Risk Analysis and Modelling (FR2208)

Coursework

Available online: Thursday 24 February 2022 Submission deadline: Friday 11 March 2022

Preface and practicalities

This coursework is about modelling and analysing the risk(s) of an equity strategy. You will be asked to perform several analyses using real-world data and present your results in a concise report. There is a total of 10 questions in this coursework, each carrying equal weight, and the maximum mark attainable is 100%. The choice of software for conducting the analyses is R(Excel is allowed if you believe it is necessary). Useful R-code is provided on page 3.

This coursework is to be solved strictly within the assigned groups. Questions can be posted on the accompanying Moodle forum. Private conversations about the coursework with anyone besides i) the members of your coursework group or ii) the lecturer will be regarded as fraud and treated as such. Any signs of conversations about the coursework across groups or copying of results across groups will result in a total mark of 0% to all members of the involved groups.

Please make specific references to the appropriate sections/pages in the module’s textbook and/or the lecture notes (‘slides’) whenever possible and adequate. The same applies to any other material (other textbooks, academic papers, newspaper articles, etc.) you use in answering the questions. Rely on dubious/folklore-based sources like Investopedia at your own risk/peril.

Some questions will ask you to comment on or interpret your results using no more than a specified maximum word count. You are welcome to use any number of words up to the maximum in your answer. However, if you use more words than the specified maximum, the mark for the entire question will be reduced by the percentage of words above the maximum allowed. Please indicate the number of words used in your answer to each question with a maximum word count adjacent to your answer of the question. Any discrepancies between the stated and the actual number of words will ber egarded as an error and will result in deduction of marks.

Please submit the coursework in a single PDF fi le. Other file formats (such as Word) will not be accepted as they do not allow for direct marking on Moodle. The maximum page count for the report itself (excluding submission forms, front pages, references, etc.) is 10 single-spaced standard pages using a font size of 12 or above. You are not required to attach appendices etc. to your report document containing code or spreadsheet screenshots etc., but you may do so if you wish to. Please indicate the group number on your report’s front page, the names and student numbers of all group members, and (if applicable) the names of any inactive or non-participating group members. Late submissions will result in a total mark of 0% to all group members.

Coursework setting

We will model and analyse the risks of an equity strategy in the context of the price momentum (MOM) anomaly: Stocks that have had relatively high returns over the previous year (i.e., the previous year’s winners) have historically outperformed (i.e., generated higher average subsequent returns than) stocks that have had relatively low returns over the previous year (i.e., the previous year’s losers) in a manner not explained by standard factor asset pricing models like the CAPM or Fama and French’s (1993) three-factor model (FF3).

Accompanying this coursework is a data file called “MOM and FF3 factors, 1969-2019.” The data file contains the monthly value (Value) in $ millions and the continuously compounded monthly return (Return) in decimals per month (so that, e.g., 0.01 means 1% per month) for a MOM strategy over the period from the end of December 1969 to the end of November 2019 (i.e., 50 years of monthly data). It also contains the monthly returns in decimals for the US stock market index in excess of the 1-month US Treasury bill rate (MKT) as well as Fama and French’s (1993) size (SMB) and value (HML) factors.

The MOM strategy buys a diversified portfolio of winner stocks and short-sells a diversified portfolio of loser stocks, but it does so within each of the small-stock and big-stock universes in the US. Specifically, at the end of each month, all US stocks are assigned to one of two portfolios, labeled small and big, based on their total market capitalisation, where the breakpoint is the median market capitalisation. At the same time, all US stocks are assigned to one of three portfolio, labeled loser, neutral, and winner, based on their cumulative stock return over the previous year, where the breakpoints are the 30th and 70th percentiles of the distribution. The portfolios are market capitalisation-weighted and rebalanced at the end of each month. The intersection gives a total of 6 portfolios; 3 within each of the two size universes:

Cumulative stock return over the previous year (30% loser, 40% neutral, 30% winner)

Size (50% small, 50% big) Small loser Small neutral Small winner

Big loser Big neutral Big winner

Within each size universe, we construct a long-short strategy that buys the winner portfolio and shorts the loser portfolio. The MOM strategy is then the simple average of the long-short strategies from each of the two size universes. Constructing the long-short strategies separately within the size universes and then averaging helps to make the MOM strategy size-neutral (i.e., it does not have a particular size tilt towards either small or big stocks).

The initial investment (the strategy’s starting value) is 1.0000 (in $ millions). The monthly strategy values cover the period from the end of December 1969 to the end of November 2019, while the monthly (continuously compounded) strategy returns cover the period from the end of January 1970 to the end of November 2019.

Useful R-code

If you are interested in learning to program and do statistical analysis in R, then this coursework is an excellent opportunity to do so. Below is a very short introduction with useful function that are relevant to the coursework and beyond.

Importing csv-files in R can be done using the function read.csv. To import a csv-file and store it as a data.frame with the name ‘D’, use the command D – read.csv(“_path_”) where _path_ is the path of the csv-file on your computer. You can obtain the path of any file by dragging the file into the R-console. To access the variable (or column) X within the data.frame D, use the command D$X. To access the first entry for the variable X within the data.frame D, use the command D$X[1]. To access the last entry for the variable X within the data.frame D, use the command D$X[nrow(D)].

