Ch08 HW Solutions

Chapter 8 Alternative Investments

Note: In the sixth edition of Global Investments, the exchange rate quotation symbols differ from previous

editions. We adopted the convention that the first currency is the quoted currency in terms of units

of the second currency.

For example, €:$ 1.4 indicates that one euro is priced at 1.4 dollars. In previous editions we used

the reversed convention $/€ 1.4, meaning 1.4 dollars per euro.

All problems in this test bank still use the old convention and have not been adapted to reflect the

new quotation symbols used in the 6th edition.

Questions and Problems

1. An investor is considering the purchase of Tata International Equity Fund (TIEF) for his portfolio.

Like many U.S.-based mutual funds today, TIEF has more than one class of shares. Although all

classes hold the same portfolio of securities, each class has a different expense structure. This

particular mutual fund has three classes of shares: A, B, and C. The expenses of these classes are

summarized in the following table:

Expense Comparison for Three Classes of TIEF

Class A Class B Class C

Sales Charge (load) on

Purchases

4%

None

None

Deferred Sales Charge

(load) on Redemptions

None 5% in the first year, declining

by 1 percentage point each

year thereafter

1% for the initial

two years

Annual Expenses:

Distribution Fee 0.10% 0.50% 0.75%

Management Fee 0.75% 0.75% 0.75%

Other Expenses 0.50% 0.50% 0.50%

1.35% 1.75% 2.00%

The time horizon associated with the investor’s objective in purchasing TIEF is three years; he

decides to specify it as just over three years. He expects equity investments with risk characteristics

similar to TIEF to earn 10% per year, and he decides to make his selection of fund share class based

on an assumed 10% return each year, gross of any of the expenses given in the table above.

Based on only the above information, determine the class of shares that is most appropriate for this

investor. Assume that expense percentages given will be constant at the given values. Assume that

the deferred sales charges are computed on the basis of net asset value (NAV).

92 Solnik/McLeavey • Global Investments, Sixth Edition

Solution

To address this question, we compute the terminal value of $1 invested at the end of year three.

The share class with the highest terminal value net of all expenses would be the most appropriate for

this investor, as all classes are based on the same portfolio and thus have the same portfolio risk

characteristics.

Class A. $1 (1 0.04) $0.96 is the amount available for investment at t 0, after paying

the front-end sales charge. Because this amount grows at 10% for three years, reduced by

annual expenses of 0.0135, the terminal value per $1 invested after three years is $0.96 1.103

(1 0.0135)3 $1.227.

Class B. Ignoring any deferred sales charge, to be paid after three years, $1 invested grows to

$1 1.103 (1 0.0175)

3 $1.262. According to the table, the deferred sales charge would be

2% just after three years; therefore, the terminal value is $1.262 0.98 $1.237.

Class C. After three years, $1 invested grows to $1 1.103 (1 0.020)

3 $1.253. There is no

deferred sales charge in the third year, so $1.253 is the terminal value.

In summary, the ranking by terminal value after three years is Class B ($1.237), Class C ($1.253),

Class A ($1.227). Class B appears to be the most appropriate for this investor with a long-time

horizon.

2. An investor is considering the purchase of CHECK Fund for his portfolio. Like many U.S.-based

mutual funds today, CHECK has more than one class of shares. Although all classes hold the same

portfolio of securities, each class has a different expense structure. This particular mutual fund

has three classes of shares: A, B, and C. The expenses of these classes are summarized in the

following table:

Expense Comparison for Three Classes of CHECK

Class A Class B Class C

Sales Charge (load) on

Purchases

3%

None

None

Deferred Sales Charge

(load) on Redemptions

None 5% in the first year, declining by

1 percentage point each year

thereafter

1% for the initial

two years

Annual Expenses:

Distribution Fee 0.25% 0.50% 0.75%

Management Fee 0.75% 0.75% 0.75%

Other Expenses 0.25% 0.25% 0.25%

1.25% 1.50% 1.75%

The time horizon associated with the investor’s objective in purchasing CHECK is five years. He

expects equity investments with risk characteristics similar to CHECK to earn 8% per year, and he

decides to make his selection of fund share class based on an assumed 8% return each year, gross of

any of the expenses given in the table above.

a. Based on only the above information, determine the class of shares that is most appropriate for

this investor. Assume that expense percentages given will be constant at the given values.

