CHAPTER 27
Managing Risk
Answers to Practice Questions
1.
Modigliani and Miller would say that both financing and risk management are
irrelevant in perfect capital markets because any changes in capital structure and
firm risk that can be created by the firm’s management can also be created in the
capital markets by individual investors. In perfect capital markets, management
can not increase firm value by adjusting capital structure and firm risk because
individual investors can accomplish these adjustments on their own.
If market imperfections exist, it may be beneficial to the firm and its stockholders if
the firm hedges the firm’s risks. A large firm can hedge many of its risks with
much greater cost effectiveness than could an individual shareholder. For
example, hedges in the commodity futures markets and in foreign currency
futures can be accomplished at much lower cost by the firm than by individual
shareholders.
2.
Insurance companies have the experience to assess routine risks and to advise
companies on how to reduce the frequency of losses. Insurance company
experience and the very competitive nature of the insurance industry result in
correct pricing of routine risks. However, BP, for example, has concluded that
insurance industry pricing of coverage for large potential losses is not efficient
because of the industry’s lack of experience with such losses. Consequently, BP
has chosen to self insure against these large potential losses. Effectively, this
means that BP uses the stock market, rather than insurance companies, as its
vehicle for insuring against large losses. In other words, large losses result in
reductions in the value of BP’s stock. The stock market can be an efficient riskabsorber for these large but diversifiable risks.
Insurance company expertise can be beneficial to large businesses because the
insurance company’s experience allows the insurance company to correctly price
insurance coverage for routine risks and to provide advice on how to minimize the
risk of loss. In addition, the insurance company is able to pool risks and thereby
minimize the cost of insurance. Rarely does it pay for a company to insure
against all risks, however. Typically, large companies self-insure against small
potential losses.
234
3.
If payments are reduced when claims against one issuer exceed a specified
amount, the issuer is co-insured above some level, and some degree of on-going
viability is ensured in the event of a catastrophe. The disadvantage is that,
knowing this, the insurance company may over-commit in this area in order to
gain additional premiums. If the payments are reduced based on claims against
the entire industry, an on-going and viable insurance market may be assured but
some firms may under-commit and yet still enjoy the benefits of lower payments.
Basis risk will be highest in the first case due to the larger firm specific risk.
4.
The list of commodity futures contracts is long, and includes:
5.
a.

