Problem Set 2

Winston Dou Fall 2024

Due: 9/26/2024

Question 1: Basic Concepts on Forward Curves (1/10)

(1) Define what a forward curve is. (1 sentence)

(2) Define what contango is. (1 sentence)

(3) Define what backwardation is. (1 sentence)

(4) Define what convenience yield is. (1 sentence)

Question 2: Basic Concepts on Currency Forward Contracts (1/10)

(1) Define what forward exchange rate is. (1 sentence)

(2) Define what carry trade is. (1 sentence)

(3) Define what covered interest rate parity is.

(4) Define what uncovered interest rate parity is.

Question 3: Non-Arbitrage and Transaction Costs (2/10)

(1) The current bid and ask prices of a share in the firm NoDividends (NDV) are $20
and $20.10 respectively. The 6-month T-Bill rate (lending rate) and the LIBOR rate
(borrowing rate) are respectively 5% and 6% (both in continuously compounded terms).
Assume that there are no other transactions costs. If the current bid and ask prices
for a 6-month forward on NDV stock are $20.35 and $20.45 respectively, is there an
arbitrage? If so, what is it? If not, explain why not.

(2) How would your answer change if there are additional transactions costs associated
with shorting NDV stock? (You do not necessarily need to do any further calculations
just clearly state your reasoning.)

(3) Re-do part (1) for the case of current bid and ask forward prices of $20.81 and $20.91
respectively.

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 (4) How would your answer to part (3) change if there are additional transactions costs
associated with shorting NDV stock? (Again, no additional calculations are required.)

Hint: Chapter 5 of McDonald’s textbook.

Table 1: DJIA Futures

DJ Industrial Average DJIA Futures Price

Jan 27, 2009 8,174.73 8,136.00
Jan 28, 2009 8,375.45 8,336.50
Jan 29, 2009 8,149.01 8,112.00

Question 4: Futures Contracts (2/10) The table above shows the daily closing values
of the Dow Jones Industrial Average index (DJIA), along with the daily prices for the DJIA
Futures contract with maturity date March 20 of the same year. The futures contract size is
$10 times the DJIA value.
The initial margin required for the futures contract is $13, 750/contract. The maintenance
margin is $11, 000/contract. The risk free rate is 1% (continuously compounded). Assume
this is the rate you earn on your margin account.

(i) Suppose you take a long position in 10 futures contracts on Jan 27. What is your
profit/loss on Jan 28 and Jan 29? What is the balance in your margin account at the
end of each of these days? Do you face any margin calls?

(ii) Repeat part (i) for a short position of 5 futures contracts.

(iii) Using the Jan 29 index value and futures price, what is the implied dividend yield on
the DJIA? (Assume that the futures price is the same as the equivalent forward price.)

(iv) For the three days shown, the index value is larger than the futures price. Why?

Question 5: Commodity Forward Contracts with Lease Rate (2/10) The current
price of crude oil is $32.00 per barrel. Forward prices for 3, 6, 9, and 12 months are $31.37,
$30.75, $30.14, and $29.54. Assuming a 2% continuously-compounded annual risk-free rate.

(i) Suppose there is an active leasing market for crude oil with continuously-compounded
annual lease rate q. What is the annualized lease rate for each maturity?

(ii) Suppose there is no leasing market or carry cost. What is the convenience yield c for
each maturity?

(iii) Is this an example of contango or backwardation?

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Question 6: Commodity Forward Contracts with Convenience Yield (2/10)
Suppose you are a trader for an investment bank. An investor comes to you and want to
enter a long forward position on one unit of soybeans. The forward price is F0,T which
is determined by you. There is no lease market, carry cost, or transaction fee. The
continuously-compounded interest rate is r.

(1) Suppose the investor has convenience yield c0 by holding the soybean and can borrow
from potential lenders who have convenience yield c00 by holding the soybean. Assume
that c0  c00 . Derive the interval of F0,T within which the investor cannot make
arbitrage profits.

(2) If c0 = c00 for the investor, what is the forward price you should set?

