When thinking of diversification, apart from thinking about the total number of investments (or eggs in the basket), it is also equally important to think about two additional aspects:
a) the probability of success vs failure (positive return vs permanent capital loss) and
b) the extent of any possible gains or losses (magnitude of the return vs magnitude of the loss)
A thought experiment will help us visualize the various possibilities and the best course of action for a prudent investor.
Assume that you have Rs 100 to invest.
First, we will examine the possibilities when this Rs 100 is invested in one investment, which has only two possible outcomes – a success with a 50% probability of occurrence or a failure also at a 50% probability of occurrence. We will also assume that in the event of success, the magnitude of success is 100% (you double your investment) and in the event of failure, the magnitude of failure is -100% (you lose your investment capital). With these assumptions, the outcomes are shown in Thought Experiment A1 (see table below).
The expected return is the average of a probability distribution of possible returns, calculated by using the following formula:
E(R)= Sum: probability (in scenario i) * the return (in scenario i)
We see that the Total Expected Return of Thought Experiment A1 is 0, with the expected returns of the individual outcomes being +50 (success) and -50 (failure).
We now extend this thought experiment with the same Rs 100 invested in 2 investments of Rs 50 each as a diversification strategy and call this as Thought Experiment A2. The probabilities of success and failure remain the same (50% each) and the magnitude of success and failure also remain the same for each investment as in Thought Experiment A1. The expected returns for this are posted under Thought Experiment A2 in the table above.
We see that the Total Expected Return in Thought Experiment A2 is is 0 (the same as in Thought Experiment A1). But the difference is in the expected returns of the individual outcomes. In 3 out of the 4 outcomes, no money is lost.
Taking Thought Experiment A2 one step further, we do Thought Experiment A3, with the same Rs 100 invested equally between 3 investments, each identical in the probabilities and magnitudes of success and failure. The outcomes are shown in Thought Experiment A3 in the table above.
We see that the Total Expected Return in Thought Experiment A3 is also 0. However there is a wider distribution in the expected returns of individual outcomes.
Analyzing the expected returns of the individual outcomes in each of the above thought experiments, we see that the main benefit of diversification is that in the event of all things going bad and the shit hitting the fan, the total expected losses are minimized. See the expected return in the case of all failures in the above 3 thought experiments. They go down from -50 in Thought Experiment A1 down to -25 in Thought Experiment A2 down to -12.5 in Thought Experiment A3. However, on the flipside, the total expected profits are also minimized in the event of everything working very well. So diversification serves as a way of attenuating the amplitude of profits and losses – with increasing diversification, you will not lose big but you also give up the upside of winning big.
The basic insight we get about diversification in investments is that the Total Expected Return does not change when you go from a concentrated portfolio to a more diversified portfolio, assuming that probabilities and magnitudes of successes and failures are identical in all the investment options. On average, you will get the same results with or without diversification – only the probability of doing extremely well or extremely badly is significantly minimized.
Let us now make things interesting – first by changing the probabilities of success vs failure from the boring 50%-50% to the more interesting 75%-25%. The same three thought experiments are conducted by investing Rs 100 equally among with 1, 2 and 3 investments as before. The expected returns are displayed in the below table under Thought Experiment B1, Thought Experiment B2 and Thought Experiment B3 respectively.
Notice the same insight that we got in the earlier ‘A’ series thought experiments. The Total Expected Return remains the same in all 3 cases of 1, 2 and 3 investments. However see the distribution of expected returns among the individual outcomes. Due to the geometric (or multiplicative) nature of probabilities (where if the probability of event X occurring is 25% and probability of event Y occurring is 25%, and X and Y are independent events, then the probability of both X and Y occurring together is 25% x 25% = 6.25%. If the probability of event X occurring is 75% and the probability of event Y occurring is 75% and X and Y are independent events, then the probability of both X and Y occurring together is 75% x 75% = 56.25%. You can see how this can work to your advantage – the worst case expected return is only -6.25% while the best case expected return is a completely asymmetric +56.25%), we see that we are able to significantly reduce the total losses in the event of complete failures, come out okay in the event of part failure and part success, and do fairly well in the event of total success.
