This exercise is based upon an equity example
in Harry Markowitz’s (1959) book Portfolio Selection.
Suppose today is January 1, 1955. Measure time t in
years and define:
[1.45]
Each accumulated value represents the value at
time 1 of an investment worth 1 USD at time 0 in the
indicated stock. Accumulated values include price changes and
dividends. Consider a portfolio with holdings
= (10,000 5,000 –1,000 2,000
–5,000 1,000 6,000)
[1.46]
Based upon data provided by Markowitz, we
construct a conditional covariance matrix
for
:
[1.47]
Calculate the portfolio’s 1-year 90% USD VaR
according to the following steps:
a. Value the vector
.
(Hint: Based upon how the problem has been presented, the
answer is trivial.)
b. Using the formula
=
,value
.
c. Specify a portfolio mapping that defines
as a linear polynomial of
.
d. Draw a schematic for your portfolio
mapping.
e. Determine the conditional standard
deviation
of
using [1.10].
f. Assume
is normally distributed with conditional mean
=
and conditional standard deviation obtained in part (e).
Calculate the .10-quantile of
with the formula
[1.48]
g. Calculate the portfolio’s 1-year 90% USD
VaR as
_small.jpg)
= (10,000 5,000 –1,000 2,000
–5,000 1,000 6,000)
for
:
.
(Hint: Based upon how the problem has been presented, the
answer is trivial.) 
=
as a linear polynomial of
of
=
has value 1 USD, so:
.gif){kind=small}
= USD 3656.
