Problem 1

In “Reflections on the Efficient Market Hypothesis: 30 Years Later” (2005), B. Malkiel concludes with

the following passage:

The evidence is overwhelming that active equity management is […] a “loser’s game.” Switching from

security to security accomplishes nothing but to increase transactions costs and harm performance. Thus,

even if markets are less than fully efficient, indexing is likely to produce higher rates of return than active

portfolio management. Both individual and institutional investors will be well served to employ indexing

for, at the very least, the core of their equity portfolio. […] And the most successful modern-day investor,

Warren Buffett, who has beaten the market over a prolonged period of time, sums up the advice in this

paper with characteristic wisdom: “Most investors, both institutional and individual, will find that the best

way to own common stocks (shares’) is through an index fund that charges minimal fees. Those following

this path are sure to beat the net results (after fees and expenses) of the great majority of investment

professionals.”

One of the arguments is the following graph of the percentage of EU-based actively managed funds

outperformed by MSCI Europe index (performance net of fees) for 3, 5 and 10 years ending 31st

December 2002 [the market and period are irrelevant, the numbers are very similar for 2010-2020)

Without focusing on whether W. Buffett is a proponent of the Efficient Market Hypothesis (he is not) or

whether the information is accurate (it is), carefully discuss the validity and strength of the argument

and its implications for the Efficient Market Hypothesis and what it entails for portfolio management.

[15 points]

Problem 2

Construct positions that fulfill the stated requirements in the following scenarios. You can use any type

and number of financial assets and instruments you like. Do not do any calculations but feel free to

assign symbols, numbers or use graphs if it helps you illustrate your answer. There may be more than

one way to hedge a position, but describing one way is enough.

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(i) An EU company that produces livestock feed imports soya beans from abroad, processes them in

their facilities (factory, silos and harbor) and sells the feed in the EU. The beans are ordered by the

producer at an agreed USD price, transported by ship, paid upon cargo delivery, stored and used into

production at a later date. The company orders quantities according to prearranged deals with clients

(no purchases for future undefined use), does not want to use cash reserves for cargo payments and is

paid upon product(livestock) delivery. The company is considered to have ownership of the soya beans

upon purchase, i.e. at placing the order and agreeing the price (ignore the captain’s liabilities during

travel). Identify the risks the company faces, hedge them and describe the transactions and assets

involved. (8 points)

(ii) It is 22nd January and a short-seller has a short position on Gamespot shares. His position is currently

profitable but during the day there has been a surge of the stock price and he is getting anxious. He is

not obliged to close the position at a certain date but does not have the funds to maintain it by the end

of the day (or a later day) if the price increases significantly during the session. On the same day, an

independent investor is thinking of investing in Gamestop shares but lacks confidence on his ability to

time the market correctly (he is worried that he might buy close to the peak or sell close to the bottom,

thus being trapped in his position) and possibly lose a great deal. He is trying to cover that uncertainty

risk by devising a strategy that will secure his level of wealth while allowing him to benefit from price

changes. Hedge the risk the short-seller faces and find a strategy that serves the independent investor’s

goal. (4 points)

(iii) A bonds investor has purchased a large number of discount bonds that mature over the next 5 years

at varying dates which do not follow a pattern or a certain frequency. The yield curve is currently flat,

but the investor is worried that large Central banks will soon take action to prevent an increase in

inflation in the post-Covid period. (3 points) [15 points]

Problem 3

Find the price of an American call on a stock that matures in two months and its price changes i) daily

ii) weekly iii) monthly (no dividends, 1 month = 20 days = 4 weeks). The exercise price is 300, the spot

price of the stock is 310, the risk-free rate is 8% per annum and stock volatility is 30% per annum. Using

the same method you used in the American call, find the price of a European call whose price changes

i) daily ii) weekly iii) monthly, with a strike price of 300, a spot price of 310, a risk-free rate of 8% per

annum and stock volatility of 30% per annum. Select one of the two calls, price it according to the BlackScholes formula, calculate its Δ (delta) and explain the Δ’s use and intuition. For the other call, calculate

its Δ based on the first monthly price change. Compare and comment on your results, with extra focus

on method applicability, accuracy and discrepancies. For visual representations, you can ignore the

daily cases. [10 points]

Problem 4

(1) A company has 5 mil shares outstanding at a price of 8 GBP. It decides to make a rights issue of one

new share for every 7 old shares. The price of the new share is 3 GBP.

