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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.
This MBTN Module covers the following concepts:
• 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
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NET PRESENT VALUE (NPV) CONCEPTS COVERED
Net Present Value (NPV) Concepts Covered
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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:
TIME VALUE OF MONEY
Time Value of Money
• 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.
Which proposal do you think Joe would prefer?
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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).
TIME VALUE OF MONEY
Time Value of Money
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.
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RISK
Risk
Let’s look at a second example that influences our Net Present Value
(NPV) calculations.
Joe has a 3rd friend, Jamie, who is contemplating starting a business
and is also looking to borrow $1000.
• 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.
In this example, although the time period of the loan is the same,
which of the loan proposals do you think Joe would prefer?
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RISK
Risk
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.
• 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).
Insight
The payment required to borrow money increases with the length and
risk of the loan.
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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.
INTEREST AND DISCOUNT RATE
Interest Rates and Discount Rates
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.
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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
BUSINESS CONTEXTS FOR NPV
Business Contexts for NPV
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.
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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
FUTURE VALUE OF AN INVESTMENT
Future Value of an Investment
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
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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
FUTURE VALUE OF AN INVESTMENT
Future Value of an Investment
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.
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Definitions
Present Value of a Future Payment =
PV = (FV) / (1 + d) ^ # periods
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?
PRESENT VALUE OF A FUTURE PAYMENT
Present Value of a Future Payment
Answer:
Present Value
= $10,000 / (1 + .08) ^ 10
= $10,000 / 2.1589 = $4,361
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Definitions
Present Value of a Series of Future Cash Flows =
PV = CF1 / (1 + d) ^ 1 + CF2 / (1 + d) ^ 2 + … + CFn / (1 + d) ^ n
Where:
CFn = Cash Flow in period n
d = appropriate discount rate for project or investment
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
PRESENT VALUE OF A SERIES OF FUTURE CASH FLOWS
Present Value of a Series of Future Cash Flows
= $500 / (1 + .05) + $500 / (1 + .05)^2 + $500 / (1 + .05)^3
= $476 + $454 + $432 = $1,362
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Definitions
Net Present Value = The present value of future cash flow less the
initial investment.
NET PRESENT VALUE
Net Present Value
NPV = CF0 + CF1 / (1 + d) ^ 1 + CF2 / (1 + d) ^ 2 + … + CFn / (1 + d) ^ n
Where:
And
CFn = Cash Flow in period n
d = appropriate discount rate for project or investment
CF0 is the cash flow in period 0 or the initial investment
which is usually negative
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%)?
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Answer:
CF0
PV of CF1
PV of CF2
PV of CF3
= -$15,000
= $6,000 / (1 + .05)
= $5,714
= $6,000 / (1 + .05)^2 = $5,442
= $6,000 / (1 + .05)^3 = $5,183
NPV
= -$15,000 + $5,714 + $5,442 + $5,183 = $1,339
NET PRESENT VALUE
Net Present Value
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.
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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?
NET PRESENT VALUE
Net Present Value
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 CF1
PV of CF2
PV of CF3
PV of CF4
PV of CF5
NPV
= $15,000 / (1 + .05)
= $15,000 / (1 + .05)^2
= $15,000 / (1 + .05)^3
= $15,000 / (1 + .05)^4
= $15,000 / (1 + .05)^5
= -$70,000 + $64,942
= $14,286
= $13,605
= $12,958
= $12,341
= $11,753
= -$5,158
Total = $64,942
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Marketing Metrics by Farris, Bendle, Pfeifer
and Reibstein, 2nd edition, pages 347-348.
FURTHER REFERENCE
Further Reference
- And MBTN Customer Lifetime Value module
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