This ‘B’ series of thought experiments is the logic behind the investment strategy of Benjamin Graham. He ensured that by having a high margin of safety in individual investments, his probability of success was much higher than the probability of failure of each individual investment. And by adding diversification into the mix and holding a collection of such ‘high probability of success’ investments, his downside was drastically minimized while he still had decent upside in the event that everything went well.
Let us now venture down a third series of thought experiments, which we will call the ‘C’ series. Here, we will go back to a 50% probability of success and a 50% probability of failure. But we will change the magnitude of success to 200% (you triple your initial investment). We will keep the magnitude of failure the same as before (-100%, or you lose your investment) since the magnitude of the loss in a majority of investment opportunities tends to be limited to the total size of the position/investment, except in a few cases such as short sales etc.
The results are presented in the table below for Thought Experiment C1 (one investment), Thought Experiment C2 (two investments) and Thought Experiment C3 (three investments).
In the above case, you are betting on the magnitude of success in each investment (how big you can win if things work out well) and not necessarily at the odds of things working out well.
But you may well wonder, and rightly so, why one should not try to take advantage of both – get great odds on successful outcomes as well as win really big in successful outcomes. The results of thought experiments along these lines are presented in the table below, under Thought Experiment BC1, Thought Experiment BC2, Thought Experiment BC3 and Thought Experiment BC4.
This combination of great odds and huge wins underscores Warren Buffett/Charlie Munger’s investment thesis, and is the key difference between the Graham and the Buffett investment philosophies.
While Ben Graham ensured that he had a high probability of success by ensuring a margin of safety in each of his investments and then let diversification protect him from downside, he did not believe it to be worth the time and effort in understanding the dynamics of the underlying business, thus ignoring the potential magnitude of the success. Warren Buffett and Charlie Munger, in addition to ensuring a margin of safety, also tried to look for businesses with extremely strong franchises and economics. The quality of the business was as important as the margin of safety.
Thought Experiment BC4 is identical to Thought Experiment BC3, except that the magnitude of success is 300% in the former case (quadruple your investment) versus 200% (triple your investment) in the latter. See how changing this one variable changes the expected returns of the individual outcomes, significantly enhancing the returns across most of the individual outcomes.
Of course, in the real world, investing is an art where you are no longer conducting thought experiments and where the only variable you have control over is the number of investments in your portfolio. The probability of success/failure of each investment and the magnitude of success/failure of each investment are both completely uncertain, unpredictable and keep changing with time. Also, in most real life investment opportunities, the odds of failure are much, much higher than the odds of success, due to the fact that any one of a number of things can go wrong and prevent success, but all things have to go right together to enable success. Thus, finding an investment where the probability of success is greater than the probability of failure is a very rare occurrence. Finding multiple such investments is an extremely low probability event, as we have just learnt, due to the geometric nature of probabilities. Our endeavour as prudent investors should be to try to create a portfolio resembling Thought Experiment BC4, with a high probability of success and a high magnitude of success for each investment. But realize that this is very, very difficult, and the odds are stacked against you in finding multiple such investments. You might have to spend a lifetime of patient searching and waiting for right opportunities to come along before you can have a portfolio of such great investments.
While I have very neatly and conveniently slotted Graham in Thought Experiment ‘B’ series and Buffett/Munger in Thought Experiments ‘BC’ series, the reality is of course that their investing styles were not uniform at all times and also evolved over time. For instance, Buffett started off emulating Graham at the beginning, but then diverged in time and with Munger’s association. The objective of this note is to provoke investors to think about the nature of probabilities, and how diversification to a limited extent can be helpful. Of course, if you go for more rampant diversification, say with 20 investments each with a 75% success probability, then the probability of the most successful outcome occurring (all 20 successes) is a very miniscule 0.32%. The art is in determining the right balance where not too much of the upside is given away in the pursuit of eliminating downside risk.
