L. Synthetic CDOs and Their Valuation
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RISK MANAGEMENT AND FINANCIAL INSTITUTIONS
From the properties of the binomial distribution, the probability of exactly k defaults
by time t, conditional on F is
P(k, T|F ) =
n!
Q(t|F )k[1 − Q(t|F )]n−k
(n − k)!k!
(L.3)
Define
nL =
␣Ln
1− R
and
nH =
␣H n
1− R
where R is the recovery rate (assumed constant). Also, define m(x) as the smallest
integer greater than x. The tranche suffers no losses when the number of defaults, k,
is less than m(nL). It is wiped out when k is greater than or equal to m(n H ). Otherwise
the tranche principal at time t is a proportion
␣H − k(1 − R)/n
␣H − ␣L
of the initial tranche principal. These results can be used in conjunction with equations (L.1), (L.2), and (L.3) to calculate the expected tranche principal at all times
conditional on F . We can then integrate over F to find the (unconditional) expected
tranche principal. This integration is usually accomplished with a procedure known
as Gaussian quadrature. (The author’s website provides the tools for integrating over
a normal distribution using Gaussian quadrature.)
It is usually assumed that defaults happen at the midpoint of the intervals
between payments. Similarly to Appendix K, we are interested in the following
quantities
1. The present value of the expected spread payments received by Party A.
2. The present value of the expected payments for tranche losses made by Party A.
3. The present value of accrual payments received by Party A.
The spread payments received by Party A at a particular time are linearly dependent
on the tranche principal at that time. The tranche loss payments made by Party A
(assumed to be at the midpoint of an interval) is the change in the principal during
the interval. The accrual payment received by Party A is proportional to the tranche
loss payments. For any assumption about spreads, all three quantities of interest can
therefore be calculated from the expected tranche principal. The breakeven spread
can therefore be calculated analogously to the way it is calculated for CDSs in
Appendix K.
Derivatives dealers calculate the implied copula correlation, , from the spreads
quoted in the market for tranches of CDOs and tend to quote these rather than
the spreads themselves. This is similar to the practice in options markets of quoting Black–Scholes–Merton implied volatilities rather than dollar prices. There is a
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Appendix L: Synthetic CDOs and Their Valuation
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correlation smile phenomenon in CDO markets similar to the volatility smile phenomenon in options markets (see Section 22.4).
The software DerivaGem, which accompanies this book and can be downloaded from the author’s website, includes a worksheet for carrying out the above
calculations.1
1
More details on the calculation can be found in J. C. Hull, Options, Futures, and Other
Derivatives, 8th ed. (Upper Saddle River, NJ: Pearson, 2012).
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Answers to Questions and Problems
CHAPTER 1
1.1 The expected return is 12.5%. The standard deviation of returns is 17.07%.
1.2 From equations (1.1) and (1.2), expected return is 12.5%. SD of return is
0.52 × 0.17072 + 0.52 × 0.17072 + 2 × 0.15 × 0.52 × 0.17072 = 0.1294
or 12.94%.
1.3
w1
w2
P
P ( = 0.3)
P ( = 1)
P ( = −1)
0.0
0.2
0.4
0.6
0.8
1.0
1.0
0.8
0.6
0.4
0.2
0.0
15%
14%
13%
12%
11%
10%
24.00%
20.39%
17.42%
15.48%
14.96%
16.00%
24.00%
22.40%
20.80%
19.20%
17.60%
16.00%
24.00%
16.00%
8.00%
0.00%
8.00%
16.00%
1.4 Nonsystematic risk can be diversified; systematic risk cannot. Systematic risk
is most important to an equity investor. Either type of risk can lead to the
bankruptcy of a corporation.
1.5 We assume that investors trade off mean return and standard deviation of
return. For a given mean return, they want to minimize standard deviation
of returns. All make the same estimates of means, standard deviations, and
coefficients of correlation for returns on individual investments. Furthermore,
they can borrow or lend at the risk-free rate. The result is that they all want to
be on the “new efficient frontier” in Figure 1.4. They choose the same portfolio
of risky investments combined with borrowing or lending at the risk-free rate.
1.6 (a) 7.2%, (b) 9%, (c) 14.4%.
1.7 The capital asset pricing theory assumes that there is one factor driving returns.
Arbitrage pricing theory assumes multiple factors.
1.8 In many jurisdictions, interest on debt is deductible to the corporation whereas
dividends are not deductible. It can therefore be more tax-efficient for a company to fund itself with debt. However, as debt increases the probability of
bankruptcy increases.
1.9 Risk decomposition refers to a procedure where risks are handled one by one.
Risk aggregation refers to a procedure where a portfolio of risks is considered.
