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  SCHOOL OF MATHEMATICAL SCIENCES                        Question 1 (Example 4b)   It is known that an investment of RM800 will increase to RM6,400 at the end of 25 years

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SCHOOL OF MATHEMATICAL SCIENCES                     

 

Question 1 (Example 4b)

 

It is known that an investment of RM800 will increase to RM6,400 at the end of 25 years. Find the sum of the present values of three payments of RM15,000 each which will occur at the end of 15, 20, and 25 years.

Question 2 (Example 4c)

 

Find the present value of RM1,000 due in 15 years if the annual effective rate of interest is 5% for each of the first five years, 6% for the next five years, 7% for the next three years and 8% for the final two years.

Question 3 (Example 5)

 

a) Money accumulates in a trust fund at an annual effective rate of interest of r. You are given two settlement options of a loan you borrowed.

 

(Option 1) A payment of RM225 immediately and another payment of RM225 at the end                    of two years

 

(Option 2) A payment of RM400 at the end of two years and another payment of RM400                    at the end of four years

Question 4

 

A two-year certificate of deposit pays an annual effective rate of 8%. The purchaser is offered two options for prepayment penalties in the event of early withdrawal.

 

Option A : A reduction in the rate of interest to 6% Option B : Loss of three months interest.

 

In order to assist the purchaser in deciding which option to select, compute the ratio of the proceeds under Option A to those under Option B if the certificate of deposit is surrendered at the end of (a) 6 months (b) 12 months (c) 18 months.

Question 5 (Example 7)

 

Money accumulates in a trust fund at an annual effective rate of interest of r during the first 6 years, and at an annual effective rate of interest of 2r thereafter. A deposit of RM1,000 is made into the fund at time 0. It accumulates to RM2,374.06 at the end of 12 years and to RM4,205.79 at the end of 18 years. Determine the value of the fund at the end of 10 years.

Question 6 (Example 9)

 

The parents of four children aged 0, 2, 4 and 6 wish to set up a trust fund that will pay RM30,000 to each child upon attainment of age 17, and RM200,000 to each child upon attainment of age 20. The fund earns effective rate of 6%. Find the amount needed now to achieve the desired fund.

Question 7 (Example 12)

 

You borrowed RM1,000 from a credit agency at an annual effective interest rate of i. You agreed to pay back RM1,000 after five years and RM1,343.50 after another five years. Three years after making your first payment, due to your better financial situation, you negotiate with the credit agency for full settlement of the loan. Determine i and the amount of rebate. 

Question 8 (Example 10)

 

You are given

 

• The sum of the present values of a payment of X at the end of 10 years and a payment of Y at the end of 20 years is equal to the present value of a payment of X ?Y at the end of 15 years
• X ?Y ? 100
• i ? 0.05

 

Calculate X and Y.

Question 9

 

The present value of RM200 paid at the end of n years, plus the present value of RM100 paid at the end of 2n years is RM200. Determine the annual effective rate of interest.

 

Question 10 (Example 15)

 

Show that   

d³     (i ?d)²

1. 2   ?       

 

(1?d)  1?v

1. d(1^? ₂ⁱ ) ?i(1? ^d₂)
2. 1? ??1? ?   m ? m ?
3. ? ? .  m m m m
4. i ? (m)

5. 1? d _m

                                        

    

   ?1 (m)  (m) i       ?         d          ?

    

    

    

6. d  ?         (m)

   Question 11 (Example 20)

    

   You deposit RM15,000 into a fund and RM25,000 twenty years later. Interest is credited at a nominal rate of discount of 6% compounded semiannually for the first 15 years and at a nominal rate of interest of 8% compounded monthly thereafter. Find the accumulated value of the fund at the end of 35 years.

   Question 12 (Example 21)

    

   You deposit RM1,000 into a bank account. The bank credit interest at a nominal rate of interest of k convertible semiannually for the first 7 years and a nominal rate of interest of 2k convertible quarterly for all years thereafter. The accumulated amount in the account at the end of 5 years is X. The accumulated amount in the account at the end of 10.5 years is 1980. Calculate X. 

    

   Question 13

    

   At time t ? 0, you deposit RM1,000 into a fund which credits interest at a nominal rate of interest of 10% compounded semiannually. At the same time, you deposit P into a different fund which credits interest at a nominal rate of discount of 6% compounded monthly. At time t ? 20, the amounts in each fund are equal. Determine the annual effective interest rate earned on the total deposit 1000 + P over the 20-year period.

   Question 14

    

   (Investment X) RM100,000 invested at a nominal rate of interest of j convertible semiannually                           accumulates to RM214,358.88 after 4 years.

   (Investment Y) RM100,000 invested at a nominal rate of discount of k convertible quarterly                           accumulates to RM232,305.73 after 2 years.

   (Investment Z) RM100,000 invested at an annual effective rate of interest of j in year one and an                           annual effective rate of discount k in year two.  Calculate the value of the Investment Z at the end of year two.

