Solution to GMS Stock Hedging1
Kate Torelli, a security analyst for Lion Fund, has identified a gold mining stock
(ticker symbol GMS) as a particularly attractive investment. Torelli believes that
the company has invested wisely in new mining equipment. Furthermore, the
company has recently purchased mining rights on land that has high potential
for successful gold extraction. Torelli notes that gold has underperformed the
stock market in the last decade and believes that the time is ripe for a large
increase in gold prices. In addition, she reasons that conditions in the global
monetary system make it likely that investors may once again turn to gold as a
safe haven in which to park assets. Finally, supply and demand conditions have
improved to the point where there could be significant upward pressure on gold
prices.
GMS is a highly leveraged company, so it is quite a risky investment by itself.
Torelli is mindful of a passage from the annual report of a competitor, Baupost,
which has an extraordinarily successful investment record. "Baupost has
managed a decade of consistently profitable results despite, and perhaps in some
respect due to, consistent emphasis on the avoidance of downside risk. We have
frequently carried both high cash balances and costly market hedges. Our results
are particularly satisfying when considered in the light of this sustained risk
aversion." She would therefore like to hedge the stock purchase — that is, reduce
the risk of an investment in GMS stock.
Currently GMS is trading at $100 per share. Torelli has constructed seven
scenarios for the price of GMS stock one month from now. These scenarios and
corresponding probabilities are shown in Table 1.
Scen. 1
Scen. 2
Scen. 3
Scen. 4
Scen. 5
Probability
0.05
0.10
0.20
0.30
0.20
GMS stock price
150
130
110
100
90
Table 1: Scenarios and Probabilities for GMS Stock in One Month
Scen. 6
0.10
80
Scen. 7
0.05
70
To hedge an investment in GMS stock, Torelli can invest in other securities
whose prices tend to move in the direction opposite to that of GMS stock. In
P. 395 in Practical Management Science (2nd ed., Winston and Albright, 2001 Duxbury Press).
Solution by David Juran, 2002.
1
particular, she is considering over-the-counter put options on GMS stock as
potential hedging instruments. The value of a put option increases as the price of
the underlying stock decreases.2 For example, consider a put option with a strike
price of $100 and a time to expiration of one month. This means that the owner of
the put has the right to sell GMS stock at $100 per share one month in the future.
Suppose that the price of GMS falls to $80 at that time. Then the holder of the put
option can exercise the option and receive $20 (= 100 - 80). If the price of GMS
falls to $70, the option would be worth $30 (= 100 - 70). However, if the price of
GMS rises to $100 or more, the option expires worthless.
Torelli called an options trader at a large investment bank for quotes. The prices
for three (European-style) put options are shown in Table 2. Torelli wishes to
invest $10 million in GMS stock and put options.
Put Option A
Put Option B
Strike price
90
100
Option price
$2.20
$6.40
Table 2: Put Option Prices (Today) for GMS Case Study
Put Option C
110
$12.50
For a brief introduction to options see, for example, Cox and Rubinstein (1985), pp.1-8, or
Jarrow and Turnbull (1996), pp. l4-18.
2
B60.2350
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Prof. Juran
Some Preliminaries
In this approach to portfolio optimization (called the scenario approach), the
universe of possible outcomes is considered to be confined to a finite number of
scenarios, as given in Table 1.
Define xi to be the number of thousands of dollars allocated to investment i.
Each investment i has a percent return under each scenario j, which we will
represent with the symbol rij. For example, under scenario 1 GMS stock has a
50% return, so if r1, 1 represents the return of GMS stock under scenario 1, then
r1, 1 = 0.50.
For GMS stock, the return under any scenario j is given by:
r1 j 
S1 j  S0
S0
Where S0 is the initial price of the GMS stock, and S1 is the final price.
For a put option i, the return under any scenario j is given by:
rij 


MAX K i  S1 j ,0  C i
Ci
Where Ki is the strike price and Ci is the cost of the option.
Using these formulas, we can expand Table 1 to include the returns on each
possible investment under each scenario.
