Good notation is more than a convention. It shapes how we think, streamlines problem solving, avoids misunderstandings and facilitates communication. Nowhere was this more evident than the introduction of Hindu-Arabic numerals into Europe during the Middle Ages. Because they made arithmetic computations so easy, the new numerals supplanted both Roman numerals and the abacus, which had been widely used in Europe. Another example comes from calculus. While Newton and Leibniz disputed who discovered the fundamental theorem of calculus, there is no dispute as to whose notation was superior. We still use Leibniz’s notation today.
One of the contributions of this book is consistent notation for expressing ideas related to value-at-risk. Once you master the notation reading the book, I encourage you to keep using it. The notation will guide your thinking and help you avoid pitfalls.
Value-at-risk draws on many branches of mathematics. Each offers its own notation conventions. Because these conflict, it is impossible to observe them all simultaneously. But the book’s notation consistently presents financial concepts related to calculus, linear algebra, probability, statistics, time series analysis, and numerical methods, drawing on existing conventions where possible.
Random quantities are indicated with capital English letters. If they are univariate random variables they are italic non-bold: Q, R, S, X, etc. If they are multivariate in some sense—random vectors, random matrices, stochastic processes—they are italic bold: Q, R, S, X, etc. Nonrandom quantities are indicated with lowercase italic letters. These are nonbold for scalars: q, r, s, x, etc. They are bold for vectors, matrices, or time series: q, r, s, x, etc.
With this notation, if a random variable is denoted X, a specific realization of that random variable may be denoted x. Such notational correspondence between random quantities and realizations of those random quantities is employed throughout the book.
Components of vectors or matrices are distinguished with subscripts. Consider the random vector
or the matrix
Time also enters the equation. To avoid confusion, I do not indicate time with subscripts. Instead, I use superscripts that precede the rest of the symbol. For example, the Australian dollar Libor curve evolves over time. We may represent its value at time t as
The value at time 3 of 1-month AUD Libor is The entire curve at time 1 is denoted 1R. The univariate stochastic process representing 1-week AUD Libor over time is represented R2. The 15-dimensional stochastic process representing the entire curve over time is denoted R. Time 0 is generally considered the current time. At time 0, current and past Libor curves are known. As nonrandom vectors, they are represented with lowercase bold letters: If time is measured in days, yesterday’s value of 12-month AUD Libor is denoted
The advantage of using preceding superscripts to denote time is clarity. By keeping time and component indices physically separate, my notation ensures one will never be confused for the other. Use of preceding superscripts is unconventional but not without precedent. Actuarial notation makes extensive use of preceding superscripts.
Much other notation is standardized, as will become evident as the book unfolds. Some frequently occurring notation is summarized below.
|n!||factorial of an integer n, which is given by the product n! = 1·2·3 ··· (n –1)·n, with 0! = 1|
|B(m, p)||binomial distribution for the number of “successes” in m trials, each with probability p of success|
|U(a, b)||uniform distribution on the interval (a, b)|
|Un(Ω)||n-dimensional uniform distribution on the region Ω|
|N(μ, σ2)||normal distribution with mean μ and variance σ2|
|Λ(μ, σ2)||lognormal distribution with mean μ and variance σ2|
|χ2(ν, δ2)||chi-squared distribution with ν degrees of freedom and
non-centrality parameter δ2
|Nn(μ, Σ)||joint-normal distribution with mean vector μ and covariance matrix Σ|
|1P||random variable for a portfolio’s value at time 1|
|0p||portfolio value at time 0|
|( 0p, 1P)||a portfolio|
|1L||random variable for portfolio loss: 0p – 1P|
|1R||random vector of key factors (key vector)|
|0r||vector of key factor values at time 0|
|1S||random vector of asset values at time 1 (asset vector)|
|0s||vector of asset values at time 0|
|Any of these might indicate a risk vector that is not a key vector.|
|E( )||unconditional expected value|
|tE( )||expected value conditional on information available at time t|
|std( )||unconditional standard deviation|
|tstd( )||standard deviation conditional on information available at time t|
|var( )||unconditional variance|
|tvar( )||variance conditional on information available at time t|
|tμ||unconditional mean of the time t term of a stochastic process|
|t|t–kμ||mean of the time t term of a stochastic process conditional on information available at time t – k|
|t Σ||unconditional covariance matrix of the time t term of a stochastic process|
|t|t–kΣ||covariance matrix of the time t term of a stochastic process conditional on information available at time t – k|
|θ||frequently used to denote a portfolio mapping function|
|frequently used to denote a (non-portfolio) mapping function|
|tφ( )||unconditional PDF of the time t term of a stochastic process|
|t|t–kφ( )||PDF of the time t term of a stochastic process conditional on information available at time t – k|
|tΦ( )||unconditional CDF of the time t term of a stochastic process|
|t|t–kΦ( )||CDF of the time t term of a stochastic process conditional on information available at time t – k|
|Tildes can be placed above or between symbols. Placed above indicates a remapping; for example, denotes a remapping of 1P = θ(1R). Placed between indicates that a random variable or random vector has a particular distribution; for example, X ~ N(0,1) indicates that random variable X is standard normal.|
|indicates that a random variable or random vector has a particular distribution, conditional on information available at time t. For example, indicates that, conditional on information available at time 0, 1X is standard normal.|
|indicates that analytic software more sophisticated than a spreadsheet may be useful in solving an exercise|
For more detailed explanations of notation, Section 2.2 addresses general mathematical notation used throughout the book. Section 4.6 elaborates on notation for time series analysis. Section 1.8 explains the notation of value-at-risk measures.
Currencies are indicated with standard codes:
Where applicable, millions are indicated as MM. For example, 3.5 million Japanese yen is indicated JPY 3.5MM.
Exchange rates are indicated as fractions, so an exchange rate of 1.62 USD/GBP indicates that one British pound is worth 1.62 US dollars.
Acronyms used include those shown in Exhibit 0.2.
|BBA||British Bankers Association|
|CAD||Capital Adequacy Directive|
|CBOT||Chicago Board of Trade|
|CDF||Cumulative Distribution Function|
|CME||Chicago Mercantile Exchange|
|CSCE||Coffee, Sugar and Cocoa Exchange|
|EICCG||explicit inversive congruential generator|
|ETL||expected tail loss|
|ICG||inversive congruential generator|
|IID||independent and identically distributed|
|IPE||International Petroleum Exchange|
|LCG||linear congruential generator|
|Libor||London Interbank Offered Rate|
|LIFFE||London International Financial Futures and Optons Exchange|
|LME||London Metals Exchange|
|MGF||moment generating function|
|MRG||multiple recursive generator|
|MSE||mean squared error|
|NYBOT||New York Board of Trade|
|NYMEX||New York Mercantile Exchange|
|NYSE||New York Stock Exchange|
|OTC||over the counter|
|P&L||profit and loss|
|probability density function|
|PMMR||probabilistic metric of market risk|
|RAROC||risk-adjusted return on capital|
|RORAC||return on risk-adjusted capital|
|SEC||Securities and exchange Commission|
|TSE||Toronto Stock Exchange|
|UNCR||Uniform Net Capital Rule|
|WCE||Winnipeg Commodities Exchange|
|Exhibit 0.2: Acronyms|