Homework answers / question archive / SCHOOL OF MATHEMATICAL SCIENCES  FINAL EXAMINATION FOR B SC (HONS) IN ACTUARIAL STUDIES, B SC (HONS) FINANCIAL ANALYSIS, B SC (HONS) FINANCIAL ECONOMICS MAT1054 THEORY OF INTEREST Question 1       (Total: 20 marks) It is your responsibility to ensure that the internet browser is ONLY to be used to download the question paper and other relevant materials and to upload your file on eLearn

SCHOOL OF MATHEMATICAL SCIENCES  FINAL EXAMINATION FOR B SC (HONS) IN ACTUARIAL STUDIES, B SC (HONS) FINANCIAL ANALYSIS, B SC (HONS) FINANCIAL ECONOMICS MAT1054 THEORY OF INTEREST Question 1       (Total: 20 marks) It is your responsibility to ensure that the internet browser is ONLY to be used to download the question paper and other relevant materials and to upload your file on eLearn

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SCHOOL OF MATHEMATICAL SCIENCES 

FINAL EXAMINATION FOR B SC (HONS) IN ACTUARIAL STUDIES, B SC (HONS)

FINANCIAL ANALYSIS, B SC (HONS) FINANCIAL ECONOMICS

MAT1054 THEORY OF INTEREST

Question 1       (Total: 20 marks)

It is your responsibility to ensure that the internet browser is ONLY to be used to download the question paper and other relevant materials and to upload your file on eLearn.

               

(a)  Based on the following graph, give TWO comparisons of the accumulation functions for the simple and compound interest.

 

 

1

+

it

(1+

i

)

t

0

t

                               a(t)

 

                      

                        

                            1

(2 marks)

 (b)   Sketch a graph that represents the accumulation functions for the simple and compound discount.

(2 marks)

 

3. Suppose you deposit a sum of money in a bank and the bank offers you either a simple discount or compound discount with the same numerical figure. Determine which one you would prefer. Justify your answer.

(3 marks)

 

4. Let i be the annual effective rate of interest. Given that the sum of the present values of 1 paid at the end of 2n years and 1 paid at the end of 4n years is 1, find the exact value of (1?i)²ⁿ.

(3 marks)

                                              

5. Let x denote the nominal rate of interest, payable m times per period. Let y denote the nominal rate of discount, payable m times per period. Show that y^?1 ? x^?1 ? m^?1.

(3 marks)

 

6. Consider 1 to be paid at the end of 25 months with an annual effective rate of discount

2.5%. Find the present value (correct to four decimal places), by applying

 

(i)      the compound discount throughout, 

1. 1. 1. mark)

 (ii) the compound discount for the first 2 years and the simple discount for the remaining fraction of the period. 

1. 1. 2. marks)

 

7. In Fund A money accumulates at a force of interest 0.01t + 0.1 for 0 ≤ t ≤ 20. In Fund B money accumulates at an annual effective rate of interest i. An amount is invested in each fund for 20 years. The value of Fund A at the end of 20 years is equal to the value of Fund B at the end of 20 years. Calculate the value of Fund B at the end of 15 years.  

(4 marks)

          

^(m) 1 m              ^mt

1. Show that a ? _m ^?t?1   v a??n  .

n

(3 marks)

 

2. An annuity-immediate has semi-annual payments of 800, 750, 700, …, 400 and 350 at i ⁽²⁾ = 16%. Find the present value of the annuity in term of a    .

108 %

                                                                                                      (3 marks)

 

3. You wish to accumulate RM100,000 in a fund at the end of 10 years. You will deposit the money at the beginning of every two months. In the first 5 years, RM200 will be deposited bimonthly, and then in the second 5 years, RM (200? X)will be deposited bimonthly. The fund earns i annual effective rate of interest. Show that 

 

50,000 ? 600??s ⁽⁶⁾

X ?        (6)                    10 i   .

3??s

5 i

 (3 marks)

 

4. Consider two perpetuities. The first perpetuity has level payments of P at the end of each year. The second perpetuity is increasing such that the payments are Q, 2Q, 3Q, 4Q, … at the end of each year. Find the rate of interest (in terms of P and/or Q) which will make the difference in the present values of these perpetuities:

 

(i)      zero,

                                                                                              (3 marks)

(ii)     a maximum. 

                                                                                              (2 marks)

 

5. There is RM400 in a fund which is accumulating at 4% per annum convertible continuously. If money is withdrawn continuously at the rate of RM2 per month, how long will the fund last?

                                                                                              (3 marks)

 

6. At time 0, you deposit P into a fund crediting interest at an annual effective interest rate of 10%. At the end of each year in Year-6 through 10, you withdraw an amount sufficient to purchase an annuity which pays X at the end of each month for Year-X, where X = 1, 2, 3, 4 and 5 (i.e., 1 to be paid at the end of each month for Year-1, 2 to be paid at the end of each month for Year-2, …, 5 to be paid at the end of each month for Year-5) at an annual effective interest rate of i. Immediately after the withdrawal at the end of Year-10, the fund value is zero. Show that 

 

1

12

5

a

 i

 

P ?   (Ia)      .                                                                         

1.1 [(1?i) ?1]   ^{510%                5 i}

(3 marks)

 

1. An investment project has the following cash flows: 

 

Year	Cash flows (in RM)
	Returns	Contribution
0	11,550	0
1	1,500	0
2	1,500	3,000
3	200	7,000
4	200	7,000

 

Assume that all cash flows occurred at year-end. Find 

 

1. 1. the net present value at interest 5% per annum, 

(2 marks) (ii)             and the internal rate of return on this investment.                 

