6  Worksheet 06: Prying open our imperfect model, Part 3

Before proceeding, make sure that you have finished the preceding worksheets.

In the previous worksheet, you were introduced to some of the inner workings of a normal random variable and the whys and hows of obtaining probabilities from such a normal random variable using traditional statistical tables, built-in R commands, and a rudimentary form of Monte Carlo simulation. Based on the previous worksheet, you should be able to trace how the normality assumption on log returns can be used to answer question about how future log returns could behave provided we believe the normality assumption.

In this worksheet, you are going to rely on a big convenience provided by the normality assumption so that we could answer questions about how log returns in the far future could behave. You will also get a chance to explore what happens if you lose normality.

Reiterating what was mentioned in the previous worksheets: We are prying open bits and pieces of a model of asset returns because it is an important step toward building in reflection as part of the model-building process.

6.1 Looking further into the future

Consider Exercise 1(b) in Section 2.5 of Ruppert and Matteson (2015) and the related exercise in the previous worksheet.

Before solving the exercise, determine first how these exercises are similar and different from each other.

Suppose that the daily log returns on a stock are independent and normally distributed with mean \(0.001\) and standard deviation \(0.015\). Suppose you buy \(\$1,000\) worth of this stock. Your task now is to find the probability that after one five trading days your investment is worth less than \(\$990\).

How will your calculations in the previous worksheet change? You can explore using the next exercise.

6.2 Exercise A

  1. Take \(\$1,000\) to be \(P_0\). What is \(P_1\) in terms of \(R_1\)? What is \(P_2\) in terms of \(P_1\) and \(R_2\)? Now, express \(P_2\) in terms of \(P_0\), \(R_1\), and \(R_2\).

  2. Extend the idea in the previous item until you get to \(P_5\). How would you interpret \(P_5\)? Why stop at \(P_5\)?

  3. Why could we not say \(P_5=990\)? Why is it useful to think of \(P_5\) as being a random variable?

  4. Express the desired probability in symbols as the probability that \(P_5\) will take values less than 990.

  5. Express the probability in terms of \(r_1, \ldots, r_5\).

6.3 The convenience that independence and normality provides

The bottom line then is to determine the distribution of \(r_1+\cdots+r_5\) in order to compute the desired probability.

In Appendix B Section B-5c of Wooldridge (2020), you will see a list of additional properties of the normal distribution.

A good point of analysis and reflection would be to determine which of the properties will apply given the setting described in Exercise 1(b) of Section 2.5 of Ruppert and Matteson (2015).

To obtain some progress on the main task, we need Property Normal.4 in Appendix B Section B-5c of Wooldridge (2020).

We should have a common understanding of what a linear combination of random variables would be. Let \(X_1,\ldots, X_n\) be \(n\) random variables. Let \(a_1,\ldots,a_n\) be constants (which could be known or unknown). A linear combination of random variables has the form \(a_1X_1+\cdots+a_nX_n\). Thus, a linear combination is formed by adding or subtracting random variables which are scaled by constants.

You should be able to show that the given information in the exercise matches the conditions of Property Normal.4. As you may have noticed, the combination of independence and normality provided a big convenience.

6.4 Exercise B

  1. If you apply Property Normal.4 as stated, what would be the distribution of \(r_1+\cdots+r_5\)? Does Property Normal.4 state anything about the parameters of that distribution? Explain.

  2. Why should we find the parameters of the distribution of \(r_1+\cdots+r_5\)? What would be the parameters we should be looking for?

  3. To find these parameters, you have to find \(\mathbb{E}\left(r_1+\cdots+r_5\right)\) and \(\mathsf{Var}\left(r_1+\cdots+r_5\right)\).

    1. Make sure that you can explain why.
    2. Now, refer to Property E.3. Check the conditions required so that you can apply the property to find \(\mathbb{E}\left(r_1+\cdots+r_5\right)\).
    3. Would independence of the log returns play a role in Property E.3?
    4. Would normality of the distribution of log returns play a role in Property E.3?

  4. Next, refer to Property VAR.4 along with the succeeding discussion. Notice that the property relies on pairwise uncorrelated random variables.

    1. Are \(r_1,\ldots, r_5\) pairiwse uncorrelated random variables? Explain why.
    2. Check the conditions required so that you can apply the property to find \(\mathsf{Var}\left(r_1+\cdots+r_5\right)\).
    3. Would independence of the log returns play a role in Property VAR.4?
    4. Would normality of the distribution of log returns play a role in Property VAR.4?

  5. At this stage, do you now have complete information to find the desired probability? If you do, compute its value using statistical tables and using pnorm(). If you do not, figure out what additional information would be needed, state these, and then proceed with the calculations.

  6. Reflect on how the assumptions on \(r_1, \ldots, r_5\) is useful in finding the desired probability.

    1. Determine at which points of the reasoning where independence would be needed.
    2. Determine at which points of the reasoning where identical normal distributions would be needed.
    3. Do you think independence and identical normal distributions are really needed in the calculations? Discuss.

6.5 Why do we have to use log returns?

Recall that the log return \(r_t\) of an asset held from time \(t-1\) to time \(t\) is given by \[r_t=\log\left(1+R_t\right).\] The logarithm is calculated with respect to base \(e\). How can this can be converted into percentage terms with a time period attached to this return?

Suppose you have a log return of \(-1\). From the definition of log return, what would be the associated value of the simple return? With some algebra and the definition of base-\(e\) logarithms, you should be able to find a simple return of about \(-63\%\). Clearly, log returns can be different from simple returns. So why use them?