Assume that the deferred sales charges are computed on the basis of NAV.

Chapter 8 Alternative Investments 93

b. Suppose that, as a result of an unforeseen liquidity need, the investor needs to liquidate his

investment at the end of the first year. Assume an 8% rate of return has been earned. Determine

the relative performance of the three fund classes, and interpret the results.

Solution

a. To address this question, we compute the terminal value of $1 invested at the end of year five

(or rather a day after the fifth year). The share class with the highest terminal value net of all

expenses would be the most appropriate for this investor, as all classes are based on the same

portfolio and thus have the same portfolio risk characteristics.

Class A. $1 (1 0.03) $0.97 is the amount available for investment at t 0, after paying the

front-end sales charge. Because this amount grows at 8% for five years, reduced by annual

expenses of 0.0125, the terminal value per $1 invested after five years is $0.97 1.085 (1

0.0125)5 $1.3384.

Class B. Ignoring any deferred sales charge, after five years, $1 invested grows to $1 1.085

(1 0.015)5 $1.3624. According to the table, the deferred sales charge would be 1% at any

time during year five; therefore, the terminal value is $1.3624 0.99 $1.3488.

Class C. After 5 years, $1 invested grows to $1 1.085 (1 0.0175)

5 $1.3452. There is no

deferred sales charge in the fifth year, so $1.3452 is the terminal value.

In summary, the ranking by terminal value after five years is Class B ($1.3488), Class C

($1.3452), and Class A ($1.3384). Class B appears to be the most appropriate for this investor

with a long-time horizon.

b. For Class A shares, the terminal value per $1 invested is $0.97 1.08 (1 0.0125) $1.0345.

For Class B shares, it is as $1 1.08 (1 0.015) (1 0.05) $1.0106, reflecting a 5%

redemption charge. For Class C shares, it is $1 1.08 (1 0.0175) (1 0.01) $1.0505,

reflecting a 1% redemption charge.

Thus, the ranking is Class C ($1.0505), Class A ($1.0345), and Class B ($1.0106). Although

Class B is appropriate given a five-year investment horizon, it is a costly choice if the fund shares

need to be liquidated soon after investment. That eventuality would need to be assessed by the

investor we are discussing. Class B, like Class A, is more attractive, the longer the holding period,

in general. Class A is more attractive than Class B if the shares are held for a very long time

because annual expenses are lower. Because Class C has the higher annual expenses, however, it

becomes less attractive the longer the holding period, in general.

3. Exchange traded funds (ETFs) are usually considered to have many interesting properties. In the list

below, indicate which statements DO NOT apply to ETFs:

a. ETFs allow to invest in a diversified portfolio.

b. ETFs are cost effective.

c. ETFs will never drop in value.

d. ETFs benefit from some attractive tax characteristics.

e. ETFs can be traded at any time during market opening.

f. ETFs are designed to take advantage of the manager’s stock picking ability.

Solution

The answers are (c) and (f):

— The share price of an ETF will drop if the underlying index tracked goes down.

— The investment objective of an ETF is to track, as well as possible, a prespecified index or basket

of securities; not to engage in stock picking within the index.


4. There are several ETFs listed on the American Stock Exchange (AMEX). One of them is iShares—

Switzerland. This ETF tracks the Morgan Stanley Capital International (MSCI)—Switzerland index,

which is based on several stocks that trade on the Swiss Exchange. The Swiss Exchange is open from

9 A.M.to 5:30 P.M. Swiss time and the AMEX is open from 9:30 A.M. to 4 P.M. U.S. Eastern Standard

Time (EST). The U.S. EST lags the Swiss time by six hours. Discuss whether the ETF price and its

NAV would fluctuate or stay the same during the time period when the AMEX is open.

