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Risk Management

Mid-term Assignment

Dear Student,

On the following three pages, you will find a total of 5 questions to be solved for the

mid-term assignment. The questions cover chapter 1 to 7 of the book Risk

Management and Derivatives by Rene M. Stulz.

The solutions to these questions have to be delivered in hard copy by Friday, 10 May,

4 pm to the letter box on the 9thfloor of building K (the Finance Departments floor).

Only copies that are delivered in time will be considered!

The final score of this course is calculated as

where F is the score on the Final exam, Athe score on this assignment, and C the

score the case study. Both the case study and this assignment are hence not

compulsory.

You can work in groups of maximum four persons on this assignment. All persons

within the group receive the same mark. There is no penalty (premium) whatsoever for

the number of students within a group.

Good luck,

Lieven Baele

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1. You are the finance director of CR7, a US based manufacturer of luxury products.

The CEO has just signed a large contract with Hotel de Paris from Monaco for a

large delivery of accessories, worth 1 million. Both delivery and payment are due

in 7 months from now. At the current exchange rate of about 1.25 $ per , this

contract is worth about $ 1.25 million.

a. Represent graphically the cash flow (in $) of CR7 in six months on a figureas a function of the $/exchange rate, supposing that CR7 remains long in

, i.e. if it does not hedge.

b. The current US interest rate is 3%, the euro area interest rate is 1%.

Calculate the arbitrage-free 7-month forward price for the $/exchange rate.

Show how this currency forward contract can be replicated by buying /

lending $ and/or s.

c. Describe the position in this forward contract if management of CR7 would

decide to fully eliminate exchange rate risk (long/short, hedge ratio).

Represent graphically the payoffs at maturity against the exchange rate for

the contract of your choice. Indicate at which ranges of the $/ exchange

rate CR7 makes a loss and at which it makes a profit on the forward contract.

d. Plot the hedged cash flow of CR7 if it adopts the position specified in (c) as

a function of the exchange rate. In this same picture, plot the unhedged cash

flow (part (a)).

e. Suppose that instead of hedging using a forward/futures contract, CR7

decides to hedge with options. Describe in detail the type of options you

would choose, and represent graphically the payoff at maturity of the

preferred option against the $/ exchange rate. Assume that the option

premium is 5% of the options strike price.

f. Represent graphically the unhedged cash flow, and the cash flows when

respectively forwards and options are used for hedging. What are thearguments in favor or against the forward / options hedging strategy?

2. You manage a portfolio worth 1 Billion for a Dutch pension fund. The portfolio

invests solely in 10 large European industry indices. The euro denominated returns

(including dividends) are given in the attached excel file. You have currently

allocated 10 percent to each of the industries, i.e. each industry gets an equal

weight.

a. Calculate the average (annualized!) returns and volatilities for the 10

industries. What industries offer the most attractive risk-return trade-off?

b. Calculate the return correlation matrix, and discuss possibilities fordiversification.

c. Calculate the average annualized portfolio return and volatility. Use the excel

function mmult in your calculations of the portfolio volatility1, and report that

formula in your text.

After typing the formula, excel requires you to jointly press ctrl-alt-enter instead of just enter to calculate

the value.

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d. Suppose that the global equity market portfolio is a good proxy for the market

portfolio in the CAPM (returns also attached in attached excel sheet).

Calculate the betas of the 10 industries with respect to the market portfolio.

Interpret the numbers you obtain.

e. Assume the expected return (on top of the riskfree rate) on the market

portfolio amounts to 6 percent. The riskfree interest rate equals 2.5%.Calculate the expected return for each of the industries. Interpret the

numbers. Calculate the expected portfolio return, and use the obtained

number in your answer of the following questions.

f. Financial theory argues that optimal weights are the result of a maximization

of expected return and a minimization of risk. Suppose that the investor

maximizes the function Portfolio Expected Return 3 times the Portfolio

Variance. Use excel solver to obtain optimal weights (again, impose no

short sales allowed and 100 % sum of weights).

g. Assume that the portfolio returns are normally distributed. Calculate the

probability that the portfolio will incur a loss of more than 5 percent over the

next month. Similarly, what is the probability that the portfolio will realize a

loss of more than 10 percent over the next three months?

h. Calculate the probability that in one year the value of the portfolio will lie

between 900M and 1050M.

i. Calculate the 1-month Value-at-Risk at the 1%, 5%, 10% level. Interpret your

results.

j. You consider liquidating 5% (absolute value) of your position in Financials

to invest it fully in Utilities. Quantify the impact of this sale on the 3-month

Value at Risk (at a 5% level) without calculating the expected return and

volatility of the new portfolio. Explain what you find.

3. You have received an additional amount of 100 million to invest in the pension

fund of question 2. Instead of investing it in Europe, you consider purchasing US

equities, as you think that an economic rebound is likely to happen first in the US.

While you are positive about the US equity market, you are less optimistic about

the evolution of the $/exchange rate.

a. Does this investor expect the euro to appreciate or depreciate with respect

to the US$?

b. Should the investor take a short or a long position in the $/ forward

contract?

c. Suppose that the current $/ exchange rate is $1.26/. Given that the $interest rate is 1 percent, and the interest rate equals 2.5 percent, what is

the arbitrage-free forward price?

d. Determine the optimal hedge ratio using the euro-denominated prices on a

broad US equity market index, and the euro to the $US exchange rate (use

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linest function in excel). Explain why or why not it is higher/lower/equal than

1. Evaluate the quality of the hedge.

4. On April 1st, the forward price on 3 month Treasury Bills of face value 1,

deliverable on July 1st and payable on September 30th is 0.98. On April 1st, the

spot price for bills of the same face value and maturing on June 30th andSeptember 30th is 0.97 and 0.945, respectively.

a. Calculate the arbitrage-free treasury-bill forward price.

b. Is there an arbitrage opportunity at these prices? If so, describe it and

compute the profits per unit (I am (also) looking for a concrete number here!)

c. You have a floating-rate loan maturing on September 30th. The interest rate

for the final quarter is set on July 1st. You are worried that interest rates will

increase in the near future, and you would want to hedge against this

increase. Describe the characteristics and position in the correctly-priced

Treasury Bill future. Show in a table that at maturity the hedge indeed works

well.

5. OKB bank, a Belgian provincial bank specializing in providing financial services to

small companies, has an investment portfolio worth 10 Million. Early March 2012,

management receives a letter from the supervisory authorities stressed after they

failed to foresee the recent subprime crisis that they want OKB bank to further

improve its risk management tools, including for relatively small positions like this

one. The returns and asset weights are given in the attached excel file, as well as

the global equity market index that serves as a market portfolio. The riskfree rate is

2 percent, and the market risk premium (in excess of riskfree rate) equals 6 percent.

a. Calculate the one-year Value-at-Risk at the 0.5 percent level (use the

normsinv function to get the 0.5 percent multiplication factor from thestandard normal distribution).

b. The risk manager has allocated 2.5 Million of capital as a buffer specifically

for potential losses on their investment portfolio. Do you consider this

sufficient?

c. Use Excel solver to find the weights that yield a Value-at-Risk at the half-a-

percent level of exactly 2.5 Million while still keeping the expected return

as high as possible. Do this by maximizing the function Expected Return

2 times the VaR(0.5%). Impose a short-sales constraints (all weights at least

0) and a sum of weights equal to a 100 percent.