Instructions

Solve the problems below. Submit your answers in a pdf, docx, or rmd file. Also submit a .R or .rmd file that contains any R code you used to get your solution. You may use the text and other resources to complete this assignment. Please do not work with other students. Your instructor will answer questions submitted by email. This exam is due by midnight, Tuesday, October 1. You may submit an Rmd file (preferred), pdf or Word in Canvas. Your submission should show all the code you used to get your solutions.

Problem 1

If C is an invertible covariance matrix for n random variables, show that C-1 is symmetric (in other words, C-1 equals its transpose, (𝐶−1)𝑇).

Covariance Matrix Symmetry: C = CT

Property of the Identity Matrix: CC-1 = I (Where I is the identity matrix.)

Transpose of Both Sides: (CC-1)T=IT

Since IT = I: (CC-1)T = I

Transpose of a Product: (CC-1)T = (C-1)TCT

Simplify: (C-1)TC = I

This shows that (C-1)T behaves as a left-inverse of C. Since C is invertible, the left-inverse and the right-inverse are the same.

Therefore: (C-1)T = C-1

Problem 2

Consider three securities with the following parameters:

\[ C = \begin{bmatrix} 0.01 & 0 & 0 \\ 0 & 0.02 & 0.02 \\ 0 & 0.02 & 0.04 \end{bmatrix} 𝝁 = \begin{bmatrix} 0.1 \\ 0.2 \\ 0.3 \end{bmatrix} \]

Find the vectors a and b described in Corollary 4.7 in the text. Use a and b to compute the portfolio, w, on the minimum variance line corresponding to m = 20%. Demonstrate that this portfolio is on the minimum variance line.

## Vector a:
##      [,1]
## [1,]   -4
## [2,]   -2
## [3,]    6
## Vector b:
##      [,1]
## [1,]  1.2
## [2,]  0.6
## [3,] -0.8
## Portfolio weights w:
##      [,1]
## [1,]  0.4
## [2,]  0.2
## [3,]  0.4
## Expected portfolio return: 0.2
## Portfolio variance: 0.012

Problem 3

Using the data in problem 2, plot the minimum variance line (MVL) in the (w1, w2) plane (in R). Consider two portfolios on the MVL with μ = 10% and μ = 20%. Find the variances of, as well as the covariance between, their returns.

## Variance of portfolio with μ = 10%: 0.008
## Variance of portfolio with μ = 20%: 0.012
## Covariance between the two portfolios: 0.004

Problem 4

Using the results from problem 3, plot the MVL in the (σ, μ) plane.

Problem 5

Using the data from problem 2, find the weights and expected return (μ) of a portfolio on the MVL with σ2 = 0.007.

## Expected return for σ^2 = 0.007: 0.15
## Portfolio weights w for σ^2 = 0.007:
##      [,1]
## [1,]  0.6
## [2,]  0.3
## [3,]  0.1

Problem 6

Determine the MVL for a portfolio consisting of the tech giants: Amazon (AMZN), Apple (AAPL), Facebook (META), Google (GOOG) and Microsoft (MSFT). Consider the time period from 1/1/2020 to now (the day you download the data). Download daily closing prices and convert to rates of return (see Lecture 4 Examples.Rmd)) for an example of how to do this.)

## [1] "AMZN" "AAPL" "META" "GOOG" "MSFT"
## Minimum Variance Portfolio Weights:
##                      [,1]
## AMZN.Adjusted  0.15440983
## AAPL.Adjusted  0.31619266
## META.Adjusted -0.08001283
## GOOG.Adjusted  0.28430918
## MSFT.Adjusted  0.32510117
## Expected Return of Minimum Variance Portfolio:  0.0008216553
## Risk (Standard Deviation) of Minimum Variance Portfolio:  0.01795854

a. Compute the vector of annualized expected returns, μ. Compute the annualized covariance matrix, C.

## Annualized Expected Returns (μ):
## AMZN.Adjusted AAPL.Adjusted META.Adjusted GOOG.Adjusted MSFT.Adjusted 
##     0.1425163     0.2455105     0.2126802     0.1894059     0.2171322
## 
## Annualized Covariance Matrix (C):
##               AMZN.Adjusted AAPL.Adjusted META.Adjusted GOOG.Adjusted
## AMZN.Adjusted    0.13163543    0.07022385    0.10172725    0.07769160
## AAPL.Adjusted    0.07022385    0.10390720    0.08374169    0.06967186
## META.Adjusted    0.10172725    0.08374169    0.21363695    0.09674062
## GOOG.Adjusted    0.07769160    0.06967186    0.09674062    0.10633616
## MSFT.Adjusted    0.07626318    0.07525774    0.08820492    0.07614348
##               MSFT.Adjusted
## AMZN.Adjusted    0.07626318
## AAPL.Adjusted    0.07525774
## META.Adjusted    0.08820492
## GOOG.Adjusted    0.07614348
## MSFT.Adjusted    0.09569259

b. Compute the weights, mean, and standard deviation of the minimum variance portfolio.

## Minimum Variance Portfolio Weights:
##                      [,1]
## AMZN.Adjusted  0.15440983
## AAPL.Adjusted  0.31619266
## META.Adjusted -0.08001283
## GOOG.Adjusted  0.28430918
## MSFT.Adjusted  0.32510117
## 
## Expected Return (Mean) of Minimum Variance Portfolio:  0.2070571
## Standard Deviation (Risk) of Minimum Variance Portfolio:  0.2850829

c. Plot the efficient frontier (hint: compute a sequence of portfolios on the MVL for a sequence of rates of return beginning with the return you got in part b. Then, compute the corresponding sequence of σ’s corresponding to the returns.)

d. Compute the weights of the market portfolio, using a risk-free rate, R, of 4%.

## Market Portfolio Weights:
##                      [,1]
## AMZN.Adjusted -0.40780217
## AAPL.Adjusted  0.79098570
## META.Adjusted  0.03511257
## GOOG.Adjusted  0.05195358
## MSFT.Adjusted  0.52975031
## 
## Expected Return of Market Portfolio:  0.2684107
## Standard Deviation (Risk) of Market Portfolio:  0.3333473

e. Compute the weights of a portfolio on the capital market line that returns 50%. Determine the proportion of each stock and the risk-free bond that would be in this portfolio. Also determine the risk (σ) you would need to take if you invested in this portfolio.

## Proportion of Market Portfolio:  2.013916
## Proportion of Risk-Free Bond:  -1.013916
## 
## Weights of Each Stock in the Portfolio:
##                      [,1]
## AMZN.Adjusted -0.82127932
## AAPL.Adjusted  1.59297878
## META.Adjusted  0.07071378
## GOOG.Adjusted  0.10463014
## MSFT.Adjusted  1.06687265
## 
## Risk (σ) of the Portfolio:  0.6713334