Question

In finance, the acronym VaR does not stand for variance; it means “value at risk.”

Imagine that you manage the $1 million portfolio of a wealthy investor. The portfolio is expected to average 10% growth over the next year with standard deviation 30%. Let’s convey the risk of this investment using the VaR that excludes the worst 5% of the scenarios. In other words, there is a 5% chance of losing a specific sum of money. What is this sum?

Hint: In this problem X is normal with µ= 0.10 and standard deviation ø= 0.30. You start by writing Pr( X≤ L) = 0.05 and find L. This quantity times the investment amount is the loss.

Answer 1

The diagram is as shown below:

mu = 0.1

Take cues from the hint, X is normal with mean = 0.10 and standard deviation as 0.30

We need to find L such that Pr(X<=z_L) = 0.05

From the z-table, z_L = -1.645

-1.645 = (L - 0.10)/0.30

L = 0.10 - 1.645*0.30 = -0.3935

VaR = Value at risk = L*Investment = 0.3935*$1 Million = $393,500

mu = 0.1