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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1802 FINANCIAL MATHEMATICS May 15, 2006 Time: 2:30 p.m.  5:30 p.m. Candidates taking examinations that permit the use of calculators may use any calculator
which fulﬁls the following criteria: (a) it should be selfcontained, silent, batteryoperated
and pocket—sized and (b) it should have numeraldisplay 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. At time 0, Peter borrows $1 from Mary and issues her a promissory note for the loan,
with payment due at time t < 1 (years) at interest rate i per annum. At time t1 < t,
Mary sells the promissory note to Carl. Carl pays Mary an amount X so that Mary’s
yield during the time she held the promissory note (i.e. from time 0 to time t1) can
be quoted as jl per annum and Carl’s yield can be quoted as jg per annum (i.e. from
time t1 to time t). All calculations are based on simple interest. (a) Derive an expression for X in terms of jl. [2 marks]
(b) Derive an expression for X in terms of jg and i. [3 marks]
(c) Show that if jl = jg then they are less than i. [5 marks]
(d) Show that if jg increases then jl decreases. [3 marks]
(e) 1ft: 0.712329, t1 = 0.243836 and i =j1 = 0.13. Find X and jg. [3 marks] [Total: 16 marks] On 1 January and 1 July each year for the next 20 years a company will pay a premium
of $200 into an investment account. In return the company will receive a level monthly
annuity for 15 years, the ﬁrst annuity payment being made on 1 January following payment of the last premium. Find the amount of the monthly annuity payment, given that it is determined on the
basis of (a) an interest rate of 12% per annum effective; [3 marks]
(b) a discount rate of 12% per annum convertible half—yearly; [3 marks]
(0) an interest rate of 12% per annum convertible monthly. [3 marks] [Total: 9 marks] S&AS: STAT1802 Financial Mathematics 2 3. Emily’s trust fund has a value of 100,000 on January 1, 2005. On April 1, 2005, 10,000
is withdrawn from the fund, and immediately after this withdrawal, the fund has a
value of 95,000. On January 1, 2006, the fund’s value is 115,000. (a) Find the timeweighted rate of investment return for this fund during 2005. [4 marks]
(b) Use Newton Raphson method to estimate (up to 4 decimal points) the annual
rate of investment return for Emily’s fund. [4 marks] ((3) Find the rate of return for Emily’s fund assuming a uniform distribution through
out the year of all deposits and withdrawals. [3 marks] [Total: 11 marks] 4. Assume that 6(t), the force of interest per annum at time 15 (years) is given by the formula
0.08 for 0 S t < 5 0.06 for 5 S t < 10
0.04 for t 2 10 6(t) (a) Derive the expression for v(t), the present value of 1 due at time t. [3 marks] (b) An investor effects a contract under which he will pay 15 premiums annually in
advance into an account which will accumulate according to the above force of
interest. Each premium will be of amount of 600 and the ﬁrst premium will be
paid at time 0. In return the investor will receive either (i) the accumulated amount of the account one year after the ﬁnal premium is
Paid; 0r [3 marks]
(ii) A level annuity payable annually for eight years, the ﬁrst payment being made
one year after the ﬁnal premium is paid. [3 marks] Find the lump sum payment under option and the amount of the annual
annuity payment under option (ii). [Total: 9 marks] 5. A loan is repaid by annual payments continuing forever, the ﬁrst one due one year
after the loan is taken out. Assume that the annual effective rate of interest is i. (a) Find the amount of the loan if i = 0.5 and the payments are 1, 2, 1, 2, . .. . [4 marks]
(b) Express the amount of the loan in terms of actuarial symbols including (I a)ﬁ and
am if the payments are 1,2,...,n, 1,2, . . .,n,.... [5 marks] (c) What is the value of the loan in part (b) if z' = 0.5 and n ——) oo? [3 marks] [Total: 12 marks] S&AS: STAT1802 Financial Mathematics 3 6. A 90,000 mortgage loan is repaid by level payments at the end of each month for the
next 25 years. The rate of interest is 11.5% convertible semiannually. (a) Find the amount of level payment. [3 marks] (b) Calculate the principal paid and the interest paid at the end of the ﬁrst month.
[3 marks] (0 Find the outstanding balance immediately after the 75th payment. [3 marks] (i.e. 76th N 94th payments). He then Wishes to increase his payments so that the
mortgage will still be paid off at the scheduled time. Find the amount of the new
level payment. [4 marks] [Total: 18 marks] 7. A 100 face value bond with annual coupon rate of 6%, redeemable at the end of 71
years at 93.04 to yield 7.5% effective per year. Find the price of a 100 face value bond
with annual coupon rate of 5%, redeemable at the end of 217, years at 104, to yield 7.5%
effective per years. [12 marks] 8. A 10—year par value bond of 1000 face amount has annual coupons which start at 200
and decrease by 20 each year to a ﬁnal coupon of 20. (a) Find the price of the bond to yield 12% per year. [5 marks] (b) Compute the yield rate (up to 4 decimal points) if the bond is purchased at its
face value. [8 marks] [Total: 13 marks] ********** ********** ...
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 Spring '08
 Dr.K.C.Yuen

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