Value at Risk(Loss Distribution vs Gains distribution)_Sep2007(i)

[Edwin]

For;

Investor 1;

P(R = r) = 0.96 if r = 1 & 0.04 if r = 0

thus R~Bernouli(p = 0.96)

Now define Y = 1000R
a) Var(Y) = 1000^2*Var(R) = 0.96*0.04*1000^2 = 38400
and the same can be done for measures b, c and d.

Investor 2

R is still the same i.e R ~ Bernoulli(p = 0.96)
But Y = R1 + R2+ ....+ R1000
i.e Y = ∑Ri ~ Bin(1000,0.96) Since Bin is a sum of INDEPENDENT Bernoulli trials !

Now the trick is with the ER. They have Y ~ Bin (1000, q =0.04). This is the probability distribution of the LOSSES i guess. Therefore to compute VaR we are concerned with the RIGHT tail of the distribution, hence P(Y>t) = 0.05.

Here VaR = 50.2 We are 95 % confident that we will not lose more than 50.2.

Note that had the ER proceded to find VaR = -t, P(Y<t) = 0.05. Then they would have used Y ~ Bin(1000,0.96) which is the probability distribution of the GAINS.

Here VaR = -949.81. We are 95% confident that we will not make less than 949.81 of profits.

Friends, please confirm if my understanding is right here, if so why did the Examiner's report use the probability distribution of Losses.

Would marks have been allocated if one worked with the distribution of gains i.e found that we are 95% confident that we will not make less than 949.81 of profits???

 

[Mike Lewry]

You say the Examiners have:

"VaR = 50.2. We are 95 % confident that we will not lose more than 50.2."

I agree with this.

You then say your approach gives:

"VaR = -949.81. We are 95% confident that we will not make less than 949.81 of profits."

I'm happy with your number, but you're interpreting it in the wrong way. This isn't a profit number, it's a portfoluio value. To get the profit, you need to compare it with the starting value of 1000. This will give you a profit of -50.19, which is the same as a loss of +50.19.

You ask:

"Friends, please confirm if my understanding is right here, if so why did the Examiner's report use the probability distribution of Losses.

Would marks have been allocated if one worked with the distribution of gains i.e found that we are 95% confident that we will not make less than 949.81 of profits???"

You can look at distribution of gains or losses - it doesn't matter. The only difference will be the sign of your final answer.

Stating you are 95% confident that we will not end up with less than 949.81 will get most of the marks, but not all as you're not stating the VaR, which is 50.19.

Stating you are 95% confident that we will not make less than 949.81 of profits is wrong and so you'll lose more marks than for the previous statement.

 

[Edwin]

Thank you Mike Lewry, atleast I know I would have worked with the distribution of Bin(1000,0.96) which is a sum of the independent Bernoulli trials for investor 1 makes more sense.

And had my interpretation been correct like yours I would have gotten the same VaR. Sometimes the ER is there to confuse!

Thanks (a lot) Mike.