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1.11
1.12
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ANSWERS TO QUESTIONS AND PROBLEMS
Risk decomposition requires an in-depth understanding of individual risks.
Risk aggregation requires an understanding of the correlations between risks.
When potential losses are large, we cannot aggregate them and assume they
will be diversified away. It is necessary to consider them one by one and handle
them with insurance contracts, tighter internal controls, etc.
This is the probability that profit is no worse than –4% of assets. This profit
level is 4.6/1.5 = 3.067 standard deviations from the mean. The probability
that the bank will have a positive equity is therefore N(3.067) where N is the
cumulative normal distribution function. This is 99.89%.
Banks are allowed to accept deposits from the general public. Companies in
retailing and manufacturing are not.
Professional fees ($10 million per month), lost sales (people are reluctant to do
business with a company that is being reorganized), and key senior executives
left (lack of continuity).
The return earned by the hedge fund manager with zero alpha would be 0.05 +
0.6 × (0.10 − 0.05) = 0.08 or 8%. Because the alpha equals 4%, the hedge
fund manager’s return was 8% plus 4% or 12%.
CHAPTER 2
2.1 The banking system has become more concentrated, with large banks having
a bigger share of the market. The total number of banks reduced from 14,483
to 6,530.
2.2 In the early twentieth century, many states passed laws restricting banks from
opening more than one branch. The McFadden Act of 1927 restricted banks
from opening branches in more than one state.
2.3 The risk is that interest rates will rise so that, when deposits are rolled over,
the bank has to pay a higher rate of interest. The rate received on loans will
not change. The result will be a reduction in the bank’s net interest income.
2.4 DLC’s loss is more than its equity capital and it would probably be liquidated.
The subordinated long-term debt holders would incur losses on their $5 million
investment. The depositors should get their money back.
2.5 The net interest income of a bank is interest received minus interest paid.
2.6 Credit risk primarily affects loan losses. Noninterest income includes trading
gains and losses. Market risk therefore affects noninterest income. It also affects
net interest income if assets and liabilities are not matched. Operational risk
primarily affects non-interest expenses.
2.7 A private placement is a new issue of securities that is sold to a small number
of large institutional investors. A public offering is a new issue of securities that
is offered to the general public. In a best efforts deal, the investment bank does
as well as it can to place securities with investors, but does not guarantee that
they can be sold. In a firm commitment deal, the investment bank agrees to
buy the securities from the issuing company for a particular price and attempts
to sell them in the market for a higher price.
2.8 The bidders when ranked from the highest price bid to the lowest are: H, C, F,
A, B, D, E, and G. Bidders H, C, and F have bid for 140,000 shares. A has bid
for 20,000. The price that clears the market is the price that was bid by A or
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2.10
2.11
2.12
2.13
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$100. H, C, and F get their orders filled at this price. Half of A’s order is filled
at this price.
A Dutch auction potentially attracts a wide range of bidders. If all interested
market participants bid, the price paid should be close to the market price
immediately after the IPO. The usual IPO situation where the price turns out to
be well below the market price should therefore be avoided. Also, investment
banks are not able to restrict purchasers to their best current and potential
clients. The Google IPO was different from a standard Dutch auction in that
Google reserved the right to choose the number of shares that would be issued,
and the percentage allocated to each bidder, when it saw the bids.
Poison pills can give management a negotiation tool, particularly if the board
has the right to overturn a poison pill or make it ineffective. When it is confronted with a potential acquirer, the poison pill can buy the company time to
bargain for a better purchase price or find other bidders. However, there is the
danger that the poison pill will discourage potential buyers from approaching
the company in the first place.
The brokerage subsidiary of a bank might recommend securities that the investment banking subsidiary is trying to sell. The commercial banking subsidiary
might pass confidential information about its clients to the investment banking
subsidiary. When a bank does business with a company (or wants to do business
with the company), it might persuade the brokerage subsidiary to recommend
the company’s shares as a “buy.” The commercial banking subsidiary might
persuade a company to which it has lent money to do a bond issue because it
is worried about its exposure to the client. (It wants the investment banking
subsidiary to persuade its clients to take on the credit risk.) These conflicts of
interest are handled by what are known as Chinese walls. They prevent the
flow of information from one part of the bank to another.
The interest is no longer accrued. The before-tax income will be reduced by
8% of $10 million or $800,000 per year.
The provision for loan losses reflects the losses the bank expects in the future.
It is updated periodically. When the provision is increased in a year by X, there
is a charge to the income statement of X. Actual loan losses, when they are
recognized, are charged against the balance in the loan loss provision account.
