MAS 261 - Lecture 21

Introduction to Portfolio Management

Author

Penelope Pooler Eisenbies

Published

November 4, 2024

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

Recall that in lecture 20 we discussed $R_{xy}$, the correlation coefficient.

The matrix shown here shows a correlation matrix for four stocks based on their 2021 - 2022 daily adjusted closing values.

A correlation matrix shows the pairwise correlation between each pair of stocks in the dataset.

Which other stock is most strongly positively correlated with Apple (AAPL)?

	AAPL	MSFT	AMZN	NFLX
AAPL	1.00	0.94	0.67	0.75
MSFT	0.94	1.00	0.64	0.77
AMZN	0.67	0.64	1.00	0.50
NFLX	0.75	0.77	0.50	1.00

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

For each of these four companies we have two year of data.
We want to know the average rate of return.
We could calculate the arithmetic mean of the adjusted close, but the conventional wisdom is that this is not ideal.
Instead, we calculate the geometric mean
The psych package in R has the geomtric.mean command to do this calculation.

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:

Stock	Arithmetic_Mean	Geo_Mean
AAPL	162.1884	161.2011
MSFT	287.3443	284.4279
AMZN	123.7405	NA
NFLX	337.3382	323.3717

Lecture 21 In-class Exercises - Q3-Q4

Question 3. What is the geometric mean for the Amazon (AMZN) stock data?

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

Question 4. 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 average rate of return of a portfolio where 80% is invested in Apple (AAPL) and 20% is invested in Netflix (NFLX).

Code

```{r echo=T}
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.

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

Stock	Variance	Std_Dev	Volatility
AAPL	318.3906	17.84350	399.3917
MSFT	1713.2753	NA	NA
AMZN	469.2115	21.66129	484.8453
NFLX	8762.2757	93.60703	2095.2089

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:

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

Stock	Variance	Std_Dev	Volatility
AAPL	318.3906	17.84350	399.3917
MSFT	1713.2753	NA	NA
AMZN	469.2115	21.66129	484.8453
NFLX	8762.2757	93.60703	2095.2089

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

Code

```{r echo=T}
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] 955.3743

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

Code

```{r echo=T}
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] 955.3743

Step 2: Calculate Volatility from Variance

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

Code

```{r echo=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

To submit an Engagement Question or Comment about material from Lecture 21: Submit it by midnight today (day of lecture).

