TUTORIAL 3– WEEK 4
ECON3107/ECON5106 – Economics of Finance
ANSWERS
1
. (i) The opportunity set describes what combinations of GA and BA can be obtained for 1PA.
Let
Q
{
states*securities
}
be the payment matrix of the two securities:
Q:
Bond
Stock
Good Weather
20
50
Bad Weather
20
25
Let
p
S
{
1*securities
}
be a vector of security prices:
ps:
Bond
Stock
18
30
The matrix of the value relatives for the two securities will indicate their coordinates in the oppor
tunity set.
VR:
Bond
Stock
Good Weather
20/18
50/30
Bad Weather
20/18
25/30
In plain words, 1PA can buy 1/18th of a bond which yields 20/18 GA and 20/18BA. The numbers
1.11 (GA) and 1.11 (BA) are the coordinates for the bond B in the opportunity set.
1PA buys
1/30th of a stock which yields 50/30GA and 25/30BA. The numbers 1.67GA and 0.83BA are the
coordinates for the stock S in the opportunity set.
The vector of the atomic prices
p
atom
can be found as
p
atom
=
p
S
·
Q
−
1
=
18
30
20
50
20
25
−
1
=
0
.
3
0
.
6
.
To summarize, 1GA costs 0.3PA, and 1BA costs 0.6PA. In other words, 3.33GA cost 1PA and
1.67BA cost 1PA. These numbers indicate the coordinates for the points where the opportunity set
cuts the GA and BA axes.
(ii) Let
c
G
B
and
c
B
B
denote the payments made by the bond in the good and bad states. Similarly,
let
c
G
S
and
c
B
S
denote the payments made by the stock in the good and bad states. Finally, let
p
B
and
p
S
denote the prices of the bond and stock. Assuming that the stock pays more in the good
state than in the bad state, then we will observe positive prices for both the atomic securities if the
following two inequalities are both satisfied:
c
G
B
p
B
<
c
G
S
p
S
,
c
B
B
p
B
>
c
B
S
p
S
.
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 '11
 valentyn
 Economics, Arbitrage, Economics of Finance

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