Estimating VaR: The Parametric Method
Rigid rules. Defined ideals. Set parameters. All of these terms describe standards that exist to define a process. And the same concept applies to the parametric method. What might be one parameter that's required for this VaR estimator?
You got it!
It's any of those choices—the parametric method uses the expected return, volatility, and covariances to calculate the VaR. It's typically done by hand, meaning that it can be computed with a calculator, so it's best to have a portfolio of limited holdings.
For example, say you've got a portfolio of two holdings, A and B. To start using the parametric method, you'd first need to establish the distribution. Typically, the parametric method relies on a normal distribution.
Why's that?
Exactly!
The normal distribution is easier to work with because it won't involve skew or kurtosis. So there's less work to do to compute the VaR.
If your two-holding portfolio has normally distributed returns, and you can determine the return and standard deviations, all that's left is to compute the z-distribution by
$$\displaystyle z = \frac{R\ -\ \mu}{\sigma}$$.
Here, _R_ equals the return, μ equals the expected value, and σ equals the standard deviation.
That's not it.
A standard distribution won't have kurtosis.
Not quite.
The standard distribution won't have skew.
Typically in standard normal distributions, VaR is measured at 5%, or 1.65 standard deviations below the expected value of 0. Other times, 1% can be used, and that's 2.33 standard deviations away.
Now that the distribution is covered, you can focus on finding the other two parameters: the expected return and the volatility of the portfolio.
In order to find the return, you'll need key data that indicates how your holdings will impact the return. What data might that be?
That's not it.
That would be helpful in determining the volatility, not the return.
Bingo!
You'll need the weightings of your two holdings, A and B, to find the expected return.
$$\displaystyle E(R_p) = w_{a}E(R_{a}) + w_{b}E(R_{b})$$
The equation is simply weighted average expected return.
No, actually.
That data would be more helpful for the volatility calculation.
Next, you'd also need to find the volatility.
$$\displaystyle \sigma _p = \sqrt{w^2_a \sigma ^2_a + w^2_b \sigma ^2_b + 2w_aw_b \rho _{a,b} \sigma _a \sigma _b}$$
Here, σ represents the standard deviation of each holding, ρ indicates the correlation between the returns, and ρ_a_,_b_σ_a_σ_b_ is the covariance between the holdings.
Now, with the volatility and return, plus the z-distribution, you can compute the VaR. But there's a catch. Those values are based on annual returns, but a one-day VaR calculation will require an adjustment
Which value needs that adjustment?
Actually, it's both!
You need to adjust the annual return by dividing it by 250. The standard deviation should be adjusted by dividing it by the square root of 250. Then, simply plug each value into the formula.
$$\displaystyle [(E(R_p) - [(\mbox{z-distribution Standard Deviation})(\sigma _p)])(-1)](\mbox{Portfolio Value})$$
To start breaking down the equation, multiply the portfolio's standard deviation by the z-score, and then subtract it from the expected return.
Then, as you can see, the next step is to take the value and multiply by -1. Why would you do that?
You got it!
No. VaR is actually the expected loss, so that's not positive.
VaR is an absolute number, so you'll need to change the sign from negative to positive. Then you simply multiply it by the portfolio value.
The parametric method is valued for its simple equations, which work best on a normal distribution. Where the parametric method struggles, however, is on nonstandard distribution, such as options, because options have a nonlinear payoff. Overall, the parametric method works really well for a small number of portfolio holdings but can easily be impacted by minor changes in the parameters.
To sum it up:
[[summary]]
The expected return
The volatility of the portfolio
The covariances of the portfolio
It's easier to work with
The distribution will show kurtosis
The skew of the distribution will be high
Weightings of each holding
Covariance between the holdings
Standard deviation of each holding
Expected return
Standard deviation
VaR is an absolute number of the expected loss
VaR can be potentially positive for the portfolio
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