FM5617 Risk Management Finals PDF

FM5617 Risk Management Finals VALUE AT RISK (VAR) VAR Confidence interval...

FM5617 Risk Management Finals VALUE AT RISK (VAR) VAR Confidence interval - estimated range; no exact figure - The expected maximum loss (or worst based on that value loss exposure at loss) over a target horizon within a given a specific confidence confidence interval level - addresses your loss exposure VAR FORMULA. - determine the maximum point of the worst scenario - Evolved through the requirements of SEC to provide a minimum capital requirement in provision for risks taken by financial institutions 2 WAYS IN DETERMINING THE VAR - Historical Simulation Approach A ly s Ph de be - Model Building Approach VAR: M, C, O, L suited for portfolio consisting short and long positions in stocks. It also follows a certain formula Confidence level Table (Inverse cumulative normal 3 COMPONENTS OF VAR distribution) MEASUREMENT Time Frame t = * - Calculated VAR may only be applicable for a period of time * Loss Amount = Value Sample Problem 1: Suppose that the gain from a Confidence Level Mas mataas yung confidence level, more portfolio during six months confidence level of O 99 or O 90 reliable yung VaR. 99% is normally distributed with a mean of $2 Financial institutions use VaR to determine how Million and a Standard deviation of $10 Million. much emergency cash they need to cover What is the VAR? potential severe losses. VaR did not emerge as a distinct concept until VaR = 2,000,000 - the late 1980s. The triggering event was the X% = 99% = a = 2.33 2.33(10,000,000) 1987 stock market crash. = -$21,300,00 u= $2 Million o= $10 Million FRM Foundations (FRM T1-02) ESTIMATED SHORTFALL vs. VAR Estimated Shortfall "We are 99% certain that we will not lose more - Pinpoint exactly how much are you than $21,300,00 in six months. willing to lose FM5617 Risk Management Finals ESTIMATED SHORTFALL FORMULA Accuracy of VaR The historical simulation approach estimates the distribution of portfolio changes from a finite number of observations. As a result, the estimates of the percentiles of the distribution are subject to error. Sample Problem 1: Suppose that the gain from a EXPECTED SHORTFALL portfolio during six months confidence level of - a measure that can produce better 99% is normally distributed with a mean of $2 incentives for traders than VaR Million and a Standard deviation of $10 Million. - To calculate expected shortfall using What is the VAR? historical simulation, we average the losses that are worse than VaR. VAR vs. EXPECTED SHORTFALL = $14,443,913.81 VaR asks the question: “How bad can things get?” ES asks: “If things do get bad, what is HISTORICAL SIMULATION APPROACH the expected loss?” Forecasting a statistical method used to estimate the value at risk (VaR) of a portfolio by BOOTSTRAP simulating its future performance based on Bootstrapping is a statistical technique historical data. Historical simulation employed to estimate the sampling involves using past data as a guide to what distribution of a statistic, particularly when will happen in the future. the data distribution is unknown or - involves using the day‐to‐day non-normal. It involves repeatedly changes in the values of market resampling the data with replacement and variables that have been observed calculating the statistic of interest for each in the past in a direct way to resample. The resulting distribution of the estimate the probability distribution statistic approximates the true sampling of the change in the value of the distribution. current portfolio between today and tomorrow The bootstrap method is a variation of the - Dependent on the sample population basic historical simulation approach aimed at calculating a confidence interval for Value at Risk. It involves creating a set of changes in the portfolio value based on historical movements in market variables FM5617 Risk Management Finals - A statistical procedure that - we first calculate the sample mean resamples a single data set to create by getting the sum of the sample many simulated samples data - It means that one available sample - next is to generate a bootstrap gives rise to many others by sample. Some observations may be resampling. selected multiple times, while