Wk 7: Value at Risk and Bank Capital

Learning Outcomes

Objectives

• Understand the concept of Value at Risk (VaR) and its application to bank portfolios.

• Compare the coherence of Value at Risk and Expected Shortfall (ES) as risk measures.

• Discuss the significance of bank capital from the perspectives of regulators and shareholders.

Market Risk Definition

Market Risk

• Defined as the potential gain or loss associated with price movements of assets over a specified time period.

• It is a systematic risk which means it cannot be diversified away

• Increased asset trading since the 1980s has prompted banks and regulators to develop market risk measurement tools.

• First mentioned by the Basel Committee in 1993.

Value at Risk (VaR) Concept

Managing Market Risk

• VaR is a fundamental technique used to manage market risk.

• Post-Global Financial Crisis (GFC), emphasis on:

  • Stressed VaR: Considers a 1-year period for losses.

  • Incremental Risk Charge: Includes default and migration risk.

Definition of Value at Risk

Understanding VaR

• VaR is expressed as: "We are X percent certain that we will not lose more than V dollars in time T."

• V is the Value at Risk of the portfolio.

• Formula: V = f(T, X).

• VaR is the loss level during a time period of length T that we are certain is X% will not be exceeded.

VaR Calculation

Ways to Calculate VaR

• If VaR is based on the distribution of gains:

  • Losses are considered negative gains.

  • VaR = - the gain at the (100 - X)th percentile.

• If VaR is based on the distribution of losses:

  • Gains are considered negative losses.

  • VaR = the loss at the Xth percentile.

Examples of VaR Calculation

Case Study 1: Normal Distribution

• Consider a portfolio with a gain that is normally distributed:Mean = £2M, Standard Deviation = £10M. Calculate the 99% VaR:

  1. Since this is a distribution of gains, VaR is the -ve gain at (100-99)th percentile which in this case is 1%.

     Next we look up the corresponding z-value at 1% which in this case is -2.33.

  2. (2 − 2.33) × 10 = − £21.3 million.

  3. The 99% VaR for this portfolio over six months is £21.3M.

Case Study 2: Uniform Distribution

• For a project with outcomes ranging uniformly from -£50M to +£50M:

  • 1% chance of exceeding a loss of £49M.

  • Therefore, the one-year VaR at 99% confidence level is £49M.

Case Study 3: Probabilistic Scenarios

• Consider a project with:

  • 98% likelihood of a £2M gain,

  • 1.5% chance of a £4M loss,

  • 0.5% chance of a £10M loss.

• Objective: Calculate the VaR at a 99% confidence level.

Expected Shortfall (ES)

Definition of Expected Shortfall

• The expected loss during time T conditional on the loss being greater than the Xth percentile of the loss distribution.

• VaR answers "How bad can things get?" while ES responds to "If things do get bad, what is the expected loss?"

• Known as conditional VaR, conditional tail expectation, or expected tail loss.

ES Distribution Comparison

Comparison of VaR and ES(include image slide)

• Distributions can show the same VaR but different Expected Shortfalls.

• Understanding these nuances is critical in risk management.

Example of Calculating Expected Shortfall

• Given:

  • X = 99

  • Time T = 10 days

  • VaR = £64M

• ES represents the average loss over 10 days, conditional on losses exceeding £64M.

Properties of Coherent Risk Measures

Characteristics of Risk Measures

• Monotonicity: when there are 2 portfolios where one is worse than the other, risk will be greater, requiring additional capital.

• Translation Invariance: Adding cash provides a buffer for losses hence risk should be reduced. risk goes down by k ( amount of invested capital)

• Homogeneity: Doubling a portfolio size should require double the capital.

• Subadditivity: Merging portfolios should not increase total risk. Risk should amount to the sum of their combined risk or less than that.

• VaR satisfies the first three conditions but not always subadditivity. (this is where ES comes in)

Evaluation of Combined Projects

Example for VaR & ES

• Two independent projects:

  • 2% probability of a £10M loss

  • 98% chance of a £1M loss

• Calculate one-year 97.5% VaR for both projects and the portfolio.

• Demonstrate VaR does not meet the subadditivity condition.

  The portfolio VaR for the combined projects is £11M:

  • VaR(project 1 + project 2) > VaR(project 1) + VaR(project 2).

• Confirms that VaR doesn't satisfy the subadditivity condition.

ES Calculation for Portfolio

Evaluating Expected Shortfall

• For a 97.5% confidence level, ES is computed:

  • 80% probability of £10M loss

  • 20% of £1M loss for each project.

• Resulting ES: £8.2M for each project.

Portfolio ES Calculation

Portfolio Expected Shortfall

• Analyzing the 2.5% tail:

  • 0.04% for a £20M loss

  • 2.46% for £11M loss

• Portfolio ES results in £11.144M, satisfying the subadditivity condition while showing ES(project 1 + project 2) < ES(project 1) + ES(project 2).

Importance of Bank Capital

Role of Bank Capital

• Central to bank regulation.

• Represents ownership interest and value of net assets.

• Essential for absorbing losses while protecting deposits to avoid loss of public confidence.

• Capital adequacy is contingent on both the volume and quality of assets.

Primary Roles of Bank Capital

Functions of Bank Capital

• Serves as a stable funding source.

• Acts as a cushion to absorb losses and prevent insolvency.

• Allows a greater proportion of defaulting assets without depleting capital.

• Protects uninsured depositors and the bank's insurance fund.

• Enhances public confidence in banking stability.

Investment Concerns with Low Capital

Issues from Insufficient Capital

• Potential for risk-shifting: Taking on overly risky projects.

• Risk of debt overhang: Avoiding otherwise positive NPV projects due to high debt burdens.

Regulatory Perspective on Capital

Views from Regulators

• Holding more capital reduces bank failure risks and enhances liquidity.

• Strengthens the overall safety of the banking system.

• In bankruptcy scenarios, common stockholders are last in line for reimbursement.

• Higher-risk banks must hold larger capital reserves compared to lower-risk banks.

2007-08 - lots of bank failures

deregulation vs regualtion