To access all but the first entry for the variable X within the data.frame D, use the command D$X[-1]. Finally, to specify the variable X within the data.frame D as a date-variable, use the command D$X – as.Date(D$X).

R has numerous built in functions for computing standard statistics and manipulating variables like mean, var, sd, sum, prod, length, exp, log (log is the natural logarithm in R, which we have denoted ln in the lecture notes), cumsum (i.e., a cumulative sum), and so on. All of these functions have arguments to exclude missing (or NA = not available) values, typically in the form of na.rm=T (i.e., setting the function’s remove NA-argument to TRUE). For instance, to calculate the mean of X from data.frame D while removing its missing values simply type mean(D$X, na.rm=T). To define a new variable M as the mean of X from the data.frame D, while removing the missing values in X before calculating its mean, use the command M – mean(D$X, na.rm=T).

Univariate and multivariate linear regressions can be done using the function lm. Getting detailed output for linear regressions (i.e., t-statistics for the intercept as well as slope-coefficients, residual standard error, R-squared values, etc.) can be prompted by using the function summary in conjunction with the function lm. For instance, to get detailed output for a linear regression of the variable Y from the data.frame D onto the variables X and Z also from the data.frame D, use summary(lm(D$Y ~ D$X + D$Z)).

You can read more about the R-functions read.csv, mean, and lm, as well as see examples of how to use them, by prompting their help-page using the commands ?read.csv, ?mean, and ?lm. The same applies to all R-functions, which all have an accompanying help-page with examples. Furthermore, there are many web forums and blogs specifically targeting R-users where you can find code examples and clever solutions to common problems. The easiest way to find these is to do a Google search for the task you are trying to accomplish or the problem you are experiencing initiated with by “R:” as in “R: doing a for-loop” or “R: adding a line to a plot.”

Questions

Question 1 (10%)

a) Estimate and report the ML parameters, measured per annum, of a BSM model fitted to the strategy. In doing so, be very clear about the estimation procedure you apply, including any distributional assumptions, the definition of the sampling times, the sampling frequency, and what the resulting estimates are measured in. Using no more than 100 words, interpret the estimates.

b) Use the estimated BSM parameters to calculate the monthly expected value of the strategy at each date (i.e., month end) in the sample period conditional on its value at the beginning of the sample period. Report the expected value of the strategy for

i) 31 December 1970, ii) 31 December 1994, iii) 31 November 2019.

c) Make a plot with the date (month-ends) on the x-axis and the strategy’s value on the y-axis. Add the expected monthly value as a function of the date to the plot and report the resulting plot. Finally, using no more than 100 words, describe and interpret the plot.

Question 2 (10%)

Suppose you are currently standing at the end of the sample period.

a) Calculate and report the expectation and standard deviation of the value in 12 months.

b) Determine and report the distribution of the continuously compounded annual return over the next month and over the next 12 months. Using no more than 50 words, comment on and explain any similarities and/or differences between these two distributions.

c) Calculate and report the return, r, such that there is only a 1% probability that the continuously compounded annual return over the next 12 months is below r.

d) Calculate and report the value, V, such that there is only a 1% probability that the value in 12 months is below V.

Question 3 (10%)

If X is a normally distributed random variable with mean m and variance v2, then it holds for a given probability 0 q 1 that

When q = 0.05 = 5%, we have N–1(0.975) = 1.96 (using NORMSINV in Excel or using qnorm in R). There is therefore a 95% probability that X is between L = m – 1.96v and U = m + 1.96v, where L and U stand for the Lower and Upper bound, respectively. The interval from L to U is therefore called a 95% confidence interval for X.

Suppose you are currently standing at the end of the sample period.

a) Calculate and report a 95% confidence interval for the continuously compounded annual return over the next 12 months.

b) Calculate and report a 95% confidence interval for the strategy’s value in 12 months.

Question 4 (10%)

If X is a normally distributed random variable with mean m and variance v2, then the conditional expectation of X given that X is below some value x is

.

Also, if Y is a log-normally distributed random variable, such that X = ln Y is normally distributed with mean m and variance v2, then the conditional expectation of Y given that Y is below some value y 0 is

.

Suppose you are currently standing at the end of the sample period.

a) Calculate and report the conditional expectation of the continuously compounded annual return over the next 12 months given that it is below –30% per annum.

b) Calculate and report the conditional expectation of the strategy’s value in 12 months given that it is below $17 million.

Question 5 (10%)

Suppose you are currently standing at the end of the sample period and that the risk-free rate for a maturity of m months is m times the current 1-month risk-free rate.

a) Calculate the current price of an at-the-money European put option on the strategy with 6 months to expiry.

b) Calculate the total loss on the put, from the viewpoint of the seller, which has only a 1% probably of being exceeded.

