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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1802 FINANCIAL MATHEMATICS May 7, 2008 Time: 2:30 p.m.  4:30 p.m. Candidates taking ea‘aminations that permit the use of calculators may use any calculator
which fulﬁls the following criteria: ( a) it should be selfcontained, silent, batteryoperated
and pocketsized and ( b ) it should have numeral—display facilities only and should be used
only for the purposes of calculation. It is the candidate’s responsibility to ensure that the calculator operates satisfactorily and
the candidate must record the name and type of the calculator on the front page of the
examination scripts. Lists of permitted/prohibited calculators will not be made available to
candidates for reference, and the onus will be on the candidate to ensure that the calculator
used will not be in violation of the criteria listed above. Answer ALL EIGHT questions. Marks are shown in square brackets. 1. Mary deposits $500 at the beginning of each year for 20 years. Simple interest at an
annual rate of i is credited to each deposit from the date of deposit to the end of
the twentyyear period. The total amount thus accumulated is $14200. If instead,
compound interest had been credited at an effective annual rate of i, What would be
the accumulated value of these deposits at the end of twenty years? [10 marks] 2. In Fund A, the accumulated value of 1 at any time t > 0 is 1 + t. In Fund B, the
accumulated value of 1 at any time t > 0 is 1 + t2. T is the time When the force of interest for Fund A is equal to the force of interest for Fund B. Calculate the value
of T. [10 marks] 3. A perpetuity with annual payments is to be payable at the end of 9 years from now.
The ﬁrst payment is $250. Each annual payment thereafter is increased by $50 until
a payment of $750 is reached. Subsequent payments remain level at $750. This
perpetuity is purchased by means of 10 annual premiums, with the ﬁrst premium of
P due immediately. Each premium after the ﬁrst is 105% of the preceding one. The
annual effective interest rates are 5% during the ﬁrst 9 years and 3% thereafter. Find
the premium P. [11 marks] S&AS: STAT1802 Financial Mathematics 2 4. The following information on an investment account is given: _ 1 January 2007 1 May 2007 1 September 2007 1 January 2008 Book value
100,000 150,000 180,000 134,000 (before deposit
or withdrawal)
_ 30,000 _—
Withdrawal —— 50.000 —
Calculate the yield rates by the dollarweighted and timeweighted methods.
[11 marks] 5. A loan is being repaid by 15 annual installments of $2000 each. Interest is at an
effective annual rate of 5%. Immediately after the ﬁfth installment is paid the loan is
renegotiated. The revised amortization schedule calls for a sixth payment of $1600,
a seventh installment of ($1600 + K), with each subsequent installment increasing
by K over the previous payment. The period of the loan is not changed. Determine
the revised amount of the last installment. [11 marks] 6. Peter purchases an annuity at a price of $2000. The annuity makes payments of $100
at the beginning of every 6 months for 20 years. The payments are reinvested in a
fund which earns interest at an annual effective rate 2'. Interest payments are received
every 6 months and reinvested at a nominal rate of 6%, convertible semiannually.
Peter realizes an overall effective annual yield of 7% on his original investment over
the 20—year period. Calculate the annual effective rate 2'. [11 marks] 7. Two nyear $1000 par value bonds are available. Bond X has 14% semiannual
coupons and a price of $14077, to yield z', compounded semiannually. Bond Y has
12% semiannual coupons and a price of $12718, to yield the same rate 2', compounded
semiannually. Find the price of Bond X to yield 7L — 1%. [11 marks] 8. Assume the eﬁective 6month interest rate is 2%, the S&R 6month forward price is
$1020, and use these premiums for S&R options with 6 months to expiration: Strike Call Put
950 120.405 51.777
1000 93.809 74.201
1020 84.470 84.470
1050 71.802 101.214 1107 51.873 137.167 S&AS: STAT1802 Financial Mathematics 3 a. Verify that you earn the same proﬁt and payoff by (1) buying the S&R index for
$1000 and (ii) buying a 950—strike S&R call, selling a 950strike S&R put, and
lending $931.37. b. Suppose the premium on a 6month S&R call is $109.20 and the premium on a
put with the same strike price is $60.18. What is the strike price? c. Suppose you buy a 1050strike 885R straddle. For which range of the S&R index
will you make a proﬁt when it expires? 01. Suppose you buy a 950—strike S&R call, sell a 1000—strike S&R call, sell a 950
strike S&R put, and buy a 1000—strike S&R put. i. What is the payoff when this portfolio expires?
ii. What is the cost of the position?
iii. What is the implicit effective interest rate over 6 months in these cash ﬂows? [Total: 25 marks] ...
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 Spring '08
 Dr.K.C.Yuen

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