Gold

Sugar

Aluminum
Buyers include jewelers.
Sellers include gold-mining companies.
Buyers include bakers.
Sellers include sugar-cane farmers.
Buyers include aircraft manufacturers.
Sellers include bauxite miners.
If the spot price falls to $525 per ounce in three months’ time, Phoenix
Motors has a loss on the futures contract equal to:
10,000  ($550 − $525) = $250,000
Phoenix Motors has locked in the cost of purchasing 10,000 ounces of
platinum at $550 per ounce, or: 10,000  $550 = $5,500,000.
Phoenix can now buy 10,000 ounces in the spot market for $525 per
ounce: 10,000  $525 = $5,250,000. The cost in the spot market plus the
loss in the futures markets gives a total cost of $5,500,000.
b.
If the spot price increases to $625, then Phoenix has a gain in the futures
market that offsets the increased cost in the spot market. The gain in the
futures market is:
10,000  ($625 − $550) = $750,000
The cost in the spot market is: 10,000  $625 = $6,250,000
The total cost is still $5,500,000.
6.
Ft = S0 (1 + rf − y)t = 21,317(1 + 0.16 − 0.04)1/4 = 21,929.59
The futures are not fairly priced.
235
7.
The counterparties are the buyer and the seller of a forward or futures contract.
Counterparty risk is the risk that one party to a forward or futures contract takes
that the other party will default on the obligation to buy or to deliver the underlying
asset. Futures exchanges eliminate counterparty risk because futures contracts
are marked to the market every day, thereby eliminating the incentive to default.
When the price of a futures contract increases on a particular day, the amount of
the increase is paid by the seller of the futures contract to the buyer of the
contract; similarly, on a day when the futures price decreases, the buyer pays the
seller. For a forward contract that is not marked to the market, an increase in the
price of the underlying asset over the life of the contract increases the motivation
of the seller to default on the contract and sell the underlying asset at the higher
price in the spot market. The opposite is true for a decrease in price.
Exchanges do not eliminate basis risk. Basis risk depends on the correlation
between the spot price of a given commodity or financial instrument and the price
in the futures market. Although the design of the futures contract can affect this
correlation, the elimination of basis risk requires a perfect correlation between
spot and futures prices.
8.
To check whether futures are correctly priced, we use the following basic
relationships for commodities and for financial futures, respectively:
Ft = S0 (1 + rf + storage costs − convenience yield)t
Ft = S0 (1 + rf − y)t
This gives the following:
a.
b.
c.
d.
e.
Actual Futures
Price
$2,728.50
0.514
78.39
6,900.00
Value of
Future
$2,728.50
0.585
78.39
7,126.18
Magnoosium
Quiche
Nevada Hydro
Pulgas
Establishment
97.54
97.54
Industries stock
f. Wine
14,200.00
13,125.00*
* Assumes surplus storage cannot be rented out. Otherwise, futures are
overpriced as long as the opportunity cost of storage is less than:
$14,200 – $13,125 = $1,075
For Establishment Industries stock, compute the total future value of the two
dividend payments, as of six months from now, and then compute y in the above
formula by dividing this total future value by the current stock price.
236
Note that for the currency futures in part (d), the futures and spot currency quotes
are indirect quotes (i.e., pulgas per dollar) rather than direct quotes (i.e., dollars
per pulga). If I buy pulgas today, I pay ($1/9300) per pulga in the spot market and
earn interest of [(1.950.5) –1] = 0.3964 = 39.64% for six months. If I buy pulgas in
the futures market, I pay ($1/6900) per pulga and I earn 7% interest on my dollars.
Thus, the futures price of one pulga should be:
1.3964/(9300  1.07) = 0.00014033 = 1/7126.18
Therefore, a futures buyer should demand 7,126.18 pulgas for $1.
Where the futures are overpriced [i.e., (f) above], it pays to borrow, buy the goods
on the spot market, and sell the future. Where they are underpriced [i.e., (b) and
(d)], it pays to buy the future, sell the commodity on the spot market, and invest
the receipts in a six-month account.
9.
We make use of the basic relationship between the value of futures and the spot
price:
Ft = S0 (1 + rf )t
This gives the following values:
(1 + rf)t
rf
10.
1
1.00437
5.37%
Contract Length (Months)
3
9
1.01663
1.05799
6.82%
7.81%
15
1.10288
8.15%
21
1.15058
8.34%
a.
The NPV of a swap at initiation is zero, assuming the swap is fairly priced.
b.
If the long-term rate rises, the value of a five-year note with a coupon rate
of 4.5% would decline to 957.30:
45
45
45
45
1045




 957.30
1
2
3
4
(1.055) (1.055)
(1.055)
(1.055)
(1.055) 5
With hindsight, it is clear that A would have been better off keeping the
fixed-rate debt. A loses as a result of the increase in rates, and the dealer
gains.
c.
A now has a liability equal to: 1,000 – 957.30 = 42.70
The dealer has a corresponding asset.
237
11.
a.
Duration =
1
{[PV(C 1 )]  1]  [PV(C 2 )]  2]  [PV(C 3 )]  3]}
V
For Security A:
VA 
40
40
40


 103.08
1
2
(1.08) (1.08)
(1.08) 3
DurationA =
  40
  40

1  40

1


2


3





   1.95 years
3
103.08  (1.08) 1   (1.08) 2
  (1.08)

For Security B:
VB 
120
 111.11
(1.08)1
DurationB =

1  120
 1 1.00 year

1
111.11  (1.08)

