Risk analysis

QUESTION 1: COMPUTING VAR AND ES
Let us consider a portfolio having a current value of 100 $. The P&L of the portfolio over the next week is
described by the following discrete random variable
-5 -4 -3 -2 -1 0 1 2 3 4 5
0.03 0.05 0.08 0.1 0.12 0.25 0.15 0.1 0.06 0.04 0.02
Table 1: Top line: possible realizations of the P&L. Bottom line: corresponding probabilities
Given the above information, compute:
The 1 week VaR at 90% confidence level.
The 1 week ES at 90% confidence level.
The 1 week VaR at 95% confidence level.
The 1 week ES at 95 % confidence level.
Explain in detail how to use Monte Carlo simulation (or any analytical method you are aware) to generate
the P&L distribution at 5 weeks horizon (you can assume that the weekly P&L are i.i.d.: i.e. the P&L over
the second week is independent on the P&L over the first week and in addition they have the same distribution
given in the Table above). Thereafter, compute the 5 weeks VaR and ES at 95% confidence level.
QUESTION 2: IS VAR A COHERENT RISK MEASURE?
1. Prove that the VaR satisfies the homogeneity property.
2. Prove that the VaR satisfies the translation invariance property.
3. Suppose we write the following two binary options:
option A pays $10,000 if the monthly return on the S&P 500 index is at least 20%;
option B pays $10,000 if the monthly return on gold is at least 20%.
Both options are sold for $1000. We assume that the returns on the S&P 500 and gold are independent
and that each has a probability of 0.025321 of returning at least 20%. Show that the sum of the 95%
VaRs on each separate position is less than the 95% VaR when the two options are taken together in a
portfolio.
QUESTION 3: COMPUTING AND BACKTESTING VAR
Data ara available via Moodle and cover the period January 2012 to December 2014.
You are required to forecast the risk of a portfolio based on a handful of US stocks (actually just three:
Microsoft, Apple and Intel) using the top down approach.
On the 2nd of January 2013 you had a capital of 1000 USD that you invested in the above mentioned
stocks (quoted at 25.91, 74.64 and 19.79), investing an equal amount of dollars in each stock. Thereafter,
you never adjusted the portfolio composition.
You are required to produce, at the end of each day, starting on 2nd of January 2013, a daily Value at
Risk (VaR) forecasts for the following day. Last VaR estimation is due on 3oth December 2014.
The main points that you have to address are:
1. Provide a preliminary statistical analysis concerning the time series of the portfolio return. A minimal
analysis requires the production of the empirical pdf versus the Gaussian one, Analysis of the QQPlot
and of autocorrelation of returns and squared returns. All these results need to be commented and your
findings supported by opportune statistical tests.
2. Using a rolling window (on 2nd of January 2013 you can use all the data of the previous year), you have
to estimate the VaR at 90%, and 99% confidence levels at the end of each day for the following day using:
the parametric Gaussian VaR: the parameters of the Gaussian distribution must be re-estimated
at the end of each day using sample moments (or other procedurs that you believe to be more
appropriate);
the parametric Student-t VaR: the parameters of the Student t distribution must be re-estimated at
the end of each day using the method of moments (or other procedurs that you believe to be more
appropriate);
the empirical quantile.
3. Given your VaR forecasts at the different confidence levels, you have to produce a Table showing for
each model and each confidence level the number of VaR violations.
4. If you perform the Kupiec test (at significance level of 95%), is the number of violations as large as
expected? Produce a Table showing for each combination of model and confidence level the value of the
likelihood ratio and your decision about the model accuracy. Which are the limits of the Kupiec test?
5. Do you see (if yes provide details on how you have detected it) any serial dependences in the occurence of
violations? Perform a conditional coverage test according to the procedure described in the Christoffersen
book.
6. On 31st December 2014 you have bought a long at-the-money put option on your portfolio. You can
price the option by using the Black-Scholes model, assuming as underlying the portfolio and as volatility
the annualized historical volatility estimated using the last 60 days. The option strike is set at 1750 $. In
addition, assume zero interest rate and zero dividend yield. The option will expiry in 30 days. Simulate
the portfolio value at the 5 days horizon, assuming that daily portfolio returns evolve according to the
EWMA model with 7» ˆ¶ 0.95. Take as starting value of variance the EWMA model the square of the
historical volatility estimated using the previous 60 days. In each simulated scenario, reprice the option
(that will have now 25 days to expire) according to the Black-Scholes model and generate the P&L
distribution of the portfolio, of the put option and of the two positions together. Then compute the 95%
VaR and expected shortfall of the single components (i.e. portfolio and option) and of the two’s together.
Discuss your findings
QUESTION 4: VAR OF A BOND
Let us consider a coupon bond with 3 years to maturity, semi-annual coupons, notional coupon at 4% (given on
a yearly basis), notional of 100$. The current price of the bond is 98 $. Compute the yield to maturity of the
bond (assume annually compounding).
Assume that the absolute daily change in the yield to maturity (ytm) is a Gaussian random variable with
mean 0 and daily volatility of 1%:
w M) -y<r> ˆ¼ N(0%,<1%)2)
(here ˆ† ˆ¶ 1 / 250). Also assume that daily changes in the ytm are i.i.d. random variables.
Provide an expression for the 10 days VaR of the bond.
QUESTION 5: SIMULATING RETURNS
Let us assume that daily log-returns are iid and drawn from the following double exponential density i.e. the
probability density function is given by
3 en if x < 0
froc) ˆ’ E X{ (27137) if x2 0
1. Explain carefully how you would simulate 10 days returns.
2. Simulate 1000 10-days returns according.
3. Out of the simulated returns compute VaR and expected shortfall.
QUESTION 6: VAR IN THE BINOMIAL MODEL
The stock price can move up or down by a factor It ˆ¶ 1.1 and d ˆ¶ 0.9 (so if today the stock is worth 100 in
one period time it will be worth 110 or 90) respectively with probability 0.55 and 0.45, (these are real-world
probability not risk-neutral ones).
You buy a portfolio of options:
long put option with strike at 80;
long call option with a strike of 120.
All the options will expiry in 20 periods (one period corresponds to 1 month).
Find the no-arbitrage price of the two options and then determine the 12 months probability distributions
of the profit and loss of your portfolio and then the corresponding 90% VaR and Expected Shortfall.