# L2.T8.1. Individual, incremental and diversified value at risk (VaR)

Discussion in 'Today's Daily Questions' started by David Harper CFA FRM, Oct 5, 2011.

1. ### David Harper CFA FRMDavid Harper CFA FRM (test)Staff Member

AIM: Define and distinguish between individual VaR, incremental VaR and diversified portfolio VaR. Discuss the role correlation has on portfolio risk. Compute diversified VaR, individual VaR, and undiversified VaR of a portfolio.

Questions:

1.1. In a portfolio where all returns are normally distributed, the diversified portfolio value at risk (VaR) of a two-asset portfolio is $76 million. The first asset has an individual VaR of$70 million. The assets have zero correlation. What is the individual VaR of the second asset?
a. $6.0 million b.$18.0 million
c. $19.4 million d.$29.6 million

1.2. A two-asset portfolio with a value of $20 million contains two equally-weighted assets (each asset has a value of$10 million). The volatility of the first asset is 10% and the volatility of the second asset is 20% (assets returns are normally distributed). What is, respectively, the 95% diversified portfolio value at risk (VaR) if (i) the assets are uncorrelated, (ii) the assets have a correlation (rho) of 0.5, and (iii) the assets are perfectly correlated?
a. $3.0, 3.9 and 4.5 million b.$3.7, 4.4 and 4.9 million
c. $3.9, 5.2 and 5.8 million d.$4.1, 5.6 and 6.3 million

1.3. A two-asset portfolio with a value of $40 million contains two equally-weighted assets (each asset has a value of$20 million). The volatility of both asset is 30%. The assets are uncorrelated (i.e., their correlation is zero). What is the incremental value at risk (VaR), assuming 95% confident VaR, if we subtract one asset from the portfolio, leaving only the remaining asset in the portfolio?
a. $4.09 million b.$6.73 million
c. $9.87 million d.$13.96 million

1.4. If computed for a portfolio where correlations are imperfect, which of the following value at risk (VaRs) will be greatest?
a. Undiversified VaR
b. Diversified VaR
c. Individual VaR
d. Incremental VaR

1.5. Which approach is most likely to find a local-valuation (delta-normal valuation) method insufficient?
a. Undiversified VaR
b. Diversified VaR
c. Individual VaR
d. Incremental VaR

1.6. Which is equal to the sum of component VaRs?
a. Undiversified VaR
b. Diversified VaR
c. Sum of individual VaRs
d. Sum of incremental VaRs

1.7. Which is equal to the sum of individual VaRs?
a. Undiversified VaR
b. Diversified VaR
c. Sum of component VaRs
d. Sum of incremental VaRs