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CHAPTER 1: THE INVESTMENT ENVIRONMENT
PROBLEM SETS
1.
Ultimately, it is true that real assets determine the material well being of an
economy. Nevertheless, individuals can benefit when financial engineering creates
new products that allow them to manage their portfolios of financial assets more
efficiently. Because bundling and unbundling creates financial products with new
properties and sensitivities to various sources of risk, it allows investors to hedge
particular sources of risk more efficiently.
2.
Securitization requires access to a large number of potential investors. To attract
these investors, the capital market needs:
1.
a safe system of business laws and low probability of confiscatory
taxation/regulation;
2.
a well-developed investment banking industry;
3.
a well-developed system of brokerage and financial transactions, and;
4.
well-developed media, particularly financial reporting.
These characteristics are found in (indeed make for) a well-developed financial
market.
Securitization leads to disintermediation; that is, securitization provides a means for
market participants to bypass intermediaries. For example, mortgage-backed
securities channel funds to the housing market without requiring that banks or thrift
institutions make loans from their own portfolios. As securitization progresses,
financial intermediaries must increase other activities such as providing short-term
liquidity to consumers and small business, and financial services.
Financial assets make it easy for large firms to raise the capital needed to finance
their investments in real assets. If Ford, for example, could not issue stocks or
bonds to the general public, it would have a far more difficult time raising capital.
Contraction of the supply of financial assets would make financing more difficult,
thereby increasing the cost of capital. A higher cost of capital results in less
investment and lower real growth.
Even if the firm does not need to issue stock in any particular year, the stock market
is still important to the financial manager. The stock price provides important
information about how the market values the firm's investment projects. For
example, if the stock price rises considerably, managers might conclude that the
market believes the firm's future prospects are bright. This might be a useful signal
to the firm to proceed with an investment such as an expansion of the firm's
business.
In addition, shares that can be traded in the secondary market are more attractive to
initial investors since they know that they will be able to sell their shares. This in
turn makes investors more willing to buy shares in a primary offering, and thus
improves the terms on which firms can raise money in the equity market.
a. No. The increase in price did not add to the productive capacity of the economy.
b.
Yes, the value of the equity held in these assets has increased.
3.
4.
5.
6.
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c.
Future homeowners as a whole are worse off, since mortgage liabilities have also
increased. In addition, this housing price bubble will eventually burst and society
as a whole (and most likely taxpayers) will endure the damage.
a. The bank loan is a financial liability for Lanni. (Lanni's IOU is the bank's
financial asset.) The cash Lanni receives is a financial asset. The new financial
asset created is Lanni's promissory note (that is, Lanni’s IOU to the bank).
b.
Lanni transfers financial assets (cash) to the software developers. In return,
Lanni gets a real asset, the completed software. No financial assets are created or
destroyed; cash is simply transferred from one party to another.
c.
Lanni gives the real asset (the software) to Microsoft in exchange for a financial
asset, 1,500 shares of Microsoft stock. If Microsoft issues new shares in order to
pay Lanni, then this would represent the creation of new financial assets.
d.
Lanni exchanges one financial asset (1,500 shares of stock) for another
($$120,000). Lanni gives a financial asset ($$50,000 cash) to the bank and gets
back another financial asset (its IOU). The loan is
since it is retired when paid off and no longer exists.
a.
Assets
Shareholders’ equityLiabilities &
7.
8.
Cash
$$ 70,000 Bank loan
$$ 50,000
Computers
30,000 Shareholders’ equity
50,000
Total
$$100,000
Total
$$100,000
Ratio of real assets to total assets = $$30,000/$$100,000 = 0.30
b.
Assets
Shareholders’ equity Liabilities &
Software
product*
$$ 70,000 Bank loan
$$ 50,000 Computers
30,000
Shareholders’ equity
50,000
Total
*Valued at cost
$$100,000
Total
$$100,000
Ratio of real assets to total assets = $$100,000/$$100,000 = 1.0
c.
Assets
Shareholders’ equity Liabilities &
Microsoft shares $$120,000 Bank loan $$ 50,000 Computers 30,000
Shareholders’ equity 100,000
Total
$$150,000
Total
$$150,000
Ratio of real assets to total assets = $$30,000/$$150,000 = 0.20
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Conclusion: when the firm starts up and raises working capital, it is characterized by
a low ratio of real assets to total assets. When it is in full production, it has a high
ratio of real assets to total assets. When the project
off for cash, financial assets once again replace real assets.
For commercial banks, the ratio is: $$140.1/$$11,895.1 = 0.0118 For non-financial
firms, the ratio is: $$12,538/$$26,572 = 0.4719 The difference should be expected
primarily because the bulk of the business of financial institutions is to make loans;
which are financial assets for financial institutions.
a. Primary- market transaction
b.
Derivative assets
c.
Investors who wish to hold gold without the complication and cost of physical
storage.
a. A fixed salary means that compensation is (at least in the short run) independent
of the firm's success. This salary structure does not tie the manager’s immediate
compensation to the success of the firm. However, the manager might view this as
the safest compensation structure and therefore value it more highly.
b.
A salary that is paid in the form of stock in the firm means that the manager
earns the most when the shareholders’ wealth is maximized. Five years of
vesting helps align the interests of the employee with the long-term performance
of the firm. This structure is therefore most likely to align the interests of
managers and shareholders. If stock compensation is overdone, however, the
manager might view it as overly risky since the manager’s career is already
linked to the firm, and this undiversified exposure would be exacerbated with a
large stock position in the firm.
c.
A profit-linked salary creates great incentives for managers to contribute to the
firm’s success. However, a manager whose salary is tied to short- term profits
will be risk seeking, especially if these short-term profits determine salary or if
the compensation structure does not bear the full cost of the project’s risks.
Shareholders, in contrast, bear the losses as well as the gains on the project, and
might be less willing to assume that risk.
Even if an individual shareholder could monitor and improve managers’
performance, and thereby increase the value of the firm, the payoff would be small,
since the ownership share in a large corporation would be very small. For example,
if you own $$10,000 of Ford stock and can increase the value of the firm by 5%, a
very ambitious goal, you benefit by only: 0.05
×
$$10,000 = $$500
In contrast, a bank that has a multimillion-dollar loan outstanding to the firm has a big
stake in making sure that the firm can repay the loan. It is clearly worthwhile for the
bank to spend considerable resources to monitor the firm.
Mutual funds accept funds from small investors and invest, on behalf of these
investors, in the national and international securities markets.
Pension funds accept funds and then invest, on behalf of current and future retirees,
thereby channeling funds from one sector of the economy to another.
9.
10.
11.
12.
13.
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Venture capital firms pool the funds of private investors and invest in start-up firms.
Banks accept deposits from customers and loan those funds to businesses, or use the
funds to buy securities of large corporations.
Treasury bills serve a purpose for investors who prefer a low-risk investment. The
lower average rate of return compared to stocks is the price investors pay for
predictability of investment performance and portfolio value.
With a “top-down” investing style, you focus on asset allocation or the broad
composition of the entire portfolio, which is the major determinant of overall
performance. Moreover, top-down management is the natural way to establish a
portfolio with a level of risk consistent with your risk tolerance. The disadvantage
of an
exclusive
emphasis on top-down issues is that you may forfeit the potential
high returns that could result from identifying and concentrating in undervalued
securities or sectors of the market.
With a “bottom-up” investing style, you try to benefit from identifying undervalued
securities. The disadvantage is that you tend to overlook the overall composition of
your portfolio, which may result in a non-diversified portfolio or a portfolio with a
risk level inconsistent with your level of risk tolerance. In addition, this technique
tends to require more active management, thus generating more transaction costs.
Finally, your analysis may be incorrect, in which case you will have fruitlessly
expended effort and money attempting to beat a simple buy-and-hold strategy.
You should be skeptical. If the author actually knows how to achieve such returns,
one must question why the author would then be so ready to sell the secret to others.
Financial markets are very competitive; one of the implications of this fact is that
riches do not come easily. High expected returns require bearing some risk, and
obvious bargains are few and far between. Odds are that the only one getting rich
from the book is its author.
Financial assets provide for a means to acquire real assets as well as an expansion
of these real assets. Financial assets provide a measure of liquidity to real assets
and allow for investors to more effectively reduce risk through diversification.
Allowing traders to share in the profits increases the traders’ willingness to assume
risk. Traders will share in the upside potential directly but only in the downside
indirectly (poor performance = potential job loss). Shareholders, by contrast, are
affected directly by both the upside and downside potential of risk.
Answers may vary, however, students should touch on the following: increased
transparency, regulations to promote capital adequacy by increasing the frequency
of gain or loss settlement, incentives to discourage excessive risk taking, and the
promotion of more accurate and unbiased risk assessment
.
14.
15.
16.
17.
18.
19.
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CHAPTER 2: ASSET CLASSES AND FINANCIAL
INSTRUMENTS
PROBLEM SETS
1.
Preferred stock is like long-term debt in that it typically promises a fixed
payment each year. In this way, it is a perpetuity. Preferred stock is also like
long-term debt in that it does not give the holder voting rights in the firm.
Preferred stock is like equity in that the firm is under no contractual obligation to
make the preferred stock dividend payments. Failure to make payments does not
set off corporate bankruptcy. With respect to the priority of claims to the assets
of the firm in the event of corporate bankruptcy, preferred stock has a higher
priority than common equity but a lower priority than bonds.
2.
Money market securities are called “cash equivalents” because of their great
liquidity. The prices of money market securities are very stable, and they can
be converted to cash (i.e., sold) on very short notice and with very low
transaction costs.
3.
(a)
A repurchase agreement is an agreement whereby the seller of a security
agrees to “repurchase” it from the buyer on an agreed upon date at an agreed
upon price. Repos are typically used by securities dealers as a means for
obtaining funds to purchase securities.
4.
The spread will widen. Deterioration of the economy increases credit risk,
that is, the likelihood of default. Investors will demand a greater premium on
debt securities subject to default risk.
5.
Corp. Bonds
Preferred Stock
Common Stock
V
oting Rights (Typically)
Yes
Contractual Obligation
Yes
Perpetual Payments
Yes
Yes
Accumulated Dividends
Yes
Fixed Payments (Typically)
Yes
Yes
Payment Preference
First
Second
Third
6.
Municipal Bond interest is tax-exempt. When facing higher marginal tax rates, a
high-income investor would be more inclined to pick tax-exempt securities.
7.
a.
You would have to pay the asked price of:
b.
c.
8.
9.
86:14 = 86.43750% of par = $$864.375
The coupon rate is 3.5% implying coupon payments of $$35.00 annually
or, more precisely, $$17.50 semiannually.
