Consider the following discrete probability distributions of payoffs for 3 securities that are held in a DI's trading portfolio (payoff amounts shown are in $millions):

$\begin{array} { | l r r | }
\text { SECURITY } & \text { PROBABILITY } & \text { PAYOFF } \
\text { Alpha } & 0.50 & 355 \
& 0.49 & 150 \
& 0.01 & - 300
\end{array}$
$\begin{array} { lcc }
\text { SECURITY } & \text { PROBABILITY } & \text { PAYOFF } \
\text {Beta } & 0.50 & 1500 \
& 0.49 & - 300 \
& 0.0025 & - 3,300
\end{array}$
$\begin{array} { lcc }
\text { SECURITY } & \text { PROBABILITY } & \text { PAYOFF } \
\text {Gamma } & 0.49 &400 \
&0.49&150 \
&0.01 &-150 \
&0.01&-2,000
\end{array}$
What is the expected payoff,the 99% value at risk (VAR)and the expected shortfall (ES)of security Gamma (in millions)?

Question

Consider the following discrete probability distributions of payoffs for 3 securities that are held in a DI's trading portfolio (payoff amounts shown are in $millions):

$\begin{array} { | l r r | } 
\text { SECURITY } & \text { PROBABILITY } & \text { PAYOFF } \
\text { Alpha } & 0.50 & 355 \
& 0.49 & 150 \
& 0.01 & - 300
\end{array}$
 $\begin{array} { lcc } 
\text { SECURITY } & \text { PROBABILITY } & \text { PAYOFF } \
\text {Beta } & 0.50 & 1500 \
& 0.49 & - 300 \
& 0.0025 & - 3,300
\end{array}$
 $\begin{array} { lcc } 
\text { SECURITY } & \text { PROBABILITY } & \text { PAYOFF } \
\text {Gamma } & 0.49 &400 \
&0.49&150 \
&0.01 &-150 \
&0.01&-2,000
\end{array}$ 
What is the expected payoff,the 99% value at risk (VAR)and the expected shortfall (ES)of security Gamma (in millions)?

Quizplus · Accepted Answer

Expected payoff of security Gamma = (0.49 x 400) + (0.49 x 150) + (0.01 x (-150)) + (0.01 x (-2,000)) = $248 million
To calculate VAR(99%), first we need to arrange the payoffs in descending order: -2,000, -150, 150, 400
Then, we need to find the portfolio payoff at the 99th percentile, which is:
-2,000 x 0.01 + (-150 x 0.02) + (150 x 0.51) + (400 x 0.46) = -$2,000 + (-3) + 76.5 + 184 = -$1,742.5 million
Therefore, VAR(99%) = $1,742.5 million
To calculate ES, we need to find the average of all portfolio payoffs that are below the VAR(99%) value. In this case, there is only one payoff below the VAR(99%) value, which is -2,000 x 0.01 = -$20 million. Therefore, ES = (-20 + (0.49 x 400) + (0.49 x 150) + (0.01 x (-150))) / 0.01 = $2,000 million. $\underline{\text{Gamma Expected Return:  }}$  11efbab3_edd2_28c4_8e22_7b7f159dc5c2_TB1314_00
Gamma 1-day, 99% VAR:
 
 Recall that VAR is a point on the probability distribution and do not take into account values beyond that point; in this case, the 99 percent level.
 VAR99% = -$2,000
 
 Gamma estimated shortfall at 1% level:
 
 At the one-percent level, the estimated shortfall is equal to the VAR:
 
 For 99% confidence level, include all values in left tail of distribution until probability is equivalent to one (1):
11efbab4_063f_d245_8e22_53b103f601d2_TB1314_00

Consider the Following Discrete Probability Distributions of Payoffs for 3