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FNCE10002 Principles of Finance Semester 2, 2023 Tutorial Questions for Topic 2 1 Department of Finance FNCE10002 Principles of Finance Semester 2, 2023 Introduction to Financial Mathematics II Tutorial Questions for Week 3 (Lecture 2) Priority questions this week: A1, B1, B4, C2 A. Short Answer Questions Provide brief responses to the following questions. A1. Outline the differences between a stated (or quoted) interest rate and an effective interest rate. A2. Your grandfather has been putting $2000 into a savings account since the day you were born on an annual basis. The account pays an interest rate of 3% p.a. How much money will be in the account on your 18th birthday immediately after your grandfather makes the deposit on that birthday? B. Multiple Choice Questions For each question pick the most reasonable response based on the information provided. B1. An investor expects to receive the following cash flows over the next four years where the cash flows are received at the beginning of each year. Beginning of Year Cash Flow 1 $20,000 2 $20,000 3 $20,000 4 $20,000 If the interest rate is 6% p.a. compound annually the present value today of this series of cash flows is closest to:

FNCE10002 Principles of Finance Semester 2, 2023 Tutorial Questions for Topic 2 2 a) $56,668. b) $69,302. c) $73,460. d) $80,000. B2. Consider the following timeline detailing a stream of cash flows: If the current market rate of interest is 8% p.a., then the future value of this stream of cash flows as at the end of the fourth year is closest to: a) $11,699 b) $10,832 c) $12,635 d) $10,339 B3. Which of the following statements regarding annuities is FALSE? a) PV of an annuity = 𝐶 × ቂ1 − ଵ (ଵା௥) ೙ ቃ b) The difference between an annuity and a perpetuity is that a perpetuity ends after some fixed number of payments. c) An annuity is a stream of N equal cash flows paid at regular intervals. d) Most car loans, mortgages, and some bonds are annuities. B4. You are head of the Gygax Family Endowment for the Environment. You have decided to fund a research school in the Melbourne area in perpetuity. Every five years, you will give the school $1 million. The first payment will occur five years from today. If the interest rate is 8% per annum compounded annually, the present value of your gift is closest to: a) $680,582. b) $2,130,697. c) $2,500,000. d) $12,500,000 C. Problems C1. It is 1 February 2023 and your child is about to start their first year of schooling at a private school. Currently, the tuition is $15,000 per year, payable at the start of the school year (i.e. 1 February) each year. You expect annual tuition increases to average 6% per year over the next 13 years (there are 13 years of school – from prep to the end of high school). Assume that your child remains in this private school through high school and that the current interest rate is 7% p.a. compounding annually. Show all calculations .

FNCE10002 Principles of Finance Semester 2, 2023 Tutorial Questions for Topic 2 3 a) Draw a timeline of the relevant cashflows and clearly label all cashflows. b) How many payments need to be made and what is the date of the final payment? c) What type of annuity are the payments? d) Write down the formula before inserting the numbers if you are to calculate the present value of the annuity identified in (c)? e) What is the PV of the annuity? Show all calculations and explanations for steps in your calculation. Explain what this number represents in terms of what is required to fund your child’s schooling. C2. Assume that you just won the lottery. Your prize can be taken either in the form of $40,000 at the end of each of the next 25 years (i.e., $1 million over 25 years) or as a lump sum of $500,000 paid immediately. a) If you expect to be able to earn 5% annually on your investments over the next 25 years, which alternative should you take? Why? b) Would your decision in part (a) be altered if you could earn 7% p.a. rather than 5% p.a. on your investments over the next 25 years? Why? D. Closing the loop with Desmos interactive graphs This week we are going to demonstrate graphically how the PV of an annuity approaches the value of a perpetuity as the number of payments increases. Open the Desmos interactive graph at the following address: https://go.unimelb.edu.au/h2hs Use the graph to answer the following questions: i. What is the PV of an ordinary annuity of 10 years of $1,000 per annum assuming a discount rate of 10% per annum? ( remember to change the values of a and b in the graph to reflect the facts of the question ) ii. What is the PV of an ordinary annuity of 40 years of $1,000 per annum assuming a discount rate of 10% per annum? iii. What is the PV of a perpetuity of $1,000 per annum assuming a discount rate of 10% per annum? iv. How would you use this graph to check your answer to B4? Note also that you may need to change the axis values displayed to see the graph generated. You do this by selecting the spanner symbol on the right hand side of the screen (graph settings) and then you can type in the range and domain of the displayed graph

