Skip to content
Quantitative Risk Management / Loss Distributions, VaR, and Coherent Risk Measures

Mapping Risks: The Loss Operator and Risk Factor Representation

Every risk measure — VaR, ES, economic capital — is computed from a loss distribution. But a portfolio is not directly a loss distribution; it is a collection of positions whose values depend on observable market quantities (equity prices, interest rates, credit spreads, FX rates). This note builds the bridge between the two: the risk factor mapping Vt=f(t,Zt)V_t = f(t, Z_t), which represents portfolio value as a function of a dd-dimensional vector of observable risk factors ZtZ_t.

Once the mapping is specified, the loss over a holding period becomes

Lt+1=−(f(t+1, zt+Xt+1)−f(t, zt)),L_{t+1} = -\bigl(f(t+1,\, z_t + X_{t+1}) - f(t,\, z_t)\bigr),
where Xt+1=Zt+1−ZtX_{t+1} = Z_{t+1} - Z_t are the risk-factor changes. The loss distribution is then entirely determined by the distribution of Xt+1X_{t+1}. This is the operational form of risk as a random variable: the abstract L∼FLL \sim F_L of Lesson 1 becomes a concrete function of observable risk factors.
The note covers: how to choose and map risk factors for stock portfolios, derivatives, and loan portfolios; the delta (linearized) approximation and when it fails; the three methods for deriving the resulting loss distribution (variance–covariance, historical simulation, Monte Carlo); and the distinction between regulatory and economic capital.
Learning objectives. By the end of this note you should be able to:
  • write down the mapping Vt=f(t,Zt)V_t = f(t, Z_t) and the loss operator for a stock portfolio, an options position, and a stylized credit portfolio;
  • compute the linearized loss L~t+1\tilde{L}_{t+1} via the delta approximation and state when this approximation is inadequate;
  • derive the distribution of the linearized loss under the multivariate normal assumption (variance–covariance method);
  • explain historical simulation as estimation of FLF_L from the empirical distribution of past risk-factor changes;
  • distinguish equity capital, regulatory capital, and economic capital.

In this lesson

    1. The balance sheet and capital concepts
    1. The risk factor mapping framework
    1. The three methods for deriving the loss distribution

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or browse the Quantitative Risk Management syllabus.