 # NPV I: Time Value of Money This module introduces the concept of the time value of money, interest rates, discount rates, the future value of an investment,

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NPV I: Time Value of Money This module introduces the concept of the time value of money, interest rates, discount rates, the future value of an investment, the present value of a future payment, and the net present value (NPV) of a future stream of payments. Author: Stu James © 2012 Stu James and Management by the Numbers, Inc.

The Time Value of Money Interest rates and Discount rates How to calculate the future value of an investment How to calculate the present value of a future payment How to calculate the NPV of a series of future cash flows N ET P RESENT V ALUE (NPV) CONCEPTS C OVERED 2 Net Present Value (NPV) Concepts Covered MBTN | Management by the Numbers This MBTN Module covers the following concepts:

T IME V ALUE OF M ONEY 3 Time Value of Money MBTN | Management by the Numbers Net Present Value (NPV) and associated calculations are based on the idea that a dollar today is worth more than a dollar in the future. Let’s consider an example to help illustrate this idea. Joe has two friends who are asking to borrow money for a business. Both are requesting to borrow \$1000. Here is a brief summary of how they plan to use the funds: Which proposal do you think Joe would prefer? John is very trustworthy, and has a business that is generating a steady stream of income, and he plans on using the \$1000 to upgrade his equipment. John promises to pay Joe back in 1 year. Jack is also very trustworthy, and has a similarly successful business, and Jack is also using the \$1000 to upgrade equipment. Jack, however, promises to pay Joe back in 5 years.

T IME V ALUE OF M ONEY 4 Time Value of Money MBTN | Management by the Numbers So, what can we say about Joe’s situation? First, setting aside the value of friendship and altruism, we might recognize that Joe would be unlikely to give either of them \$1000 without some additional payment for delaying being able to use the \$1000 himself. This payment would be considered interest. Second, we can also say without some additional payment, that Joe should prefer to lend to John over Jack because John will back him back sooner (1 year instead of 5). Both of these considerations speak to the time value of money. a)People and organizations expect some payment, “interest”, in exchange for allowing the use of their money for a period of time. b)The longer the time period (all other factors being equal), the greater the payment expected.

R ISK 5 Risk MBTN | Management by the Numbers Let’s look at a second example that influences our Net Present Value (NPV) calculations. Joe has a 3 rd friend, Jamie, who is contemplating starting a business and is also looking to borrow \$1000. In this example, although the time period of the loan is the same, which of the loan proposals do you think Joe would prefer? Jamie has borrowed tools and CDs from Joe from time to time and never returns them. Jamie has an idea to make a flying lemonade stand and needs the \$1000 to get the idea “off the ground”. Jamie promises to pay Joe back in 1 year. Recall John’s situation of a business that is generating a steady stream of income, and where John plans on using the \$1000 to upgrade his equipment. John also promises to pay Joe back in 1 year.

R ISK 6 Risk MBTN | Management by the Numbers How are Jamie and John’s requests different? While both Jamie and John make the same promise to pay in the same amount of time, there is a considerable difference in how Joe evaluates the two loans. Insight The payment required to borrow money increases with the length and risk of the loan. First, Jamie has demonstrated that his reliability leaves something to be desired. Therefore, Joe doesn’t feel as confident that Jamie will actually repay the loan. Second, although we don’t know the specifics about the various businesses, John’s business seems to be a better bet because it has a successful track record and because he is using the \$1000 to purchase an asset (which presumably will still have value next year).

I NTEREST AND D ISCOUNT R ATE 7 Interest Rates and Discount Rates MBTN | Management by the Numbers For many investments, however, there is no interest rate, but instead a series of future cash flows. To calculate the current value of those future cash flows, we have to discount those by an implied or expected rate of return. This is called the discount rate, and this rate is used to discount future cash flows to the present. It is a similar concept to interest rates, but used in Net Present Value analysis to convert future cash flows to current values. How to determine the appropriate discount rate is beyond the scope of this module. As previously described, interest is the term for what we call the payments for the use of money over time. Normally the interest rate will be provided as percentage rate for a particular period of time. For example, a 10% annual interest rate would mean that the borrower would owe the lender 10% of the amount borrowed each year. Interest is also generally something that accrues over time going forward.

B USINESS C ONTEXTS FOR NPV 8 Business Contexts for NPV MBTN | Management by the Numbers NPV analysis is commonly used in business to evaluate the value of future cash flows in today’s dollars. You may also hear people refer to this as Discounted Cash Flow Analysis (DCF). Examples include: Stock, bond and business valuation Analysis of equipment purchases (such as productivity increases due to equipment vs. the cost to purchase or lease payments) Lifetime value of a customer (CLV) Real estate investments Insight NPV analysis is a useful tool for the valuation of any future stream of cash flows, whether regular or irregular. However, the quality of the analysis is very much dependent on the quality of the projections: the two critical ones being the amount and timing of the cash flows and the appropriate discount rate. Often these are difficult to estimate.

