Homework answers / question archive /   SCHOOL OF MATHEMATICAL SCIENCES                          Question 1   Prove that           (–1)                                                          (+1)                       1         1                             1         1                     |_____|_____|_______________|_____|_____|            0         1         2                            n – 1   n       n + 1           a      a??                                                 s      ??s         n                     n                      n                      n                    a   +1                                                        s           –1 n?1                     n?1   Question 2   You are given an annuity immediate with 11 annual payments of RM100 and a final balloon payment at the end of 12 years

  SCHOOL OF MATHEMATICAL SCIENCES                          Question 1   Prove that           (–1)                                                          (+1)                       1         1                             1         1                     |_____|_____|_______________|_____|_____|            0         1         2                            n – 1   n       n + 1           a      a??                                                 s      ??s         n                     n                      n                      n                    a   +1                                                        s           –1 n?1                     n?1   Question 2   You are given an annuity immediate with 11 annual payments of RM100 and a final balloon payment at the end of 12 years

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SCHOOL OF MATHEMATICAL SCIENCES                     

 

 

Question 1

 

Prove that

 

        (–1)                                                          (+1)

                      1         1                             1         1        

            |_____|_____|_______________|_____|_____|

           0         1         2                            n – 1   n       n + 1

          a      a??                                                 s      ??s        

n                     n                      n                      n

                   a   +1                                                        s           –1

n?1                     n?1

 

Question 2

 

You are given an annuity immediate with 11 annual payments of RM100 and a final balloon payment at the end of 12 years. At an annual effective rate of interest of 3.5%, the present value at time 0 of all payments is RM1,000. Using an annual effective rate of interest of 1%, calculate the present value at the beginning of the ninth year of all remaining payments.

 

Question 3

 

An investment requires an initial payment of RM10,000 and an annual payments of RM1,000 at the end of each of the first 10 years. Starting at the end of the eleventh year, the investment returns five equal annual payments of X. Determine X to yield an annual effective rate of interest of 10% over the 15-year period.

Question 4

 

At time 0, you deposit RM P into a fund crediting interest at an annual effective interest rate of 8%. At the end of each year in Year-6 through 20, you withdraw an amount sufficient to purchase an annuity due of RM100 per month for 10 years at a nominal rate of interest of 12% compounded monthly. Immediately after the withdrawal at the end of year 20, the fund value is zero. Calculate P.

Question 5

 

Two brothers Alan and Brian shared equally in an inheritance. Using his inheritance, Alan immediately bought a 10-year annuity due with annual payments of RM2,500 each. Brian puts his inheritance in an investment fund earning an annual effective rate of interest of 9%. Two years later, Brian bought a 15-year annuity immediate with annual payment of Z. The present value of both annuities was determined using an annual effective interest rate of 8%. Calculate Z.

Question 6

 

Alan took a RM2,000,000 construction loan, disbursed to him in three installments. The first installment of RM1,000,000 is disbursed immediately and this is followed by two RM500,000 installments at six month intervals. The interest on the loan is calculated at a nominal rate of 15% compounded semiannually and accumulated to the end of the second year. At that time, the loan and accumulated interest will be replaced by a 30-year mortgage at 12% convertible monthly. The amount of the monthly mortgage payment for the first 5 years will be X and from the sixth years and later years will be 2X. The first mortgage payment is due exactly two years after the initial disbursement of the construction loan. Calculate X. 

Question 7

 

An annuity provides a payment of n at the end of each year for n years. The annual effective rate of

?         ⁿ

interest is ¹_n   . Show that the present value of the annuity is n² ?1?^?_? ^{n ?}_? ? ? . 

Question 8

 

The present value of an n-year annuity due of 1 per year plus a final payment of X at time n?k?1 is

n?k

1?v

given by       . Show that X = s . d       k

Question 9

 

Prove that a ? a = vⁿa (deferred annuity)

m?n                       n                      m

 

Question 10 (Based on Question 9)

 

You are given a = 10.00, and a = 24.40 . Determine a .

n                 3n                   4n

Question 11

 

Dolly receives payments of X at the end of each year for n years. The present value of her annuity is 493. Eugene receives payments of 3X at the end of each year for 2n years. The present value of his annuity is 2748. Both present values are calculated at the same annual effective rate of interest. Determine vⁿ .

Question 12

 

An annuity pays 2 at the end of each year for 18 years. Another annuity pays 2.5 at the end of each year for 9 years. At an annual effective rate of interest of i, the present values of both annuities are equal. Calculate i.

Question 13

 

At an annual effective rate of interest of i, you are given

 

• The present value of an annuity immediate with annual payments of 1 for n years is 40
• The present value of an annuity immediate with annual payments of 1 for 3n years is 70

 

Calculate the accumulated value of an annuity immediate with annual payments of 1 for 2n years.

Question 14

 

Eric receives RM12,000 yearly bonus from his company. He uses the fund to purchase two different annuities each costing RM6,000. The first annuity is a 24-year annuity immediate paying RM  K per year to himself. The second annuity is an 8-year annuity immediate paying RM 2K per year to his son. Both annuities are based on an annual effective rate of interest of i. Determine i.

Question 15

 

At an annual effective rate of interest of i, both of the following annuities have a present value of X. 

 

• A 20-year annuity immediate with annual payments of 55
• A 30-year annuity immediate with annual payments that pays 30 per year for the first 10 years, 60 per year for the second 10 years, and 90 per year for the final 10 years

 

Calculate X.

Question 16

 

Jess deposits RM1,000 into a fund at the beginning of each year for 10 years. At the end of 15 years, she makes an additional deposit of RM X. At the end of 20 years, Jess uses the accumulated balance in the fund to buy a perpetuity immediate with annual payments of RM2,000 per years for 10 years, and RM1,000 per year thereafter. Interest is credited at an annual effective rate of 5%. Calculate X. 

