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Course FM Manual
by Dr. Krzysztof Ostaszewski, FSA, CFA, MAAA
May 2007 Edition
Errata
Posted May 22, 2007
Problem No. 26 in Practice Examination No. 8 should be:
An investor buys a six-months call on an asset, with the exercise price of the call being
85. The premium paid for the call is 5. At the same time, this investor buys a put with the
exercise price of 90, also expiring in six months, and pays a premium of 2. What is the
least amount of combined gain this investor will earn, including the cost of both options
purchased?
A. –2
B. 0
C. 2
D. 5
E. Cannot be calculated
Solution.
If the asset price in six months is P then the payoff ! of the position is
if P # 90,
$ P " 85,
( $ P " 85, if P # 90,
&
& &
! = %( P " 85 ) + ( 90 " P ) , if 90 > P # 85, ) = % 5,
if 90 > P # 85,
& 90 " P,
&
&
if P < 85.
'
* '90 " P, if P < 85.
This shows that for the price in excess of 90, this investor will earn more than 5, and for
the price under 85, the investor will also earn more than 5. Therefore, the minimum
earned on the option position is 5, while the total cost of the options is 5 + 2 = 7, for a
total net gain of 5 – 7 = –2.
Answer A.
Posted May 22, 2007
Problem No. 26 in Practice Examination No. 6 should be:
The current price of a stock is $50. The stock value either increases by 6% or decreases
by 5% in six months. The risk-free interest rate is 4% per annum with continuous
compounding. Determine the value of a six-month European call option with a strike
price of $52.
A. $0.63
B. $0.64
C. $0.65
D. $0.66
E. $0.67
Solution.
The risk-neutral probability of the up move is
1
!0.04
e2
" 0.95
p=
# 0.638194.
1.06 " 0.95
In the case of the up move the stock price will be $50 !1.06 = $53, and in the case of the
down move it will be $50 ! 0.95 = $47.50. The call will pay $1 in the case of the up
move, and $0 in the case of the down move. The value of the call is therefore
p ! $1! e
Answer A.
"0.04!
1
2
# $0.63.
Posted May 14, 2007
Problem No. 26 in Practice Examination No. 8 should be (previous version used the
word “profit” in a manner inconsistent with the definition used in McDonald’s
textbook):
An investor buys a six-months call on an asset, with the exercise price of the call being
85. The premium paid for the call is 5. At the same time, this investor writes a put with
the exercise price of 90, also expiring in six months, and receives a premium of 2. What
is the least amount of combined profit this investor will earn?
A. –4
B. 0
C. 2
D. 5
E. Cannot be calculated
Solution.
If the asset price in six months is P then the payoff ! of the position is
if P # 90,
$ P " 85,
( $ P " 85, if P # 90,
&
& &
! = %( P " 85 ) + ( 90 " P ) , if 90 > P # 85, ) = % 5,
if 90 > P # 85,
& 90 " P,
& &90 " P, if P < 85.
if P < 85.
'
* '
This shows that for the price in excess of 90, this investor will earn more than 5, and for
the price under 85, the investor will also earn more than 5. Therefore, the minimum
earned on the option position is 5, while the net cost of the options is 5 – 2 = 3, for a total
net gain of 5 – 3 = 2.
Answer C.
Posted April 11, 2007
The section on cross hedging should be shortened to just say this:
Cross-Hedging
There are situations when hedging is achieved not by a position in a security identical or
directly related to the originally position held, but a more remotely related security. This
means practically that the movements of prices of the hedged security will not be
mimicked by the hedge, but will only be related somehow. A position consisting of a
portfolio currently held and a cross hedge associated with it has some risk left in it, due to
possible divergence of the value of the portfolio hedged and the value of the hedge used.
This kind of risk is called the basis risk.
The material presented previously went beyond what is needed for the exam.
Posted April 9, 2007
Problem No. 26 in Practice Examination No. 1 should read (the problem previously
used went beyond what is covered on the exam):
You are given that an investment of 100 made at time 0 will produce the same interest
income in the eleventh year, as an investment of 50 will produce in the seventeenth year.
The rate of interest is the same every year, and interest is compounded annually.
Calculate the nominal annual rate of discount compounded semiannually, denoted by
d ( 2 ) , which gives this result.
A. 11.89%
B. 10.78%
C. 9.79%
D. 7.73%
E. 5.45%
Solution.
Let us write i for the effective annual interest rate. Then we have
10
16
100 ! (1 + i ) ! i = 50 ! (1 + i ) ! i,
so that
(1 + i )6 = 2,
and
1
i = 2 6 ! 1.
From this we get
# 1
&
# 1
&
d ( 2 ) = 2 ! % 2 6 " 1( = 2 ! % 2 12 " 1( ) 11.892619%.
$
'
$
'
Answer A.
Posted April 9, 2007
Problem No. 28 in Practice Examination No. 8 should read:
May 2005 Casualty Actuarial Society Course 8 Examination, Problem No. 22(c)
(multiple choice answers added)
An investor contacts her broker to enter into five long futures contracts on bushels of
corn. Each contract is for the delivery of 5,000 bushels. The current futures price is $2.40
per bushel. The initial margin is $2,000 per contract. The maintenance margin is $1,500
per contract. Calculate the futures price of corn at which the investor could withdraw
$2,000 from the margin account.
A. $2.38
B. $2.40
C. $2.42
D. $2.44
E. $2.46
Solution.
Five contracts for 5,000 bushels total 25,000 bushels. The current futures price is $2.40
per bushel, for a total futures price for 25,000 bushels for the entire position of
25, 000 ! $2.40 = $60, 000.
The investor posts initial margin of five times $2,000, i.e., $10,000. Following that,
maintenance margin is five times $1,500, i.e., $7.500. This means that once the trade is
placed, $2500 will become available immediately. But for $2000 to be available, the
investor will have to wait till the end of the first day of trading. The investor will be able
to withdraw $2000 if a loss of $500 is produced at the end of the day. For that loss to be
produced, futures price must decrease by
$500
= $0.02.
25, 000
Thus the futures price must be $2.38.
Answer A.
Posted February 11, 2007
In Exercise 6.2, the value of duration should be 6.92463175 years. The rest of the
exercise is not materially affected by this correction.
Posted December 4, 2006
In the Problem 29, Practice Examination 4, the solution is correct, but the answer
choice should be D.
Posted December 14, 2006
In Practice Examination 7, the Problem 18 should be:
28. The premium on a two-month call option on the TECHWIG index of the Warsaw
Stock Exchange in Poland with an exercise price of 1050 is 9.30 Polish Złoty (PLN).
Zbigniew, a well-known Polish political analyst, and a shrewd investor, borrows just
enough to buy 100 such call options. The interest rate charged on the loan is the current
risk-free rate of 0.20% per month compounded continuously. After two months, the index
spot price is 1072 PLN, and Zbigniew closes the position, paying off the loan with the
proceeds of the transaction. Calculate the amount of profit realized by Zbigniew at the
close of the entire transaction.
A. PLN 1232 B. PLN 1266 C. PLN 1299 D. PLN 1325 E. PLN 1378
Solution.
Zbigniew borrows PLN 930, and will have to pay off the accumulated value of the loan
in the amount (in PLN) 930 ! e2!0.002 " 933.72. The call payoff will be PLN 1072 – PLN
1050 = PLN 22 per option, for a total of PLN 2200 for one hundred options. The net
profit of Zbigniew is PLN 2200 – PLN 933.72 = PLN 1266.28.
Answer B.