Solution

When the AMEX market opens at 9:30 A.M., the time in Switzerland is 3:30 P.M. and the Swiss exchange

is still open. The ETF shares in the United States would open at a price based on the NAV in

Switzerland at that time and the prevailing dollar to Swiss franc exchange rate. From 9:30 A.M. to

11:30 A.M., both the AMEX and Swiss exchange are open, and the NAV in Switzerland would

continue to change. The price of the ETF in dollars would also be changing consistent with the

changes in NAV and exchange rate. Subsequently, from 11:30 A.M. to 4 P.M., the AMEX is open but

the Swiss exchange is closed. So, the NAV in Swiss francs would remain the same. However, the

price of the ETF in the United States would change based on the changes in expectations about the

future stock prices in Switzerland and the changing exchange rate.

5. A real estate company has prepared a simple hedonic model to value houses in a specific downtown

area. A summary list of the houses’ characteristics that can affect pricing are:

The number of main rooms.

The surface of the garden (if any).

The construction material (wood or brick).

The distance to a subway station.

A detailed statistical analysis of a large number of recent transactions in the area allowed to derive

the following slope coefficients:

Characteristics

Units

Slope Coefficient in

Euros per Unit

Number of Rooms Number 30,000

Surface of the Garden Square meters 200

Construction Material (bricks) 0 or 1 30,000

Distance to Subway Station In meters 100

A typical house in the area has 5 main rooms, a garden of 500 square meters, constructed with bricks,

and a distance of 300 meters to the nearest subway station. The transaction price for a typical house was €250,000.

a. You wish to value a house that has 7 rooms, a small garden of 100 square meters, constructed in

wood, and a distance of 100 meters to the nearest subway station. What is the appraisal value

based on this sales comparison approach of hedonistic price estimation?

b. You wish to value a house that has 7 rooms, a garden of 1,000 square meters, constructed in

brick, and a distance of 1 kilometer to the nearest subway station. What is the appraisal value

based on this sales comparison approach of hedonistic price estimation?


Solution

a. The appraised value is given by the equation:

Value 30,000 (# Rooms) 200 (Garden surface) 30,000 (Bricks) – 100

(Distance to subway station).

This specific house has an appraised value of €220,000.

b. The appraised value is given by the equation:

Value 30,000 (# Rooms) 200 (Garden surface) 30,000 (Bricks) – 100

(Distance to subway station).

This specific house has an appraised value of €340,000.

6. An investor wants to evaluate an apartment building using the income approach. She gathers the

following data on the apartment complex; all income items are on an annual basis.

Investment under Consideration

Gross Potential Rental Income $100,000

Estimated Vacancy and Collection Losses 5%

Insurance and Taxes $ 8,000

Utilities $ 5,000

Repairs and Maintenance $ 12,000

Depreciation $ 15,000

Interest on Proposed Financing $ 6,000

Two apartment buildings have recently been sold in the area. Building A had a sales price of $5 million

with an annual net operating income of $500,000. Building B had a sales price of $1 million with an

operating income of $95,000. Except for size, both buildings have characteristics (location, age,

quality, . . .) similar to that of the apartment building under consideration.

According to the income approach, what is the value of the apartment complex?

Solution

The net operating income for the apartment complex is given by gross potential rental income minus

estimated vacancy and collection costs, minus insurance and taxes, minus utilities, minus repairs and

maintenance.

NOI 100,000 – (0.05 100,000) 8,000 5,000 12,000 70,000.

Buildings A and B are comparable properties.

The capitalization rate of Building A:

NOI/(transaction price) 500,000/5,000,000 0.10.

The capitalization rate of Building B:

NOI/(transaction price) 95,000/1,000,000 0.095.


Applying these two capitalization rates to the apartment building under consideration gives two

different appraisal prices of:

NOI/(capitalization rate) 70,000/0.10 $700,000,

and

NOI/(capitalization rate) 70,000/0.095 $736,842.

Buildings A and B have characteristics similar to the apartment building under consideration, except

for size. The considered investment is apparently a building of small size (low price), hence the

capitalization rate of building B is probably more relevant. Hence, an appraisal price in the order of

$736,842 is a reasonable estimation.

Note that we do not use the financing costs to determine the net operating income (NOI), as we wish

to appraise the value of the property independently of its financing. Neither do we use the depreciation,

which is not an expense but would only be used for tax calculation. The implicit assumption is that

repairs and maintenance will allow to keep the building in good condition forever.

