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HEC Paris

MBA Program

Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Financial Markets

Prof. Laurent E. Calvet

Fall 2010

MIDTERM EXAM

90 minutes

Open book

The exam will be graded out of 100 points.

Points for each question are shown in brackets.

There are 5 questions carrying equal weight. Answer all five questions.

You are allowed to use a calculator. All other electronic devices arestrictly prohibited.

Good luck!

Q1 Q2 Q3 Q4 Q5 Total

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Problem 1. Decide whether the following statements are true or false. Noexplanations are necessary.

(a) Forwards and futures are examples of derivative contracts.

(True)(b) You short a stock if you believe that it will go up.

(False)

(c) You are financially better off if you receive $1 in five years than if youreceive $1 today.(False)

(d) The IRR rule always gives the same answer as the NPV rule.

(False)

(e) A futures contract is typically settled daily.(True)

(f) A forward contract is typically settled daily.(False)

(g) In order to trade a futures contract, you need to deposit funds in a marginaccount.(True)

(h) Consider an interest rate that is quoted as 10% per annum with semian-nual compounding. The equivalent rate with continuous compounding is9.758% per annum.(True)

(i) According to the liquidity premium hypothesis (also called liquidity prefer-ence theory), long-term interest rates are typically higher than short-term

interest rates.(True)

(j) A zero-coupon bond typically trades above its face value prior to maturity.(False)

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Problem 2. You want to buy an apartment in Versailles, which costs 700,000euros. You can put 300,000 euros down, and for the rest you get a 20-yearfixed rate mortgage from your bank. The annual percentage rate is 5% peryear, compounded monthly. How big is your monthly payment? You assume

that there are no other taxes and fees involved.

Solution: Denote the unknown monthly payment amount by C. You areliable an annuity with monthly cash flows C, and you know that the fair valueis 400,000 euros. There are T = 240 monthly cash flows and the monthlyinterest rate is r = 5%/12.

The annuity formula implies that:

400, 000 = Cr

1 1(1 + r)T

or equivalently:

C =400, 000 r

1 (1 + r)T.

So the amount you have to pay each month is

C =400, 000 0.05/12

1 (1 + 0.05/12)240= 2, 639.82 euros.

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Problem 3. A riskless coupon bond is offered in the market at a price of$124.73. It has coupon payments of $10 in one year, $10 in two years, $10in three years, and a coupon and principal payment of $110 in four years. Inthis problem, we assume that all yields and interest rates are compounded

annually.

(a) Compute the YTM based on the market offer price.

Solution: The YTM satisfies the equation:

$124.73 =$10

1 + Y T M+

$10

(1 + Y T M)2+

$10

(1 + Y T M)3+

$110

(1 + Y T M)4.

We check by trial and error (or with a financial calculator) that Y T M =3.30%.

(b) Using the yield curve of zero coupon bonds, price this bond and determineif it the offer in the market is a fair price. The annualized yields on zero-coupon bonds are given below.

Maturity Annualized Yield

1 year 1.00%

2 years 2.00%3 years 3.00%4 years 3.50%

Solution: The bond is worth:

P0 =$10

1.01+

$10

(1.02)2+

$10

(1.03)3+

$110

(1.035)4= $124.52.

The offer price is slightly higher than the price implied by zero yields.

(c) The zero yields are now 5% per annum for all maturities. What is thebond worth?

Solution: The bond is worth:

P =$10

1.05+

$10

(1.05)2+

$10

(1.05)3+

$110

(1.05)4= $117.73.

The higher interest rates negatively impact the bond price.

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Problem 4. You are asked to compute the price of the following fowardcontracts.

(a) Suppose that you enter into a 6-month forward contract on a non-dividendpaying stock when the stock price is $30 and the risk-free interest rate(with continuous compounding) is 12% per annum. What is the forwardprice?

Solution: The price of the 3-month futures contract is

F0 = $30e0.120.5 = $31.86.

(b) A stock index currently stands at $350. The risk-free interest is 8% perannum (with continuous compounding) and the dividend yield on theindex is 4% per annum. What should the futures price for a 4-monthcontract be?

Solution: The price of the 3-month futures contract is:

F0 = $350e(0.080.04)4/12 = $354.70.

(c) The spot price of silver is $9 per ounce. The storage costs are $0.06 perquarter payable in advance. Assuming that interest rates are 10% perannum for all maturities, calculate the forward price of silver for delivery

in 9 months.Solution: The present value of the storage costs is:

$0.06 + $0.06e0.10.25 + $0.06e0.10.5 = $0.1756.

The forward price is therefore:

F0 = ($9 + $0.1756)e0.19/12 = $9.89.

The forward contract can be replicated as follows.

At date t = 0, we borrow $9.06, purchase 1 ounce of silver on thespot market, pay the storage cost, and store our purchase.

In 3 months, we borrow $0.06 and pay the storage cost.

In 6 months, we borrow $0.06 and pay the storage cost.

In 9 months, we pay back our debt, which now amounts to:

$9.06e0.19/12 + $0.06e0.16/12 + $0.06e0.13/12 = $9.89,

and take the silver out of storage.

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Problem 5. A stock is expected to pay a dividend of $2 per share in 2 monthsand in 5 months. The stock price is $50, and the risk-free rate of interest is8% per annum with continuous compounding for all maturities. An investorhas just taken a short position in a 6-month forward contract on the stock.

(a) What are the forward price and the initial value of the forward contract?

Solution: The present value of the dividends is

I = $2e0.082/12 + $2e0.085/12 = $3.9079.

The forward price is therefore

F0 = (50 3.9079)e0.086/12 = $47.97.

The value at origination of a forward contract is zero.

(b) Three months later, the price of the stock is $48 and the risk-free rate ofinterest is still 8% per annum. What is the value of the short position inthe forward contract?

Solution: The present value of the future dividend is now

I1 = $2e0.082/12 = $1.9735.

The forward price is now

F1 = ($48 $1.9735)e0.083/12 = $46.96.

The value of the short position is therefore

(F0 F1)e0.083/12 = $1.00

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