Market Risk : Estimate VaR using a Parametric Approach

What is the Parametric Approach?

The parametric approach assumes that portfolio returns follow a specific type of mathematical distribution (like a bell-shaped curve for normal distribution). Using some simple formulas, we calculate how much you could lose at a certain confidence level (like 95% or 99%).

Steps to Estimate VaR

1. Get the Data

• You need:
  • Portfolio value (how much your portfolio is worth today).
  • Mean return (average return, usually based on past data).
  • Volatility (how much the returns fluctuate, measured as a standard deviation).
  • Confidence level (how certain you want to be about the worst-case loss, like 95% or 99%).

2. Use a Formula

• VaR is calculated as: $$\text{VaR} = z \cdot \text{Volatility} \cdot \text{Portfolio Value}$$ Here:
  • $z$ is a number from statistics that depends on the confidence level:
    • For 95%, $z = 1.645$.
    • For 99%, $z = 2.33$.
  • Multiply $z$, volatility, and portfolio value to get the potential loss.

3. Interpret VaR

• Example: If your portfolio is worth $1,000,000, has a volatility of 2% per day, and you choose a 95% confidence level: $$\text{VaR} = 1.645 \cdot 0.02 \cdot 1,000,000 = 32,900.$$ This means: “With 95% confidence, the worst-case loss in one day won’t exceed $32,900.”

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Normal vs. Lognormal Distributions

Normal Distribution

• Assumes returns can be positive or negative.
• Works for most portfolios with small fluctuations.
• Formula is simple and directly uses $z$, volatility, and portfolio value.

Lognormal Distribution

• Assumes prices cannot go below zero (useful for stock prices).
• Adjusts returns to account for compounding (because percentages grow over time).
• The formula adds a bit of complexity but is better for modeling stocks.

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Key Differences

Feature	Normal Distribution	Lognormal Distribution
Can returns be negative?	Yes	No (prices can't be negative).
Best for:	General portfolios	Stock prices or investments that can't drop below zero.

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Simplified Example

Imagine your portfolio:

• Is worth $1,000,000.
• Has a daily volatility of 2%.
• Uses a 95% confidence level.

For Normal Distribution:

• Use $z = 1.645$.
• VaR = $1.645 \cdot 0.02 \cdot 1,000,000 = 32,900$.
  "You might lose up to $32,900 in a day with 95% confidence."

For Lognormal Distribution:

• Adjust for compounding effects 
• If you calculate, you might get a similar but slightly smaller loss, say $31,000.
  "The lognormal VaR accounts for compounding, so the loss might be a bit smaller."

Lognormal Distribution

The lognormal distribution is used when we’re dealing with things like stock prices that:

1. Cannot go below zero.
2. Grow over time through compounding (e.g., a 5% return today increases the base for future returns).

Instead of working directly with regular returns, we calculate the logarithmic returns (returns based on the natural logarithm). This gives more accurate results for assets like stocks.

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How to Estimate VaR with Lognormal Distribution?

1. Adjust the Inputs

We start with the expected return ($\mu$) and volatility ($\sigma$), but we make a small adjustment because of compounding.

• Adjusted mean ($\mu_{\text{log}}$):
  It’s the regular mean adjusted for compounding effects.

2. Find the Worst-Case Scenario

For a chosen confidence level (e.g., 99%), we use a statistical value ($z$) to find the worst-case return in the left tail of the distribution.

3. Convert Logarithmic Returns Back

Since we calculated logarithmic returns, we convert them back to regular returns using an exponential formula.

4. Calculate the Loss

We multiply the worst-case return by the portfolio value to get the VaR.

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Example

Let’s calculate lognormal VaR step by step for a simple portfolio:

Portfolio Details:

• Portfolio value: $1,000,000.
• Expected return ($\mu$): 5% annually.
• Volatility ($\sigma$): 20% annually.
• Confidence level: 99%.

Step 1: Convert Annual to Daily

Since we’re estimating daily VaR, divide annual values:

• $\mu_{\text{daily}} = \frac{5\%}{252} = 0.000198$
  
• $\sigma_{\text{daily}} = \frac{20\%}{\sqrt{252}} = 0.0126$
  

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Step 2: Adjust the Mean for Compounding

The adjusted mean ($\mu_{\text{log}}$) accounts for compounding effects:

$$\mu_{\text{log}} = \ln(1 + \mu_{\text{daily}}) - \frac{\sigma_{\text{daily}}^2}{2}.$$

Plugging in the numbers:

$$\mu_{\text{log}} = \ln(1 + 0.000198) - \frac{(0.0126)^2}{2} \approx 0.000119.$$

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Step 3: Find the Worst Logarithmic Return

For a 99% confidence level, $z = -2.33$:

$$R_{\text{log}} = \mu_{\text{log}} + z \cdot \sigma_{\text{daily}}.$$

Substitute the values:

$$R_{\text{log}} = 0.000119 + (-2.33 \cdot 0.0126) \approx -0.0293.$$

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Step 4: Convert Back to Regular Return

Convert $R_{\text{log}}$ (logarithmic return) back to a regular return:

$$R_{\text{actual}} = e^{R_{\text{log}}} - 1.$$

Substitute $R_{\text{log}} = -0.0293$:

$$R_{\text{actual}} = e^{-0.0293} - 1 \approx -0.0289.$$

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Step 5: Calculate VaR

Finally, multiply the worst-case return by the portfolio value:

$$\text{VaR} = \text{Portfolio Value} \cdot |R_{\text{actual}}|.$$

Substitute $R_{\text{actual}} = -0.0289$ and portfolio value = $1,000,000:

$$\text{VaR} = 1,000,000 \cdot 0.0289 = 28,900.$$

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Interpretation

At a 99% confidence level:

• The portfolio may lose up to $28,900 in a single day.

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Key Points to Remember

1. Lognormal VaR is just like normal VaR, but it adjusts for compounding returns.
2. The steps involve adjusting the mean and using the exponential function to account for compounding effects.
3. It’s especially useful for assets like stocks that cannot have negative prices.