7. An analyst is evaluating a real estate investment project using the discounted cash flow approach.

The net purchase price is $10 million, which is financed 20% by equity and 80% by a five-year

mortgage loan. The loan carries an interest rate of 7%. Annual interest expenses on the $8 million

loan are $560,000 and the loan is repaid in full after the fifth year.

The net operating income for the first year is estimated at $800,000 and is expected to grow annually

at a 3% growth rate. Using straight-line depreciation over 50 years, the annual tax depreciation of the

real estate project is equal to $200,000. It is expected that the property will be sold in five years

(just after the end of the fifth year) at a net price of $11 million.

The marginal income tax rate for this project is 30%. The capital gains tax rate is 20%.

The investor’s cost of equity for projects with level of risk comparable to this real estate investment

project is 15%.

a. Compute the after-tax cash flows resulting from the operating income for each of the first

five years.

b. Compute the after-tax cash flow for the fifth year, taking into account the resale value.

c. Compute the expected net present value (NPV) of the project and its internal rate of return (IRR).

d. State whether the investor should decide to invest in the project.

e. Compute the expected NPV and IRR of the project if the resale price is expected to be only

$10 million.

f. State whether the investor should decide to invest in this project under this new scenario.

Solution

a. After-tax net income in year one (NOI – Depreciation – Interest) (1 – Marginal tax rate)

($800,000 – $200,000 – $560,000) (1 – 0.30) $28,000.

After-tax cash flow After-tax net income Depreciation

$28,000 $200,000 $228,000.

New NOI in year two 1.03 $800,000 $824,000.

After-tax net income in year two (NOI – Depreciation – Interest) (1 – Marginal tax rate)

($824,000 – $200,000 – $560,000) (1 – 0.30) $44,800.



$44,800 $200,000 $244,800.

New NOI in year three 1.03 $824,000 $848,720.

After-tax net income in year three (NOI – Depreciation – Interest) (1 – Marginal tax rate)

($848,720 – $200,000 – $560,000) (1 – 0.30) $62,104.


$62,104 $200,000 $262,104.

New NOI in year four 1.03 $848,720 $874,182.

After-tax net income in year four (NOI – Depreciation – Interest) (1 – Marginal tax rate)

($874,182 – $200,000 – $560,000) (1 – 0.30) $79,927.


$79,927 $200,000 $279,927.

New NOI in year five 1.03 $874,182 $900,407.

After-tax net income in year five (NOI – Depreciation – Interest) (1 – Marginal tax rate)

($900,407 – $200,000 – $560,000) (1 – 0.30) $98,285.

After-tax cash flow on operating income After-tax net income Depreciation

$98,285 $200,000 $298,285.

b. Ending book value Original purchase price – Total depreciation during five years

$10,000,000 – 5 $200,000 $9,000,000.

Capital gains tax 0.20 ($11,000,000 – $9,000,000) $400,000.

After-tax cash flow from property sale Net sales price – mortgage loan – Capital gains tax

$11,000,000 – $8,000,000 – $400,000 $2,600,000.

The total after tax cash flow for the fifth year is $298,285 $2,600,000 $2,898,285.

c. The investment requires equity of 0.20 $10,000,000 $2,000,000. At a cost of equity of 15%,

the net present value of the after-tax cash flows in years one through five is as follows:

$2,000,000 $228,000/1.15 $244,800/1.152 $262,104/1.15

3 $279,927/1.15

4

$2,898,285/1.155 $136,271.

The internal rate of return using a spreadsheet is equal to IRR 17.1%.

d. The recommendation based on NPV would be to accept the project, as the NPV is positive. Also,

the IRR is above the cost of equity.

e. Assuming a resale price of $10,000,000, the expected NPV is $209,591 and the IRR is 11.3%.

f. The project should not be accepted.