--- title: "MAS 261 - Lecture 21" subtitle: "Introduction to Portfolio Management" author: "Penelope Pooler Eisenbies" date: last-modified toc: true toc-depth: 3 toc-location: left toc-title: "Table of Contents" toc-expand: 1 format: html: code-line-numbers: true code-fold: true code-tools: true execute: echo: fenced --- ## Housekeeping ```{r setup, echo=FALSE, warning=F, message=F, include=F} #| include: false # this line specifies options for default options for all R Chunks knitr::opts_chunk$set(echo=F) # suppress scientific notation options(scipen=100) # install helper package that loads and installs other packages, if needed if (!require("pacman")) install.packages("pacman", repos = "http://lib.stat.cmu.edu/R/CRAN/") # install and load required packages pacman::p_load(pacman,tidyverse, magrittr, olsrr, shadowtext, mapproj, knitr, kableExtra, countrycode, usdata, maps, RColorBrewer, gridExtra, ggthemes, gt, mosaicData, epiDisplay, vistributions, psych, tidyquant, dygraphs) # verify packages # p_loaded() ``` - 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](https://posit.cloud/plans/free){target="_blank"} - 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](https://penelope2040.quarto.pub/mas-261/#installing-r-and-rstudio){target="_blank"} ## ### Lecture 21 In-class Exercises - Q1 :::::: columns ::: {.column width="48%"} Recall that in lecture 20 we discussed $R_{xy}$, the correlation coefficient. The matrix shown here shows a correlation matrix for four stocks based on their 2021 - 2022 daily adjusted closing values. A correlation matrix shows the pairwise correlation between each pair of stocks in the dataset. **Which other stock is most strongly positively correlated with Apple (AAPL)?** ::: ::: {.column width="4%"} ::: ::: {.column width="48%"} ```{r, include=F} #|label: import stocks getSymbols("AAPL", from = "2022-01-01", to = "2024-01-01") getSymbols("MSFT", from = "2022-01-01", to = "2024-01-01") getSymbols("AMZN", from = "2022-01-01", to = "2024-01-01") getSymbols("NFLX", from = "2022-01-01", to = "2024-01-01") ``` ```{r} #|label: correlation matrix and dygraph adj_cl <- merge(AAPL$AAPL.Adjusted, MSFT$MSFT.Adjusted, AMZN$AMZN.Adjusted, NFLX$NFLX.Adjusted) names(adj_cl) <- c("AAPL.adj", "MSFT.adj", "AMZN.adj", "NFLX.adj") dy <- dygraph(adj_cl, main="2022 Daily Adj. Closing Stock Values") |> dyShading(from = "2022-01-01", to = "2024-01-01", color = "beige") |> dySeries("AAPL.adj", label="AAPL", color= "purple") |> dySeries("MSFT.adj", label="MSFT", color= "darkturquoise") |> dySeries("AMZN.adj", label="AMZN", color= "orange") |> dySeries("NFLX.adj", label="NFLX", color= "tomato") |> dyRangeSelector() |> dyAxis("y", label = "Adjusted Close", drawGrid = FALSE) |> dyAxis("x", label = "Date", drawGrid = FALSE) adj_close <- adj_cl |> fortify.zoo() |> as_tibble(.name_repair = "minimal") |> rename("date" = "Index", "AAPL" = "AAPL.adj", "MSFT" = "MSFT.adj", "AMZN" = "AMZN.adj", "NFLX" = "NFLX.adj") cor(adj_close[,2:5]) |> round(2) |> kable() ``` ::: :::::: ## ### Examining Trends in these Four Stocks ```{r} dy ``` ## ### 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 :::::: columns ::: {.column width="48%"} - For each of these four companies we have two year of data. - We want to know the average rate of return. - We could calculate the arithmetic mean of the adjusted close, but the conventional wisdom is that this is not ideal. - Instead, we calculate the [geometric mean](https://www.investopedia.com/articles/investing/071113/breaking-down-geometric-mean.asp) - **The `psych` package in R has the `geomtric.mean` command to do this calculation.** ::: ::: {.column width="4%"} ::: ::: {.column width="48%"} ![](img/geometric_mean.png) ::: :::::: ## ### 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:** ```{r} Arithmetic_Mean <- c(mean(adj_close$AAPL), mean(adj_close$MSFT), mean(adj_close$AMZN), mean(adj_close$NFLX)) Geo_Mean <- c(geometric.mean(adj_close$AAPL), geometric.mean(adj_close$MSFT), NA, geometric.mean(adj_close$NFLX)) Stock <- c("AAPL", "MSFT", "AMZN", "NFLX") tibble(Stock, Arithmetic_Mean, Geo_Mean) |> kable() ``` ## ### Lecture 21 In-class Exercises - Q3-Q4 ::: fragment **Question 3. What is the geometric mean for the Amazon (AMZN) stock data?** ::: - Use the `geometric.mean` command from the `psych` package. - Round answer to two decimal places. ::: fragment **Question 4. 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 average rate of return of a portfolio where 80% is invested in Apple (AAPL) and 20% is invested in Netflix (NFLX). ```{r echo=T} w1 <- .8 w2 <- .2 gm_aapl <- geometric.mean(adj_close$AAPL) gm_nflx <- geometric.mean(adj_close$NFLX) w1*gm_aapl + w2*gm_nflx ``` ## ### 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. ::: fragment **Variances, standard deviations, and volatilities for each of these stocks:** ```{r} Variance <- c(var(adj_close$AAPL), var(adj_close$MSFT), var(adj_close$AMZN), var(adj_close$NFLX)) Std_Dev <- sqrt(Variance) Volatility <- Std_Dev*sqrt(501) SV <- tibble(Stock, Variance, Std_Dev, Volatility) SV[2,3:4] <- NA SV |> kable() ``` ::: ## ### 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: $$ Volatility = SD \times \sqrt{T} = \sqrt{VAR \times T} $$ ```{r} SV |> kable() ``` ## ### 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 ```{r echo=T} 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 ``` ## ### 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 ```{r echo=T} 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 ``` **Step 2: Calculate Volatility from Variance** $$ Volatlity = SD \times \sqrt{T} = \sqrt{VAR \times T}$$ ```{r echo=T} (portfolio_volatility <- sqrt(portfolio_var*501)) ``` ## ### 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 ```{r} adj_close <- adj_close |> mutate(PTFL1 = .8*AAPL + .2*NFLX, PTFL2 = .6*AMZN + .4*MSFT, PTFL3 = .7*AMZN + .3*AAPL) ptfl_xts <- xts(x=adj_close[,6:8], order.by=adj_close$date) (dy_ptfl <- dygraph(ptfl_xts, main="Portfolio Comparison") |> dyShading(from = "2022-01-01", to = "2024-01-01", color = "beige") |> dySeries("PTFL1", label="80%AAPL+20%NFLX", color= "purple") |> dySeries("PTFL2", label="60%AMZN+40%MSFT", color= "darkturquoise") |> dySeries("PTFL3", label="70%AMZN+30%AAPL", color= "tomato") |> dyRangeSelector() |> dyAxis("y", label = "Adjusted Close", drawGrid = FALSE) |> dyAxis("x", label = "Date", drawGrid = FALSE)) ``` ## ### 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 ::: fragment **To submit an Engagement Question or Comment about material from Lecture 21:** Submit it by midnight today (day of lecture). :::