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FNCE10002 Principles of Finance Semester 2, 2023 Suggested Answers to Tutorial Questions for Topic 2 1 Department of Finance FNCE10002 Principles of Finance Semester 2, 2023 Introduction to Financial Mathematics II Tutorial Questions for Week 3 (Lecture 2) A. Short Answer Questions A1. Generally, a stated (or quoted) interest rate per period is the rate provided without considering/incorporating the frequency of compounding per period whilst the effective interest rate incorporates the frequency of compounding per period. For example, if the stated rate is 12 % per annum and the frequency of compounding per year is one then the stated rate is the same as the effective rate per annum. If the frequency of compounding is greater than one then the effective rate per annum is greater than the stated rate per annum (e.g. 12% p.a. compounding semi-annually is equivalent to an effective rate of 12.36% p.a.). A2. Timeline: 0 1 2 3 18 2,000 2,000 2,000 2,000 2,000 We first calculate the present value of the deposits at date 0. The deposits are an 18-year annuity due: PV = 2,000 + ଶ,଴଴଴ ଴.଴ଷ ቂ1 − ଵ (ଵ.଴ଷ) భఴ ቃ = $29,507.02 Now, we calculate the future value of this amount: FV = 29,507.02 ×(1.03) 18 = $50,233.74 B. Multiple Choice Questions B1. C is correct. Since the cash flows are received at the beginning of each year we have an annuity due. The present value of an annuity due is computed as: P 0 = 𝐶𝐹 + ஼ி ௥ ቂ1 − ଵ (ଵା௥) ೙షభ ቃ = 20,000 + ଶ଴,଴଴଴ ଴.଴଺ ቂ1 − ଵ (ଵ.଴଺) య ቃ =$73,460.24

FNCE10002 Principles of Finance Semester 2, 2023 Suggested Answers to Tutorial Questions for Topic 2 2 B2. A is correct. FV = 1000(1.08) 4 + 2000(1.08) 3 + 3000(1.08) 2 + 4000(1.08) 1 = $11,699 B3. B is the correct answer. A perpetuity never ends while an annuity does. B4. B is correct. We’re given the annual rate but the cash flows occur every five years. So, we first need the effective five-year interest rate, which is: (1.05) 5 ̶ 1 = 0.46933 = 46.933% per five years. As the five-year cash flow is a perpetuity, the present value is ($1m/0.46933) = $2,130.697. (Note that if you do not round the effective interest rate your answer will be $2,130,706. Either answer is fine because this amount is closest to alternative B.) C. Problems C1. a) Draw a timeline of the relevant cashflows and clearly label all cashflows. b) How many payments need to be made and what is the date of the final payment? 13 payments need to be made and the final payment is at t=12 years which is on the 1 February 2035. This is the start of the 13 th year of schooling. c) What type of annuity are the payments? This is a growing annuity due. The payments are growing for a fixed period of time rather than into perpetuity it is a growing annuity rather than growing perpetuity. It is due rather than ordinary. d) Write down the formula before inserting the numbers if you are to calculate the present value of it? Students should put down the following: 𝑃𝑉 = 𝐶𝐹 + ஼ி(ଵା௚) ௥ି௚ ቂ1 − (ଵା௚) ೙షభ (ଵା௥) ೙షభ ቃ e) What is the PV of the annuity? Show all calculations and explanations for steps in your calculation. Explain what this number represents in terms of what is required to fund your child’s schooling 𝑃𝑉 = $15,000 + $ଵହ,଴଴଴(ଵ.଴଺) ଴.଴଻ି଴.଴଺ ቂ1 − (ଵ.଴଺) భమ (ଵ.଴଻) భమ ቃ = $184,431.52 This amount represents what you would need to deposit in an account earning a fixed rate of 7% p.a. today (as at 1 Feb 2023) in order to have the exact amount of money required to pay fees of $15,000 p.a. if they grow at the expected rate of 6% p.a. C2.

FNCE10002 Principles of Finance Semester 2, 2023 Suggested Answers to Tutorial Questions for Topic 2 3 (a) P 0 = $ସ଴,଴଴଴ ଴.଴ହ ቂ1 − ଵ (ଵ.଴ହ) మఱ ቃ = $563,758 At 5%, taking the award as an annuity is better because its present value of $563,578 is larger than the $500,000 lump-sum amount. (b) P 0 = $ସ଴,଴଴଴ ଴.଴଻ ቂ1 − ଵ (ଵ.଴଻) మఱ ቃ = $466,144 At 7%, taking the award as a lump sum is better because the present value of the annuity of $466,144 is less than the $500,000 lump-sum payment.

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