THEORY RATIONALES

THEORY RATIONALES

FOUNDATION

We start by underpinning the foundations of time as temporal to equidistance in which investment rollover and the return expected as a percentage of risk-anticipated against the opportunity costs of immobilization. Two factors determine the premises of the variable called "Return Expectation", of which the first factor labelled Time/Return [T/R Factor] determines the risks in terms of time over return as a fraction of anticipated exposure, and the second factor is the level of T/R Factor against the Indifference Frontier Level [IFL] in which linear functions determine the equidistance of exposure in terms of Opportunity Frontiers [OP].

Second, we determine a set of variables on the scale denominated LYSCALE RISKGRADE SYSTEM to apply non-arbitrary numerical magnitudes under which risk exposures are identified. Risk Exposures go from minus infinity to plus infinity. Zero level risk is set as Risk Neutral Zone [RNZ] which oscillates between positive and negative magnitudes and is located in four (4) different parametric positions. Each position encompasses a set of two-rays of evolution lines departing from point zero, that is, Risk Neutral Zone evolving simultaneously towards equidistant and symmetrical set of bundles that ascend in magnitudes.

The RNZ is the zone where risk is virtually inexistent. In real world, such zone is hard to define since any action or investment has a risk exposure in terms, at least, of return expected. For ease of articulation, RNZ is the parameter in which initial investment is recuperated after any given test-of-time. That suggests that for any investment, the minimum return equals to the investment itself allowing excess in compensation for inflationary fluctuations over time.

LYSCALE RISKGRADE SYSTEM TABLE INCLUDING KEY VARIABLES INDICATORS [RISK SCALE MAGNITUDES; DISCOUNTING FACTORS, RISK INCIDENCES, RISK INTERFERENCES & OPTIMAL RISK RATIOS]
RISK SCALE MAGNITUDES [RM]	DISCOUNTING FACTOR [DF]	RISK INCIDENCE [RIC]	RISK INTERFERENCE [RIF]	OPTIMAL RISK RATIO [ORR]
RM 1	0.05	5%	.95	5.2
RM 2	0.10	10%	.90	11.1
RM 3	0.15	15%	.85	17.6
RM 4	0.20	20%	.80	25.0
RM 5	0.25	25%	.75	33.3
RM 6	0.30	30%	.70	42.8
RM 7	0.35	35%	.65	53.8
RM 8	0.40	40%	.60	66.6
RM 9	0.45	45%	.55	81.8
RM 10	0.50	50%	.50	100.0
RM 11	0.55	55%	.45	122.2
RM 12	0.60	60%	.40	150.0
RM 13	0.65	65%	.35	185.7
RM 14	0.70	70%	.30	233.3
RM 15	0.75	75%	.25	300.0
RM 16	0.80	80%	.20	400.0
RM 17	0.85	85%	.15	566.6
RM 18	0.90	90%	.10	900.0
RM 19	0.95	95%	.05	1900.0
RM 20	1.00	100%	.01	10,000.0

The Discounting Factor point is computed by producing a ratio exposing the magnitude set point ranging from 0.1 to 2.0 over the total set of risk magnitude set of magnitude point levels on the LYSCALE RISKGRADE SYSTEM. For example at RM 3, the computation is as follows: 0.3 / 2.0 = 0.15 discounting effect on the whole scale to offset exposure uncertainty over the lifetime of the action, venture, investment or custody of the asset, securities or financial vehicle.

The Risk Incidence variable arises by the combination of the Discounting Factor and the level achieved on the Optimal Risk Line as compared to the distance separating the point of origin where the Risk Incidence equates the Risk Neutral Zone, denoted RNZ = 0

The Risk Interference is the result of the total exposure minus the Discounting Factor.

The Optimal Risk Ratio computes the Discounting Factor against the Risk Interference Variable and sets it as a percentage on the exposure of the return expected. Technically speaking, if an asset is graded at magnitude 7 on the LYSCALE RISKGRADE SYSTEM, the return expected is exposed to adverse risk interference at 53.8%

Please note that Risk Intereference computed at RM 20 is computed arbitrary in order to provide a positive approximation for the column of Optimal Risk Ratio. If the variable was set at nil, the Optimal Risk Ratio would equate zero altogether, dispaying a dysfunctional pattern in rapport of Risk Interefence against Risk Incidence valued at 100% of the total assets at risk and the Discounting Effect registered at RM 20, which is the highest level on LYSCALE RISKGRADE SYSTEM. Note that the Discounting Factor is 1.00 over 1.00 making it plausible that at this risk magnitude, no expected return is anticipated.

