Chapter 2Time Value of Money

Chapter 2Time Value of Money

Short Questions

B3C02T2Q001eng

Alice is shopping ata supermarket. She wants to buy sushi, ice cream and chocolate. However, she can only afford one of them. She ranks these three choices as follows: 1. ice cream;2. chocolate; 3. sushi. What is her opportunity cost if Alice chooses sushi? Explain your answer. (3 marks)

Answer:

The opportunity cost of choosing sushi is ice cream.Sincethe opportunity cost is the highest-valued alternative forgone, and Alice ranks ice cream as the food which she likes the most, the opportunity cost of choosing either chocolate or sushi is the same, that is, ice cream. (3 marks)

B3C02T2Q002eng

Emily is considering buying a flat. She has two choices:

1A flat in Tai Po which costs $800,000. Its value will probably increase by 10% per year over the coming three years.

2A flat in Prince Edward which costs $2,000,000. It will probablyappreciate by 2% per year over the next three years.

Which apartment will appreciate more (in monetary terms) after three years?

(Calculations to the nearest dollar)(4 marks)

Answer:

The future value of the flat in Tai Po: $800,000  (1 + 10%)3 = $1,064,800(1 mark)

The future value of the flat in Prince Edward: $2,000,000  (1 + 2%)3 =$2,122,416(1 mark)

The appreciation of the flat in Tai Po: $1,064,800$800,000 = $264,800(1 mark)

The appreciation of the flat in Prince Edward: $2,122,416$2,000,000 = $122,416(1 mark)

Therefore, the flat in Tai Po appreciates more in monetary terms after three years.

B3C02T2Q003eng

Julia wants to buy a bicycle. Her mother asked her to save a fixed amount of her red packet money each year and promised to give her 5% annual interest on the total amount. Suppose the bicycle costs $1,490 and Julia plans to buy it after three years. How much should she save each year? (3 marks)

Answer:

Assume that the amount Julia should save is equal to X, X(1 + 5%)2+ X(1 + 5%)+ X = $1,490

X= $473

Therefore Julia should save $473 per year.(3 marks)

B3C02T2Q004eng

Loki plans to buy a piece of equipmentwhich costs $100,000 for his plastics factory. He expects that the equipment will be used for six years and then be sold for $20,000. What is the minimum present value of all annualcash inflows generated so that its financial benefit will outweigh the cost? The opportunity cost of capital is 8%. (Calculations to the nearest dollar) (4 marks)

Answer:

Assume that the minimum present value of all cash inflowsis equal to X,

The present value of the equipment at the end of the sixth year: $20,000(1 + 8%)6

= $12,603(1 mark)

NPV=PV of all annual cash inflows of using the equipment + PV of the equipment at the end of Year 6Initial cost of the equipment

0 =X + $12,603$100,000

X=$87,397(3 marks)

B3C02T2Q005eng

Suppose you have received two leaflets from two securities firms.

Securities Firm A:Investment plan A guarantees a10% rate of return annually.

Securities Firm B:Investment plan B guarantees an 8% rate of return annually and the interest is compounded semi-annually.

Your neighbour, Mrs Lo, thinks that investment plan A generates a greater return. Do you agree with her? Explain your answer. (4 marks)

Answer:

Mrs Lo is correct. Although the nominal interest rate of plan A is higher than thatof plan B, the compounding frequencies of the two plans are different. Thus, we shouldcompare their effective interest rates. Assume that the principal is $100 and the investment is held for one year. The total amount of plan A after one year is $100(1 + 10%) = $110 while that of plan B is $100  (1 + 4%)2= $108.2. Therefore, plan A generatesthe greater return. (4 marks)

B3C02T2Q006eng

Victor wants to invest$100,000 for three years. He is going to choose among three financial instruments:a mutual fund, a bond and shares of stock. The annual rate of return for each option is as follows:

(a)How much will Victor have from investing in each of the above financial instruments?(3 marks)

(b)Though the return onthe shares is the highest, Victor finally invests in a mutual fund. Give one reason for his decision. (3 marks)

(c)Suppose the rates of return on the shares for the first year, the second year and the third year are 8%, -5%, and 10%, respectively. How much will Victor have after three years if he invests in the shares? (2 marks)

(d)Suppose Victor decides to invest in either a mutual fund or a bond. If the annual rate of return onthe bond increases from 5% to 8% and the interest is compounded semi-annually,calculate the future value of the bond three years later. Based on your calculation, which financial instrument should Victor choose? (3 marks)

(Calculations to the nearest dollar)

Answer:

