Introduction

This is an exercise to understand more about the potential performance of a stock, its risk and its relationship to the market. We will use the histogram, descriptive statistics, including some assessment of the reliability of these statistics, and (if there is time) discuss bootstrapping and Monte Carlo simulation.

We would like to know what sort of performance we can expect from a firm’s share price. We could apply this technique to any financial asset. We use the past as a sample of the whole population that includes the future. Like any sample, it must be representative. This usually means getting as much data as possible.

1. Data download

  1. Go to The Bloomberg Terminal and download as much daily price data for Bank of America as you can.

  2. Combine the data with that for the S&P 500 stock market index.

  3. Explain what a stock market index is and how it is calculated.

2. Data distribution

If we have a representative sample it will show the likelihood of various outcomes. A histogram will allow us to view these potential outcomes and get an idea of what to expect.

  1. Create two histograms to compare Bank of America returns with S&P 500 returns. What do you see?

  2. Calculate the daily mean, median, standard deviation, maximum, minimum, skew and kurtosis for the daily returns of Bank of America and the S&P 500 index. What does it tell you about investment in Bank of American and the S&P 500 index?

  3. Given that this is a sample, how would we expect these numbers to change if we were using different samples? Try a different sample.

3. Market risk

The beta will assess the amount of market risk that a stock contains.

  1. Run a regression of Bank of America returns relative to those of the market.

  2. Assess the results and their implications for an investment in Bank of America.

4. Random walk

If the market is efficient, it will react to each new item. If there is good news, it will go up; if there is bad news, it will go down. This means that stocks should move as a random walk.

\[P_t = P_{t-1} + \varepsilon_t\]

where \(P_t\) is the price level at time t and \(\varepsilon_t\) is a random vari able.

  1. What is a random variable? What does it represent in this case?

  2. Calculate the 95% confident intervals for your estimate of the mean daily return. What does this tell you about performance?

  3. Calculate the covariance and correlation between the returns for successive days. What does the Efficient Market Hypothesis suggest would be the result? What do you find?

5. Monte Carlo simulation and boostrapping

A Monte Carlo simulation will attempt to simulate the performance of a variable based on some parameters of a random variable. In our case, these stock returns are a random variable. Bootstrapping will use an existing sample to get more samples. Draw randomly from your sample to get a new sample and calculate new descriptive statistics.

  1. Generate 10 Monte Carlo simulations of Bank of America stock price. What is the 1st decile performance and what is the 9th decile performance? Compare simulated BAC to the real thing, what do you notice? How do you explain the difference?

  2. Create 10 samples of Bank of America returns from your existing sample. Re-calculate the key descriptive statistics and assess the ninety percent confidence intervals.