SOLUTIONS TO
Problem Set 2
Introduction to Econometrics
prepared by
Prof. Marcelo J. Moreira and Seyhan E Arkonac, PhD
for all sections
Spring 2010
1.
Suppose you have some money to invest – for simplicity, $1 – and you are planning to put a
fraction w into a stock market mutual fund and the rest, 1 – w, into a bond mutual fund.
Suppose that $1 invested in a stock fund yields R
s
after one year and that $1 invested in a
bond fund yields R
b
, that R
s
is random with mean
s
and variance
2
, and that R
b
is random
with mean
b
and the same variance
2
.
The covariance between R
s
and R
b
is
sb
. If you
place a fraction w of your money in the stock fund and the rest, 1 – w, in the bond fund, then
the return on your investment is
R = wR
s
+ (1 – w)R
b
a)
Provide an expression for the expected return, E(R), in terms of w,
s
, and
b
.
E
(
R
) =
E
[
wR
s
+
(1 –
w
)
R
b
] =
wE
(
R
s
)
+
(1 –
w
)
E
(
R
b
) =
w
s
+ (1 –
w
)
b
b)
Provide an expression for the variance of the return, var(R), in terms of w,
2
, and
sb
.
var(
R
) = var[
wR
s
+
(1 –
w
)
R
b
]
=
w
2
var(
R
s
) + (1 –
w
)
2
var(
R
b
) + 2
w
(1 –
w
)cov(
R
s
,
R
b
)
=
w
2
2
+ (1 –
w
)
2
2
+ 2
w
(1 –
w
)
sb
c)
Suppose that
sb
= 0.
Compute the variance of two different portfolios: one with all the
money in stocks, and one with 50% in stocks and 50% in bonds.
Which (if either) has the
lower variance?
Provide an intuitive explanation.
Because the covariance is zero, var(
R
) =
w
2
2
+ (1 –
w
)
2
2
.
For the 50/50 portfolio,
w
= ½ so
var(
R
) = (½)
2
2
+ (1 – ½)
2
2
= ½
2
.
For the allstocks portfolio,
w
= 1 and var(
R
) =
2
.
The
50/50 portfolio has the lower variance.
The variance of the 50/50 portfolio is less than then all
stocks portfolio because in the 50/50 portfolio, a high return for one asset is offset, on average,
by an unrelated (sometimes high, sometimes low) return for the other asset, so the variance of the
combined return is smaller.
This is the fundamental idea of diversification – reducing the risk of
a portfolio by diversifying among more than one asset.
d)
Now suppose instead that
sb
is the greatest it can be.
In this case, what is var(R) as a
function of w?
Recompute the variances for the two portfolios in (c) (100% stocks, and
50/50 stocks and bonds).
How do the two variances compare in this case?
Explain.
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View Full DocumentThe covariance satisfies cov(
R
s
,
R
b
)
var(
)var(
)
sb
RR
; because var(
R
s
) = var(
R
b
) =
2
,
sb
2
,
so the greatest
sb
can be is
2
.
Substituting this into the expression from part (b) yields,
var(
R
) =
w
2
2
+ (1 –
w
)
2
2
+ 2
w
(1 –
w
)
2
= [
w
2
+ (1 – 2
w
+
w
2
) – (2
w
2
– 2w)]
2
=
2
.
When cov(
R
s
,
R
b
) is at its greatest, then the variance of the return on the portfolio for any
w
is the
same as the variance of the return on stocks (or bonds, they have the same variance in this
example).
In particular, the 50/50 and 100/0 portfolios have the same variance,
2
.
Thus there is
no variance reduction from diversification.
This makes sense:
in this case the assets covary
perfectly, so when stock returns are high, bonds returns are high and vice versa.
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 Fall '11
 SeyhanArkonac
 Econometrics, Linear Regression, Regression Analysis, Standard Deviation, Variance, Mean squared error

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