2. Future value Aa Aa The principal of the time value of money is probably the single most important concept in financial management. One of the most frequently encountered applications involves the calculation of a future value The process for converting present values into future values is called knowledge of the values of three of four time-value-of-money variables. Which of the following is not one of these variables? This process requires O The interest rate (I) that could be earned by deposited funds The trend between the present and future values of an investment O The duration of the deposit (N) O The present value (PV) of the amount deposited All other things being equal, the numerical difference between a present and a future value corresponds to the amount of interest earned during the deposit or investment period. Each line on the following graph corresponds to an interest rate: 0%, 12%, or 24%. Identify the interest rate that corresponds with each line. VALUE (Dollars) 0 1 2 3 45 6 789 10 TIME (Years) Line A: Line B: Line C Investments and loans base their interest calculations on one of two possible methods: the the rate, and the investment or deposit period-to the amount deposited or invested in order to compute the amount of interest. However, the two methods differ in their relationship between the variables. interest and interest methods. Both methods apply three variables-the amount of principal, the interest Assume that the variables I, N, and PV represent the interest rate, investment or deposit period, and present value of the amount deposited or invested, respectively. Which equation best represents the calculation of a future value (FV) using: Compound interest? Simple interest? O FV-PV x (1+1)N O FV PV (1 I)N Identify whether the following statements about the simple and compound interest methods are true or false Statement True False After the end of the second year and all other factors remaining equal, a future value based on compound interest will exceed a future value based on simple interest. All other factors being equal, both the simple interest and the compound interest methods wl not generate the amount of earned interest by the end of the first year The process of earning compound interest allows a depositor or investor to earn interest on any interest earned in prior periods Laura is willing to invest $35,000 for six years, and is an economically rational investor. She has identified three investment alternatives (L, M, and P) that vary in their method of calculating interest and in the annual interest rate offered. Since she can only make one investment during the six-year investment period, complete the following table and indicate whether Laura should invest in each of the investments. Note: When calculating each investment's future value, assume that all interest is earned annually. The final value should be rounded to the nearest whole dolla Make this investment? Investment Interest Rate and MethodExpected Future Value Yes No 5% compound interest 4% simple interest 7% compound interest 3. Present value Aa Aa Finding a present value is the reverse of finding a future value. Which of the following is true about finding the present value of cash flows? Finding the present value of cash flows tells you how much you need to invest today so that it grows to a given future amount at a specified rate of return. Finding the present value of cash flows tells you what a cash flow will be worth in future years at a specified rate of return. Which of the following investments that pay will $17,000 in 12 years will have a lower price today? The security that earns an interest rate of 10.00% O The security that earns an interest rate of 15.00%. Eric wants to invest in government securities that promise to pay $1,000 at maturity. The opportunity cost (interest rate) of holding the security is 6.80%. Assuming that both investments have equal risk and Eric's investment time horizon is flexible, which of the following investment options will exhibit the lower price? An investment that matures in six years O An investment that matures in seven years Which of the following is true about present value calculations? Other things remaining equal, the present value of a future cash flow increases if the discount rate increases. Other things remaining equal, the present value of a future cash flow decreases if the discount rate increases. 4. Finding the interest rate and the number of years Aa Aa The future value and present value equations also help in finding the interest rate and the number of years that correspond to present and future value calculations. If a security currently worth $12,800 will be worth $17,983.08 three years in the future, what is the implied interest rate the investor will earn on the security-assuming that no additional deposits or withdrawals are made? 7.12% 0 1.41% 9.60% 12.00% If an investment of $50,000 is earning an interest rate of 4.00%, compounded annually, then it will take for this investment to reach a value of $56,243.20-assuming that no additional deposits or withdrawals are made during this time. Which of the following statements is true-assuming that no additional deposits or withdrawals are made? 0 If you invest $5 today at 15% annual compound interest for 82.3753 years, you'll end up with $100,000. If you invest $1 today at 15% annual compound interest for 82.3753 years, you'll end up with $100,000. 