DEFINITION OF LYSCALE RISKGRADE SYSTEM STATISTICAL METHODOLOGIES

Lyscale Riskgrade System is made of numerical magnitudes for risk exposures according to variables spanning several determinants. The scale is scientific in that all parameters are defined according to their specific risk factors as well as the ambivalent underlyiung triggers as enacted within a given operational environment. Geographical distributions plays a non-determinant role since the most important factor that sets non-arbitrary variances between entities located in different settings is the return on investments observed and the expected rate of asset turnover defined on a country-by-country basis, industrial average basis, banking return retention and capture basis or sector-defined (observed) growth potential.

Risk factors are determined irrespective of subjective ratings provided by agencies that based their research on non-scientific rationales to grading entities including countries. As a pioneer, a risk storming exercise is based on objective patterns of observations that are matched against hard realities of investment outcomes readily available in the public domain. LYSCALE RISKGRADE SYSTEM does not concern itself to a rating tool, but better, it defines the way in which investment actions are taken to produce sufficient and real return generated under the constraint of the investment itself plus environmental factors such as location, time and space, culture and performance, all variables subjected to their own disparate and interchanging constraints. Special emphasis is given to culture becaouse when we mean "culture", we refer to the "Managerial stance of culture" as opposed to perceived ethnic or racial divergence. Managerial culture is much diversified and differentiated and the amalgamated effect of the sum of all these divergences amount to a single valued added technique-spread that possesses many advantages rather than disadvantages.

LYSCALE RISKGRADE SYSTEM has magnitudes starting from 0.1 inclusive to 2.0 inclusive. The magnitude 0.00 is inexistent since it is the level by which risk exposure equate to nil. This starting point assumes that Zero Risk or Risk Neutral Positioning is utopia if not impossible to materialize in any investing action. By increasing order, the magnitude evolves in showing the degree of risk exposure for action taken in order to increase asset turnover and return deployment and capture. Taking the magnitude as the percentage value of underwriting premium would suggest reasonable value at which risk analyst and insurers can accpet subscriptions from operatinf entities involved in risk business, if not in all businesses.

Under the perfect frictionless market assumptions of the Black-Scholes model, the thoery of replication gives a universal methodology for pricing risk using the LYSCALE RISKGRADE SYSTEM. One such rationale is to base pricing decisions on an assumption that agents seek to maximize expected utility and simulatenously seek to minimize expected risk exposures. Under this framework, analysts and risk managers need to illustrate their methodologies in writing claims upon the underlying value of matching assets or liabilities or their equivalents residual value that discounts features of goodwill and intangibles. The risk manager will agree to pay Price P for the claim if his expected utility on buying the claim p is at least as great as his utility from not buying the claim. In that way, it is claimed that he can use the theory of utility maximization to find a preference-dependent bid price for a contingent claim.

In algebraic terms, the LYSCALE RISKGRADE SYSTEM can appear on the form:

F(dr) = P[D<dr], which is the risk-distribution function of disaster-risk set of elements.

N(dr) = P[D>dr] = 1-F(dr), which is the Non-claim function of disaster-risk set of elements.

We often need to deal with disaster-risk greater than 0.1 on the magnitude of the LYSCALE RISKGRADE SYSTEM in order to dismiss the Risk Neutral Zone [RNZ] whereby risk exposure R = 0, bearing in mind that the maximum exposure R_max = 2.00 (20 times the benchmark level of R_minimax = 0.1)

Time-span of effective cover for the underwriter is denoted C = 365 days, we therefore denote Cx as being the future lifetime after date x, where x is the corresponding day where disaster strike, hence the day where claim for compensation is effected. This assumes the end of the cover at x, provided that the claimant can renew a fresh cover at date x^new. Note that x and x^neware given as number of days such that [1 < x < 365] and

[x < x^new< 365].

