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Problc,"s
15.440 f.dvanc,," Topics in final1ciel Markets and Institutions
Fischer lllack
Spring, 1982
Capital Marbt Eoul1i t-t'it:m
1. Assume thet all the assumptions of the capital asset pricing model
hold, except that tr.en is a tax on interest income at a constant
rate, and a subsidy on interest expense at the same rate. If there
is a riskless real asset, the returns on that asset are not taxed.
How will th~ equilibrium t-~ changed? Now aSsume that there is a tax
on the excesS return on an investor's portfolio, whether realized or
unrealized, with a subsidy at the same rate when the excess return
is negative. Assume the revenues from the tax or the cost of the
subsidy are distributed evenly across investors. How will the
equilibrium be changed this ti~e?
2. Assume tha t securities prices are always correct and are always
equal to tne present value, properly computed, of the joint distribution
of possible future dividends. Assume that t:,2 ret,,:n on the
market portfolio is lognormal, with constant expected return and
variance ra tee ~'ri tc a discrete ti me model for di vide~ds on the
market portfoEo that will give a standard deviation for percentage
changes in dividends strictly less than the: standard deviation for
percentage char.ges i" the level of the market. \';c.a~ are the key
features of your modf'l? ""at mignt cause divideuGs to behave in
this >Tay?
3. Assume that individuals hav(· identical time-se~rable constant
relative risk aversion uti;'! ty functions on total cons'''::!'tion (but
not leisure). Assume that the~r utility functions ".re stateindependent.
':here are two reA.l investtlent opp"rtun~ti~s: a
riskless asset "dth constant rc'!:urns to scale. ar:~ ("'. risky asset
with constant stochastic rett:rns to scale •. The return on the
riskless Esset is constant, but th,= expec.ted excess ret'.:rr. per unit
standard deviation 0:: the risky asset is stat~ del'.endent. Ca:o we
estimate the .Jegree of risk aversior. in the utility furc:; on from
the average excess return on te.e market? Givell the tice discount
rate and degree of risk aversion in the utility ;ur.cticQ, is there
any pcssiblE: feature of the time series c:' total consulO?tim: that
cannot be matched usin& the right: cUte dependence in the OPPOl"tunity
set?
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4.
5.
6.
7.
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Assume tha t tastes and technology are the same as in problem 3,
except that the utility function is allowed to be state-dependent.
Is there any possible feature of the time series of total consumption
that cannot be matched using the right state-dependence in both
the utility function and the opportunity set? Can you think of any
observable state variables for tastes? For technology?
Describe some possible tests of the simple capital asset pricing
model (with constant parameters). What are the advantages and disadvantages
of each? How accurate do you expect the best test to
be? Now suppose that firms in the aerospace business have higher
expected returns on average than predicted by the CAPM. Will the
best test pick that up? Is there another test that .ill pick it up?
Assume a one-perio:>d world in which all the assumptions of the
capital asset pricing model hold but one: the supplies of available
assets are not fixed. Each person starts with an endOl;ment of
wheat, and can plant it using any of a number of technologies.
There is a riskless technology that returns 1.03 bushels :or each
bushel planted. Asset i is the total investment in risky technology
i. The joint distribution of the returns on invest~ents in
the different technologies is independent o'f the number of bushels
planted in each technology. After all investment decisions are
made, the portfolio of all risky assets has an expected return of
12%. (The market portfolio may contain both risky and riskless
assets.) Will a version of the capital asset priCing model hold in
this world? If so, how much can you say about the relation between
the expected return on asset i and its S ?
Assume a one-period world like the one in problem 6, except that the
expected return on the portfolio of all risky assets is not known.