 (2 marks)

 

2. A project requires an initial investment of RM100 and it produces net cash flows of RM500 one year from now and RM200 two years from now. Show that there is a unique internal rate of return. 

(2 marks)

 

3. Consider a loan of RM20,000 to be repaid by amortization method with level monthly payments in one year. 

 

1. 1. Create an amortization schedule if the interest is i⁽¹²⁾ ?6%.             

(4 marks)

1. 2. Find the monthly repayment if the interest is 6% effective annually.

(2 marks)

 

4. You invest 1 at the beginning of the year in Fund A which earns an annual effective rate of interest of i₁. You then invest half of each interest payment earned in Fund A every end of the year equally into Fund B and Fund C, which earn annual effective rates of interest of i₂ and i₃, respectively. Each interest payment earned from Fund B and C are both reinvested into Fund D which guarantees an annual effective rate of interest of i₄. Determine the total accumulation for Funds A, B, C and D at the end of 10 years in terms of i₁, i₂, i₃, and (Is) .

9 i4

 (3 marks)

 

5. You borrow RM12,000 for 10 years and agree to make semi-annual payments of RM1,000. The lender receives 12% convertible semi-annually on the investment each year for the first 5 years and 10% convertible semi-annually for the second 5 years. The balance of each payment is invested in a sinking fund earning 8% convertible semiannually. Find the amount by which the sinking fund is short of repaying the loan at the end of the 10 years. Give you answer to the nearest integer.

(5 marks)

          

 (a) You own a RM1,000 par value 10% bond with semi-annual coupons. The bond will mature at par at the end of 10 years and you will immediately buy a new 8-year bond. The current yield rate for both bonds is 7% convertible semi-annually. You use the proceeds from the sale of the 10% bond to purchase the new 6% bond with semi-annual coupons, maturing at par at the end of 8 years. Find the par value of the 8-year bond and provide the answer to the nearest integer.

(4 marks)

 

2. Consider two 20-year bonds, each having semi-annual coupons and each maturing at par. The yield rate for both bonds is the same. One bond has a par value of RM380 and a coupon of RM45. The other bond has a par value of RM1,000 and a coupon of RM30. Calculate the difference of the bonds’ price if the yield rate is 5% convertible semiannually.
   3. marks)

 

3. Consider a RM1,000 par value two-year 8% bond with semi-annual coupons bought to yield 6% convertible semi-annually. Compute the flat price, accrued coupon, and market price 8 months after the purchase of the bond. Use all three methods, i.e., the theoretical method, the practical method, and the semi-theoretical method.

 (9 marks)

4. A RM1,000 callable bond with 9% coupons payable semi-annually was redeemed for RM1,100. The bond was bought for RM918 to yield a nominal rate of 10% convertible semi-annually. Calculate the number of years the bond was held. Provide your answer to the nearest integer.
   4. marks)

 

 

 

 

 

 

                                  

 

1. Consider an annuity-due with a term of n periods in which the first payment is 1 and successive payments increase in geometric progression with common ratio 1 + k. The effective interest rate for each period is i where i > k. Show that the present value of this annuity is a?? .

n ₁i?_?kk

(4 marks)

 

2. A common stock pays annual dividends at the beginning of each year. The dividends per share for the most recent year is 1. Assume that the dividends increase at the rate of 5% for the first 20 years and 0% thereafter. Suppose the yield rate is 7%. Use the formula in (a) to find the theoretical price of the stock. Leave your answer in terms of

a?? and a where j and i are rate of interests. 

20 j       ? i

 (3 marks)

 

3. In a specific term structure of interest rates, the spot rate is defined by s_t = 0.1(0.9)^t for t = 1, 2, 3, 4, 5. Compute the 

 

1. 1. 2-year forward rate, one year from now, 

(1 mark)

1. 2. 3-year forward rate, two years from now.                                                         

(1 mark)

 

4. A 50-year loan is to be repaid with payments of t at the end of month-t for t = 1, 2, …, 600. Given that the nominal rate of interest is 5% compounded monthly, compute the

          

1. 1. loan amount,                                                                                                  

(2 marks)

1. 2. modified duration,                                                                                

                                                                                                            (2 marks) (iii)           Macaulay duration,                                                                                  

                                                                                                            (2 marks)

(iv)    convexity.                                                                                              

                                                                                                            (2 marks)

 

5. Based on part (d), if the nominal rate of interest increases from 5% to 5.1% (convertible  monthly), find the new loan amount’s

 

1. 1. exact answer.

(1 mark)  (ii)  approximation using the modified duration obtained in (d)(ii).

(1 mark)

          

(iii)  approximation using the modified duration and the convexity obtained in (d)(ii)   and (d)(iv).

(1 mark)

~ END OF PAPER ~