8. An investor estimates that investing €5 million in a particular venture capital project can return

$40 million at the end of five years if it succeeds; however, she realizes that the project may fail at

any time between now and the end of the fifth year. The investor is considering an equity investment

in the project and her cost of equity for a project with this level of risk is 15%. In the table below are

the investor’s estimates of certain probabilities of failure for the project. First, 0.30 is the probability

of failure in year one. The probability that project fails in the second year, given that it has survived

through year one, is 0.25; and so forth:

Year 1 2 3 4 5

Failure Probability 0.30 0.25 0.20 0.20 0.20

a. Determine the expected net present value of the project.


b. Recommend whether the project should be undertaken.

Solution

a. The probability that the project survives to the end of the first year is (1 0.30) 0.70. The

probability that it survives to the end of second year is the product of the probability it survives

the first year times the probability it survives the second year, or (1 0.30) (1 0.25). Using this

pattern, the probability that the firm survives to the end of the fifth year is (1 0.30) (1 0.25)

(1 0.20)3 (0.70) (0.75) (0.80)

3 0.2688 or 26.88%.

There are two ways to compute the probability-weighted net present value.

A first approach is to weight each cash flow by its probability of occurrence. The initial

investment of 1 million is certain (probability of 100%), the cash flow in five years has a

probability of 26.88%. Hence the NPV is equal to:

€€ €

5

40 million 26.88%NPV 5 million 345,644.

1.15

A second approach is to compute the NPV assuming first failure and then no failure, and to

weight these NPVs by the probability of occurrence.

— The net present value of the project given that it survives to the end of the fifth year and

thus earns €40 million equals €14.8871 million €5 million €40 million/1.155.

— The net present value of project given that it fails is €5 million.

Thus the project’s expected NPV is a probability-weighted average of these two amounts, or

(0.2688)(€14.8871 million) (0.7312)( €5 million) €345,644.

Both approaches yield the same NPV.

b. Based on its positive net present value, the recommendation is to undertake the investment.

9. A hedge fund currently has assets of $500 million. The annual fee structure of this fund consists of a

fixed fee of 1% of portfolio assets plus a 20% incentive fee. The fund applies the incentive fee to the

gross return each year in excess of the portfolio’s previous high watermark, which is the maximum

fund value in the past two years. The fund is closed to new investors and the maximum value that the

fund has achieved in the past two years was $520 million. Compute the fee that the manager will earn,

in dollars, if the return on the fund in the coming year turns out to be:

a. 30%

b. 2%

c. 2%

Solution

a. Fixed fee 1% of $500 million $5 million.

If the return is 30%, new value of the fund would be $500 million 1.30 $650 million. This

new value would be $650 million – $520 million $130 million above the high watermark.

So, the incentive fee 20% $130 million $26 million.

Total fee $5 million $26 million $31 million.

b. Fixed fee 1% of $500 million $5 million.

If the return is 2%, the new value of the fund would be $500 million 1.02 $510 million. Since

this new value is below the high watermark of $520 million, no incentive fee would be earned.

Total fee $5 million.


c. Fixed fee 1% of $500 million $5 million.

If the return is –2%, no incentive fee would be earned.

Total fee $5 million.

10. A hedge fund has a capital of €100 million and invests in a market neutral long/short strategy on the

European equity market. Shares can be borrowed from a primary broker. The arrangement with the

primary broker is that the hedge fund deposits as guarantee securities with an equivalent market value

at time of lending, plus an additional cash margin deposit equal to 10% of the value of the shares. The

primary broker keeps any interest earned on the margin and charges a fee equal to an annual rate of

0.5% of the value of the shares borrowed. The hedge fund believes that European value stocks will

outperform European growth stocks. The hedge fund expects that value stocks will outperform growth stocks by 5% over the year. The hedge fund wishes to retain a cash cushion of €10 million

for unforeseen events. The short-term euro interest rate is 3%.

a. What market-neutral strategy would you suggest that would take full advantage of this scenario?

b. What is the expected return according to the funds’ expectations?

c. Assume now that the European stock index appreciates by 20% over the year, but that value stocks

underperform growth stocks by 10%. Compute a likely market value of the fund at year’s end.