(i) Find the ex-rights price of the stock and the value of the right.

(ii) Is there an increase in portfolio value for the current shareholders? Also, what price change are they

willing to accept before they become indifferent to having a right?

(iii) Show that, for the same number of rights and target amount to be raised, a higher or lower price

for the new shares does not make a difference for the shareholders.

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(iv) What is the effect on shareholder portfolio value if the same amount was to be raised via a cash

offer at the same price (3 GBP)? Show numerically. (5 points)

(2) Individual investors, asset management firms and institutional investors are subject to 40%, 10%

and 5% taxation on dividends and 20%, 25% and 10% on capital gains respectively. They all invest in

growth stocks, neutral stocks and blue chips. Growth stocks are expected to pay dividends of 2 USD and

have capital gains of 30 every year perpetually. The same forecasts for neutral stocks are 7 and 7 and

for blue chips 40 and 5. The institutional investors are the marginal investors and can determine the

price of the stocks. Every category of investor maximises their after-tax income.

(i) Suppose that institutional investors require a 10% after-tax return. What are the prices of the

growth, neutral, and bluechip stocks?

(ii) Calculate the after-tax returns of the three types of stocks for each investor category. (5 points) [10

points]

Problem 5

(1) Two investors have purchased shares of an all-equity firm which has 70,000 shares outstanding at

15 GBP each. A owns shares of a total value of 50,000 GBP and has borrowed 15,000 GBP. B owns shares

of a total value of 90,000 GBP and has lent 20,000 GBP. The firm decides to repurchase 25% of its shares

by raising perpetual debt at the risk-free rate of 6%.

(i) Assume that the first Modigliani – Miller proposition holds and the investor positions are optimal.

Find the new positions of the investors.

(ii) Consider the alternative case where the firm issues debt and repurchases equity but there is a

corporate tax rate of 20%. Is there any change in wealth for the investors and in the value of the firm?

(5 points)

(2) A company has been reliably following a policy of one annual dividend payment of 5$ per share,

where all earnings are distributed. The company does not grow and there are 200,000 shares priced at

$60. This year, the company decides to use its earnings to repurchase shares instead of issuing a

dividend. There are no taxes and no conclusions can be drawn on profitability or business risk from the

decision. The number of stocks is rounded at the unit digit.

(i) What happens to the stock price immediately after the decision is announced, and how many shares

will be repurchased?

(ii) Project the stock prices and annual rates of return for stockholders under each policy in the future

and discuss your findings. When will the share price exceed $1000 for the repurchase policy? When will

the company be left with no shares (rounded to unit) to be repurchased? (5 points) [10 points]

Problem 6

(1) A discount bond with a face value of 100 maturing in one period has a yield of 5%. A stock currently

priced at 170 USD is expected to either increase to 213 if the annual report (to become public next

period) is profitable or decrease to 152 if it reports losses. Construct the replicating portfolio of a call

and the replicating portfolio of a put, both with a strike price of 170, and find the prices of both options

if the spot price is 170 and if it is 160. Verify the prices of the puts using a different method. (5 points)

(2) (i) The lending (deposit) and borrowing market rates for GBP loans are 0.5% and 0.625% for one

year respectively, and 5.375% and 5.5% for EUR loans (all rates annualised). The spot GBP/EUR

exchange rates are 1 (“buy GBP at”) and 1.005 (“sell GBP at”). A trader quotes a one-year forward rate

of -100 points (buys GBP at 0.99 and sells GBP at 0.955 respectively)

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Explain in full detail why there is an arbitrage opportunity, the position that generates it and calculate

the arbitrage profits. Eliminate it by changing (a) the GBP borrowing or lending rate (b) the EUR

borrowing or lending rate (c) the trader’s forward rates.

