Firm-wide, Corporate Firm-wide, Corporate Risk ManagementRisk Management

Risk Management

Prof. Ali Nejadmalayeri,

a.k.a. “Dr NDr 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 transformed into a standard normal variable!

• This is quite handy when we want to compute VaR:

zStd

zEzu

zEzStdzofquantileFifth 65.1

Computing VaR

• If portfolio returns, ri, is normally distributed with zero mean and volatility, σi, then the 5% VaR of the portfolio is:

• In general, an α% VaR can be computed by:

ValuePorfolioVaR i 65.1

ValuePorfoliouNVaR i )(%

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:

– 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

ti

ittt SRVaRSVaRMax

t

59

1601 10%,1;10%,1

1Day for Capital Required

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-pointValue

TransactionalValue-at-Risk

(with volatilizes)

Portfolio Value-at-Risk,Enterprise Risk(with volatilitiesand correlations)

Increasing Sophistication

Increasing Sophistication

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

Scenarios

Interest Rates

Vega

ForeignExchange

CreditSpread

Equity

Swap Spread

Commodity

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