This note was inspired by the Risk Profile table considering investment alternatives in this paper.
a) the probability of success vs failure (positive return vs permanent capital loss) and
b) the extent of any possible gains or losses (magnitude of the return vs magnitude of the loss)
A thought experiment will help us visualize the various possibilities and the best course of action for a prudent investor.
Assume that you have Rs 100 to invest.
First, we will examine the possibilities when this Rs 100 is invested in one investment, which has only two possible outcomes – a success with a 50% probability of occurrence or a failure also at a 50% probability of occurrence. We will also assume that in the event of success, the magnitude of success is 100% (you double your investment) and in the event of failure, the magnitude of failure is -100% (you lose your investment capital). With these assumptions, the outcomes are shown in Thought Experiment A1 (see table below).
The expected return is the average of a probability distribution of possible returns, calculated by using the following formula:
E(R)= Sum: probability (in scenario i) * the return (in scenario i)
We see that the Total Expected Return of Thought Experiment A1 is 0, with the expected returns of the individual outcomes being +50 (success) and -50 (failure).
Thought experiment No A1
| |||||
Investment amount
|
100
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
1
|
Case 1
|
1 success
|
50
| |
Probability of success
|
50%
|
Case 2
|
1 failure
|
-50
| |
Probability of failure
|
50%
|
Total Expected return
|
0
| ||
Magnitude of success
|
100%
| ||||
Magnitude of failure
|
-100%
| ||||
Thought experiment No A2
| |||||
Investment amount
|
50
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
2
|
Case 1
|
2 successes
|
25
| |
Probability of success of each investment
|
50%
|
Case 2
|
1 success, 1 failure
|
0
| |
Probability of failure of each investment
|
50%
|
Case 3
|
1 failure, 1 success
|
0
| |
Magnitude of success of each investment
|
100%
|
Case 4
|
2 failures
|
-25
| |
Magnitude of failure of each investment
|
-100%
|
Total Expected return
|
0
| ||
Thought experiment No A3
| |||||
Investment amount
|
33.33
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
3
|
Case 1
|
3 successes
|
12.50
| |
Probability of success of each investment
|
50%
|
Case 2
|
1 success, 1 success, 1 failure
|
4.17
| |
Probability of failure of each investment
|
50%
|
Case 3
|
1 success, 1 failure, 1 success
|
4.17
| |
Magnitude of success of each investment
|
100%
|
Case 4
|
1 failure, 1 success, 1 success
|
4.17
| |
Magnitude of failure of each investment
|
-100%
|
Case 5
|
1 success, 1 failure, 1 failure
|
-4.17
| |
Case 6
|
1 failure, 1 success, 1 failure
|
-4.17
| |||
Case 7
|
1 failure, 1 failure, 1 success
|
-4.17
| |||
Case 8
|
3 failures
|
-12.50
| |||
Total Expected return
|
0
| ||||
We now extend this thought experiment with the same Rs 100 invested in 2 investments of Rs 50 each as a diversification strategy and call this as Thought Experiment A2. The probabilities of success and failure remain the same (50% each) and the magnitude of success and failure also remain the same for each investment as in Thought Experiment A1. The expected returns for this are posted under Thought Experiment A2 in the table above.
We see that the Total Expected Return in Thought Experiment A2 is is 0 (the same as in Thought Experiment A1). But the difference is in the expected returns of the individual outcomes. In 3 out of the 4 outcomes, no money is lost.
Taking Thought Experiment A2 one step further, we do Thought Experiment A3, with the same Rs 100 invested equally between 3 investments, each identical in the probabilities and magnitudes of success and failure. The outcomes are shown in Thought Experiment A3 in the table above.
We see that the Total Expected Return in Thought Experiment A3 is also 0. However there is a wider distribution in the expected returns of individual outcomes.
Analyzing the expected returns of the individual outcomes in each of the above thought experiments, we see that the main benefit of diversification is that in the event of all things going bad and the shit hitting the fan, the total expected losses are minimized. See the expected return in the case of all failures in the above 3 thought experiments. They go down from -50 in Thought Experiment A1 down to -25 in Thought Experiment A2 down to -12.5 in Thought Experiment A3. However, on the flipside, the total expected profits are also minimized in the event of everything working very well. So diversification serves as a way of attenuating the amplitude of profits and losses – with increasing diversification, you will not lose big but you also give up the upside of winning big.