For consistency with S, the risk-distribution function of random variable S (x) must satisfy the following relationships

Fx (dr) = P [Sx < Cx] = P [S < x + Cx | S > x] = F (x + Cx) - F (x) / S (x)

A quantity which plays a central role in a disaster model is the force of occurence of any one claim. We denote the force of occurence at age x (1 < x < 365 ) by Mx and define as:

Mx = 1/Cx P [S<x+Cx | S > X], subject to limit whereb y Cx > 1

The interpretation of Mx is very important. The probability P [S<x+Cx | S > x] is from definition above Fx (Cx) is the rate of disaster occurence RDO. For small Cx, we can ignore the limit and write:

RDO = Cx . Mx

In other words, the probability of disaster occurence in a small time Cx after date x is roughly proportionate to Cx, the constant of probability being Mx.

THE MECHANISM FOR EFFECTIVE RISK ISOLATION USING RANDOM RISK VARIABLES

The mechanism of determining risk starts by demarcating it from uncertainty. This is called Risk Isolation. To isolate risk, one needs to define patterns of Effective Risk Identification [ERI].

Effective Risk Isolation is defined as a function of moment which is just a particular expected value that summarizes features of a risk-distribution in terms of occurences. Moments about the mean are called central moments.

The risk weight applicable to a fully-guaranteed expsure . i.e. whre the nominal amount of the credit protection equals that of the exposure is:

r* = w x r + (1.w) x g

where x* is the effective risk weight of the position taking into account the risk reduction from the guarantee/credit derivative

r is the risk weight of the obligor

w is the weight applied to the underling exposure

g is the risk weight of the guarantor/protector provider

For a credit-protected exposure, the risk weighted assets will be:

E x r* = (E. GA) x r + GA x [w x r + (1.w) x g ]

where,

E is the value of the exposure (e.g. nominal amount of loan or investment);

GA is the nominal amount of the cover (adjusted if necessary for foreign exchange risk)

r* is the effective rist weight of the position taking into account the risk reduction from the credit/investment protection purchased;

w is the residual risk factor; and

g is the risk weight of the protection provider.

In the case of a full guarantee/credit protection, the equation becomes the following:

E x r* = E x [w x r +(1. w) x g ]

The derivative of risk weights is dependent on estimates of the Probability Default (PD), the Loss Given Default (LGD) and, in some cases, maturity (M), that are attached to an exposure.

Throughout this section, PD, LGD and EAD are expressed as whole numbers rather than decimals, except where explicitly noted otherwise. For example, LDG of 100% would br input as 100. The exception is in the context of the benchmark risk weight (BRW) and the maturity slope. In these equations, PD is measured as a decimal (e.g. a 1% probability of default would expressed as 0.01).

Where there is no explicit maturity dimension in the foundation approach, corporate exposures will receive a risk weight that depends on the probability of default and loss given default (after recongnising any credit enhancements from collateral, guarantees or credit derivatives). The average maturity of all exposures will be assumed to be three years.

Thus, an exposure's risk weight, RWC, can be expressed as a function of PD and LGD according to the following formula:

RWC = (LGD/50) x (BRWC (PD) or 12.5 x LGD, whichever is smaller.

In this expression, RWC denotes the risk weight associated with given values of PD and LGD for corporate exposures, while BRWC denotes the corporate benchmark risk weight associated with a given PD, which is calibrated to an LGD of 50% as consistent with Basel Capital Accord wisdom. The BRWC is assigned to eah exposure reflecting the PD of the exposure based on the following equation: In this section, PD is expressed as a decimal . e.g. a PD of 10% would be inputed as 0.1

BRWc (PD) = 976.5 x N (1.118 x G (PD) + 1.288) x (1+0.0470 x (1-PD) / PD^0.41)^zG

where N(x) denotes the cumulative distribution function for a standard normal random variable (i.e. the probability that a normal random variable with mean zero and variance of one is less than or equal to x), and where (zG) denotes cumulative distribution function for a standard normal random variable (i.e. the value x such that (x N = z)

E(X) = (MU) is the first-order moment providing information on the average value of risk demarcated from uncertainty. E(X) and E(X²), moment up to the second-order provides information on spread uselful to determine standard deviation occurring on events that are risk-prone.

E(X), E(X²) and E(X³), moments up to thrid-order, provide information on one aspect of the shape of a distribution, namely skewness. The third central moment expands using the rules for the use of expectation operator E ( ) noted earlier. The sign of the Third-Order Moment called Micron gives the direction of the skew.

The usual moment of skew based on moments is the coefficient of skewness given by the fraction between Micron and a dimensionless measure denoted as Teta. Positive skew corresponds to Micron superior to nil while symmetry corresponds to Micron equals nil.