6.6 Exercise C

  1. Suppose you have a simple return of \(-1\%\), what would be the log return? Are they reasonably close? What if the simple return were \(-10\%\)? \(-20\%\)? \(-30\%\)? \(10\%\)? \(20\%\)?
  2. You can visualize the previous item by plotting the functions \(f(x)=x\) and \(g(x)=\log(1+x)\). Explain why these are the relevant functions.
  3. Run the R commands curve(1*x, from = -1, to = 1, lty = 2, ylab = "") and curve(log(1+x), from = -1, to = 1, ylab = "", add = TRUE). Explain how the resulting graphs are related to the previous items. What conclusion can you come to about log returns and their connection to simple returns?

6.7 Exercise D

The previous exercise provides a justification for using log returns from an approximation perspective. Approximations may or may not be very good, but for a relevant range for simple returns, log returns are a good approximation.

Unfortunately, knowing this is not enough for justifying why we impose distributional assumptions on log returns. The remaining piece of the answer is related to something you have seen in Section 6.2.

  1. What would be \(\log P_1\) in terms of \(P_0\) and \(r_1\)?
  2. What about \(\log P_5\)?
  3. Using some algebra, compute \(\log\left(P_1/P_0\right)\) and \(\log\left(P_5/P_0\right)\). What do you notice?
  4. Compare with \(P_1/P_0\) and \(P_5/P_0\). Had we assumed normality of the simple returns rather than the log returns, discuss how you would be able to calculate the desired probabilities for Exercise 1 of Ruppert and Matteson (2015)?

6.8 Can we ditch normality?

You will find out later on that the normality assumption on log returns may not be supported by the data. Interestingly, even if normality of log returns is not supported by the data, it is still used as a baseline for financial models. You should be able to be convinced by now why it is such a convenient assumption and what happens when someone uses or applies ideas that are too convenient.

As a first step toward removing normality, we will preserve the independence assumption for now but change the distribution of log returns to a distribution you encountered before.

Let log returns at every trading day take on three possible values \(-1\), \(0\), and \(1\), with corresponding probabilities \(0.1\), \(0.8\), and \(0.1\).

6.9 Exercise E

To understand how to draw random numbers from this distribution, refer to the following lines of R code:

logret <- sample(x = c(-1, 0, 1), 
       size = 100, 
       prob = c(0.1, 0.8, 0.1), 
       replace = TRUE)
table(logret)
mean(logret == -1)
  1. How are the results in table(logret) connected to prob = c(0.1, 0.8, 0.1)?
  2. How are the results in mean(logret == -1) connected to prob = c(0.1, 0.8, 0.1)?
  3. Change size = 100 to size = 10^4. Compare with the results to your findings for size = 100?
  4. What happens if you change replace = TRUE to replace = FALSE? replace = TRUE
  5. How is independence incorporated into the random number generation here?

6.10 Exercise F

Suppose that the daily log returns on a stock are independent and normally distributed with mean 0.001 and standard deviation 0.015 follows the distribution described in Section 6.9. Suppose you buy \(\$1,000\) worth of this stock.

  1. Your task now is to find the probability that after one trading day your investment is worth less than \(\$990\). Revisit your calculations in Section 5.13. Which parts of your calculation will change? Can you proceed with the calculation of the desired probability? If you can, do it. If you cannot, what additional information would be needed and compute the desired probability given that additional information. Take note that if you supply additional information, it should not conflict with the given information.

  2. Your task now is to find the probability that after five trading days your investment is worth less than \(\$990\). Revisit your calculations in Section 6.4.

    1. Would the distribution of \(r_1+\cdots+r_5\) be the same as the distribution of any of the individual log returns \(r_1,\ldots, r_5\)? Explain why or why not.
    2. To find \(\mathbb{P}\left(r_1+\cdots+r_5 <\log(0.99) \right)\), which probabilities from the distribution of \(r_1+\cdots+r_5\) would be relevant for the calculation?
    3. Can you find these probabilities by hand? Show how you produced these calculations. Can you think of step-by-step instructions to use Monte Carlo simulation in this case?

6.11 Exercise G

You are now going to work on the following lines of R code:

temp <- replicate(10^4, sample(x = c(-1,0,1), 
       size = 5, 
       prob = c(0.1,0.8,0.1), 
       replace = TRUE))
sums <- colSums(temp)
table(sums)
mean(sums < log(0.99))
  1. Adjust 10^4 in the replicate() command to a smaller number so that you can figure out what exactly that line is doing.
  2. What do you think is sums <- colSums(temp) for? What does it shed light on?
  3. What information is being provided by the remaining commands? How are they relevant for finding the desired probability?
  4. Reflect on how Monte Carlo simulation was helpful in finding the desired probability. What is the price we have to pay for using this method instead of using the distribution of \(r_1+\cdots+r_5\) directly? Discuss with a classmate.

6.12 Stocks for the long run

Work with a classmate and explore why many have recommended to buy stocks and to hold them for very long periods of time. This strategy for stocks can be explored for the distributions considered in this worksheet. How would you and your classmate evaluate this strategy? Discuss your results with each other.

References

Ruppert, David, and David S. Matteson. 2015. Statistics and Data Analysis for Financial Engineering: With r Examples. 2nd ed. Springer Texts in Statistics. Springer. https://doi.org/10.1007/978-1-4939-2614-5.
Wooldridge, Jeffrey M. 2020. Introductory Econometrics: A Modern Approach. 7th edition. Cengage Learning.