A very wealthy person takes so much of a liking to a special risky
asset that he is willing to hold a huge long positicu in it, even
though it has a negative expected return and would not have been
included in the portfolio of all risky assets. All other investors
are willing to hold short positions in the special asset. If we
omit the wealthy person's holding", the market portfolio w11l
contain a lar:e negative position in the special asset. Will total
holdings of the special asset be positive, so that it is part of the
overall market portfolio? From the point of Yiew of all other
investors, will a version of the capital asset pricing codel hold?
What kind of e will the special asset have? Are the other
investors made better off or worse off by the existence of the very
wealthy person?
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8. Assume that all the assumptions of the capital asset pricing model
hold except that only one person is allowed to borrow. How will the
expected returns on securities differ from what is predicted by the
capital asset pricing model? How will the interest rate be
sffected? What kinds of portfolios will most individuals who cannot
borrow hold? What kind of portfolio will the individual who can
borrow hold if he is very tolerant of risk? What kind will he hold
if he is very averse to risk? How will fluctuations in the degree
to which borrowing is restricted affect the economy?
9. Suppose that the government wants to prevent people from taking too
much risk, because it believes that they don't fully understand the
losses they may incur. Assume that real investments are fluid, so
society can shift between safe and risky investments easily. The
government sets a limit on the total risk, measured in dollars or
percentages, that each individual can take. Will the capital asset
pricing model hold in this world (assuming that it would hold without
these restrictions)? What kinds of portfolios. will people
hold? Will the restrictions affect the interest rate or the
expected return on the market portfolio of all real investments?
10. Assume that securities markets are efficient, and that changes in
the money stock are associated with both past and future changes in
the general level of economic activity. Assume that the current
level of the money stock is widely known, without error. .~at relationships
might we find between changes in the money stock and past,
present, or future returns on the market portfolio?
11. Assume that markets are efficient in the sense that all investors
have the same information, including information about the process
the managers of a firm are going through in making investment
decisions. The managers of one firm decide to save thecsel'/es the
effort of making investment decisions in the following way. Each
time an investment is being considered, they will an~ounce a
decision to make that investment. If the firm's stock price goes
up, they will go ahead with the investment. If the firm's stock
price does not go up, they will reverse the decision. Will the firm
be able to make optimal investment decisions this way? Will they be
optimally timed?
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Corporate Finance
12. As a second approximation, what factors might influence a firm's
optimal and actual asset and liability structure?
13. Assume no inflation-indexing is allowed. Can you think of a set of
sufficient conditions for the before-tax real interest rate to be
independent of the rate of inflation? Under your conditions, \lill
the rate of inflation affect the optimal capital structu=e? \/ha t
kinds of taxes can be used in your world?
14. \/hy is it commOn for a steel firm to buy coal under a long term
contract? Why doesn't it just buy what it needs in the spot market?
15. Assume that depositors in a mutual thrift institution receive the
same interest rates as depositors in a stock institution, and that
they have no control over the affairs of the institution. \/ho bears
the risk in a mutual institution? Who owns the e~uity? What are
the advantages and disadvantages of the mutual format? ;,nat' s the
natural way to allo\l a conversion from mutual to stock format?
16. Assume that a firm has two classes of stock outstanding that are
identical except that class A pays a cash dividend and class B pays
a stock dividend \lith shares worth the same as the cash dividend.
Assume that shares of class A and clsss B sell for the same price
just before each ex-dividend date; but that class A shares fallon
the ex-date by 70% of the amount of the dividend, while class B
shares fall On the ex-dete by 100% of the amount of the dividend.
What interpretations of these facts are possible?
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17. Suppose that there are two kinds of assets and two kinds of investors:
tax-exempt and taxable. If a taxable investor holds a
taxable asset, he pays a 50% tax on its return. All assets are
riskless. Write Rl and R2. for the before-tax returns on
tax-exempt and taxable assets. AssUlIle that the government limits
tax arbitrage by prohibiting short sales of (a) taxable securities
or (b) tax-exempt securities. Consider the possiblity for each case
that the initial wealth of taxable investors (i) exceeds or (11)
falls short of the initial amount of tax-exempt securities. 1.1Ia t
will investor positions look like in each of the four cases? What
will the relation between Rl and R2 be? Look for an equilibrium
in which every investor is happy with his assets and liabilities,
given Rl, R2 , and the constraints.