The basic insight we get about diversification in investments is that the Total Expected Return does not change when you go from a concentrated portfolio to a more diversified portfolio, assuming that probabilities and magnitudes of successes and failures are identical in all the investment options. On average, you will get the same results with or without diversification – only the probability of doing extremely well or extremely badly is significantly minimized.
Let us now make things interesting – first by changing the probabilities of success vs failure from the boring 50%-50% to the more interesting 75%-25%. The same three thought experiments are conducted by investing Rs 100 equally among with 1, 2 and 3 investments as before. The expected returns are displayed in the below table under Thought Experiment B1, Thought Experiment B2 and Thought Experiment B3 respectively.
Thought experiment No B1
| |||||
Investment amount
|
100
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
1
|
Case 1
|
1 success
|
75
| |
Probability of success
|
75%
|
Case 2
|
1 failure
|
-25
| |
Probability of failure
|
25%
|
Total Expected return
|
50
| ||
Magnitude of success
|
100%
| ||||
Magnitude of failure
|
-100%
| ||||
Thought experiment No B2
| |||||
Investment amount
|
50
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
2
|
Case 1
|
2 successes
|
56.25
| |
Probability of success of each investment
|
75%
|
Case 2
|
1 success, 1 failure
|
0
| |
Probability of failure of each investment
|
25%
|
Case 3
|
1 failure, 1 success
|
0
| |
Magnitude of success of each investment
|
100%
|
Case 4
|
2 failures
|
-6.25
| |
Magnitude of failure of each investment
|
-100%
|
Total Expected return
|
50
| ||
Thought experiment No B3
| |||||
Investment amount
|
33.33
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
3
|
Case 1
|
3 successes
|
42.18
| |
Probability of success of each investment
|
75%
|
Case 2
|
1 success, 1 success, 1 failure
|
4.69
| |
Probability of failure of each investment
|
25%
|
Case 3
|
1 success, 1 failure, 1 success
|
4.69
| |
Magnitude of success of each investment
|
100%
|
Case 4
|
1 failure, 1 success, 1 success
|
4.69
| |
Magnitude of failure of each investment
|
-100%
|
Case 5
|
1 success, 1 failure, 1 failure
|
-1.56
| |
Case 6
|
1 failure, 1 success, 1 failure
|
-1.56
| |||
Case 7
|
1 failure, 1 failure, 1 success
|
-1.56
| |||
Case 8
|
3 failures
|
-1.56
| |||
Total Expected return
|
50
| ||||
Notice the same insight that we got in the earlier ‘A’ series thought experiments. The Total Expected Return remains the same in all 3 cases of 1, 2 and 3 investments. However see the distribution of expected returns among the individual outcomes. Due to the geometric (or multiplicative) nature of probabilities (where if the probability of event X occurring is 25% and probability of event Y occurring is 25%, and X and Y are independent events, then the probability of both X and Y occurring together is 25% x 25% = 6.25%. If the probability of event X occurring is 75% and the probability of event Y occurring is 75% and X and Y are independent events, then the probability of both X and Y occurring together is 75% x 75% = 56.25%. You can see how this can work to your advantage – the worst case expected return is only -6.25% while the best case expected return is a completely asymmetric +56.25%), we see that we are able to significantly reduce the total losses in the event of complete failures, come out okay in the event of part failure and part success, and do fairly well in the event of total success.
This ‘B’ series of thought experiments is the logic behind the investment strategy of Benjamin Graham. He ensured that by having a high margin of safety in individual investments, his probability of success was much higher than the probability of failure of each individual investment. And by adding diversification into the mix and holding a collection of such ‘high probability of success’ investments, his downside was drastically minimized while he still had decent upside in the event that everything went well.