Current yield = Annual coupon income/price
= $$35.00/$$864.375 = 0.0405 = 4.05%
P = $$10,000/1.02 = $$9,803.92
The total before-tax income is $$4. After the 70% exclusion for preferred stock
dividends, the taxable income is: 0.30
×
$$4 = $$1.20
Therefore, taxes are: 0.30
×
$$1.20 = $$0.36
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After- tax income is: $$4.00 – $$0.36 = $$3.64
Rate of return is: $$3.64/$$40.00 = 9.10%
a.
You could buy: $$5,000/$$67.32 = 74.27 shares
b.
c.
d.
11.
a.
Your annual dividend income would be: 74.27
×
$$1.52 = $$112.89
The price-to-earnings ratio is 11 and the price is $$67.32. Therefore:
$$67.32/Earnings per share = 11
?
Earnings per share = $$6.12
General Dynamics closed today at $$67.32, which was $$0.47 higher than
yesterday’s price. Yesterday’s closing price was: $$66.85
At t = 0, the value of the index is: (90 + 50 + 100)/3 = 80
At t = 1, the value of the index is: (95 + 45 + 110)/3 = 83.333
b.
The rate of return is: (83.333/80)
?
1 = 4.17%
In the absence of a split, Stock C would sell for 110, so the value of the
index would be: 250/3 = 83.333
After the split, Stock C sells for 55. Therefore, we need to find the
divisor (d) such that: 83.333 = (95 + 45 + 55)/d
?
d = 2.340
The return is zero. The index remains unchanged because the return for
each stock separately equals zero.
Total market value at t = 0 is: ($$9,000 + $$10,000 + $$20,000) = $$39,000
Total market value at t = 1 is: ($$9,500 + $$9,000 + $$22,000) = $$40,500
Rate of return = ($$40,500/$$39,000) – 1 = 3.85%
The return on each stock is as follows:
r
A
= (95/90) – 1 = 0.0556
r
B
= (45/50) – 1 = –0.10 r
C
= (110/100) – 1 = 0.10
The equally-weighted average is:
[0.0556 + (-0.10) + 0.10]/3 = 0.0185 = 1.85%
13.
14.
The after-tax yield on the corporate bonds is: 0.09
×
(1 – 0.30) = 0.0630 = 6.30%
Therefore, municipals must offer at least 6.30% yields.
Equation (2.2) shows that the equivalent taxable yield is: r = r
m
/(1 – t)
a.
b.
c.
15.
4.00%
4.44%
5.00%
10.
c.
12.
a.
b.
d.
5.71%
In an equally- weighted index fund, each stock is given equal weight regardless of
its market capitalization. Smaller cap stocks will have the same weight as larger
cap stocks. The challenges are as follows:
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?
Given
equal
weights
placed
to
smaller
cap
and
larger
cap,
equalweighted indices (EWI) will tend to be more volatile than their
market-capitalization counterparts;
?
It
follows that EWIs
are not
good reflectors of the broad market
which they represent; EWIs underplay the economic importance of
larger companies;
?
Turnover rates will tend to be higher, as an EWI must be rebalanced
back
to
its
original
target.
By
design,
many
of
the
transactions
would be among the smaller, less-liquid stocks.
The higher coupon bond.
The call with the lower exercise price.
The put on the lower priced stock.
You bought the contract when the futures price was $$3.835 (see Figure
2.10). The contract closes at a price of $$3.875, which is $$0.04 more than the
original futures price. The contract multiplier is 5000. Therefore, the gain
will be: $$0.04
×
5000 = $$200.00
Open interest is 177,561 contracts.
Since the stock price exceeds the exercise price, you exercise the call.
The payoff on the option will be: $$21.75
?
$$21 = $$0.75
The cost was originally $$0.64, so the profit is: $$0.75
?
$$0.64 = $$0.11
If the call has an exercise price of $$22, you would not exercise for any
stock price of $$22 or less. The loss on the call would be the initial cost:
$$0.30
Since the stock price is less than the exercise price, you will exercise the
put.
16.
a.
b.
c.
17.
a.
18.
b.
a.
b.
c.
19.
The payoff on the option will be: $$22
?
$$21.75 = $$0.25
The option originally cost $$1.63 so the profit is: $$0.25 ? $$1.63 = ?$$1.38
There is always a possibility that the option will be in-the-money at some time
prior to expiration. Investors will pay something for this possibility of a positive
payoff.
Value of call at expiration Initial Cost
a.
0
4
-4
b.
0
4
-4
c.
0
4
-4
d.
5
4
1
e.
10
4
6
Value of put at expiration
Initial Cost
a.
10
6
4
Profit
20.
Profit
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b.
5
6
-1
c.
0
6
-6
d.
0
6
-6
e.
0
6
-6
21.
A put option conveys the
right
to sell the underlying asset at the exercise price.
A short position in a futures contract carries an
obligation
to sell the underlying
asset at the futures price.
22.
A call option conveys the
right
to buy the underlying asset at the exercise price.
A long position in a futures contract carries an
obligation
to buy the underlying
asset at the futures price.
CFA PROBLEMS
1.
(d)
2.
3.
4.
The equivalent taxable yield is: 6.75%/(1
?
0.34) = 10.23%
(a) Writing a call entails unlimited potential losses as the stock price rises.
a.
The taxable bond. With a zero tax bracket, the after-tax yield for the
taxable bond is the same as the before-tax yield (5%), which is greater than the
yield on the municipal bond.
b.
The taxable bond. The after-tax yield for the taxable bond is:
0.05
×
(1 – 0.10) = 4.5%
c.
You are indifferent. The after-tax yield for the taxable bond is:
0.05
×
(1 – 0.20) = 4.0%
The after-tax yield is the same as that of the municipal bond.
d.
The municipal bond offers the higher after-tax yield for investors in tax
brackets above 20%.
If the after-tax yields are equal, then: 0.056 = 0.08
×
(1 – t) This implies that t =
0.30 =30%.
5.
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CHAPTER 3: HOW SECURITIES ARE TRADED
PROBLEM SETS
1.
Answers to this problem will vary.
2.
The dealer sets the bid and asked price. Spreads should be higher on inactively
traded stocks and lower on actively traded stocks.
3.
a.
In principle, potential losses are unbounded, growing directly with increases in
the price of IBM.
b.
If the stop-buy order can be filled at $$128, the maximum possible loss per
share is $$8, or $$800 total. If the price of IBM shares goes above $$128, then the
stop-buy order would be executed, limiting the losses from the short sale.
4.
(a) A market order is an order to execute the trade immediately at the best possible
price. The emphasis in a market order is the speed of execution (the reduction of
execution uncertainty). The disadvantage of a market order is that the price it will be
executed at is not known ahead of time; it thus has price uncertainty.
5.
(a) The advantage of an Electronic Crossing Network (ECN) is that it can execute
large block orders without affecting the public quote. Since this security is illiquid,
large block orders are less likely to occur and thus it would not likely trade through
an ECN.
Electronic Limit-Order Markets (ELOM) transact securities with high trading
volume. This illiquid security is unlikely to be traded on an ELOM.
6.
a.
The stock is purchased for: 300
×
$$40 = $$12,000
The amount borrowed is $$4,000. Therefore, the investor put up equity, or
margin, of $$8,000.
b.
If the share price falls to $$30, then the value of the stock falls to $$9,000. By
the end of the year, the amount of the loan owed to the broker grows to: $$4,000
×
1.08 = $$4,320
Therefore, the remaining margin in the investor’s account is: $$9,000
?
$$4,320 = $$4,680
The percentage margin is now: $$4,680/$$9,000 = 0.52 = 52% Therefore,
the investor will not receive a margin call.
c.
The rate of return on the investment over the year is:
(Ending equity in the account
?
Initial equity)/Initial equity
7.
= ($$4,680
?
$$8,000)/$$8,000 =
?
0.415 =
?
41.5%
a.
The initial margin was: 0.50
×
1,000
×
$$40 = $$20,000
As a result of the increase in the stock price Old Economy Traders loses:
$$10
×
1,000 = $$10,000
Therefore, margin decreases by $$10,000. Moreover, Old Economy Traders
must pay the dividend of $$2 per share to the lender of the shares, so that the
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margin
in
the account
decreases by an additional
$$2,000. Therefore, the
remaining margin is:
b.
c.
$$20,000 – $$10,000 – $$2,000 = $$8,000
The percentage margin is: $$8,000/$$50,000 = 0.16 = 16% So there will be a
margin call.
The equity in the account decreased from $$20,000 to $$8,000 in one year, for a
rate of return of: (
?
$$12,000/$$20,000) =
?
0.60 =
?
60%
a.
The buy order will be filled at the best limit-sell order price: $$50.25
b.
The next market buy order will be filled at the next-best limit-sell order price:
$$51.50
c.
You would want to increase your inventory. There is considerable buying
demand at prices just below $$50, indicating that downside risk is limited. In
contrast, limit sell orders are sparse, indicating that a moderate buy order could
result in a substantial price increase.
a.
You buy 200 shares of Telecom for $$10,000. These shares increase in value by
10%, or $$1,000. You pay interest of: 0.08
×
$$5,000 = $$400
8.
9.
b.
The rate of return will be: $$1,000
?
$$400
=
0.12
12%
=
$$5,000
The value of the 200 shares is 200P. Equity is (200P – $$5,000). You will
receive a margin call when:
= 0.30
?
when P = $$35.71 or lower
10.
a.
Initial margin is 50% of $$5,000 or $$2,500.
b.
Total assets are $$7,500 ($$5,000 from the sale of the stock and $$2,500 put up for
margin). Liabilities are 100P. Therefore, equity is ($$7,500 – 100P). A margin
call will be issued when:
$$7,500
?
100P
= 0.30
?
when P = $$57.69 or higher
100P
11.
The total cost of the purchase is: $$40
×
500 = $$20,000
You borrow $$5,000 from your broker, and invest $$15,000 of your own funds. Your
margin account starts out with equity of $$15,000.
a.
(i)
Equity increases to: ($$44
×
500) – $$5,000 = $$17,000
Percentage gain = $$2,000/$$15,000 = 0.1333 = 13.33% (ii)
With price unchanged, equity is unchanged.
Percentage gain = zero
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(iii) Equity falls to ($$36
×
500) – $$5,000 = $$13,000
Percentage gain = (–$$2,000/$$15,000) = –0.1333 = –13.33%
The relationship between the percentage return and the percentage change in
the price of the stock is given by:
% return = % change in price
×
Total investment
= % change in
price
×
1.333
Investor's initial equity
For example, when the stock price rises from $$40 to $$44, the percentage change
in price is 10%, while the percentage gain for the investor is:
$$20,000
% return = 10%
×
= 13.33% $$15,000
The value of the 500 shares is 500P. Equity is (500P – $$5,000). You will
receive a margin call when:
= 0.25
?
when P = $$13.33 or lower
The value of the 500 shares is 500P. But now you have borrowed $$10,000
instead of $$5,000. Therefore, equity is (500P – $$10,000). You will receive a
margin call when:
= 0.25
?
when P = $$26.67 or lower
d.