18. How will proportional trading costs affect th~ strategy of realizing
losses immediately and gains before they become losses? How will
fixed trading costs affect the strategy?
19. Assume that a regula ted utility is the only source of a service in.
its area. Demand for the service is inelastic for the life of the
utility's plant, but falls to zero when the plant is nO longer
useful. The operating cost of providing the service is zero. The
.-A'rate at which service is prOVided follows a geotaetric random walk
d until the plant life is over. The regulators have mapped out a
(~ depreciation schedule for the plant, and want to choose the price of
the service so that the market price of the utility's stock will
always be equal to its depreciated book value. All operating
revenues are to be paid out as dividends. There is no regulatory
lag. How will changes in the demand for the service affect (a) the
price of the service, and (b) the utility's stock price?
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,. Accounting and Economic Earnings
20. Assume that when a firm announces its earnings for the most recent
accounting period, its stock price often moves up or dolro by an
unusually large amount. In other words, the variance of the return
is larger around earnings announcement dates than at other times.
Is this consistent with market efficiency? Is it consistent with
the notion that earnings figures don't matter? If a different set
of accounting rules "'ere in place. would the price change at the
time of the earnings announcement be different?
21. EConomists and financial analysts fre~uently define "rate of return"
for a country or a firm as earnings o'/er book value. If both book
value and earnings give estimates of market value, "'hat information
is there in the rate of return for a country or firm?
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Underwriting
22. Assume that underwriting syndicates compete to offer the highest
price to the issuer of a security and the lowest price to the
buyers. Assume that underwriters receive no indirect benefit when
the issue is underpriced, sells out quickly, and goes to a premium,
so the underwriters receive a maximum fee equal to the full spread.
Assume that existing shareholders have the right to buy the new
shares in proportion to their holdings if they wish. In this
situation, how will the spread that a syndicate proposes be related
to the price it offers the issuer? What other factors will
influence the spread?
23. Assume that people have different information about a stock, and
divide into optimists and pessimists of varying degrees. There is
also a group of people ignorant about a stock who are not potential
stockholders. Dumping more stock on the market drives down the
price, even when the event is known, because informed investors will
hold more only if the expected return goes up. Underwriting that
expands the group of informed investors, even if the information
given is balanced, will bring the stock closer to its correct
price. When should a firm use underwriters, and when should a firm
use rights? When should an investor buy the stock of a firm that
habitually uses rights? Will the stockholders vote to oaximire the
current market price of the firm's shares, or the future marke t
price?
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Trading and Investment Companies
24. Imagine that American Airlines has long held 20% of Howard Johnson's
Common stock. The Federal Trade Commission has decided that this is
likely to reduce potential competition between American Airlines and
Howard Johnson's, and has ordered that American eliminate its
holdings of Howard Johnson's within a year. The decision is
appealed. A distinguished economist from the University of Pennsylvania
has testified that selling this large a block of stock is
likely to depress the price at least 5%, so that the shareholders of
American will be cheated out of 5% of the value of their investment
if forced to sell. You have been hired by the government to evaluate
his testimony. Does the cost to American's shareholders depend
on the strategy that American uses to eliminate its holdings of
Howard Johnson's? What is the best strategy? What are the factors
that might influence the effect on American's shareholders of using
the best strategy?
25. Suppose that a firm has 8 million authorized and outstanding common
shares, and that the outstanding short position in the stock is
2 million shares. How many votes might be cast if all proxies were
sent in for the annual meeting? Why is it sometimes hard to borrow
stock during a contest for control of the firm?