In the originate-to-distribute model, a bank originates loans and then securitizes them so that they are passed on to investors. This was done extensively with household mortgages during the seven-year period leading up to
July 2007. In July 2007, investors lost confidence in the securitized products
and banks were force to abandon the originate-to-distribute model, at least
temporarily.
CHAPTER 3
3.1 Term life insurance lasts a fixed period (e.g., 5 years or 10 years). The policyholder pays premiums. If the policyholder dies during the life of the policy, the policyholder’s beneficiaries receive a payout equal to the principal
amount of the policy. Whole life insurance lasts for the whole life of the policy
holder. The policy holder pays premiums (usually the same each year) and the
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3.3
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ANSWERS TO QUESTIONS AND PROBLEMS
policyholder’s beneficiaries receive a payout equal to the principal amount of
the policy when the policyholder dies. There is an investment element to whole
life insurance because the premiums in early years are high relative to the expected payout in those years. (The reverse is true in later years.) Tax on the
investment income is deferred until death.
Variable life insurance is whole life insurance where the policyholder can specify how the funds generated in early years (the excess of the premiums over
the actuarial cost of the insurance) are invested. There is a minimum payout
on death, but the payout can be more than the minimum if the investments
do well. Universal life insurance is whole life insurance where the premium
can be reduced to a specified minimum level without the policy lapsing. The
insurance company chooses the investments (generally fixed income) and guarantees a minimum return. If the investments do well, the return provided on
the policyholder’s death may be greater than the guaranteed minimum.
Annuity contracts have exposure to longevity risk. Life insurance contracts
have exposure to mortality risk.
The lifetime annuity created from an accumulated value was calculated using
an interest rate that was the greater of (a) the market interest rate and (b) a
prespecified minimum interest rate.
The probability that the woman will die during the first year is 0.003255. The
probability that the woman will die during the second year is 0.003517 × (1 −
0.003255) = 0.003506. Suppose that the breakeven premium is X. We must
have
1,000,000 × (0.003255 + 0.003506) = X + (1 − 0.003255)X
so that X = 3,386. The breakeven premium is therefore $3,386.
3.6 The probability of a male surviving to 30 is 0.97147. The probability of a male
surviving to 90 is 0.15722. The probability of male surviving to 90 conditional
that 30 is reached is therefore 0.15722/0.97147 = 0.16184. The probability of
a female surviving to 90 conditional that 30 is reached is 0.27333/0.98466 =
0.27759.
3.7 The biggest risks are those arising from catastrophes such as earthquakes and
hurricanes and those arising from liability insurance (e.g., claims related to
asbestos in the United States). This is because there is no “law of large numbers” working in the insurance company’s favor. Either the event happens
and there are big payouts or the event does not happen and there are no
payouts.
3.8 CAT bonds (catastrophe bonds) are an alternative to reinsurance for an insurance company that has taken on a certain catastrophic risk (e.g., the risk of a
hurricane or an earthquake) and wants to get rid of it. CAT bonds are issued
by the insurance company. They provide a higher rate of interest than riskfree bonds. However, the bondholders agree to forego interest, and possibly
principal, to meet any claims against the insurance company that are within a
prespecified range.
3.9 The CAT bond has very little systematic risk. Whether a particular type of
catastrophe occurs is independent of the return on the market. The risks in the
CAT bond are likely to be largely “diversified away” by the other investments
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Answers to Questions and Problems
3.10
3.11
3.12
3.13
3.14
3.15
in the portfolio. A B-rated bond does have systematic risk that cannot be
diversified away. It is likely therefore that the CAT bond is a better addition to
the portfolio.
In Canada and the United Kingdom, health care is provided by the government.
In the United States, publicly funded health care is limited and most individuals
buy private health care insurance of one sort or another. In the United Kingdom,
a private health care system operates alongside the public system.
Both moral hazard and adverse selection are potential problems. The insurance
might lead to an individual not trying to keep a job as much as he or she
otherwise would. Indeed, an individual might purposely lose his or her job to
collect the insurance payout! Also, individuals that are most at risk for losing
their jobs would be the ones that choose to buy the insurance.
The payouts of property–casualty insurers show more variability than the payouts of life insurers. This is because of the possibility of catastrophes such as
earthquakes and hurricanes and liability insurance claims such as those related
to asbestos in the United States.
The loss ratio is the ratio of payouts to premiums in a year. The expense
ratio is the ratio of expenses (e.g., sales commissions and expenses incurred
in validating losses) to premiums in a year. The statement is not true because
investment income is usually significant. Premiums are received at the beginning
of a year and payouts are made during the year or after the end of the year.