(ii) The 3-month interest rate is 1% in Mexico and 3% in Brazil (continuously compounded), while the

spot price of the Mexican peso is 1.05 real. The price of a 3-month future is the same. Are there arbitrage

opportunities? Explain in detail. (5 points) [10 points]

Problem 7

A trader holds two calls and one put on the same share, all of which expiring in three months. The

exercise price of both calls is $70 and the exercise of the put is $80. Each option is sold as a 100-share

contract.

(i) Find the payoff at the expiration date if the stock sells for $65 and if it sells for 90$. Draw the payoff

diagram (profile) for your option position. (3 points)

(ii) Using the above example as a base case (strike prices, long-short, type of assets etc), turn the

position into (a) a short straddle and a long strangle (show both) (b) a butterfly with calls (c) a bear put

spread (d) a collar (e) butterfly with calls (f) iron condor by making any changes you see fit. You can

add or remove any assets you need but try to make as few changes as possible. Explain in detail the

changes you make and the reasoning behind them. Construct the payoff profile for each position and

explain its usefulness and aims. (7 points) [10 points]

Problem 8

It is 12/31/00 and a company is considering two all-equity financed projects on how to construct and

price the same product. The first project requires a new facility that will cost $10 million to purchase.

It will be available and fully operational on 1/1/01, with a production capacity of 10,000 units in the

first and second years after purchase. The market price is $1,000 and per unit costs (labour and raw

materials) are $300. At the end of the second year, production stops and the facility will be salvaged for

$2 million. The second project requires a plant that costs $8 million to purchase, which will also be fully

operational on 1/1/01 and has a production capacity of 7,742 machines in the first and second years

after the purchase. The market price is $1,200 per machine, the per unit costs (labour and raw

materials) are $400. At the end of the second year, production stops and the plant is salvaged for $1

million. All figures are given in nominal terms. The firm has a corporate tax rate of 35 percent and uses

straight line depreciation. The nominal opportunity cost of capital for this type of project is 12 percent.

The nominal debt rate is 6 percent. Assume all cash flows occur at year’s end and that the firm has

profitable ongoing operations.

(i) Assume that in every year the optimal debt capacity of the firm is increased by 30 percent of the

project’s base-case PV and this is the only financing side effect. Which project would you select based

on their NPV and which on their APV at 12/31/00 (express your answer in millions and with 2 decimals,

i.e. ignore small differences)

(ii) In light of the previous calculations, the company is adjusting both projects in the following ways

(all figures nominal). (a) Each year, 10% of the units produced remain unsold and are stored. The

storage costs are $100,000 per year, and at the end of the fourth year the entire stock can be sold at its

production cost. (b) instead of salvaged, the plant is sold for $2 million. (c) The projects run for four

years instead of two. Which project should be selected based on APV? [10 points]

Problem 9

A product manager is evaluating a new product with a 5-year production span. The forecasted annual

sales are 15.000 units, real annual production costs are 12,000 and the real per unit price is constant at

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2 at t=0 (today). The production requires the purchase today of 45,000 (real) of new machinery, which

will be fully straight-line depreciated over the next 5 years, the use of currently owned equipment

which has been fully depreciated and would make a net after tax profit of 25,000 if sold today and land

of 100,000 (nominal) which will be sold at the end of production for 100,000 (nominal). Overheads and

general expenditures are estimated at $5,000 (real) in the first year, increasing by 5% per year in real

terms. The tax rate is 40%, the required return on equity is 14%, the nominal debt rate is 5%, inflation

is 7% and the target debt ratio is 30% and depreciation is straight-line. All cash flows occur at year’s

end and the firm has other profitable ongoing operations.

(b) Find the firm’s weighted average cost of capital.

(c) Suppose that the land purchase is financed via a 5-year loan at the nominal debt rate whose principal

is repaid in full at maturity by using resources from the other activities of the firm. What is the APV of

the project? [10 points]

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