113

Finance 320
Problem Set #6
1. Which two parameters completely specify a normal distribution? If the returns on a portfolio are normally distributed, then the following simple equation may be used to compute Value at Risk: VaR = zW. Explain what each of these three variables represents.
2. Your portfolio has a mean return of 15% and a standard deviation of 25%. What portfolio return is two standard deviations below the mean?
3. What are the probabilities that a normal random variable is further than zstandard deviations below its mean for values of zequal to 1.65, 1.96, and 2.33?
4. Explain the meaning of Value at Risk in terms of expected loss and the probability of such a loss.
5. Effigy Co. stock has a mean annual return of 15% and a standard deviation of 30%. What is the largest expected percentage loss in the coming year with a probability of 95%? What is the corresponding VaR for a $200,000 portfolio comprised solely of Effigy stock?
6. Your stock portfolio is currently worth $350,000. Your portfolio has an annual standard deviation of 20%. What is the daily standard deviation in percent and in dollars? What is the 10-day 99% VaR?
7. Barings Bank has estimated its 1-day 99% VaR as $1 Billion. What is the 10-day VaR?
8.
Last National Bank currently has $2.77 million in capital reserves.
Government regulations require
that banks keep 3 times their 10-day 99% VaR in capital reserves.
The standard deviation of Last
National’s entire portfolio has been determined to be 16% per annum, and the current value of the
portfolio is $14.68 million.
Is the bank meeting its current capital reserve requirement?
How much
more is required?
9.
Your portfolio currently consists of $50, $50, $100, and $200 in each of four stocks.
The standard
deviations of the stocks are 50%, 20%, 30%, and 40%, respectively.
The return correlations among all
four stocks are zero.
What is your greatest expected loss in the coming year with 95% probability?
What is your greatest expected loss in the coming year with 99% probability?
10. Shares of Chumbucket Restaurants Inc. sell for $100 and have a standard deviation of 40%.
Consider
a portfolio that is long 50 shares of stock, long 100 call options with strike price $110, and short 100
put options with strike price $90.
A graph of the distribution of annual portfolio returns based on a
Monte Carlo simulation appears below.
From the simulation, the standard deviation of changes in
portfolio value over the next year is found to be $5,835.
If we use the normal-linear Value at Risk
model to compute a 95% 1-year VaR, we would get VaR = 1.65 * 5,835 = $9,628.
However, we would
not expect this number to be very accurate.
Why not?
What is the actual 95% 1-year VaR statistic for
this portfolio (the data from the simulation is provided below)?
Distribution of Annual Changes in Portfolio Value
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
-10,601
-6,822
-3,042
737
4,517
8,296
12,075
15,855
Change in Value
Frequency
Percentiles
Change in
Portfolio
Value
0%
-11,602
5%
-7,031
10%
-5,850
15%
-4,952
20%
-4,173
25%
-3,458
30%
-2,767
35%
-2,091
40%
-1,415
45%
-955
50%
-717
55%
-466
60%
-199
65%
414
70%
1,385
75%
2,495
80%
3,814
85%
5,462
90%
7,733
95%
11,522
100%
49,956
114

Finance 320
Problem Set #7