An insurance company currently has a security portfolio with a market value of $243 million. The daily returns on the company’s portfolio are normally distributed with a standard deviation of 1.4%. Using the table below, determine which of the following statements are TRUE.

zcritical

Alpha

One-tailed

Two-tailed

10%

1.28

1.65

2%

2.06

2.32

One-day VAR(1%) for the portfolio on a percentage basis is equal to 3.25%.

One-day VAR(10%) for the portfolio on a dollar basis is equal to $5.61 million.

One-day VAR(6%) > one-day VAR(10%).

A) II and III only.

B) I only.

C) I and III only.

D) I, II, and III.

View Answer and Explanation

To find the appropriate zcritical value for the VAR(1%), use the two-tailed value from the table correspondnig to an alpha level of 2%. Under a two-tailed test, half the alpha probability lies in the left tail and half in the right tail. Thus the zcritical 2.32 is appropriate for VAR(1%). For VAR(10%), the table gives the one-tail zcritical value of 1.28. Calculate the percent and dollar VAR measures as follows:

VAR(1%)

= z1% × σ

= 2.32 × 0.014

= 0.03248 ≈ 3.25%

VAR(10%)

= z10% × σ × portfolio value

= 1.28 × 0.014 × $243 million

= $4.35 million

Thus, Statement I is correct and Statement II is incorrect. For Statement III, recall that as the probability in the lower tail decreases (i.e., from 10% to 6%), the VAR measure increases. Thus, Statement III is correct.

Source:Scheweser

My doubt is why they have used one tail for VAR 10% but two tail for Var 2%

zcritical

Alpha

One-tailed

Two-tailed

10%

1.28

1.65

2%

2.06

2.32

One-day VAR(1%) for the portfolio on a percentage basis is equal to 3.25%.

One-day VAR(10%) for the portfolio on a dollar basis is equal to $5.61 million.

One-day VAR(6%) > one-day VAR(10%).

A) II and III only.

B) I only.

C) I and III only.

D) I, II, and III.

View Answer and Explanation

To find the appropriate zcritical value for the VAR(1%), use the two-tailed value from the table correspondnig to an alpha level of 2%. Under a two-tailed test, half the alpha probability lies in the left tail and half in the right tail. Thus the zcritical 2.32 is appropriate for VAR(1%). For VAR(10%), the table gives the one-tail zcritical value of 1.28. Calculate the percent and dollar VAR measures as follows:

VAR(1%)

= z1% × σ

= 2.32 × 0.014

= 0.03248 ≈ 3.25%

VAR(10%)

= z10% × σ × portfolio value

= 1.28 × 0.014 × $243 million

= $4.35 million

Thus, Statement I is correct and Statement II is incorrect. For Statement III, recall that as the probability in the lower tail decreases (i.e., from 10% to 6%), the VAR measure increases. Thus, Statement III is correct.

Source:Scheweser

My doubt is why they have used one tail for VAR 10% but two tail for Var 2%

Hey Anbu

The formula to calculate VAR(X%) is given by:

VAR(X%) = z(X%) x σ

In this formula, z(X%) corresponds to the critical z value based on alpha X % using ONE TAIL normal distribution.

So, in the above question, VAR (10%) = z(10%) x σ

Here, z(10%) corresponds to critical value of z for 10% alpha using ONE TAIL.

Now, to calculate VAR (1%), we need critical value of z for 1% alpha using ONE TAIL. And since this value is not directly given in the question, we use the following concept:

Under a two-tailed test, half the alpha probability lies in the left tail and half in the right tail. That means z critical using 2% alpha TWO TAIL is equal to z critical using 1% alpha ONE TAIL. Thus, the z critical 2.32 is appropriate for VAR(1%).

The formula to calculate VAR(X%) is given by:

VAR(X%) = z(X%) x σ

In this formula, z(X%) corresponds to the critical z value based on alpha X % using ONE TAIL normal distribution.

So, in the above question, VAR (10%) = z(10%) x σ

Here, z(10%) corresponds to critical value of z for 10% alpha using ONE TAIL.

Now, to calculate VAR (1%), we need critical value of z for 1% alpha using ONE TAIL. And since this value is not directly given in the question, we use the following concept:

Under a two-tailed test, half the alpha probability lies in the left tail and half in the right tail. That means z critical using 2% alpha TWO TAIL is equal to z critical using 1% alpha ONE TAIL. Thus, the z critical 2.32 is appropriate for VAR(1%).

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