(a)Mutual fund: $100,000 (1 + 10%)3= $133,100(1 mark)

Bond: $100,000(1 + 5%)3 = $115,763(1 mark)

Shares: $100,000(1 + 12%)3 = $140,493(1 mark)

(b)A mutual fund allows Victor to diversify his risks because it actually invests in various securities. This helps reduce the market risk to the minimum. Although the return onthe share is the highest, its price often fluctuates sharply. Thus, Victor may suffer a loss if the share price suddenly drops. (3 marks)

(c)The value of the share after three years is: $100,000 (1 + 8%)(1 - 5%)  (1 + 10%) = $112,860(2 marks)

(d)The future value of the bond: $100,000 x (1 + 4%)6 = $126,532(2 marks)

Since the future value of the mutual fund ($133,100) is higher than that of the bond ($126,532), Victor should invest inthe mutual fund. (1 mark)

B3C02T2Q007eng

Becky is the owner of a printing company. She plans to buy a printing machine for business expansion purposes. Three options are available:

1Machine A is the most advanced model and has outstanding printing performance.

2Machine B has average quality and efficiency but its price is lower than Machine A.

3Machine C is the cheapest but its performance may not be satisfactory.

Becky decides to buy Machine A, but the manager, Roger, prefers Machine B.

(a)Assuming that the cost of capital is 10%, calculate the net present value of each machine. Which machine should the company buy? (6 marks)

(b)Suppose a new model, Machine D, is now available. It costs $8 million and it can generate a cash inflow of $1.8 million per year. It is estimated that the machine can be used for six years and will be sold for $2 million thereafter. Should the company buy Machine D or Machine B? Explain your answer. (2 marks)

(c)Rank the priority of the four machines based on their net present values. Suppose the company finally chooses Machine D. What is the opportunity cost of this choice? (3 marks)

(Calculations to the nearest dollar)

Answer:

(a)NPV of Machine A = 2M/(1 + 10%)6 + 2M/(1 + 10%)5 + 2M/(1+ 10%)4 + 2M/(1 + 10%)3 + 2M/(1 + 10%)2 + 2M/(1 + 10%) +2M/(1 + 10%)6$10M = -$160,531 (1.5 marks)

NPV of Machine B = 1.5M/(1 + 10%)4 + 1.5M/(1 + 10%)3 + 1.5M/(1 + 10%)2 + 1.5M/(1 + 10%) + 1M/(1 + 10%)4 5M = $437,812 (1.5 marks)

NPV of Machine C = 1.1M/(1 + 10%)2 + 1.1M/(1 + 10%) +0.1M/(1 + 10%)22M = -$8,264(1.5 marks)

Since the NPV of Machine B is the highest and is greater than 0, the company should buy Machine B.(1.5 marks)

(b)NPV of MachineD = 1.8M/(1 + 10%)6 + 1.8M/(1 + 10%)5 + 1.8M/(1 + 10%)4 + 1.8M/(1 + 10%)3 + 1.8M/(1 + 10%)2 + 1.8M/(1 + 10%) + 2M/(1 + 10%)6 8M = $968,417 (1 mark)

As the NPV of Machine D is higher than that of Machine B ($437,812), the company should buy Machine D. (1 mark)

(c)The priority of the four machines is:

1Machine D

2Machine B

3Machine C

4Machine A(1.5 marks)

If the company chooses Machine D, the opportunity cost is Machine B because it is the highest-valued alternative forgone. (1.5 marks)

B3C02T2Q008eng

Ivy has won third prize in the Mark Six. She wants to invest the $100,000 amount for two years. There are three investment plans available and she will choose one of them. The annual rates of return and the frequencies of compounding are listed below:

(a)Should Ivy compare the nominal interest rates of these three plans when she makes a choice? Explain your answer. (2 marks)

(b)Calculate the future values for each of the three plans. Based on your calculations, which investment plan should Ivy choose? (4 marks)

(c)Suppose Ivy plans to get married after three years and the wedding will cost her $200,000. If she picks Plan B, how much should Ivy invest now so that she can have enough money after three years? (2 marks)

(d)Refer to (c). What is the opportunity cost if Ivy chooses Plan B? Suppose Ivy finally switches to Plan C. Does the opportunity cost change? (3 marks)

(Calculations to the nearest dollar)

Answer:

(a)No. As the compounding frequencies of the three plans are different, Ivy should compare their effective interest rates instead of their nominal interest rates. (2 marks)

(b)FV of plan A: $100,000(1 + 10%)2 = $121,000(1 mark)

FV of plan B: $100,000(1 + 4%)4 = $116,986(1 mark)