 

Question 17

 

Karen receives RM500,000 at her retirement. She invests RM(500,000 – X) in an annual payment 10-year annuity immediate and RM X in an annual payment perpetuity immediate. Each payment received during the first 10 years is twice as large as each received thereafter. The annual effective rate of interest is 6%. Calculate X.

Question 18

 

Larry purchased a perpetuity due paying (him) RM500 annually. He deposits the payments into a savings account earning an annual effective rate of interest of 10%. Ten years later, before receiving the eleventh payment, Larry sells the perpetuity based on an annual effective rate of interest of 10%. Using the proceeds from the sale plus the money in the savings account, Larry purchases an annuity due paying RM X per year for 20 years at an annual effective rate of interest of 10%. Calculate X.
Question 19
A loan is to be repaid by annual payments continuing forever, the first one due one year after the loan is made. Find the amount of the loan if the payments are 1, 2, 3, 4, 1, 2, 3, 4, … assuming an annual effective rate of interest of 10%.
Question 20
At an annual effective rate of interest of i, the present value of a perpetuity paying 10 at the end of each 3-year period, with the first payment at the end of year 6, is 32. At the same annual effective rate of interest of i, the present value of a perpetuity immediate paying 1 at the end of each 4-month period is X. Calculate X. 
Question 21 
 

A perpetuity pays 1 at the end of every year plus an additional 1 at the end of every second year. The

3?i

present value of the perpetuity is K. Prove that K =  . i(2 ?i)

Question 22

 

A sum P is used to buy an n-year deferred perpetuity due of 1 payable annually. The annual effective rate of interest is i. Show that n =1? ^ln(_?^iP) .
Question 23
You wish to accumulate RM60,000 in a fund at the end of 25 years. You plan to deposit RM80 into the fund at the end of each of the first 120 months. You then plan to deposit RM(80 + X) into the fund at the end of each of the last 180 months. The fund earns interest at an annual effective rate of interest of 3.66%. Determine X.
Question 24
Tom and Jerry are planning to retire on 1 Jan 2025. Their goal is to have enough money in savings to be able to withdraw RM3,000 per month beginning one month after retirement and continuing for 25 years after retirement. They earn an annual effective rate of interest of 10% on their account. Determine the minimum amount needed in their savings account on 1 Jan 2025, to accomplish their goal.
Question 25
A car dealer offers to sell a car for RM10,000. The current market loan rate is a nominal rate of interest of 12% compounded monthly. As an inducement, the dealer offers 100% financing at an effective rate of interest of 5%. The loan is to be repaid in equal installments at the end of each month over a four-year period. Calculate the cost to the dealer of this inducement.

Question 26

 

You win RM1,000,000 in a lottery and will be paid 20 equal annual installments of RM50,000 with the first payment due today. A bank offers to exchange your winning for a perpetuity of RM X per month with the first payment due today. Find the value of X (to the nearest RM) assuming an annual effective rate of interest of 10%. 

Question 27

 

Kathryn deposits RM100 into an account at the beginning of each 4-year period for 40 years. The account credits interest at an annual effective rate of interest of i. The accumulated amount in the account at the end of 40 years is RM X, which is 5 times the accumulated amount in the account at the end of 20 years. Calculate X. 

Question 28

 

You are given

 

• The present value of a 6n-year annuity immediate of 1 at the end of every year is 9.996
• The present value of a 6n-year annuity immediate of 1 at the end of every second year is

4.760

• The present value of a 6n-year annuity immediate of 1 at the end of every third year is X

 

Calculate X.

Question 29

 

You need an amount on 1 Jan, 2045 to provide for a lump sum of RM50,000 and a 15-year annuity due with semiannual payments of RM K. The annual payout is based on a nominal rate of interest of 3% compounded semiannually. The amount will be accumulated by 25 annual deposits of RM K beginning 1 Jan, 2020. The deposits accumulate at a nominal rate of interest of 4% compounded

Question 30

 

You have an annuity which pays RM1,200 every two years. The first payment is two years from now and the last payment is ten years from now. You can trade that annuity for another annuity of equivalent present value, which pays RM 180 per quarter starting today. The interest rate for both annuities is 4% per annum convertible quarterly. If you took the second annuity, how many quarterly payments would you receive? The last payment may be less than RM180 but not more than RM180.

[Hint :

and

]

Question 31  Prove that 

 

(a)  

?         a ?nvⁿ ? ?      a ?nvⁿ   ?

(Ia)n ? (Da)n   =?a_n ? ^{n i}           ?????na

n

n

?      ⁱ           ?? = (n ?1)a_n  

 

?

 

(b)  

 

 

a ?nv ⁿ                         _n

 

(c) (Ia)

 

1

n

n

?

= (d) (Ia) ?v (Da) = a a??

 

^d                                             

 

 

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1

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1

(

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1

1

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n

n

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n

n

n

n

n

a

n

v

a

Ia

i

n

v

a

ia

i

v

n

a

d

a

v

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d

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=

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=

 

 

Question 32

 

An annuity provides for 10 annual payments. The first of these payments is RM100 and each subsequent payment is RM10 higher than the one preceding it. Find the present value of this annuity at the time one year prior to the first payment if the annual effective rate of interest is 10%.

Question 33 

 

You inherited a perpetuity that will pay you RM10,000 at the end of the first year increasing by RM 10,000 per year until a payment of RM150,000 is made at the end of the ^[1]th year. Payments remain level after the 15^th year at RM150,000 per year. Determine the present value of this annuity assuming the annual effective rate of interest of 7.5%.