Overview

The task you are given is to estimate the market risk for a hypothetical holding of 10,000 NAB shares, held on August 1, 2016 (you are working out the risk position assuming that you own these shares at the open of trading that day).  You will do this by estimating the Value-at-Risk for the stock using historical data.  You will then analyse two hedging strategies using derivatives that will alter the risk of holding these shares.

Description

You will be asked to calculate the following;

•  10 day VaR for the portfolio of NAB shares at confidence levels of 99%.

Note: This risk estimate applies to the 10 days from August 1, 2016 until August 12, 2016 (i.e. – it should be a forecast of risk).

Based on what you have learnt from EFB344, you realized that you have several options for how to compute this risk measure.  You are considering using a VaR based on a) the normal distribution based on a 252 day rolling window, b) the normal distribution with a variance estimated by the EWMA, or c) historical simulation based on a window of 252 days. 

To aid in the decision of which to use, you are going to consider the recent historical performance of the three models in calculating 1 day VaR at the confidence levels of 99%.  You will do so by first examining the frequency of instances when the VaR was exceeded by the observed return using the last five years of data (e.g. – from 1/08/2011 to 29/07/2016). 

 

Next, you will look at the possibility of beta hedging the portfolio of NAB shares using S&P200 Index futures contracts.  To do this you will need to calculate the following

• The number of futures contracts required to eliminate beta from the portfolio.
• The 10 day VaR for the joint holding of the NAB shares and the S&P 200 index futures at a confidence level of 99%.

To undertake this part of the task, you will use the historical data obtained for NAB as well as historical data for the S&P 200 Index (ticker code AXJO).  To calculate the historical beta for NAB, you should use the formula[1]

βNAB=Cov(RetNAB,RetS&P200)σS&P2002

 

To calculate the components of this beta, you should use the method you selected as the best for calculating the VaR for NAB shares (i.e. – rolling window/historical simulation or EWMA).  Under the assumption that returns for NAB shares and the S&P200 Index are jointly normally distributed, you can then calculate the VaR.  To simplify the analysis, you can make two further assumptions.

1. There is no margin requirement.
2. Changes in the futures price are perfectly correlated with the spot price for the S&P200 Index over short horizons (e.g. – over 10 day windows).

 

Finally, you will look at the possibility of hedging your risk over the next three months using an American put option written on NAB shares that expires in exactly three months. 

 

The current stock price as of the close of trading is $26.54.  Your portfolio consists of 10,000 shares.   The risk-free rate is currently 2% p.a. continuously compounding.  The annual variance of NAB shares can be estimated from historical data.

 

1. Noting that each put option covers 100 shares, determine how many put options you would need to buy to hedge your risk.  Next, determine the appropriate price for each of these put options when the strike price if $26.  Use a six-step binomial tree for this task.
2. Plot a profit diagram that shows the profit from the stock holding, option holding and the combined portfolio (stocks and options) as a function of the share price at maturity (assume that you didn’t exercise the options early and you were able to buy them at the price from part a).

 

Presenting your results

• You will then write a brief report of no more than twelve pages including graphs that present your results.   A rough outline of what your report could include is
  • Present the final VaR(10, 99%) for August 1 based on your preferred model.  You must also include some explanation of how these final estimates should be interpreted (what does this VaR number mean).
  • A brief description of the final/preferred VaR model you used.  This includes the length of any subsamples of data or selection of any fixed parameters.
  • Presentation of the main results in the back-testing study.  The raw results should be accompanied with an evaluation of the relative performance of the various models and a clear justification of which model is superior. With this should be a brief description of data (start/end dates, number of observations and details of where you got the data, what the data was (e.g. – closing prices?), how the data was cleaned (e.g. -  non-trading days) and how you calculated returns.
  • A section on the beta hedging strategy.  Present the details of the hedge, how you have formed your estimates, what risk is being removed and what risk remains.
  • A section on the option hedging strategy.  Present the profit diagram representing the hedge, how you priced the option, what risk is being removed and what risk remains.

 

• You will also be asked to submit the excel file in which you conducted your analysis.  It should be formatted in a reasonably clear way, so that someone who was given the job after you was able to understand and replicate what you had done.
  • I would ask that you use multiple worksheet with the excel file.  The first should include the raw data (with dates) as given to you by your source.  All working and final results can be compiled on subsequent worksheets.

 

Additional Notes and Instructions

• You will need to source the data yourself.  I recommend using Yahoo Finance as was outlined in lecture 2.
• For the backtest, you need only start with five years of data when you begin your analysis.  This is does not mean that you will have five years of VaR numbers and exceedances to compare (you will have less than this).  This will make sense once you begin.
• Your excel spreadsheet should contain the formulas that you have used for all calculates (i.e. – don’t paste the values of the main calculations).  This does not apply to the steps you undertook to clean the data.