Risk Management and Value Creation in ... - Arabictrader.com

188 RISK MANAGEMENT AND VALUE CREATION IN FINANCIAL INSTITUTIONS
ity of their occurrence. Therefore, VaR has evolved as the standard methodology
for measuring market risk. It is defined as the loss to the portfolio
due to an adverse and unexpected move in one or more market risk factors
that is only exceeded with a given probability α% over a predetermined
period of time. In the case of trading units, the time interval typically chosen
is a one-day (holding) period. VaR is therefore the α-quantile of the cumulative
probability distribution of the value changes in the portfolio over the
measurement period H, and α is typically 2.5% or 1%. Hence, there is a
(1 – α)% probability that the critical threshold loss will not be exceeded. 242
Mathematically, we can express the probability that a loss will exceed
VaR over a predetermined period of time and that this will occur less than
α% of the time as:
p( [ ∆VH – E( RH) ]> VaRH) ≤ α% ⇔ f ( RH) dRH
≤ α % (5.27)
where p = Probability
∆V H
= Change in the portfolio value V over period H, which
equals R H
, the return of the portfolio over the same
time horizon
E(R H
) = Expected (or mean) return of the portfolio over time
horizon H
VaR H
= Value at risk for period H, which is a negative number
α = Confidence level for not exceeding threshold VaR H
f(R H
) = Assumed distribution of the portfolio returns over time
horizon H
Note that this definition so far does not make any assumptions about
the distribution function of the value changes. We assume now that the (daily)
returns are (as mostly assumed in modern finance theory and especially for
tradable market instruments) normally distributed 243 with mean return µ
and standard deviation of the return σ R
, that is f(R) ~ N(µ; σ R
). We can then
rearrange terms 244 and eventually derive the dollar amount for VaR as: 245
VaR = ([Φ -1 (1 – α)⋅σ R
]–µ) ⋅ V = (c⋅σ R
– µ) ⋅ V (5.28)
where Φ -1 = Inverse standard normal cumulative density function
VaR
∫
– ∞
242 See Stulz (2000), pp. 4-9–4-10, Hirschbeck (1998), p. 143, Dowd (1998), p. 41,
Jorion (1997), Guldimann et al. (1995).
243 See Dowd (1998), p. 42.
244 By using the property that we can transform R to a standard normal variable by
the following transformation [(R–µ)/σ].
245 See Hirschbeck (1998), p. 154.