   Question 15 (Example 24)

    

   You are given a loan on which interest is charged over a 4-year period, as follows

    

7. an effective rate of interest of 7% for the first year
8. a nominal rate of interest of 6% compounded quarterly for the second year
9. a nominal rate of discount of 5% compounded every two years for the third year ? a force of interest of 4% for the fourth year

10.  

    Calculate the annual effective rate of interest over the 4-year period.

    Question 16 (Example 25)

     

    Two funds, X and Y, start with the same amount. You are given

     

11. Fund X accumulates at a force of interest of 5%
12. Fund Y accumulates at a nominal rate of interest j, compounded monthly
13. At the end of 10 years, the accumulated values of both Fund X and Fund Y are equal

14. Calculate j.

     

    Question 17 

     

    You are given

     

15. • Investment X will triple in 87.88 years at a constant force of interest, ?
    • Investment Y will quadruple in t years at a nominal rate of interest numerically equivalent to ? and convertible one every four years

16.  

    Calculate t.

    Question 18 (Example 26b)

     

    At time t ? 0 , RM100 is deposited into each Fund X and Fund Y. You are given

17. • Fund X accumulates at a force of interest ?_t ?0.5(1?t)^?2
    • Fund Y accumulates at an annual effective rate of interest of i 
    • At time t ? 9 , the accumulated values of both Fund X and Fund Y are equal

      Question 19 (Example 27)

       

      At time t ? 0 , RM1 is deposited into each Fund X and Fund Y. You are given

      t

    • • Fund X accumulates at a force of interest ?_t ?       

    • 6

    • • Fund Y accumulates at a nominal rate of interest of 12% convertible monthly 
      • At time t ? T , the accumulated values of both Fund X and Fund Y are equal

        Question 20

         

        You are given

      • • Fund X accumulates at a force of interest ?_t ? a ?bt
        • Fund Y accumulates at a force of interest ?_t ? g ?ht
        • a ? g ? 0, b ? h ? 0
        • At time t ? 0 , Fund X = Fund Y
        • At time t ? n, Fund X = Fund Y

      •  

        2(a ? g)

        Show that n ?    . h ?b

        Question 21 (Example 28)

         

        At time t ? 0 , RM100 is deposited into a fund. You are given

         

      • • For the first three years, the fund pays interest at a nominal rate of discount d compounded monthly 

      • 1

      • • Beginning at time t ? 3, interest is credited at a force of interest ?_t ?       

      • 2?t

      • • At time t ? 6, the accumulated value of the fund is RM250

      •  

        Question 22 (Example 29)

         

        You are given

         

      • At time t ? 0 , Aaron deposited RM1,000 into a fund crediting interest at a nominal rate of interest of i compounded quarterly
      • At time t ? 3, Bala deposited RM1,000 into another fund crediting interest at a force of

      • 4 ?t

      • At time t ? 15, the accumulated value of both funds are equal

        Question 23 

         

        Carmen deposited RM100 into a fund that pays an annual effective rate of discount of 20% for the

        2t

        first two years and a force of interest ?_t ? ₂ , 2 ? t ? 4, for the next two years. At the end of four t ?8

         

        years, the amount in Carmen’s account is the same as what it would have been if she had deposited RM100 into an account crediting interest at a nominal rate of interest of i, compounded quarterly, for four years. Calculate i.

        Question 24 (Example 32)

         

        You are given two loans, with each loan to be repaid by a single payment in the future. Each payment includes both principal and interest. Loan #1 is repaid by a RM3,000 payment at the end of four years. Interest is accrued at 10% per annum compounded semi-annually. Loan #2 is repaid by a RM4,000 payment at the end of five years. The interest is accrued at 8% per annum compounded semi-annually. These two loans are to be consolidated by two equal instalments of RM X, with interest at 12% per annum compounded semi-annually. The first payment is due immediately and the second payment is due one year from now. Calculate X. 

        Question 25 (Example 33)

         

        Carol deposits RM10,000 into a bank account that pays an annual effective rate of interest of 4% for ten years. If a withdrawal is made during the first five and one-half years, a penalty of 5% of the withdrawal amount is charged. Carol withdraws K at the end of each of the years 4, 5, 6, and 7. The balance in the account at the end of year 10 is RM10,000. Calculate K.

        Question 26 (Example 34)

         

        A loan of RM1,000 is made at a nominal rate of interest of 12% compounded quarterly. The loan is to be repaid with three payments of RM400 at the end of the first year, RM800 at the end of the fifth year, and a final payment at the end of the tenth year. Calculate the amount of the final payment.

        Question 27 (Example 35)

         

        Joanna agrees to pay an amount of 2X at the end of 3 years and an amount of X at the end of 6 years. In return, she will receive RM2,000 at the end of 4 years and RM3,000 at the end of 8 years. At an annual effective rate of interest of 8%, determine X.

        Question 28 (Example 36)

         

        On January 1, 2015, Mabel has the following two options for repaying a loan

         

      • 60 monthly payments of RM100 commencing February 1, 2015
      • A single payment of RM6,000 at the end of K months

      • 1

        interest ?_t ?  

         

18.