P1.T4.28. Value at Risk (VaR)

[Suzanne Evans]

AIM: Define the Value-at-Risk (VaR) measure of risk, discuss assumptions about return distributions and holding period ...

Questions:

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

28.2. Because the asset price is $20.00, a long position in 100 call option contracts (i.e., 10,000 options) has a current notional value of $200,000. The options are at-the-money (ATM) with percentage (per option) delta of 0.60. The asset has a volatility of 18.0% per annum. If we assume returns are i.i.d. normal and a year contains 250 trading days, which is nearest to the 99.0% 10-day value at risk (VaR) under a delta-normal assumption; i.e., if assume only delta's linear approximation (delta-Normal) and ignore gamma?

a. $7,605
b. $10,066
c. $15,252
d. $21,018

28.3. A two-asset 130/30 long/short portfolio has $130 million invested in an asset with 24.0% volatility per annum hedged with a $30 million short position in an asset with a 16.0% volatility per annum. The correlation between asset returns, which are i.i.d. normal, is 0.30. Which of the following is nearest the portfolio's 99.0% confident 10-day relative value at risk (VaR)?

a. $8.33 million
b. $10.52 million
c. $14.03 million
d. $30.11 million

Answers:

• Here in forum

 

[Sushant]

28.1 Answer
Duration of zero coupon bond is same as their maturity.
Hence
50 * 3 * 2.33 * 1% * (10)^1/2 + 50 * 7 * 2.33* 1% * (10)^1/2
=> 50 * 2.33 * 1% * 10 *(10)^1/2
= 36.84

 

[David Harper CFA FRM]

Hi Sushant, I like your answer except for one detail: the Macaulay duration is the same as the maturity of the zero.

But whereas Mac duration is the weighted maturity of the bond, we generally are looking for the modified duration (sensitivity, VaR, hedging with dollar duration). So the annual compounding assuming is meaningful and the modified duration of the 3-year zero (eg) = Mac duration/(1+yield/k periods per year) = 3/(1+6%/1) ... so my answer agrees with your if you replace your Mac durations with modified durations. Thanks!

 

[vakshay]

For the third question, how would you calculate the volatility of the portfolio?
Would you consider the weighting of the assets in this portfolio?

 

[abhinav0131]

@vakshay

Yes, we should consider the weightings. The vol would be : sqrt(130%^2*24%^2 + (-30%)^2*16%^2 + 2*130%*(-30%)*24%*16%*0.3)