| Expected Return | 25% | 20% | |||||
| Standard Deviation | 45% | 10% | |||||
| Covariance | -4.50% | ||||||
| Correlation Coefficient | -1.00 |
Pf Var
| Portfolio Variance & Standard Dev | ||
| Stock | PF % | σ |
| L | 50% | 45% |
| U | 50% | 10% |
| Covariance | -4.50% | |
| Portfolio Variance | 0.030625 | |
| Portfolio Standard Dev | 17.50% |
BETA
| Expected vs Required Return | ||||
| Stock | E(R) | Beta | Req R | |
| A | 14% | 1.3 | 13.4% | Undervalued |
| B | 10% | 0.8 | 11.1% | Overvalued |
| Assume: | Market Return = | 12.0% | ||
| Risk-free Rate = | 7.5% |
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