5. Companies and growth rates Aa Aa Future value and present value concepts are applied in various ways, such as calculating growth rates, earnings per share, expected sales and revenues in the future, and so forth Consider the following case: Pharmacist John S. Pemberton invented a soft drink in 1886 that eventually became not only an integral part of everyday life in the United States but also a symbol of consumerism worldwide. In 1929 the first Coca-Cola vending machines were installed in Germany, and in 1930, the German branch of the Coca-Cola Co. opened in Essen Germany was a growing market for Coca-Cola, along with other countries in Europe, before World War II With the previous data given, calculate the company's sales growth rate for each time period in the following table Growth Rate Coca-Cola sales in Germany were 243,000 cases in 1934, 1 million cases in 1936, and 4.5 million cases in 1939 Years 1934-1936 1936-1939 1934-1939 (Source: "Coca-Cola GmbH and World War Two," www.gettherealfacts.co.uk/docs/gmbh.pdf) During World War II, Coca-Cola Co. cut off all syrup sales to Germany in 1940, reslting in no sales from 1943 to 1945. If Coca-Cola's sales had grown from 1939 to 1945 at the same rate that they grew between 1934 and 1939, its sales in 1945 would have been approximately present value.) cases. (Hint: Use sales data from 1939 as the Coca-Cola's worldwide sales as of December 31, 2011, was 26.7 billion cases. Assume the following sales distribution: Unit Case Volume Eastern Europe Germany Spain Great Britain Italy France Other 20% 16% 14% 12% 9% 8% 21% If Coca-Cola's worldwide growth were to continue at the same growth rate as it did in Germany between 1939 and 2008, when sales grew at the rate of 10.26%, its hypothetical sales in Germany in 2053 (42 years from 2011) would be approximately billion cases. (Note: For this question, ignore other factors that affect sales.) 7. Present value of annuities and annuity payments Aa Aa The present value of an annuity is the sum of the discounted value of all future cash flows. You have the opportunity to invest in several annuities. Which of the following 10-year annuities has the greatest present value (PV)? Assume that all annuities earn the same positive interest rate O An annuity that pays $1,000 at the end of each year OAn annuity that pays $500 at the end of every six months O An annuity that pays $1,000 at the beginning of each year An annuity that pays $500 at the beginning of every six months An ordinary annuity selling at $8,979.12 today promises to make equal payments at the end of each year for the next eight years (N). If the annuity's appropriate interest rate (1) remains at 8.00% during this time, the annual annuity payment (PMT) will be You just won the lottery. Congratulations! The jackpot is $10,000,000, paid in eight equal annual payments. The first payment on the lottery jackpot will be made today. In present value terms, you really won assuming annual interest rate of 8.00%. 8. Implied interest rate and period Aa Aa Consider the case of the following annuities, and the need to compute either their expected rate of return or duration Raul inherited an annuity worth $9,048.50 from his uncle. The annuity will pay him eight equal payments of $1,400 at the end of each year. The annuity fund is offering a return of Raul's friend, Nicholas, wants to go to business school. While his father will share some of the expenses, Nicholas stl needs to put in the rest on his own. But Nicholas has no money saved for it yet. According to his calculations, it will cost him $14,895 to complete the business program, including tuition, cost of living, and other expenses. He has decided to deposit $4,500 at the end of every year in a mutual fund, from which he expects to earn a fixed 10% rate of return. It will take approximately school. years for Nicholas to save enough money to go to business 9. Perpetuities Aa Aa Perpetuities are also called annuities with an extended, or unlimited, life. Based on your understanding of perpetuities, answer the following questions. Which of the following are characteristics of a perpetuity? Check all that apply The value of a perpetuity cannot be determined. The current value of a perpetuity is based more on the discounted value of its nearer (in time) cash flows and less by the discounted value of its more distant (in the future) cash flows. A perpetuity is a stream of unequal cash flows. The value of a perpetuity is equal to the sum of the present value of its expected future cash flows. Your grandfather wants to establish a scholarship in his father's name at a local university and has stipulated that you will administer it. As you've committed to fund a $10,000 scholarship every