The normal distribution is the probability distribution that plots all of its values in a symmetrical fashion with most of the results situated around the probability's mean.

## Normal (Bell Curve) Distribution

Data sets (like the height of 100 humans, marks obtained by 45 pupils in a class, etc.) tend to have many values at the same data point or within the same range. This distribution of data points is called the normal or bell curve distribution.

For example, in a group of 100 individuals, 10 may be below 5 feet tall, 65 may stand between 5 and 5.5 feet and 25 may be above 5.5 feet. This range-bound distribution can be plotted as follows:

Similarly, data points plotted in graphs for any given data set may resemble different types of distributions. Three of the most common are left aligned, right aligned and jumbled distributions:

Note the red trendline in each of these graphs. This roughly indicates the data distribution trend. The first, “LEFT Aligned Distribution,” indicates that a majority of the data points falls in the lower range. In the second “RIGHT Aligned Distribution” graph, the majority of data points fall in the higher end of the range, while the last, “Jumbled Distribution,” represents a mixed data set without any clear trend.

There are a lot of cases wherein the distribution of data points tends to be around a central value, and that graph shows a perfect normal distribution—equally balanced on both sides, with the highest number of data points concentrated in the center.

Here is a perfect, normally distributed data set:

The central value here is 50 (which has the most number of data points), and distribution tapers off uniformly toward extreme end values of 0 and 100 (which have the fewest number of data points). The normal distribution is symmetrical around the central value with half the values on each side.

A lot of real-life examples fit the bell curve distribution:

- Toss a fair coin many times (say 100 times or more) and you will get a balanced normal distribution of heads and tails.
- Roll a pair of fair dice many times (say 100 times or more) and the result will be a balanced, normal distribution centered around the number 7 and uniformly tapering towards extreme-end values of 2 and 12.
- The height of individuals in a group of considerable size and marks obtained by people in a class both follow normal patterns of distribution.
- In finance, changes in the log values
of forex rates, price indices, and stock prices are assumed to be normally distributed.

## Risk and Returns

Any investment has two aspects: risk and return. Investors look for the lowest possible risk for the highest possible return. The normal distribution quantifies these two aspects by the mean for returns and standard deviation for risk.

### Mean or Expected Value

A particular mean change of a share's price could be 1.5% on a daily basis—meaning that, on average, it goes up by 1.5%. This mean value or expected value signifying return can be arrived at by calculating the average on a large enough dataset containing historical daily price changes of that stock. The higher the mean, the better.

### Standard Deviation

Standard deviation indicates the amount by which values deviate on average from the mean. The higher the standard deviation, the riskier the investment, as it leads to more uncertainty.

Here is a graphical representation of the same:

Hence, the graphical representation of normal distribution through its mean and standard deviation enables the representation of both returns and risk within a clearly defined range.

It helps to know (and be assured with certainty) that if some data set follows the normal distribution pattern, its mean will enable us to know what returns to expect, and its standard deviation will enable us to know that around 68% of the values will be within 1 standard deviation, 95% within 2 standard deviations and 99% of values will fall within 3 standard deviations. A dataset which has a mean of 1.5 and standard deviation of 1 is much riskier than another dataset having a mean of 1.5 and a standard deviation of 0.1.

Knowing these values for each selected asset (i.e. stocks, bonds, and funds) will make an investor aware of the expected returns and risks.

It’s easy to apply this concept and represent the risk and return on one single stock, bond or fund. But can this be extended to a portfolio of multiple assets?

Individuals start trading by buying a single stock or bond or investing in a mutual fund. Gradually, they tend to increase their holdings and buy multiple stocks, funds or other assets, thereby creating a portfolio. In this incremental scenario, individuals build their portfolios without a strategy or much forethought. Professional fund managers, traders and market-makers follow a systematic method to build their portfolio using a mathematical approach called modern portfolio theory (MPT) that is founded on the concept of “normal distribution.”

## Modern Portfolio Theory

Modern portfolio theory (MPT) offers a systematic mathematical approach which aims to maximize a portfolio’s expected return for a given amount of portfolio risk by selecting the proportions of various assets. Alternately, it also offers to minimize risk for a given level of expected return.

To achieve this objective, the assets to be included in the portfolio should not be selected solely based on their own individual merit but instead on how each asset will perform relative to the other assets in the portfolio.

In a nutshell, MPT defines how to best achieve portfolio diversification for the best possible results: maximum returns for an acceptable level of risk or minimal risk for a desired level of returns.

## The Building Blocks

The MPT was such a revolutionary concept when it was introduced that its inventors won a Noble Prize. This theory successfully provided a mathematical formula to guide diversification in investing.

