We use throughout this book the notation introduced in S.A. Zenios, Practical Financial Optimization: Decision Making for Financial Engineers, (Blackwell Publishing, Cambridge, MA, 2007), abbreviated as PFO. To the extent possible the same notation is adopted in the accompanying software. Cross-references to chapters, sections, models, and so on from PFO are given using the notation PFO-m.n.p where m.n.p refers to PFO labels. For instance, “Section PFO-6.3” refers to Section 3 of Chapter 6, and “Model PFO-6.4.2” refers to the second model of the fourth section of Chapter 6. Readers will easily locate the cited material in PFO.

U = {1, 2, . . . , n} index set of available financial instruments or asset classes.

T = {0, 1, ..., τ, ... T} set of time periods, from today (0) until maturity (T ). Unless stated otherwise in the text all time periods are of equal duration, which is typically taken to be one month.

K = {1, 2, . . . , κ, . . . , K} index set of risk factors.

Σ_t = {1, 2, . . . , S_t } index set of states at period t .

Ω = {1, 2, . . . , N } index set of scenarios.

i index of instrument or asset class from the set U .

t index of time periods from the set T .

j index of risk factor from the set K.

l index of scenario from the set Ω.

x n-dimensional vector of investments in assets, with elementsx_i . The units are in percentages of the total asset value or amounts in face value; the choice of units depends on the model and is made clear in the text.

b₀ n-dimensional vector of initial holdings in assets, with elements b_0i .

cash invested in short-term deposits at period t .

cash borrowed at short-term rates at period t .

v₀ initial holdings in risk-free asset (cash).

p^l statistical probability assigned to scenario l.

n-dimensional random vector of asset returns, with elements_i.

r^l n-dimensional vector of asset returns in scenario l, with elements r_i ..

_t n-dimensional random vector of asset returns at period t , with elements_ti .

t n-dimensional vector of asset returns at period t in scenario l, with elements

r_ft spot rate of return of the risk-free asset at period t.

n-dimensional random vector of cashflows from assets, with elementsi.

F^l n-dimensional vector of cashflows from assets in scenario l, with elements

_Ft n-dimensional random vector of cashflows at period t , with elements_ti.

n-dimensional vector of cashflows from the assets at period t in scenario l, with elements

n-dimensional random vector of prices of assets, with elements_i.

P^l n-dimensional vector of prices of assets in scenario l, with elements

=_t n-dimensional random vector of prices at period t , with elements_ti

n-dimensional vector of prices of assets at period t in scenario l, with elements

n-dimensional random vector of ask prices at period t, with elementsIn order to buy an instrument the buyer has to pay the price asked by traders.

n-dimensional random vector of bid prices at period t, with elementsIn order to sell an instrument the owner must accept the price at which traders are bidding.

n-dimensional vector of ask prices at period t in scenario l, with elements

n-dimensional vector of bid prices at period t in scenario l, with elements

random variable of the total return of a benchmark portfolio or a market index.

I^l total return of a benchmark portfolio or a market index in scenario l.

_t random variable liability due at period t.

value of the liability in scenario l.

Q a conformable covariance matrix.

σ_ii’ covariance of random variables indexed by i and i’.

ρ_ii’ correlation of random variables indexed by i and i’.

maximum holdings in asset i.

x_i minimum holdings in asset i.

ε[] expectation of the random variable or vectorwith respect to the statistical probabilities p^l assigned to scenarios l ∈ Ω.

E_Ρ [] or E_λ[] expectation of the random variable or vectorwith respect to the probability distribution Ρ or the probabilities λ ∈ Ρ .

U(a) utility function with arguments over the real numbers a.

mean value of a random variable or vector.

R(x;) portfolio return as a function of x with parameters.

V (x;) portfolio value as a function of x with parameters. max[a,b] the maximum of a and b.

Prob (= r) the probability that the random variable argumenttakes the certain value r .

I a conformable identity matrix.

1 conformable vector with all components equal to 1.

ALM Asset and liability management.

APT Arbitrage Pricing Theory.

CAPM Capital Asset Pricing Model.

CBO Collateralized bond obligation.

CEexROE Certainty equivalent excess return on equity.

CEROE Certainty equivalent return on equity.

CLO Collateralized loan obligation.

CRO Chief risk officer.

CVaR Conditional Value-at-Risk.

EWRM Enterprise-wide risk management.

FHA Federal Housing Association.

LTCM Long Term Capital Management.

MAD Mean absolute deviation.

MBS Mortgage-backed security.

OAP Option adjusted premium.

OAS Option adjusted spread.

PFO Practical Financial Optimization: Decision Making for Financial Engineers, Blackwell Publishing, Cambridge, MA, 2007, by S.A. Zenios.

PSA Public Securities Association.

ROE Return on equity.

SPDA Single premium deferred annuities.

VaR Value-at-Risk.

GAMS General Algebraic Modeling System

IDE Integrated Development Environment.