year beginning one year from tomorrow, you'll want to set aside the money for the scholarship immediately. At tomorrow's meeting with your grandfather and the bank's representative, you will need to deposit can fund the scholarship forever, assuming that the account will earn 5.50% per annum every year. rounded to the nearest whole dollar) so that you Oops! The bank representative just reported that he misquoted the available interest rate on the scholarship's account. Your account should earn 3.50%. The amount of your required deposit should be revised to relationship between the interest rate earned on the account and the present This suggests there is value of the perpetuity. Identify whether the situations described in the following table are examples of uneven cash flows or annuity payments: Uneven Cash Flows Annuity Payments Description Shania bought a new dress for her brother's wedding for $450. She negotiated a deal with the retailer in which she would pay for the dress in three installments of $250, $100, and $100 over the next three months. You signed up to make a monthly payment of $10 for one year for a lifetime tosubscription to your favorite magazine. British consols are British government bonds that promise to make payments of a specified amount at regular intervals to the bearer forever. You receive interest earnings from variable deposits in a regular interest-bearing savings account. 12. Mortgage payments Aa Aa Mortgages, loans taken to purchase a property, involve regular payments at fixed intervals and are treated as reverse annuities. Mortgages are the reverse of annuities, because you get a lump-sum amount as a loan in the beginning, and then you make monthly payments to the lender. You've decided to buy a house that is valued at $1 million. You have $150,000 to use as a down payment on the house, and want to take out a mortgage for the remainder of the purchase price. Your bank has approved your $850,000 mortgage, and is offering a standard 30-year mortgage at a 9% fixed nominal interest rate (called the loan's annual percentage rate or APR). Under this loan proposal, your mortgage payment will be month. (Note: Round the final value of any interest rate used to four decimal places.) per Your friends suggest that you take a 15-year mortgage, because a 30-year mortgage is too long and you will pay a lot of money on interest. If your bank approves a 15-year, $850,000 loan at a fixed nominal interest rate of 9% (APR), then the difference in the monthly payment of the 15-year mortgage and 30-year mortgage will be ? (Note: Round the final value of any interest rate used to four decimal places.) It is likely that you won't like the prospect of paying more money each month, but if you do take out a 15-year mortgage, you will make far fewer payments and will pay a lot less in interest. How much more total interest will you pay over the life of the loan if you take out a 30-year mortgage instead of a 15-year mortgage? $1,074,172.64 O $1,165,204.22 O $910,315.80 O $1,256,235.80 Which of the following statements is not true about mortgages? O The ending balance of an amortized loan contract will be zero O Mortgages are examples of amortized loans. O If the payment is less than the interest due, the ending balance of the loan will decrease. O Every payment made toward an amortized loan consists of two parts-interest and repayment of principal. 14. More on the time value of money Aa Aa Lloyd is a divorce attorney who practices law in Florida. He wants to join the American Divorce Lawyers Association (ADLA), a professional organization for divorce attorneys. The membership dues for the ADLA are $650 per year and must be paid at the beginning of each year. For instance, membership dues for the first year are paid today, and dues for the second year are payable one year from today. However, the ADLA also has an option for members to buy a lifetime membership today for $7,000 and never have to pay annual membership dues Obviously, the lifetime membership isn't a good deal if you only remain a member for a couple of years, but if you remain a member for 40 years, it's a great deal. Suppose that the appropriate annual interest rate is 5.9%. what is the minimum number of years that Lloyd must remain a member of the ADLA so that the lifetime membership is cheaper (on a present value basis) than paying $650 in annual membership dues? (Note: Round your answer up to the nearest year.) O 13 years O 14 years 19 years O 16 years In 1626, Dutchman Peter Minuit purchased Manhattan Island from a local Native American tribe. Historians estimate that the price he paid for the island was about $24 worth of goods, including beads, trinkets, cloth, kettles, and axe heads. Many people find it laughable that Manhattan Island would be sold for $24, but you need to consider the future value (FV) of that price in more current times. If the $24 purchase price could have been invested at a 5% annual interest rate, what is its value as of 2012 (386 years later)? $4,784,698,910.93 $3,624,771,902.22 $4,168,487,687.55 $3,081,056,116.89