# A Statistical Approach To Price Targets

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Includes: MSFT
by: Michele Manganelli

## Summary

Price Target vs Fair Value.

Theoretical foundation of stock simulation: the Law of Large Numbers.

How to build an effective model.

Analysis of Microsoft Corporation.

The importance of Fair Value.

In this article I will give a new prospective on Price Targets through the introduction of a statistically based approach for targets projections, compared with the traditional Earnings-based calculations. After a theoretical introduction and the overview of the model's implications, I will show its application to Microsoft Corp. (NASDAQ:MSFT)

What is a Price Target and its differences with Fair Value

The price target for a stock is the projected future price level over a certain period. Its main purpose is to help investors to make better decision by setting a concrete investment objective and by helping to evaluate the potential risk/reward for each stock. Usually price targets are a 1-year projection of current price, mainly based on Earnings and multiples forecasts.

An important distinction should be done between Price Target and Intrinsic Value. While targets are usually earnings-dependent, the value of a stock is a function of its future cash-flow generating ability and, sometimes, we can notice evident discrepancies between an analyst price target and the current fair value of the stock. These differences derive from two main aspects:

1. The way the two values are calculated (fair value usually implies some sort of DCF analysis)
2. The time frame: price targets are generally calculated with a one-year time horizon, while the intrinsic value is derived by discounting future cash flows over a longer period

Stock Simulation: Price Targets & The Law of Large Numbers

The most common way to calculate targets is by relying on stock's fundamentals such as EPS forecasts and average price multiples. Instead, the stock simulation approach is based on historical data analysis in order to come up with reasonable estimates.

The conceptual difference is quite evident: through market simulations, what we are trying to achieve is the modeling of human behavior utilizing a data-intensive method. The theoretical foundation of this approach is the Law of Large Numbers (LLN), which states that the average results obtained from a large enough number of trials, should be close to the Expected Value.

In its formulation, the Strong LLN states that the sample average converges almost surely to the Expected Value

with

According to the Dow Theory, the market moves in trends, which are a visual representation of human behavior for a particular asset class. A trend has two main components that constitute the core of stock price simulation:

1. The trend direction: it depends on how strong the upward or downward pressure is, and it is determined by market returns.
2. The trend volatility: according to the Efficient Market Hypothesis, the greater the volatility is, the higher the returns are

However, high volatility levels have a negative impact on stock returns: the idea is that volatility erodes market returns over the long-term. We can think of volatility as a double-edged sword: it provides the possibility to obtain higher capital gains, but at the same time it erodes long-term returns by reducing the arithmetic average return to a geometric average.

Build an effective model

The first step to create an effective model, is to understand how the law of continuous compounding affects stock prices. The amount function for compound interest is an exponential function

When n tends to infinity we can apply the limit for the Euler's number e

Continuous compounding can be seen as making the compounding period infinitesimally small by solving the limit as n goes to infinity.

We can model the stock price according to the law of continuous compounding

In order to define the stock's future growth rate, we need to consider two assumptions:

1. Stock returns are eroded by volatility over the long term
2. Stock returns follow a Normal distribution

The rate of return can be written as the sum of two components

• The first component is known as the Stochastic Drift: the average growth rate is diminished by half-variance
• The second component shows that volatility follows a Normal distribution, where x is random number between 0 and 1, and Norm.inv represents the Normal deviate for x

By generating random x values, we can simulate future price levels according to the continuous compounding model. In order to come up with a one-year simulated price, we assume 252 trading days in a Year and we calculate the simulated price for each trading day.

The second step is to repeat the process over a large number of trials (in Microsoft Corp. analysis, I will utilize a 100000 trials simulation). The arithmetic average of these n prices will represent the final one-year price target.

Microsoft Corporation : a concrete example

In order to understand the theory presented, I will apply the simulation analysis to the well known Microsoft Corp. and I will calculate its one-year price target according to the above model. After gathering 10-year worth of adjusted daily closes, we come up with the following values for Microsoft:

The starting price for the stock is set at \$64.5. We then have to simulate the stock price for the next 252 trading days (one trading year)

By repeating this process for n=100000 trials, we calculate the arithmetic average of all the simulated prices and we come up with a one-year price target of \$69.60.

Here, I plot the price target distribution for Microsoft:

Starting from our final data, we can also derive the probability that the returns will assume certain values by studying the Cumulative Probability graph

For instance, we may calculate the probability that the investment in Microsoft Corp. will produce a positive return over the next year. For the stock, this probability is equal to:

The importance of a catalyst

The statistical approach to price target does not take in account any EPS projection or analyst estimates. In order to have a wider view on the stock, I suggest to implement the Fair Value in your analysis: I typically select stocks that are trading under their Fair Value and have a long-term upside potential and only then I apply a statistical stock analysis to have an idea of a possible one-year scenario.

Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.