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 are
strictly prohibited.
• Good luck!
Q1
Q2
Q3
Q4
1
Q5
Total
Problem 1. Decide whether the following statements are true or false. No
explanations 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 you
receive $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 margin
account.
(True)
(h) Consider an interest rate that is quoted as 10% per annum with semiannual compounding. The equivalent rate with continuous compounding is
9.758% per annum.
(True)
(i) According to the liquidity premium hypothesis (also called liquidity preference 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 700,000
euros. You can put 300,000 euros down, and for the rest you get a 20-year
fixed rate mortgage from your bank. The annual percentage rate is 5% per
year, 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 are
liable an annuity with monthly cash flows C, and you know that the fair value
is 400,000 euros. There are T = 240 monthly cash flows and the monthly
interest rate is r = 5%/12.
The annuity formula implies that:
1
C
400, 000 = × 1 −
r
(1 + r)T
or equivalently:
400, 000 × r
.
1 − (1 + r)−T
So the amount you have to pay each month is
C=
C =
400, 000 × 0.05/12
= 2, 639.82 euros.
1 − (1 + 0.05/12)−240
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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, $10
in three years, and a coupon and principal payment of $110 in four years. In
this 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
$110
$10
$10
+
+
.
+
1 + Y T M (1 + Y T M )2 (1 + Y T M )3 (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 determine
if it the offer in the market is a fair price. The annualized yields on zerocoupon 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
$10
$110
$10
+
+
+
= $124.52.
1.01 (1.02)2 (1.03)3 (1.035)4
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 the
bond worth?
Solution: The bond is worth:
P =
$10
$10
$10
$110
+
+
+
= $117.73.
1.05 (1.05)2 (1.05)3 (1.05)4
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 foward
contracts.
(a) Suppose that you enter into a 6-month forward contract on a non-dividend
paying stock when the stock price is $30 and the risk-free interest rate
(with continuous compounding) is 12% per annum. What is the forward
price?
Solution: The price of the 3-month futures contract is
F0 = $30e0.12×0.5 = $31.86.
(b) A stock index currently stands at $350. The risk-free interest is 8% per
annum (with continuous compounding) and the dividend yield on the
index is 4% per annum. What should the futures price for a 4-month
contract be?
Solution: The price of the 3-month futures contract is:
F0 = $350e(0.08−0.04)×4/12 = $354.70.
(c) The spot price of silver is $9 per ounce. The storage costs are $0.06 per
quarter payable in advance. Assuming that interest rates are 10% per
annum 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 + $0.06e−0.1×0.25 + $0.06e−0.1×0.5 = $0.1756.
The forward price is therefore:
F0 = ($9 + $0.1756)e0.1×9/12 = $9.89.
The forward contract can be replicated as follows.
• At date t = 0, we borrow $9.06, purchase 1 ounce of silver on the
spot 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:
$9.06e0.1×9/12 + $0.06e0.1×6/12 + $0.06e0.1×3/12 = $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 months
and in 5 months. The stock price is $50, and the risk-free rate of interest is
8% per annum with continuous compounding for all maturities. An investor
has 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 = $2e−0.08×2/12 + $2e−0.08×5/12 = $3.9079.
The forward price is therefore
F0 = (50 − 3.9079)e0.08×6/12 = $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 of
interest is still 8% per annum. What is the value of the short position in
the forward contract?
Solution: The present value of the future dividend is now
I1 = $2e−0.08×2/12 = $1.9735.
The forward price is now
F1 = ($48 − $1.9735)e0.08×3/12 = $46.96.
The value of the short position is therefore
(F0 − F1 )e−0.08×3/12 = $1.00
6