FV of plan C: $100,000(1 + 1%)8 = $108,286(1 mark)

The FV of plan A is the highest, so Ivy should choose plan A.(1 mark)

(c)The amount Ivy should invest: $200,000/(1 + 4%)6 = $158,063(2 marks)

(d)Since the return on plan A is the highest, it is the highest-valued alternative forgone if Ivy chooses Plan B. Therefore the opportunity cost is Plan A. Ifshe switches to Plan C, the highest-valued alternative forgone remains unchanged, that is, Plan A. The opportunity cost is not changed. (3 marks)

B3C02T2Q009eng

Franklin is the CEO of an automobile component manufacturer. The company sold $10 million products on credit last year to an automobile manufacturer, which promised to pay off the amount this year. However, it now wants to delay payment and it offers Franklinthree suggestions

Franklin and the majority of shareholders prefer choice 1, but the CFO, Joyce, thinks that the company can benefit most from choice 2.

(a)Identify the the automobile component manufacturer’s stakeholders.(2 marks)

(b)For each stakeholder mentioned in (a), explain its relationship with the automobile component manufacturer.(3 marks)

(c)Assume the cost of capital is 10%. Do you support Joyce’s suggestion? Explain your answer.(2 marks)

(d)Suppose there is a business opportunity which requires the automobile component manufacturer to pool $10 million in capital immediately. It is expected that the company can obtain a lump sum of $18 million after five years. Should the company invest in this project? Explain your answer. (3 marks)

(e)Suppose the automobile component manufacturer decides to invest in the project mentioned in (d). Which payment plans should it choose now? Explain your answer. (3 marks)

Answer:

(a)Shareholders(0.5 mark)

Customers: automobile manufacturer(0.5 mark)

Employees: Joyce and Franklin(1 mark)

(b)Shareholders are the owners of the company. They provide capital to the company and may receive dividends when the company makes a profit. (1 mark)

The automobile manufacturer buys products from the company. It can obtain the raw materials needed while the company has cash inflows. (1 mark)

Employeeswork for the company. They earna salary by providing their expertise to the company.(1 mark)

(c)PV of Choice 1: $10M(0.5 mark)

PV of Choice 2: $11.2M/(1 + 10%) = $10.18M(0.5 mark)

PV of Choice 3: $12/(1 + 10%)2 = $9.92M(0.5 mark)

The PV of choice 2 is the highest. Therefore the company should choose choice 2 as suggested by Joyce.(0.5 mark)

(d)NPV of the project: $18M/(1 + 10%)5– $10M= $1.18M(1.5 marks)

Since the NPV is greater than zero, the company should invest in this project.(1.5 marks)

(e)It should choose choice 1 which requires the automobile manufacturer to pay back $10 million now. The company can then use the money to invest in the project. (3 marks)

B3C02T2Q010eng

Your neighbour, Mr Ng, is comparing different investment plans. He will choose the one which generates the highest returns. However, as interest rates and compounding frequencies vary among these plans, Mr Ng finds it difficult to compare them. Now he asks for your help.

(a)Suppose Mr Ng will invest $80,000. How much can he get from Plan B and Plan C, respectively?(2 marks)

(b)For Plan A, investors can withdraw as much as 40% of the total amount of the principal and interest at the end of the second year, while the remaining balance would be compounded in the third year. Assume Mr Ng will invest $80,000 in Plan A and will not withdraw any money within the investment period. How much will he have after three years? (1 mark)

(c)Based on the results of (a) and (b), which investment plan should Mr Ng choose?

(1 mark)

(d)Suppose Mr Ng decides to invest $75,000 in Plan A. At the end of the second year, he withdraws 40% of the total amount from the investment. What is the total amount he will have in three years? (3 marks)

(Calculations to the nearest dollar)

Answer:

(a)FV of Plan B: $80,000  (1 + 4%)6 = $101,226(1 mark)

FV of Plan C: $80,000  (1 + 1.5%)12 = $95,650(1 mark)

(b)FV of Plan A: $80,000  (1 + 5%)2  (1 + 10%) = $97,020(1 mark)

(c)Since Plan B has the highest FV, Mr Ng should choose Plan B.(1 mark)

(d)The amount Mr Ng withdraws at the end of the second year: $75,000(1 + 5%)2  40% = $33,075(1 mark)

The amount Mr Ng would receive at the end of the third year: [$75,000(1 + 5%)2$33,075] (1 + 10%) = $54,574 (1 mark)

The total amount MrNg would receive after three years = $33,075 + $54,574 = $87,649(1 mark)