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
(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
(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
(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.