Notes: This is an optional assignment that carries 5 “extra” points. The marking will be Good (complete or almost complete, 5 Points), Satisfactory (at least 75%, 2.5 points), not satisfactory (0).
- You are required to upload two files one showing your empirical work, and another explaining each question in detail.
- Assume you are a financial analyst and you are provided with the following questions by your employer and that you can get help for any online source but you cannot ask your employer. Please don’t ask me for any help as it will be unfair to others if I do provide any.
- This is an individual assignment, plagiarizing in the empirical work or writeups will be dealt with in accordance to University of Toronto guidelines.
Qestion-1: Use Yahoo Finance to download daily closing prices for Microsoft and Google for Year 2020.
- Calculate the returns for each stock. Graph the histogram of daily returns. Do the returns follow the normal distribution very well? Does the histogram have heavy tails?
- Are daily returns for each stock independent? (an easy way is to graph the scatter plot of the returns and make a visual inspection, you can also calculate the autocorrelation).
- Use File-1 (GBM) to simulate the weekly returns for year 2020. (Note, I have simulated the weekly prices of a stock with a mean of 10%, volatility of 30%, and initial price of 100)
Question-2: Google is listed S = 2000, with one-month ( = T – t = 1) options as below. Suppose that r = 2% (C. C.), and that = 30% per year, and consider the following options portfolio:
Number of shares (minus sign indicates option written)
Strike price Calls Puts
1950 –300 –200
2000 400 375
2150 0 350
- Find the net delta, gamma, and vega exposures of the portfolio?
- How many shares must be traded to achieve a delta-neutral position? What is the value of those shares?
- What will happen to the value of the options portfolio if the stock price immediately increases to 2050? What happens to the value of the shares you found in part (b)? What is the net gain/loss on the delta-hedged position?
- Now suppose that the portfolio owner decides to make the portfolio both delta and gamma neutral by modifying the number of shares as well as the number of 1200strike price call options in the portfolio. How many of these calls should be bought or sold to achieve a gamma-neutral position? What is the new value for the number of shares that should be bought or sold for delta-neutrality?
- What will be the change in the value of the portfolio in (d) if the stock price increases immediately from 2000 to 2100? Compare this change to your answer in (c) for the portfolio that was only delta-hedged.
- Repeat (e) for a stock price decrease from 2000 to 1900.
- Use my excel file “Gamma-hedge” to simulate your results.
Question-3 (VaR-1): This question is intended to help you understand how to estimate Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). In parts (a) through (e), please use the variance-covariance approach. In parts (f), you need to use the historical simulation approach. You will need to use your MSFT/GOOG dataset of the prior question.
- Using the variance-covariance approach, estimate the one-day VaR at the 99% confidence level for a $1,000,000 investment in MSFT What would be the one-day VaR for a $1,000,000 investment in GOOG?
- Compute the correlation coefficient between the returns on MSFT and GOOG.
- Using your estimates in parts (a) and (b), find the standard deviation of an equallyweighted portfolio that invests in MSFT and GOOG.
- Using the variance-covariance approach, estimate the one-day VaR at the 99% confidence level for a $1,000,000 investment in the equally-weighted portfolio.
- Repeat parts (a) and (d) using the historical simulation approach. You may use the “SORT” function under the DATA menu in Excel.
- Using the historical simulation approach, estimate: (i) the one-day CVaR for a $1,000,000 investment in MSFT; (ii) the one-day CVaR for a $1,000,000 investment
in GOOG; and (iii) the one-day CVaR for a $1,000,000 investment in the equallyweighted portfolio.
Question-4 (VaR-2): You are in charge of managing a portfolio for a Canadian Bank. The portfolio consists of three companies listed on TSX index:
|Stocks # of shares (Millions)
||Price (Feb 1, 2021)||Value||Weight|
|Bank of Montreal (BMO) 10||96.21||962.1||36.5%|
|Suncor (Su) 5||22.24||112.2||4.2%|
|Shopify (Shop) 1||1562||1562||59.3%|
The bank wishes to introduce Value at Risk (VaR) as one of its risk management tools and has asked you to investigate this using your portfolio as a test case with a view to introducing it across all bank investments.
You have decided to calculate the value at risk at a 99% confidence level over a 10-day period using two different methods:
- Historical simulation based on the data for the past two years.
- Monte Carlo simulation for the portfolio as a whole. (You can use GBM-2 to simulate a Geometric Brownian Motion process)
For the second method you know that you will need to calculate the volatility of the daily returns. You decided to use the volatilities from the actual data up to and including Feb 1 2021 for [calendar] periods of 1 quarter (= 3 months) and one year.
Prepare a report for the directors of the Bank evaluating the results of your tests and assessing the introduction of VaR across the bank as a whole.
Question-5 (Credit-Risk): You are holding 1 unit of a zero bond issued by XYZ. The face value and the recovery rate are respectively $100 and 40%. The term to maturity is 5 years. If XYZ defaults at any time in the coming five years, the recovery rate will be paid at the maturity day. The price of the bond is $60.
- If the bond is Ba-rated, find the probability of defaults using the table of historical default rate.
- Suppose a four-year zero coupon US treasury bond with face value of $100 is selling at $93, what would be the default probability of the junk bond implied from the market price?
- Using each of the probabilities obtained from a) & b) calculate the expected value and variance of your credit loss.
- (Ignore parts a to c) As a risk analyst, you are trying to estimate the C-VaR (Credit VaR) for a portfolio that includes two risky bonds. We define C-VaR as the maximum unexpected loss with a very high confidence (99.9%) in a rather short time (typically 1 month). Suppose each of the two bond are worth $1,000,000 one month forward, and that the one-year cumulative default probability is 4% for each of the two bonds. Find the CVaR of this portfolio assuming no default correlation and no zero recovery rate.
(Note: For this part is an application of finding default probability using bond prices)