Probability
GMS stock price
Return on GMS stock (r1)
Return on Option A (r2)
Return on Option B (r3)
Return on Option C (r4)
Scen. 1
0.05
$150
50%
-100%
-100%
-100%
Scen. 2
0.10
$130
30%
-100%
-100%
-100%
Scen. 3
0.20
$110
10%
-100%
-100%
-100%
Scen. 4
0.30
$100
0%
-100%
-100%
-20%
Scen. 5
0.20
$90
-10%
-100%
56%
60%
Scen. 6
0.10
$80
-20%
355%
213%
140%
Scen. 7
0.05
$70
-30%
809%
369%
220%
Note: We don’t know what the return would be under any scenario unless we
know how much money was invested in each of the four instruments.
B60.2350
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Prof. Juran
The return on the portfolio is represented by the random variable R.
4
R   ri x i
i 1
The portfolio return under any scenario j is given by:
4
R j   rij x i
i 1
(Note that we are using r as a percent and R as thousands of dollars.)
Let Pj represent the probability of scenario j occurring. The expected value of R is
given by:
7
 R   R j Pj
j 1
The standard deviation of the portfolio’s return is given by:
R 
 R
7
j 1
j

  R 2 Pj
Example: Let’s say Kate buys $7 million worth of GMS stock, and $1 million
worth of each put option. This means that
(x1, x2, x3, x4) = (7000, 1000, 1000, 1000).
Under scenario 3, her return would be
r3
4
  ri 3 x i
i 1
 r13 x 1  r23 x 2  r33 x 3  r43 x 4
 0.107000   1.001000   1.001000   1.001000 
 700  1000  1000  1000
 $2300
Using the same procedure, it can be shown that for this particular allocation of
assets, the seven scenarios would have returns as follows:
Scenario
1
2
3
4
5
6
7
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4
Return
500
-900
-2,300
-2,200
-538
5,670
11,878
Prof. Juran
Therefore, the expected return on this particular allocation of assets is calculated as
follows:
R
7
  R j Pj
j 1
 R1 P1  R2 P2  R3 P3  R4 P4  R5 P5  R6 P6  R7 P7
 500 0.05    900 0.1   2300 0.2    2200 0.3    538 0.2   5670 0.1  11878 0.05 
 25    90    460    660    108   567   594 
 132
Finally, to calculate the standard deviation of the returns under this particular
allocation of assets:
R

 R
7
j 1



j

  R 2 Pj
R1  R 2 P1  R2  R 2 P2  R3  R 2 P3  R4  R 2 P4  R5  R 2 P5  R6  R 2 P6  R7  R 2 P7
500   132 2 0.05   900   132 2 0.1   2300   132 2 0.2    2200   132 2 0.3   538   132 2 0.2   5670   132 2 0.1  11878   132 2 0.05
632 2 0.05    768 2 0.1   2168 2 0.2    2068 2 0.3    406 2 0.2   5802 2 0.1  12010 2 0.05 
 19 ,942  59 ,054  940 ,449  1,283,565  32 ,962  3,366 ,307  7 ,211,937
 12 ,914 ,216
 3,594
So, in English, if Kate buys $7 million worth of GMS stock, and $1 million worth
of each put option, then her return on investment in dollar terms is a random
variable with an expected value of about -$132,000 and a standard deviation of
about $3,594,000.
B60.2350
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Prof. Juran
1.
Based on Torelli's scenarios, what is the expected return of GMS stock?
What is the standard deviation of the return of GMS stock?
The expected one-month return is 2%. The standard deviation is 18.33%. Here is
a spreadsheet model for the calculations:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
GMS price
B
C
100
D
=(B5-$B$1)/$B$1
Scenarios for GMS stock in one month
Scenario GMS price
Probability
1
150
0.05
2
130
0.10
3
110
0.20
4
100
0.30
5
90
0.20
6
80
0.10
7
70
0.05
=SUMPRODUCT(D5:D11,C5:C11)
Mean return
Stdev of return
B60.2350
Return
50%
30%
10%
0%
-10%
-20%
-30%
E
F
=(D6-$B$13)^2
SqDev
0.2304
0.0784
0.0064
0.0004
0.0144
0.0484
0.1024
=B13*10000000
0.0200 $
200,000.00
0.1833 $ 1,833,030.28
=B14*10000000
=SQRT(SUMPRODUCT(E5:E11,C5:C11))
6
Prof. Juran
2.