Diversification is a risk management technique, which removes the “all eggs in one basket” risk by investing in non-correlated stocks, sectors, or asset classes. Ideally, the positive performance of one asset in the portfolio will cancel the negative performance of other assets.

To take the average return of the portfolio that has *n* different assets, the proportion-weighted combination of the constituent assets' returns is calculated.

Due to the nature of statistical calculations and normal distribution, the overall portfolio return (R_{p}) is calculated as:

$R_p=\sum{w_iR_i}$

The sum (∑), where w_{i }is the proportionate weight of asset i in the portfolio, R_{i }is the return (mean) of asset i.

The portfolio risk (or standard deviation) is a function of the correlations of the included assets, for all asset pairs (with respect to each other in the pair).

Due to the nature of statistical calculations and normal distribution, the overall portfolio risk (Std-dev)_{p} is calculated as:

$\begin{aligned} &\left(Std-dev\right)_p=\\ &sqrt\left[\sum_i\sum_j{w_i}{w_j}\left(std-dev\right)_i\left(std-dev\right)_j\left(cor-cof_{ij}\right)\right]\\ \end{aligned}$

Here, cor-cof is the correlation coefficient between returns of assets i and j, and sqrt is the square-root.

This takes care of the relative performance of each asset with respect to the other.

Although this appears mathematically complex, the simple concept applied here includes not just the standard deviations of individual assets, but also the related ones with respect to each other.

A good example is available here from the University of Washington.

## A Quick Example of MPT

As a thought experiment, let's imagine we are a portfolio manager who has been given capital and is tasked with how much capital should be allocated to two available assets (A & B) so that the expected return is maximized and risk is lowered.

We also have the following values available:

R_{a} = 0.175

R_{b} = 0.055

(Std-dev)_{a} = 0.258

(Std-dev)_{b} = 0.115

(Std-dev)_{ab} = -0.004875

(Cor-cof)_{ab} = -0.164

Starting with equal 50-50 allocation to each asset A & B, the R_{p} calculates to 0.115 and (Std-dev)_{p} comes to 0.1323. A simple comparison tells us that for this 2 asset portfolio, return as well as risk is midway between individual values of each asset.

However, our aim is to improve the return of the portfolio beyond the mere average of either individual asset and reduce the risk, so that it is lower than that of the individual assets.

Let’s now take a 1.5 capital allocation position in asset A, and a -0.5 capital allocation position in asset B. (Negative capital allocation means shorting that stock and capital received is used to buy the surplus of the other asset with positive capital allocation. In other words, we are shorting stock B for 0.5 times of capital and using that money to buy stock A for amount 1.5 times of capital.)

Using these values, we get R_{p} as 0.1604 and (Std-dev)_{p} as 0.4005.

Similarly, we can continue to use different allocation weights to asset A & B, and arrive at different sets of Rp and (Std-dev)p. According to the desired return (Rp), one can choose the most acceptable risk level (std-dev)p. Alternately, for the desired risk level, one can select the best available portfolio return. Either way, through this mathematical model of portfolio theory, it is possible to meet the objective of creating an efficient portfolio with the desired risk and return combination.

The use of automated tools allows one to easily and smoothly detect the best possible allocated proportions easily, without any need for lengthy manual calculations.

The efficient frontier, the Capital Asset Pricing Model (CAPM) and asset pricing using MPT also evolve from the same normal distribution model and are an extension to MPT.

## Challenges to MPT (and Underlying Normal Distribution)

Unfortunately, no mathematical model is perfect and each has inadequacies and limitations.

The basic assumption that stock price returns follow normal distribution itself is questioned time and again. There is sufficient empirical proof of instances where values fail to adhere to the assumed normal distribution. Basing complex models on such assumptions may lead to results with large deviations.

Going further into MPT, the calculations and assumptions about correlation coefficient and covariance remaining fixed (based on historical data) may not necessarily hold true for future expected values. For example, the bond and stock markets showed a perfect correlation in the UK market from 2001 to 2004 period, where returns from both assets went down simultaneously. In reality, the reverse has been observed over long historical periods prior to 2001.

Investor behavior is not taken into consideration in this mathematical model. Taxes and transaction costs are neglected, even though fractional capital allocation and the possibility of shorting assets is assumed.

In reality, none of these assumptions may hold true, which means realized financial returns may differ significantly from expected profits.

## The Bottom Line

Mathematical models provide a good mechanism to quantify some variables with single, trackable numbers. But due to the limitations of assumptions, models may fail.

The normal distribution, which forms the basis of portfolio theory, may not necessarily apply to stocks and other financial asset price patterns. Portfolio theory in itself has lots of assumptions which should be critically examined, before making important financial decisions.