FRM - 11 - Problem Set (IV) Solved PDF

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Chapter 11 solved exercise class...

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UC3M

11 – Problem Set (IV) (Solved) Financial Risk Management Pedro Agudo

Problem Set (IV) (Solved) Normal (0,1) Values

Portfolio Volatility

 x  y  2x  2y  2 x  y

Parametric VaR = position * daily volatility

Z -2.330 -1.960 -1.644 0,000 1,644 1,960 2,330

N(x) 0,010 0,025 0,050 0,500 0,950 0,975 0,990

Problem Set (IV) (Solved) 1.  is the variance covariance matrix of three assets. Calculate the volatility of each asset, the correlation coefficients between them and the volatility of two portfolios formed by the weights (30% 40% 30 %) and (10% 60% 30 %)

Problem Set (IV) (Solved)  112  12     222     21      n1  n2 

Volatilities 0,09 = 0,3 0,10 = 0,316 0,01 = 0,1

Correlations 𝜌, =

𝜌, =

0,03 = 0,316 (0,3 × 0,316) , (,×,)

 1n    2n 

 

   nn2 

 

 𝜎 = 0,3 0,4 0,3

𝜎 =

 𝜎

0,09 0,03 0,007 0,03 0,1 −0,02 0,007 −0,02 0,01

0,3 0,4 = 0,02866 0,3

0,02866 = 16,929%

0,09 0,03 0,007 0,03 0,1 −0,02 = 0,1 0,6 0,3 0,007 −0,02 0,01

=0,230

−0,02 = −0,63 (0,316 × 0,1)

w1     ...  wn 

Portfolio Volatilities:

𝜎 = 𝜌, =

   '  2 p

0,03462 = 18,606%

0,1 0,6 = 0,03462 0,3

Problem Set (IV) (Solved) 2. Suppose that a random variable follows a Normal distribution with mean 48.5 and standard deviation of 10. What percentage of the distribution is aprox between 71.8 and 65? a. b. c. d.

4% 8.96 % 18.15 % 24.17 %

(,) (,)

99%-95,05%=4%

Problem Set (IV) (Solved) 3. You have invested 360,000 € in shares of a company. The annual volatility of that stocks is a 32% per annum (assume 252 working days in one year). The 1 day VaR at 95% confidence level is: a. b. c.

€11,937 751 € 189,487 € 

Problem Set (IV) (Solved) 4. You have invested 360,000 € in shares of a company €. The annual volatility of that stocks is a 32% per annum (assume 252 working days in one year). The 10 days VaR at 95% confidence level is: a. b. c.

€11,937 €37,746 599,211 € 

Problem Set (IV) (Solved) 5. Consider a position of €100,000 invested in the asset A and €100,000 in another active B. Both assets have a daily volatility of 1% and a correlation between their returns of 0.3 . What is the not diversified VaR or sum of VaR in a horizon of 1 day at 99 %? a. b. c.

4,652.69 2,236.34 3,751.12

; Asset A B

Position 100,000.00 100,000.00

Z=99%

 1% 1%

VaR

2.326348 2.326348

2,326.35 2,326.35

Undiversified VaR

4,652.70

Problem Set (IV) (Solved) 6. Consider a position of €100,000 invested in the asset A and €100,000 in another active B. Both assets have a daily volatility of 1% and a correlation between their returns of 0.3. What is the VaR of the portfolio or diversified VaR in a horizon of 1 day at 99 %? a. b. c.

4 652.69 2 236.34 3 751.12

σ =

σ + σ + 2𝜌σ σ

σ = =

50% × 1% +50% × 1% + 2 × 0,3 × 50% × 1% × 50% × 1% = 0,8%

Asset Portfolio

Position 200,000.00

 0.008062

Z=99% 2.326348

VaR 3,751.12

Problem Set (IV) (Solved) 7. Consider an equally weighted portfolio invested in two assets with two identical volatility and perfect negative correlation (-1), therefore the portfolio is completely hedged such that each movement of an asset is offset by the other asset movement. The VaR of the portfolio is: a) It is necessary to know the parameters: volatility and confidence level b) Zero c) The sum of the individual VaR σ =

σ + σ + 2𝜌σ σ =

σ = 2𝜎 − 2𝜎  = 0 𝑉𝑎𝑅 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 × 𝜎 × 𝑍 ×

σ + σ + 2 −1 σ σ and because σ = σ

𝑇 ; Var=0

Problem Set (IV) (Solved) 8. If today a particular Market Index is located at 8 500 points and the annual volatility of the index is 20 %. Assuming normality in returns. What will be the daily volatility of the index? Assume that one year has 252 working days. a. b. c. d.



 1.259 % 1.6 % It is not possible to determine None of the above 

 

1.259%

Problem Set (IV) (Solved) 9. The 90% confidence level VaR of a portfolio is $5 000, therefore one could expect that: a) One of each 10 days the portfolio loses more than 5 000 $ b) One of each 90 days the portfolio loses more than 5 000 $ c) c. One of each 10 days the portfolio earns more than 5 000 $ d) d. One of each 90 days the portfolio earns more than 5 000 $

Problem Set (IV) (Solved) 10. Suppose that you are the risk manager of a fund. You are estimating the VaR using historical returns and find that the 10 worst returns in the past 100 days of negotiation have been : -1 %, -3 %, -0.6 %, -0.2 %, -2.7 %, -0.7 %, -2.9 %, 0.1 %, 1.1 %, -3 %. Approximately, what is the daily VaR at 95% confidence level? Tail of the distribution Value

a. b. c. d.

-0.6% -0.7% -1% -3%

-3.0% -3.0% -2.9% -2.7% -1.1% -1.0% -0.7% -0.6% -0.2% 0.1%

% Obs 100% 99% 98% 97% 96% 95% 94% 93% 92% 91%...