After a cursory examination of the put option prices, Torelli suspects
that a good strategy is to buy one put option A for each share of GMS
stock purchased. What are the mean and standard deviation of return
for this strategy?
Torelli’s strategy will have an expected return of 1.76% with a standard deviation
of 15.59%. In other words, it is less risky than the stock by itself, but also has a
lower expected return.
Here is the spreadsheet model. Note the use of “IF” functions in cells D5:D11 to
incorporate the returns on the put option. This could also be done using the
“MAX” function.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A
GMS price
B
C
D
E
F
G
100
=(B5-$B$1+IF(B5-$B$15>0,0,$B$15-B5)-$B$16)/($B$1+$B$16)
Scenarios for GMS stock in one month
Scenario GMS price
Probability
1
150
0.05
2
130
0.10
3
110
0.20
4
100
0.30
5
90
0.20
6
80
0.10
7
70
0.05
Portfolio
Return
Sqdev
47% 0.202588 =(D5-$B$19)^2
27% 0.064721
8% 0.003447
-2% 0.001532
-12% 0.018765
-12% 0.018765
-12% 0.018765
Put options on GMS stock that expire in one month
Option
A
B
C
Strike price
90
100
110
Option price
$2.20
$6.40
$12.50
=SUMPRODUCT(D5:D11,C5:C11)
Portfolio with one unit of GMS stock and one put A
Mean
0.0176 $ 176,125.24
Stdev
0.1559 $ 1,559,430.28
B60.2350
=SQRT(SUMPRODUCT(E5:E11,C5:C11))
7
Prof. Juran
3.
Assuming that Torelli's goal is to minimize the standard deviation of
the portfolio return, what is the optimal portfolio that invests all $10
million? (For simplicity, assume that fractional numbers of stock shares
and put options can be purchased. Assume that the amounts invested
in each security must be nonnegative. However, the number of options
purchased need not equal the number of shares of stock purchased.)
What are the expected return and standard deviation of return of this
portfolio? How many shares of GMS stock and how many of each put
option does this portfolio correspond to?
Managerial Formulation
Decision Variables
Torelli needs to invest $10 million in some combination of GMS stock and three
types of put options.
Objective
Minimize risk (standard deviation of the portfolio’s return).
Constraints
All $10 million must be invested.
No shorting.
Mathematical Formulation
Decision Variables
The decision variables are four amounts: x1, x2, x3, and x4, representing GMS
stock, Put Option A, Put Option B, and Put Option C, respectively.
Objective
Minimize Z =  R 
 R
7
j 1
j

  R 2 Pj
Constraints
4
x
i 1
i
 10 ,000
x i  0 for all investments i.
B60.2350
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Prof. Juran
Solution Methodology
Here’s the spreadsheet model:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
GMS price
B
C
D
E
F
G
H
I
J
K
L
100
Scenarios for GMS stock in one month Returns from one unit of
Scenario GMS price Probability
GMS
Put A
1
150
0.05
50%
-100%
2
130
0.10
30%
-100%
3
110
0.20
10%
-100%
4
100
0.30
0%
-100%
5
90
0.20
-10%
-100%
6
80
0.10
-20%
355%
7
70
0.05
-30%
809%
=(B11-$B$1)/$B$1
each investment
Put B Put C
-100% -100%
-100% -100%
-100% -100%
-100%
-20%
56%
60%
213% 140%
369% 220%
Portfolio
Return
Sqdev
500
398835
-900
590540
-2300
4702244
-2200
4278551
-538
164808
5670
33663072
11878 144238735
Returns here are in
thousands of dollars
=(H11-$K$19)^2
=(IF($B11>B$15,0,B$15-$B11)-B$16)/B$16
Put options on GMS stock that expire in one month
Option
A
B
C
Strike price
90
100
110
Option price
$2.20
$6.40
$12.50
=SUMPRODUCT($B$20:$E$20,D11:G11)
Investment decision (thousands of dollars spent on each investment)
GMS
Put A
Put B
Put C
Total
7000
1000
1000
1000 10000
=
Budget
10000
Return from portfolio ($1000)
Mean
-132
Stdev
3594
Units of investments purchased (shares for GMS, number of puts for options)
GMS
Put A
Put B
Put C
70000
454545
156250
80000
The decision variables are in B20:E20.
The objective function is in K20.
