We have two independent investments. Each of them may have a 1% chance of a loss of 10 million dollars, a 2% chance of a loss of 5 million, a 3% chance of a loss of 1 million, and a 94% chance of a profit of 1 million.
(Express all the following answers in million dollars: 100 million = 100, 1.3 million = 1.3)
1. The VaR0.95 for each single investment is = ?
2. The VaR0.98 for each single investment is = ?
3. The ES0.95 for each single investment is = ?
4. What is the VaR for a portfolio consisting of the two investments when the confidence level is 95%?
5. What is the expected shortfall for a portfolio consisting of the two investments when the confidence level is 95%? (round off to the second decimal place)
Remember: VaR0.95 means Value-at-Risk at 95% confidence level, ES means Expected Shortfall.
Var0.95 = 1
Aar0.98 = 5
confidence level is 95% = 4
shortfall for a portfolio consisting of the two investments when the confidence level is 95% = 6.18
Var0.95 = 1
Var 0.98= 5
Es 0.95 = 4.4
a)1
b)5
c)4.4
d)4
e)6.14
1. The VaR0.95 for each single investment is:
Since there is a 94% chance of a profit of 1 million and only a 6% chance of a loss, the VaR0.95 for each single investment would be 0 million.
2. The VaR0.98 for each single investment is:
Since there is a 94% chance of a profit of 1 million and only a 6% chance of a loss, the VaR0.98 for each single investment would still be 0 million.
3. The ES0.95 for each single investment is:
The ES0.95 is the expected shortfall, which is the average of the losses beyond the VaR0.95. However, since the VaR0.95 is 0 million for each single investment, the ES0.95 would also be 0 million.
4. The VaR for a portfolio consisting of the two investments when the confidence level is 95%:
In order to calculate the VaR for a portfolio, we need to consider the correlation between the two investments. However, since the investments are stated to be independent, which means they have no correlation, the VaR for the portfolio would simply be the sum of the VaRs for each investment. Therefore, it would still be 0 million.
5. The expected shortfall for a portfolio consisting of the two investments when the confidence level is 95%:
Similar to the previous question, since the investments are independent and have a VaR0.95 of 0 million, the expected shortfall would also be 0 million.
To calculate the Value-at-Risk (VaR) and Expected Shortfall (ES) for each single investment as well as the portfolio, we need to follow these steps:
1. Calculate the weighted probabilities for each loss and profit scenario for the single investments:
- For a loss of 10 million: Probability (1%) x Weight (-10 million)
- For a loss of 5 million: Probability (2%) x Weight (-5 million)
- For a loss of 1 million: Probability (3%) x Weight (-1 million)
- For a profit of 1 million: Probability (94%) x Weight (1 million)
2. Sort the weighted probabilities in descending order.
3. Calculate the cumulative probabilities by summing the sorted weighted probabilities.
4. Determine the VaR at the desired confidence level:
- For example, to calculate VaR0.95, find the cumulative probability nearest to 0.95. The associated loss value will be the VaR.
Now, let's calculate the values you requested:
1. VaR0.95 for each single investment:
- Calculate the weighted probabilities for each scenario:
- Loss of 10 million: 1% x -10 million = -0.1 million
- Loss of 5 million: 2% x -5 million = -0.1 million
- Loss of 1 million: 3% x -1 million = -0.03 million
- Profit of 1 million: 94% x 1 million = 0.94 million
- Sort the weighted probabilities: -0.1, -0.1, -0.03, 0.94
- Calculate the cumulative probabilities: -0.1, -0.2, -0.23, 0.71
- VaR0.95 is the loss associated with a cumulative probability nearest to 0.95: -0.23 million
2. VaR0.98 for each single investment:
- Calculate the cumulative probabilities as explained in step 1.
- VaR0.98 is the loss associated with a cumulative probability nearest to 0.98.
3. ES0.95 for each single investment:
- Calculate the weighted probabilities for each scenario as explained in step 1.
- Sort the weighted probabilities in descending order.
- Calculate the cumulative probabilities as explained in step 1.
- ES0.95 is the average of the losses associated with cumulative probabilities greater than or equal to 0.95.
4. VaR for the portfolio when the confidence level is 95%:
- Add up the weighted probabilities for each scenario from both investments.
- Sort the weighted probabilities in descending order.
- Calculate the cumulative probabilities.
- VaR0.95 for the portfolio is the loss associated with a cumulative probability nearest to 0.95.
5. Expected shortfall for the portfolio when the confidence level is 95%:
- Calculate the weighted probabilities for each scenario from the portfolio as explained in step 4.
- Sort the weighted probabilities in descending order.
- Calculate the cumulative probabilities.
- ES0.95 for the portfolio is the average of the losses associated with cumulative probabilities greater than or equal to 0.95.
I hope this helps you understand how to calculate the VaR and ES for the given investment scenarios.