Solution

a. The hedge fund would sell short growth stocks and use the proceeds to buy value stocks at an

equal amount. Some capital needs to be invested in the margin deposit. Furthermore, the hedge fund wishes to retain €10 million. The hedge fund has €90 million that can be used for cash

margin with the primary broker. So the hedge fund can borrow up to €900 million worth of

shares. The hedge funds could take long/short positions for €900 million:

Keep €10 million in cash.

Borrow €900 million of growth stocks from a broker.

Deposit €90 million in margin (10% 900 million).

Sell those growth shares for €900 million in cash.

Use the sale proceeds to buy €900 million worth of value stocks.

The ratio of invested assets to equity capital is roughly 9:1. If expectations materialize, the return

for investors in the hedge fund will be large.

b. The long/short portfolio of shares should have a gain over the year of 5% on €900 million

whatever the movement in the general market index. This is a €45 million gain. Furthermore,

the hedge fund will earn 3% €10 million €0.3 million on its cash position. The total gain of

€45.3 million will translate into an annual return before fees of 45.3% over the invested capital

of €100 million. This calculation assumes that the dividends on longs will offset the dividends on

shorts.

c. Under the new scenario, the long/short portfolio of shares should have a loss over the year of 10% on €900 million whatever the movement in the general market index. This is a €90 million

loss. However, the hedge fund will earn 3% €10 million €0.3 million on its cash position.

The total loss of €89.7 million will translate into an annual return before fees of –89.7% over the

invested capital of €100 million. The market value of the hedge fund will drop to €10.3 million.

This calculation assumes that the dividends on longs will offset dividends on shorts and that there

will not be any call from the broker to mark up the margin.


11. Survivorship bias is a serious potential problem in drawing conclusions from historical track records.

Show why the following statements can be misleading:

a. “There are today 100 Type-A hedge funds in operation. Their average return over the past two years is 20%. Hence, they have outperformed the stock market (return of 15%)” [actually, some 50 funds disappeared during these two years].

b. “The Poupou commodity index has been back-calculated from 1970 to 1990 using the leading commodity futures contracts; by leading we mean those that have been most active. The Poupou commodity index had a remarkable performance from 1970 to 1995” [actually, several commodity futures contracts have been removed from the commodity exchange or have experienced a drop in trading activity].

Solution

a. The bad performance of the 50 funds, which have disappeared is not included in the average

performance of 20%, but it should.

b. There is a selection bias based on past historical performance. Back in 1970, it was not possible to guess with certainty which commodities would experience a drop in activity. This is a serious problem because there is a correlation between the activity in a commodity futures contract and its price movement. All contracts, even those, which later disappeared, should be included in a back-calculation of an index.

12. The SOL Group specializes in hedge funds invested on the Paf stock market. Over the year 1999, the Paf stock market index went up by 20%. The SOL Group had three hedge funds with very different investment strategies. As expected, the 1999 returns on the three funds were quite different. Here are the performances of the three funds before and after management fees set at 20% of gross profits:

Fund Gross Return Net Return

SOL A 50% 40%

SOL B 20% 16%

SOL C 10% 12%

The average gross performance of the three funds is exactly equal to the performance on the Paf stock index. At year-end, most clients had left the third fund, and SOL C was closed. At the start of 2000, the SOL group launched an aggressive publicity campaign among portfolio managers, stressing the remarkable return on SOL A. If potential clients asked whether the SOL Group had other hedge funds invested in Paf, the SOL Group mentioned the only other fund, SOL B, and claimed that their average gross performance during 1999 was 35%.

What do you think of this publicity campaign?

Solution

This publicity campaign is misleading because of selection bias. Only the funds that survive because

of their good performance are included in the track record.

13. Let’s assume that you are a U.S. investor who wants to invest $10,000 in gold. The current price of gold is $400, and you expect it to go up by 10% in the very short-term. You consider buying shares of gold mines; you are debating whether to invest in Bel Or or Schoen Gold. Your broker gives you the following information:

Bel Or Schoen Gold

Production Cost per Ounce $147 $340

Gold beta ( ) 1.6 6


The gold is obtained by running a regression of the percentage price movements in the gold mine

stock on the percentage price movements in gold bullion. It indicates the stock market price sensitivity

to gold.

a. Explain why a gold mine with a high production cost should have a value that is more sensitive

to gold price movements than a gold mine with low production costs.

b. Which mine would you buy and why?

c. What is your expected return, given this scenario?

