In MG, the underlyings were short positions in long-term forward contracts to deliver oil. The hedge was a stack-and-roll hedge: long positions in short-term futures contracts that were rolled over consecutively. The strategy depended on the continuation of (i) stable or gently increasing spot oil prices and (ii) backwardation
SaR is value at risk (VaR) for a pension fund.
The best hedge is based on portfolio volatility in the mean-variance framework. Specifically, 1. Given a current portfolio with value (W), and 2. Given an asset (A) with correlation (rho) to the portfolio, 3. What is the trade that produces the minimum volatility for the new portfolio (W+a)?
RAPMs are variations of: return per unit of risk. Treynor and Sharpe are similar: both are excess return per unit of risk. Treynor defines risk as systematic risk (beta) and is therefore appropriate to well-diversified portfolios (i.e., into such portfolios idiosyncratic risk is eliminated); Sharpe defines risk as total risk (volatility). Jensen’s alpha is outperformance relative to expected performance under CAPM.
The security market line (SML) plots the expected return of an asset (or portfolio) as a function of the asset's beta.
The capital market line is determined by a mix of: the riskfree asset and the market portfolio. The market portfolio, in turn, consists of all risky assets (this example has only two assets).
This is a review which follows Jorion's (Chapter 7) calculation of marginal value at risk (marginal VaR). Marginal VaR requires that we calculate the beta of a position with respect to the portfolio.
The very traditional (mean-variance) two asset portfolio volatility is largely a function of asset correlation/covariance.
The next building block is mapping transitional probabilities to standard normal variables; then using a bivariate normal to capture joint probabilities of default
The bivariate normal distribution (common in credit risk) gives the joint probability for two normally distributed random variables