For Security C:
VC 
10
10
110


 105.15
1
2
(1.08) (1.08)
(1.08) 3
DurationC =
b.
  10
  110

1  10
 1  
 2  
 3   2.74 years

1
2
3
105.15  (1.08)
  (1.08)
  (1.08)

DurationA = [(X) (DurationB)] + [(1 – X) (DurationC)]
1.95 = 1.0X + [(1 – X) (2.74)]  X = 0.454 and (1 – X) = 0.546
Therefore, the following position would immunize the investment:
a short position of $4,540,000 in Security B and a short position of
$5,460,000 in Security C.
c.
DurationB = [(X) (DurationA)] + [(1 – X) (DurationC)]
1.0 = 1.95X + [(1 – X) (2.74)]  X = 2.203 and (1 – X) = –1.203
Therefore, the following position would immunize the investment:
a short position of $22,030,000 in Security A and a long position of
$12,030,000 in Security C.
238
12.
Suppose you own an asset A and wish to hedge against changes in the value of
this asset by selling another asset B. In order to minimize your risk, you should
sell delta units of B; delta measures the sensitivity of A’s value to changes in the
value of B.
In practice, delta can be measured by using regression analysis, where the value
of A is the dependent variable and the value of B is the independent variable.
Delta is the regression coefficient of B. Sometimes considerable judgement must
be used. For example, it may be that the hedge you wish to establish has no
historical data that can be used in a regression analysis.
13.
Gold Price
Per Ounce
$280
$300
$320
14.
(a)
Unhedged
Revenue
$280,000
$300,000
$320,000
(b)
Futures-Hedged
Revenue
$301,000
$301,000
$301,000
(c)
Options-Hedged
Revenue
$298,000
$298,000
$318,000
Standard & Poor’s index futures are contracts to buy or sell a mythical share,
which is worth $500 times the value of the index. For example, if the index is
currently at 400, each ‘share’ is worth: $500  400 = $200,000
Legs’ portfolio is equivalent to five such ‘shares.’
If Legs sells five index futures contracts, then, in six months, he will receive:
5  $500  price of futures
If the relationship between the futures price and the spot price is used, this is
equivalent to receiving:
5  500  (spot price of index)  (1 + rf)1/2 = $1,000,000  (1 + rf)1/2
This is exactly what he would receive in six months if he sold his portfolio now and
put the money in a six-month deposit. Of course, when he sells the futures, Legs
also agrees to hand over the value of a portfolio of five index ‘shares.’ So, at the
end of six months, he can sell his portfolio and use the proceeds to settle his
futures obligation. Thus, by hedging his portfolio, Legs can ‘cash in’ without
selling his portfolio today.
239
15.
16.
a.
0.75  $100,000 = $75,000
b.
 = 0.75
c.
You could sell (1.2  $100,000) = $120,000 of gold (or gold futures) to
hedge your position. However, since the R2 is less (0.5 versus 0.6 for
Stock B), you would be less well hedged.
a.
For the lease:
Year
1
2
3
4
5
6
7
8
Ct
PV(Ct)
at 12%
2
1.7857
2
1.5944
2
1.4236
2
1.2710
2
1.1349
2
1.0133
2
0.9047
2
0.8078
V = 9.9353
Proportion Proportion of
of Total
Total Value
Value
Times Year
0.1797
0.1797
0.1605
0.3210
0.1433
0.4299
0.1279
0.5117
0.1142
0.5711
0.1020
0.6119
0.0911
0.6374
0.0813
0.6504
Duration =
3.9131
For the 6-year debt (value $8.03 million):
Year
1
2
3
4
5
6
Ct
PV(Ct)
at 12%
120 107.14
120
95.66
120
85.41
120
76.26
120
68.09
1120 567.43
V = 1000.00
Proportion Proportion of
of Total
Total Value
Value
Times Year
0.1071
0.1071
0.0957
0.1913
0.0854
0.2562
0.0763
0.3050
0.0681
0.3405
0.5674
3.4046
Duration =
4.6048
The duration of the one-year debt (value $1.91 million) is one year.
Therefore, the average duration of the debt portfolio is:

