At the end of three years the machine will be worn-out, but the equipment will still be useful to a milling company in Mexico. Quaker plans to sell it to Molino Grande at a price of $10,000.
THE TIMING OF CASH FLOWS QUAKER OATS FILLING MACHINE PROJECT
(in thousands of dollars)
In this example the timing of cash flows is critical to determining the project’s value. A commonly used representation of the timing of cash flows is a bar graph. Each period, cumulative cash flow is reflected either below the line for cash investments or above the line for returns. Our Quaker example is shown in the following bar graph:
Suppose that the cash flows are the same but the timing of the cash is advanced as follows:
or prolonged as follows:
These diagrams raise the critical issue of the “value” of timing.
ACCUMULATED VALUE
When the milling project produces cash, Quaker reinvests it rather than let it remain idle. Therefore if Quaker receives $51,000, $51,000, and $61,000 as described earlier, the company earns income with the cash for two more years in Scenario A than in Scenario B.
If the company has investment opportunities that yield 10 percent, then Scenario A will produce $34,230 more than Scenario B.
The flows have an accumulated value at the end of three years of $163,000 plus earned interest of $34,230 that equals $197,230. Scenario A is clearly the better scenario.
A simpler calculation is to use the formula for the accumulated value or future value of a dollar:
Future Value of a $ in x periods = ($ today) × (1 + Reinvestment Rate)Number of Periods
At 10% the factor for 1 year = $1 × (1 + .10)1 = 1.10.
You don’t have to memorize the factors or calculate them each time—you can use the tables provided in the appendix, or any basic business calculator. (The best calculator in my opinion is the Hewlett-Packard. Owning an HP, an MBA icon, also sends a strong signal to others that you are serious about numbers.)
Per the tables in the Appendix, the accumulated value factors for varying rates and investment periods at 10 percent are:
ACCUMULATION FACTORS
$1 today = $1 today
$1 invested = $1.100 in 1 year
$1 invested = $1.210 in 2 years
Using our factors on Scenario A, on the $163,000 received at the end of Year 1 and invested for two years till the end of Year 3, the accumulated value is:
Year 1’s $163,000 × 1.2100 = $197,230 in 2 years or $197,230 − $163,000 = $34,230 of reinvestment income
ACCUMULATED VALUE CALCULATION
When evaluating projects or investments that extend into the future, it is not only the magnitude of the cash flows that is important, but also the timing and the subsequent use to which those flows can be reinvested.
NET PRESENT VALUE (NPV)
Accumulated value analysis is a good tool to determine how much a retiree needs to invest today to have an adequate pension in thirty years, but it doesn’t solve the problem of evaluating investments and projects today. Investments need to be evaluated in today’s dollars. How much is Quaker’s milling project worth today? How does it compare to a similar piece of equipment that costs $150,000 but lasts four years?
Cash flow analysis determines the flows and the NPV technique values them in today’s dollars. In that way different projects can be compared regardless of timing.
If Apple Computer, for example, knows that a new Tangerine computer will be a surefire $5 billion hit, but that it will take ten years to develop, it may not make sense to invest in this project. Not only will that $5 billion be worth less due to inflation, but Apple could also use the money to invest in robotics, which will save Apple production costs today. Even if an NPV analysis justifies the Tangerine, there may be strategic reasons that overshadow it. That’s where the MBA must use management judgment.
Securities analysts see stocks and bonds as an equipment purchase. The stocks provide dividend payments and bonds provide interest payments in the future. The securities’ values lie in the present value of their future cash flow. Just as Quaker Oats uses NPV analysis to evaluate the merits of buying a new piece of production equipment, corporations evaluate new factories, and the worth of increased advertising. Lawyers involved in wrongful death suits can use net present value techniques to value the total of an individual’s future earnings when considering a settlement. The basic idea to remember is: A dollar today is worth more than a dollar received in the future.
The Quaker Oats project yielded $163,000 over three years (51 + 51 + 61). As previously calculated, the $163,000 in cash flows would yield an additional $34,230 if the cash produced was reinvested at a rate of 10 percent in other company projects or an interest-bearing investment. Would you pay $163,000 for $163,000 to be received over three years? Of course not! You would be giving up the time value of money, or $34,230.
Using this simple logic, NPV analysis takes future cash flows and discounts them to their present-day value. This is the inverse of accumulated value. The formula is as follows:
NPV = ($ in Future) × (1 + Discount Rate) − Number of periods
One dollar received one year from now, with a discount rate of 10 percent, would be worth:
$1 × (1 + .10)−1 = .90909
Using this formula, tables of discount factors tell the NPV of $1 at varying rates and for varying periods. Considering the 10 percent reinvestment opportunities available and the project’s riskiness, $1 in the future is worth the following amounts today per the formula and tables:
DISCOUNT FACTORS
$1 today = $1 today
$1 in 1 year = $.90909 today
$1 in 2 years = $.82645 today
$1 in 3 years = $.75131 today
The cash flows of the Quaker project would be valued in the following way:
FUTURE CASH × DISCOUNT FACTOR = NPV
Start Year 0 − $102,000× 1 = −$102,000 today
in year 1 $51,000 × .90909 = $46,363.59 today
in year 2 $51,000 × .82645 = $42,148.95 today
in year 3 $61,000 × .75131 = $45,829.91 today
The Quaker project NPV = $32,342.45 today
The evaluation of any project depends on the magnitude of the cash flows, the timing, and the discount rate, 10 percent in our case.
