Last night, an angel came into my dreams. She has a very beautiful face, glittering eyes, and wore red attire. Her dark hairs make her exceedingly gorgeous. I was stunned by looking at her. She spoke in her soft voice, Aman, How are you? I am great. Thanks.
Angel: Why is there so much unhappiness these days?
Me: People are worried about money.
Angel: Does money buy happiness?
Me: Not having money will make our life miserable but having money is not going to make us happy. Money is indispensable only to satisfy the basic needs of our life.
Angel: Are people really making smart decisions about investing their money?
Me: Human beings are not as rational as they think about themselves. We are oblivious that our brain is taking shortcuts to reach a conclusion that makes us flawed thinkers.
Angel: Are you a rational thinker?
Me: No, I am not. Many times I fall in the trap of various psychological biases.
Angel: Let's play a game to check your decision-making heuristics.
Me: Go ahead.
Angel: You have two devices namely A and B out of which you can pick only one. You can insert any money into it in the beginning and hold the device for the next 20 years. Device A gives an annual return of 14% per annum whereas Device B gives a 40% return for half the years and -10% return for another half years. Which one will you pick?
(Don't call the police to complain about the devices, It's just a thought experiment which is happening in my dreams.)
Me: Device A gives 14% per annum. Device B gives on average a 15% return per annum. How?
= (40% - 10%) / 2
= 30% / 2
= 15%
I have used the Arithmetic Mean to compute the average value. Device B gives a higher return. So, I will pick Device B.
Angel: Are you sure? She is mildly smiling.
Me: I can't figure out why is she smiling. I reached this conclusion through my logic. So, I stayed with my decision.
Angel: Rethink about this problem.
Me: Let's take 50,000 as the initial money.
Money given by Device A at the end of 20 years:
= 50,000 * (1.14)^20
= 6,87,174.
(Symbol ^ means power. Eg. 2^3 = 2* 2*2 = 8)
Money given by Device B at the end of 20 years:
= 50,000 * (1.40)^10 * (0.90)^10
= 5,04,284.
How is this possible?
According to my logic, Device B has to give high returns compared to Device A but here the opposite happens. I start thinking where is the flaw in my logic.
Angel: Do you know any other way to compute the average value?
Me: There are two ways to compute the average value.
Arithmetic Mean
Geometric Mean
Arithmetic Mean is the sum of N terms divided by the total number of terms.
A.M = (x1 + x2 + x3 + ...... + xn) / n
Geometric Mean is the product of N terms and derives the nth root.
G.M = (x1 * x2 * x3 * ...... * xn) ^ (1/n)
Angel: Why did you use Arithmetic Mean over Geometric Mean?
Me: Geometric Mean doesn't come to my mind.
Angel: Think about the problem from the standpoint of Geometric Mean.
Me: Device A gives a 14% return per annum.
Device B gives (1.40 * 0.90) ^ (1/2) = 12.24% return per annum.
Angel: Aman, Money doesn't grow linearly. It compounds exponentially. This is the flaw in your logic. In exponential situations, think geometrically not arithmetically. It gives you a more accurate picture.
Me: I know about Geometric mean than why doesn't it come into my mind?
Angel: Human beings are not evolved to think in terms of Square root, cube root, or Nth root.
Is there any relation between Arithmetic Mean and Geometric Mean?
Me: Yes, there exists a relationship between them in the form of Arithmetic-Geometric Inequality.
Angel: What's that?
Me: Arithmetic-Geometric Mean inequality states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the Geometric Mean of the same list.
The Proof is simple:
~ (x - y)^2 >= 0
~ x^2 + y^2 - 2xy >= 0
~ x^2 + y^2 - 4xy + 2xy >= 0
~ x^2 + y^2 + 2xy >= 4xy
~ (x + y)^2 >= 4xy
~ (x + y) >= (4xy) ^ (1/2)
~ (x + y) / 2 >= (xy) ^ (1/2)
~ A.M >= G.M
Angel: Can you use it in this game?
Me: Oh! I see. Now, I started connecting the dots. A moment ago, you are smiling at my foolish logic. According to inequality, Arithmetic Mean always overestimates the money compared to the Geometric Mean. Hence, it nudges me in the wrong direction. So, I ended up picking the wrong option.
Thanks, Angel for showing me how to think geometrically. You have helped me to see the problems in a different light.
Don't look at me like this, Angel.
The sweet glances from your eyes make my heart jump. You are a perfect example of beauty with the brain.
On hearing my words, She smiles, blushed, and then vanished like a breeze of spring. The rays of sunshine and the soothing voices of birds awakened me from the sleep with a comprehensive view of averages.
Darrell Huff in How to Lie with Statistics explains:
Statistics is much an art as it is a science. A great many manipulations and even distortions are possible with the bounds of propriety. Often the statistician must choose among methods, a subjective process, and find the one that he will use to represent the facts.
Take the simplest example, let's say that milk cost Rs 100 a quart last year and bread was also Rs 100 a loaf. This year milk is down to Rs 50 and bread has gone up to Rs 200. Now, what would you like to prove? Cost of living up? Cost of living down? or No change?
Consider last year as the base period, making the prices of that time 100 percent. Since the price of milk has since dropped to half (50 percent) and the price of the bread has doubled (200 percent) and the average of 50 and 200 is 125, the price has gone up 25 percent.
To prove that the cost hasn't changed at all we simply switch to a geometric average. [...] Take last year as the base and call its price level 100. Actually, you multiply the 100 percent of each item together and take the root, which is 100. For this year, milk being at 50 percent of last year and bread at 200 percent, multiply 50 by 200 to get 10,000. The square root, which is the geometric average, is 100. Prices haven't gone up or down.
Learning is of no use unless we apply it in everyday life. Replace angel with the stock market and devices with various alternatives of stock can help us to use the above experiment in finance.
Key Takeaways that can be taken from the above thought experiment:
Emotions play an important role in decision making. Extreme emotions cloud our judgment. They drive us crazy by propelling us in irrational behavior. Emotional stability is one of the most important skills one needs to develop in his/her life to make more sensible decisions.
I don't want to be at the mercy of my emotions. I want to use them, enjoy them, and to dominate them. Oscar Wilde
Most of the time we use our logical thinking to reach a conclusion but the assumption we use to reach it is wrong. It is clearly visible in our thought experiment where we use erroneous assumptions like Arithmetic Mean to reach the conclusion. Rationality is generally served by broader and more comprehensive frames and joint evaluation is always better than a single evaluation. Using Geometric Mean and identifying the relation between Arithmetic Mean and Geometric Mean shows us the whole picture.
Always try to look at the problem from as many angles as possible. It broadens your framework of mind to make more reasonable decisions. Assume you are on a ship and talking with your friend. Your friend is at rest with respect to you but is in motion with respect to the fish who is looking at you from the sea. You say your friend is at rest but the fish says, No, it is in motion. According to your standpoints, both of you are right.
One must understand the difference between the desire to be right and the desire to have been right. Separating them will help us to see reality as it is, not the way we want it to be. The desire to be right is the thirst for truth. The desire to have been right is God complex which means an unshakable belief characterized by consistently inflated feelings of personal ability, privilege, or infallibility.
Conclusion: Have a strong opinion over your belief but hold them weakly. Don't be rigid in your thinking. Update your apriori beliefs according to the shreds of evidence. Identify the cause and effect relationships. Be more flexible and adopt the growth mindset thinking. A rational thinker will engage in broader framing but humans are narrow framers by nature. Look out for broader aspects, not the narrower ones.
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