The conditions under which experiments are performed and data collected allow the data to be considered as samples from specific, well-defined probability distributions.

LYSCALE RISKGRADE methodology encompasses both discrete and continuous distributions. Of discrete risk distribution, Uniform Distribution, Bernoulli, Binomial, Geometric, Negative Binomial, Hyper-geometric, Poisson distributions are of particluar essence in the overall deployment of LYSCALE RISKGRADE SYSTEM. Of Continuous Risk-Distributions, our methodology focuses on Uniform, Gamma (special case 1 & 2), Beta and Normal Distributions.

DISCRETE RISK DISTRIBUTIONS AS APPLIED WITHIN LYSCALE RISKGRADE SYSTEM

The distributions considered here are all models for the number of something - e.g. number of successes, number of trials, number of insurance claims, catastrophes, sovereign defaults, political intereferences, connectivity-gap shortcomings, ambivalence determination, etc... The values assumed by the variables are integers in the set {0,1,2,3,...., k...} - such variables are often referred to as counting variables. Considering an Uniform distribution for Effective Risk Identification [ERI] required sample space S = {1,2,3,....K}, where probability measure equals assignment (1/k) to all outcomes i.e. all outcomes equally likely. The random variable X defined by X(i) = i, i = 1,2,3,...., k.

The distribution for ERI being in the form,

P(X=x) = 1/k, x = 1,2,3,...., k.

Moments beinf therefore,

Micron = E(X) = (1+2+3+...+k) / k = { k (k+1) / 2}/k = (k+1)/2

E(X²) = (1²+2²+3²+...+K²) / k = {k(k+1) (2k+1)/6}/k = (k+1) (2k+1) / 6

Teta = E(X²)²= (k²-1) / 12

Effective Risk Identification [ERI] can also be determined by using Bernoulli risk-distribution, using an experiment which has (or can be regarded as having) only two possible outcomes s (success) or f (failure). The sample S = {s,f}, bearing in mind that the words "success" or "failure" are merely labels - they do not carry with them the ordinary meaning of the words. In this case, the probability measure denominated P ({S}) = ¬, P ({f}) = 1 - ¬

The Risk Random Variable X defined by X(s) = 1, X(f) = 0. X is the number of successes which occur (0 or 1).

Distribution: P(X=x) = ¬

A Bernoulli variable is also called an "indicator" variable - Its value can be used to indicate whether or not some specified event, A says, occurs. Set X = 1 if A occurs, 0 if A does not occur. If P(A) = ¬ then X has the above Bernoulli distribution.

The event A could, for example, be the survival of an assured life over one year, an assured investment expected return over a day at the London Stock Exchange or even an assured Bilateral Loan Default of a sovereign country against expectation to another sovereign country.

In Risk Analytics perspective, it is crucial to define sound distribution techniques so tat risk can levied accurately. Bernoulli approach is rigorous yet unsophisticated.

Unlike Bernoulli, LYSCALE RISKGRADE use of Binomial risk-distribution is risk analytics is quite straightforward but highly dense in terms of alleviation. Considering a sequence of n Bernoulli trials as above such that:

the trials are independent of one another i.e. the outcome of any trial does not depend on the outcomes of any othertrials, and
the trials are identical i.e. at each trial P ({s}) = ¬.

Such a sequence is called a "sequence of n independent, identical, Bernoulli (¬) trials" or, for short, "a sequence of n Bernoulli (¬) trials".

The independence allows the probability of a joint outcome involving two or more trials to be expressed as the product of the probabilities of the outcomes associated with each separate trial concerned.

Sample space S; the joint set of outcomes of all n trials

Probability measure: as above for each trial.

Geometric risk-distribution can also be applied in determinig Effective Risk Identification by considering a sequence of independent, identica Bermoulli trials with P {(s)} = ¬. The variable of interest now is the number of trials which have to be performed until the first success occurs. Because trials are performed one after the other and a success is awaited, this distribution is one of a class of distributions called waiting-time distributions.

Random variable X: number of the trial on which the first success occurs.

Distribution: For X = x there must run of (x-1) failures followed by a success, so

P(X=x) = ¬ (1-¬)^x-1, providing that x=1,2,3,...; 0 < ¬ < 1

Consider the conditional probability P (X > x+n |X >n ).