The discount rate is highly subjective. The higher the rate or hurdle rate, the less a dollar in the future would be worth today (see the appendix). It is called a hurdle rate because a project with a higher discount rate must generate more cash in the future to be worth the same value of today. The project thus has a higher hurdle to jump to stay even. In cases where the outcome of an investment is risky, as in our oil well example, a higher discount rate is appropriate. If the outcome of an investment is certain, as in the investment in a laborsaving device or in a U.S. Treasury bond, a lower rate is warranted. Companies not guided by an MBA’s expertise will use only one hurdle rate for all investment decisions and thereby ignore the relative riskiness of projects. They end up rejecting sure things, and chasing high-risk projects. Under no circumstances should the interest rate of a company’s bank debt be the rate that is used, unless it is just coincidence. The risk of the project should determine the discount rate. Stable companies can borrow at low interest rates, but they can invest in risky projects.
INTERNAL RATE OF RETURN (IRR)
IRR is a derivative of NPV. Simply stated: The internal rate of return of an investment is the rate at which the discounted cash flows in the future equal the value of the investment today.
To find the IRR one must try different discount rates until the NPV equals zero. (Of course the HP calculator yields the IRR at a push of a button!) For the Quaker project the IRR is 26.709 percent. To confirm that number we can calculate the following:
USING “26.709% DISCOUNT FACTORS”
today 1.00 × −102,000 = −$102,000
1 year .78920 × $51,000 = $40,250
2 years .62285 × $51,000 = $31,765
&n
bsp; 3 years .49155 × $61,000 = $29,985
NPV = −0−
Using IRR to rank projects is useful, but it does not consider the magnitude of the values. A small investment with proportionately large returns would be ranked higher than large investments with adequate returns. If General Electric has a billion dollars allocated to research, it needs to deploy large sums of money to large projects that may have lower IRRs.
Ranking by IRR also neglects the hurdle rates or discount factors used in NPV analysis. Those hurdle rates, as I said, adjust for risk. All things being equal, the investment in equipment by Quaker may have a lower IRR than highly speculative research into a Swedish cancer cure by Merck, but the Quaker project could have a higher NPV. The equipment project’s smaller cash flows would be discounted at a 10 percent rate because of the lower risk involved. This could result in higher NPV. The cancer research would be assessed using a high discount rate of 50 percent. Remember, the higher the discount rate, the less the cash is worth today and the more risk is implied.
PROBABILITY THEORY
Probability theory is a nice term for statistics, the subject that creates fear in the hearts of even the brightest CPAs in business schools. Actually, probability theory is a more accurate term because it describes how statistics are used to solve problems. Given the probabilities of striking oil, what should Sam do? Out of eight hundred married MBA students in the Top Ten programs, how many spouses are likely to be ignored during the first year of the MBA program? It’s all probability theory. Because most businesspeople shy away from statistics, here is an opportunity for MBAs to excel. I took a statistics course as an undergraduate and learned virtually nothing because I was taught theory, not problem solving. MBA programs concentrate on the practical aspects of statistics and tend to leave theory for mathematicians to sort out. If you are not familiar with statistics, do not skip this section. I cannot make you statistically proficient in a few pages, but if you give it a chance with some patient reading, I promise you that you will have enough working knowledge of the discipline to ask for help whenever appropriate. Preparing students by giving them a working knowledge of different subjects is the main thrust of an MBA education. In only two years, professors do not expect their students to become technical experts, but they expect them to recognize where they should seek the help of an expert to solve a particular problem.
PROBABILITY DISTRIBUTIONS
In situations where multiple outcomes are possible, the result is a distribution of outcomes. Each possibility is assigned a probability. Through careful analysis, intuition, and judgment, all the possible outcomes of any event add up to 100 percent, like the event fork of a decision tree. The graph that shows a distribution of outcomes is called a probability mass or density function. If there are many possible outcomes, the curve is smooth and is called a probability density function. If there are only a few, an uneven curve is drawn, called a probability mass function.
A Rainfall Example. Rainfall in Seattle is an event resulting in a probability distribution. Seattle’s rainfall, using hypothetical data, looked as follows in a table and in probability distribution charts below.
SEATTLE DAILY RAIN MEASUREMENTS MARCH 2010
PROBABILITY MASS FUNCTION OF RAIN MEASURES
daily rainfall in seattle
march 2010 (31 days)
THE BINOMIAL DISTRIBUTION
Flipping a coin results in two possibilities, heads or tails. Therefore the distribution of outcomes of two coin flippings could have several possible outcomes if you consider “heads” a success.