26. How can you automate the role of the specialist? Could a corporation
with a computer take over the function of the specialist for
all the stocks listed on an exchange? What rules would you ·use for
changing the bid and asked prices as trades are executed? How would
you handle orders for traders who are not information-motivated?
27.
28.
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Asslltle that you are a "buy-and-hold" investor in a no load mutual···
fund. You plan to hold your shares for a very long time~ . Other·
investors in the fund, however, are doing short-term trading to ·take
advantage of the way the fund prices its shares. It bases its
calculations of net asset value per share on the aost recent price
at which each asset has traded, even though the asset may not have
traded for several days. This means that an increase in value of
the fund's shares is more likely to be followed by another increase
than by a decrease. It means that the fund is issuing and redeeming
ahares at prices that are sometimes higher and sometimes lower than
the current market value of the fund's assets per share. Are you
hurt by the trading of these active investors? If you are hurt,
what policies can the fund adopt to protect you? Are buy-and-hold
investors in stock hurt by the actions of short term traders?
Assume that the managers of a closed end investment company generate
expenses for the shareholders that are 1.5% of the assets per year,
while the expenses of a comparable open end fund are 0.5% of the
assets per year. The shareholders put up with this because a closed
end fund does not redeem its shares continuously at net asset value
per share, so the shareholders as a group can withdraw their money
only as the fund pays dividends to them. Assume that the fund pays
diVidends of exactly 4% of the assets per year, and that tax factors
are not important. lfuat discount would you expect for this fund?
(How will the value of its liabilities compare with the value of its
assets?) Why?
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Options and Futures
29. Make all the assumptions leading to the simple call option pricing
model except: (a) assume that options can be exercised st any time;
snd (b) assume that the underlying stock pays quarterly divid~nds.
At what points in time may there be early exercise or massive
closing out of option positions? Assuming that you have upper
bounds on all the dividends, can you give a series of lower bounds
for the value of the option? Can you give a condition under which
you can be sure that no early exercise will take place over the life
of the option?
30. Assume that everyone is allowed to lend freely, but that each person
can borrow no more than SlOO. Assume that exeryone can buy or sell
stocks freely, and will receive the proceeds of any call options
that are sold or stocks that are sold short. E:xcept for the
borrowing restriction, make all the assumptions that lead to the
capital asset pricing model and the simple option pricing model.
How will the pricing of options be affected by the borrowing
restriction? How will the pricing of stocks be affected? Wha t
portfolios will people at different levels of risk aversion hold?
How will your answers change if short sales of stocks are also
restricted?
31. Assume that we are 1n a world with no trading costs where
individuals and firms pay taxes at the same rate, and capital
structure is a matter of indifference. Common stocks pay no
dividends, and are not taxed. Options are not taxed either.
Pension funds are not taxed on any of their income. In this world,
will the interest rate in the option pricing formula be the beforetax
rate or the after-tax rate? Will the gain from having bonds in
the pension fund be there if the pension fund holds out-of-the-money
call options too? Now do your answers change in a world with no
personal taxes or bankruptcy costs, but with a maximum amount of
debt for each firm derived from a maximum total risk for its stock?
32. What is the relation between forward contracts and futures contracts?
Why are the exchange-traded contracts generally futures
contracts? What assumptions must we make to derive a relation
between forward prices (not the value of a forward contract) and
futures prices?
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33. Write lC for the current value of a pure discount goverr:.::.ent bond
with a face value of 1.0. Write t for the current time, and t*
for the maturity of the bond. What are the values of (a) a European
call option, (b) an American call option, (c) a European put option,
and (d) an American put option, where all the options are options on
the bond that expire at time t* and have exercise price c 1 What
is the futures price with maturity t* for the bond? Wr~t are the
values of options (a), (b), (c). and (d) if they are futures options
rather than bond options, where the underlying futures contract
calls for delivery of the bond at time t*?
34. For what reasons should the relation between the futures price on a
market index and the currently quoted value of the index vary? For
what reasons should the relation between the near and far futures
prices On a market index differ from the relation between the near
and far futures prices on a geometric index like Value Line's?