  8.03

1.91

 1  
 4.6048   3.9121 years
 1.91 8.03   1.91 8.03

This is equal to the duration of the lease (within a rounding error).
240
b.
See the table below. Potterton is no longer fully hedged. The value of the
liabilities ($14.022 million) is now less than the value of the asset ($14.039
million). A one percent change in interest rates affects the value of the
asset more than the value of the liabilities. To maintain the hedge, the
financial manager would adjust the debt package to have the same
duration as the lease. Note, however, that the mismatch is negligible and
should not give the manager sleepless nights.
Lease
Yield Value Change
2.5% 14.340 +2.144%
3.0% 14.039
3.5% 13.748
-2.073%
(a) $8.03 million face value
(b) $1.91 million face value
17.
6-Year Debt
Price
Value
152.33 12.232a
148.75 11.945
145.29 11.667
1-Year Debt
Debt Package
Price
Value
Value Change
109.27
2.087b 14.319 +2.118%
108.74
2.077 14.022
108.21
2.067 13.734 -2.054%
Assume the current price of oil is $24 per barrel, the futures price is $26, and the
option exercise price is $26.
Oil Price
Per Barrel
Futures-Hedged
Expense
$24
$26
$28
$26
$26
$26
Options-Hedged
Expense
$24
$26
$26
The advantages of using futures are that risk is eliminated and that the hedge,
once in place, can be safely ignored. The disadvantage, compared to hedging
with options, is that options allow for the possibility of a gain. Hedging with
options has a cost (i.e., the cost of the option).
18.
a.
To calculate the six-month futures price, we use the following basic
relationships for commodities and for financial futures, respectively:
Ft = S0 (1 + rf + storage costs − convenience yield)t
Ft = S0 (1 + rf − y)t
Thus, the six-month futures prices are:
Magnoosium: 2,800  (1.03 – 0.02) =
Oat Bran:
0.44  (1.03 – 0.03) =
Biotech:
140.2  1.03 =
Allen Wrench: 58.00  [1.03 – (1.20/58.00)] =
5-Year T-Note: 108.93  [1.03 – (4.00/108.93)] =
Ruple:
*
241
$2,828 per ton
$0.44 per bushel
$144.41
$58.54
$108.20
3.017 ruples/$
*Note that, for the currency futures (i.e., the Westonian ruple), the spot currency
quote is an indirect quote (i.e., ruples per dollar) rather than a direct quote (i.e.,
dollars per ruple). If I buy ruples today in the spot market, I pay ($1/3.1) per ruple
in the spot market and earn interest of [(1.120.5) –1] = 0.0583 = 5.83% for six
months. If I buy ruples in the futures market, I pay ($1/X) per ruple (where X is
the indirect futures quote) and I earn 6% interest on my dollars. Thus, the futures
price of one ruple should be:
1.0583/(3.1  1.03) = 0.33144 = 1/3.017
Therefore, a futures buyer should demand 3.017 ruples for $1.
b.
The magnoosium producer would sell 1,000 tons of six-month magnoosium
futures.
c.
Because magnoosium prices have fallen, the magnoosium producer will
receive payment from the exchange. It is not necessary for the producer to
undertake additional futures market trades to restore its hedge position.
d.
No, the futures price depends on the spot price, the risk-free rate of
interest, and the convenience yield.
e.
The futures price will fall to $48.24 (same calculation as above, with a spot
price of $48):
48.00  [1.03 – (1.20/48.00)] = $48.24
f.
First, we recalculate the current spot price of the 5-year Treasury note.
The spot price given ($108.93) is based on semi-annual interest payments
of $40 each (annual coupon rate is 8%) and a flat term structure of 6% per
year. Assuming that 6% is the compounded rate, the six-month rate is:
(1 + 0.06)1/2 – 1 = 0.02956 = 2.956%
Incorporating similar assumptions with the new term structure specified in
the problem, the new spot price of the 5-year Treasury note will be
$118.16. Thus, the futures price of the 5-year T-note will be:
118.16  [1.02 – (4.00/118.16)] = $116.52
The dealer who shorted 100 notes at the (previous) futures price has lost
money.
g.
The importer could buy a three-month option to exchange dollars for
ruples, or the importer could buy a futures contract, agreeing to exchange
dollars for ruples in three months’ time.
242
19.
