Published on 21 Nov 2012

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Actsc 372 - Assignment 1 - Fall 2012

Due on Friday September 28 by 5pm in my ofﬁce (M3 3112).

1. Suppose the economy has N= 2 assets and let ρdenote the correlation between the two

assets.

(a) Prove that the relationship between µpand σpis linear when ρ= 1.

(b) Derive the weights w1and w2that produce the minimum variance (risk) portfolio

(for any |ρ| ≤ 1).

(c) Prove that when ρ=−1, it is possible to ﬁnd weights w1and w2such that σp= 0

and provide expressions for these weights.

2. For this question, you will use the spreadsheet ’a1data.xlsx’ available on the course web-

page. The data provided in the spreadsheet is an excerpt of Table 10.1 of the course

textbook. More speciﬁcally, you are provided with the yearly returns data for Canadian

stocks, long bonds and U.S. stocks over the period 2000-2009. Given a set of weights

(wC, wB, wU), the spreadsheet outputs the portfolio’s expected return and return vari-

ance (Note that the subscript ’C’ stands for Canadian stocks, ’B’ for long bonds and ’U’

for U.S. stocks respectively). You should review the formulas in the spreadsheet to see

how the outputs are calculated.

(a) Using the formula seen in class, determine the weights (wC, wB, wU)that produce

the minimum variance portfolio.

(b) Use Solver in Excel to ﬁnd the minimum variance portfolio. For this question, you

must provide either a screenshot of the Solver dialog or provide the cell references

in the Solver dialog (objective, variables and constraints).

(c) If you desire a portfolio that yields an expected return of 9%, using Solver, compute

the portfolio risk and provide the corresponding weights. (As in the previous part,

you must provide either the cell references in the Solver dialog or a screenshot.)

(d) If you now desire a portfolio that yields an expected return of 12%, using Solver,

compute the portfolio risk and compare it to that in the previous part. (As in the

previous part, you must provide either the cell references in the Solver dialog or a

screenshot.)

(e) i. Using any software, compute z∗

(using the expression provided in class) and

show that (z∗)Te= 0. (This part requires some matrix algebra.)

ii. Recall that the optimal weights are given be w∗=wM I N +τz∗

. Provide an

expression for µw∗

in terms of the risk tolerance τ.

iii. Determine the risk tolerance level τfor a portfolio with 9% expected return.

What is the risk tolerance level if the expected return is instead 12%? Compare

your results.

(f) Suppose now that there exists a risk-free asset that yields a return of 1%. If you

desire a portfolio that yields an expected return of 9%, using Solver, compute the

portfolio risk and provide the corresponding weights in the risk-free asset, Cana-

dian stocks, long bonds and U.S. stocks. Modify the spreadsheet (’a1data.xlsx’) as

required and describe your method of ﬁnding the optimal weights. You will be

asked to submit this spreadsheet via a dropbox on the course webpage.

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