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Corporate Assets and Liabilities
35. Assume that the market for borrowing and lending stock is fully competitive,
and that there are no transaction costs or administrative
costs or government restrictions. On what terms might a loan of
stock be made? What would the borrower pay the lender? Who would
bear the risk of default?
36. In the options market, you sell short by writing a newly created
option. There is no need to borrow an option before selling it
short. Could there be an analogous way to sell cOl!llllon stocks
short? How could voting rights be handled? What changes in
procedures for trading on stock exchanges would be needed?
37. A firm worth 360 million before any debt payments has a debt payment
of $10 million due shortly, and a final payment of $100 million due
in one year. AsSuming no debt payments are due, the value of a:t
option to buy the firm for $100 million in one year will be $9
million if the firm's value is ~60 million. If the firm is otherwise
similar but is worth only $50 million, the value of an option
to buy the firm for ~100 million in one year will be $6 million.
What will the value of the firm's equity and debt be if both debt
~ayments arc due, and (a) it is allowed to sell assets to pay the
$10 million, or (b) it is required to sell stock to pay the $10
million.
38. Miller and Modigliani ha.ve said, under certain assumptions, that the
announcement of a change in a corporation's dividend policy will not
affect the price of its stock. Assume that there are no taxes, that
the dividend policy change is a response to well-known events, and
that investors are indifferent between diVidends and capital gains,
but that the firm has risky debt outstanding. Can the Hil1erModig1iani
theorem be true under these conditions? Conside= various
possible sources for the funds used to pay an increased dividend.
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Banking and Monetary Theory
39. A firm maintains average working balances with a bank of $5 million,
and receives no interest on those balances, largely because the
government prohibits the direct payment of interest on demand
deposits. Now the firm wants to take out a loan of 35 million
(without changing its average working balances). The short term
riskless interest rate is 6%, and the rate on this loan would be 8%
if the firm had no balances. There are no reserve requirements.
Assume that the bank wants to begin paying full interest on the
firm's working balances by using them as compensating bala~ces, and
that the bank has the right to use the balances to payoff part or
all of the loan if the firm is in trouble. What will the interest
rate on the loan be?
40. Suppose that lenders are divided into two groups, LJ. and Lz.
Suppose that borrowers are divided into two groups, Bl and
B2' There are two kinds of debt securities, short-terCl and longterm.
What pattern of expected returns will we find if (a) lenders
in Ll hold only short-term securities, lenders in L2 hold
only long term securities, borrowers in Bl issue only short-term
securities, and borrowers in B2 issue only long-term securities;
(b) lenders in both groups can hold both kinds of securities, but
borrowers in Bl and B2 can issue only short-term and longterm
securities, respectively; (c) lenders in Ll and L2 can
hold only short-term and long-term securities, respectively, bu t
borrowers in both groups can issue both kinds of securities; and (d)
lenders in both groups can hold both kinds of securities, and
borrowers in both 'groups can issue both kinds of securities. State
the assumptions you need, ,to reach your conclusions.
41. Suppose that the government succeeds in controlling the aClount of
credit in the economy through such devices as margin requirements,
restrictions on the kinds of loans that can be made by banks and
other intermediaries, and taxes on interest income without an
equivalent subsidy for interest expense. Perhaps the government
restricts the extent to which financial institutions can issue
securities or borrow to raise the money to make loans, or applies
non-interest-bearing reserve requirements to loans. !.'bat effects
will these policies have on the pricing of risky securities? What
effects will they have on the composition and total risk of the
market portfolio of real assets? What effects will they have on
real investment decisions by firms? What effects will they have on
aggregate consumption? Focus on the effects of a tax on interest
income whose proceeds are given back in a way that avoids significant
redistribution of income.