Given that there have already been n trials without a success, what is the probablity that more x additional trials are required to get a success?

The intersection of the events "X>n" and "X>x+n" is just "X>x+n", so

P (X > x+n | X>n) = P(X>x+n) / P (X>n)

= (1 - ¬)^x+n/ (1 - ¬)ⁿ

= (1 - ¬)^x

= P(X > x)

i.e. just the same as the original probablity that more than x trials are required.

The lack of success on the first n trials is irrelevant - under this model the chances of success are not any better because there has been a run of bad luck.

This characteristic - a reflection of the "independent ,identicial trials" structure - is important, and is referred to as the "memoryless" property.

Another formulation of the geometric risk-distribution is sometimes used.

Let Y be the number of failures before the first success.

Then P (Y=y) = ¬(1-¬)^y, y = 0,1,2,3.... with mean

U = (1 - ¬) / ¬. Y = X-1, where X is defined above.

However, the negative binomial risk-distribution is the generalization of the geometric distribution which is uselful in deetermining the random variable X, number of the trial on which the k-th success occurs, where k is a positive integer. Note that in applying this model, the value of k is known.

Note: The mean and variance are just k times those for the geometric (¬) variable, which is itself a special case of this random variable (with k = 1). Further, the negative binomial variable can be expressed as the sum of k geometric variables n^thnumber of trials to the first success, plus the number of additional trials to the second success, plus...to the (k+1)^th success, plus the number of additonal trials to the k^thsuccess.

Another formulation of the negative binomial risk-distribution is sometimes used:

Let Y be the number of failures the k^th success. Eith mean U = K(1-¬) / ¬.

Y = X - k, where X is defined as above.

Furthermore, we can consider the Hyper-geometric risk-distribution that can be defined as the finite population equivalent of the binomial distribution, in the following context. Suppose objects are selected at random, one after another, without replacement, from a finite population consisting of k "successes" and N-k "failures". The trials are not independent, since the result of one trial (the selection of a success or a failure) affects the make-up of the population from which the next selection is made.

The details of the derivation of the mean and variance (or indeed the formula for the variance) of the number of successes are not reuired for the Risk Analyst. The mean is given by U = nk/N, which parallels the "U = n¬" result for the binomial distribution - the initial proportion of successes here being k/N.

The binomial, with ¬ = k/N, provides a good approximation to the hyper-geometric in many situations.

In such a context, LYSCALE RISKGRADE, recognizes that the binomial is the appropriate model for selecting with replacement, which is equivalent to selecting from an infinite population for which P(SUCCESS) = ¬ = k/N.

The Risk Manager or the Investment Manager of a typical Hedge Fund or Asset Management institution would be best suited in using the mdel referred to as the Poisson risk-distribution. This model concerns the number of events which occur in a specified interval of time, when the events occur one after another in time in a well-defined manner. This manner presumes that the events occur singly, at a constant rate, and that the numbers of events which occur in separate (i.e. non-overlapping) time intervals are independent of one another. These conditions can be described loosely by saying that the events occur "randomly", at a rate of... per ...", and such events are said to occur according to Poisson process.

Another approach to the Poisson risk-distribution uses arguments which appears at first sight to be unrelated to the above. Consider a sequence of binomial (n, ¬) distributions as n moves towards infinity and ¬ moves towards zero together, such that the mean n¬ is held constant at the value g.

Distribution: P (X = x) = g / x P(X = x-1)

Moments: Since the binominla mean is held constant at g through the limited process, it is reasonable to suggest that the distribution of X (the limiting distribution) also has mean g - this is in fact the case - The binomial variance is n¬(1-¬) = n(g/n) (1-g/n) = g(1-g/n) moves towards g as n moves towards infinity. This suggests that X has variance g - this is in fact also the case.

So U = g

The Poisson risk-distribution provides a very good approximation to the binomial when n is large and ¬ small - typical applications have n =100 and ¬ = 0.05 or less. The approximation depends only on the product n¬ (=g) - the individual values of n and ¬ are irrelevant. So, for example, the value of P(X=x) in the case n =200 and ¬ = 0.02 is effectively the same as the value of P(X=x) in the case where n = 400 and ¬ = 0.01.

When dealing with large numbers of opportunities for the occurence of "rare" events such as the calamity of September 11, 2001 (under "binomial assumptions"), the distribution of the number which occur depends only on the expected number which occur.

