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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1802 FINANCIAL MATHEMATICS May 23, 2007 Time: 9:30 a.m. — 12:30 p.m. Candidates taking examinations that permit the use of calculators may use any calculator which fulﬁls the following criteria: (a) it should be self—contained, silent, batteryoperated
and pocket—sized 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 SIX questions. Marks are shown in square brackets. 1. Find the following amounts. (a) The accumulated value of $150 at the end of 9 years at a nominal annual
rate of interest of 7% compounded semi—annually for the ﬁrst 4 years and 5%
compounded quarterly for the next 5 tears. [4 marks] (b) The present value of $750 due 7 years from now, at a nominal annual rate of
interest of 6% compounded monthly. [4 marks] (c) The accumulated value of $1,000 at the end of 6 years, at a nominal annual
rate of discount of 5% compounded semiannually. [4 marks] ((1) You are given, a(t) Kt2+Lt+M for 0932 H a(0) = 100
a(1) = 110
a(2) = 136.
Determine the force of interest at time t = [6 marks] [Total: 18 marks] S&AS: STAT1802 Financial Mathematics 2 2. (a) What effective rate of interest is the square root of its equivalent effective rate of discount? . [4 marks]
(b) Given a?) = 10, (52%) = 15. Determine i. [4 marks]
(c) An annuity—immediate consists of 10 payments of 5,8,11,...,32 at times
3, 6, 9, . . . , 30 at an annual effective rate 2'. Express the present value in terms
of annuity functions. [4 marks] [Total: 12 marks] 3. (a) Bank A makes a loan of $80,000 repaid in 30 annual payments of $12,184 at
the end of the year. What is the yield rate? [3 marks] (b) Suppose that, in part (a), Bank A sells the loan after 10 years to Bank B at a
price to yield a rate of 12% for Bank B. Show that the price is $91,000, to the
nearest $100. [4 marks] (c) In parts (a) and (b), what is the yield rate for Bank A? [3 marks] (d) Peter has assets as follows: December 31, 2005 : Asset 2 $2,878,000;
December 31, 2006 : Asset = $2,898,000; Only one net withdrawal of $198,000 was make on July 1, 2006.
Calculate, (i) the dollarweighted rate of return using the exact method, (ii) the dollar—weighted rate of return using the approximate method. [4 marks] (e) David’s stock portfolio has had the following history since he began investing Date Value
1/1/2002 $100,000
1/1/2003 $115,000, followed by immediate deposit of $18,000
1 / 1 / 2004 $145,000, followed by immediate deposit of $23,000
1/1/2005 $185,000, followed by immediate withdrawal of amount X
1/1/2006 $100,000 The annualized equivalence of the timeweighted rate of return of the 4 year
period is 10%. What is the amount of the withdrawal X ? [4 marks] [Total: 18 marks] S&AS: 4. (a) STAT1802 Financial Mathematics 3 A loan of $12,000 is being repaid with payments of $1,500 at the end of each
year for 10 years. These payments can earn interest at an effective rate of 6%
per annum. At the end of the year, this interest is reinvested at the annual
effective rate 5% for the ﬁrst 5 years and only 4% for the second 5 years. Find
the yield rate over the 10year period. [5 marks] A loan of $50,000 is to be repaid by 10 annual payments which are made at the
beginning of each year. The ﬁrst 5 payments are $K each and the remaining 5
payments are $2K each. Find the payments, interests paid, principals repaid
and outstanding balances for the 4th and 5th year if the annual effective interest
rate is 6.0%. [5 marks] A loan bears interest at 6% per annum effective, the interest being payable at the end of each year. The loan is repayable by means of a sinking fund in 10 years. The sinking fund deposits will be made at the end of each year. The sinking fund yields an effective interest rate of 5% per annum for the ﬁrst ﬁve years and 7% per annum thereafter. Suppose the amount of loan is $100,000.
(i) Find the amount of annual outlay. (ii) How much interest Will be earned in the sinking fund for the 7th year?
[6 marks] [Total: 16 marks] Using the bond pricing formula notation, show that
n—l P+iZBt =nFr+C.
t=0 [6 marks] A tenyear $1,000 bond with a coupon rate of 9% payable semiannually is
redeemable at $1,100 at the end of ten years. (i) If Peter is expecting an annual yield rate of not less than 6% convertible
semiannually in his investment, ﬁnd the range of prices of the bond at the
time of issue such that Peter is Willing to buy the bond. [4 marks] (ii) if Peter purchases the bond at $1,250, ﬁnd his expected annual yield rate
convertible semiannually and construct the bond amortization schedule for
the ﬁrst two years. [4 marks] (iii) After Peter has received the fourth coupon payment at the end of second
year, he immediately sells the bond at $1,200. Find the annual yield rate convertible semiannually during this two—year investment period.
[4 marks] [Total: 18 marks] S&AS: STAT1802 Financial Mathematics 4 6. Assume the effective 6month interest rate is 2%, the BBC 6month forward price
is $1,020, and use the following premiums for BBC options with 6 months to expi—
ration. Strike($) Call($) Put ($) 950 120.405 51.777
1,000 93.809 74.201
1,020 84.470 84.470
1,050 71.802 101.214
1,107 51.873 137.167 (a) Suppose you short the BBC for $1,000 and buy a. 950strike call. (1) Construct payoff and proﬁt diagrams for this position. (ii) You can obtain the same payoﬂ‘ and proﬁt by borrowing $X and buying a
950strike put. What is the value of X? [6 marks] (b) Suppose the premium on a 6—month BBC call is $109.20 and the premium on a
put with the same strike price is $60.18. What is the strike price? [4 marks] (0) Suppose you invest in the BBC for $1,000, buy a 950strike put, and sell a
1050—strike call. Draw a proﬁt diagram for this position. What is the net
option premium? [4 marks] (d) If you wanted to construct a zerocost collar keeping the put strike equal to
$950, in what direction would you have to change the call strike? Explain. [4 marks] [Total: 18 marks] ********** END OF PAPER ********** ...
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 Spring '08
 Dr.K.C.Yuen
 Interest Rates, Effective Interest Rate

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