Portfolio Risk and Return

Grades and Auditing

• Grades seen are not final.
• Tutors mark first, followed by auditing.
• Grades may increase or decrease slightly after auditing.
• Auditing ensures consistent marking standards across tutors.

Midterm and AI Usage

• Do not use AI for quizzes.
• Quizzes are open book, but work must be original.
• AI usage leaves traces in answers.
• Evidence of AI use will be reported to the academic integrity committee.
• Students under review will have grades put on hold.
• Communication about grades on hold is possible, but decisions may take weeks.

Open Book Exam

• Open book allows textbook usage but prohibits copying.
• Reading the textbook aids learning.
• Final exam will be closed book.
• Final exam venue may be cold, so dress warmly.

Upcoming Quiz and Topics

• One last quiz will be announced in several weeks.
• Five weeks of teaching remain, including today.
• Today covers the last topic for time value issues.

Review of Previous Topics

• Previous topics included risk and return.
• Calculations involved future and historical returns.
• Focus was on single assets like stocks.

Portfolio Risk and Return

• Today's focus is on portfolio risk and return.
• A portfolio contains multiple investment assets (stocks, bonds, options).
• Each asset has its own risk and return.
• The goal is to determine the overall risk and return of the portfolio.
• Calculate portfolio return and risk.
• Understand the impact of diversification.
• Diversification can lead to the same return with less risk.
• Systematic risk will be discussed.

Systematic vs. Unsystematic Risk

• Systematic risk = market risk = undiversifiable risk
• Unsystematic risk = individual risk = diversifiable risk
• Capital Asset Pricing Model (CAPM) will be used to calculate systematic risk.

Portfolio Returns

• Portfolio return is the weighted average of individual asset returns.

Calculating the Weight

• Weight is calculated as investment in an asset divided by total investment.
• Example: Invest 100 in a share, total investment 300. Weight = 100/300 = 1/3.

Weighted Average Return

• Multiply each asset's weight by its expected return and sum them up.

Formulas for Return

• Two equations are provided for return calculation.
• Both are weighted averages but used in different scenarios.
• One is based on historical data and the other on expected future returns (denoted by 'e').
• The Equations: \text{Historical Return} = \sum (\text{Weight}i \times \text{Return}i), \text{Expected Return} = \sum (\text{Weight}i \times \text{Expected Return}i)

Portfolio Risk

• Volatility or standard deviation represents risk.
• Simply replacing returns with volatility to calculate weighted average risk is technically possible but provides the maximum possible risk.
• Risk can be reduced by including different assets in the portfolio.

Example: Airline Stocks and Texas Oil

• Three stocks: North Air, West Air (airline companies), and Texas Oil.
• Historical data shows each stock has an average return of 10%.
• Volatility (standard deviation) is calculated.
• Volatility for North Air is 13.4\%. Standard deviation is used as units are provided, unlike variance.

Creating Artificial Portfolios

• Portfolio 1: 50% North Air, 50% West Air (same industry).
• Portfolio 2: 50% West Air, 50% Texas Oil (different industries).

Returns

• Both portfolios have an average return of 10%.

Standard Deviation

• Portfolio 1 standard deviation: 12.1\%.
• Portfolio 2 standard deviation: 5.1\%.
• Combining similar companies reduces risk slightly, but combining different industries reduces risk more significantly.

Movement of Individual Shares vs. Portfolio

• Portfolio A (North Air & West Air): Both move in the same direction.
• If West Air increases, North Air increases and vice versa. The price movement is the same because they are in the same business thus will be affected by the same events.
• Portfolio B (West Air & Texas Oil): They move in opposite directions.
• If Texas Oil benefits from higher oil prices, airline companies suffer.
• Portfolio B's return is always above zero because the assets offset each other.

Diversification

• Diversification reduces risk as assets offset each other.
• Volatility is the fluctuation of price, so offsetting the price movement reduces volatility.
• Combine different assets that move in different ways to reduce volatility or risk.

Measuring Co-movement: Correlation Coefficient

• Correlation coefficient measures co-movement.
• Values range from -1 to +1.
• Positive Correlation: Stock A and Stock B move in the same direction.
• Stock A increase by 5%, Stock B increase 5% as well.
• Negative Correlation: Stock A and Stock B move in opposite directions.
• Stock A increase by 5%, Stock B decrease 5%.
• Zero Correlation: No relationship between Stock A and Stock B.
• Stock A increase by 10%, Stock B could increase or decrease by anything with an average of zero.

Examples of Correlation

• Coca Cola and Netflix actually has no relationship at all. The factors that affect the consumption of Coca Cola will not affect Netflix.
• Coca Cola and McDonald have a high positive correlation because McDonald's sells Coca-Cola products.
• Gold and bonds could have a negative correlation with stocks.