When events are described as occuring "as a poisson process with rate g per unit time" then the number of events which occur in a time period of length t has a Poisson distribution with mean gt.

CONTINUOUS RISK DISTRIBUTIONS UNDER LYSCALE RISKGRADE SYSTEM

Under uniform risk distribution, X takes values between two specified numbers a and b say,

Probability Density Function: FX(x) = 1/(b-a), b<x<a.

Moments: U = (a+b)/2, by symmetry, the mid-point of the range of possible values.

In this model, the total probability of 1 is spread "evenly" between the two limits, so that sub-intervals of the same length have the same probability.

Under Gamma (including exponential and chi-square) distributions, the family of distributions has two positive parameters and is a versatile family. The probability density function can take different shapes depending on the vlaues of the parameters. The range of the variable is {x:x>0}.

First note that the Gamma Function ¬(a) is defined for a > 0 as follows.

¬(a) = (y^a-1) (e^-y) dy

The PDF of the gamma risk-distribution is defined by

FX(x) = g^a/¬(a) x^a-1e^-¬xfor x > 0.

Moments: U = a /¬ and U²= a / ¬²

The exponential risk-distribution is ised as a simple model for the lifetimes of certain types of equipments. Very importantly, it also gives the distribution of the waiting time, T, from one event to the next in a Poisson process with rate ¬.

P[T>t] = P[0 events in time t]

= P[X=0] where X ~ Poisson (¬t)

= e^-¬t

In fact the time from any specified starting point (not necessarily the time at which the last event occured) to the next event occuring the distribution has this exponential distribution. This property can alos be expressed as the "memoryless" property.

A gamma variable with parameters a=k (a positive integer) and ¬ can be expressed as the sum of k exponential variables, each with parameter ¬. This gamma distribution of risk patterns is in fact the model for the time from any specified starting point to the occurence of the k-th event in a poisson process with rate ¬.

Another special case is to use chi-square risk-distribution where gamma with a = v/2 where v is a positive integer, and b = 2.

Moments: U = v, U² = 2v

Note however that A X²variable with v = 2 is the same as an exponential variable with mean 2.

Beta risk-distribution is another versatile family of distributions with 2 positive parameters. The range of the variable is

{x: 0<x<1}.

First note that the Beta function B (a,b) is defined by

x^a-1(1-x)^b-1 dx = ¬(a) . ¬(b) / ¬(a+b)

The PDF of a beta distribution:

FX(x) = ¬(a+b) / ¬(a) . ¬(b) x^a-1 (1-x)^b-1 for 0<x<1.

The uniform risk-distribution on (0,1) is a special case (with a=b=1)

Normal distribution is if fundamental importance in both statistical theory and practice, with its symmetrical "bell-shaped" density curve.

It is a good model for the distribution of measurements which occur in practicew in a wide variety of different situations. It provides good approximations to various other distributions - in particular it is alimiting form of the binomial (n,¬).
It provides a model for the sampling risk-distributions of various statistics used in LYSCALE RISKGRADE SYSTEM such as country modules, central banks, financial centres, stock exchnages and GMOS data aggregator.
Much of large sample statistical inference is based on it, and some procedures require an assumption that a variable is normally distributed.
It is a building block for many other distributions.

The distribution has two parameters, which can conveniently be expressed directly as the mean U and the standard deviation of the distribution. The distribution is symmetrical about U.

The PDF of the normal risk-distribution is defined by

F (x) = 1/gV2Pi . e

A linear function of a normal variable is also normal variable, i.e. if X is normally distributed, so is

Y = Ax + b.

It is not possible to find an explicit expression for F X(x) = P(X< or =x), so tables have to be used. These are provided for the risk distributions of Z = (X-u) /g, which is the standard normal variable - it has mean 0 and standard deviation 1. The risk-distribution is symmetrical about 0.

The x-values u, u+g, u+2g, u+3g correspond to the z-values 0,1,2,3 respectively, and so on. These are provided for the distribution of Z = (X-u)/g. The z-values measures how many standard deviations the corresponding x value is above or below the mean. For example the value x = 30 from a normal risk-distributions with mean 20 and standard deviation 5 has z-value +2 (30 is 2 standard deviations above the mean of 20).

The calculation of a probability for a normal variable is always done the same way - transform to standard normal via z = (x-u)/g and look at the tables.

95% and 99% intervals:

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