# Portfolio Theory and its Applications

You will find on canvas a .csv file, labelled with your SID, containing the annualised daily returns on 20 assets, *r _{i,t}*, (

*i*= 1

*,…,*20) together with the returns on two factor portfolios,

*f*

_{1,t},

*f*

_{2,t }and the constant risk free rate,

*r*. For the purposes of this assignment, you may assume that the returns are generated via the factor model

_{f}(1)

where Cov{*f*_{1,t}*,f*_{2,t}} = 0 and Cov= Cov= Cov

0 for *i,j *= 1*,*2*,…,*20.

Using your unique set of returns, you are to compute the required quantities. The cell references for the location of each answer are provided in square brackets at the end of each question.

- The value of
*α*for each stock [C3:V3]. - The value of
*β*_{1 }for each stock [C4:V4] and*β*_{2 }for each stock [C5:V5]. - The idiosyncratic risk () for each stock [C6:V6].
- The expected return for each stock using your answers for 1) and 2). You may assume that the sample mean is an appropriate value for E{
*f*},_{j,t}*j*= 1*,*2 [C7:V7]. - The
*R*^{2 }for each regression [C8:V8]. - The covariance matrix for all 20 assets using the factor model (1) and its assumptions [C10:V29].
- The allocation vector for the global minimum variance portfolio. You are to use your answer to 6) as an input [C31:V31].
- The allocation vector for the tangency portfolio. You are to use your answer to 4) and 6) as inputs [C32:V32].
- The allocation vector for the tangency portfolio if the minimum allowed weight in each asset is 1% [C33:V33].
- Assuming the tangency portfolio computed in 8) is the market portfolio, compute the CAPM beta (
*β*) for each asset [C34:V34]._{M} - The t-statistic for a regression coefficient estimate,
*β*^{ˆ}, is given by

where *s.e*(*β*^{ˆ}) is the standard error in the estimate and *β*_{0 }is a constant value for which we are testing the hypothesis

*H*0 : *β**M *= *β*0

*H*_{1 }: *β _{M }*6=

*β*

_{0}

Provide the t-statistic relevant to the hypothesis that estimate of *β _{M }*is statistically different from 1 [C35:V35].

# Submission Requirements

You are to compute these quantities and place your results in the solution template provided which is in the form of a .xlsx excel file. The location for your answers is shaded in grey and cell references for the answer are provided in the question. All values entered should be reported to at least 6 decimal places. After completing your assignment, you are to save the solutions template as a .xlsx (excel) file with the title “Solution SID”.

For example, if you SID was “123456789”, then you will save a file named “Solution 123456789.xlsx”. There must be no spaces. The underscore must be used to separate the word “Solution” from your SID. You are to then upload this file to canvas via the submission link. It is essential that these instructions are followed as the assignments will be computer marked.

Your answers will be compared to the correct answers, both rounded to 4 decimal places. The only exception is question 9) which will be marked correct if it is sufficiently close my computed answer. If all solutions provided for a question are correct, 2 marks will be awarded. If some of the answers are correct then 1 mark will be awarded. If a mistake is made in a previous part and carried through to a later question, this will be accounted for. If all answers are incorrect then no marks will be awarded.

Attention to detail and the ability to follow the provided instructions is critical. All students will receive a different set of returns so you must use your individual return series otherwise all your answers will be deemed incorrect. If I have to manually go through your assignment because you have not followed instructions appropriately this will incur a penalty of 20% of maximum marks.