p1.T4 Measures of Financial Risk (chpt 2 Dowd)

[skoh]

For question 30.4, why do we have to multiply

50% × 0 + 50% × E[loss|loss event]

I mean why should 50% be the average of the 10% tail?

 

[David Harper CFA FRM]

Hi skoh,

Source is here @ http://forum.bionicturtle.com/threads/p1-t4-30-expected-shortfall-es.5700/ (I will copy my answer to here as well)

The 90% expected shortfall is the (conditional) average of the 10% tail. In this case, from 90% to 95% is still part of the "no loss" where loss = 0. In the final 5%, we have the three outcomes. From the perspective of the parent (overall) distribution, the worst 10% tail includes:

• 5% prob of loss = 0
• 20%*5% = 1% prob of $10 million loss
• 50%*5% = 2.5% prob of $18 million loss, and
• 30*5% = 1.5% prob of $25 million loss

50% × 0 + 50% × E[loss|loss event] is the same as the conditional average of this 10% loss tail, which is just to take the weighted average:
5%/10%*0 + 1%/10%*$10 + 2.5%/10%*$18 + 1.5%/10%+$25 = 50%*0 + 10%*$10 + 25%*$18 + 15%+$25 = 9.25 million

 

[[email protected]]

for 29.2 question,
One of the options says that
If the return distribution is normal, then VAR is sub-additive.

So VAR is sub-additive for normal distributions?

Regards,
Aman

 

[[email protected]]

Hi David,
This is Regarding
28.1. A portfolio consists of two zero-coupon bonds, each with a current value of $50.0 million; the first maturing in 3.0 years the second maturing in 7.0 years. The yield curve is flat, with all yields at 6.0%. The daily volatility is 1.0% and assumed to be i.i.d. normally distributed. Using only duration's linear approximation (not convexity) and assuming annual compounding, which is nearest to the portfolio's 99.0% 10-day value at risk (VaR)?
a) $11.6 million
b) $27.5 million
c) $34.7 million
d) $36.8 million
Answers:
28.1. C. $34.7 million
Portfolio duration = [$50*3.0/(1+6%) + $50*7.0/(1+6%)]/100 = 4.71698 years
10-day VaR ($) = $100*1%*SQRT(10/1)*2.33*4.71698 = $34.755 (or $34.7007 if using exact deviate).

I didnt get the solution here. Could you please explain which formula is it used(duration,standard deviation and var relationship)

Many Thanks,
Aman

 

[coquin22]

ist that 98 oder 99% if 99 then 2,33 is the wrong to apply
for 98% you better use 2,326
So i would receive 34,6955

 

[David Harper CFA FRM]

Hi Aman,

I happened to recently expand on 28.1 at the source here: http://forum.bionicturtle.com/threads/p1-t4-28-value-at-risk-var.5669/#post-20652

The Macaulay duration of the portfolio is 5.0 years, but we need the modified duration of 5.0(1+6%) = 4.17 years for the linear sensitivity used in VaR.
This tells use that the price changes approximately 4.17 basis point for each basis point change in the yield (or 4.17% for each 1.0%, if you like).

The daily yield volatility is 1.0%, so under i.i.d., the 10-day volatility = SQRT(10)*1.0% = 3.162%,
but for VaR we want the worst expected yield change with 99% confidence, so as usual, we scale volatility by the deviate: 99% 10-day yield VaR = 1.0%*SQRT(1)*2.33 = 7.368% is the worst expected 10-day yield shock; i.e., 10-day VaR(dy).

How much will that impact the portfolio's price? As yield change * duration ~= % price change, we estimate that a 7.368 yield shock will change the price by approximately 7.368% * 4.17 years = 34.755%, which for a $100 value portfolio is $34.75 million.

This is highly testable, must know this, please see how each step explains the final product:

• 10-day VaR ($) = $100*1%*SQRT(10/1)*2.33*4.71698; i.e.,
• 10-day VaR ($) = Value * daily volatility * scale volatility to 10-days * scale or stress the volatility to its 99% worst outcome * multiply by modified duration to translate this yield shock into an estimated price shock in %