(3) Suppose here are many investors come to you with convenience yields c1 < · · · < cI ,
and each investor i 2 {1, · · · , I} has the same convenience yield ci with her potential
lenders. What is the minimum and maximum forward prices you would o↵er?

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A forwardcurve is agraph showinghow forwardpriiesare set fordifferentmaturities

contango occurswhenforwardpricesincreaseas termto maturityincreases r q

backwardation osiers whenforwardprices delrease as termto maturity

increases req

a convenience yieldis the benefit youget whenphysicalinventory of theunderlyingis valuable

a forwardexchange rate is the rate at which twocurrencies aretradedat or what thevalue

ofone currency is interms ofanothercurrenty

The currencycarrytradeis where a trader borrows a currencyfroma country with a lowinterestrate

converts it into a currencyin a country with a high rate and invests it

idea that the basis traded forwardrate theoreticalforwardrate equals or is closeto

Hythe

rates eaval the futureexpectedspotrate but

I itemp a with

in e c
ask you
buy at

assumed so 20
2210 20.0s
2
Fo 20.0s e 20.56
20.56 20.45so thereis anarbitrage opportunity
Youshould shortsell Nov andreceive920 fromthesale Then

Keep 20.506 20.45 9.06in profit

 Then it woulddependon theadditional costs If returnsfrom
frominvesting proceedsfrom the shortsale AND after
additional costs no longer exiledthetraded forwardpulle
it is no longer worth it

assumeds zn.es s
Fo 20.56 220.81 and20.91

portunity exists you ran shortforward at to

20.81
borrow 20.10 to
buy NDV settle forward 6 mths later by
selling NAV at 20.81 pay bank loan of 20 10x e se20.71 so you
earn the difference of 20.81 20.71 10

to make
sure the forwardprice those costs exceed what you owe foryourloan

contraits

amazon.soasainonm.it
Jan28
ii
r 1 also earnedon
13750 10 137500 initial balante marginaccount
3
balance 13 500 e 137503.77
mustmaintain110000

balance 15 553
si i.ii.iIfiis
e 01
1
365 157558.09
endof
157558.09 22450 1135108.097
gang
we are any both above 110 no
 5 10 8136 8336.5 10025
13750 5 68750 initial balance
11000 S 55000 maintenancemargin
1
balance 68 so e 01 365 68751.88 no margincall
91 8 68751.88 10025 158726.81 isneededbecause
me

g
58728.49 11225
496
69953

Foot Saxe
r 9 5 Jan 29 mar 20
g
01 9 5936s
112 8149.01 e
9 0432 14.322
since the dividend yield isgreater than theinterest rate
the futures priceamounts for thefact that youwon'tbe
receiving that large dividend if youlongthefuture anddontown
theunderlying

50
327 s e't
d T

Fo 31.37 31.37 328 q 099s 9.95

For61,2 30.75 30.75 3228.02 9 a 0997 9.97
9 g 98
Fosa 30.14 30.14 3281.02 q 0998
02 9 10
For 29.54 29.54 32e a 09999
2010

r
 1 0
rte c le
Foie s e
Fo31 31.37 31.3 32
02 c 4 0995 9,45 lo
c 0997
Fo 61,2 30.75 30.75 3228.02 9.97
c 9.98
Fosa 30.14 go.ly 3281.02 c 0998
02 c 10
For 29.54 29.54 32e 09999
2010

This is backwardation because towardprices decrease as timetomaturityincreases

maximum
r St
cash larry trade possible unless Soe for
If Fort is bigger it assumes a lower c Investorcan long
the soybean incurring c's and short theforward at its highprice
meaning they can sell the soybean at that talked in higher
price earning the difference betweentoss and cost of holdingtheasset
r e minimum
reverse sasht larry trade possibleunless so e fog
otherwise they can short the soybean 1 and compensate c to lender
and short theforward at the lowerprice earning

nonarbitrage region Soe't c's fast soeer cast

T
then for Soe C c c

maximum for s e r cist investor w lowest CI

minimum Speir cast investor w highest e

for
e suppose investors 1 1 I