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Homework answers / question archive / SCHOOL OF MATHEMATICAL SCIENCES Question 1 In order to develop a new product and place it in the market for sale, a company has to invest RM50,000 now and then RM20,000 for each of the next three years

__SCHOOL OF MATHEMATICAL SCIENCES __

__Question 1__

In order to develop a new product and place it in the market for sale, a company has to invest RM50,000 now and then RM20,000 for each of the next three years. The product is made available for sale in the fourth year. For that, a contribution of RM10,000 must be made in the fourth year. The company incurs maintenance expenses of RM3,000 in each of the next five years. The project is expected to provide an investment return of RM15,000 at the end of the fourth year, RM25,000 at the end of the fifth year, RM40,000 at the end of the sixth year, RM30,000 at the end of the seventh year, RM25,000 at the end of the eighth year, RM15,000 at the end of the ninth year and RM10,000 at the end of the tenth year. After the tenth year, the product is withdrawn from the market.

- Create a chart to describe the cash flows of this project
- Find the net present value of the project
__Question 2__An investment project has the following cash flows

Year

Contributions C

_{t}Returns R

_{t}Net Cash Flow c

_{t }? R_{t }?C_{t}0

100,000

0

– 100,000

1

5,000

0

– 5,000

2

4,000

10,000

6,000

3

2,000

10,000

8,000

4

0

20,000

20,000

5

0

40,000

40,000

6

0

60,000

60,000

7

0

80,000

80,000

Calculate

- the net present value at 15%.
- the internal rate of return IRR (i.e. the yield rate) on this investment.
__Question 3__You are considering building a new factory. The factory will require an investment of RM100,000 immediately. It will also require an additional investment of RM15,000 at the end of Year 2 to initiate production. Finally, maintenance costs for the factory will be RM5,000 per year at the end of Years 3 through 6. The factory is expected to generate profits of RM10,000 at the end of Year 1, RM15,000 at the end of Year 2, RM20,000 at the end of Year 3, and RM30,000 at the end of Years 4 through 6. Calculate the internal rate of return on the potential factory.

__Question 4__Eric deposits 15 into a fund at time 0 and an additional 15 into a same fund at time 10. The fund credits interest at an annual effective rate of interest of i. Interest is payable annually and reinvested at an annual effective rate of interest of j = 0.75i. At time 20, the accumulated amount of the reinvested interest payments is 80. Calculate i.

__Question 5__Victor invests RM300 into a bank account at the beginning of each year for 20 years. The account pays out interest at the end of every year at an annual effective rate of interest of i. The interest is reinvested at an annual effective rate of interest of 0.5i. The yield rate on the entire investment over the 20-year period is 8% annual effective. Calculate i.

__Question 6__Caren invests RM2,000 at the beginning of the year in a fund which credits interest an annual effective rate of interest of 9%. She invests each interest payment in a separate fund, accumulating at an annual effective rate of interest of 8%. The interest payments from this fund accumulate in a bank account that guarantees an annual effective rate of interest of 7%. Determine the sum of the principal and interest at the end of 10 years.

__Question 7__Mary invests RM1,000 at the end of each year for 5 years at an annual effective rate of interest of 9% and reinvests the interest at an annual effective rate of interest of 9%. At the end of 5 years, her investment has a value of X. John invests RM1,000 at the beginning of each year for 5 years at an annual effective rate of interest of 10% and reinvests the interest at an annual effective rate of interest of 8%. At the end of 5 years, his investment has a value of Y. Calculate Y – X.

__Question 8__An investor pays P for an annuity which provides payments of 100 at the beginning of each month for 10 years. These payments are invested at a nominal rate of interest of 12% convertible monthly. Monthly interest payments are reinvested at a nominal rate of interest of 6% convertible monthly. The annual yield rate over the 10-year period is 8% effective. Calculate P.

__Question 9__Esther invests 100 at the end of each year for 12 years at an annual effective rate of interest of i. The interest payments are reinvested at an annual effective rate of interest of 5%. The accumulated value at the end of 12 years is 1748.40. Calculate i.

__Question 10__At the beginning of the year an investment fund was established with an initial deposit of RM1,000. A new deposit of RM500 was made at the end of four months. Withdrawals of RM200 and RM100 were made at the end of six months and eight months, respectively. The amount in the fund at the end of the year is RM1,272. Find the approximate effective rate of interest earned by the fund during the year using the dollar-weighted rate of interest formula.