Question 6 (10%)

Let be the strategy’s (continuously compounded) monthly return per month for month t

(specifically, from the end of month t-1 to the end of month t). Recall that the strategy is long-short,

i.e., its return is the difference between the returns of two portfolio positions. According to the Fama and French three-factor (FF3) model, such a return can be described by the linear regression

MKTt + sSMBt + hHMLt +?t,

where ??t has mean zero and standard deviation ???? for all t.

a) Using no more than 50 words, explain why we do not subtract the risk-free rate from the strategy’s return (i.e., from the left-hand side) in the above regression.

b) Estimate and report the linear regression parameters of a FF3 model fitted to the strategy along with the model’s R2-value. Also, report the t-test statistics for the estimated intercept and slope.

c) Using no more than 100 words, describe and interpret these results.

Question 7 (10%)

A prime application of factor asset pricing models like the CAPM or FF3 is the fact that they, in principle, provide a recipe for optimally hedging a particular asset or strategy against systematic risk(s) [or, more agnostically, against the common movements in stocks captured by the factor(s)]. We will now study the performance of a hedged version of the long-short strategy, where the hedging is done according to the FF3 model fitted in Question 6.

According to the FF3 model, the hedged strategy’s (continuously compounded) monthly return per month for month t is given by

MKTt sSMBt hHMLt.

That is, we hedge the strategy by taking offsetting positions in the MKT, SMB, and HML factors, where the size of each offsetting position is the negative of the strategy’s estimated loading (slope) on the corresponding factor.

a) Prove that another expression for the hedged strategy’s (continuously compounded) monthly return per month for month t is given by

.

Using no more than 200 words, interpret this expression and discuss its implications.

b) Calculate the hedged strategy’s continuously compounded monthly return per month for each month-end in the sample period. Report it for:

i) 31 December 1970, ii) 31 December 1994, iii) 31 November 2019.

c) Given rth, the hedged strategy’s value at the end of any month t 0 is

Sth = Sth 1erth.

Since also Sth 1 = Sth 2erth 1, we can write the above as

.

Continuing this recursively, we get

,

i.e., the hedged strategy’s value at the end of month t 0 is the initial investment, , continuously compounded forward to month t 0 at a rate equal to the cumulative sum of the hedged strategy’s monthly continuously compounded return per month up to month t 0.

Calculate the monthly value of the hedged strategy for each month-end in the sample period starting with an initial investment of 1 (in $ millions). Report the value of the hedged strategy for:

i) 31 December 1970, ii) 31 December 1994, iii) 31 November 2019.

Add the value of the hedged strategy as a function of the date to the plot you made in Question 1 and report the resulting plot (adjust the axes as you see fit). Finally, using no more than 100 words, describe and interpret the plot.

Question 8 (10%)

Consider again the hedged strategy defined in Question 7.

a) Suppose that ??t is normally distributed with mean zero and standard deviation ???? for all t.

Determine and report the distribution of the hedged strategy’s (continuously compounded) monthly return per month.

b) Report the ML parameters, measured per annum, of a BSM model fitted to the hedged strategy. Using no more than 100 words, compare the estimates for the hedged strategy with the ones for the unhedged strategy from Question 1, and discuss the implications.

c) Use the estimated BSM parameters from Question 8b) to calculate the monthly expected value of the hedged strategy at each date (month-end) in the sample period conditional on its value at the beginning of the sample period. Add this expected monthly value as a function of the date to the plot you made in Question 7c) and report the plot (adjust the axes as you see fit). Also,

report the expected value of the hedged strategy on

i) 31 December 1970, ii) 31 December 1994, iii) 31 November 2019.

Finally, using no more than 100 words, describe and interpret the plot.

Question 9 (10%)

A commonly used measure of a strategy’s downside risk is the so-called maximum drawdown, which can be defined as the maximum loss between a strategy’s most recent peak value (i.e., its latest highest value) to a subsequent valley (i.e., its latest low). Specifically, a strategy’s drawdown (DD) at time t can be defined as its current percentage loss relative to its most recent peak value, i.e.,

Vt VtPeak

DDt = VtPeak ,

where Vt is the strategy’s value at time t and VtPeak is the strategy’s most recent peak value. From this, the strategy’s maximum drawdown (MDD) at time t is defined as the latest maximum of the negative of DD up to time t, i.e.,

MDDt =max{ DDs}.

s t

a) Calculate the peak, drawdown, and maximum drawdown for each of the unhedged and hedged strategies at each date (month-end) during the sample period. Make a plot with the date (monthends) on the x-axis and the unhedged strategy’s monthly maximum drawdown on the y-axis. Add the hedged strategy’s monthly maximum drawdown as a function of the date to the plot in a different colour and report the resulting plot. Report the peak, drawdown, and maximum drawdown for each of the unhedged and hedged strategies on the following dates:

i) 31 December 1970, ii) 31 December 1994, iii) 31 November 2019.

b) Using no more than 100 words, describe and interpret the plot and results.

Question 10 (10%)

Using no more than 200 words, conclude on the performance of the hedged vs. unhedged strategy.

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