Let us now venture down a third series of thought experiments, which we will call the ‘C’ series. Here, we will go back to a 50% probability of success and a 50% probability of failure. But we will change the magnitude of success to 200% (you triple your initial investment). We will keep the magnitude of failure the same as before (-100%, or you lose your investment) since the magnitude of the loss in a majority of investment opportunities tends to be limited to the total size of the position/investment, except in a few cases such as short sales etc.
The results are presented in the table below for Thought Experiment C1 (one investment), Thought Experiment C2 (two investments) and Thought Experiment C3 (three investments).
Thought experiment No C1
| |||||
Investment amount
|
100
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
1
|
Case 1
|
1 success
|
100
| |
Probability of success
|
50%
|
Case 2
|
1 failure
|
-50
| |
Probability of failure
|
50%
|
Total Expected return
|
50
| ||
Magnitude of success
|
200%
| ||||
Magnitude of failure
|
-100%
| ||||
Thought experiment No C2
| |||||
Investment amount
|
50
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
2
|
Case 1
|
2 successes
|
50
| |
Probability of success of each investment
|
50%
|
Case 2
|
1 success, 1 failure
|
12.5
| |
Probability of failure of each investment
|
50%
|
Case 3
|
1 failure, 1 success
|
12.5
| |
Magnitude of success of each investment
|
200%
|
Case 4
|
2 failures
|
-25
| |
Magnitude of failure of each investment
|
-100%
|
Total Expected return
|
50
| ||
Thought experiment No C3
| |||||
Investment amount
|
33.33
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
3
|
Case 1
|
3 successes
|
25.00
| |
Probability of success of each investment
|
50%
|
Case 2
|
1 success, 1 success, 1 failure
|
12.50
| |
Probability of failure of each investment
|
50%
|
Case 3
|
1 success, 1 failure, 1 success
|
12.50
| |
Magnitude of success of each investment
|
200%
|
Case 4
|
1 failure, 1 success, 1 success
|
12.50
| |
Magnitude of failure of each investment
|
-100%
|
Case 5
|
1 success, 1 failure, 1 failure
|
0.00
| |
Case 6
|
1 failure, 1 success, 1 failure
|
0.00
| |||
Case 7
|
1 failure, 1 failure, 1 success
|
0.00
| |||
Case 8
|
3 failures
|
-12.50
| |||
Total Expected return
|
50
| ||||
In the above case, you are betting on the magnitude of success in each investment (how big you can win if things work out well) and not necessarily at the odds of things working out well.
But you may well wonder, and rightly so, why one should not try to take advantage of both – get great odds on successful outcomes as well as win really big in successful outcomes. The results of thought experiments along these lines are presented in the table below, under Thought Experiment BC1, Thought Experiment BC2, Thought Experiment BC3 and Thought Experiment BC4.
Thought experiment No BC1
| |||||
Investment amount
|
100
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
1
|
Case 1
|
1 success
|
150
| |
Probability of success
|
75%
|
Case 2
|
1 failure
|
-25
| |
Probability of failure
|
25%
|
Total Expected return
|
125
| ||
Magnitude of success
|
200%
| ||||
Magnitude of failure
|
-100%
| ||||
Thought experiment No BC2
| |||||
Investment amount
|
50
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
2
|
Case 1
|
2 successes
|
112.5
| |
Probability of success of each investment
|
75%
|
Case 2
|
1 success, 1 failure
|
9.375
| |
Probability of failure of each investment
|
25%
|
Case 3
|
1 failure, 1 success
|
9.375
| |
Magnitude of success of each investment
|
200%
|
Case 4
|
2 failures
|
-6.25
| |
Magnitude of failure of each investment
|
-100%
|
Total Expected return
|
125
| ||
Thought experiment No BC3
| |||||
Investment amount
|
33.33
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
3
|
Case 1
|
3 successes
|
84.37
| |
Probability of success of each investment
|
75%
|
Case 2
|
1 success, 1 success, 1 failure
|
14.06
| |
Probability of failure of each investment
|
25%
|
Case 3
|
1 success, 1 failure, 1 success
|
14.06
| |
Magnitude of success of each investment
|
200%
|
Case 4
|
1 failure, 1 success, 1 success
|
14.06
| |
Magnitude of failure of each investment
|
-100%
|
Case 5
|
1 success, 1 failure, 1 failure
|
0.00
| |
Case 6
|
1 failure, 1 success, 1 failure
|
0.00
| |||
Case 7
|
1 failure, 1 failure, 1 success
|
0.00
| |||
Case 8
|
3 failures
|
-1.56