With less equity in the account, you are far more vulnerable to a margin call.
By the end of the year, the amount of the loan owed to the broker grows to:
$$5,000
×
1.08 = $$5,400
The equity in your account is (500P – $$5,400). Initial equity was $$15,000.
Therefore, your rate of return after one year is as follows:
(i)
(ii)
(iii)
= 0.1067 = 10.67%
= –0.0267 = –2.67%
= –0.1600 = –16.00%
b.
c.
The relationship between the percentage return and the percentage change in
the price of Intel is given by:
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% return =
% change in price
×
initial equity
?
8%
×
borrowedinitial equity
For example, when the stock price rises from $$40 to $$44, the percentage change
in price is 10%, while the percentage gain for the investor is:
10%
×
$$20,000
Investor'Total investments
Investor'Fundss
?
8%
×
$$5,000
=10.67%
e.
$$15,000
$$15,000
The value of the 500 shares is 500P. Equity is (500P – $$5,400). You will
receive a margin call when:
= 0.25
?
when P = $$14.40 or lower
12.
a.
The gain or loss on the short position is: (–500
×
ΔP) Invested funds = $$15,000
Therefore: rate of return = (–500
×
ΔP)/15,000
The rate of return in each of the three scenarios is:
(i)
rate of return = (–500
×
$$4)/$$15,000 = –0.1333 = –13.33%
(ii)
rate of return = (–500
×
$$0)/$$15,000 = 0%
(iii)
rate of return = [–500
×
(–$$4)]/$$15,000 = +0.1333 = +13.33%
Total assets in the margin account equal:
$$20,000 (from the sale of the stock) + $$15,000 (the initial margin) = $$35,000
Liabilities are 500P. You will receive a margin call when:
$$35,000
?
500P
= 0.25
?
when P = $$56 or higher
500P
With a $$1 dividend, the short position must now pay on the borrowed shares:
($$1/share
×
500 shares) = $$500. Rate of return is now:
[(–500
×
ΔP) – 500]/15,000
(i)
rate of return = [(–500
×
$$4) – $$500]/$$15,000 = –0.1667 = –16.67%
b.
c.
(ii)
rate of return = [(–500
×
$$0) – $$500]/$$15,000 = –0.0333 = –3.33%
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(iii)
rate of return = [(–500)
×
(–$$4) – $$500]/$$15,000 = +0.1000 = +10.00%
Total assets are $$35,000, and liabilities are (500P + 500). A margin call
will be issued when:
= 0.25
?
when P = $$55.20 or higher
The broker is instructed to attempt to sell your Marriott stock as soon as the Marriott
stock trades at a bid price of $$20 or less. Here, the broker will attempt to execute,
but may not be able to sell at $$20, since the bid price is now $$19.95. The price at
which you sell may be more or less than $$20 because the stop-loss becomes a market
order to sell at current market prices.
a.
$$55.50
b.
$$55.25
c.
The trade will not be executed because the bid price is lower than the price
specified in the limit sell order.
d.
The trade will not be executed because the asked price is greater than the price
specified in the limit buy order.
a.
In an exchange market, there can be price improvement in the two market orders.
Brokers for each of the market orders (i.e., the buy order and the sell order) can agree
to execute a trade inside the quoted spread. For example, they can trade at $$55.37,
thus improving the price for both customers by $$0.12 or $$0.13 relative to the quoted
bid and asked prices. The buyer gets the stock for $$0.13 less than the quoted asked
price, and the seller receives $$0.12 more for the stock than the quoted bid price.
b.
Whereas the limit order to buy at $$55.37 would not be executed in a dealer
market (since the asked price is $$55.50), it could be executed in an exchange
market. A broker for another customer with an order to sell at market would
view the limit buy order as the best bid price; the two brokers could agree to
the trade and bring it to the specialist, who would then execute the trade.
a.
You will not receive a margin call. You borrowed $$20,000 and with another
$$20,000 of your own equity you bought 1,000 shares of Disney at $$40 per share. At
$$35 per share, the market value of the stock is $$35,000, your equity is $$15,000, and
the percentage margin is: $$15,000/$$35,000 = 42.9% Your percentage margin exceeds
the required maintenance margin.
b.
You will receive a margin call when:
13.
14.
15.
16.
= 0.35
?
when P = $$30.77 or lower
17.
The proceeds from the short sale (net of commission) were: ($$21
×
100) – $$50 =
$$2,050 A dividend payment of $$200 was withdrawn from the account.
Covering the short sale at $$15 per share costs (with commission): $$1,500 + $$50 =
$$1,550
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Therefore, the value of your account is equal to the net profit on the transaction:
$$2,050 – $$200 – $$1,550 = $$300
Note that your profit ($$300) equals (100 shares
×
profit per share of $$3). Your net
proceeds per share was:
$$21
selling price of stock
–$$15
repurchase price of stock
–$$ 2
dividend per share
–$$ 1
2 trades
×
$$0.50 commission per share
$$ 3
CFA PROBLEMS
1.
a. In addition to the explicit fees of $$70,000, FBN appears to have paid an implicit
price in underpricing of the IPO. The underpricing is $$3 per share, or a total of
$$300,000, implying total costs of $$370,000.
b.
No. The underwriters do not capture the part of the costs corresponding
to the underpricing. The underpricing may be a rational marketing
strategy. Without it, the underwriters would need to spend more resources
in order to place the issue with the public. The underwriters would then
need to charge higher explicit fees to the issuing firm. The issuing firm
may be just as well off paying the implicit issuance cost represented by
the underpricing.
2.
(d) The broker will sell, at current market price, after the first transaction at $$55 or
less.
3.
(d)
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CHAPTER 4: MUTUAL FUNDS AND
OTHER INVESTMENT COMPANIES
PROBLEM SETS
1.
The unit investment trust should have lower operating expenses. Because the
investment trust portfolio is fixed once the trust is established, it does not have to
pay portfolio managers to constantly monitor and rebalance the portfolio as
perceived needs or opportunities change. Because the portfolio is fixed, the unit
investment trust also incurs virtually no trading costs.
2.
a.
Unit investment trusts
: diversification from large-scale investing, lower
transaction costs associated with large-scale trading, low management fees,
predictable portfolio composition, guaranteed low portfolio turnover rate.
b.
Open-end mutual funds
: diversification from large-scale investing, lower
transaction costs associated with large-scale trading, professional management
that may be able to take advantage of buy or sell opportunities as they arise,
record keeping.
c.
Individual stocks and bonds
: No management fee, realization of capital gains or
losses can be coordinated with investors’ personal tax situations, portfolio can be
designed to investor’s specific risk profile.
3.
Open-end funds are obligated to redeem investor's shares at net asset value, and
thus must keep cash or cash- equivalent securities on hand in order to meet potential
redemptions. Closed-end funds do not need the cash reserves because there are no
redemptions for closed-end funds. Investors in closed-end funds sell their shares
when they wish to cash out.
4.
Balanced funds keep relatively stable proportions of funds invested in each asset
class. They are meant as convenient instruments to provide participation in a range
of asset classes. Life-cycle funds are balanced funds whose asset mix generally
depends on the age of the investor. Aggressive life-cycle funds, with larger
investments in equities, are marketed to younger investors, while conservative
lifecycle funds, with larger investments in fixed-income securities, are designed for
older investors. Asset allocation funds, in contrast, may vary the proportions
invested in each asset class by large amounts as predictions of relative performance
across classes vary. Asset allocation funds therefore engage in more aggressive
market timing.
5.
Unlike an open-end fund, in which underlying shares are redeemed when the fund
is redeemed, a closed-end fund trades as a security in the market. Thus, their prices
may differ from the NA
V
.
6.
Advantages of an ETF over a mutual fund:
?
ETFs are continuously traded and can be sold or purchased on margin
?
There are no Capital Gains Tax triggers when an ETF is sold (shares are just
sold from one investor to another)
?
Investors buy from Brokers, thus eliminating the cost of direct marketing to
individual small investors. This implies lower management fees
Disadvantages of an ETF over a mutual fund:
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?
Prices can depart from NA
V (unlike an open-end fund)
7.
?
There is a Broker fee when buying and selling (unlike a no-load fund)
The offering price includes a 6% front-end load, or sales commission, meaning that
every dollar paid results in only $$0.94 going toward purchase of shares. Therefore:
NA
V
$$10.70
= $$11.38
=
1
?
load
1
?
0.06
NA
V = offering price
×
(1 – load) = $$12.30
×
.95 = $$11.69
Stock Value held by fund
A
$$
7,000,000
B
12,000,000
C
8,000,000
D
15,000,000
Total
$$42,000,000
Offering price =
8.
9.
Net asset value = $$42,000,000
?
$$30,000 = $$10.49
4,000,000
10.
Value of stocks sold and replaced = $$15,000,000
Turnover rate =
11.
$$15,000,000
= 0.357 = 35.7%
$$42,000,000
a.
NA
V
=
$$200,000,000
?
$$3,000,000
=
$$39.40
5,000,000
Price
?
NA
V
$$36
?
$$39.40
b.
Premium (or discount) =
=
= –0.086 = -8.6%
NA
V
$$39.40
The fund sells at an 8.6% discount from NA
V
.
12.
NAV
1
?
NAV
0
+
Distributions
=
$$12.10
?
$$12.50
+
$$1.50
=
0.088
8.8%
=
NAV
0
$$12.50
a.
Start-of-year price: P
0
= $$12.00 × 1.02 = $$12.24
End-of-year price: P
1
= $$12.10 × 0.93 = $$11.25
Although NA
V increased by $$0.10, the price of the fund decreased by: $$0.99
13.
Rate of return =
P P
1
?
+
0
Distributions
=
$$11.25
?
$$12.24
+
$$1.50
=
0.042
4.2%
=
b.
P
0
$$12.24
An investor holding the same securities as the fund manager would have earned
a rate of return based on the increase in the NA
V of the portfolio:
NAV
1
?
NAV
0
+
Distributions
=
$$12.10
?
$$12.00
+
$$1.50
=
0.133
13.3%
=
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NAV
0
$$12.00
14.
a.
Empirical research indicates that past performance of mutual funds is not
highly predictive of future performance, especially for better-performing funds.
While there
may
be some tendency for the fund to be an above average performer
next year, it is unlikely to once again be a top 10% performer.
b.
On the other hand, the evidence is more suggestive of a tendency for poor
performance to persist. This tendency is probably related to fund costs and
turnover rates. Thus if the fund is among the poorest performers, investors
would be concerned that the poor performance will persist.
15.