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42. Assume that monetary policy is passive, and that banks and othEr
financial institutions are initially unregulated. Assume that the
riskless interest rate is 8%, and that banks pay 8% on deposits
transferrable by check. Now suppose (a) that the government imposes
a 25% non-interest-bearing reserve requirement on such deposits.
What impact will this have on deposit rates? On loan rates? On the
nature of banks' funds transfer business? Alternatively, suppose
(b) that the government imposes a 25% non-interest-bearing reserve
requirement on all loans, whether made by banks or other firms that
specialize in lending. What effects will this have on deposit
rates? On loan rates? On the nature of banks' loan business?
43. Assume that money is supplied by the Federal Reserve System in the
U.S. entirely passively. For example, assume that the System
borrows currency or reserves freely at the federal funds rate, and
lends currency or reserves freely against llO~ collateral in the
form of government bonds at the federal funds rate. The federal
funds rate is kept in line with other interest rates. Give as many
factors as you can that will influence the stock of coney under
these conditions. How will the rate of inflation and the short term
interest rate be determined?
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I~.ernational Economics
44. Assume that the government has two kinds of liabilities: interestbearing
liabilities in the form of deposits at the Federal Reserve
Banks, and non-interest-bearing liabilities in the form of currency.
Assume that the private sector is not regulated, and that
interest-bearing government deposits may be held by individuals,
financial institutions, and non-financial firms. Assume that there
are no government bonds, and no reserves in the sense of government
deposits that don't bear interest. In this world, how can the
government adopt a passive monetary policy? How should it set the
interest rate on government deposits? If two governments adopt
passive monetary policy in this way, is it possible for the e%change
rate between the two currencies to be fixed?
45. Assume that there are two countries in the world, A and B.
Country A is fixing the exchange rate between the two countries.
For various reasons, the government of country A decides to devalue.
A year after the devaluation, the consumer price index in
A is the same as it was just before the devaluation. In other
words, the devaluation did not produ.ce inflation in country A.
Give three possible reasons for this.
46. Assume that the world is divided into two countries, A and B,
whose residents have the same tastes, wealth, and income, but
different opportunities. The countries are identical in size,
population, and total risk, but the correlation between the return
on the market in country A and the return on the market in country
B is zero. Even the interest rates and rates of inflation are the
same in the two countries, though all trade and investment between
the two countries have been banned for centuries. Now suppose that
trade and investment are allowed. What kinds of flows of securities
do you expect between the countries? What kinds of trade flows do
you expect? Do you expect any changes in the capital asset pricing
model? Do you expect any changes in real investments?
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47. Suppose that country A is fixinz its exchange rate with country
B. Continuing to maintain the sam~ exchange rate, country A
imposes a 10% tariff on all. imports, and an equi",aleilt subsidy on
all exports. This action was a surprise, but is expected to be
permanent. What will happen to the price levels in countries A
and B? What will happen to the trade balance? What has happened
to owners of foreign capital in each country?
48. Suppose that countries A and B both produce many goods and
services, some of which are traded. The exchange rate between the
two countries is floating. Now suppose that country A bposes a
10% tax on both imports and exports. What is likely to ha?pen to
the trade balance between A and B? To residents of one country
who own capital in the other?
49. Imagine a world of certainty in which there is just one good that is
produced, consumed, and used as a factor of production. Assume that
the world is divided into two countries, A and B, and that
people cannot move between countries. Goods, though, can move
freely between countries with no transport costs. Country A has a
5% tax on any assets in country A . held by residents of country
B. Country B has no such tax, nnd there are no other taxes.
Assume that initially, the residents of country A hold some of the
assets 1n country B. Country B' s residents, hOl·:ever, are
thriftier. They gradually become wealthier than the residents of
country A. In the end, they hold some of the assets of country
A. State how the relation between the interest rates in A and
B changes over time.
50. What are the advantages of a fixed exchange rate regime? What are
the advantages of a floating exchange rate regime?
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