A defined contribution plan is a plan where the contributions of each employee
(together with contributions made by the employer for that employee) are
kept in a separate account and invested for the employee. When retirement
age is reached the accumulated amount is usually converted into an annuity.
In a defined benefit plan, all contributions for all employees are pooled and
invested. Employees receive a pre-defined pension that is based on their years
of employment and final salary. At any given time, a defined benefit plan may
be in surplus or in deficit.
The employee’s wages are constant in real terms. Suppose that they are X
per year. (The units for X do not matter for the purposes of our calculation.)
The pension is 0.75X. The real return earned is zero. Because employees work
for 40 years, the present value of the contributions made by one employee
is 40XR where R is the contribution rate as a percentage of the employee’s
wages. The present value of the benefits is 20 × 0.75X = 15X. The value of R
that is necessary to adequately fund the plan must therefore satisfy
40XR = 15X
The solution to this equation is R = 0.375. The total of the employer and
employee contributions should therefore be 37.5% of salary.
CHAPTER 4
4.1 The number of shares of an open-end mutual fund increases as investments in the fund increase and decreases as investors withdraw their funds.
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4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
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A closed-end fund is like any other corporation with a fixed number of shares
that trade.
The NAV of an open-ended mutual fund is calculated at 4 P.M. each day
as the value of the assets held by the fund divided by the number of shares
outstanding.
The investor is deemed to have made capital gains of $300 and $100 in 2012
and 2013, respectively. In 2014, the investor is deemed to have made a capital
loss of $200.
An index fund is a fund that is designed so that its value tracks the performance
of an index such as the S&P 500. It can be created by buying all the stocks (or
a representative subset of the stocks) that underly the index. Sometimes futures
contracts on the index are used.
The front-end load is the amount an investor pays, as a percentage of his or
her investment, when shares of the fund are purchased. The back-end load is
the amount an investor pays, as a percentage of his or her investment, when
shares of the fund are redeemed.
An exchange-traded fund (ETF) that tracks an index is created when an institutional investor deposits a portfolio of shares that is designed to track the
index and receives shares in the ETF. Institutional investors can at any time
exchange shares in the ETF for the underlying shares held by the ETF, or
vice versa. The advantage over an open-end mutual fund that tracks the index is that the fund can be traded at any time, the fund can be shorted, and
the fund does not have to be partially liquidated to accommodate redemptions. The advantage over a closed-end mutual fund is that there is very little
difference between the ETF share price and the net asset value per share of
the fund.
The arithmetic mean of a set of n numbers is the sum of the numbers divided
by n. The geometric mean is the nth root of the product of the numbers.
The arithmetic mean is always greater than or equal to the geometric mean.
The return per year realized when an investment is held for several years is
calculated using a geometric, not an arithmetric mean. (The procedure is to
calculate the geometric mean of one plus the return in each year and then
subtract one.)
Late trading is the illegal practice of putting in an order to buy or sell an openend mutual fund at the 4 P.M. price after 4 P.M. Market timing is a practice
where favored clients are allowed to buy and sell a mutual fund frequently
to take advantage of the fact that some prices used in the calculation of the
4 P.M. net asset value are stale. Front running is the practice of trading by
individuals ahead of a large institutional trade that is expected to move the
market. Directed brokerage describes the situation where a mutual fund uses
a brokerage house for trades when the brokerage house recommends the fund
to clients.
Mutual funds must disclose their investment policies; their use of leverage is
limited; they must calculate NAV daily; their shares must be redeemable at any
time.
If a hedge fund is making money out of trading convertible bonds, it must be
doing so at the expense of its counterparties. If most of the traders are hedge
funds, they cannot all be making money.
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4.11 Hurdle rate is the minimum return necessary for an incentive fee to be applicable. High-water mark refers to the previous losses that must be recouped
before incentive fees are applicable. Clawback refers to investors being able to
apply a percentage of incentive fees against a percentage of future losses.
4.12 If the return is X (Ͼ 2%), the investors pay 0.02 + 0.2(X − 0.02) in fees. It
must therefore be the case that
X − 0.02 − 0.2(X − 0.02) = 0.2
so that 0.8X = 0.216 or X = 0.27. A return of 27% is necessary.
4.13 Short-term gains and losses do matter if the hedge fund is highly leveraged.
Short term losses can lead to margin calls that destroy the hedge fund.
4.14 Prime brokers decide on the amount of leverage that they are prepared to let
hedge funds have. This influences the risks that hedge funds can take.
CHAPTER 5
5.1 When a trader enters into a long forward contract, she is agreeing to buy the
underlying asset for a certain price at a certain time in the future. When a trader
enters into a short forward contract, she is agreeing to sell the underlying asset
for a certain price at a certain time in the future.