| |||
Total Expected return
|
125
| ||||
Thought experiment No BC4
| |||||
Investment amount
|
33.33
|
S. No
|
Outcomes
|
Expected return
| |
No of investments
|
3
|
Case 1
|
3 successes
|
126.55
| |
Probability of success of each investment
|
75%
|
Case 2
|
1 success, 1 success, 1 failure
|
23.44
| |
Probability of failure of each investment
|
25%
|
Case 3
|
1 success, 1 failure, 1 success
|
23.44
| |
Magnitude of success of each investment
|
300%
|
Case 4
|
1 failure, 1 success, 1 success
|
23.44
| |
Magnitude of failure of each investment
|
-100%
|
Case 5
|
1 success, 1 failure, 1 failure
|
1.56
| |
Case 6
|
1 failure, 1 success, 1 failure
|
1.56
| |||
Case 7
|
1 failure, 1 failure, 1 success
|
1.56
| |||
Case 8
|
3 failures
|
-1.56
| |||
Total Expected return
|
200
| ||||
This combination of great odds and huge wins underscores Warren Buffett/Charlie Munger’s investment thesis, and is the key difference between the Graham and the Buffett investment philosophies.
While Ben Graham ensured that he had a high probability of success by ensuring a margin of safety in each of his investments and then let diversification protect him from downside, he did not believe it to be worth the time and effort in understanding the dynamics of the underlying business, thus ignoring the potential magnitude of the success. Warren Buffett and Charlie Munger, in addition to ensuring a margin of safety, also tried to look for businesses with extremely strong franchises and economics. The quality of the business was as important as the margin of safety.
Thought Experiment BC4 is identical to Thought Experiment BC3, except that the magnitude of success is 300% in the former case (quadruple your investment) versus 200% (triple your investment) in the latter. See how changing this one variable changes the expected returns of the individual outcomes, significantly enhancing the returns across most of the individual outcomes.
Of course, in the real world, investing is an art where you are no longer conducting thought experiments and where the only variable you have control over is the number of investments in your portfolio. The probability of success/failure of each investment and the magnitude of success/failure of each investment are both completely uncertain, unpredictable and keep changing with time. Also, in most real life investment opportunities, the odds of failure are much, much higher than the odds of success, due to the fact that any one of a number of things can go wrong and prevent success, but all things have to go right together to enable success. Thus, finding an investment where the probability of success is greater than the probability of failure is a very rare occurrence. Finding multiple such investments is an extremely low probability event, as we have just learnt, due to the geometric nature of probabilities. Our endeavour as prudent investors should be to try to create a portfolio resembling Thought Experiment BC4, with a high probability of success and a high magnitude of success for each investment. But realize that this is very, very difficult, and the odds are stacked against you in finding multiple such investments. You might have to spend a lifetime of patient searching and waiting for right opportunities to come along before you can have a portfolio of such great investments.
While I have very neatly and conveniently slotted Graham in Thought Experiment ‘B’ series and Buffett/Munger in Thought Experiments ‘BC’ series, the reality is of course that their investing styles were not uniform at all times and also evolved over time. For instance, Buffett started off emulating Graham at the beginning, but then diverged in time and with Munger’s association. The objective of this note is to provoke investors to think about the nature of probabilities, and how diversification to a limited extent can be helpful. Of course, if you go for more rampant diversification, say with 20 investments each with a 75% success probability, then the probability of the most successful outcome occurring (all 20 successes) is a very miniscule 0.32%. The art is in determining the right balance where not too much of the upside is given away in the pursuit of eliminating downside risk.
This note was inspired by the Risk Profile table considering investment alternatives in this paper.
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