2 successes, Heads/Heads
1 success/1 failure, Heads/Tails, Tails/Heads
2 failures, Tails/Tails
Coin flipping gives rise to the most basic of distributions, called a binomial distribution. There are two outcomes in a binomial distribution, success and failure, each with an equal likelihood of occurring.
Seemingly arcane binomial distribution theories can be applied to such practical pursuits as stock market analysis. Success in a stock analysis would be a positive return on a stock in a month, and failure would be a loss or break even. In a historical study of the old AT&T share prices from 1957 to 1977, each month was examined to determine the rate of positive returns. It was found that 56.7 percent of the time there was a success.
The months studied were grouped into periods of three months each (quarters). Researchers noted that the frequency of actual successes was as follows:
A coin-flipping mathematician created tables of numbers to solve all binomial distribution problems. In the AT&T case, the information needed to use a binomial table is:
r = number of successes possible = 0 to 3
n = number of trials = 3 (3 months in a quarter)
p = probability of success = 56.7%
Using this information, a binomial table predicts that the expected outcomes should be:
Surprisingly, the binomial distribution matches rather well the actual results of the AT&T case. Given a guess of probability of success (p), the probability of positive monthly returns in a given quarter could be read off the table. Binomial distribution therefore has practical applications for assessing probabilities for portfolio managers, sales directors, and research analysts.
THE NORMAL DISTRIBUTION: THE MYSTERY OF THE BELL CURVE
The normal distribution is the most widely used distribution and is most commonly known as the bell curve. At Harvard the bell curve is used to determine grades. The curve dictates that 15 percent of the class receive “Low Passes” (“loops”). At the Darden School, the professors use their judgment to dole out unsatisfactory marks of C or F. The result is two campuses with vastly different competitive environments.
When a probability mass function is based on many trials, the curve tends to fill in and become bell-shaped. We call this a probability density function. Such was the case with the two graphs describing rainfall in Seattle. The hump in the middle is caused by the Central Limit Theorem. It states that “the distribution of averages of repeated independent samples will take the form of the bell-shaped normal distribution.” Why? Simply because a large number of independent samples tend to a central average.
THE BELL-SHAPED GRADING CURVE
The concept of “averages of samples” is pretty vague. In case applications the definition expands to include any large group of data. Why? Because the normal distribution is so easy to use and closely approximates reality anyway. Stock prices are the result of many market fluctuations that culminate in a return (profit or loss). The return can be considered an “average” of those market fluctuations. Just about anything can be rationalized as an average, hence the usefulness of normal distributions.
PROBABILITY DENSITY FUNCTION OF RAIN MEASUREMENTS
daily rainfall in seattle
1970–2010 (14,600 days)
Measures of the Normal Curve. The bell-shaped curve is described by two terms, the mean and its standard deviation (SD). The mean (µ) is the center of the curve. The mean is commonly called the average. It is the result of adding up the data and dividing by the number of data points. The standard deviation (σ) is how wide the curve appears. The SD can also be described as a measure of the “variability from the mean.” These two terms are central to most probability concepts.
Other less-used measures of averages for a set of data are the median, the item in the middle of the list if sorted by size, and the mode, the item occurring most frequently in a data set.
PROBABILITY DENSITY FUNCTIONS WITH CURVES OF DIFFERENT STANDARD DEVIATIONS
As with the binomial distribution, the sum of all the outcomes as represented by the region under the curve equals 100 percent. What makes the normal distribution’s curve special is that for any given SD measure away from the mean or the center, the same probability exists for an event despite the normal distribution’s shape.
Normal Distribution Retailing Example. Al Bundy, a shoe store owner, wants to make sure he has enough stock for all size requests. He
purchased a study of ladies’ shoe sizes from the Academy of Feet and received a stack of research data from survey responses.
He plotted the data on graph paper and it appeared as a normal distribution. He also entered the series of sizes in his calculator and hit the “Standard Deviation” key. The answer was 2. Al also took the average or mean of all the survey’s respondents’ sizes and found it to be 7. Looking at the graph he created, he saw that it looked like our trusty normal distribution.
NORMAL DISTRIBUTION OF SHOE SIZES
Just by recognizing the shape, Al could apply the laws of the normal distribution curve. The laws governing the area under all normal curves are the following:
1 SD = .3413
2 SD = .4772
3 SD = .49865
4 SD = .4999683
Using these rules, if Mr. Bundy stocks sizes 5 to 9 he has covered .6826 (2 × .3413) of the population. Increasing the sizes to 3 to 11, he has covered .9544 of the feet out there. If Al stocked sizes 1 to 13, .9973 of customers at his store would be satisfied with his selection. He can always special-order for those feet beyond sizes 1 to 13.
Of course normal distribution tables have been developed to determine the probability for any specific point on the curve (noninteger SDs away from the mean). To use the tables, a Z value must be calculated.
A NORMAL CURVE FINANCE EXAMPLE
The Ten-Day MBA 4th Ed. Page 16