__Question 11__An association had a fund balance of 85 on 1 Jan and 50 on 31 Dec. At the end of every month during the year, the association deposited 10 from membership fees. There were withdrawals of 5 on 28 Feb, 15 on 30 Apr, and 50 on 15 Sep, and 25 on 30 Nov. Calculate the dollar-weighted rate of interest for the year.

__Question 12__On January 1, an investment account is worth RM100,000. On May 1, the account value has increased to RM112,000, and RM30,000 of new principal is deposited. On November 1, the account value has decreased to RM125,000, and RM42,000 is withdrawn. On January 1 of the following year, the account value is RM100,000. Compute the yield rate using (a) the dollar-weighted method and (b) the time-weighted method.

__Question 13__Suppose that an investor makes a series of payments and withdrawals, as follows

Date

Flow

Balance before

Balance after

1 Jan 2019

0

100,000

100,000

30 June 2019

+500,000

70,000

570,000

31 Dec 2019

0

870,000

870,000

Compute the time-weighted rate of interest.

__Question 14__On 1 January, an investment account is worth 100. On 1 May, the value has increased to 120 and D is deposited. On 1 November, the value is 100 and 40 is withdrawn. On 1 January of the following year, the investment account is worth 65. The time-weighted rate of interest is 0%. Calculate the dollar-weighted rate of interest.

__Question 15__On 1 January, Lucy deposits 90 into an investment account. On 1 April, when the amount in the account is equal to X, a withdrawal of W is made. No further deposits or withdrawals are made to the account for the remainder of the year. On 31 December, the amount in the account is 85. The dollarweighted rate of return over the 1-year period is 20% and the time-weighted rate of return over the 1year period is 18%. Calculate W and X.

__Question 16__You are given the following table of interest rates in %. You deposited RM100 on January 2002. Find the accumulated amount on January 2004 under the Portfolio Method and Investment Year Method.

y

i

_{1 }^{y}i_{2}^{y}i_{3}^{y}i_{4}^{y}i^{y+4}Portfolio Year1998

1999

2000

2001

2002

2003

2004 2005

7.0 6.5 6.0 5.8 5.9 2002

6.4 6.1 5.8 5.9 6.0 2003

6.2 6.0 5.9 6.0 6.2 2004

6.1 5.9 6.1 6.4 6.6 2005

6.0 6.1 6.3 6.6

6.4 6.6 6.7

6.8 7.0

7.5

__Question 17__The following table lists the interest rate credited under an Investment Year Method of crediting interest. Brian invests RM1,000 on 1 January 2019 and will add RM500 on 1 January 2023. How much money does Brian have on 1 January 2025?

y

i

_{1 }^{y}i_{2}^{y}i_{3}^{y}i_{4}^{y}i_{5}^{y}i^{y+5}2019

2020

2021

2022

2023

2024

7.00 6.75 6.50 6.25 6.00 5.50

6.00 5.50 5.25 5.10 5.00

5.00 4.80 4.60 4.30

4.00 3.75 3.50

3.00 3.20

4.00

__Question 17__The following table lists the interest rate credited under an Investment Year Method of crediting interest. Brian invests RM1,000 on 1 January 2019 and will add RM500 on 1 January 2023. How much money does Brian have on 1 January 2025?

y

i

_{1 }^{y}i_{2}^{y}i_{3}^{y}i_{4}^{y}i_{5}^{y}i^{y+5}2019

2020

2021

2022

2023

2024

7.00 6.75 6.50 6.25 6.00 5.50

6.00 5.50 5.25 5.10 5.00

5.00 4.80 4.60 4.30

4.00 3.75 3.50

3.00 3.20

4.00

__Question 18__You are given the following chart of interest rates in %.

y

i1 y

i2y

i3y

iy+3

2015

3.7

3.6

3.5

6.0

2016

3.2

3.1

3.0

5.5

2017

2.7

2.6

2.5

5.0

2018

2.2

2.1

2.0

4.5

2019

1.7

1.6

1.5

4.0

A deposit of RM100 is made at the beginning of 2017. - How much interest was credited during 2018?
- What is the accumulated value at the end of 2022?

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