Since electricity can not be stored, the formula connecting spot and futures prices
does not work for electricity. The formula depends on the storage cost and
convenience yield for a commodity that is held in inventory. In general, futures
prices do not help in forecasting commodity prices because, as the formula makes
clear, futures prices are dependent on current spot prices, not expected future
prices. However, the exception to this conclusion is a ‘commodity’ like electricity
that can not be stored. For electricity, there is no connection between current
spot price and expected future price, so electricity futures prices do provide
information about the expected future price of electricity.
20.
Think of Legs Diamond’s problem (see Practice Question 13). If futures are
underpriced, he will still be hedged by selling futures and borrowing, but he will
make a known loss (the amount of the underpricing). If, for example, he hedges
by selling seven-month futures, he not only needs to know that they are fairly
priced now but also that they will be fairly priced when he buys them back in six
months. If there is uncertainty about the fairness of the repurchase price, he will
not be fully hedged.
Speculators like mis-priced futures. For example, if six-month futures are
overpriced, speculators can make arbitrage profits by selling futures, borrowing
and buying the spot asset. This arbitrage is known as ‘cash-and-carry.’
243
Challenge Questions
1.
a.
Phillips is not necessarily stupid. The company simply wants to eliminate
interest rate risk.
b.
The initial terms of the swap (ignoring transactions costs and the dealer’s
profit) will be such that the net present value of the transaction is zero.
Phillips will borrow $20 million for five years at a fixed rate of 9% and
simultaneously lend $20 million at a floating rate two percentage points
above the three-month Treasury bill rate which is currently a rate of 7%.
c.
Under the terms of the swap agreement, Phillips is obligated to pay $0.45
million per quarter ($20 million at 2.25% per quarter) and, in turn, receives
$0.40 million per quarter ($20 million at 2% per quarter). That is, Phillips
has a net swap payment of $0.05 million per quarter.
d.
Long-term rates have decreased, so the present value of Phillips’ long-term
borrowing has increased. Thus, in order to cancel the swap, Phillips will
have to pay the dealer. The amount paid is the difference between the
present values of the two positions:


The present value of the borrowed money is the present value of $0.45
million per quarter for 16 quarters, plus $20 million at quarter 16,
evaluated at 2% per quarter (8% annual rate, or two percentage points
over the long-term Treasury rate). This present value is $20.68 million.
The present value of the lent money is the present value of $0.40
million per quarter for 16 quarters, plus $20 million at quarter 16,
evaluated at 2% per quarter. This present value is $20 million, as we
would expect. Because the rate floats, the present value does not
change.
Thus, the amount that must be paid to cancel the swap is $0.68 million.
2.
If the plant runs at full capacity, month in and month out, then both the quantity of
electricity produced and the quantity of natural gas required are known in
advance. The cost of production can be hedged by buying natural gas futures
contracts in the appropriate quantity, and the price of the electricity produced can
be hedged by selling electricity futures contracts in the appropriate quantity.
If the plant is to be shut down when the spark spread is negative, then the hedge
described above should be put in place only for months when the spark spread
indicated by the difference in the futures prices is positive. If this difference is
negative for a particular date (e.g., month) in the future, then the hedge should not
be put in place today. If the spread remains negative at that future date, then the
plant should be shut down at that future date. If, at that future date, the spark
spread turns positive, then the plant should be operated.
244