NA
V
0
= $$200,000,000/10,000,000 = $$20
Dividends per share = $$2,000,000/10,000,000 = $$0.20 NA
V
1
is based on the 8% price gain, less the 1% 12b-1 fee: NA
V
1
= $$20
×
1.08
×
(1 – 0.01) = $$21.384
Rate of return = $$21.384
?
$$20
+
$$0.20 = 0.0792 = 7.92%
$$20
16.
The excess of purchases over sales must be due to new inflows into the fund.
Therefore, $$400 million of stock previously held by the fund was replaced by new
holdings. So turnover is: $$400/$$2,200 = 0.182 = 18.2%
Fees paid to investment managers were: 0.007
×
$$2.2 billion = $$15.4 million Since
the total expense ratio was 1.1% and the management fee was 0.7%, we conclude
that 0.4% must be for other expenses. Therefore, other administrative expenses
were: 0.004
×
$$2.2 billion = $$8.8 million
18.
As an initial approximation, your return equals the return on the shares minus the
total of the expense ratio and purchase costs: 12%
?
1.2%
?
4% = 6.8%
But the precise return is less than this because the 4% load is paid up front, not at
the end of the year.
17.
To purchase the shares, you would have had to invest: $$20,000/(1
?
0.04) = $$20,833
The shares increase in value from $$20,000 to: $$20,000
×
(1.12
?
0.012) = $$22,160
The rate of return is: ($$22,160
?
$$20,833)/$$20,833 = 6.37%
19.
Assume $$1000 investment
Yearly Growth
1 Year (@ 6%)
3 Years (@ 6%)
10 Years (@ 6%)
20.
a.
=
$$10
Loaded-Up Fund
(1
+ ? ?
r
.01
.0075)
$$1,042.50
$$1,133.00
$$1,516.21
Economy Fund
(.98)
× + ?
(1
.0025)
$$1,036.35
$$1,158.96
$$1,714.08
r
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b. The redemption of 1 million shares will most likely trigger capital gains taxes
which will lower the remaining portfolio by an amount greater than $$10,000,000
(implying a remaining total value less than $$440,000,000). The outstanding shares
fall to 43 million and the NA
V drops to below $$10.
21.
Suppose you have $$1,000 to invest. The initial investment in Class A shares is $$940
net of the front-end load. After four years, your portfolio will be worth:
$$940
×
(1.10)
4
= $$1,376.25
Class B shares allow you to invest the full $$1,000, but your investment performance
net of 12b-1 fees will be only 9.5%, and you will pay a 1% back-end load fee if you
sell after four years. Your portfolio value after four years will be:
$$1,000
×
(1.095)
4
= $$1,437.66
After paying the back-end load fee, your portfolio value will be:
$$1,437.66
×
.99 = $$1,423.28
Class B shares are the better choice if your horizon is four years.
With a fifteen-year horizon, the Class A shares will be worth:
$$940
×
(1.10)
15
= $$3,926.61
For the Class B shares, there is no back- end load in this case since the horizon is
greater than five years. Therefore, the value of the Class B shares will be:
$$1,000
×
(1.095)
15
= $$3,901.32
At this longer horizon, Class B shares are no longer the better choice. The effect of
Class B's 0.5% 12b-1 fees accumulates over time and finally overwhelms the 6%
load charged to Class A investors.
22.
a.
After two years, each dollar invested in a fund with a 4% load and a portfolio
return equal to r will grow to: $$0.96
×
(1 + r – 0.005)
2
Each dollar invested in the
bank CD will grow to: $$1
×
1.06
2
If the mutual fund is to be the better investment, then the portfolio return (r)
must satisfy:
0.96
×
(1 + r – 0.005)
2
> 1.06
2
0.96
×
(1 + r – 0.005)
2
> 1.1236
(1 + r – 0.005)
2
> 1.1704
1 + r – 0.005 > 1.0819
1 + r > 1.0869
Therefore: r > 0.0869 = 8.69%
b.
If you invest for six years, then the portfolio return must satisfy:
0.96
×
(1 + r – 0.005)
6
> 1.06
6
= 1.4185
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(1 + r – 0.005)
6
> 1.4776
1 + r – 0.005 > 1.0672
r > 7.22%
The cutoff rate of return is lower for the six-year investment because the “fixed
cost” (the one-time front- end load) is spread over a greater number of years.
c. With a 12b-1 fee instead of a front-end load, the portfolio must earn a rate of return
(r) that satisfies:
1 + r – 0.005 – 0.0075 > 1.06
In this case, r must exceed 7.25% regardless of the investment horizon.
23.
The turnover rate is 50%. This means that, on average, 50% of the portfolio is sold
and replaced with other securities each year. Trading costs on the sell orders are
0.4% and the buy orders to replace those securities entail another 0.4% in trading
costs. Total trading costs will reduce portfolio returns by: 2
×
0.4%
×
0.50 = 0.4%
24.
For the bond fund, the fraction of portfolio income given up to fees is:
0.6%
= 0.150 = 15.0%
4.0%
For the equity fund, the fraction of investment earnings given up to fees is:
0.6%
= 0.050 = 5.0%
12.0%
Fees are a much higher fraction of expected earnings for the bond fund, and
therefore may be a more important factor in selecting the bond fund.
This may help to explain why unmanaged unit investment trusts are concentrated in
the fixed income market. The advantages of unit investment trusts are low turnover,
low trading costs and low management fees. This is a more important concern to
bond-market investors.
25.
Suppose that finishing in the top half of all portfolio managers is purely luck, and
that the probability of doing so in any year is exactly ?. Then the probability that
any particular manager would finish in the top half of the sample five years in a row
is (?)
5
= 1/32. We would then expect to find that [350
×
(1/32)] = 11 managers
finish in the top half for each of the five consecutive years. This is precisely what
we found. Thus, we should not conclude that the consistent performance after five
years is proof of skill. We would expect to find eleven managers exhibiting
precisely this level of
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CHAPTER 5: LEARNING ABOUT RETURN AND RISK FROM
THE HISTORICAL RECORD
PROBLEM SETS
1.
The Fisher equation predicts that the nominal rate will equal the equilibrium
real rate plus the expected inflation rate. Hence, if the inflation rate increases
from 3% to 5% while there is no change in the real rate, then the nominal rate
will increase by 2%. On the other hand, it is possible that an increase in the
expected inflation rate would be accompanied by a change in the real rate of
interest. While it is conceivable that the nominal interest rate could remain
constant as the inflation rate increased, implying that the real rate decreased as
inflation increased, this is not a likely scenario.
2.
If we assume that the distribution of returns remains reasonably stable over the
entire history, then a longer sample period (i.e., a larger sample) increases the
precision of the estimate of the expected rate of return; this is a consequence of
the fact that the standard error decreases as the sample size increases.
However, if we assume that the mean of the distribution of returns is changing
over time but we are not in a position to determine the nature of this change,
then the expected return must be estimated from a more recent part of the
historical period. In this scenario, we must determine how far back,
historically, to go in selecting the relevant sample. Here, it is likely to be
disadvantageous to use the entire dataset back to 1880.
3.
The true statements are (c) and (e). The explanations follow.
Statement (c): Let
σ
= the annual standard deviation of the risky investments
and
σ
1
= the standard deviation of the first investment alternative over the two-
year period. Then:
σ
1
=
2
×
σ
Therefore, the annualized standard deviation for the first investment
alternative is equal to:
σ
=<
σ
2
Statement (e): The first investment alternative is more attractive to investors
with lower degrees of risk aversion. The first alternative (entailing a
sequence of two identically distributed and uncorrelated risky investments) is
riskier than the second alternative (the risky investment followed by a risk-
free investment). Therefore, the first alternative is more attractive to
investors with lower degrees of risk aversion. Notice, however, that if you
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mistakenly believed that ‘time diversification’ can reduce the total risk of a
sequence of risky investments, you would have been tempted to conclude that
the first alternative is less risky and therefore more attractive to more risk-
averse investors. This is clearly not the case; the two-year standard deviation
of the first alternative is greater than the two-year standard deviation of the
second alternative.
4.
For the money market fund, your holding period return for the next year depends
on the level of 30-day interest rates each month when the fund rolls over maturing
securities. The one-year savings deposit offers a 7.5% holding period return for
the year. If you forecast that the rate on money market instruments will increase
significantly above the current 6% yield, then the money market fund might result
in a higher HPR than the savings deposit. The 20-year Treasury bond offers a
yield to
maturity of 9% per year, which is 150 basis points higher than the rate on
the one-year savings deposit; however, you could earn a one- year HPR much less
than 7.5% on the bond if long-term interest rates increase during the year. If
Treasury bond yields rise above 9%, then the price of the bond will fall, and the
resulting capital loss will wipe out some or all of the 9% return you would have
earned if bond yields had remained unchanged over the course of the year.
a.
If businesses reduce their capital spending, then they are likely to decrease
their demand for funds. This will shift the demand curve in Figure 5.1 to the left
and reduce the equilibrium real rate of interest.
b.
Increased household saving will shift the supply of funds curve to the right
and cause real interest rates to fall.
c.
Open market purchases of U.S. Treasury securities by the Federal Reserve
Board are equivalent to an increase in the supply of funds (a shift of the
supply curve to the right). The equilibrium real rate of interest will fall.
a.
The “Inflation-Plus” CD is the safer investment because it guarantees the
purchasing power of the investment. Using the approximation that the real rate
equals the nominal rate minus the inflation rate, the CD provides a real rate of
1.5% regardless of the inflation rate.
b.
The expected return depends on the expected rate of inflation over the next
year. If the expected rate of inflation is less than 3.5% then the conventional
CD offers a higher real return than the Inflation-Plus CD; if the expected rate
of inflation is greater than 3.5%, then the opposite is true.
c.
If you expect the rate of inflation to be 3% over the next year, then the
conventional CD offers you an expected real rate of return of 2%, which is
0.5% higher than the real rate on the inflation-protected CD. But unless you
know that inflation will be 3% with certainty, the conventional CD is also
riskier. The question of which is the better investment then depends on your
attitude towards risk versus return. You might choose to diversify and invest
part of your funds in each.
5.
6.
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No. We cannot assume that the entire difference between the risk-free
nominal rate (on conventional CDs) of 5% and the real risk-free rate (on
inflation-protected CDs) of 1.5% is the expected rate of inflation. Part of the
difference is probably a risk premium associated with the uncertainty
surrounding the real rate of return on the conventional CDs. This implies that
the expected rate of inflation is less than 3.5% per year.
E(r) = [0.35 × 44.5%] + [0.30 × 14.0%] + [0.35 × (–16.5%)] = 14%
σ
2
= [0.35 ×
(44.5 – 14)
2
] + [0.30 × (14 – 14)
2
] + [0.35 × (–16.5 – 14)
2
] = 651.175
σ
=
25.52%
The mean is unchanged, but the standard deviation has increased, as the
probabilities of the high and low returns have increased.
