Lecture 21 - Introduction to Portfolio Management

Housekeeping

• Today’s plan 📋

  • Comments and Questions from Engagement Questions or about R

  • Upcoming Dates

  • Review Questions

  • Comparing Portfolios of Stocks

    • Profitability of sne 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

Review: R and RStudio 🪄

• Review: You have two options to facilitate your introduction to R and RStudio:

  • Option 1: Create Posit Cloud account and download and install R and RStudio on your laptop.
  • Option 2: Start with free Posit Cloud account and use that and later transition to using R/Rstudio on your laptop.

• If you are comfortable with coding: Start with Option 1, but still sign up for Posit Cloud account.

  • We will use Posit Cloud for Quizzes.

• If you are nervous about coding: Choose Option 2.

• For both options: I can help with download/install issues during office hours.

• What I do: I maintain a Posit Cloud account for helping students but I do most of my work on my laptop.

• NOTE: We will use R and RStudio in class during MOST lectures

  • You can use either Posit Cloud or your laptop.

Upcoming Dates

• HW 7 is due 11/8 (Grace period ends 11/10)

  • Demo videos are posted on Blackboard

• Test 2 is on November 14th and will include material up through Lecture 20

  • Some Practice Questions will be posted before class on 11/9

  • Updates to Practice Questions and some demo videos will be posted by this weekend.

  • Zoom Review on Monday 11/13 at 7:00 PM

• Today:

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

💥 Lecture 21 In-class Exercises - Q1 💥

Examining Trends in these Four Stocks

Calculating Average Rate of Return for Each Stock

Calculating the 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 - Q2 and Q3 💥

Question 2. What is the geometric mean for the Amazon (AMZN) 21-22 stock data?

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

Question 3. Which of these four stocks shows the largest disparity between the geometric and arithmetic mean?

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 21-22 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_st$AAPL) 
gm_nflx <- geometric.mean(adj_st$NFLX) 

w1*gm_aapl + w2*gm_nflx

[1] 194.1367

💥 Lecture 21 In-class Exercises - Q4 and Q5 💥

Round answer to both questions below to two decimal places.

Question 4: What is the 21-22 average rate of return of a portfolio with 60% investment in Amazon (AMZN) and 40% investment in Microsoft (MSFT)?

Question 5: What is 21-22 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: \(Volatility = SD \times \sqrt{T} = \sqrt{VAR \times T}\)

where T = number of time periods. In 21-22, there were 503 trading days, T = 503.

Variances, standard deviations, and volatilities for each of these stocks:

💥 Lecture 21 In-class Exercises - Q6 💥

Question 6: 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_st$AAPL)                                  # Var 1
var2 <- var(adj_st$NFLX)                                   # Var 2
cov12 <- cov(adj_st$AAPL, adj_st$NFLX)                     # Cov 1 & 2
(portfolio_var <- w1^2*var1 + w2^2*var2 + 2*w1*w2*cov12)   # Part 1 + Part 2 + Part 3

[1] 1032.6

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

[1] 1032.6

Step 2: Calculate Volatility from Variance

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

(portfolio_volatility <- sqrt(portfolio_var*503)) 

[1] 720.6927

💥 Lecture 21 In-class Exercises - Q7 and Q8 💥

Round answer to both questions below to two decimal places.

Question 7: What is the 21-22 volatility of a portfolio with 60% investment in Amazon and 40% investment in Microsoft?

Question 8: What is the 21-22 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