others - It is a powerful tool for risk may not be selected at all. management, as it can be used to - calculate the bootstrap mean for assess the potential losses a each sample portfolio could su er under various scenarios. MONTE CARLO When to apply the bootstrap method - construct confidence intervals Monte Carlo simulation is a computational - estimate the sampling distribution of method that relies on repeated random a statistic sampling to obtain numerical results, - perform hypothesis testing particularly useful for complex problems - estimate standard errors that cannot be solved analytically. It can be used to estimate VaR, the probability of Bootstrap method in risk management: financial crises, or option prices. - bootstrap method helps us estimate the likelihood that a portfolio will Monte Carlo Casino in Monaco experience a specific amount of - Computational technique used to financial loss over a certain time model and analyze complex systems frame. or problems. - Bootstrap method in risk - Model and analyze complex systems management determines the or problems that involve uncertainty worst-case scenario for a portfolio and randomness over a particular time frame while - Conducts repeated random maintaining a certain level of sampling confidence. - It provides multiple calculations to - risk managers need to understand get the possible outcome how a portfolio would perform under - Results provides an estimate once various market conditions, the simulation is done bootstrapping allows them to simulate di erent market scenarios Risk Assessment by resampling historical data Assess and quantify various types of risk: - Market risk BOOTSTRAP METHOD COMPUTATION: - Credit risk estimate the mean of the underlying - Operational risk distribution and integrate the given confidence interval Portfolio Optimization - Multiple asset classes Key Differences: Bootstrap focuses on resampling from what you have, while Monte Carlo uses random simulations to explore possibilities. FM5617 Formula: Risk Management Step 1: D1 = (Ln (SP/StrikeP) + (RFR + ((SD^2)/2) * T)) / SD * Sqt(T) D2 = D1 - (SD * Sqt(T)) Step 2: get the normal distribution of D1 and D2 Finals Step 3: C = (SP * nD1) - (StrikeP * e^(-RFR*T)*nD2) Step 4: Principal amount - Call Option or S3 - Risk levels - Return expectations European Options American Options FORMULA: (Si - Sn) / Sn - cannot be exercised - can be exercised at where: or assigned before the any time before the Si = present day’s price Sn = previous price expiration date expiration date - BSM does not provide - BSM is an accurate STEPS USING EXCEL: an accurate price for determinant in pricing American options. 1. DATA 4. DATA TABLE of European options 2. DATA TAB 5. EMPTY CELL 3. WHAT IF ANALYSIS VOLATILITY SKEW - measure is the representation of PROS CONS SIGMA - Black-Scholes assumes stock prices - Data-Driven Decision -Assumption follow a lognormal distribution Making Sensitivity - Handling Complex - Limited Precision - Asset prices are observed to have significant right skewness and some Systems degree of kurtosis. -Incorporating - The assumption of lognormal Uncertainty underlying asset prices should show that implied volatilities is similar for BLACK SCHOLES MODEL each strike price according to the Black-Scholes model The Black-Scholes model is a mathematical model for pricing options. It assumes a log-normal di usion process for the underlying asset and no transaction costs or frictions. The Black-Scholes model is complex but has exhibited accuracy in empirical studies. - BLACK SCHOLES MERTON (with innovations and improvements Put-Call Parity because of BSMs limitations) * call price must be close to spot price - used in pricing of an option contract. Used for valuation of option price. (Applicable for European option) Limitations of Black Scholes Model: The Black-Scholes model requires five input - Does not take into consideration all variables: types of options (Limited to - option’s strike price = Call option price European Market) - underlying stock price - May lack cashflow flexibility based - time remaining until expiration = time kung kailan mo bibilhin on the future projections of a - risk-free rate = common sense na lang security - volatility a.k.a Standard