Probability distribution of price and one-year holding period return for a 30year
U.S. Treasury bond (which will have 29 years to maturity at year’s end):
d.
7.
8.
Capital
Coupon
Economy
Probability
YTM
Price
Gain
Interest
HPR
Boom
0.20
11.0%
$$ 74.05
?
$$25.95
$$8.00
?
17.95%
Normal Growth
0.50
8.0%
$$100.00 $$ 0.00
$$8.00
8.00%
Recession
0.30
7.0%
$$112.28 $$12.28
$$8.00
20.28%
9.
E(q) = (0 × 0.25) + (1 × 0.25) + (2 × 0.50) = 1.25
σ
q
= [0.25 × (0 – 1.25)
2
+ 0.25 × (1 – 1.25)
2
+ 0.50 × (2 – 1.25)
2
]
1/2
= 0.8292
10.
(a) With probability 0.9544, the value of a normally distributed variable will fall
within two standard deviations of the mean; that is, between –40% and 80%.
11.
From Table 5.3 and Figure 5.6, the average risk premium for the period was:
(11.63%
?
3.71%) = 7.92% per year
Adding 7.92% to the 3% risk-free interest rate, the expected annual HPR for
the S&P 500 stock portfolio is: 3.00% + 7.92% = 10.92%
12.
The average rates of return and standard deviations are quite different in the sub
periods:
STOCKS
Standard
Mean
Skewness
Kurtosis
Deviation
1926 – 2005
12.15%
2
0.26%
-
0.3605
-0.0673
1976 – 2005
13.85%
1
5.68%
-
0.4575
-0.6489
1926 – 1941
6.39%
3
0.33%
-0.0022
-1.0716
BON
DS
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Standard
Skewness
Kurtosis
Deviation
1926 – 2005 5.68% 8.09% 0.9903 1.6314 1976 – 2005 9.57%
10.32% 0.3772 -0.0329
1926 – 1941
4.42%
4.32%
-0.5036
0.5034
The most relevant statistics to use for projecting into the future would seem to
be the statistics estimated over the period 1976-2005, because this later period
seems to have been a different economic regime. After 1955, the U.S.
economy entered the Keynesian era, when the Federal government actively
attempted to stabilize the economy and to prevent extremes in boom and bust
cycles. Note that the standard deviation of stock returns has decreased
substantially in the later period while the standard deviation of bond returns
has increased.
Mean
13.
a
r
%
b.
r
≈
R
?
i = 80%
?
70% = 10%
Clearly, the approximation gives a real HPR that is too
high.
14.
From Table 5.2, the average real rate on T-bills has been: 0.70%
a.
T-bills: 0.72% real rate + 3% inflation = 3.70%
b.
Expected return on large stocks:
3.70% T-bill rate + 8.40% historical risk premium = 12.10%
c.
The risk premium on stocks remains unchanged. A premium, the difference
between two rates, is a real value, unaffected by inflation.
15.
Real interest rates are expected to rise. The investment activity will shift the
demand for funds curve (in Figure 5.1) to the right. Therefore the equilibrium
real interest rate will increase.
16.
a.
Probability Distribution of the HPR on the Stock Market and Put:
STOCK
PUT
HPR
Ending Value
State oEconomy f the
Probability Ending Price + Dividend
HPR
Excellent
0.25
Poor
Crash
$$ 131.00
31.00%
$$ 0.00
?
100% Good
0.45
$$ 20.25
$$ 64.00
$$ 114.00
68.75%
433.33%
14.00%
$$ 0.00
?
100%
0.25
$$ 93.25
?6.75%
0.05
$$ 48.00
?
52.00%
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Remember that the cost of the index fund is $$100 per share, and the cost
of the put option is $$12.
The cost of one share of the index fund plus a put option is $$112. The
probability distribution of the HPR on the portfolio is:
b.
State of the
Economy
Probability Ending Price + Dividend Put +
HPR
Excellent
0.25
$$ 114.00
$$ 131.00
17.0%
= (131
?
112)/112 Good
0.45
1.8%
= (114
?
112)/112
Poor 0.25 $$ 113.50 1.3% = (113.50
?
112)/112 Crash 0.05 $$ 112.00 0.0% = (112
?
112)/112
c.
Buying the put option guarantees the investor a minimum HPR of 0.0%
regardless of what happens to the stock's price. Thus, it offers insurance
against a price decline.
17.
The probability distribution of the dollar return on CD plus call option is:
State of the Economy
Probability Ending Value of CD Ending Value of Call Combined
Value
Excellent
0.25
$$ 114.00
$$16.50
$$130.50
Good
0.45
$$ 114.00
$$ 0.00
$$114.00
Poor
0.25
$$ 114.00
$$ 0.00
$$114.00
Crash
0.05
$$ 114.00
$$ 0.00
$$114.00
CFA PROBLEMS
1.
The expected dollar return on the investment in equities is $$18,000 compared to the
$$5,000 expected return for T-bills. Therefore, the expected risk premium is $$13,000.
2.
3.
E(r) = [0.2 × (
?
25%)] + [0.3 × 10%] + [0.5 × 24%] =10%
E(r
X
) = [0.2 × (?20%)] + [0.5 × 18%] + [0.3 × 50%] =20% E(r
Y
) = [0.2 × (?15%)]
+ [0.5 × 20%] + [0.3 × 10%] =10%
4.
σ
X
2
= [0.2 × (– 20 – 20)
2
] + [0.5 × (18 – 20)
2
] + [0.3 × (50 – 20)
2
] = 592
σ
X
=
24.33%
σ
Y
2
= [0.2 × (– 15 – 10)
2
] + [0.5 × (20 – 10)
2
] + [0.3 × (10 – 10)
2
] = 175
σ
Y
= 13.23%
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5.
6.
E(r) = (0.9 × 20%) + (0.1 × 10%) =19%
?
$$1,900 in returns
The probability that the economy will be neutral is 0.50, or 50%.
Given
a neutral
economy, the stock will experience poor performance 30% of the time. The
probability of both poor stock performance and a neutral economy is therefore:
0.30
×
0.50 = 0.15 = 15%
E(r) = (0.1 × 15%) + (0.6 × 13%) + (0.3 × 7%) = 11.4%
7.
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CHAPTER 6: RISK A
VERSION AND
CAPITAL ALLOCATION TO RISKY ASSETS
PROBLEM SETS
1.
(e)
2.
(b) A higher borrowing rate is a consequence of the risk of the borrowers’ default.
In perfect markets with no additional cost of default, this increment would equal
the value of the borrower’s option to default, and the Sharpe measure, with
appropriate treatment of the default option, would be the same. However, in
reality there are costs to default so that this part of the increment lowers the
Sharpe ratio. Also, notice that answer (c) is not correct because doubling the
expected return with a fixed risk-free rate will more than double the risk premium
and the Sharpe ratio.
3.
Assuming no change in risk tolerance, that is, an unchanged risk aversion
coefficient (A), then higher perceived volatility increases the denominator of the
equation for the optimal investment in the risky portfolio (Equation 6.7). The
proportion invested in the risky portfolio will therefore decrease.
4.
a.
The expected cash flow is: (0.5
×
$$70,000) + (0.5
×
200,000) = $$135,000 With
a risk premium of 8% over the risk-free rate of 6%, the required rate of return is
14%. Therefore, the present value of the portfolio is:
b.
$$135,000/1.14 = $$118,421
If the portfolio is purchased for $$118,421, and provides an expected cash
inflow of $$135,000, then the expected rate of return [E(r)] is as follows:
$$118,421
×
[1 + E(r)] = $$135,000
Therefore, E(r) =
14%. The portfolio price is set to equate the expected rate
of return with the required rate of return.
If the risk premium over T-bills is now 12%, then the required return is:
6% + 12% = 18%
The present value of the portfolio is now:
d.
$$135,000/1.18 = $$114,407
For a given expected cash flow, portfolios that command greater risk premia
must sell at lower prices. The extra discount from expected value is a
penalty for risk.
c.
5.
When we specify utility by U =
E(r) – 0.5A
σ
2
, the utility level for T-bills is:
0.07 The utility level for the risky portfolio is:
U = 0.12 – 0.5 × A × (0.18)
2
= 0.12 – 0.0162 × A
In order for the risky portfolio to be preferred to bills, the following must hold:
0.12 – 0.0162A > 0.07
?
A < 0.05/0.0162 = 3.09
A must be less than 3.09 for the risky portfolio to be preferred to bills.
6.
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Points on the curve are derived by solving for E(r) in the following equation:
U = 0.05 = E(r) – 0.5Aσ
2
= E(r) – 1.5σ
2
The values of E(r), given the values of
σ
2
, are therefore:
σ
σ
2
E(r)
0.00
0.0000
0.05000
0.05
0.0025
0.05375
0.10
0.0100
0.06500
0.15
0.0225
0.08375
0.20
0.0400
0.11000 0.25
0.0625
0.14375
The bold line in the graph on the next page (labeled Q6, for Question 6) depicts
the indifference curve.
7.
Repeating the analysis in Problem 6, utility is now:
U = E(r) – 0.5A
σ
2
= E(r) – 2.0
σ
2
= 0.05
The equal-utility combinations of expected return and standard deviation are
presented in the table below. The indifference curve is the upward sloping line in
the graph on the next page, labeled Q7 (for Question 7).
σ
0.00
0.05
0.10
0.15
0.20
0.25
σ
2
0.0000
0.0025
0.0100
0.0225
0.0400
0.0625
E(r)
0.0500
0.0550
0.0700
0.0950
0.1300
0.1750
The indifference curve in Problem 7 differs from that in Problem 6 in slope.
When A increases from 3 to 4, the increased risk aversion results in a greater
slope for the indifference curve since more expected return is needed in order to
compensate for additional σ.
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E(r)
U(Q7,A=4)
U(Q6,A=3)
5
U(Q8,A=0)
U(Q9,A<0)
8.
The coefficient of risk aversion for a risk neutral investor is zero. Therefore, the
corresponding utility is equal to the portfolio’s expected return. The
corresponding indifference curve in the expected return-standard deviation plane
is a horizontal line, labeled Q8 in the graph above (see Problem 6).
9.
A risk lover, rather than penalizing portfolio utility to account for risk, derives
greater utility as variance increases. This amounts to a negative coefficient of risk
aversion. The corresponding indifference curve is downward sloping in the graph
above (see Problem 6), and is labeled Q9.
10.