F UTURE VALUE OF AN I NVESTMENT Definition Future Value of an Investment = PV * (1 + i) ^ n Where i = interest rate per period PV = Present value of the investment N = number of periods 9 Future Value of an Investment MBTN | Management by the Numbers Question 1: Alise deposits 5000 Euros into a 3 year CD that pays 4% interest compounded annually. How much will the CD be worth in 3 years if the interest earned is automatically reinvested in the CD? Answer: Future Value= 5000 * (1 +.04) ^ 3 = 5000 * 1.125 = € 5,624.32

F UTURE VALUE OF AN I NVESTMENT 10 Future Value of an Investment MBTN | Management by the Numbers Question 2: An on-line bank is offering a 3 year CD at 4% interest that compounds monthly. Which is the better investment for Alise (presuming same risk and automatic reinvestment)? Answer: Future Value= PV * (1 + i) ^ n Since the CD compounds monthly, first convert the annual rates and periods to monthly. Monthly Interest=.04 / 12 =.00333 Periods = 3 * 12 = 36 Future Value = 5000 * (1 +.00333) ^ 36 = 5000 * 1.27 = € 5,636.36 5,636 > 5,624, so the investment that compounds more frequently is better.

P RESENT V ALUE OF A F UTURE P AYMENT Definitions Present Value of a Future Payment = PV = (FV) / (1 + d) ^ # periods 11 Present Value of a Future Payment MBTN | Management by the Numbers Question 3: Nicole’s friend Arthur offers to purchase her antique map collection for \$10,000 in 10 years. The appropriate discount rate for this investment is 8%. What is the present value of Arthur’s offer? Answer: Present Value= \$10,000 / (1 +.08) ^ 10 = \$10,000 / 2.1589 = \$4,361

P RESENT V ALUE OF A S ERIES OF F UTURE C ASH F LOWS Definitions Present Value of a Series of Future Cash Flows = PV = CF 1 / (1 + d) ^ 1 + CF 2 / (1 + d) ^ 2 + … + CF n / (1 + d) ^ n Where: CF n = Cash Flow in period n d = appropriate discount rate for project or investment 12 Present Value of a Series of Future Cash Flows MBTN | Management by the Numbers Question 4: Joe Loppy’s Auto Service is considering leasing a machine that will save the company \$500 per year over the 3 year term of the lease (including lease payments). What is the present value of that investment if the appropriate discount rate is 5%? Answer: Present Value= \$500 / (1 +.05) + \$500 / (1 +.05)^ 2 + \$500 / (1 +.05)^ 3 = \$476 + \$454 + \$432 = \$1,362

N ET P RESENT V ALUE Definitions Net Present Value = The present value of future cash flow less the initial investment. NPV = CF 0 + CF 1 / (1 + d) ^ 1 + CF 2 / (1 + d) ^ 2 + … + CF n / (1 + d) ^ n Where: CF n = Cash Flow in period n d = appropriate discount rate for project or investment And CF 0 is the cash flow in period 0 or the initial investment which is usually negative 13 Net Present Value MBTN | Management by the Numbers Question 5: Joe Loppy’s Auto Service is also considering purchasing the equipment outright instead of leasing. The equipment costs \$15,000 and is estimated to save \$6,000 per year over its 3 year life. What is the NPV of the investment in the equipment and which is the better approach, lease or purchase (assume discount rate of 5%)?

N ET P RESENT V ALUE 14 Net Present Value MBTN | Management by the Numbers Answer: CF 0 = -\$15,000 PV of CF 1 = \$6,000 / (1 +.05) = \$5,714 PV of CF 2 = \$6,000 / (1 +.05)^ 2 = \$5,442 PV of CF 3 = \$6,000 / (1 +.05)^ 3 = \$5,183 NPV = -\$15,000 + \$5,714 + \$5,442 + \$5,183 = \$1,339 Since the NPV is positive, the investment should be considered from a financial point of view. However, since the leased approach has a PV of \$1,362 which is higher than the purchase NPV \$1,339, the lease would be the preferred option.

N ET P RESENT V ALUE 15 Net Present Value MBTN | Management by the Numbers Question 6: Fred Snerdsly, the new Service Manager at Joe Loppy’s tells Joe that there’s an even better piece of equipment that will save \$15,000 / year, has an estimated life of 5 years. Fred thinks this is even better because while it costs \$70,000, it will save a total of \$5,000 at the end of 5 years. What do you think? Answer: While Fred is correct that in non-discounted dollars, the machine will save \$5,000 at the end of 5 years, (\$15,000 * 5 - \$70,000), NPV analysis shows otherwise, presuming the same discount rate of 5%. PV of CF 1 = \$15,000 / (1 +.05) = \$14,286 PV of CF 2 = \$15,000 / (1 +.05)^ 2 = \$13,605 PV of CF 3 = \$15,000 / (1 +.05)^ 3 = \$12,958 PV of CF 4 = \$15,000 / (1 +.05)^ 4 = \$12,341 PV of CF 5 = \$15,000 / (1 +.05)^ 5 = \$11,753 Total = \$64,942 NPV = -\$70,000 + \$64,942 = -\$5,158

Marketing Metrics by Farris, Bendle, Pfeifer and Reibstein, 2 nd edition, pages 347-348. - And - MBTN Customer Lifetime Value module F URTHER R EFERENCE 16 Further Reference MBTN | Management by the Numbers

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