deviation *gagamitin mo lang si BSM if constant si RFR and volatility FM5617 Risk Management Finals - May make inaccurate assumptions CONDITIONAL PROBABILITY It I event = (Relies on a number of other is the probability of an event of which would assumptions) a ect or be a ected by another event. In other words, a conditional probability, as the name implies, comes with a condition. HAZARD RATE Fail Quickly o = Failure Rate , The hazard rate represents the P(B|A) = P(A and B) / P(A) instantaneous probability of an event occurring at a given time. It is often used to First, the probability of drawing a blue model the risk of death, default, or marble is about 33% because it is one equipment failure. possible outcome out of three. Assuming - The Hazard Rate is the probability this first event occurs, there will be two that a component will fail quickly marbles remaining, with each having a 50% given it's up to that point in time chance of being drawn. So, the chance of divided by the unit of time. drawing a blue marble after already - The Hazard Rate is a one metric that drawing a red marble would be about 16.5% is typically used to determine the (33% x 50%). proper probability distribution of a given mechanism. UNCONDITIONAL PROBABILITY I vent - I @ - The Hazard Rate is sometimes called refers to a probability that is una ected by the failure rate and it only applies to previous or future events. The unconditional things that are irreparable. probability of an event happening is simply the probability of the event itself. It does h(t) = f(t)/R(t) not have a condition. WHERE: F(T) IS THE PROBABILITY DENSITY FUNCTION (PDF) P(A) = Number of Times ‘A’ Occurs/ R(T) IS THE SURVIVAL FUNCTION OR THE Total Number of Possible Outcomes PROBABILITY THAT SOMETHING WILL SURVIVE PAST A CERTAIN TIME (T). A stock can either be a winner, which earns a positive return, or a loser, which has a PURPOSE OF HAZARD RATE: It calculates negative return. Say that stocks A and B are the subject's probability of survival at a winners out of five stocks, while stocks C, D, specific moment in time. and E are losers. What, then, is the unconditional probability of choosing a BATHTUB HAZARD RATE CURVE winning stock? Since two outcomes out of a used to characterize the failure rate of a possible five will produce a winner, the system over time. Plotting the hazard rate unconditional probability is 2 successes or failure rate of a system against time divided by 5 total outcomes (2 / 5 = 0.4), or yields a graphical representation that is 40%. commonly referred to as a "bathtub" shape. FM5617 Risk Management Greater than 3.0 = unlikely to default Finals Between 2.7 and 3.0 = "on alert" Between 1.8 and 2.7 = Good chance of default Less than 1.8 = Financial embarrassment ALTMAN’S Z-SCORE a financial distress prediction model that assesses a company's creditworthiness. It is based on various financial ratios, including Z = 1.2 (0.254) + 1.4 (0.448) + 3.3 (0.0896) + profitability, liquidity, and leverage. 0.6 (1.583) + 0.999 (3.284) Altman's Z-score is relatively simple and Z=5.46 e ective in predicting bankruptcy. - A financial model used to predict a company’s likelihood of declaring bankruptcy - Weighs five ratios, and getting their sum - The computed z-score is compared to a scale - Does not work with new companies long TV - Output for credit strength test POWER LAW = SCALING LAW - Provides investors with an overview of a company's financial health a statistical distribution characterized by a Five Accounting Ratios used for Altman's large number of small events and a small Z‐Score: number of large events. It is often used to X1 : Working capital/Total assets model phenomena like income distribution, X2 : Retained earnings/Total assets city sizes, and earthquakes. X3 : Earnings before interest and taxes/Total assets - Power law provides an alternative to X4 : Market value of equity/Book value of total assuming normal distributions. liabilities - Law describing the tails of many X5 : Sales/Total assets probability distributions that are Sample Prob. encountered in practice. Consider a company for which working capital is - It is also called SCALING LAW 170,000, total assets are 670,000, earnings before - Power laws are very important interest