The portfolio expected return and variance are computed as follows:
W(1)
Bills
r(2)
Bills
W(3)
Index
r(4)
Index
(1)×(2)+(3)×(4)r
Portfolio
20%
σ
2 Portfolio
(3)
σ
Portfolio
×
0.0400
0.0256
0.0144
0.0064
0.0016
0.0000
0.0
0.2
0.4
0.6
0.8
1.0
5%
5%
5%
5%
5%
5%
1.0
0.8
0.6
0.4
0.2
0.0
13.0% 13.0% = 0.130
20% = 0.20
13.0% 11.4% = 0.114
16% = 0.16
13.0% 9.8% = 0.098
12% = 0.12
13.0% 8.2% = 0.082
8% = 0.08
13.0% 6.6% = 0.066
4% = 0.04
13.0% 5.0% = 0.050
0% = 0.00
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11.
Computing utility from U = E(r) – 0.5
×
A
σ
2
= E(r) –
σ
2
, we arrive at the values
in the column labeled U(A = 2) in the following table:
W
Bills
0.0
0.2
0.4
0.6
0.8
1.0
W
Index
1.0
0.8
0.6
0.4
0.2
0.0
r
Portfolio
σ
0.130
0.114
0.098
0.082
0.066
0.050
Portfolio
σ
2Portfolio
0.20
0.16
0.12
0.08
0.04
0.00
0.0400
0.0256
0.0144
0.0064
0.0016
0.0000
U(A = 2)
0.0900
0.0884
0.0836
0.0756
0.0644
0.0500
U(A = 3)
.0700
.0756
.0764
.0724
.0636
.0500
The column labeled U(A = 2) implies that investors with A = 2 prefer a portfolio that
is invested 100% in the market index to any of the other portfolios in the table.
12.
The column labeled U(A = 3) in the table above is computed from:
U = E(r) – 0.5A
σ
2
= E(r) – 1.5
σ
2
The more risk averse investors prefer the portfolio that is invested 40% in the
market, rather than the 100% market weight preferred by investors with A = 2.
13.
Expected return = (0.7
×
18%) + (0.3
×
8%) = 15% Standard deviation = 0.7
×
28% = 19.6%
14.
Investment proportions:
30.0% in T-bills
0.7 × 25% = 17.5% in Stock A
0.7 × 32% = 22.4% in Stock B
0.7 × 43% = 30.1% in Stock C
15.
Your reward-to-volatility ratio:
S
=
0.3571
0.3571
Client's reward-to-volatility ratio:
S
=
30
25
20
E(r)
15
10
5
0
0
10
20
16.
CAL (Slope = 0.3571)
Client
P
30
40
%
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σ
(%)
17.
a.
E(r
C
) = r
f
+ y × [E(r
P
) – r
f
] = 8 + y × (18
?
8) If the expected return
for the portfolio is 16%, then:
16% = 8% + 10% × y
?
y
=
0.8
Therefore, in order to have a portfolio with expected rate of return equal to
16%, the client must invest 80% of total funds in the risky portfolio and 20%
in T-bills.
b.
Client’s investment proportions:
20.0% in T-bills
0.8 × 25% =
20.0% in Stock A
0.8 × 32% = 25.6% in Stock B
0.8 × 43% = 34.4% in Stock C
c.
σ
C
= 0.8
×
σ
P
= 0.8
×
28% = 22.4%
18.
a.
σ
C
= y
×
28%
If your client prefers a standard deviation of at most 18%, then:
y = 18/28 = 0.6429 = 64.29% invested in the risky portfolio
b.
E r
(
C
)
=
.08
+
.1
×
y
=
.08
(0.6429
+
.
1)
× =
14.429%
19.
a.
y*
=
E(rA
P
)
?
2
r
f
=
=
=
0.3644
σ
P
Therefore, the client’s optimal proportions are: 36.44% invested in the risky
portfolio and 63.56% invested in T-bills.
b.
E(r
C
) = 8 + 10 × y* = 8 + (0.3644
×
10) = 11.644%
σ
C
= 0.3644
×
28 = 10.203%
20.
a.
If the period 1926 - 2009 is assumed to be representative of future
expected performance, then we use the following data to compute the
fraction allocated to equity: A = 4, E(r
M
) ? r
f
= 7.93%, σ
M
= 20.81% (we
use the standard deviation of the risk premium from Table 6.7). Then y
*
is given by:
y*
=
E(r )A
M
σ
2
M
?
r
f
4
× =
2
0.4578
b.
That is, 45.78% of the portfolio should be allocated to equity and 54.22%
should be allocated to T-bills.
If the period 1968 - 1988 is assumed to be representative of future expected
performance, then we use the following data to compute the fraction
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allocated to equity: A = 4, E(r
M
) ? r
f
= 3.44%, σ
M
= 16.71% and y* is given
by:
y*
=
E(r )A
M
σ
2
M
?
r
f
4
× =
2
0.3080
Therefore, 30.80% of the complete portfolio should be allocated to equity and
69.20% should be allocated to T-bills.
c.
In part (b), the market risk premium is expected to be lower than in part (a)
and market risk is higher. Therefore, the reward-to- volatility
ratio
is
expected to be lower in part (b), which explains the greater proportion
invested in T-bills.
21.
a.
E(r
C
) = 8% = 5% + y × (11% – 5%)
?
y
=
0.5
σ
C
= y × σ
P
= 0.50
×
15% = 7.5%
The first client is more risk averse,
allowing a
smaller standard deviation.
22.
Johnson requests the portfolio standard deviation to equal one half the market
b.
c.
portfolio standard deviation. The market portfolio
σ
M
=
20%which implies
σ
P
=
10%. The intercept of the CML equals
r
f
=
0.05and the slope of the CML equals
the Sharpe ratio for the market portfolio (35%). Therefore using the CML:
?
r
f
E r
(
P
)
= +
r
f
E r
(
M
)
σ
P
= +
0.05
0.35
×
0.10
=
0.085
=
8.5%
σ
M
23.
Data: r
f
= 5%, E(r
M
) = 13%, σ
M
= 25%, and r
f
B
= 9% The CML and indifference
curves are as follows:
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E(r)
borrow
lend
13
9
5
P
CAL
CML
25
σ
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24.
For
y
to be less than 1.0 (that the investor is a lender), risk aversion (A) must be
large enough such that:
y
=
E(rA
M
σ)
2
M
?
r
f
<
1
?
A
>
0.05
2
=
1.28
For
y
to be greater than 1 (the investor is a borrower), A must be small enough:
y
=
E(rA
M
σ
)
2M
?
r
f
>
1
?
A
<
=
0.64
For values of risk aversion within this range, the client will neither borrow nor lend,
but will hold a portfolio comprised only of the optimal risky portfolio:
y = 1 for 0.64 ≤ A ≤ 1.28
25.
a.
The graph for Problem 23 has to be redrawn here, with: E(r
P
) = 11% and σ
P
=
15%
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E(r)
M
13
11
9
5
CML
CAL
F
15
25
σ
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b.
For a lending position: A
For a borrowing position: A
Therefore, y = 1 for 0.89 ≤ A ≤ 2.67
26.
The maximum feasible fee, denoted f, depends on the reward-to-variability ratio.
For y < 1, the lending rate, 5%, is viewed as the relevant risk-free rate, and we
solve for f as follows:
.11
?
?
.05
f
=
.13
?
.05
?
f
=?
.06
.15
×
==
.08
.012
1.2%
.15
.25
.25
For y > 1, the borrowing rate, 9%, is the relevant risk-free rate. Then we notice that, even
without a fee, the active fund is inferior to the passive fund because:
`
More risk tolerant investors (who are more inclined to borrow) will not be clients of
the fund. We find that f is negative: that is, you would need to
pay
investors to
choose your active fund. These investors desire higher risk-higher return complete
portfolios and thus are in the borrowing range of the relevant CAL. In this range,
the reward-to-variability ratio of the index (the passive fund) is better than that of
the managed fund.
27.
a.
Slope of the CML
=
The diagram follows.
18
16
14
Exp
12
ect
ed
10
Ret
8
run
6
4
2
0
0
0.20
CML and CAL
CAL: Slope = 0.3571
CML: Slope = 0.20
10
Standard Deviation
20
30
b.
My fund allows an investor to achieve a higher mean for any given standard deviation
than would a passive strategy, i.e., a higher expected return for any given level of risk.
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28.
a.
With 70% of his money invested in my fund’s portfolio, the client’s expected
b.
29.
a.
return is 15% per year and standard deviation is 19.6% per year. If he shifts
that money to the passive portfolio (which has an expected return of 13% and
standard deviation of 25%), his overall expected return becomes:
E(r
C
) = r
f
+ 0.7 × [E(r
M
) ? r
f
] = .08 + [0.7
×
(.13 – .08)] = .115 = 11.5% The
standard deviation of the complete portfolio using the passive portfolio
would be:
σ
C
= 0.7
×
σ
M
= 0.7
×
25% = 17.5%
Therefore, the shift entails a decrease in mean from 15% to 11.5% and a
decrease in standard deviation from 19.6% to 17.5%. Since both mean return
and
standard deviation decrease, it is not yet clear whether the move is
beneficial. The disadvantage of the shift is that, if the client is willing to
accept a mean return on his total portfolio of 11.5%, he can achieve it with a
lower standard deviation using my fund rather than the passive portfolio. To
achieve a target mean of 11.5%, we first write the mean of the complete
portfolio as a function of the proportion invested in my fund (
y
):
E(r
C
) = .08 + y × (.18 ? .08) = .08 + .10 × y
Our target is: E(r
C
) = 11.5%. Therefore, the proportion that must be invested in
my fund is determined as follows:
.115 = .08 + .10 × y
?
y
=
0.35
The standard deviation of this portfolio would be:
σ
C
= y
×
28% = 0.35
×
28% = 9.8%
Thus, by using my portfolio, the same 11.5% expected return can be achieved
with a standard deviation of only 9.8% as opposed to the standard deviation of
17.5% using the passive portfolio.
The fee would reduce the reward-to-volatility ratio, i.e., the slope of the CAL.
The client will be indifferent between my fund and the passive portfolio if the
slope of the after-fee CAL and the CML are equal. Let f denote the fee:
Slope of CAL with fee
=
.18
?
?
.08
f
=
.10
?
f
.28
.28
Slope of CML (which requires no fee)
=
0.20
Setting these slopes equal we have:
.10
?
f
=
0.20
? =
f
0.044
4.4
=
% per year
.28
The formula for the optimal proportion to invest in the passive portfolio is:
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y*
=
E(rA
M
σ
)
?
r
f
2M
Substitute the following: E(r
M
) = 13%; r
f
= 8%; σ
M
= 25%; A = 3.5:
0.13
0.08
y*
=
=
0.2286 = 22.86% in the
passive portfolio 3.5
0.25
b.
The answer here is the same as the answer to Problem 28(b). The fee that you
can charge a client is the same regardless of the asset allocation mix of the
client’s portfolio. You can charge a fee that will equate the reward-to-volatility
ratio
of your portfolio to that of your competition.
CFA PROBLEMS
1.
Utility for each investment = E(r) – 0.5 × 4 ×
σ
2
We choose the investment with the highest utility value, Investment 3.
