Understanding Portfolio Risk Measures | CFA Level I Quantitative Methods

In this lesson, we’ll touch on the measures of portfolio risk, including Roy’s safety-first ratio, and more.

Mean-Variance Analysis and Shortfall Risk

Analysts use z-values to evaluate portfolio risk. For example, Portfolio A has an expected return of 10% and a standard deviation of 5%. An analyst views a return of less than 5% as unacceptable (threshold return). The probability of a return falling below this level is the shortfall risk.

Calculating shortfall risk using the z-table:

1. Find the z-value: (5 – 10) / 5 = -1.0
2. Look up the z-table: F(-1.0) = 0.1587, so the shortfall risk is 15.87%

Comparing two portfolios, B and A:

• Portfolio B: mean of 15%, standard deviation of 7.5%, z-value of -1.33, shortfall risk of 9.18%
• Portfolio A: mean of 10%, standard deviation of 5%, z-value of -1.0, shortfall risk of 15.87%

Despite Portfolio B having a higher standard deviation, it has a lower shortfall risk than Portfolio A. Risk evaluation is complex and requires analyzing the best method based on the investor’s objectives and risk tolerance.

Roy’s Safety First Ratio

A shortcut to evaluate the suitable portfolio is to use the Roy’s safety first ratio. The safety first ratio is the reverse of the top 2 terms of the z-formula: (Expected return – Threshold return) / Standard deviation. Due to the symmetric property of the normal distribution, the shortfall risk remains the same when using the safety first ratio. With this ratio, we select portfolios with higher values.

In summary, the z-transform and Roy’s safety first ratio are useful tools in evaluating portfolios based on their risk characteristics. Remember that risk evaluation is complex, and it’s essential to consider the investor’s objectives and risk tolerance when selecting a suitable portfolio.