and taxes is 60,000, sales are 2,200,000, the because they reveal an underlying market value of equity is 380,000, total liabilities is 240,000, and retained earnings is 300,000. regularity in the properties of systems. Solution: - They use the power law to estimate X1 : Working capital/Total assets = 170,000/670,000 = the losses at an extreme level 0.254 X2 : Retained earnings/Total assets = 300,000/670,000 = 0.448 Extreme Value Theory X3 : Earnings before interest and taxes/Total assets = 60,000/670,000 = 0.0896 - science of estimating the tails of a X4 : Market value of equity/Book value of total distribution liabilities = 380,000/240,000 = 1.583 - used to Improve the VaR estimates X5 : Sales/Total assets = 2,200,000/670,000 = 3.284 FM5617 Risk Management Finals - to help in situations where analysts - We can also say that 84.56% probability that want to estimate VaR with a very losses will not exceed to $54 million. high confidence level Power law supports that the determination USE OF POWER LAW or probably limiting your expectations on - Measures operational risk how much losses after delivering or coming - Power law holds up well for the large up with the VAR on what you think would be operational risk losses experienced the possibility of having exceeded this by banks. values that you had already calculated. - Loss with the lowest alpha defines the extreme tail of the total operational risk loss distribution. VASICEK MODEL a stochastic volatility model that describes the behavior of short-term interest rates. It : assumes mean-reverting interest rates, V: Actual value of the variable x: Possible value in the right-hand tail of the distribution meaning they tend to gravitate towards a K: Scale factor long-term average. The Vasicek model is α: Alpha Symbolizes anything in excess (excess in profit) relatively simple and has shown accuracy in empirical studies. Sample Prob. A risk manager in PGP Company has established - one of the most influential models in that there is an 85% probability quantitative finance. that losses next year will not exceed $70 million. - a one-period model used to Noting that the power law parameter is 0.11, construct default indicators of calculate the probability of the loss exceeding correlated exposures over a given $54 million time horizon - a model of default correlation based Solution Substitute to the Power Law Formula: on the Gaussian Copula Model 0.15 = K70^-0.11 - allows more restrictive assumptions 1. “Alpha + Calc” to have equal sign 2. “Alpha + )” to get x sign to formulate the specific model with 3. “Shift + Cal” to solve the x and press equal sign which his name is associated. (2x) WHEN TO APPLY? Given: - to predict the extreme percentile of 𝛂 = 0.11 (power law parameter) 𝔁 = $54M the loss distribution. K =.2394 - regulatory capital requirements for internationally active banks Substitute to the Power Law Formula: Prob (v - calculating high percentiles of the >54) = (.2394)(54)^-0.11 distribution of the default rate =.154369 or 15.44% WORST CASE DEFAULT RATE Interpretation - It shows that there is a 15.44% probability that the company will exceed a $54 million loss. FM5617 Risk Management Finals Sample Prob. Suppose that a bank has a large number of loans to retail customers. The one‐year probability of default for each loan is 2% and the copula correlation parameter, in Vasicek's model, is estimated as 0.1. N-1 = NORMSINV N = NORMSDIST Sample Prob. PD = 0.02 P = 0.1 X = 0.999 Gootowmna Corp., a food and beverage company, is interested in 2 Philippine consumer stocks. The company decided to invest $25M in Emperador Inc. with a 3% daily volatility and $15M in Jollibee Foods Corporation with a 2% daily volatility. The returns on = 99% worst case one-year default is 12.8% the two shares have a correlation of 0.5. Assuming the changes in the portfolio value are normally distributed with a bivariate normal distribution, compute for the TWO-ASSET CASE 15-day 96% VaR for the portfolio a simplified model of the relationship between two assets, assuming linearly PVEMI = $25 PVJBFCF = $15 o = 0.5 Odaily = 3% correlated returns. It illustrates the concept t = 15 days Odaily = 2% of portfolio diversification, where combining x = 96% assets can reduce overall portfolio risk. Solve for the absolute volatility by using the daily Model-building approach volatility: - Also Known as the 25,000,000 x 0.03 = 750,000 variance-covariance approach 15,000,000 x 0.02 = 300,000 - Used as an alternative for calculating risk measure 750, 000^2 + 300, 000^2 + 2(0. 