Expected return of the portfolio is E(Rp)=0.000384.
Volatility of the portfolio is σp=0.009960.
Value of the portfolio is $150,000,000
A 5% VaR is obtained by identifying the point on the distribution that lies 1.65 standard deviations to the left of the mean.
Using the parametric method to estimate VaR.
Solution 1: (according to book 6 on page 309)
Step 1 Multiply the portfolio standard deviation by 1.65. 0.009960 × 1.65 = 0.016434
Step 2 Subtract the answer obtained in Step 1 from the expected return. 0.000384 – 0.016434 = –0.016050
Step 3 Because VaR is expressed as an absolute number (despite representing an expected loss), change the sign of the value obtained in Step 2. Change –0.016050 to 0.016050
Step 4 Multiply the result in Step 3 by the value of the portfolio. $150,000,000 × 0.016050 = $2,407,500
Solution 2:
Step 1 Multiply the portfolio standard deviation by 1.65. 0.009960 × 1.65 = 0.016434
Step 2 Multiply the result in Step 1 by the value of the portfolio. $150,000,000 × 0.016434 = $2465100
Which one is correct? I think the second one is correct because the final result is the value of expected loss which is value at risk. However, the result of the first one is the return of the portfolio after the loss was subtracted.
“For a 5 percent yearly VAR, we have μP – 1.65σP = 0.0955 – 1.65(0.1487) = –0.1499. Then the VAR is $150,000,000(0.1499) = $22.485 million.”
(This is from my Level III book when we had VAR as part of Level III curriculum).
So why would anyone use Method 2… answers in paragraph below
“Some approaches to estimating VAR using the analytical method assume an expected return of zero. This assumption is generally thought to be acceptable for daily VAR calculations because expected daily return will indeed tend to be close to zero. Because expected returns are typically positive for longer time horizons, shifting the distribution by assuming a zero expected return will result in a larger projected loss, so the VAR estimate will be greater. Therefore, this small adjustment offers a slightly more conservative result and avoids the problem of having to estimate the expected return, a task typically much harder than that of estimating associated vol- atility.”
So Method 2 is used - when a) you are trying to convert e.g. a yearly VaR estimate to a Daily VaR estimate.
or b) when you need a more conservative estimate of the loss
In your example 24.651 Million of Loss is higher than 24.075 Million … so a more conservative estimate of loss (because it is a higher loss).
If you are given a mean return for the portfolio - then you would use Method 1 - always…
A normal distribution with expected value μ and standard deviation σ can be converted to a standard normal distribution, which is a special case of the normal distribution in which the expected value is zero and the standard deviation is one. A standard normal distribution is also known as a z-distribution. If we have observed a return R from a normal distribution, we can convert to its equivalent z-distribution value by the transformation:
z=(μ−R)/σ (μ greater than R) then R=μ-zσ
For method one, VaR=$150,000,000*(μ-zσ)=$150,000,000*R
R is the rate of return when the loss is expected to occur. Then $150,000,000*R is the return of the portfolio when the loss is expected occur.
However, by definition, VaR is the the minimum loss that would be expected, which should be $150,000,000*μ subtracts $150,000,00*R, which is $150,000,00*(μ−R) = $150,000,00*zσ.
So, by definition, VaR is an estimation of expected loss, which should be the difference between the mean return of the portfolio and the actual return of the portfolio.
no. That is not what the definition of parametric VaR says.
μ is the expected return of the portfolio under normality assumption with 0 standard deviation. When zσ is subtracted from that model (z being the normal score 1.65 for the 5% significance level) you are obtaining the return of the portfolio with the 5% times standard deviation applied - so you get the max loss possible under 5% assumption.
So now (μ-zσ) * VALUE of the Portfolio = VALUE at Risk (Possibility of Loss) for the Portfolio.