Scatter Plots

• Scatter plots visually represent correlation.
• Zero correlation: Upside and downside offset each other.
• Positive correlation: Clear trend.

Calculating Portfolio Risk and Volatility

• Return part, we are already learning.

Expected Portfolio Return Formula

• E(Rp) = WA \times E(RA) + WB \times E(R_B)
• E stands for expected, R stands for the return, and p denotes for the portfolio.

Risk of the Portfolio

• The equation only works for two asset portfolios.

Portfolio Variance Formula

• Var(Rp) = WA^2 \times \sigmaA^2 + WB^2 \times \sigmaB^2 + 2 \times WA \times WB \times \rho{A,B} \times \sigmaA \times \sigmaB
• Var(R_p) stands for is a's risk.
• \rho_{A,B} actually shows the correlation coefficient between asset a's return and b's return.
• \sigma is the standard deviation.
• The common mistake is the decimal points.

Key Points

• The value for WA^2 \times \sigmaA^2 and WB^2 \times \sigmaB^2 will be positive.
• The only one that could potentially be negative is just the \rho_{A,B}.
• If \rho_{A,B} is negative, can reduce the overall variance.

Total Risk, Individual Risk, and Market Risk

• Calculate the total risk of portfolio or the maximum possible risk of the portfolio by adding different assets.

Risk Reduction with Added Assets

• The X axis showing how many different assets you are adding to the portfolio.
• The Y axis is the portfolio's total risk or volatility.
• When add assets, there will be sharp decrease of the green area.
• There will be a limit for the diversification effect.
• Normally, add 30 different assets, then can achieve a major effect of diversification.

The Trick for the Diversification

• Diversification reduces the fatter only affect very limited number of stocks.
• Diversification can only reduce the individual risk.
• If a FATTER can affect all the assets in the market, the market portfolio can never ever help you to reduce any risk.

Two Conclusions from this figure

• Yes. When we add more access into the portfolio, you will reduce the risk a bit more.
• The diversification can only reduce the individual weeks, but not for the systematic risk.

Market Rewards Only Systematic Risk

• Diversify portfolio, otherwise, going to be punished by the market.

How to Calculate Systematic Risk

• What is the systematic risk? (Risk factor that can affect nearly all the companies or assets in the market).

Logic Steps

1. Define a marketplace portfolio.
2. Market portfolio will include every possible asset that you can potentially invest in.
3. Use the stock Index to as a kind of proxy for it.
4. Compare the return of the stock a and the perfect market portfolio, to see what is the relationship between them.

• Draw a trend line.
• The equation will be y = a + beta * b.
• The beta or the slope represent the geometric represents the line or the measure of the systematic risk.
• Use beta to measure the systematic risk.
• Stock has one unit of the systematic risk.
• The higher of the beta indicates the stock bearing more systematic risk compared to the market portfolio.
• Then we can make a decision that the stock actually bear with less than the one market portfolio's risk.

Examples for Using Different Stocks

• Total area of the bard represent the total risk.
• Compare the Boston Bear with Microsoft company to look at something basic.
• If I'm the weeks averse investor, I probably will say Microsoft will be safer.

CAPM Model

• Assume the Cisco stock to have a standard deviation 30% and a beta of 1.2.
• The UNICODE stock to have a standard deviation of 41% and beta of 0.6. What is expected return next year?

Key Points

• We are using beta to match the systematic weeks, and Cisco's beta is 1.2, which is higher than the unicose beta.

CAPM

• Capital asset pricing model or CAPAM model can help us to define what is the appropriate expected return.

CAPM Formula

• E(Ri) = Rf + \beta \times (E(Rm) - Rf)

• Rf stands for the the rate free investment.

• Beta is how many units that you're gonna bear with the systematic weeks.

• Market portfolio bear with one unit of systematic risk for UTs systematic multi risk.

• If the stock bear with more market risk, which means the beta is higher, in general, the expected return is higher as well.

Relationship between Total Risk and the Expected Return

• If there's a relationship, we don't have to use CAPM.
  • If you put the return and the risk level, you might see yeah. It might be some trend there, a positive trend, higher of the weeks, higher of the return.
    We're using the beta, the systematic weeks to calculate return expected return, but not the standard deviation.

Example of Kaplan

• Assume the economy has 60% chance of the market return being 15% next year and 40% chance of the market return being 5% next year.
  Given Microsoft's beta is 1.18, what is expected return next year?
• Once, see the expected return, you use the percentage times the expected return under that expectation and sum the other expectations return is given.
  We need to look at the number and pick from this.
  Finish the calculation, you will see the expected return for this stock, Microsoft is 11.9\%.
  Week premium already being equal to the amount minus its street rate.

Weighted Average

• Formula: You can literally use exactly the same weighted average to calculate the return and get weighted beta. So if each stock's beta and you know the weight, use the weight average and that is for you.