5)(750, 000)(300, 000) - Supposes simultaneous changes in = 936,749.6998 several market variables - Extension of Portfolio Theory of Solve for the one-day 96% VaR = 936,749.6998 x 1.75 Harry Markowitz = 1,639,311.98 Limitation: Two-asset case VAR formula - Challenging to use in portfolios made of non-linear products. - The assumption that returns are normal is di cult to abandon N-1 (X) is the inverse cumulative normal distribution without a significant increase in Op is the standard deviation of the portfolio calculation duration. √t is the time horizon Two-asset case Formula Sample Prob. O = 936,749.6998 = 0.5. t = 15 days N-1 (X) is the inverse cumulative normal distribution FM5617 Risk Management Finals VARIABLES THAT ARE TAKEN INTO Compute for the 15-day VaR of the Portfolio CONSIDERATION N-1 (96%) = 1.75 - Options’ strike price VaR =1.75 x 936,749.6998 x √15 VaR = 6,349,027.98 - Present underlying prices - Risk -free interest rate Portfolio Diversification - Amount of time before expiration - is the process of investing into various securities in order to MERTON MODEL ASSUMPTIONS minimize or lessen the risk of the - Continuous-time framework: value of portfolio. a company’s assets and debt continuously evolving Sample Prob. - Geometric Brownian motion: This EMI=₱750,000(₱25 ×3%) assumption implies asset prices have JFC=₱300,000(₱15 ×2%) log-normal distributions with 𝑁 96%=1.75 𝑡=15𝑑𝑎𝑦s constant expected returns and volatility (σ). VaREMI=1.75x750,000x√15 - Tradeable debt and equity: debt and VaREMI = 5,083,290.6420 equity of a company are tradeable. VaRJFC=1.75x300,000x√15 And have observable market prices VaR JFC = 2,033,316.2570 - Risk-neutral valuation: market participants are risk-neutral and VaR P = 6,349,027.98 make decisions based on risk-free interest rates (5,083,290.6420+2,033,316.2570) -6,349,027.98 - No taxes or transaction costs: = 767, 578. 92 focuses on the fundamental dynamics of credit risk MERTON MODEL - Constant asset volatility: volatility of a credit risk model that estimates the a company’s asset value (σ) remains probability of company default. It assumes consistent over time default occurs when the company's asset value falls below a certain threshold. The Merton model is complex but has proven DD = distance to the default accurate in empirical studies. V = market value of a company’s assets - a mathematical technique used to D = market value of the company’s debt assess a company’s credit default In = natural logarithm r = risk-free interest rate risk σ = volatility of the company’s asset value - Myron Scholes and Fischer Black T = time to maturity of the debt later developed the Merton Model, which known as the Black-Scholes Sample Prob. pricing model. AD Corporation’s assets (V) market value: $100 million. Black Scholes AD Corporation’s debt (D) market value is $60 million. to Merton model Risk-free interest rate (r): 3%. The volatility of AD Corporation’s asset value (σ): 0.2. FM5617 Risk Management Finals Time to maturity of the debt (T): 2 years. Derivagem calculated the following figures for 100,000 shares, how would you interpret the following? Using these assumptions, we can calculate the distance to default (DD) using the Merton Model formula: DD = (ln(V / D) + (r + σ²/2) × T) / (σ × √T) Substituting the given values into the formula: DELTA BALANCING or delta hedging is a risk management DD = (ln($100 million / $60 million) + (0.03 + strategy used in financial markets to reduce 0.2²/2) × 2) / (0.2 × √2) the impact of price fluctuations on a portfolio. It involves establishing and Simplifying the calculation: managing positions in both the underlying DD = (ln(1.67) + (0.03 + 0.02) × 2) / (0.2 × 1.41) asset and its corresponding options. = (0.51 + 0.05 × 2) / (0.2 × 1.41) - A measure of how much the option = (0.51 + 0.2) / (0.2 × 1.41) = 0.61 / 0.2828 can be expected to gain or lose per ≈ 2.1570 or 2.16 $1 move in the stock price - it measures the sensitivity of the In this example, the