Estimating Market Risk Measures: A Quick Review (FRM Part 2, Book 1, Market Risk)

Summary

This script delves into market risk estimation, exploring time series analysis of profits/losses and dividends, and adjusting for time value of money. It discusses calculating arithmetic and geometric returns, and introduces historical simulation and parametric methods for Value at Risk (VaR) estimation. The script also covers coherent risk measures, expected shortfall, and general risk measures, emphasizing VaR's limitations and the importance of sub-additivity. It concludes with standard error estimation and QQ plots for distribution fitting and outlier detection.

Takeaways

• 📈 **Time Series Analysis**: The script discusses the importance of time series data for financial instruments, including crisis data for listed instruments and MTM valuations for exotic products.
• 💹 **Profit and Loss (P&L) Calculation**: It explains how P&L is calculated as the difference between the end-of-period receipts and the initial investment at the beginning of the period.
• 💰 **Time Value of Money**: The script covers two methods to adjust P&L for the time value of money: discounting future payments or capitalizing past payments using the risk-free rate.
• 🔢 **Arithmetic vs. Geometric Returns**: It differentiates between arithmetic returns (average return proxy) and geometric returns (considering income reinvestment) and how they relate.
• 📊 **Loss Time Series (LP)**: Introduces the concept of creating a loss time series as the negated version of the P&L time series, important for understanding risk.
• 📚 **Historical Simulation VAR**: Describes the historical simulation technique for VAR estimation, which involves sorting P&L observations and selecting a value at a certain confidence level.
• 📐 **Parametric VAR Methods**: Discusses three parametric approaches to VAR estimation: normal distribution on P&L, normal distribution on returns, and log-normal distribution on returns.
• 🔗 **Coherent Risk Measures**: Defines coherent risk measures and their four conditions, emphasizing the importance of sub-additivity for margin calculations and regulatory capital.
• 📉 **Expected Shortfall (ES)**: Introduces ES as a risk measure that provides insight into tail losses beyond the VAR threshold, calculated as a weighted average of tail losses.
• 🌐 **Generalized Risk Measures**: Explains how VAR and ES can be part of a broader class of risk measures, the generalized risk measures, which are weighted averages of quantiles.
• 📊 **QQ Plots**: Describes the use of QQ plots for comparing empirical distributions with benchmark distributions, identifying outliers, and estimating distribution parameters.

Q & A

What are the two types of time series inputs required for a War estimation approach?

The two types of time series inputs required are the time series of profits/losses and the time series of dividends/coupons, which can be referred to as the income time series.

How is Profit and Loss (P&L) computed in the context of War estimation?

Profit and Loss (P&L) is computed as the receipts at the end of any period t minus the initial investment at the beginning of the period.

What are the two methods to adjust P&L for time value of money?

The two methods to adjust P&L for time value of money are by discounting the payments received at time T or by capitalizing the payments made at time t-1, both using the risk-free rate R.

What is the difference between arithmetic and geometric returns?

Arithmetic returns are calculated by dividing the P&L by the initial investment and are used as a proxy for average or expected return. Geometric returns consider reinvestment of income and a chosen compounding frequency over a designated time period.

How is Value at Risk (VaR) estimated using the historical simulation technique?

In the historical simulation technique, VaR is estimated by sorting the P&L observations in increasing order and picking the VaR at a certain confidence level as the N alpha plus 1/8 observation from the left.

Why is the sub additive property important for risk measures?

The sub additive property is important for risk measures because it is crucial for margin calculations, regulatory capital estimation, and setting a conservative upper bound to the combined risk of multiple positions.

What is the Expected Shortfall (ES) and how is it different from VaR?

Expected Shortfall (ES) is the probability-weighted average of tail losses and provides insight into how bad things can get if losses exceed VaR. Unlike VaR, which only indicates a threshold loss number, ES gives a measure of the average loss in the tail beyond the VaR level.

How can the standard error of an estimate be computed?

The standard error of an estimate can be computed by creating a thin strip around the VaR with a width of H and calculating three probabilities: to the left of the strip, to the right of the strip, and in the middle of the strip. These probabilities are then used in a formula to compute the standard error.

What is a QQ plot and how is it used in risk estimation?

A QQ plot is a quantile versus quantile plot that compares quantiles from an empirical distribution with quantiles from a specified benchmark distribution. It is used to identify if the benchmark distribution is a good fit for the empirical data, to identify outliers, and to estimate parameters such as location and scale.

Why is the normal distribution assumption used in parametric VaR approaches?

The normal distribution assumption is used in parametric VaR approaches to simplify calculations and because it is a commonly used distribution in statistics. However, it may not always accurately represent the true distribution of P&L or returns, especially in the tails.

What is a spectral risk measure and why is it considered coherent?

A spectral risk measure is a special subset of general risk measures where the weighting function satisfies three conditions: non-negative weights, weights sum to one, and weights assigned to more extreme losses are not less than those assigned to less extreme losses. It is considered coherent because it satisfies all four conditions of coherent risk measures, including sub additivity.