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  • Probability, Statistics and the Time Value of Money

    TEACHER: Hello, Student. Tell me, please... are there many possible future events you can clearly foresee?

    STUDENT: There are a few, but I’d rather not think about them!

    TEACHER: You are a wise person. I assume what you have in mind is the old saying: "the only events surely to happen in life are death and taxes." And I don’t want to think about those either.

    But in fact, uncertainty is the most prevalent condition in life. No matter which type of decision me must make, we are really almost never sure of the outcome.

    STUDENT: I know. I recall your mentioning before that "forecasting is difficult, and forecasting the future is impossible!" A joke, of course.

    TEACHER: A joke with some element of truth. Well, although in most cases it is impossible to be completely sure of what will happen if we make a given decision, it is possible to estimate the likeliness of a certain outcome. What we can do is estimate the probability of future events to happen.

    STUDENT: What is the exact definition of "probability"?

    TEACHER: Probability is the chance that something will happen. To determine the probability of something occurring, the first thing we have to do is collecting data and organize them: this process is called statistics.

    Using these two tools, probability and statistics, we may quantify the uncertainty before taking a decision. When issuing a weather report, forecasters rarely say: "it will rain later today". They might say "75% chances of rain later today". Can you guess what this means?

    STUDENT: I assume that they are able to say this because they have collected statistics for a long time in a given area, and they know that when current weather conditions were present, 75% of the time it rained.

    TEACHER: Close. Actually, they know that under those conditions it rained in 75% of the given geographical area. But in any case, the point is that it's up to you now to make a decision whether to go on a picnic today. You can't be sure, but you know that you are running a high risk of getting wet!

    STUDENT: You said you need to produce statistics before calculating probabilities.

    TEACHER: Exactly. I will now discuss examples and definitions of statistics.

    Let's imagine that you go target shooting. The target has 20 divisions, the center with a value of 100, the outside one a value of 5.

    You keep track (statistics) of your scores. Assume that you shoot 5 times, scoring 20, 20, 45, 75, and 90.

    In statistical terms, each shot is an observation and each score is the value of that observation. Understood?

    STUDENT: Yes. In the example, we have 5 observations, with the respective values being 20, 20, 45, 75, and 90.

    TEACHER: Exactly. Now, the range is the difference in value from the highest to the lowest one. What would be the range in our example?

    STUDENT: In our example the range is 70, because 90-20=70.

    TEACHER: Correct. Now let me tell you that your average or mean score is the sum of the values (20+20+45+75+90 = 250) divided by the number of observations (5). What is your average (or mean) score?

    STUDENT: Not exactly rocket science, Teacher. Since the average (mean) in this example is 250 divided into 5, my mean score is 50.

    TEACHER: You are good at computing! And now, I’ll add that your median score is 45, because you have exactly the same number of scores (2) above it (observations 75 and 90) and below it (observations 20 and 20).

    STUDENT: Let me recap. My range is 70, my average or mean is 50, and the median is 45. But I also frequently hear the word mode in statistics.

    TEACHER: Mode means "fashion" in Latin, indicating something that is common, frequent. In the example, your most frequent score was 20 (you scored it twice); therefore, your mode score was 20.

    Mean (average), Median and Mode are measures of central tendency.

    There are also other measures that express values of dispersion, the "spread" of the values of the observations.

    Your shooting scores (observations) in the example showed a lot of dispersion, with a range from of 70 (20 to 90). You could have scored the same average (50) with less dispersion by shooting 5 times again and scoring say, 40 40 45 60 65.

    STUDENT: Evidently the value of the average is not an indication of dispersion.

    TEACHER: It is not. But the range we mentioned before is one measure of dispersion (the spread or narrowness of the distribution). While in the first example (scoring 20, 20, 45, 75, and 90) your range was 70, in your second scoring (40, 40, 45, 60, 65) your range was 25, the difference (distance) from the lowest score of 40 to the highest of 65.

    STUDENT: Obviously, in the second case I had a lower "dispersion".

    TEACHER: Sure. Other measures of dispersion are:

    * The variance: "the average of the squared distances from the mean to the individual observations".
    * The standard deviation (square root of the variance).

    Standard deviation is a measure of uncertainty: the narrower the spread, the more certainty we have.

    Now let me ask you a question: if the values of the observations were 2, 4, 6, 8, and 10, what would the value of the variance be?

    STUDENT: Let me see.

    The variance is "the average of the squared distances from the mean to the individual observations".

    * Obviously the mean is 6 because the observations are 2, 4, 6, 8, and 10; the mean is the sum of the observations (30) divided into the number of observations (5), equals 6.
    * In the example the values of the distances of the observations from the mean (6) are 6-2=4, 6-4=2, 6-6=0, 8-6=2 and 10-6=4.
    * The respective squares are 16, 4, 0, 4, 16 ; sum total of the squares is 40.
    * The average of the squares is calculated this way: sum total of 40 divided into 5 observations , equals 8 (the variance).

