Sta 2191: Financial Mathematics I Question Paper

Sta 2191: Financial Mathematics I

Course:Bachelor Of Science In Actuarial Science

Institution: Dedan Kimathi University Of Technology question papers

Exam Year:2012

KIMATHI UNIVERSITY COLLEGE OF TECHNOLOGY
University Examinations 2011/2012
FIRST YEAR SPECIAL/SUPPLEMENTARY EXAMINATION FOR THE DEGREE
OF BACHELOR OF SCIENCE IN ACTUARIAL SCIENCE
STA 2191: FINANCIAL MATHEMATICS I
DATE: 1ST MARCH 2012 TIME: 2 HOURS
Instructions: Answer QUESTION ONE and any other TWO QUESTIONS.
QUESTION ONE (30 marks) (COMPULSORY)
(a). An investment is discounted for 28 days at a simple rate of discount of 4.5% per
annum. Calculate the annual eective rate of interest. [3 marks]
(b). For a rate of interest of 7% per annum, convertible monthly, calculate:
(i). the equivalent rate of interest per annum convertible half yearly, and
(ii). the equivalent rate of discount per annum convertible monthly
(ii). the equivalent force of interest per annum. [6 marks]
(c). Calculate s(12)
5:5 at an eective rate of interest of 13% per annum. [3 marks]
(d). If i(p) denotes the eective rate of interest and d(p) denotes the eective rate of
discount for p-thly payments, show that i(p) ?? d(p) = 1
p i(p)d(p). [3 marks]
(e). A sum of

$100 is accumulated at a nominal rate of discount of 71 2 p.a. convertible quarterly for 1 year, and then at a nominal rate of interest of 71 2 p.a. convertible quarterly for 1 year. Calculate the accumulated amount of the investment after 2 years. [4 marks] (f). The force of interest is given by: (t) = 0:05 + 0:001t + 0:0001t2 0 < t 10 Calculate (i). the total at time 10 of the accumulated proceeds of an investment of$ 100 at
time 0 plus an investment of

$100 at time 5. [4 marks] (ii). the equivalent constant force of interest earned on the transaction.[3 marks] 1 (g). In an annuity the rate of payment per unit of time is continuously increasing so that at time t it is t. Show that the value at time 0 of such an annuity payable until time n, which is represented by (Ia)10 , is given by (an ?? nvn)= where is the force of interest per unit of time and an = Z n 0 e??tdt [4 marks] QUESTION TWO (20 marks) (a). A loan of$ 80; 000 is repayable over 25 years by level monthly instalments in arrears
of capital and interest. The repayments are calculated using an eective rate
of interest of 8% per annum.
Calculate:
(i). (a). The capital repaid in the rst monthly instalment.
(b). The total amount of interest paid during the last six years of the loan.
(c). The interest included in the nal payment. [9 marks]
(ii). Explain how your answer to (i)(b) would alter if, under the original terms of
the loan, repayments had been made less frequently than monthly.[3 marks]
(b). A fund had a value of

$21; 000 on 1 July 2003. A net cash ow of$ 5; 000 was
received on 1 July 2004 and a further net cash
ow of

$8; 000 was received on 1 July 2005. Immediately before receipt of the rst net cash ow, the fund has a value of$ 24; 000, and immediately before receipt of the second net cash
ow the
fund had a value of

$32; 000. The value of the fund on 1 July 2006 was$ 38; 000.
(i). Calculate the annual eective money weighted rate of return earned on the
fund over the period 1 July 2003 to 1 July 2006. [3 marks]
(ii). Calculate the annual eective time weighted rate of return on the fund over
the period 1 July 2003 to 1 July 2006. [3 marks]
(iii). Explain why the values in (i) and (ii) dier. [2 marks]
QUESTION THREE (20 marks)
(a). The force of interest, , is 6%. Calculate the value of (Ia)10 . [4 marks]
(b). Assuming a rate of 6% p.a., nd the present value as at 1 January 2009 of the
following annuities, each with a term of 25 years:
(i). an annuity payable annually in advance from 1 January 2010 of

$3000 pa increasing by$ 500 pa on each subsequent 1 January
(ii). an annuity as in (i), but only 10 increases are to be made, the annuity then
remaining level for the remainder of the term. [8 marks]
(c). An investor is to receive a special annual annuity for a term of 10 years such that
payments are increased by 5% compound each year to allow for in
ation. The rst
payment is to be

$1000 on 1 November 2010. Find the accumulated value of the annuity payments as at 31 October 2027 if the investor invests at an e ective rate of interest of 4% per half year. [8 marks] 2 QUESTION FOUR (20 marks) (a). A 90-day government bill is purchased for$ 96 at the time of issue and is sold after
45 days to another investor for

$97:90. The second investor holds the bill until maturity and receives$ 100.
Determine which investor receives the higher rate of return. [2 marks]
(b). The force of interest, (t), is a function of time and at any time t, measured in
years, is given by the formula:
(t) =
8>><
>>:
0:06 0 t 4
0:10 ?? 0:01t 4 < t 7
0:01t ?? 0:04 7 < t
Calculate
(i). the value at time t = 5 of

$1; 000 due for payment at time t = 10.[6 marks] (ii). the constant rate of interest per annum convertible monthly which leads to the same result as in (i) being obtained. [2 marks] (iii). the accumulated amount at time t = 12 of a payment stream, paid continuously from time t = 0 to t = 4, under which the rate of payment at time t is (t) = 100e0:02t. [6 marks] (c). The market value of a small pension fund''s assets was$ 2:7m on January 2010 and

$3:1m on 31 December 2010. During 2010 the only cash ows were: { bank interest and dividends totalling$ 125; 000 received on 30th June.
{ a cash payment of

$100; 000 received on 1 August when a block of shares was sold. { a lump sum retirement bene t of$ 75; 000 paid on 1 May
{ a contribution of

$50; 000 paid by the company on 31 December. Calculate the money-weighted rate of return. [4 marks] QUESTION FIVE (20 marks) (a). Two projects A and B have the following expected cash ows: Project A Project B Initial outlay:$ 170; 000

$200; 000 Other expenses:$ 20; 000 at the end of year 1 -

$10; 000 at the end of year 2 - Income:$ 20; 000 at the end of year 1

$14; 000 at the end$ 20; 000 at the end of year 2 of each of the rst 6 years

$200; 000 at the end of year 3$ 200; 000 at the end of year 6.
3
(i). Calculate the internal rate of return (IRR) (correct to 1 decimal place) for
each project. [4 marks]
(ii). Calculate the net present value of each project using a risk discount rate of
6% per annum. [6 marks]
(iii). If funds for the project can be raised by borrowing from a bank, determine
the interest rate charged by the bank above which each project become unpro
table. Mention any other factors that should be taken into account when
deciding between the projects. [4 marks]
(b). Find an approximate value for i, if P = 75, I = 5, R = 125 and n = 10.
. [6 marks]
4

Sta 2191: Financial Mathematics I Question Paper

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