MAS 261 - Lecture 21

Housekeeping

• Today’s plan 📋

  • Comments and Questions from Engagement Questions or about R

  • Upcoming Dates

  • Review Questions

  • Comparing Portfolios of Stocks

    • Profitability of one stock using geometric mean

    • Weighted Mean of two or more stocks

    • Volatility of One Stock using adjusted standard deviation

    • Volatility of weighted combination of two or more stocks

Upcoming Dates

• HW 7 is due Wed. 11/6 at midnight.

• Test 2 is on November 12th and will include material up through Lecture 20.

• Practice Questions are now available and demo videos will be posted by Saturday.

• Lecture 21 - Intro to Portfolio Management will be on Final Exam, not on Test 2.

R and RStudio

• In this course we will use R and RStudio to understand statistical concepts.

• You will access R and RStudio through Posit Cloud.

  • Sign up for a Free Posit Cloud Account

• I will post R/RStudio files on Posit Cloud that you can access in provided links.

• I will also provide demo videos that show how to access files and complete exercises.

• NOTE: The free Posit Cloud account is limited to 25 hours per month.

  • I demo how to download completed work so that you can use this allotment efficiently.

  • For those who want to go further with R/RStudio:

    • I have added a new page to the MAS 261 website, Installing R and RStudio

💥 Lecture 21 In-class Exercises - Q1 💥

Examining Trends in these Four Stocks

💥 Lecture 21 In-class Exercises - Q2 💥

Given the non-linear trends in stock prices, does it make sense to calculate the correlation, \(R_{xy}\), between pairs of stocks?

Calculating Average Rate of Return for Each Stock

Geomtric Mean and Arithmetic Mean

If the Stocks are relatively stable, these values will be similar.

The geometric mean is more reliable because of the compounded interest.

The table below shows the geometric means of three of the stocks:

💥 Lecture 21 In-class Exercises - Q3-Q4 💥

• Use the geometric.mean command from the psych package.
• Round answer to two decimal places.

Calculating the Mean Rate of Return of a Portfolio

• A primary concern when investing is “not putting all of your eggs in one basket”.

• In other words, it is important to diversify your portfolio by investing in multiple stocks so that you have some protection if one stock crashes.

• We can calculate the rate of return of a portfolio

• To do this we calculate a weighted average of the individual stock geometric means:

  • \(W_{1}\times Geo.Mean_{1} + W_{2}\times Geo.Mean_{2}\)
  • \(W_{1}\) and \(W_{2}\) are the percentage of the portfolio invested in each stock.

• In our simple examples, we will look at 2 stock portfolios, but the same principles apply to larger portfolios.

• This weighted average the stock portfolio is referred to as it’s Expected Value.

Average Rates of Return of Portfolio Options

Example: Calculate the average rate of return of a portfolio where 80% is invested in Apple (AAPL) and 20% is invested in Netflix (NFLX).

w1 <- .8
w2 <- .2

gm_aapl <- geometric.mean(adj_close$AAPL) 
gm_nflx <- geometric.mean(adj_close$NFLX) 

w1*gm_aapl + w2*gm_nflx

[1] 193.6352

💥 Lecture 21 In-class Exercises - Q5-Q6 💥

Round answer to both questions below to two decimal places.

Question 5: What is the average rate of return of a portfolio with 60% investment in Amazon (AMZN) and 40% investment in Microsoft (MSFT)?

Question 6: What is average rate of return of a portfolio with 70% investment in Amazon (AMZN) and 30% investment in Apple (AAPL)?

Volatility of a single Stock Rate of Return

Volatility is a measure of variability and risk associated with a stocks rate of return over time.

Volatlity for a single stock: \(Volatility = SD \times \sqrt{T} = \sqrt{VAR \times T}\)

• T = number of time periods.

• In these two years, there were 501 trading days, T = 501.

  • Examine imported data to verify T.

💥 Lecture 21 In-class Exercises - Q7 💥

Question 7: What is the volatility of the Microsoft (MSFT) stock?

Round answer to two decimal places.

Volatlity for a single stock:

\[ Volatility = SD \times \sqrt{T} = \sqrt{VAR \times T} \]

Variance of a Two Stock Portfolio

Recall our portfolio where 80% is invested in Apple and 20% is invested in Netflix

Calculating the volatility of a portfolio little more complex, but it is easier if we break it down into steps.

Step 1: Calculate variance of portfolio of stocks 1 and 2, which is the sum of three parts:

Part 1: \(W_{1}^2 \times Variance_{1}\)
Part 2: \(W_{2}^2 \times Variance_{2}\)
Part 3: \(2 \times W_{1} \times W_{2} \times COV_{1,2}\)

Portfolio Variance = Part 1 + Part 2 + Part 3

w1 <- .8
w2 <- .2
var1 <-  var(adj_close$AAPL)                                  # Var 1
var2 <- var(adj_close$NFLX)                                   # Var 2
cov12 <- cov(adj_close$AAPL, adj_close$NFLX)                     # Cov 1 & 2
(portfolio_var <- w1^2*var1 + w2^2*var2 + 2*w1*w2*cov12)   # Part 1 + Part 2 + Part 3

[1] 955.3744

Volatility of Two Stock Portfolio

Step 1: Calculate variance of portfolio of stocks 1 and 2, which is the sum of three parts:

Part 1: \(W_{1}^2 \times Variance_{1}\)
Part 2: \(W_{2}^2 \times Variance_{2}\)
Part 3: \(2 \times W_{1} \times W_{2} \times COV_{1,2}\)

Portfolio Variance = Part 1 + Part 2 + Part 3

w1 <- .8
w2 <- .2
var1 <-  var(adj_close$AAPL)                                  # Var 1
var2 <- var(adj_close$NFLX)                                   # Var 2
cov12 <- cov(adj_close$AAPL, adj_close$NFLX)                     # Cov 1 & 2
(portfolio_var <- w1^2*var1 + w2^2*var2 + 2*w1*w2*cov12)   # Part 1 + Part 2 + Part 3

[1] 955.3744

Step 2: Calculate Volatility from Variance

\[ Volatlity = SD \times \sqrt{T} = \sqrt{VAR \times T}\]

(portfolio_volatility <- sqrt(portfolio_var*501)) 

[1] 691.84

💥 Lecture 21 In-class Exercises - Q8-Q9 💥

Round answer to both questions below to two decimal places.

Question 8: What is the volatility of a portfolio with 60% investment in Amazon and 40% investment in Microsoft?

Question 9: What is the volatility of a portfolio with 70% investment in Amazon and 30% investment in Apple?

Understanding These Results Visually

Key Points from Today

• Stock portfolios are linear combinations of stocks

• The geometric mean is average rate of return of a single stock

  • Also referred to as the expected value of the stock.

• The average rate of return of a portfolio is the weighted average of the individual stock means.

  • Also referred to as the expected value of the stock.

• Volatility of a stock or a portfolio is \(SD \times \sqrt{T}\)

• To find volatility of a portfolio, first find the variance

  • Portfolio \(Volatility = \sqrt{VAR \times T}\)

• We will review these concepts and calculations after Quiz 2