Solution

a. Given the production cost of a mine, C, we can compute the operating leverage of the two mines,

as the ratio of the gold price per ounce, G, to the profit margin per ounce, G – C:

Operating leverage G/(G C).

Hence we get:

Bel Or Schoen Gold

Production Cost per Ounce $147 $340

Profit Margin $253 $ 60

Operating Leverage 1.58 6.67

Schoen Gold is much more sensitive to gold price movements than Bel Or. This can be primarily

explained by its cost structure. The information derived from the operating leverage confirms the

information derived from the regression .

b. Given my expectation of a 10% gold price increase, I should buy Schoen Gold.

c. With an expected 10% appreciation in the price of gold, I expect to make some 60% on my

investment in Schoen Gold.

14. G.O. Bug wants to invest $12,000 in gold. In December, the spot price of gold is $400 per ounce.

Bug is very confident that gold will appreciate by at least 10% before the end of January and is

willing to assume fairly risky positions to maximize the return on this forecast. Bug is considering

several alternatives:

Gold bullion. Bug could buy 30 ounces, or roughly 1 kilogram.

Gold futures. Bug could buy February futures. These contracts trade at $413 per ounce, with an

initial margin of $1,500 per contract of 100 ounces. Therefore, Bug could buy eight contracts

(12,000/1,500).

Gold options. Bug considers two February call options with different strike prices. Each option

contract covers 100 ounces. The February 410 call quotes at $8 per ounce; the February 430 call

quotes at $4 per ounce. Therefore, Bug could buy fifteen contracts of the first option or thirty

contracts of the second option.

Two gold mines. Mines A and B have the same stock price: $10 per share. A British broker has

estimated the gold of both mines using a discounted cash flow model as well as historical

regression analysis. Mine A is a rich mine with a gold equal to 2; mine B has much higher

production costs with a gold equal to 5. Bug could buy 1,200 shares of one of the gold mines.

Bug quickly rules out investing directly in bullion, which does not offer enough leverage.

a. Assuming that Bug’s expectations are realized by the end of February, compute the realized

returns on the various alternative strategies considered. Simulate various values of the spot price

of gold in February (320, 360, 380, 400, 420, and 480).

b. Which investment strategy would you suggest to Bug?


Solution

a. Assuming that G.O. Bug’s expectations are realized by the end of February, the realized returns

on his various alternatives are calculated below:

Performance of Various Gold Investments Following a 10% Gold Price Appreciation

Investment Initial Value ($) Final Value ($) Performance

Asset

Quantity

Market

Price

Total

Value

Market

Price

Total

Value

Profits

($)

Return

(%)

Bullion 30 ounces 400 12,000 440 13,200 1,200 10

Mine A 1,200 shares 10 12,000 12 14,400 2,400 20

Mine B 1,200 shares 10 12,000 15 18,000 6,000 50

Futures 8 contracts or

800 ounces

413 12,000

(margin)

440 33,600 21,600 180

Feb. 410

Option

15 contracts or

1,500 ounces

8 12,000 30 45,000 33,000 275

Feb. 430

Option

30 contracts or

3,000 ounces

4 12,000 10 30,000 18,000 150

A direct investment in gold bullion would yield 10%, or $1,200 when the gold price moves to $440. The return on gold mining shares is somewhat uncertain, even if the price of gold does move up 10% from $400 to $440. If the gold betas of the mines are reliable estimates of their gold price elasticity, the stock price of Mine A should move up by 2 10 or 20% and that of Mine B by 5 10 or 50% (the stock price return is equal to beta times the percentage movement in gold price). Therefore, the stocks should be worth respectively $12 and $15. The expected profit on an investment in Mine A is equal to $2,400 while the profit would be $6,000 on Mine B.