calculated distance to default (DD) for ABC Corporation is value of an option to changes in the approximately 2.51. The DD represents the price of the underlying stock, company’s bu er or cushion before it reaches assuming all other variables remain the default threshold. A higher DD indicates a unchanged lower probability of default, suggesting that ABC Delta is a risk metric that estimates the Corporation has relatively lower credit risk. change in the price of a derivative, such as an option contract. TRADERS’ MANAGEMENT OF EXPOSURES - It is denoted by ⃤ THE GREEKS PUT DELTA Since Put premiums increase as the price of a stock increases, Put Deltas are negative - Deep ITM Puts have Deltas of approx. - 100, act like short stock - ATM Puts have Deltas of approx. -50 - Far OTM Puts have Deltas of 0 CALL DELTA Since Call premiums increase as the price of Given that the stock price is $49, the strike price is $5o, a stock increases, Call Deltas are positive. the risk-free is 5%, the stock price volatility is 20% and - Deep ITM Calls have Deltas of the time to exercise is 20 weeks, assuming that the approx. 100, act like long stock FM5617 Risk Management Finals STATIC DELTA-HEDGING - involves establishing and maintaining a fixed delta position. IMPORTANCE OF DELTA By rebalancing the portfolio, - Significant variable related to the investors can control their exposure directional risk of an option. to price movements. - It can help investors and traders forecast how option pricing will DYNAMIC DELTA-HEDGING change as the underlying security's - takes into account changing market price varies. conditions and adjusts the delta - Delta can also be utilized for position over time. This strategy hedging purposes. The neutral delta involves actively managing the technique is a popular hedging portfolio, and making adjustments in strategy, as it entails owning a response to price fluctuations and number of options so that the delta market volatility. in aggregate is equal to or extremely close to 0. DELTA NEUTRAL - this portfolio evens out the response ADVANTAGES to market movements for a certain - It allows traders to hedge the risk of range to bring the net change of the constant price fluctuations in a position to zero. portfolio. - It protects profits from an option or LINEAR PRODUCT stock position in the short term while protecting long-term holdings. A product whose value is linearly dependent - Protects against price fluctuations on the value of the underlying asset price or market variable. DISADVANTAGES - When the value of the delta is - Traders must continuously monitor constant for any change in the and adjust the positions they enter. underlying asset. Depending on the equity volatility, - Examples: future contracts, forward the investor would need to buy and contracts, stocks sell securities to avoid being under or over-hedged. where: - Large expenses P is the resulting change in a portfolio - Delta is not constant S is a small increase in the value of the variable Sample Prob. TYPES OF DELTA HEDGING The value of the portfolio is currently -$5,683,000. STRATEGIES There is a small increase in the price of gold from $1,300 per ounce to $1,300.10 per ounce. Suppose that this $0.10 increase in the price of gold decreases the value of the portfolio by $100 from -$5,683,000 to -$5,683,500. FM5617 Risk Management Finals −100 DELTA = $0.10 = -$1000 NONLINEAR PRODUCT Often have complex payo s that may involve options, contingent claims, or other structures with nontrivial dependencies on various market variables. - Examples: convertible bonds and variance swaps where: S = Current Price K = Strike Price r = Risk-free-rate q = Dividend yield σ = Volatility T = Time Sample Prob. A financial institution has just sold 1,000 seven-month European call options on the Japanese yen. Suppose that the spot exchange rate is 0.80 cent per yen, the exercise price is 0.81 cent per yen, the risk-free interest rate in the United States is 8% per annum, the risk-free interest rate in Japan is 5% per annum, and the volatility of the yen is 15% per annum. Calculate the delta call option. = 0.1016 =0.5250 Interpretation: As the spot price increases, the value of an option to buy one yen increases by 0.5250 times the same amount.