Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs

Problem Set I: Interest Rates and Present Value Calculations Problem 1: The Power of Compounding and Interest Rate Quotes a) An amount of EUR 8,000 is invested at 5% per year. (i) What is the balance in the account after 3 years? After 13 years? (ii) How long does it take for the balance to reach EUR 32,000? b) How much should you have deposited in a bank account 5 years ago in order to have EUR 50,000 today, given that the interest rate has been 5% per year over the period? c) Which do you prefer: a bank account that pays 5% per year for three years or (i) An account that pays 2.5% every six months for three years? (ii) An account that pays 7.5% every 18 months for three years? (iii) An account that pays 0.5% per month for three years? d) You have found three investment choices for a one-year deposit: (i) 10% annual percentage rate (APR) compounded monthly; (ii) 10% APR compounded annually, and (iii) 9% APR compounded daily. Assuming that there are 365 days in the year, compute the effective annual rate (EAR) for each investment choice. e) Your bank account pays interest with an EAR of 5%. What is the APR quote for this account based on semiannual compounding? What is the APR with monthly compounding? f) An amount $1000 earns interest at 5% per year. What this amount has grown to after 10 years when interest is compounded (i) yearly; (ii) monthly; (iii) continuously?

Problem 2: The NPV Decision Rule a) You are considering a unique investment opportunity. If you invest LVL 10,000 today, you will receive LVL 500 one year from now, LVL 1500 two years from now, and LVL 10,000 ten years from now. (i) What is the net present value (NPV) of the investment opportunity if the interest rate is 6% per year? Should you undertake the opportunity? (ii) Re-calculate the NPV if the interest rate is 2% per year? Should you undertake the opportunity?

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b) Your company has identified three potential investment projects. The projects and their respective cash flows are shown in the following table:

Project

Cash Flow Today ($)

Cash Flow in One Year ($)

A

-10

20

B

5

5

C

20

-10

Assume that all cash flows are certain and the risk-free interest rate is 10% per annum. (i) What is the NPV of each project? (ii) If the firm can choose only one of these projects, which project should it undertake? (iii) If the firm can choose any two of these projects, which projects should it undertake?

Problem 3: Valuing Perpetuities a) Vanja, a class of 1998, has just graduated from SSE Riga and wants to endow an annual graduation party at his alma mater. He wants the event to be a memorable one, so he budgets LVL 30,000 per year forever for the party! If the interest rate is 8% per year, and if the first party is in one year's time, how much does Vanja need to donate to endow the party? b) Before accepting the money, the SSE Riga Student Association has asked Vanja to increase the donation to account for the effect of inflation on the cost of the party in future years. Although LVL 30,000 is adequate for the next year's party, the students estimate that the party's cost will rise by 4% per year thereafter. How much does Vanja need to donate now to satisfy their request?

Problem 4: Valuing Annuities a) When Armands purchased his new flat, he took out a 30-year annual-payment mortgage with an interest rate of 6% per year. The annual payment on the mortgage is EUR 1200. He has just made a payment and has decided to pay the mortgage off by repaying the outstanding balance. What is the payoff amount if Armands has lived in the house for 12 years (so there are 18 years left on the mortgage)? b) You work for a pharmaceutical company RigaFarm that has developed a new drug against the AH1N1 virus. The patent for the drug will last for 17 years. You expect that the drug's profits will be LVL 2 million in its first year and then this amount will grow at a rate of 5% per year for the next 17 years. Once the patent expires, other pharmaceutical companies will be able to produce the same drug and competition will likely drive profits to zero. What is the present value of the new drug if the interest rate is 10% per year?

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Problem 5: Some Further Considerations... a) You are thinking of purchasing a house which costs EUR 350,000. You have EUR 50,000 in cash that you can use as a down payment on the house, but you need to borrow the rest of the purchase price. SwedBank is offering you a 30-year mortgage that requires annual payments and has an interest rate of 7% per year. (i) What will your annual payment be if you sign up for this mortgage? (ii) Suppose you would like to buy the house and take the mortgage described. Unfortunately, you can afford to pay only EUR 23,500 per year. SwedBank agrees to allow you to pay this amount each year, yet still borrow EUR 300,000, on condition that at the end of the mortgage (in 30 years), you must make a balloon payment; that is, you must repay the remaining balance on the mortgage. How much will this balloon payment be? b) Taavi is saving for his retirement. To live peacefully and comfortably, he decides to save up $2 million by the time he is 65. Today is his 30th birthday, and he decides, starting today and continuing on every birthday up to and including his 65th birthday, that he will put the same amount into his savings account. The interest rate is 5%. (i) How much should Taavi set aside each year to make sure that he will have $2 million in the account on his 65th birthday? (ii) Instead assume that Taavi expects his income to increase over the lifetime; hence, for him it will be more rational to save less now and more later. Instead of putting the same amount aside each year, he decides to let the amount that he sets aside grow by 7% per year. How much will he put into account today? Notice that Taavi is planning to make the first contribution today! c) Your spouse bought an annuity from BATVA Life Insurance Company for LVL 200,000 when she retired. In exchange for LVL 200,000, the company will pay her LVL 25,000 per year until she dies. The interest rate is 5%. How long must she live after the day she retired to come out ahead; that is, to get more in value than what she paid in? d) Veiko is running a hot Estonian Music Company. Analysts predict that its earnings will grow at 30% per year for the next five years. After that, as competitors from the South enter, earnings growth is expected to slow down to 2% per year and continue at that level forever. Veiko’s company has just announced earnings of EEK 1 million. Assuming that all cash flows occur at the end of the year, what is the present value of all future earnings if the interest rate is 8%?

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Additional Problems for Your own Study Problem 6: Tricky but Interesting a) You have just turned 30 years old and you have accepted your first job - congratulations! Now you must decide how much money to put into your retirement plan which works as follows: Every dollar in the plan earns 7% per year. You cannot make withdrawals until you retire on your sixty-fifth birthday. After that point, you can make withdrawals as you like. You decide that you will plan to live up to 100 and work until you turn 65. You estimate that to live comfortably in retirement, you will need EUR 100,000 per year starting at the end of the first year of retirement and ending on your one hundredth birthday. (i) If you plan to contribute the same amount to the plan at the end of every year that you work, how much do you need to contribute each year to fund your retirement? (ii) The setup of your retirement plan in a) is not very realistic because most retirement plans do not allow you to specify a fixed amount to contribute every year. Instead, you are required to specify a fixed percentage of your salary that you want to contribute. Assume that your starting salary is EUR 75,000 per year and it will grow 2% per year until you retire. Assuming that everything else stays the same as in a) above, what percentage of your income do you need to contribute to the plan every year in order to fund the same retirement income? b) Kenneth Hornbill (KHO) is thinking of making an investment in a new restaurant in Helsinki. The restaurant will generate revenues of $1 million per year for as long as KHO maintains it. He expects that the maintenance cost will start at $50,000 per year and will grow at 5% per year thereafter. Assume that all revenue and maintenance costs occur at the end of the year. KHO intends to run the restaurant, which can be built and become operational immediately, as long as it continues to make a positive cash flow. If the restaurant costs $10 million to build, and the interest rate is 6% per year, should KHO invest in the restaurant? Answer:

a)

i 

C  EUR 9,366.29

 ii 

p  9.948%

b) Yes because NPV  3,995, 073.97  0

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