B60.2350
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Prof. Juran
Here’s the optimized spreadsheet:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
GMS price
B
C
D
E
F
G
H
I
J
K
L
100
Scenarios for GMS stock in one month Returns from one unit of each investment
Portfolio
Scenario GMS price Probability
GMS
Put A
Put B Put C Return
Sqdev
1
150
0.05
50%
-100% -100% -100%
2737
6903774
2
130
0.10
30%
-100% -100% -100%
1039
863485
3
110
0.20
10%
-100% -100% -100%
-660
591400
4
100
0.30
0%
-100% -100%
-20%
-302
169098
5
90
0.20
-10%
-100%
56%
60%
56
2852
6
80
0.10
-20%
355%
213% 140%
414
92663
7
70
0.05
-30%
809%
369% 220%
772
438530
Put options on GMS stock that expire in one month
Option
A
B
C
Strike price
90
100
110
Option price
$2.20
$6.40
$12.50
Investment decision (thousands of dollars spent on each investment)
GMS
Put A
Put B
Put C
Total
8491
0
0
1509 10000
=
Budget
10000
Return from portfolio ($1000)
Mean
109
Stdev
795
Units of investments purchased (shares for GMS, number of puts for options)
GMS
Put A
Put B
Put C
84913
0
0 120694
Conclusions
Kate should buy $8,491,000 worth of GMS stock, and $1,509,000 worth of Put
Option C. This portfolio will have an expected one-month return of $109,000
(1.09%) and a standard deviation of $795,000 (7.95%).
B60.2350
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Prof. Juran
4.
Suppose that short selling is permitted — that is, the nonnegativity
restrictions on the portfolio weights are removed. Now what portfolio
minimizes the standard deviation of return?
Here we simply remove the nonnegativity constraint (by unchecking the box in
the Solver Options that says “assume nonnegative”).
Here is the new optimal solution:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
GMS price
B
C
D
E
F
G
H
I
J
K
L
100
Scenarios for GMS stock in one month
Scenario GMS price Probability
1
150
0.05
2
130
0.10
3
110
0.20
4
100
0.30
5
90
0.20
6
80
0.10
7
70
0.05
Returns from one unit of each investment
GMS
Put A
Put B
Put C
50%
-100%
-100%
-100%
30%
-100%
-100%
-100%
10%
-100%
-100%
-100%
0%
-100%
-100%
-20%
-10%
-100%
56%
60%
-20%
355%
213%
140%
-30%
809%
369%
220%
Portfolio
Return
Sqdev
2446
5201674
786
385978
-873
1077810
198
1068
230
4237
225
3543
219
2912
Now the nonnegativity conditions for the
changing cells is removed, and the investor
sells short on the put A and B options. This
lowers the standard deviation of the portfolio
(and also increases its mean).
Put options on GMS stock that expire in one month
Option
A
B
C
Strike price
90
100
110
Option price
$2.20
$6.40
$12.50
Investment decision (thousands of dollars spent on each investment)
GMS
Put A
Put B
Put C
8297
-8
-665
2376
Total
10000
=
Return from portfolio ($1000)
Mean
165
Stdev
718
Budget
10000
Units of investments purchased (shares for GMS, number of puts for options)
GMS
Put A
Put B
Put C
82972
-3798
-103843
190058
Kate should buy $8,297,000 worth of stock and $2,376,000 worth of Put Option C,
and she should short sell $8,000 worth of Put Option A and $665,000 worth of
Put Option B.
This portfolio will have an expected monthly return of 1.65% and a standard
deviation of 7.18%. In other words, it will be more profitable and less risky than
the portfolio without shorting.
A possible final step: We introduce a constraint on the portfolio’s expected
return, and use the right-hand side of this constraint as the input cell for
SolverTable, allowing us to create this chart:
GMS Risk vs. Return
$600
Efficient Frontier with Shorting
Expected Return (x 1000)
$500
$400
Efficient Frontier - No Shorting
$300
Minimum Risk with Shorting
$200
Minimum Risk - No Shorting
$100
GMS Stock Only
"One-for-One"
$$-
$200
$400
$600
$800
$1,000
$1,200
$1,400
$1,600
$1,800
$2,000
Std Dev of Return (x 1000)
B60.2350
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Prof. Juran