Financial risks faced by investors and financial institutions

CAPM: E(rp) = rf − bp (E(rm) − rf)
Beta: βi = Cov(ri,rM) / Var (rM)
Mean – Variance:
Sharp Ratio: Measures the expected return per unit of risk.
rp − rf / σp, the greater the value of the Sharpe ratio, the more attractive the investment is.
Treynor Ratio: Performance metric for determining how much excess return was generated for each unit of risk taken on by a portfolio, rp – rf / bp
Major Risk Measures used in Portfolio Optimization and associated Mathematical Formulations of the Optimization Models (2/3)
Figure 1: : “Panorama of financial products and their betas”
Figure 2: “Variance vs Absolute Deviation”
Figure 3: “Representations of VaR and cVaR”

• Assets having b>1 are more exposed to risk than the market is, thus should provide higher returns than the market to get investors interested in higher risk exposures.
• Assets having b<1 are less exposed to risk than the whole market portfolio, so, their returns should be less than the market returns
• Assets having b=1 are exposed to the same level of risk as the market and their returns should be equal the market return.
• The figure shows the risk and return trade off in accordance with beta for a variety of assets.

Major Risk Measures used in Portfolio Optimization and associated Mathematical Formulations of the Optimization Models (3/3)
Alternative measure of risk that illustrates a more accurate outcome for the amount of losses in a portfolio given that losses will occur.
Coherent risk measure because satisfies the four axioms of coherent measures.
Expected Shortfall
A target return is specified and the portfolio figures gains only whether the actual return exceeds this target return.
Maximize Σpsys+
Subject to: Σpsys+
ys+ − ys− − (R(x,rs) − ts) = 0 , ∀s∈Ω
ys+ ≥ 0, ∀ s ∈ Ω