The futures and option contracts are held until expiration, therefore, their final price is exactly known once the spot price of gold is known. The final February price of the February futures is $440 per ounce if the spot gold price is $440 at that time; so the investor makes a profit of $27 per ounce ($440 – $413). The total profit for eight contracts is 800 $27 or $21,600. Furthermore, G.O. Bug recovers this initial margin investment of $12,000. The total value of his futures investment is equal to $33,600. This $21,600 profit, compared to an initial investment of $12,000 translates into a 180% rate of return. The calculation for the option is somewhat different. At expiration, the options are equal to their intrinsic value, the difference between the gold spot price and the strike price. If the gold price is $440, the expiration value of the February 410 option is $30 ($440 – $410), and that of the February 430 option, $10 ($440 – $430). Hence, the final value of the investment in the fifteen February 410 contracts is equal to 1500 30 or $45,000. This implies a $33,000 profit and a rate of return of 275%. The profit on a February 430 call is only $18,000.

b. Given G.O. Bug’s expectations, the February 410 option offers the best alternative, with an expected return on the $12,000 equal to 275%. In practice, taxes and transaction costs should also be taken into account. Both the commissions and bid–ask spread could be large relative to the option premiums, especially for out-of-the money options. An investor should seriously consider these costs when engaging in active trading strategies.

His second choice would be to buy futures on 800 ounces (12,000/15 800 ounces or eight contracts of 100 ounces). And his third choice would be to buy February 430 call options on 3,000 ounces (12,000/4 3,000 ounces, or thirty contracts of 100 ounces).


Of course, the outcomes would be quite different if G.O. Bug’s expectations of a 10% gold price

appreciation were not realized. Gold and stocks are long-term income-producing assets, whereas

options and futures are short-term leveraged investments. The risks are quite different if the

forecast is not realized. To get a feeling for the risks involved, we calculate the rates of return for

a range of simulated gold price variations in the exhibit below. Only the three most-profitable

investments are considered. All returns are a calculated gross of transaction costs and taxes.

From the table, we see that options are poor investments compared to futures, where the price of

gold moves very little. This is because, as mentioned above, the cost of the insurance premium is

implicit in the option value. That is why, as shown in the graph, the futures contract has a higher

return than the February 430 options, where the gold price at expiration is the range of $400 to

$440. Above that range, the 430 option has a higher return, and below that range both options

allow an investor to limit this loss. In general, the larger the gold price movement, the more

attractive out-of-the-money options are, because they provide an investor with more leverage.

A speculator must select the best contract for him- or herself depending on how much he or she

expects the price of gold will change.

Simulation of the Rate of Return on Three Speculative Investments

Gold Price, End of

Feb. ($/ounce)

320

360

380

400

420

440

480

Bullion Performance –20% –10% –5% 0 5% 10% 20%

Feb. Futures Perf. –620% –353% –220% –87% 47% 180% 447%

Feb. 410 Perf. –100% –100% –100% –100% 25% 275% 775%

Feb. 430 Perf. –100% –100% –100% –100% –100% 150% 1150%


15. Bel Or Mine issues a five-year Eurobond with the following characteristics:

Par value 100 gold ounces. Each bond is issued and repaid in dollars at the market value of

100 ounces of gold.

Annual coupon payment of the dollar market value of 3 ounces of gold.

Maturity of five years with no early redemption.

A few days after issue, the yield on straight dollar Eurobonds, for issuers of the quality of Bel Or

Mine, is 10%. The price of gold is $400 per ounce. The gold-linked bond sells for $35,000. What

can you say about the market expectations of gold prices?

Solution

The Bel Or issue is a bond that is reimbursed in full at maturity in the form of 100 gold ounces

(or rather, its dollar value) and pays an annual coupon worth 3 ounces of gold. Therefore, the value

of this bond verifies the following equation:

3 5 51 2 4

2 3 4 5 5

3 3 1003 3 3$35,000

(1.1) (1.1) (1.1) (1.1) (1.1) (1.1)

G G GG G G

where Gt is the expected price of gold at time t. Here we assume that investors also use a 10%

discount rate.

Let’s further assume a simple expectation model for gold price, with an expected yearly price

appreciation rate of %. Gt may be written as

400 (1 )ttG

hence

55

1

1 1$35,000 1,200 40,000

1.1 1.1

3.81%.

t

t

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