Investment Expected return E(r) deviation Standard
σ
U
tility U
2.
3.
4.
5.
6.
7.
8.
0.12
0.30
-0.0600
0.15
0.50
-0.3500 3
0
.21
0.16
0.1588
4
0.24
0.21
0.1518
When investors are risk neutral, then A = 0; the investment with the highest utility is
Investment 4 because it has the highest expected return.
(b)
Indifference curve 2
Point E
(0.6 × $$50,000) + [0.4 × (
?
$$30,000)]
?
$$5,000 = $$13,000
(b)
Expected return for equity fund = T-bill rate + risk premium = 6% + 10% = 16% Expected
rate of return of the client’s portfolio = (0.6
×
16%) + (0.4
×
6%) = 12%
Expected return of the client’s portfolio = 0.12 × $$100,000 = $$12,000
(which implies expected total wealth at the end of the period = $$112,000)
Standard deviation of client’s overall portfolio = 0.6
×
14% = 8.4%
1
2
9.
Reward-to- volatility ratio =
=
0.71
CHAPTER 6: APPENDIX
1.
By year end, the $$50,000 investment will grow to: $$50,000 × 1.06 = $$53,000
Without insurance
, the probability distribution of end-of-year wealth is:
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No fire
Fire
Probability
Wealth
0.999
$$253,000
0.001
$$ 53,000
For this distribution, expected utility is computed as follows:
E[U(W)] = [0.999 × ln(253,000)] + [0.001 × ln(53,000)] = 12.439582 The
certainty equivalent is:
W
CE
= e
12.439582
= $$252,604.85
With fire insurance
, at a cost of $$P, the investment in the risk-free asset is:
$$(50,000 – P)
Year-end wealth will be certain (since you are fully insured) and equal to:
[$$(50,000 – P) × 1.06] + $$200,000 Solve
for P in the following equation:
[$$(50,000 – P) × 1.06] + $$200,000 = $$252,604.85
?
P = $$372.78
This is the most you are willing to pay for insurance. Note that the expected loss is
“only” $$200, so you are willing to pay a substantial risk premium over the expected
value of losses. The primary reason is that the value of the house is a large
proportion of your wealth.
2.
a.
With insurance coverage for one-half the value of the house, the premium
is $$100, and the investment in the safe asset is $$49,900. By year end,
the investment of $$49,900 will grow to: $$49,900 × 1.06 = $$52,894 If
there is a fire, your insurance proceeds will be $$100,000, and the
probability distribution of end-of- year wealth is:
No fire
Fire
Probability
Wealth
0.999
$$252,894
0.001
$$152,894
For this distribution, expected utility is computed as follows:
E[U(W)] = [0.999 × ln(252,894)] + [0.001 × ln(152,894)] = 12.4402225 The
certainty equivalent is:
W
CE
= e
12.4402225
= $$252,766.77
b.
With insurance coverage for the full value of the house, costing $$200, end-
ofyear wealth is certain, and equal to:
[($$50,000 – $$200) × 1.06] + $$200,000 = $$252,788
Since wealth is certain, this is also the certainty equivalent wealth of the fully
insured position.
With insurance coverage for 1? times the value of the house, the premium
is $$300, and the insurance pays off $$300,000 in the event of a fire. The
investment in the safe asset is $$49,700. By year end, the investment of
c.
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$$49,700 will grow to: $$49,700 × 1.06 = $$52,682 The
probability distribution of end-of-year wealth is:
No fire
Fire
Probability
Wealth
0.999
$$252,682
0.001
$$352,682
For this distribution, expected utility is computed as follows:
E[U(W)] = [0.999 × ln(252,682)] + [0.001 × ln(352,682)] = 12.4402205 The
certainty equivalent is:
W
CE
= e
12.440222
= $$252,766.27
Therefore, full insurance dominates both over- and under-insurance.
Overinsuring creates a gamble (you actually gain when the house burns
down). Risk is minimized when you insure exactly the value of the house.
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CHAPTER 7: OPTIMAL RISKY PORTFOLIOS
PROBLEM SETS
1.
(a) and (e).
2.
(a) and (c). After real estate is added to the portfolio, there are four asset classes in
the portfolio: stocks, bonds, cash and real estate. Portfolio variance now includes a
variance term for real estate returns and a covariance term for real estate returns
with returns for each of the other three asset classes. Therefore, portfolio risk is
affected by the variance (or standard deviation) of real estate returns and the
correlation between real estate returns and returns for each of the other asset classes.
(Note that the correlation between real estate returns and returns for cash is most
likely zero.)
3.
(a) Answer (a) is valid because it provides the definition of the minimum variance
portfolio.
4.
The parameters of the opportunity set are:
E(r
S
) = 20%, E(r
B
) = 12%, σ
S
= 30%, σ
B
= 15%, ρ = 0.10
From the standard deviations and the correlation coefficient we generate the
covariance matrix [note that
Cov r
(
S
,
r
B
)
=
ρ
σ
σ
× ×
S B
]:
Bonds
Stocks
Bonds
225
45
Stocks
45
900 The minimum- variance portfolio
is computed as follows:
w
Min
(S) =
=
0.1739
w
Min
(B) = 1
?
0.1739 = 0.8261
The minimum variance portfolio mean and standard deviation are:
E(r
Min
) = (0.1739
× .
20) + (0.8261
× .
12) = .1339 = 13.39%
σ
Min
= [w
S2
σ
S2
+
w
2B
σ
2B
+
2w
S
w
B
Cov(r
S
,r
B
)]
1/ 2
= [(0.1739
2
×
900) + (0.8261
2
×
225) + (2
×
0.1739
×
0.8261
×
45)]
1/2
= 13.92%
5.
Proportion
in stock fund
0.00%
17.39%
20.00%
40.00%
45.16%
Proportion
in bond fund
100.00%
82.61%
80.00%
60.00%
54.84%
Expected
return
12.00%
13.39%
13.60%
15.20%
15.61%
Standard
Deviation
15.00%
13.92%
13.94%
15.70%
16.54%
minimum variance
tangency portfolio
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60.00%
80.00%
100.00%
40.00%
20.00%
0.00%
16.80%
18.40%
20.00%
19.53%
24.48%
30.00%
Graph shown below.
25.00
INVESTMENT OPPORTUNITY SET
20.00
CML
Tangency
Portfolio
15.00
Efficient frontier
of risky assets
10.00
r
f
=
8.00
5.00
Minimum
Variance
Portfolio
0.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
6.
7.
The above graph indicates that the optimal portfolio is the tangency portfolio with
expected return approximately 15.6% and standard deviation approximately 16.5%.
The proportion of the optimal risky portfolio invested in the stock fund is given by:
w
S
=
[ (
E r
)
?
r
]
×
σ
2
[ (
E r
+
[ (
E r
S
)
?
B
r
)
f
?
]
×
r
f
σ
]
×
B
2
σ
?
S
[ (
2
E r
?
[ (
E r
B
)
?
S
)
r
f
?
+
]
r
×
f
Co vr rE
)
r
((
SB
,)
B
?
r
f
]
×
Co vr r
(
S
,
B
)
S
f
B
=
=
0.4516
w
?
=
B
1
0.4516
=
0.5484
The mean and standard deviation of the optimal risky portfolio are:
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E(r
P
) = (0.4516
× .
20) + (0.5484
× .
12) = .1561
= 15.61% σ
p
= [(0.4516
2
×
900) + (0.5484
2
×
225) + (2
×
0.4516
×
0.5484
×
45)]
1/2
8.
= 16.54%
The reward-to-volatility ratio of the optimal CAL is:
E r
(
p
)
?
r
f
=
.1561
?
.08
0.4601
σ
9.
p
.1654
a.
If you require that your portfolio yield an expected return of 14%, then you can
find the corresponding standard deviation from the optimal CAL. The equation for
this CAL is:
+
E r
(
C
)
= +
r
f
=
p
?
r
f
σ
C
.08 0.4601
σ
C
E r
( )
σ
P
If E(r
C
) is equal to 14%, then the standard deviation of the portfolio is 13.04%.
b.
To find the proportion invested in the T-bill fund, remember that the mean of
the complete portfolio (i.e., 14%) is an average of the T-bill rate and the
optimal combination of stocks and bonds (P). Let y be the proportion invested
in the portfolio P. The mean of any portfolio along the optimal CAL is:
E r
(
C
)
= ?
(1
y
)
× + ×
r
f
+
y
×
(.1561
?
.08)
y
E r
(
P
)
= +
r
f
y
×
[ (
E r
P
)
?
=
r
f
]
.08
Setting E(r
C
) = 14% we find: y = 0.7884 and (1 ? y) = 0.2116 (the proportion
invested in the T-bill fund).
To find the proportions invested in each of the funds, multiply 0.7884 times
the respective proportions of stocks and bonds in the optimal risky portfolio:
Proportion of stocks in complete portfolio = 0.7884
×
0.4516 = 0.3560
Proportion of bonds in complete portfolio = 0.7884
×
0.5484 = 0.4324
10.
Using only the stock and bond funds to achieve a portfolio expected return of 14%,
we must find the appropriate proportion in the stock fund (w
S
) and the appropriate
proportion in the bond fund (w
B
= 1 ? w
S
) as follows:
.14 = .20 × w
S
+ .12 × (1 ? w
S
) = .12 + .08 × w
S
?
w
S
= 0.25
So the proportions are 25% invested in the stock fund and 75% in the bond fund.
The standard deviation of this portfolio will be:
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σ
P
= [(0.25
2
×
900) + (0.75
2
×
225) + (2
×
0.25
×
0.75
×
45)]
1/2
= 14.13% This
is considerably greater than the standard deviation of 13.04% achieved using Tbills
and the optimal portfolio.
11.
a.
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25.00
Optimal CAL
20.00
P
15.00
Stocks
10.00
5.00
0.00
0
10
20
Gold
30
40
Standard Dev iation(%)
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Even though it seems that gold is dominated by stocks, gold might still be an
attractive asset to hold as a
part
of a portfolio. If the correlation between gold
and stocks is sufficiently low, gold will be held as a component in a portfolio,
specifically, the optimal tangency portfolio.
b.
If the correlation between gold and stocks equals +1, then no one would hold
gold. The optimal CAL would be comprised of bills and stocks only. Since
the set of risk/return combinations of stocks and gold would plot as a straight
line with a negative slope (see the following graph), these combinations would
be dominated by the stock portfolio. Of course, this situation could not
persist. If no one desired gold, its price would fall and its expected rate of
return would increase until it became sufficiently attractive to include in a
portfolio.
12.
Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio
can be created and the rate of return for this portfolio, in equilibrium, will be the
risk-free rate. To find the proportions of this portfolio [with the proportion w
A
invested in Stock A and w
B
= (1 – w
A
) invested in Stock B], set the standard
deviation equal to zero. With perfect negative correlation, the portfolio standard
deviation is:
σ
P
= Absolute value [w
A
σ
A
?
w
B
σ
B
]
0 = 5 × w
A
? [10
×
(1 – w
A
)]
?
w
A
= 0.6667
The expected rate of return for this risk-free portfolio is:
E(r) = (0.6667 × 10) + (0.3333 × 15) = 11.667%
Therefore, the risk-free rate is: 11.667%
13.
False. If the borrowing and lending rates are not identical, then, depending on the
tastes of the individuals (that is, the shape of their indifference curves), borrowers
and lenders could have different optimal risky portfolios.
14.
False. The portfolio standard deviation equals the weighted average of the
component-asset standard deviations
only
in the special case that all assets are
perfectly positively correlated. Otherwise, as the formula for portfolio standard
deviation shows, the portfolio standard deviation is
less
than the weighted average
of the component-asset standard deviations. The portfolio
variance
is a weighted
sum
of the elements in the covariance matrix, with the products of the portfolio
proportions as weights.
15.
The probability distribution is: Probability
Rate of Return
0.7
0.3
100%
?50%
Mean = [0.7
×
100%] + [0.3 × (-50%)] = 55%
Variance = [0.7
×
(100 ? 55)
2
] + [0.3
×
(-50 ? 55)
2
] = 4725
Standard deviation = 4725
1/2
= 68.74%
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16.
σ
P
= 30 = y × σ = 40 × y
?
y = 0.75 E(r
P
) = 12 + 0.75(30 ? 12) = 25.5%
17.
The correct choice is c. Intuitively, we note that since all stocks have the same
expected rate of return and standard deviation, we choose the stock that will result
in lowest risk. This is the stock that has the lowest correlation with Stock A.
More formally, we note that when all stocks have the same expected rate of return,
the optimal portfolio for any risk-averse investor is the global minimum variance
portfolio (G). When the portfolio is restricted to Stock A and one additional stock,
the objective is to find G for any pair that includes Stock A, and then select the
combination with the lowest variance. With two stocks, I and J, the formula for the
weights in G is:
w
Min
(I)
=
w
Min
(J)
=
1
?
w
Min
(I)
Since all standard deviations are equal to 20%:
Cov r r
(
I
,
J
)
=
ρ
σ
σ
I
J
400
ρ
and
=
w
Min
( )
I
=
w
Min
( )
J
0.5
=
This intuitive result is an implication of a property of any efficient frontier, namely,
that the covariances of the global minimum variance portfolio with all other assets
on the frontier are identical and equal to its own variance. (Otherwise, additional
diversification would further reduce the variance.) In this case, the standard
deviation of G(I, J) reduces to:
σ
Min
(
G
)
=
[200
(
1
× +
ρ
IJ
)]
1/2
This leads to the intuitive result that the desired addition would be the stock with the
lowest correlation with Stock A, which is Stock D. The optimal portfolio is equally
invested in Stock A and Stock D, and the standard deviation is 17.03%.
18.
No, the answer to Problem 17 would not change, at least as long as investors are not
risk lovers. Risk neutral investors would not care which portfolio they held since all
portfolios have an expected return of 8%.
19.
Yes, the answers to Problems 17 and 18 would change. The efficient frontier of
risky assets is horizontal at 8%, so the optimal CAL runs from the risk-free rate
through G. This implies risk-averse investors will just hold Treasury Bills.
20.
Rearranging the table (converting rows to columns), and computing serial
correlation results in the following table:
Nominal Rates
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1920s
1930s
1940s
1950s
1960s
1970s
1980s
1990s
Serial Correlation
Small company
stocks
-3.72
7.28
20.63
19.01
13.72
8.75
12.46
13.84
0.46
Large
company
stocks
18.36
-1.25
9.11
19.41
7.84
5.90
17.60
18.20
-0.22
Long-term
government
bonds
3.98
4.60
3.59
0.25
1.14
6.63
11.50
8.60
0.60
Intermed-term
government
bonds
3.77
3.91
1.70
1.11
3.41
6.11
12.01
7.74
0.59
9.00
5.02
0.63
Treasury bills
Inflation
-1.00
-2.04
5.36
2.22
2.52
7.36
5.10
2.93
0.23
3.56 0.30
0.37 1.87
3.89
6.29
For
example:
to
compute
serial
correlation
in
decade
nominal
returns
for
largecompany
stocks,
we
set
up
the
following
two
columns
in
an
Excel
spreadsheet. Then, use the Excel function “CORREL” to calculate the correlation
for the data.
Decade Previous
1930s
-1.25% 18.36%
1940s
9.11%
-1.25%
1950s
19.41%
9.11%
1960s
7.84% 19.41%
1970s
5.90%
7.84%
1980s
17.60%
5.90%
1990s
18.20% 17.60%
Note that each correlation is based on only seven observations, so we cannot arrive
at any statistically significant conclusions. Looking at the results, however, it
appears that, with the exception of large-company stocks, there is persistent serial
correlation. (This conclusion changes when we turn to real rates in the next
problem.)
21.
The table for real rates (using the approximation of subtracting a decade’s average
inflation from the decade’s average nominal return) is:
Real Rates
1920s
1930s
1940s
1950s
1960s
1970s
1980s
1990s
Serial Correlation
Small company
stocks
-2.72
9.32
15.27
16.79
11.20
1.39
7.36
10.91
0.29
Large
company
stocks
Long-term
government
bonds
Intermed-term
government
bonds
4.77
5.95
-
3.66
-1.11
0.89
-
1.25
6
.91
4.81
-0.27
0.38
0.11
Treasury bills
4.56
2.34
-4.99
-0.35
1.37
-1.07
3.90
2.09
0.00
19.36
4.98
0.79
3
.75
6.64
-1.77
17.19
-1.97
5.32
-
1.46
12.50
-1.38
15.27
-0.73
6.40
5.67
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While the serial
correlation
in
decade
nominal
returns seems
to
be positive, it
appears that real rates are serially uncorrelated. The decade time series (although
again too short for any definitive conclusions) suggest that real rates of return are
independent from decade to decade.
CFA PROBLEMS
1.
a.
Restricting the portfolio to 20 stocks, rather than 40 to 50 stocks, will increase
the risk of the portfolio, but it is possible that the increase in risk will be minimal.
Suppose that, for instance, the 50 stocks in a universe have the same standard
deviation (
σ
)
and the correlations between each pair are identical, with correlation
coefficient ρ. Then, the covariance between each pair of stocks would be ρσ
2
, and
the variance of an equally weighted portfolio would be:
σ
2P
=
1
σ
2
+
n
?
1
ρ
σ
2
n
n
The effect of the reduction in n on the second term on the right-hand side
would be relatively small (since 49/50 is close to 19/20 and
ρσ
2
is smaller than
σ
2
), but the denominator of the first term would be 20 instead of 50. For
example, if σ = 45% and ρ = 0.2, then the standard deviation with 50 stocks
would be 20.91%, and would rise to 22.05% when only 20 stocks are held.
Such an increase might be acceptable if the expected return is increased
sufficiently.
Hennessy could contain the increase in risk by making sure that he maintains
reasonable diversification among the 20 stocks that remain in his portfolio.
This entails maintaining a low correlation among the remaining stocks. For
b.
2.
3.
example, in part (a), with
ρ
= 0.2, the increase in portfolio risk was minimal.
As a practical matter, this means that Hennessy would have to spread his
portfolio among many industries; concentrating on just a few industries would
result in higher correlations among the included stocks.
Risk reduction benefits from diversification are not a linear function of the number
of issues in the portfolio. Rather, the incremental benefits from additional
diversification are most important when you are least diversified. Restricting
Hennessy to 10 instead of 20 issues would increase the risk of his portfolio by a
greater amount than would a reduction in the size of the portfolio from 30 to 20
stocks. In our example, restricting the number of stocks to 10 will increase the
standard deviation to 23.81%. The 1.76% increase in standard deviation resulting
from giving up 10 of 20 stocks is greater than the 1.14% increase that results from
giving up 30 of 50 stocks.
The point is well taken because the committee should be concerned with the
volatility of the entire portfolio. Since Hennessy’s portfolio is only one of six
welldiversified portfolios and is smaller than the average, the concentration in fewer
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issues might have a minimal effect on the diversification of the total fund. Hence,
unleashing Hennessy to do stock picking may be advantageous.
4.
d.
Portfolio Y cannot be efficient because it is dominated by another portfolio. For
example, Portfolio X has both higher expected return and lower standard
deviation.
5.
c.
6.
d.
7.
b.
8.
a.
9.
c.
10.
Since we do not have any information about expected returns, we focus exclusively
on reducing variability. Stocks A and C have equal standard deviations, but the
correlation of Stock B with Stock C (0.10) is less than that of Stock A with Stock B
(0.90). Therefore, a portfolio comprised of Stocks B and C will have lower total
risk than a portfolio comprised of Stocks A and B.
11.
Fund D represents the single
best
addition to complement Stephenson's current
portfolio, given his selection criteria. Fund D’s expected return (14.0 percent) has
the potential to increase the portfolio’s return somewhat. Fund D’s relatively low
correlation with his current portfolio (+0.65) indicates that Fund D will provide
greater diversification benefits than any of the other alternatives except Fund B.
The result of adding Fund D should be a portfolio with approximately the same
expected return and somewhat lower volatility compared to the original portfolio.
The other three funds have shortcomings in terms of expected return enhancement
or volatility reduction through diversification. Fund A offers the potential for
increasing the portfolio’s return, but is too highly correlated to provide substantial
volatility reduction benefits through diversification. Fund B provides substantial
volatility reduction through diversification benefits, but is expected to generate a
return well below the current portfolio’s return. Fund C has the greatest potential to
increase the portfolio’s return, but is too highly correlated with the current portfolio
to provide substantial volatility reduction benefits through diversification.
12.
a.
Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the
new portfolio.
i.
E(r
NP
) = w
OP
E(r
OP
) + w
ABC
E(r
ABC
) = (0.9
×
0.67) + (0.1
×
1.25) =
OP
×
σ
ABC
= 0.40
×
2.37
×
2.95 = 2.7966
?
2.80 iii.
0.728% ii. Cov = ρ
×
σ
σ
NP
= [w
OP2
σ
OP2
+ w
ABC2
σ
ABC2
+ 2 w
OP
w
ABC
(Cov
OP , ABC
)]
1/2
= [(0.9
2
×
2.37
2
) + (0.1
2
×
2.95
2
) + (2
×
0.9
×
0.1
×
2.80)]
1/2
b.
= 2.2673%
?
2.27%
Subscript OP refers to the original portfolio, GS to government securities, and
NP to the new portfolio.
-
-
-
-
-
-
-
-
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