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Firm-wide, Corporate Risk Management Risk Management Prof. Ali Nejadmalayeri, a.k.a. “Dr N” Value at Risk • The dollar loss that will be exceeded with a given probability during some period. Usually, 1%, 5% or 10% probabilities are used to defined VaR. Basis of VaR • Formally, VaR at (100 – z) level of confidence is the value that satisfies Prob[loss > VaR] = z. – z is the probability that loss is greater than VaR. – Ordinarily, we use z = 5% • How to measure VaR? – Straightforward if we assume that returns are normal, because for a standard normal distribution: • Probability of values lower than – 1.65 is 5% • Any normally distributed variable, z, can be u z E z transformed into a standard normal variable! Std z • This is quite handy when we want to compute VaR: Fifth quantile of z 1.65 Stdz Ez Computing VaR • If portfolio returns, ri, is normally distributed with zero mean and volatility, σi, then the 5% VaR of the portfolio is: VaR 1.65 i Porfolio Value • In general, an α% VaR can be computed by: VaR % N (u ) i Porfolio Value Computing VaR with Excel – We can use Excel to compute any VaR. Function NORMSINV can generate N(u ≤ α). Just enter the α% and the function computes the N(u ≤ α)! Banks and VaR • Example of VaR can be readily found in bank risk capital management. Basel Accord 1988 and its subsequent amendments requires: Required Capital for Day t 1 59 Max VaRt 1%,10; St 601 VaRt i 1%,10 SRt i 1 – Where St is multiplier and SRt is an additional change for idiosyncratic risk. • St is determined based on whether the bank’s 1% VaR has been accurate over the past 250 days or not – Exceeding VaR by no more than 4 times, St is set to 3 – Exceeding VaR by more than 10 times, St is to 4 VaR in Practice • RiskMetrics, a former division of JPMorgan, has devised complex techniques to evaluate the VaR for any bank – Challenge for a bank with thousands of clients and thousands of transactions is not only compute each position VaR but to account for cross correlations to find firm-wide VaR! – The solution is to map assets into major asset classes, e.g., country indexes, and then compute the volatilities, correlations and VaRs. VaR & Fundamentals • To compute VaR analytically, we need to assume returns are normal or that values are log-normal! • Otherwise we need to estimate VaR! Cash Flow at Risk • For non-financial, the important element is cash flows and not per se value. So we need to define a measure to capture same intuition as VaR, or CaR! • CaR at p% reports the least cash shortfall with probability of p%. • Formally, CaR at p % is defined as: Prob[E(C) – C > CaR] = p% VaR Impact of a Project • The VaR impact of a project is the change in VaR brought about by the project. – Vol. impact of trade = (βip – βjp) Δw Vol(Rp) • VaR impact of trade = – (E(Ri)– E(Rj)) Δw W + (βip – βjp) 1.65 Vol(Rp) Δw W • Expected gain of trade net of increase in total cost of VaR = Expected return impact of trade Portfolio value – Marginal cost of VaR per unit VaR impact of trade Example • Ibank’s $100M portfolio consist of 3 equal size positions. Expected returns are 10%, 20%, & 15%. Volatilities are 10%, 40%, & 60%. – We know that: Portfolio volatility is 0.1938. – We know that: Portfolio VaR is $16.977, or 16.977% of value • 0.1500 – 1.65 (0.1938) = – 16.977 • Now consider a trade in which we sell security 3 and buy security 1 to the tune of 1% of the portfolio. – The dollar change is (0.10 – 0.15) 0.01 = – 0.0005 – We also know that betas for 1 & 3 are 0.0033/0.19832 = 0.088 and 0.088/0.19832 = 2.343 – So VaR impact of the trade is (0.10 – 0.15) 0.01 $100M + (0.088 – 2.343) 1.65 0.1983 0.01 $100M = – $671,081 CaR Impact of a Project • The CaR impact of a project is the change in CaR brought about by the project. • Imagine CaR without the project: – CaRE = 1.65 Vol(CE) • CE is the cash flow from existing operations • Then, after the project, CaR is: CaR = 1.65 Vol(CE+CN) = = 1.65 [Var(CE) + Var(CN) + 2 Cov(CE,CN)] ½ Example • A firm generates $80M cash flows with $50M volatility. A project requires $50M investments and has $50M volatility. The project has 0.50 correlation with the firm. Its beta is 0.25 with market portfolio. The expected payoff before CAPM cost is $58M. If risk-free rate is 4.5% and the market risk premium is 6%, then COC is 6%. – NPV = $58/1.06 – $50M= $4.72M – Total volatility after the project is (502 + 502 + 2 0.5 50 50) ½ = 86.6025 – CaR before the project was 1.65 $50M = $82.5M – CaR after the project is 1.65 $86.6025M = $142.894M – If CaR has a 0.10 cost, then the project has a negative NPV based on CaR cost adjustments: 4.72M – 0.10 ($142.894M – $82.5M) = – $1.32M Measures of Risk • Traditional and new measures of risk Notional Value Basis-point Value Transactional Value-at-Risk (with volatilizes) Portfolio Value-at-Risk, Enterprise Risk (with volatilities and correlations) Notional Amount • Literally taking into account the notional value of positions. For instance, saying that $1M US T-bond is at risk, so risk capital is equal to $1M. • Shortcomings: – No distinction between assets with high and low probabilities of capital loss – No distinction for offsetting positions. For instance, an option market maker has $20M call options on SP100 and $18M puts on SP100. In notional value sense, the market maker has $38M risk capital whereas in reality she has only $2M at risk! Basis-Point Approach • For every basis-point change in fundamentals what happens to value? – Bonds and options risks are reported in these terms – In case of bonds, interest rates are the key – In case options, the “Greeks” are the key • • • • • Delta, or price risk Gamma, or convexity risk (how delta changes) Vega, or volatility risk Theta, or time decay risk Rho, or discount rate risk Value-at-Risk • • Based on distribution of value, find out what is the minimum loss in rare events Where to get the distributions? 1. Selection of Risk Factors • Factors that drive value; such as exchange rates, interest rates, volatilities, etc. 2. Selection of Methodology • • Analytical covariance-variance Historical Simulation – Random draws from past results (random sampling) • Monte Carlo Simulation – Forecast evolution of risk factors Stress Testing Envelopes • Seven Major Components Interest Rates % Swap Spread Equity Scenarios % Vega Foreign Exchange Credit Spread Commodity