5.2 A trader is hedging when she has an exposure to the price of an asset and takes
a position in a derivative to offset the exposure. In a speculation, the trader has
no exposure to offset. She is betting on the future movements in the price of
the asset. Arbitrage involves taking a position in two or more different markets
to lock in a profit.
5.3 In the first case, the trader is obligated to buy the asset for $50. (The trader
does not have a choice.) In the second case, the trader has an option to buy the
asset for $50. (The trader does not have to exercise the option.)
5.4 Selling a call option involves giving someone else the right to buy an asset from
you for a certain price. Buying a put option gives you the right to sell the asset
to someone else.
5.5 (a) The investor is obligated to sell pounds for 1.7000 when they are worth
1.6900. The gain is (1.7000 − 1.6900) × 100,000 = $1,000.
(b) The investor is obligated to sell pounds for 1.7000 when they are worth
1.7200. The loss is (1.7200 − 1.7000) × 100,000 = $2,000.
5.6 (a) The trader sells for 50 cents per pound something that is worth 48.20 cents
per pound. Gain = ($0.5000 − $0.4820) × 50,000 = $900.
(b) The trader sells for 50 cents per pound something that is worth 51.30 cents
per pound. Loss = ($0.5130 − $0.5000) × 50,000 = $650.
5.7 You have sold a put option. You have agreed to buy 100 shares for $40 per
share if the party on the other side of the contract chooses to exercise the right
to sell for this price. The option will be exercised only when the price of stock
is below $40. Suppose, for example, that the option is exercised when the price
is $30. You have to buy at $40 shares that are worth $30; you lose $10 per
share, or $1,000 in total. If the option is exercised when the price is $20, you
lose $20 per share, or $2,000 in total. The worst that can happen is that the
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5.9
5.10
5.11
5.12
5.13
5.14
5.15
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ANSWERS TO QUESTIONS AND PROBLEMS
price of the stock declines to almost zero during the three-month period. This
highly unlikely event would cost you $4,000. In return for the possible future
losses, you receive the price of the option from the purchaser.
The over-the-counter (OTC) market is a telephone- and computer-linked network of financial institutions, fund managers, and corporate treasurers where
two participants can enter into any mutually acceptable contract. An exchangetraded market is a market organized by an exchange where traders either meet
physically or communicate electronically and the contracts that can be traded
have been defined by the exchange.
One strategy would be to buy 200 shares. Another would be to buy 2,000
options. If the share price does well, the second strategy will give rise to greater
gains. For example, if the share price goes up to $40, you gain [2,000 × ($40 −
$30)] − $5,800 = $14,200 from the second strategy and only 200 × ($40 −
$29) = $2,200 from the first strategy. However, if the share price does badly,
the second strategy gives greater losses. For example, if the share price goes
down to $25, the first strategy leads to a loss of 200 × ($29 − $25) = $800,
whereas the second strategy leads to a loss of the whole $5,800 investment.
This example shows that options contain built in leverage.
You could buy 5,000 put options (or 50 contracts) with a strike price of $25
and an expiration date in four months. This provides a type of insurance. If,
at the end of four months, the stock price proves to be less than $25, you can
exercise the options and sell the shares for $25 each. The cost of this strategy
is the price you pay for the put options.
A stock option provides no funds for the company. It is a security sold by one
trader to another. The company is not involved. By contrast, a stock when it is
first issued is a claim sold by the company to investors and does provide funds
for the company.
Ignoring the time value of money, the holder of the option will make a profit
if the stock price in March is greater than $52.50. This is because the payoff
to the holder of the option is, in these circumstances, greater than the $2.50
paid for the option. The option will be exercised if the stock price at maturity is greater than $50.00. Note that, if the stock price is between $50.00
and $52.50, the option is exercised but the holder of the option takes a loss
overall.
Ignoring the time value of money, the seller of the option will make a profit if
the stock price in June is greater than $56.00. This is because the cost to the
seller of the option is in these circumstances less than the price received for the
option. The option will be exercised if the stock price at maturity is less than
$60.00. Note that if the stock price is between $56.00 and $60.00 the seller of
the option makes a profit even though the option is exercised.
A long position in a four-month put option can provide insurance against the
exchange rate falling below the strike price. It ensures that the foreign currency
can be sold for at least the strike price.
The company could enter into a long forward contract to buy 1 million Canadian dollars in six months. This would have the effect of locking in an exchange
rate equal to the current forward exchange rate. Alternatively, the company
could buy a call option giving it the right (but not the obligation) to purchase 1
million Canadian dollars at a certain exchange rate in six months. This would