    TEACHER: Very well. And what about the standard deviation?

    STUDENT: Since the standard deviation is the square root of the variance, the value is 2.8284 (approximate square root of 8).

    TEACHER: Correct. Now let’s discuss....

    Probability Analysis

    We said before that probability analysis is the tool used to evaluate the chance that something might occur.

    A simple example: if you throw a dice, you have one chance (possibility) in six to get a given number (say, a 5).

    So, the probability of your rolling a 5 in a single throw, is one in six. One divided into six: 1/6= 0.1666 (We can express it as a probability of 16.66 percent).

    In this case the distribution of the probabilities for each number is even, because you have exactly the same chance of getting any number between 1 and 6. This is a uniform distribution.

    STUDENT: True for dice throwing, but I guess that in general, probability distributions are not even.

    TEACHER: True. But it is also true that in many instances (but certainly not all) the distributions are similar; there is something called a normal distribution. A graphic display of a normal distribution looks like a bell: most occurrences are around the center, with less and less away from the center. The standard deviation is low (a narrow spread).

    STUDENT: Example, please?

    TEACHER: One example of normal distribution is the height of people. Let's assume that you go to a cocktail party where there are 100 people. Chances are that most people present are close to average height; only a few will be very tall or very short.

    Now, if you were to enter the party with your eyes closed and open them after you began talking to any one of the persons present, there is a high probability that he or she would be close to the average height of the population. It is highly improbable that when you opened your eyes you’d see that you were talking to a very short or a very tall person. So, what is the conclusion, Student?

    STUDENT: The conclusion is that in a normal distribution, since the standard deviation is very low (a narrow spread), the certainty of a given outcome is high (in the example, that the first person I talk to after entering with my eyes closed happens to be very close to average height).

    TEACHER: Very well. And now let’s change the subject and talk about...

    The time value of money

    Intuitively we all know that receiving money now is better than receiving it in the future. More rationally, we can make specific calculations.

    If we can get a bank to pay us 10% interest a year on a deposit of $1,000, we would have $1,100 a year from now. It follows that, arithmetically, the present value of $1,100 to be received a year from now is $1,000. Why is that, Student?

    STUDENT: Because in both cases, we would end up with $1,100 a year from today.

    TEACHER: Right. The calculation of the present value of money to be received in the future is necessarily related to the interest rate taken as basis of the calculation. What would be the present value of $1,100 received a year from now if the interest rate were only 5% instead of 10%?

    STUDENT: Considering that today we’d need to deposit $1,047.62 at 5% interest rate in order to receive $1,100 a year from now... the present value of $1,100 to be received after one year if the current interest rate is 5%... amounts to $1,047.62!

    TEACHER: Correct. Five percent of $1,047.62 is $52.38; after one year you’d have in the bank your original capital of $1,047.62 plus the interest of $52.38: a total of $1,100. This example helps us to notice that the lower the prevalent interest rate, the higher the present value of a given sum to be collected at a given time.

    Now let’s talk about Investment Project Evaluation (IPE).

    STUDENT: Are there different systems for IPE?

    TEACHER: Yes, but we will discuss the discounted cash flow method, which is the one most frequently used. Applying the concept of the present value of money to be received in the future to project evaluation, we have a useful tool for decisions.

    As I said, the most popular one is the discounted cash flow method (DCF), also called the time adjusted return method.

    The DCF method consists in calculating at which interest rate the aggregate present value of future income equals the initial investment.

    The value we are looking for is that particular interest rate; when we find it, we call it the internal rate of return.

    A simple example to illustrate the concept:

    By trial and error we have found that 20% is the discount rate at which the present value of an income of $1,200 (that is, $1,200 to flow in -to be received) a year from now, equals the cash outflow of $1,000 today.

    STUDENT: Let me see: the present value at 20% of $1,200 to flow in a year from now is $1,000. Since this sum equals the initial outflow of $1,000, it follows that the internal rate of return is 20%.

    TEACHER: Exactly. By comparing the respective internal rates of return we may choose rationally among alternative projects. Of course, this information must also be compared with the probability that the estimated inflows actually will occur.

    If we deposited the $1,000 in the bank for one year we would end up with $1,080, an internal rate of return of only 8%. Obviously, it's better to invest the $1,000 in Project X, which gives is a return of 20%. But the question is, Student: can we be sure of this conclusion?

    STUDENT: Well, at the bank the future inflow of $1,080 is practically a certainty. On the other hand, the future inflow of $1,200 with Project X may not be so certain.

    TEACHER: Very well. What if in Project X there is a 25% probability of the inflow being just $1,000, a 0% rate of return? Which alternative would you choose?

    STUDENT: Teacher, I am only a student. I don’t know!

    TEACHER: Well, this is what managers were put on the earth for: to make tough decisions!


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