Transcript Risk Management

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  Stdz   Ez 
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
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•
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